Lunds tekniska högskolakurslab-atom.fysik.lth.se/FED4MedOpt/JorgeR.pdf · A c kno wledgemen ts Man yare thosewhom I m ust thank for their help and supp ort during the progress of

INSTITUTO DE CIENCIA DE MATERIALES DE MADRID

C. S. I. C.

LIGHT DIFFUSION IN TURBID MEDIA

WITH BIOMEDICAL APPLICATION

Presented in the Department of Condensed Matter Physics of the Science

Faculty of the Universidad Autónoma de Madrid by

Jorge Ripoll Lorenzo

Supervisor: Manuel Nieto Vesperinas

Tutor: Juan José Sáenz

Madrid 2000

Acknowledgements

Many are those whom I must thank for their help and support during the progress of this thesis,

and naming each one may result not only a di�cult chore, but a dangerous one, since there is

no doubt I will leave someone out. In order to maintain the friends I have, you will permit me

to thank you all in general, and acknowledge in particular those who played a direct role in

this work.

First of all, I would like to thank the supervisor of this thesis, Prof. Manuel Nieto Vesperinas,

for all learned from him, not only as regards physics, but about research in general. Specially,

his knowledge and interest in the new technics and advances in the �eld of optics (amongst

which we may count his contributions) have rendered every single step of the work presented

in this thesis as extremely exciting. In this context, I also owe special gratitude to my tutor,

Prof. Juan José Sáenz, since his interest and vitality not only distinguishes him, but I may say

that both are contagious, and he transmits his ideas and knowledge with special ability.

I would like to thank Prof. Luisa Bausá, since she was who introduced me into research in

the last stages of my undergraduate studies. Her help and constant interest have been most

important in my formation. In a similar manner, I would like to extend this gratitude to Prof.

José García Solé and the C-IV section of the UAM in general, for their support.

I would like to thank Prof. Arjun Yodh and his family, for his hospitality and support

during my visits at UPENN, where the experiments presented in this thesis took place. In a

similar manner, I would like to thank the group under his supervision, Turgut, Joe, Vasilis,

Teodore, and Monica, for the support I have received from them.

I would like to thank Prof. Simon Arridge and his family, for the hospitality and friendly

treatment I received from him, from whom I have learned much during my visit at UCL, and

with whom I wish to maintain a long-lasting collaboration. Part of the work presented in this

thesis was developed with him and Hamid Dehghani.

My thanks to Dr. Rémi Carminati, whose help in the �rst stages of this thesis has been

fundamental, from whom I have learned much, and I know will still learn more. A part of the

work presented in this thesis was developed with his collaboration.

I would also like to thank Vasilis Ntziachristos, who with Joe Culver performed the experi-

ments shown in this thesis, and whom I consider a great scientist and friend.

I would like to thank Dr. Joaquim Fortuny, for his hospitality during my visit to JRC, in

Ispra.

1

2

I would also like to thank Andrea Bonanomi and his family for all the help and support I

received from them during my time in Ispra and Bergamo.

I would like to thank Charles, Maryjoe, Richard Aleja, Leni, Ma Angeles, Antonio, Rosarr,

David, Tiiito, Karsten, Patrick, Marcelo, Alberto, and Ramón.

Above all, my most sincere gratitude to my father, mother, brother and grandmother for

the support I have always recieved from them, and to O�� (�� v� �o� �!; ��!). To them

I dedicate this thesis.

Finally, I wish to acknowledge a grant from the Ministerio de Educación y Cultura of Spain,

received in the duration of this thesis.

Contents

1 Introduction 6

2 The Di�usion Equation 11

2.1 Speci�c Intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.1 Re�ection and transmission of the speci�c intensity . . . . . . . . . . . . 13

2.2 Radiative Transfer Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2.1 Radiative Transfer in a Non-Scattering Medium . . . . . . . . . . . . . . 19

2.2.2 Invariance of the speci�c intensity . . . . . . . . . . . . . . . . . . . . . . 20

2.3 Di�usion Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3.1 Angular dependence of the speci�c intensity . . . . . . . . . . . . . . . . 21

2.3.2 Derivation of the Di�usion Equation . . . . . . . . . . . . . . . . . . . . 22

2.3.3 Approximations taken in the Di�usion Equation . . . . . . . . . . . . . . 25

2.3.4 Improving the Di�usion Equation . . . . . . . . . . . . . . . . . . . . . 31

2.3.5 Other ways of reaching the Di�usion Equation . . . . . . . . . . . . . . . 35

3 Propagation of Di�use Photon Density Waves 36

3.1 Di�use Photon Density Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.1.1 Solution for in�nite homogeneous media . . . . . . . . . . . . . . . . . . 38

3.2 Angular Spectrum Representation . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.2.1 Evanescent Di�use Photon Density Waves? . . . . . . . . . . . . . . . . . 41

3.2.2 Angular spectrum of a point source . . . . . . . . . . . . . . . . . . . . . 42

3.3 Transfer Function and Impulse Response . . . . . . . . . . . . . . . . . . . . . . 44

3.4 Di�raction and interference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4 Integral Equations for Di�use Photon Density Waves. 51

4.1 Derivation of the Scattering Equations . . . . . . . . . . . . . . . . . . . . . . . 51

4.1.1 Rayleigh scattering for di�usive waves . . . . . . . . . . . . . . . . . . . 58

4.1.2 Source Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.2 Relationship with the angular spectrum . . . . . . . . . . . . . . . . . . . . . . . 61

4.3 Multiple Volumes of Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.4 Solving coupled integral equations . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3

CONTENTS 4

5 Spatial resolution of Di�use Photon Density Waves 73

5.1 Spatial Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.1.1 Propagating Scalar Waves . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5.1.2 Di�use Photon Density Waves . . . . . . . . . . . . . . . . . . . . . . . . 75

5.2 The Electrostatic Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.3.1 Scattering Integral equations for N bodies . . . . . . . . . . . . . . . . . 80

5.3.2 Numerical Results for Two Cylinders . . . . . . . . . . . . . . . . . . . . 81

5.4 E�ects of Noise on Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5.4.1 Filtering out the Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.5 Back-propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

6 Index matched di�usive/di�usive interfaces 93

6.1 Re�ection and Transmission coe�cients . . . . . . . . . . . . . . . . . . . . . . . 94

6.1.1 Wave Scattered from a Plane Interface . . . . . . . . . . . . . . . . . . . 97

6.1.2 Zero re�ectivity frequency . . . . . . . . . . . . . . . . . . . . . . . . . . 98

6.1.3 Frequency independent coe�cients . . . . . . . . . . . . . . . . . . . . . 101

6.2 Detection of buried objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

6.2.1 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

6.2.2 Rough Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

6.2.3 Rough Interface in the Presence of an Object . . . . . . . . . . . . . . . 107

7 Index mismatched di�usive/di�usive interfaces 115

7.1 Boundary Conditions for Index Mismatched media . . . . . . . . . . . . . . . . . 116

7.2 Re�ection and Transmission coe�cients . . . . . . . . . . . . . . . . . . . . . . . 119

7.2.1 Characterization of di�usive media . . . . . . . . . . . . . . . . . . . . . 121

7.3 Integral Equations for index mismatched di�usive media . . . . . . . . . . . . . 122

7.4 Numerical results for index mismatched media . . . . . . . . . . . . . . . . . . . 123

7.4.1 Comparison with Monte Carlo simulations . . . . . . . . . . . . . . . . . 123

7.5 Approximate Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 127

7.5.1 Approximate re�ection and transmission coe�cients . . . . . . . . . . . . 132

7.5.2 Approximate surface integrals . . . . . . . . . . . . . . . . . . . . . . . . 133

7.6 Rough Di�usive/Di�usive Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . 135

7.6.1 Discretization of Curved Boundaries . . . . . . . . . . . . . . . . . . . . 137

8 Di�usive/non-di�usive interfaces 140

8.1 Boundary conditions for plane interfaces . . . . . . . . . . . . . . . . . . . . . . 142

8.2 Re�ection and transmission coe�cients . . . . . . . . . . . . . . . . . . . . . . . 145

8.2.1 Black interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

8.2.2 Zero re�ection frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

CONTENTS 5

8.2.3 Characterization of Di�usive Media . . . . . . . . . . . . . . . . . . . . . 149

8.3 Boundary conditions at arbitrary interfaces . . . . . . . . . . . . . . . . . . . . . 150

8.3.1 Boundary conditions in two dimensions . . . . . . . . . . . . . . . . . . . 156

8.4 Scattering Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

8.4.1 Smooth boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

8.4.2 Rough surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

8.5 Non-di�usive volume versus di�usive object . . . . . . . . . . . . . . . . . . . . 169

9 Multi-layered di�usive media 172

9.1 Expression for a Slab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

9.2 Solving multiple layered media . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

9.3 Smoothly varying parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

10 Experimental Results 181

10.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

10.1.1 Treating the CCD images . . . . . . . . . . . . . . . . . . . . . . . . . . 183

10.1.2 Preliminary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

10.2 Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

10.2.1 The incident �eld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

10.2.2 Two layer expression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

10.2.3 One layer expression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

10.2.4 Semi-in�nite expression . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

10.2.5 Fitting the data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

10.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

10.4 Fitting independent measurements . . . . . . . . . . . . . . . . . . . . . . . . . 196

11 Conclusions 198

11.1 Future Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

A List of Symbols 202

B Summary of boundary conditions 204

B.1 Di�usive/di�usive interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

B.1.1 Approximate boundary conditions . . . . . . . . . . . . . . . . . . . . . . 205

B.1.2 Index matched conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 205

B.2 Di�usive/Non-di�usive interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . 206

B.2.1 Convex or plane surfaces (no light re-entering) . . . . . . . . . . . . . . . 207

B.2.2 Non-di�usive within di�usive volumes . . . . . . . . . . . . . . . . . . . . 207

Chapter 1

Introduction

Figure 1.1: Multiple scattering of light as it travels through the medium from the source to thedetector. As seen, the information of the object that may be present in the detected signal is�hidden� behind the multiple scattering processes.

Up to date, many techniques are available to diagnose lesions inside the human body, each

presenting its own advantages and hindrances. Among them, we �nd x-ray imaging, and

ultrasound [1]. Although these methods yield high spatial resolution and provide accurate

structural information, there is one main common drawback which distinguishes them: they

cannot provide optical information, i.e. they fail to characterize the exhibited structure. For

instance, it is not possible to determine by the use of these techniques alone whether a certain

spot corresponds to a tumor and whether this tumor is benign or malign. A technique which to

some degree can detect speci�c chemicals, is magnetic resonance imaging (MRI) but it cannot

detect some important elements, such as oxygen [2], and the need of super-conducting magnets

makes this technique highly expensive. Most importantly, techniques such as x-ray imaging

employ ionizing radiation, and it is suspected that 0:2% of the breast cancers1 detected are

induced by the use of x-rays in mammography [3]. Hence, x-rays cannot be employed as a

1This percentage must be regarded considering that x-ray screening is usually performed after the age of 40.

6

CHAPTER 1. INTRODUCTION 7

regular screening procedure in young patients. Therefore, an inexpensive method capable of

providing both structural and optical information, preferably in real time, by probing tissue

non-invasively, is needed. That is, a technique that may detect, localize and characterize a

hidden object within biological media, without altering the surrounding medium. A possible

answer that may ful�ll these requirements is the use of near infrared light [2, 3, 4].

Figure 1.2: Multiple scattering medium approximated to an in�nite homogeneous di�usivemedium of parameters, �a0 and D0.

The study of light transport through highly scattering media, such as living tissue, has been

the focus of recent research in the biophysics and medical physics communities2, mainly due

to its application to medical diagnosis [2, 3, 4, 5]. Many di�erent techniques that probe tissue

with infrared light are available, their main di�erence being the photon time of �ight detected:

ballistic (i.e. coherent light), weakly scattered, and multiple scattered or di�usive photons [2, 6].

It is the di�usive component of the detected light, that with which this thesis is concerned, the

main reason being that although all these techniques involve multiple scattering [7, 8, 9] (see

Fig. 1.1), light transport is accurately described by a simple equation in the case of di�usive

photons, namely, the di�usion equation [9]. It is in this case when light transport within

the turbid medium is better described as an statistical process, in which the magnitudes to

determine are the average intensity and the �ux density, rather than the electromagnetic vector

wavefunction. Therefore, the complex system depicted in Fig. 1.1, can be approximated by the

homogeneous medium shown in Fig. 1.2 described by a di�usion coe�cient, D0, an absorption

coe�cient �a0, and a refractive index n0. In a similar manner, the object will be characterized

by the corresponding parameters D1, �a1 and n1. A very interesting situation occurs when the

intensity of the photon source is modulated, thus there being created a di�use photon density

wave [10]. This signal has a well de�ned complex wave number and a wavelength3 in the

2See Ref. [3] for a historical review on the subject.3A common error is to assume that x-rays can detect breast cancer in its primary stages more e�ectively

than optical techniques, since the latter can detect microcalci�cations (� 100�m), when di�use photon densitywaves have wavelengths in the range of � 5cm. The truth is, that when microcalci�cations appear, a greatdeal of the surrounding tissue is receiving much more oxygen than normal [3], constituting an object optically


range of � 5cm, its propagation being governed by the Helmholtz equation. The use of di�use

photon density waves (DPDWs) has given rise to a new �eld in optical imaging, where one can

apply all the known aspects of the theory for scalar waves. In spite that we have reduced the

very complex (and, in general, unsolvable up to date) integro-di�erential equations to simple

di�usion equations, we are faced with new questions concerning these DPDWs: How do these

waves interact with interfaces? Up to what extent are multiple re�ections of these waves

important? What are the expressions and typical values of their corresponding re�ection and

transmission coe�cients? What is their spatial resolution content and, is characterization of the

buried object possible with these waves? Also, we �nd ourselves faced with some fundamental

questions such as: What is the correct way to describe the photon source? Since tissues are

highly absorbing media, what are the limits of validity of the di�usion approximation? What

happens in the proximity of non-di�usive volumes? Although a great deal of experiments have

been carried out to answer such questions, we have found that a rigorous theoretical scheme is

needed. Such is the purpose of this thesis.

Figure 1.3: Typical experimental con�guration for detection of breast lesions.

An example of applications of the use of di�use light to probe biological media is depicted in

Fig. 1.3. The patient's breast is placed in a basin �lled with a known di�usive medium, such as

intralipid, bounded by a resin of well known di�usive parameters. A collection of light emitting

�bers is placed at the basin rear, their intensity being modulated at a certain frequency !. A

�rst question arises concerning the optimum wavelength employed. In Fig. 1.4 the absorption

spectra of the main components in tissue is shown, following Ref. [4]. As seen, there exists a

spectral window in the 700� 900nm region (i.e. near infrared), where we �nd low absorption

values. From this �gure we see that di�erent wavelengths can also be employed to monitor

blood oxygenation [5]. Also, in terms of the modulation frequency !, we may distinguish two

regimes: the DC or constant illumination, and the AC or frequency dependent. The di�usive

component of pulsed sources, namely, in the time domain, must also be considered, but since

by Fourier transform of the data the theory for these sources reduces to that of modulated

visible, with sizes of the order of several mm. The main objective of the optical techniques is to detect theselesions before the microcalci�cations appear.


Figure 1.4: Absorption spectra of the main components of tissue, namely, hemoglobin and water,following Ref. [4].

sources, it will thus not be considered in this thesis.

In Fig. 1.3 we may distinguish the following interfaces: 1) Di�usive/di�usive interfaces

between the intralipid and the breast and resin, and between the breast and the object. 2)

Di�usive/non-di�usive interfaces between the resin and the outer medium, typically air. 3)

A Di�usive/non-di�usive interface between the intralipid and a possible �air bubble� (shown

in Fig. 1.3 as a clear region within the intralipid), i.e. a non-di�usive volume within the

di�usive medium. The DPDW will interact di�erently with these interfaces, and the refractive

index will play an important role. Therefore, an analysis of the detected image at the plane

of measurement (a CCD in the example shown in Fig. 1.3), requires an accurate knowledge of

the interaction of the DPDWs with these interfaces.

This thesis is organized as follows: a �rst part is devoted to the propagation of DPDWs,

a second part pertains to the interaction of these waves with the di�erent possible interfaces

aforementioned above, and a third part deals with the characterization of di�usive media. In

Chapter 1 the di�usion equation is derived from the radiative transfer equation, and a study of

the approximations involved in deriving it, as well as their implications, are presented. Then,

in Chapter 3, the theory of the propagation of these DPDWs is presented, and a study of

their di�raction, illustrated with appertures, is put forward. In Chapter 4 the scattering in-

tegral equations of DPDWs in complex geometries are derived, and a method to rigorously

solve these equations is presented. In Chapter 5, the limits of spatial resolution of DPDWs

are derived, taking into account the e�ect of noise in the detected signals. In Chapter 6, we

study the interaction of DPDWs with index matched di�usive/di�usive interfaces, presenting

the corresponding re�ection and transmission coe�cients, and performing a numerical study

of the detection of buried objects within di�usive media with rough interfaces. In Chapter

7, we study the e�ect of index mismatch in di�usive/di�usive arbitrary interfaces, we derive


the corresponding re�ection and transmission coe�cients, and present an approximation to the

boundary conditions. The limits of this approximation are studied, contrasting the results with

Monte Carlo simulations. In Chapter 8, di�usive/non-di�usive interfaces are considered, deriv-

ing the corresponding re�ection and transmission coe�cients. Then, the boundary conditions

for non-di�usive volumes within di�usive media are derived within the di�usion approxima-

tion, and results are compared with Monte Carlo simulations for di�erent geometries. This

has the aim of simulating light transport in the brain. In Chapter 9, a method for rigorously

solving multiple layered di�usive media by means of the re�ection and transmission coe�cients

is presented, studying the case of smoothly varying di�usive parameters. In Chapter 10, all

the theory presented in this thesis is put to test, characterizing experimental data by means

of a novel and accurate method which employs the re�ection and transmission coe�cients of

DPDWs. Finally, the conclusions of this thesis and future perspectives are presented in Chapter

11.

Chapter 2

The Di�usion Equation

In this chapter, basic concepts such as speci�c intensity, and quantities related to it, are pre-

sented. This is done in order to derive the expression for the di�usion equation and gain a

better insight of the approximations that it implicitly involves. Even though great part of this

section can be found in Chapters 7 and 9 of Ref. [9], which deal with the RTE, for the sake of

clarity they are here rewritten, using the same notation. Also, in Ref. [9] the results are derived

from the time-independent RTE, whereas here they are derived from the time-dependent RTE.

In any case, due to the inherent complexity of the concept of speci�c intensity, I found out that

following and understanding all the steps presented was a di�cult matter. Hence, I decided to

revise and present in this chapter, not only the derivation of the di�usion approximation, but

also many of the concepts which took me a long time to comprehend, in hope of helping those

who �nd themselves in the same position. The basic concepts and formulae shown in this sec-

tion are key to the derivation of the boundary conditions which shall be employed throughout

the present work, and are an extremely useful tool not only for improving the present boundary

conditions, but for �nding new ones.

2.1 Speci�c Intensity

A quantity to characterize the light �ow of energy and its interaction with the medium is the

speci�c intensity I(r; s)1, which represents the average power �ux at point r which �ows in the

direction s, and therefore has units2 [Wcm�2sr�1] (see Fig. 2.1).

Let us consider the energy density u(r). The amount of energy in time dt leaving an area

dS in a direction normal to it within a solid angle d = d� sin �d�, is I �dS �d �dt. This energyshould occupy a volume v � dS � dt where v is the wave speed. Therefore, the energy density du

1The speci�c intensity can be envisaged as a statistical average of the randomly varying Poynting vector.Even so, this quantity represents the averaged power �ow and no consideration will be given to the wave�uctuations associated to this power �ow.

2Usually I(r; s) and all the related quantities are de�ned per unit frequency interval , i.e. I(r; s) would haveunits [Wcm�2sr�1Hz�1]. This is useful when dealing with non-monochromatic waves, but will be dropped inthe present study since we shall allways work with monochromatic waves.

11

CHAPTER 2. THE DIFFUSION EQUATION 12

Figure 2.1: Representation of the speci�c intensity I(r; s) at the point r �owing in the directions within the solid angle d.

within d is given by:

du =I � dS � d � dtv � dS � dt =

I(r; s)d

v; (2.1)

which has units [Jcm�3sr�1], and therefore, adding the energy due to the radiation in all

directions, the energy density is:

u(r) =n

c

Z4�I(r; s)d ; (2.2)

which has units of [Jcm�3], and where n is the refractive index of the medium being c the speed

of light in vacuum. In terms of the speci�c intensity, the average intensity U , and the total �ux

density J are de�ned as3:

U(r) =Z4�I(r; s)d ; (2.3)

J(r) =Z4�I(r; s)sd ; (2.4)

where both U and J have units [W=cm2]. The total �ux density that �ows through an area

element dS = ndS then is:

Jn(r) = J(r) � n =Z4�I(r; s)s � nd ; (2.5)

where n is the unit outward normal to the area element dS. The total �ux passing through

dS can also be de�ned as a sum of the upward �ux, J+, and the downward �ux, J�, these

quantities being:

J+(r; n) =Z(2�)+

I(r; s)s � nd ; (2.6)

J�(r; n) =Z(2�)�

I(r; s)s � (�n)d ; (2.7)

where, as shown in Fig. 2.2, (2�)+ stands for integration in 0 � � � �=2 and (2�)�corresponds

3When de�ning Average Intensity, it is also common to normalize it to the unit solid angle 4� (see Ref.[9]).


to integration in �=2 � � � 3�=2 . In terms of the upward and downward �ux, the total �ux

density that traverses dS is expressed as:

Jn(r) = J+(r)� J�(r) : (2.8)

Figure 2.2: Schematic view of the geometry with the upward (J+) and downward (J�) density�ux at an interface.

Therefore, the amount of power dp emitted from dS into a solid angle d can be written

as:

dp(r; s) = I(r; s)n � sdSd : (2.9)

The total power emitted from dS would therefore be:

dP (r) =Z4�dp(r; s) =

Z4�I(r; s)n � sdSd = Jn(r)dS : (2.10)

Both dp(r; s) and dP (r) are measured in [W ].

Let us assume that there is an isotropic point source at point rs. Therefore, the speci�c

intensity will have no angular dependence, i.e. I(r; s) = I(r). In that case, we can rewrite Eq.

(2.9) as:

dp(rs; �) = I(rs) cos �dSd ; (2.11)

where cos � = n � s. This relationship (2.11) is Lambert's cosine law [11], and is frequently used

to describe light emerging from a strong scatterer into a non-scattering medium.

2.1.1 Re�ection and transmission of the speci�c intensity

Let the area S separate two di�erent media with di�erent refractive indices (see Fig. 2.3): an

upper medium with index n0 and a lower medium with index n1. Then, the total downwardJ�

and upward J+ �uxes through dS by means of Eq. (2.4) are:

J+(r) =Z(2�)+

I t1!0(r; s)s � nd ; (2.12)

J�(r) =Z(2�)�

I t0!1(r;�s)(�s � n)d ; (2.13)


Figure 2.3: Schematic view of the geometry with the upward (J+) and downward (J�) density�ux at an index-mismatched interface.

where I ti!j represents the speci�c intensity transmitted from medium i into medium j.

Returning to Eq. (2.10), we can now de�ne the upward and downward �uxes in Eqs. (2.12)

and (2.13) as:

J+(r) =dP t

1!0(r)

dS=

1

dS

Z2�dpt1!0(r; s) ; (2.14)

J�(r) =dP t

0!1(r)

dS=

1

dS

Z2�dpt0!1(r; s) : (2.15)

Therefore, the upward and downward �ux through dS can be considered as the transmitted

power in those directions, per unit area. In terms of energy conservation, we then write the

total incident power at dS as a sum of the re�ected and transmitted powers, i.e.:

dpi(r; si) = dpr(r; sr) + dpt(r; st) ; (2.16)

where [see Fig. 2.4],

dpr(r; st) = R(�i)dpi(r; si) ; (2.17)

dpt(r; st) = T (�i)dpi(r; si) : (2.18)

In Eqs. (2.17) and (2.18) R is the power re�ectivity and T represents the power transmissivity .

We shall �rst �nd the expressions of R and T in terms of the Fresnel coe�cients[11],

rk =n0 cos �t � n1 cos �in0 cos �t + n1 cos �i

; tk =2n0 cos �i

n0 cos �t + n1 cos �i; (2.19)

r? =n0 cos �i � n1 cos �tn0 cos �i + n1 cos �t

; t? =2n0 cos �i

n1 cos �t + n0 cos �i;

where k and ? denote TM and TE polarization, respectively. For a plane electromagnetic wave

Ei incident on a plane boundary, we have that r = Er=Ei, and t = Et=Ei. Therefore, we would


expect the re�ected speci�c intensity to hold the relationship:

Ir = jrj2Ii : (2.20)

Now, we must �nd the link between It and Ii; since this is not4 It = jtj2Ii. In these cases, r

and t denote rk or r?, and tk or t?, respectively, depending on whether the polarization is TM

or TE. If the wave is completely unpolarized, then we have that I = (Ik + I?)=2, i.e. :

jrj2 = 1

2(jrkj2 + jr?j2) ; (2.21)

jtj2 = 1

2(jtkj2 + jt?j2) : (2.22)

From now on, we shall always consider light completely unpolarized. R and T are de�ned in

Eqs. (2.17) and (2.18) by the ratio of the transmitted and re�ected power to the incident power

normal to the surface:

T =dpt

dpi=

It(r; st) cos �tdSdt

Ii(r; si) cos �idSdi

=�n0n1

�2 It(r; st)Ii(r; st)

; (2.23)

R =dpr

dpi=

Ir(r; sr) cos �rdSdr

Ii(r; si) cos �idSdi= jr2j ; (2.24)

where we have made use of Eq. (2.20), and jrj2 is as stated in Eq. (2.21), in the case of

unpolarized light. In Eqs. (2.23) and (2.24) we have made use of the relationship between di

and dr;t :

di = dr ;

di =�n1n0

�2 cos �tcos �i

dt ; (2.25)

which is a direct consequence of Snell's Law:

n0 sin �i = n1 sin �t ;

n0 cos �id�i = n1 cos �td�t : (2.26)

and the fact that di = sin �id�id�i, dt = sin �td�td�t, noting that d�i = d�t. Now, since from

Eq. (2.16), and Eqs. (2.17) and (2.18) we have that:

R + T = 1 ; (2.27)

by means of Eq. (2.24) we obtain T = 1� jrj2. If we introduce the values given in Eqs. (2.21)

4The reason for this is that the speci�c intensity is related to the Pointing vector, and not directly to theintensity as de�ned in electromagnetic theory, i. e. jEj2: Therefore, what we are looking at is transmission ofpower.


and Eqs. (2.19), after some basic algebra we obtain:

T =n1 cos �tn0 cos �i

jtj2 ; (2.28)

R = jrj2 : (2.29)

Figure 2.4: Re�ection and transmission of speci�c intensity.

In order to �nd the relationship between the speci�c intensity and R and T , Eq. (2.16) can

be rewritten in terms of the speci�c intensity by means of Eq. (2.9) as (see Fig. 2.4):

Ii(r; si)dS cos �idi = Ir(r; sr)dS cos �rdr + It(r; st)dS cos �tdt ; (2.30)

from which, by means of Eqs. (2.25) and (2.26), we obtain the following relationship:

Ii(r; si)

n20=

Ir(r; sr)

n20+It(r; st)

n21; (2.31)

that is,

Ii(r; si) = Ir(r; sr) +�n0n1

�2It(r; st) : (2.32)

Since Ir = jrj2Ii, by means of Eqs. (2.28) and (2.29) we obtain:

It(r; st) =�n1n0

�2T (�i)Ii(r; si) =

�n1n0

�3 cos �tcos �i

jtj2Ii(r; si) ; (2.33)

Ir(r; sr) = R(�i)Ii(r; si) = jrj2Ii(r; si) : (2.34)

By means of the relationship shown in Eqs. (2.17) and (2.18), we can rewrite Eqs. (2.14)

and (2.15) as:

J+(r) =1

dS

Z2�[1� R1!0(�i)]dp

i1!0(r; si); (2.35)


J�(r) =1

dS

Z2�[1� R0!1(�i)]dp

i0!1(r; si) ; (2.36)

where we have made use of Eq. (2.27), and therefore, on introducing the expression for dp from

Eq. (2.9), the upward and downward �uxes are expressed as:

J+(r) =Z2�[1� R1!0(�i)]I1(r; si)si � ndi ; (2.37)

J�(r) =Z2�[1� R0!1(�i)]I0(r; si)si � ndi ; (2.38)

where I0 and I1 represent the speci�c intensities incident on S from medium 0 and 1, respec-

tively. It should be noticed that, due to the relationships between dpt and dpi through the

Fresnel coe�cients, Eqs. (2.17) and (2.18), the integration in Eqs. (2.37) and (2.38) is per-

formed over the incident angles, whereas in Eqs. (2.12) and (2.13) the integration is done

over the transmitted angles. Namely, in Eqs. (2.37) and (2.38), Ri!j(�i) represents the power

re�ectivity corresponding to the angle of incidence �i going from medium i to medium j, and

di = d�i sin �id�i. Eqs. (2.37) and (2.38) are general for the RTE, since the scattering or

absorbing speci�c properties of media 0 and 1 have not yet been introduced.

2.2 Radiative Transfer Equation

Once we have de�ned the speci�c intensity and all the quantities of interest related to it, the

time-dependent equation which describes the propagation of light intensity, i.e. the radiative

transfer equation[9, 7, 12] (RTE) is5:

n

c

@I(r; s)

@t= �s � rI(r; s)� �tI(r; s) +

�t4�

Z4�p(s; s0)I(r; s0)d0 + �(r; s) ; (2.39)

where �t is the total macroscopic-cross section (also called the transport coe�cient , or total

attenuation coe�cient), with units of [cm�1], and is de�ned as:

�t = ��t = �(�a + �s) : (2.40)

In Eq. (2.40) � is the density of scatterers, and �a, �s are the absorption and scattering cross-

section, respectively, both measured in [cm2]. In Eq. (2.39) we have omitted the temporal

dependence of I, and assumed �xed stationary scatterers; �(r; s) is the power radiated by

the medium per unit volume and per unit solid angle in direction s, and p(s; s0) is the phase

function6. By means of Eq. (2.40) �t can therefore be separated into �t = �s + �a, where �s5A detailed derivation of this equation can be found in many textbooks, see for example Ch. 7 of [9] or [12].6The name �phase function� has its origin in astrophysics, and is not directly related to the �phase� as de�ned

in electromagnetism.


and �a are the scattering and absorption coe�cients, respectively7. In terms of �s, we de�ne

the scattering mean free path lsc (or just mean free path ) as [13]:

lsc =1

�s; (2.41)

which is the characteristic distance between two scattering events. Note that the de�nition

of mean free path does not include absorption. The reason can be found in the average time

of �ight , i.e the mean time between two scattering events, tsc, de�ned as tsc = lscn=c. This

quantity has two contributions [13, 14]: the travelling time from one scatterer to the next, and

the dwell time spent in the neighborhood of one scatterer. That is to say, absorption does not

change the average time of �ight, it just diminishes the intensity8 . For detailed information

on the statistics of the optical pathlength in tissues, we refer to Ref. [15]. In terms of the

absorption coe�cient �a, we can de�ne the absorption length9 as la = 1=�a, which is the

distance at which the light intensity decreases by a factor of e, i.e. I / exp[��ajrj].In Eq. (2.39) the phase function p(s; s0) holds the following relationships:

p(s; s0) =4�

�tjF(s; s0)j2 ; (2.42)

1

4�

Z4�p(s; s)d = W0 =

�s�t

; (2.43)

where W0 is the albedo10 of a single particle, and F(s; s0) is the scattering amplitude. The

phase function can also be de�ned in such a way that its solid angle integral Eq. (2.43) is equal

to one, i.e. normalize p(s; s0) to the albedo. In this case the expression for the RTE, Eq. (2.39),

would have a factor �s instead of a factor �t in the term involving the phase function. The

albedo, phase function, and other related quantities are very well explained in [11] and [16]

among others, quantities which are basic for the description of scattering events.

7At this point it should be remarked that the value that should be introduced for n in Eq.(2.39) is still amatter of debate. There is an opinion that its value should be the average refractive index of the homogeneousmedium and the particles. As presented in Eq. (2.39), it is quite clear that n=c represents the speed of light inpropagation between scattering events. Therefore, n can only represent the refractive index of the homogeneousmedium, i.e. the medium in which the particles are embedded, and we shall therefore always use this value.Another matter which is still not clear pertains to the relationship between both �t and �a with the imaginarypart of the refractive index.

8Other authors de�ne the mean free path as lt = 1=�t, i.e. it includes absorption. In any case it is just amatter of di�erent terminology, but when lt = 1=�t is de�ned I �nd more complicated to understand what ist = lt=c, since it cannot be the time between scattering events, now that it includes absorption, but the time of�ight needed for the intensity to decay a factor of e in the forward direction. For example, Ishimaru [9] leaveseverything in terms of the cross-sections, which has a more understandable signi�cance.

9Care must be taken not to confuse this term with di�usion length (sometimes also called absorption length)de�ned in Sec. 2.3.2, which relates the decrease of the average intensity with distance.

10from the Latin albus, which means white, i.e. it represents the �whiteness� of a particle, therefore itscapability to scatter light.


If we integrate over all 4� of solid angle in the RTE (2.39), we obtain11:

1

c

@

@t

Z4�I(r; s)d = �r �

Z4�I(r; s)sd� �a

Z4�I(r; s)d +

Z4��(r; s)d ; (2.44)

which by means of Eq.(2.3) and Eq.(2.4), results in the following equation for the total �ux

density, which is the equation of �ux conservation:

1

c

@U(r)

@t+r � J(r) + �aU(r) = E(r) : (2.45)

In Eq. (2.45) we de�ne:

E(r) =Z4��(r; s)d ; (2.46)

measured in [Wcm�3], as the power generated per unit volume, and where

Ea(r) = �a

Z4�I(r; s)d = �aU(r) ; (2.47)

is the total absorbed power per unit volume, also measured in [Wcm�3].

2.2.1 Radiative Transfer in a Non-Scattering Medium

The expression for the RTE in a non-scattering medium is the equivalent to Eq. (2.39) in the

absence of scattering particles:

n

c

@I(r; s)

@t+ s � rI(r; s) + �aI(r; s) = �(r; s) ; (2.48)

where �a is the absorption coe�cient of the non-scattering medium, n is its refractive index,

and c is the speed of light in vacuum. For a continuous stationary source located at r0, the

solution to Eq. (2.48) is:

I(r; ur�r0) = �(r0; ur�r0) exp[��ajr� r0j] ; (2.49)

where

ur�r0 =r� r0jr� r0j : (2.50)

In the case of a source whose intensity is modulated with frequency !, the solution to Eq.

(2.48) is:

I(r; ur�r0) = �(r0; ur�r0) exp��a + i

!n

c

�jr� r0j

�: (2.51)

Care must be taken not to confuse the intensity modulation frequency ! with the frequency

of the photon source �. Consider the following example: a typical experimental con�guration

11In order to reach this expression we have made use of the following properties: s � rI(r; s) = r � [sI(r; s)] ;�t

4�

R4�

R4�p(s;s0)I(r; s0)d0d = �t

4�U(r)R4�p(s; s)d = �sU(r).


would consist of modulating at 200MHz the intensity of a laser tuned at 780nm. The frequency

of the photon source would therefore be � = c=780nm ' 3:8� 1014Hz. A di�erence of 6 orders

of magnitude can therefore be observed between ! and �.

2.2.2 Invariance of the speci�c intensity

The spatial invariance of the speci�c intensity directly arises from the conservation of power,

Eq. (2.9). As seen from Fig. 2.5, by means of Eq. (2.9) the power emitted by dS1 into d1

in terms of I at r1 (I1), is expressed as dpem = I1(n1 � ur2�r1)dS1d1 = I1 cos �dS1d1, where

ur2�r1 is as de�ned in Eq. (2.50). The power received by dS2 through d2 in terms of I at r2

(I2), is dprec = I2(n2 � ur1�r2)dS2d2 = I2dS2d2. In a non-absorbing free space, dpem = dprec:

Figure 2.5: Spatial invariance of the speci�c intensity.

Now, since the solid angles can be written as:

d1 =dS2n2 � ur2�r1jr2 � r1j2 =

dS2

jr2 � r1j2 ;

d2 =dS1n1 � ur1�r2jr1 � r2j2 =

dS1 cos �

jr1 � r2j2 ; (2.52)

then dpem = dprec = I1 cos �dS1dS2=jr2 � r1j2 = I2dS2dS1 cos �=jr1 � r2j2, which yields I1 = I2.

Hence, we have invariance of the speci�c intensity along the ray in free space. If we consider

such a relationship in a medium with absorption �a, we must make use of Eq. (2.49). In this

case, the power received at dS2, emitted from dS1, would be:

dprec(r2) = I(r1; ur2�r1) exp[��ajr2 � r1j] cos �dS1d1 : (2.53)


2.3 Di�usion Approximation

Let us address the special case in which there is a high concentration of scatterers, so that

the contribution of single-scattered light is very small. In this case light transport within

the medium can be accurately described from the assumption that all contributions are from

multiple-scattered light, and thus the intensity can be considered di�use12.

2.3.1 Angular dependence of the speci�c intensity

Figure 2.6: Angular distribution of the speci�c intensity in the di�usion approximation (thevalue of J is enhanced for the sake of clarity).

We shall assume that this di�use intensity encounters many particles and is scattered almost

uniformly in all directions, and therefore its angular distribution is almost uniform. It cannot

be constant because then the �ux J would be zero. We will de�ne as sJ the direction of the

di�use �ux vector J, i.e. J(r) = J(r)sJ. In this manner, writing I(r; s) in terms of s � sJ up to

�rst order, we obtain:

I(r; s) ' �U(r) + �J(r)s � sJ + ::: ; (2.54)

We can determine the constant � by introducing into Eq. (2.3) the expression for I given

in Eq. (2.54):

U(r) = �U(r)Z4�d + �J(r)

Z4�s � sJd ;

U(r) = �U(r)4� ;

which yields � = 1=4�. Proceeding in a similar way, on introducing Eq. (2.54) into Eq. (2.4)

12Di�use light conveys a wider signi�cance than just that from the di�usion approximation. This can be seenin Ref. [9], where the di�usion approximation is a particular case of di�use light. In general, di�use light isconsidered all highly incoherent light whose direction of propagation is best determined statistically, as opposedto coherent light.


for J :

J(r) = J(r) � sJ = �U(r)Zs � sJd + �J(r)

Z4�[s � sJ]2d ;

J(r) = �J(r)4�

3;

which yields the value � = 3=4�. Therefore, in the di�usion approximation the speci�c intensity

is written as:

I(r; s) ' U(r)

4�+

3

4�J(r)s � sJ : (2.55)

2.3.2 Derivation of the Di�usion Equation

We shall assume that the phase function p(s; s0) only depends on the angle between s and s0,

i.e. p(s; s0) = p(s � s0) (see Sec. 2.3.3). If we substitute Eq.(2.55) into Eq.(2.39) we obtain:

n

c

1

4�

@U(r)

@t+n

c

3

4�

@[J(r) � s]@t

= � 1

4�s � rU(r)� 3

4�s � r[J(r) � s]

� 1

4��tU(r)� 3

4��tJ(r) � s + 1

4��tW0U(r) +

3

4��tp1J(r) � s + �(r; s) ; (2.56)

where p1 is:

p1 =1

4�

Z4�p(s � s0)s � s0d0 ; (2.57)

and represents the averaged forward scattering (s � s0 > 0) minus the backward scattering

(s � s0 < 0) of a single particle. By using Eq.(2.43), we can rewrite p1 as p1 = W0g, where g:

g =< cos � >=

R4� p(s � s0)s � s0d0R

4� p(s � s0)d0; (2.58)

is the average cosine of the scattering angle. Therefore g is a quantity which expresses the

anisotropy of the scattered light on interaction with the particle, and, as such, is called the

anisotropy factor .

The phase function can often be approximated by the following form involving the albedo

W0 and g:

p(cos �) =W0(1� g2)

(1 + g2 � 2g cos �)3=2: (2.59)

Eq. (2.59) is the well-known Henyey-Greenstein [17] formula, and constitutes the most com-

monly used approximation for the phase function in biological media.

By introducing the new expression for p1 into Eq. (2.56) and grouping terms:

n

c

@U(r)

@t+ 3

n

c

@[J(r) � s]@t

= �s � rU(r)� 3s � r[J(r) � s]�(�t � �s)U(r)� 3(�t � �sg)J(r) � s+ 4��(r; s) ;


on multiplying by s and integrating over 4� we obtain13 :

rU(r) = �3(�0s + �a)J(r)� 3n

c

@J(r)

@t+Z�(r; s)sd ; (2.60)

where we have de�ned:

�0s = �s � (1� g) ; (2.61)

�0s being the reduced scattering coe�cient . In terms of �s, we shall de�ne the transport mean

free path ltr [see Eq. (2.41)][13]:

ltr =1

�s0=

lsc1� g

; (2.62)

which takes into account the anisotropy of the scattered light. This term can be understood

as follows: the characteristic distance between two scattering events is lsc: Now suppose that

scattering is rather ine�ective, so that at each scattering event the direction of propagation

does not change much. This implies that the overall average transport length within which the

radiation gets lost must be larger than lsc: Therefore, the factor (1�g) in Eq. (2.62) takes care

of this fact, since if scattering were isotropic, then g = 0 and ltr = lsc: On the other hand, if

scattering were highly anisotropic, g � 1 and ltr ! 1, i.e. radiation does not get lost due to

scattering processes in the direction of light propagation 14.

Let us examine closely Eq. (2.60) before proceeding with the next approximation. This

equation can be rewritten as:

rU(r) = �3�0s"n

c�0s

@

@t+ �a

c

n

!+ 1

#J(r) +

Z�(r; s)sd : (2.63)

In Eq. (2.63) we can see two characteristic times:

ttr =n

c�0s=

n

cltr ;

which is the average time required to travel the transport mean free path distance, and

ta =n

c�a;

which is the characteristic time of �ux J change due to absorption. At this point, we shall

make the next approximation: we will assume that variations in the di�use total �ux take

place over a time scale much larger than the lapse between scattering events on particles of the

medium, and also, that the time of change of the total �ux due to absorption is much larger

13For any vector A,R4� s � (s �A)d = 4�

3 A, andR4� s � [s �r(A � s)]d = 0. Also

R4�

@U@tsd = @U

@t

R4� sd = 0

14At this point we have introduced a change with respect to Ishimaru [9] in the de�nitions, since he de�nesthe transport coe�cient �tr = �0s + �a , which as de�ned in Eq. (2.62), would be �tr = �0s. We shall not usethis notation since it would cause confusion with the term ltr since as we have de�ned it, it is not ltr = 1=�tr,but ltr = 1=�0s.


than the time between scattering events. This means that in Eq. (2.63) we can neglect the

term (@=@t + �ac=n)[18, 19]. This gives us:

J(r) = � 1

3�0srU(r) + JE ; (2.64)

where JE is the density �ux of power E generated by the medium [see Eq. (2.46)]:

JE =1

3�0s

Z�(r; s)sd : (2.65)

The coe�cient in Eq. (2.64) that relates rU and J is called the di�usion coe�cient. It is

de�ned as:

D =1

3�0s=

1

3�s(1� g)=

ltr3; (2.66)

and has units of [cm]. It is still a matter of debate whether D should include �a or not. The

reasons for this will be discussed in Sec. 2.3.4. In any case, this does not a�ect the present work,

since everything will be derived in terms of D. It is also usual to �nd the di�usion coe�cient

de�ned as D = ltrv=3, where v is the speed of light in the medium. This is commonly used

when deriving the di�usion approximation in terms of the energy density instead of the average

intensity, since they are related by u = U=v [see Eq. (2.2)]. Speci�cally, this de�nition is used

in particle transport such as neutrons [20], charge transport [21], and also in studies of light

transport within index-matched regions [13]. It is not useful, however, when dealing with index

mismatched regions, since it signi�cantly complicates the formulation15.

On substituting the expression for J in the di�usion approximation, Eq. (2.64), into the

general expression for �ux conservation, Eq. (2.45) (see page 19) we obtain the following

di�erential equation for the di�use average intensity:

1

c

@U(r)

@t+r[�DrU(r)] + �aU(r) = E(r) +rJE ; (2.67)

where we have explicitly left D inside the brackets, since when D is r dependent, the gradient

operator must be correctly applied. In the right hand side of Eq. (2.67), E and rJE can be

envisaged as the monopolar and dipolar moments of the source, respectively [23].

For the sake of simplicity, we shall assume that the sources are isotropic, i.e. the incident

intensities in Eq. (2.46) do not depend on s. Then, in Eq. (2.64), JE = 0. This means that

Eq. (2.64) transforms into:

J(r) = �DrU(r) ; (2.68)

15This of course does not occur when particle transport is addressed, since particle hits are not ruled byFresnel's coe�cients. In order to convert any of the quantities related to the speci�c intensity into somethingrelated to quantities expressed by Boltzmann's transport equation [22], simply substitute the energy unit Joulefor photon=h�, where h is plank's constant, and � is the photon's frequency. In this manner, the energy densityu(r) in Eq. (2.2) measured in [Jcm�3] becomes the photon density N(r) = u(r)=h�, measured in [photons=cm3].


which is Fick's law for di�usion of the average intensity, and Eq. (2.67) transforms into the

di�usion equation:

1

c

@U(r)

@t�Dr2U(r) + �aU(r) = E(r) +rD � rU(r) : (2.69)

In an in�nite homogeneous medium in which both D and �a are constant throughout the

medium, the di�usion equation reduces to the most common expression16:

1

c

@U(r)

@t�Dr2U(r) + �aU(r) = E(r) : (2.70)

At this point we shall de�ne a new quantity. Suppose we have a continuous source of photons

at some point rs. In this case, the solution to Eq. (2.70) is:

U(r) / exp[�jr� rsjLd

] ;

where we have de�ned the di�usion length17 Ldas:

Ld =

sD

�a(2.71)

The di�usion length Ld, as opposed to the absorption length la (see Sec. 2.2), is the distance at

which the average intensity decreases by a factor of e. It is important to understand that this

quantity is basic for the de�nition of the di�usion coe�cient, since, in terms of Ld it is de�ned

as D = L2d�a (see Ref. [24] for more details on this subject). Ref. [12] gives the dispersion

relation that determines Ld for isotropic scattering, used in Ref. [24]:

W0Ld�t2

lnLd�t + 1

Ld�t � 1= 1 : (2.72)

2.3.3 Approximations taken in the Di�usion Equation

Many studies have been performed in order to �nd the limits of validity of the di�usion approx-

imation (DA) as regards the experimental setups [25, 26], and the fundamental limits of the

DA [27, 28]. One key aspect in understanding the DA is that concerning the approximations

and assumptions involved in its derivation. It is quite unusual to �nd these approximations

explicitly, and therefore here they are rewritten. Even though some of them appear in Refs.

[9, 29], the assumptions and their implications are not explained in detail.

16The di�usion equation as shown here was derived for 3D media. Its derivation for a 3D medium with a 2Dgeometry, would requiere to go back to the RTE Eq. (2.39) and derive it from the start, taking great care withthe solid angle integrals, since their conversion to 2D is not straightforward. In that case, the expression for thedi�usion coe�cient Eq. (2.66) would change, since it was derived for 3D. A conversion to 2D is done within thespeci�c intensity formulation in Sec. 8.3.1 for the case of a di�use/non-di�use interface.

17In Ref. [13], Ld is de�ned as the absorption length.


Generally, as stated in Ref. [13], the length scales in which the di�usion approximation

holds, are:

�� ltr � L� Ld ;

where � is the wavelength of the incident �eld, ltr is the transport mean free path, L is the

system's length, and Ld is the di�usion length. These length scales play a fundamental role in

the approximations used. We here enumerate all the approximations taken in order to derive

Eq. (2.70) and discuss their implications:

1. First order angular dependence: This is the basic assumption of the di�usion ap-

proximation. It is taken when we consider �rst order angular dependence of the speci�c

intensity, see Eq. (2.55) and Fig. 2.6:

I(r; s) =1

4�U(r) +

3

4�J(r)sJ � s ; (2.73)

which is equivalent to a Taylor's expansion of I in terms of sJ � s = cos �, truncated up to

�rst order. It can also be seen as the truncation to �rst order of the series in Legendre

polynomials:

Pn(cos �) =nX

m=0

an cosm� = a0 + a1 cos � + ::: (2.74)

By virtue of Eq. (2.74), Eq. (2.73) is also called the P1 approximation to the RTE.

Corrections to the DA consist of adding terms to the series, then calling each successive

approximation P2, P3, etc. The derivation of the Pn approximations is explained in detail

in Refs. [29, 30]. In fact, when dealing with highly scattering media such as most biological

tissues, the P1 approximation is more than adequate to describe photon transport in most

practical cases. In order for this approximation to remain valid, the following relationship

must hold:

U(r)� 3J(r) : (2.75)

Eq. (2.75) is extensively used to estimate the limits of validity of the DA. It can be

rewritten in terms of Eq. (2.68) as:

ltrjrU(r)j � U(r) ; (2.76)

which means that the spatial change of the di�use average intensity U must be su�ciently

small within the range of the transport mean free path ltr; this is the condition that has

been accepted as that of applicability for the di�usion approximation [31]. The basic

situations which make this approximation break down are:

� Photon transport near a non-di�usive region18,

18�near�, whenever speaking in the DA context, means of the order of ltr.


� System lengths L of the order of ltr[27],

� Photon transport at distances of the order of ltr from the source.

In Ref. [27] it is experimentally demonstrated that for system lengths of 10ltr, the average

time of photon arrival is about 0:9 times the one predicted by di�usion theory. The same

conclusion was reached in Ref. [32], where a study of the distance from the source at

which the photons can be considered di�use, and its model, is presented. In all these

cases in which the DA is violated, higher orders of the Pn approximation are commonly

used, since in these cases the angular dependence of the speci�c intensity is not accurately

expressed by the DA. Even so, as it will be demonstrated in Sec. 8.3, in some practical

con�gurations the DA can still be accurate in some of these limiting cases.

2. Angular dependence of the phase function: As seen in Sec. 2.3.2, the following

approximation in the phase function was taken:

p(s; s0) = p(s � s0) ; (2.77)

which means that the scattered amplitude measured at s0 when inciding from s, only

depends on the angle between both vectors, i.e. cos � = s � s0. In order for this to be

true, the particles must be spherical. This approximation breaks down whenever the

scattering amplitude of the particle di�ers from that of a sphere.

3. Small particle sizes: Once the angular dependence of the phase function has been

established as in Eq. (2.77), the way of modeling the interaction of the incident wave

with the particle requires the average cosine g, Eq. (2.58). One way of describing this

interaction is through the Henyey-Greenstein function, Eq. (2.59). If this interaction

were explained as Rayleigh scattering, we would use instead the phase function:

p(s � s) = 3

4(1 + cos2 �) ; (2.78)

which only scatters the TM component of the wave. In any case, in order that both Eq.

(2.59) and Eq. (2.78) be correct requires particle sizes much smaller than the wavelength

�; i.e. a� � , where a is the particle radius. Therefore this approximation breaks down

for particle sizes of in the order of �: Then the phase function has a much more complex

expression, and cannot be described in terms of Eq. (2.59) or Eq. (2.78). A theoretical

and experimental study of the limits of validity of the DA in terms of the particle size

can be found in Ref. [33]. As seen in this reference, particle sizes of the order of a � �=5

reach very good agreement with the DA.

4. Small values of g: Values of g � 1 mean that the particle is almost transparent and all

incident light is practically transmitted in the forward direction. Values of g � 0 mean


that the particle scatters isotropically in all directions, whatever the angle of incidence.

Even so, as stated in Ref. [33], small particles compared to the wavelength can scatter

isotropically (i.e. have values of g � 0) even when they have a low value of �s: For

example, in Ref. [33] particles with radius a � �=6 give rise to g � 0:09, whereas the

same particles with a � 3� show values g � 0:9.

Figure 2.7: (a) Values of the phase function p(s�s0), Eq. (2.57), in terms the scattering angle, aspredicted by the Henyey-Greenstein formula, Eq. (2.59), for di�erent values of g. (b) Scatteringpattern predicted by Eq. (2.59) for di�erent values of g. As g tends to unity, the backscatteredintensity tends to zero, and most energy is transmitted into the forward direction.

Therefore, this approximation breaks down the less scatterer the sphere is. This case

is represented by values g � 1. An example on the relation between g and the scattering

pattern in terms of the Henyey-Greenstein function, Eq. (2.59), can be seen in �g. 2.7.

The reason that the DA breaks down at these values is that, in order for Eq. (2.73) to

hold, light must su�er multiple scattering events. If the particles have low albedo values

W0, see Eq. (2.43), i.e. are poor scatterers, light does not behave di�usively, and higher

orders of Eq. (2.74) must be used. A simple way of writing this approximation would be:

W0 =�s�t� 1 ; (2.79)

which implies �s � �a ; (see Sec. 2.2) and at the same time we must have that:

g < 8:5 : (2.80)

The limiting value for g in the DA has been studied in Ref. [34], where it is shown

that the DA is no longer valid for values of g � 8:5: The reason that both W0 � 1 and

g < 8:5 together are needed is that for non-absorbing particles,W0 � 1 and we could have

particles with very low values of �s. Therefore, the relation most commonly employed to


estimate if a medium would be adequately described by the DA or not is:

�0s � �a : (2.81)

5. Slow variations of the total �ux and low absorption rate: This approximation

was taken in Eqs. (2.63) of Sec. (2.3.2) and is expressed as [18, 19]:

n

c�0s

@

@t+ �a

c

n

!� 1 ; (2.82)

and means that:

(a) The variations in the total �ux occur in a time scale much larger than the time

between scattering events. This can be clearly seen in the frequency domain, where

we have U(r; t) = U(r) exp[�i!t]:

n

c�0s

@

@t= tsc

@

@t� 1 ) ! � �sc : (2.83)

It is easily understood by comparing the frequencies involved in the changes in U ,

and typical frequencies in photon transport, which are given by:

�sc = t�1sc = �0sc

n� 1011Hz ; (2.84)

whereas for the changes in U it is !, whose value is in the order ! � 108Hz: This

approximation breaks down with the DA, since the only way of reducing the value

of �sc is lowering the value of �s, thus reducing multiple scattering.

(b) The variations of the total �ux due to absorption occur in a time scale much larger

than the time between scattering events:

�ac=n

�0sc=n=

tscta� 1) �a � �sc ;

where �a is the absorption rate �a = t�1a , whose value is in the order19 of �a =

�ac=n � 108Hz.

In both cases, we see that there are di�erences of at least two orders of magnitude. As will

be shown in Sec. 2.3.4, if one wishes to obtain a better approximation to light transport

in these scattering media, the complete expression for J, Eq. (2.63), can be used.

6. Isotropic Sources: The approximation involved in assuming that the photon sources are

isotropic was taken in Eq. (2.67) of Sec. 2.3.2, and signi�cantly simpli�es the resultant19It can be seen that for values of high absorption, we obtain values of �a in the order of �a � 109Hz: It is in

these cases that the application of this approximation is questionable.


di�usion equation. This approximation implies that:

JE(r) = DZ4��(r; s)sd = 0 ;

where � is the power generated in the medium per unit volume [see Eq. (2.46)]. It should

be remarked that this approximation is not necessary in order for the DA approximation

to hold, and can thus be overcome by including the value of JE in the �complete� di�usion

equation Eq. (2.67). Such approach has been taken for example in Refs. [23, 35], in which

an angular dependent term is included in the source function.

7. �Hidden� approximations: There are other approximations that must be taken into

consideration, which are �hidden� in the RTE. These approximations involve a much

greater issue: the relationship between Maxwell's equations and the RTE. Two main

approximations are taken when deriving the RTE:

(a) Absence of Interference e�ects: So far, to our knowledge, these e�ects have been

included in free-space (see Ref. [36] and references therein), but their inclusion is

yet to be determined in the general case, Eq. (2.39), as Mandel and Wolf note

in Ref. [37]:�In spite of its extensive use, no satisfactory derivation of the RTE

from electromagnetic theory or even from scalar wave theory has been obtained up

to now, except in some special cases�. Ishimaru in Ref. [9], Ch. 7, also points out:

�... The development of the theory (RTE) is heuristic and it lacks the mathematical

rigor of the analytical theory. Even though di�raction and interference e�ects are

included in the description of the scattering and absorption characteristics of a single

particle, transport theory itself does not include di�raction e�ects. It is assumed

in transport theory that there is no correlation between �elds, and therefore, the

addition of powers rather than the addition of �elds holds�. Therefore, there are

some fundamental aspects of multiple scattering which the RTE as presented in Eq.

(2.39) cannot account for, such as interference, coherence, e�ects which take place

in multiple scattering media, such as localization, backscattering enhancement (see

Refs. [38, 39] for details on these e�ects), resonances[16], etc20.

(b) Short wavelength limit: Actually, the fact that the RTE cannot account for interfer-

ence e�ects is due to this condition, as can be seen in Ref. [36], since the expression

in free-space for the RTE [see Eq. (2.48)] which includes interference e�ects is:

r2I(r; s)� 2iks � rI(r; s) = 0 ; (2.85)

where k = 2�=�, is the wavenumber of the incident �eld. In order for Eq. (2.85)

20An updated review on all these e�ects in connection with multiple scattering by small particles can be foundin Ref. [13].


to reduce to Eq. (2.48), we must have k !1, and therefore � must be very small

compared to the source and detector typical sizes, thus reducing Eq. (2.85) to the

interference-free equation Eq. (2.48).

2.3.4 Improving the Di�usion Equation

As several approximations were taken to reach the di�usion equation (DE) (see Sec. 2.3.3), there

are many di�erent modi�cations to Eq. (2.69), each with its own limits of validity, depending on

the approximations involved. In Sec. 2.3.3 we have already mentioned the Pn approximations,

and here we will just deal with the di�erent modi�cations to Eq. (2.69) that can be reached

by just considering di�erent approximations to Eq. (2.63). These di�erent expressions have

been a motive of controversy in the past [18, 19, 40, 41], and still continue so in the present

[24, 42, 43, 44]. In the past, the main discussion was centered around the limits of validity

of each approximation, specially in time-domain. One of the main reasons of this controversy

was based on the fact that there were di�erent de�nitions of �di�usion approximation�, and

therefore di�erent limits of validity.

In the present, the main discussion is centered around the dependence on absorption of the

di�usion coe�cient. This new discussion arose from the fact that biological tissues are highly

absorbing, and photons probe regions buried very deep into tissue. It is in these cases that the

precise value of the di�usion coe�cient is needed whenever characterizing biological media.

I shall �rst derive the expressions that arise from the di�erent approximations to Eq. (2.63),

and, later on, I shall try to explain the reasons of such controversies. In order to derive the

di�erent expressions, isotropic sources will be considered in all cases.

1. Time independent di�usion equation: In this case, Eq. (2.63) reduces to:

rU(r) = �3�0s"n

c�0s

��a

c

n

�+ 1

#J(r) : (2.86)

At this point two steps can be taken:

(a) No approximation: This means that we directly introduce Eq. (2.86) into the equa-

tion for �ux conservation, Eq. (2.45), thus obtaining the di�usion equation:

D1r2U(r) + �aU(r) = E(r) ; (2.87)

D1 =1

3(�0s + �a):

Eq. (2.87) is exactly the equation derived by Ishimaru in Ref. [9], Ch. 9.

(b) Low absorption rate: As in Eq. (2.83), we can estimate tsc � ta in Eq. (2.86), which


introduced in Eq. (2.45) gives us:

D2r2U(r) + �aU(r) = E(r) ; (2.88)

D2 =1

3�0s:

Eq. (2.88) is the time-independent case of Eq. (2.70).

2. Time dependent di�usion equation: In this case, there are three di�erent approxi-

mations that can be done to Eq. (2.63):

(a) Slow �ux variation: In this case [see Eq. (2.82)], we have Eq. (2.86), and the

resulting di�usion equation is:

n

c

@U(r)

@t+D1r2U(r) + �aU(r) = E(r) ; (2.89)

D1 =1

3(�0s + �a):

This is the equation presented by Patterson in Ref. [45], and one of the most

commonly used expressions.

(b) Slow �ux variation and low absorption rate: In this case we apply conditions (2.82)

and (2.83) to Eq. (2.63), which result in the di�usion equation:

n

c

@U(r)

@t+D2r2U(r) + �aU(r) = E(r) ; (2.90)

D2 =1

3�0s;

which is the expression for the di�usion equation as derived in Sec. 2.3.2. This is

the expression obtained by Furutsu in Ref. [41].

(c) No approximation: In this case, in order to �nd the di�usion equation, it is easier

if take the divergence of Eq. (2.63), and substitute the value for r � J from the

equation of �ux conservation, Eq. (2.45), into this expression. In doing so, after

some straightforward calculations, we obtain:

3D3

�n

c

�2 @2U(r)@t2

+n

c

@U(r)

@t�D3r2U(r) (2.91)

+3(�0s + �a)�aD3U(r) = q(r) ;

D3 =1

3(�0s + 2�a);

where q(r) = 3(�0s + �a)D3E(r) + 3D3(n=c)@E=@t. Eq. (2.91) has been derived for

a homogeneous medium, i.e. with �0s and �a constant. A more rigorous derivation


can be found in Ref. [29]. Eq. (2.91) is also called the telegraph equation, and is not

in the strict sense a di�usion equation, but one that includes both propagation and

di�usion. This is the equation derived by Ishimaru in Ref. [40].

There exists yet another two modi�cations to the DE, put forward by Durian and Rudnick in

Ref. [46], and by Polishchuk et al in Ref. [47]. The one presented in Ref. [46] consists of a

modi�cation of the telegraph equation Eq. (2.91). The one presented by Polishchuk et al is

derived by using higher angular moments. These two theories are not presented here since they

do not arise from approximations taken in Eq. (23). In any case, they can barely be considered

as �di�usion approximations�, and neither can Eq. (2.91).

Comparison of the theories

� Time independent: First of all, let us compare the expressions obtained for the time-

independent di�usion equation, Eqs. (2.87) and (2.88). As stated by Furutsu in Refs.

[48, 49], later on by Durduran et al [42], and experimentally presented Bassani et al in Ref.

[44], the di�usion coe�cient in the time-independent di�usion equation is independent

of the absorption coe�cient, being this absorption coe�cient subject to the constraint

�a � �0s, discussed in Eq. (2.81). The reasons they put forward are as those exposed in

order to derive Eq. (2.70) in Sec. 2.3.2.

Other authors such as Aronson in Ref. [24] and Durian in Ref. [43], defend that the

di�usion coe�cient is dependent of the absorption coe�cient. Their argument is not as

straightforward as the one presented in Sec. 2.3.2, but it is more robust. It is basically

as follows: let us go back to Eq. (2.72) where we have the dispersion relation for the

di�usion length Ld, in terms of the albedo W0, and the total scattering cross-section �t,

in the isotropic case. If we expand21 Eq. (2.72)in terms of (1�W0) we obtain:

1

L2d�

2t

= 3(1�W0)�1� 4

5(1�W0) +

4

175(1�W0)

2 + :::�: (2.92)

Now, if we consider W0 � 1 as in Eq. (2.79), we can truncate the quantity inside the

brackets in Eq. (2.92) at zeroth order, thus obtaining D = 1=3�s, as in Eq. (2.88). On

the other hand, if we take up to the �rst order term, we obtain:

D =1

3(�s + 0:2�a):

This solution is for isotropic sources and cases of low absorption. Depending on these

two factors and the order of truncation, there exist other dispersion relations for Ld,

and di�erent expressions for D: In any case, as put forward in Refs. [24, 43] the di�u-

sion coe�cient does depend on absorption. The problem is that this dependence is not

21See Ref. [24] and references therein.


straightforward, and it seems to be that the expression for D is neither Eq. (2.88) nor

(2.87) but something in between:

D =1

3(�0s + a�a); (2.93)

where the typical values for a are always in the order of a � 0:5 or lower, depending on g

and W022. A similar result is put forward by Durian [43], where the value he presents is

a = 1=3. The reason we do not use this expression, and use Eq. (2.88) instead, is because

the value of a in Eq. (2.93) does not have a compact expression in terms of g and W0.

We therefore use the value of D = 1=3�0s (i.e. a = 0), since, as shown in Refs. [42, 48, 49],

it yields better results than the value D = 1=3(�s0 + �a) (i.e. a = 1) as in Eq. (2.87).

� Time dependent:

In the time-dependent case, we have in total 5 di�erent expressions to compare: 1) Eq.

(2.89), which is the most commonly used, 2) Eq. (2.90) also very commonly used, and

defended by Furutsu, 3) Eq. (2.91) defended by Ishimaru, 4) Polishchuk et al, and 5)

Durian and Rudnick's. All these �ve cases have been put to test and compared in Ref.

[34], yielding all very similar results within the DA limits established in Sec. 2.3.3. The

expression that demonstrated the best performance when tested at limiting conditions was

the one put forward by Durian and Rudnick in Ref. [46]. A comparison between Ishi-

maru's and Furutsu's expressions can be found in Ref. [18], where the main conclusion is

that Ishimaru's expression has wider limits of validity than Furutsu's, but that Ishimaru's

theory cannot be considered to be within the di�usion approximation. If we stay within

the de�nition of di�usion approximation23, then the expressions to be contrasted are Eqs.

(2.89) and (2.90). In this case, the results put forward for the time-independent case

can also be applied, since the time-independent case is obtained by integrating the time-

dependent case. One of the drawbacks of Eq. (2.90), and of all di�usion equations which

are similar, is that it has no propagation time, and therefore a pulse di�uses instantly

[18]. This is pointed out in Ref. [31], when studying heat di�usion, where they note

that the di�usion approximation predicts that the temperature will rise instantaneously

everywhere, and therefore that the di�usion equation is correct only after a su�ciently

long time has elapsed.

The major drawback when employing complex expressions as the ones used by Polishchuk et

al [47], and Durian and Rudnick [46], is that in those cases the expression for the boundary

conditions is more cumbersome. Therefore, whenever complex geometries and interfaces are

22Some derived values for D in terms of g and W0 can be found in Ref. [24].23As implied in the de�nition in Ref. [31], see Eq. (2.76), in order to be within the di�usion approximation we

must have that ltrjrU j � U , which implies that J has to obey Fick's law, Eq. (2.68). In the cases presented byIshimaru [40], Polishchuck et al [47], and Durian and Rudnick [46], J obeys Eq. (2.63) rather than Eq. (2.68).


addressed, it is always more convenient to work with Eq. (2.90), as long as the parameters

which de�ne the regions under study do not push the DA too far.

2.3.5 Other ways of reaching the Di�usion Equation

Basically, we can distinguish three di�erent ways of reaching the DE, Eq. (2.69): 1) deriving

it from the RTE; 2) deriving it from electromagnetic theory; 3) deriving it heuristically. The

�rst case is the one shown in Sec. 2.3.2. The third case is as presented for example in Ref.

[50], where the DA is derived directly from Fick's law, just on taking into account that the

change of particle density with time is equal to the production minus the absorption and the

escape of particles. In any case, it is the derivation of the DA from electromagnetic theory

that really presents an alternative to the RTE24. The way of performing this derivation is by

means of iterative methods, in particular diagramatic expansions. In these cases, equations

such as Dyson's equation and the Bethe-Salpeter equation appear. Dyson's equation applies

to the electromagnetic �eld in a medium with scattering particles, whereas the Bethe-Salpeter

equation is the equivalent for the mean intensity. These equations and their derivations are

very well explained in Refs. [8, 51].

Once the Bethe-Salpeter equation is derived, by a series of approximations one reaches

the di�usion equation. This derivation can be found in Refs. [13] and [39], for example.

Mathematically, the derivation of the DE from electromagnetic theory is more formal than its

derivation from the RTE. In any case the RTE is much easier to work with, and whenever

deriving boundary conditions within multiple scattering media, it is always simpler with the

RTE. Of course, as noted in Sec. 2.3.3, interference e�ects cannot be taken into consideration

with the RTE. On the other hand, in electromagnetic theory, the DA can be improved and

new terms which can explain complex interference phenomena such as the enhancement of the

backscattering peak or localization, can be included.

24Actually, we should say that the RTE presents an alternative to the derivation from electromagnetic theory,and not the other way around.

Chapter 3

Propagation of Di�use Photon Density

Waves

In this Chapter, di�use photon density waves (DPDWs) are introduced. The quantities that

characterize them, such as their wavelength, group velocity, and decay length are put forward.

In terms of the angular spectrum, we shall de�ne the propagation of these waves, and expres-

sions for the transfer function and the impulse response will be presented. We shall derive

the angular spectrum representation for a point source. We shall then make a study of the

di�raction of DPDWs by slits and circular apertures, comparing them with these known e�ects

for propagating scalar waves (PSWs).

3.1 Di�use Photon Density Waves

In the particular case in which Eq. (2.70) from Sec. 2.3.2 is applied to a light source, modulated

with a frequency !, E(r; t) = S0(r)(1 + exp[�i!t]), we shall expect the solution to be written

as U(r; t) = UDC(r) + UAC(r) exp[�i!t], being UDC and UAC the continuous and frequency

dependent components. In this manner, the general di�usion equation for the AC component

is:

r2U(r) + �20(r)U(r) = �S0(r)

D(r)� rD(r)

D(r)� rU(r) ; (3.1)

where �0 is the wave number of the DPDW:

�0(r) =

vuut��a(r)D(r)

+ i!n

cD(r); (3.2)

n being the refractive index of the di�usive medium, and c the speed of light in vacuum.

It must be remarked that the second term in the right in Eq. (3.1) must always be taken

into consideration in regions with di�erent values of D [52, 53, 54]. This will be studied in

greater detail in Sec. 4.1. It has been shown by Arridge [29] how Eq. (3.1) can be reduced to

36

CHAPTER 3. PROPAGATION OF DIFFUSE PHOTON DENSITY WAVES 37

a Helmholtz equation by making � =pDU :

r2�(r) + �2(r)�(r) = � S0(r)qD(r)

; (3.3)

where:

�(r) =

0@�r2 [D(r)]12q

D(r)+ �20(r)

1A 12

: (3.4)

As stated in Ref. [29], the Helmholtz form, Eqs. (3.5) and (3.3), allows the problem of

recovering the parameters of hidden objects to be directly related to inverse scattering problems

in di�raction tomography.

In an in�nite homogeneous medium Eq. (3.1) reduces to:

r2U(r) + �20U(r) = �S0(r)

D: (3.5)

In, Eq. (3.2) �0 can be rewritten as �0 = �<e + i�=m, where [55, 56, 57]:

�<e =1p2

vuuut !2n2

D2c2+

�2aD2

! 12

� �aD

; (3.6)

�=m =1p2

vuuut !2n2

D2c2+

�2aD2

! 12

+�aD

: (3.7)

Eqs. (3.6) and (3.7) show that for low values of ! we have that �<e � 0, �=m � Ld (see Eq.

(2.71) in Sec. 2.3.2), whereas for high values of ! we have that �<e ' �=m. In terms of �<e

and �=m, we can de�ne a wavelength �0, a phase velocity Vp, and a frequency dependent decay

length1 L!, of the DPDW as:

�0 =2�

�<e; (3.8)

Vp =!

�<e; (3.9)

L! =1

�=m: (3.10)

From Eqs. (3.8)-(3.10) we see that for ! = 0 we recover L! = Ld. The values for �0; Vp ; L!; �<e;

and �=m are shown in Fig. 3.1 for the case �a = 0:025cm�1, �0s = 10cm�1, n = 1:333, for

di�erent values of the modulation frequency f , where f = !=2�. As shown, �<e; and �=m are

constant practically up to f � 108Hz since for those cases !=c � �a. Therefore, for values

1The term L! can also be called the di�usion length, since it is related to Ld, but we prefer to just refer toit as a decay length to avoid confusion.


f < 108, the following approximations are very accurate:

�<e =!

2Ld

1

�ac=n; �=m =

1

Ld; (3.11)

�0 =2Ld

f�a

c

n; Vp = 2Ld�a

c

n;

where we also have L! = Ld.

Figure 3.1: Values of �0; Vp ; L!; �<e; and �=m versus the modulation frequency f , where f =!=2�. In all cases �a = 0:025cm�1, �0s = 10cm�1, and n = 1:333. Note that Vp is plotted inunits of the speed of light c.

Experimentally, the values of the modulation frequency f usually lie near f = 200MHz;

which, as seen in Fig. 3.1, yield �0 � 10cm, and L! � 1cm. Therefore, the intensity is greatly

attenuated before completing one cycle, decreasing by a factor of � exp(�10) after travelingone wavelength. The fact that L! � �0 constitutes a general characteristic of DPDWs. With

this assumption, we expect interference e�ects in DPDWs to be very low. Also, as shown in

Fig. 3.1, the phase velocity is of the order of c=20, thus implying that although these waves are

greatly attenuated, they propagate at high speeds.

3.1.1 Solution for in�nite homogeneous media

Before going further, it is very useful to derive the solution to Eq. (3.5) corresponding to

an in�nite homogeneous di�usive medium. In this case, the corresponding equation held by


Green's function is [58]:

r2G(�0jr� r0j) + �20G(�0jr� r0j) = �4�Æ(r� r0) : (3.12)

The solution to Eq. (3.12) in 3D is therefore (see Ref. [58]):

G(�0jr� r0j) = exp[i�0jr� r0j]jr� r0j :

For a frequency modulated point source, located at rs, the solution to Eq. (3.5) is:

U(r) =1

4�

Z S0Æ(r0 � rs)D

G(�0jr� r0j)dr0 = S0 exp[i�0jr� rsj]4�Djr� rsj : (3.13)

From now, on the solution U(r) for a point source, Eq. (3.13), will always be referred to as

U (i)(r). The expressions for the Green's functions and their derivatives in 3D, 2D and 1D can

be found in Ref. [58], but are here rewritten for convenience:

� In 3D:

G(3D)(�0jr� r0j) = exp[i�0jr� r0j]jr� r0j ; (3.14)

rG(3D)(�0jr� r0j) = exp[i�0jr� r0j]jr� r0j

i�0 � 1

jr� r0j!ur�r0 ; (3.15)

� In 2D:

G(2D)(�0jr� r0j) = �iH(1)0 (�0jR�R0j) ; (3.16)

rG(2D)(�0jr� r0j) = �i�0H(1)1 (�0jR�R0j)uR�R0 ; (3.17)

� In 1D:

G(1D)(�0jr� r0j) = 2�i

�0exp[i�0jz � z0j] ; (3.18)

rG(1D)(�0jr� r0j) = �2� exp[i�0jz � z0j]uz�z0 ; (3.19)

where r = (R; z), and H(1)0;1 are the zeroth and �rst order Hankel functions of the �rst kind.

Care must be taken since these are the Green's functions corresponding to a 3D medium with

2D or 1D symmetries, and therefore they are not necessarily solutions to 2D or 1D media,

since Eq. (3.1) may di�er with the number of dimensions.


Figure 3.2: Geometry used for the angular spectrum representation. At z < 0 there is acollection of objects and sources, and at z > 0 is a di�usive half-space.

3.2 Angular Spectrum Representation

Let us consider a homogeneous multiple-scattering medium separated into two half-spaces z < 0

and z > 0 (see Fig. 3.2). It is assumed that the domain z < 0 contains sources and scatterers

(hidden objects), whereas the domain z > 0 is source free. At any plane z = constant, with

z > 0, we can express the scattered wave U(r) by its Fourier transform as:

U(r) =Z +1

�1

~U(K; z) exp[iK �R]dK : (3.20)

If we substitute Eq. (3.20) into Eq. (3.5), considering it is a source free region, one obtains:

@2 ~U(K; z)

@z2+ (�20 � jKj2) ~U(K; z) = 0 : (3.21)

The solution to Eq. (3.21) is:

~U(K; z) = A(K) exp[iq(K)z] ; (3.22)

where:

q(K) =q�20 � jKj2 : (3.23)

Therefore the scattered wave U(r) can be expressed by its angular spectrum representation of

plane waves, namely, by a superposition of such waves of amplitude A(K) and wave vector


k = (K; q), with jkj = �0 [59, 60, 37] as:

U(R; z) =Z +1

�1A(K) exp[iK �R+ iq(K)z]dK ; (3.24)

where R = (x; y), jKj2 + q2 = �20, i.e., K = (Kx; Ky). jKj and q can also be expressed as:

jKj = �0 sin � ; (3.25)

q(K) = �0 cos � ;

� being complex. Since �0 is a complex number for DPDWs, q(K) = q<e + iq=m is always

complex, namely, q=m 6= 0 (as seen in Eq. (3.7), �=m 6= 0). In Eq. (3.24), we choose q<e > 0

and q=m > 0 so that the �eld satis�es the Sommerfeld's radiation condition at in�nity:

limr!1

r

@U(r)

@r� i�0U(r)

!= 0 ; (3.26)

which simply states that at r = jrj ! 1, U(r) behaves as an outgoing spherical wave. q=m > 0

implies that the wave is absorbed on propagation, i.e. not ampli�ed, and q<e > 0 means it is an

outgoing wave. The total �ux density that traverses any plane z = z0, is given by Fick's law,

Eq. (2.68), and in terms of the angular spectrum representation [see Eq. (3.22)] is written as:

~Jn(K; z0) = �D @ ~U(K; z)

@z

��z=z0

= �iDq(K) ~U(K; z0) ; (3.27)

where the surface normal n at z = z0 is pointing in the +z direction, i.e. n = (0; 0; 1). When

it points inward, Eq. (3.27) must change its sign.

3.2.1 Evanescent Di�use Photon Density Waves?

Let us go back to Eq. (3.23) and the fact that for DPDWs q=m 6= 0. Taking into consideration

that for propagating scalar waves (PSWs) in the same geometry (the background turbid medium

being replaced by a transparent dielectric), �0 = �<e is real we have:

q(K) = q<e =q�20 � jKj2 jKj � �0 ;

for homogeneous components, and:

q(K) = iq=m = iqjKj2 � �20 jKj � �0 ;

for evanescent components. This di�erence between DPDWs and PSWs is essential for spatial

resolution. For PSWs in a transparent medium, high spatial frequencies jKj are exponentiallyattenuated, whereas low spatial frequencies always propagate. The cut-o� frequency �0 is well-


Figure 3.3: Values of (a) q<e and (b) q=m, normalized to �<e: PSWs [ �] ; DPDWs [Æ].

de�ned, and underlines Rayleigh's di�raction resolution limit in optical imaging. On the other

hand, for DPDWs in a multiple scattering medium, even without absorption, there is always

both propagation and attenuation at any spatial frequency. This di�erence is illustrated in

Fig. 3.3, where we plot q<e [Fig. 3.3(a)] and q=m [Fig. 3.3(b)] versus jKj for both DPDWs

and PSWs. The values taken for �<e and �=m correspond to breast tissue illuminated with

light at a wavelength of 780nm [4] and a modulation frequency f = 200MHz, with parameters

�0 = 7:53cm; L! = 0:066cm, �a = 0:035cm�1, �0s = 15cm�1, and n = 1:333. Fig. 3.3 shows that

in the case of PSWs, the regions of propagation (jKj � �r) and attenuation (jKj > �r) regimes

are clearly separated, whereas for DPDWs, q<e and q=m are of the same order of magnitude

for jKj ' �<e, so that there is no sharp transition between these two regimes. Nevertheless, as

shown by these �gures, the values of q for DPDWs asymptotically behave like for PSWs when

jKj � �<e. Therefore, we conclude that there exist no evanescent components for DPDWs,

at least in the sense as for PSWs. This main di�erence between DPDWs and PSWs and its

consequences will later be shown.

3.2.2 Angular spectrum of a point source

At this point, in order to de�ne the propagation of DPDWs, it is convenient to derive �rst the

angular spectrum A(K) for a point source, Eq. (3.13). From Eq. (3.24) in terms of the average

intensity, A(K) is de�ned as:

A(K) =1

4�2

Z +1

�1U(R; z = 0) exp[�iK �R]dR : (3.28)

In the case of a point source at zs = 0:

U (i)(R; z = 0) =S0

4�D

exp[i�0jRj]jRj : (3.29)


Although the solution to Eq. (3.28) is well known (see Refs. [37, 61]), it will be derived here

since most of the steps involved will prove very useful in the following chapters. Introducing

Eq. (3.29) into Eq. (3.28) we obtain:

A(K) =C

4�2

Z 1

0

Z 2�

0

exp[i�0R]

Rexp[�iKR cos �]Rd�dR ; (3.30)

where we have made dR = RdRd� and C = S0=4�D. Grouping terms, we obtain:

A(K) =C

4�2

Z 1

0exp[i�0R]

�Z 2�

0exp[�iKR cos �]d�

�dR : (3.31)

The quantity in brackets in Eq. (3.31) is exactly the expression for the Bessel function of the

�rst kind and zero order J0[58]:

Z 2�

0exp[�iKR cos �]d� = 2�J0(KR) ; (3.32)

which introduced in Eq. (3.31) gives:

A(K) =C

4�22�Z 1

0

exp[i�0R]

RJ0(KR)RdR : (3.33)

Eq. (3.33) is the Hankel transform of zeroth order of Eq. (3.29). The Hankel transforms are

de�ned as:

H[f(R)] = 2�Z 1

0f(R)J0(KR)RdR ; (3.34)

H�1[ ~f(K)] = 2�Z 1

0

~f(K)J0(KR)KdK : (3.35)

The properties of some Hankel transforms can be found for example in Ref. [62]. Eqs. (3.34)

and (3.35) are used for cylindrical symmetry. As seen, the main advantage of the Hankel

transform is that of being a kernel that performs the integration in one dimension, whereas the

Fourier transform, Eq. (3.28), applies in two dimensions. This means a di�erence of a power

of two in computer memory and calculation time.

Eq. (3.33) can be solved directly, since it is a well-know transform formula[62], thus obtain-

ing:

A(i)(K) = Ci

2�q(K); (3.36)

which is the angular spectrum of a point source. Therefore, at any z�plane, the expression for

the average intensity in Fourier space is:

~U(K; z) =S0

4�D

i

2�q(K)exp[iq(K)z] ; (3.37)

where we have made use of Eq. (3.22). Eq. (3.37) is extremely useful, and the reason for


this can be understood on taking into consideration that the expression for a point source, Eq.

(3.37), is that for the Green's function, Eq. (3.12):

~G(K; z � z0) =i

2�q(K)exp[iq(K)(z � z0)] ; (3.38)

and therefore:

G(�0jr� r0j) =Z +1

�1

~G(K; z � z0) exp[iK � (R�R0)]dK : (3.39)

After introducing Eq. (3.38), Eq. (3.39) is the Weyl representation of a diverging spherical

wave, and is a very useful tool to localize and characterize hidden objects in di�usive media, as

can be seen from Refs. [63, 64]. We can still take Eq. (3.38) further if we realize that [37, 59]:

@ ~G(K; z � z0)

@z0=

1

2�exp[iq(K)(z � z0)] ; (3.40)

and therefore Eq. (3.24) can be rewritten as:

U(R; z) = 2�Z +1

�1

~U(K; z0 = 0)@ ~G(K; z � z0)

@z0exp[iK �R]dK : (3.41)

Eq. (3.41) is the �rst Rayleigh-Sommerfeld integral formula, and will be considered in the next

Chapter.

3.3 Transfer Function and Impulse Response

From Eq. (3.24), one obtains:

A(K) exp[iq(K)z] =1

4�2

Z +1

�1U(R; z) exp[�iK �R]dR : (3.42)

Eq. (3.42) shows that A(K) exp[iq(K)z] is the two-dimensional Fourier transform of the wave-

�eld U(K; z) in the plane z = constant. The spatial frequency �lter:

F (K; z) = exp[iq(K; z)] ; (3.43)

constitutes the propagation transfer function.

The amplitude and phase of F (K; z) are represented in Figs. 3.4 and 3.5, respectively, for

a two-dimensional geometry, namely, K = (K; 0), at several propagation distances z. Both

transfer functions for PSWs (left column) and DPDWs (right column) are shown. For DPDWs,

the values of �<e and �=m correspond to the breast tissues parameters as in Fig. 3.3.

For PSWs, the propagating and attenuation regions are clearly visible. For jKj � �r, the

transfer function constitutes only a phase factor, whereas for jKj > �<e, it is a real low-pass

�lter. For large jKj, one has q(K) ' jKj, so that the transfer function is exp[�jKjz], and high


Figure 3.4: Amplitude [(a) PSWs and (b) DPDWs] of F (K; z) for di�erent values of z: z = �[�]; z = �=2 [Æ]; z = �=10 [�]; z = �=20 [u].

Figure 3.5: Phase [(a) PSWs and (b) DPDWs] of F (K; z) for di�erent values of z: z = � [�];z = �=2 [Æ]; z = �=10 [�]; z = �=20 [u].

spatial frequencies are exponentially attenuated. Thus, for PSWs, a given spatial frequency K

has a decay length 1=jKj, and the cut-o� frequency in the plane z = constant is 1=z. These

properties are well-known and exploited in near-�eld optics microscopy[65, 66, 67].

However, for DPDWs, the behavior of the transfer function is substantially di�erent from

that of PSWs. As already noticed in Fig. 3.4, now there no longer exist two separated prop-

agation and attenuation regions. For a given observation distance z, the amplitude has its

maximum at K = 0, and decreases for jKj > 0. The peak value at K = 0 tends monotonically


to zero as z increases, due to the factor exp[��=mz], while its width broadens. The phase varies

less abruptly than in the case of PSWs.

In order to discuss the propagation of DPDWs, a description in direct space is also useful.

This can be done by introducing the impulse response H(R; z), namely, the Fourier transform

with respect to K of F (K) = exp[iq(K)z]:

H(R; z) =Z +1

�1F (K; z) exp[iK �R]dK : (3.44)

In terms of this impulse response, from Eq. (3.42) with z = 0 and from Eq. (3.24), the

wavefunction can be written as:

U(R; z) =Z +1

�1H(R�R0; z)U(R0; z = 0)dR0 : (3.45)

By means of Eq. (3.40), we can write the transfer function F in terms of ~G as:

F (K) = 2�@ ~G(K; z � z0)

@z0;

and therefore Eq. (3.45) reduces to:

U(R; z) = 2�Z +1

�1

@G(�0jr� r0j)@z0

��z=0

U(R0; z0)jz=0 dR0 ; (3.46)

which again is the �rst Rayleigh-Sommerfeld integral formula, as in Eq. (3.41). It is obvious

from Eq. (3.46) that the impulse response H is related to G by:

H(R�R0; z � z0) = 2�@G(�0jr� r0j)

@z0:

From Eq. (3.45) we obtain that the total wave�eld measured at a certain distance z from

the object or collection of sources, is the convolution of the impulse response with the total

wave�eld that emerges from the object or collection of sources at z = 0: U(R; z) = H(R �R0; z) ? U(R0; z = 0).

The full width at half maximum (FWHM) of the amplitude of the impulse response indicates

how the original information U(R; z = 0) will be modi�ed on propagation to the plane z =

constant. As an illustration, the amplitude of H(R; z) for a two-dimensional con�guration

[R = (x; 0)] is shown in Fig. 3.6, at di�erent values of z, for both DPDWs and PSWs. As z

tends to zero, H tends to a delta function. Therefore, the narrower the impulse response is,

the larger amount of spatial information is conveyed from z = 0 to z > 0 . Of course, as seen

in Fig. 3.6, this only occurs when measures are performed very close to the object or collection

of sources, i.e. z = �=20. It is also important that, at a given z, the width of this function is

smaller for DPDWs than for PSWs. The consequences of this fact are discussed in Ch. 5.


Figure 3.6: Amplitude [(a) PSWs and (b) DPDWs] of H(x; z) for di�erent values of z: z = �[�]; z = �=2 [Æ]; z = �=10 [�]; z = �=20 [u].

3.4 Di�raction and interference

Another important issue related to propagation, conveys the manner in which these DPDWs

di�ract, and the resulting interference patterns. As an illustration, by means of the formulation

presented in Sec. 3.2 we shall derive exact expressions for di�raction of DPDWs by a slit, and

by a circular aperture, presenting numerical simulations of di�raction by a slit. Interference

patterns will be addressed for two slits at di�erent separation distances. These results will

be compared with those for PSWs, thus applying results derived in section 3.3. The most

illustrative case for the �eld di�racted from a slit or a circular aperture, is by considering an

incident plane wave:

U (i) = S0 exp[i�0zs] ;

generated at a plane zs. In the case of a rectangular aperture of width 2a and height 2b , the

angular spectrum, see Eq. (3.28), at z = 0 is2:

A(K) =S0

4�2

Z +b

�b

Z +a

�aexp[i�0zs] exp[�i(xKx + yKy)]dxdy ; (3.47)

which gives:

A(K) =S0

4�2exp[i�0zs]

sin(aKx)

aKx

sin(bKy)

bKy

: (3.48)

In Eq. (3.48), the sin(k)=k is usually referred to as the sinc function, sinc(k). On introducing

Eq. (3.48) into Eq. (3.24), we obtain the �eld generated by a rectangular aperture. In the

case of a slit, we simply substitute b ! 1. In Fig. 3.7 we plot the values of the wave�eld

at a distance z = � for di�erent slit widths. As seen, there are no interference fringes for

DPDWs, in contrast with PSWs. Also for DPDWs we see that once the slit is wider than a

2As will be shown in the following chapter, this is the Kirchho� approximation.


Figure 3.7: Amplitude U(x; z) at z = � for PSWs (a), and DPDWs (b), for di�erent slit widths:w = �=2 [solid line]; w = � [dotted line]; w = 2� [�]; w = 5� [Æ]. In (b) the parameters are:�0s = 10cm�1, �a = 0:025cm�1, n = 1:333, f = 200MHz.

wavelength, the �eld measured is approximately constant. Even so, if we take a close look at

the plot corresponding to the w = 5� case in Fig. 3.7, we see that near the border there is some

interference. This is shown in Fig. 3.8, were the pattern corresponding to di�raction from a

border is recognized. This pattern is greatly damped due to absorption and therefore only one

maximum can be appreciated.

The case corresponding to two slits of width w = �=2 is plotted in Fig. 3.9 for several

slit distances. Again, we see that there is practically no interference of DPDWs, and that the

total �eld amplitude is approximately jU j ' jU (1)j + jU (2)j, i.e. the incoherent superposition

of both contributions. Therefore we may conclude that there is practically no interference of

DPDWs, thus being a good approximation the assumption that the total �eld is the sum of the

amplitudes3. The only way to manage strong interference patterns would be by generating two

incident �elds with opposed phases, as in Refs. [68, 69].

3We cannot say in any case that there is no di�raction of DPDWs, since no di�raction implies that the �eldmeasured at a plane z > 0 would simply be the geometrical projection of the aperture (i.e. its shadow). Asseen from Fig. 3.7 it is by no means what occurs.


Figure 3.8: Amplitude U(x; z) at z = � for DPDWs, for a slit w = 5� [Æ]. Inset: Detail of theplot. The parameters are: �0s = 10cm�1, �a = 0:025cm�1, n = 1:333, f = 200MHz.

Figure 3.9: Amplitude U(x; z) at z = � for PSWs (a), and DPDWs (b), for di�erent slitdistances of width w = �=2: d = � [solid line]; d = 2� [dotted line]; d = 5� [�]; d = 10� [Æ]. In(b) the parameters are: �0s = 10cm�1, �a = 0:025cm�1, n = 1:333, f = 200MHz.


In the case in which we have a circular aperture of radius a, Eq. (3.47) must be written as:

A(K) =S0

4�2

Z a

0

Z 2�

0exp[i�0zs] exp[�iKR cos �]Rd�dR ; (3.49)

which is the Hankel transform of R exp[i�0zs], see Eq. (3.34). This transform can be found in

Ref. [62], for example, and yields:

A(K) =S0�a

2

4�2exp[i�0zs]

2J1(Ka)

Ka; (3.50)

where 2J1(k)=k is usually called the Besinc function, besinc(k). It is interesting to note that

both in the case of a circular aperture Eq. (3.50), and of a rectangular aperture, Eq. (3.48), the

frequency �lter generated by the obstacle is the same as in the corresponding electromagnetic

case.

Chapter 4

Integral Equations for Di�use Photon

Density Waves.

In this Chapter, we derive the integral equations for the scattering of DPDWs. This accounts

for the contribution of object boundaries and interfaces between media. We formulate this

theory [11, 59] from �rst principles, and in particular, we show how the extinction and the

Helmholtz-Kirchho� theorems for DPDWs appear, as well as the relationship between volume

and surface integrals. In connection with Ch. 3, we shall demonstrate the relationship between

the surface integrals and the angular spectrum representation. We shall put forward a method

to numerically solve any kind of boundary and address a di�usive object buried in a di�usive

medium in the presence of an arbitrary interface. Numerical results for this particular case will

be shown later on, in Ch. 5. For the sake of clarity, all the derivations in this Chapter will be

performed for index matched media, and the expression of the integral equations in the case of

index mismatch will be presented in Ch. 7.

4.1 Derivation of the Scattering Equations

Let us consider a di�usive inhomogeneous medium of volume V , with parameters �1, D1, and

�a1, embedded in an isotropic and homogeneous di�usive medium of volume ~V , as shown in

Fig. 4.1, in which the parameters �0, D0 and �a0 are constants. Then, ��2(r), and D(r) will

be de�ned as (see Fig. 4.2):

D(r) =

8<: D0 ; r 2 ~V ;

D1(r) ; r 2 V ;(4.1)

��2(r) =

8<: 0 ; r 2 ~V ;

�21(r)� �20 ; r 2 V :(4.2)

Notice that ��2(r) is the equivalent to the scattering potential, de�ned in potential scatter-

51

CHAPTER 4. INTEGRAL EQUATIONS FOR DIFFUSE PHOTON DENSITY WAVES. 52

Figure 4.1: Scattering geometry corresponding to a di�usive medium of volume V embedded ina semi-in�nite isotropic di�usive medium.

Figure 4.2: Arbitrary pro�le of D(r), corresponding to a di�usive inhomogeneous medium ofvolume V embedded in an isotropic and homogeneous di�usive medium of volume ~V .

ing theory. The di�usion equation, see Sec. 3.1, can therefore be expressed as:

r2U(r) + �20U(r) = �S0(r)

D(r)��2(r)U(r)� rD(r)

D(r)� rU(r) ; (4.3)


and the Green's function corresponding to this equation satis�es, as in Eq. (3.12):

r2G(�0jr� r0j) + �20G(�0jr� r0j) = �4�Æ(r� r0) : (4.4)

Let us now derive the scattering integral equations. On multiplying Eq. (4.3) by G, Eq.

(4.4) by U , subtracting both, performing a volume integral and applying Green's theorem to

both U(r) and G(r; r0):Zv(Ur2G�Gr2U)d3r =

Zs(UrrG�GrrU) � ds ; (4.5)

and integrating Eq. (4.5) over the volume ~V one has:

� r0 2 ~VZ~VÆ(r� r0)U(r0)d3r0 = � 1

4�

Z~V

�S0(r0)

D0G(�0jr� r0j)d3r0

� 1

4�

Z~SfU(r0)rr0G(�0jr� r0j)�G(�0jr� r0j)rr0U(r

0)g � dS0 : (4.6)

where the integral of [rD(r)=D(r)] � rU(r) over the volume ~V vanishes because D(r) = D0 =

constant. On the other hand, proceeding as in Eq. (4.6), on applying Green's theorem to both

U(r) and G(r; r0), and integrating over volume V one obtains:

� r0 2 V

ZVÆ(r� r0)U(r0)d3r0 = � 1

4�

ZV

�S0(r0)

D1(r0)G(�0jr� r0j)d3r0

+1

4�

ZVG(�0jr� r0j)��2(r0)U(r0)d3r0

� 1

4�

ZVG(�0jr� r0j)

"�rr0D1(r

0)

D1(r0)

#� rr0U(r

0)d3r0

� 1

4�

ZSfU(r0)rr0G(�0jr� r0j)�G(�0jr� r0j)rr0U(r

0)g � dS0 : (4.7)

In Eqs. (4.6) and (4.7), ~S and S denote the surfaces delimiting volumes ~V and V , respectively

(see Fig. 4.1). Now let us de�ne the boundary conditions at the interface. First of all, the total

density �ux traversing the interface must be equal when considered from volume V or ~V , and

therefore (see Eq. (2.68) in Sec 2.3.2):

�D0 n(r) � rU+(r)��S= �D1(r) n(r) � rU�(r)

��S

(4.8)

Secondly, since we have index matched media, we must have that I0(r; s) = I1(r; s) at the

interface (see Sec. 2.1.1). Therefore, by applying Eq. (4.8) into the expression for the di�usion


approximation, Eq. (2.55) from Sec. 2.3.1, we obtain:

U+(r)��S= U�(r)

��S

(4.9)

where we have de�ned the di�usive wave in volumes ~V and V as U+ and U�, respectively,

and n(r) is the outward surface normal at the point r. If D(r) varies slowly, i.e. there is

no sharp boundary for D, the total current density condition reduces to n(r) � rU+(r)jS =

n(r) � rU�(r)jS, most commonly used when using the Finite Element Method (FEM) [70, 71,

72].

We can express the di�erent contributions of the surface ~S to the surface integral in Eq.

(4.6) as: Z~SdS0 =

ZS(1)

dS0 +ZS[�n(r0)]dS 0 ; (4.10)

where S(1) is the surface of a sphere at in�nite enclosing the in�nite volume ~V as shown in

Fig. 4.3.

Figure 4.3: (a) Surface S enclosing volume V and its normal dS. (b) Surface ~S enclosingvolume ~V , which has two contributions: ~S � S

SS(1). Notice the surface normal to surface S

pointing into volume V .

Considering in Eqs. (4.6) and (4.7) the possible locations of r and r0 either in V or ~V , these

vectors being denoted by r> or r< according to whether they belong to ~V or V , respectively,

the �nal coupled integral equations are (see Fig. 4.4):

� r0; r 2 ~V

U+(r>) = U (i)(r>) +1

4��S(+)(r>) +

1

4��(1)(r>) ; (4.11)

� r0 2 ~V ; r 2 V

0 = U (i)(r<) +1

4��S(+)(r<) +

1

4��(1)(r<) ; (4.12)

� r0; r 2 V


U�(r<) =1

4�

ZVG(�0jr< � r0j)rD1(r

0)

D1(r0)� rU�(r0)d3r0 (4.13)

+1

4�

ZVG(�0jr< � r0j)��2(r0)U�(r0)d3r0 � 1

4��S(�)(r<) ; (4.14)

� r0 2 V; r 2 ~V

0 =1

4�

ZVG(�0jr> � r0j)rD1(r

0)

D1(r0)� rU�(r0)d3r0

+1

4�

ZVG(�0jr> � r0j)��2(r0)U�(r0)d3r0 � 1

4��S(�)(r>) ; (4.15)

where, denoting the normal derivative by @=@n = n(r) � rr, �S(+;�)(r) stands for the surface

integrals, schematically represented in Fig. 4.4:

�S(+;�)(r) =ZS

(U (+;�)(r0)

@G(�0jr� r0j)@n0

�G(�0jr� r0j)@U(+;�)(r0)

@n0

)dS 0 : (4.16)

Figure 4.4: The di�erent volumes involved in the formulation and the surface integrals thatenclose them.

By using the saltus conditions expressed in Eqs. (4.9) and (4.8), it is easy to see that both

�S(+)(r) and �S(�)(r) verify the following relationship:

�S(�)(r) = �S(+)(r) +ZS

1� D0

D(r0)jS

!G(�0jr� r0j)@U

+(r0)

@n0dS 0 ; (4.17)


that is, when there are saltus for the normal derivative @U=@n accross S, the surface integrals

�S(+) and �S(�) are not the same. A similar relationship can be found in Ref. [73], where the

equations are solved in the �rst order Born approximation. Eq. (4.17) can be written in terms

of the total �ux Jn that traverses the surface S as:

�S(�)(r) = �S(+)(r) +ZS

�1

D1� 1

D0

�G(�0jr� r0j)Jn(r0)dS 0 : (4.18)

We will assume that �a0 > 0, and therefore (see Fig. 4.4):

�(1)(r) =ZS(1)

(U+(r0)

@G(�0jr� r0j)@n0

�G(�0jr� r0j)@U+(r0)

@n0

)dS 0 = 0 : (4.19)

The term U (i)(r) in Eqs. (4.11) and (4.12) is the incident wave:

U (i)(r) =1

4�

Z~V

�S0(r0)

D0G(�0jr� r0j)d3r0 ; (4.20)

where we have assumed that the photon source in located in volume ~V . In analogy with the

electromagnetic and potential scattering cases [59, 74], one can consider Eq. (4.12) as the

extinction theorem for DPDWs, namely:

1

4��S(+)(r<) = �U (i)(r<) ; (4.21)

where we have made use of Eq. (4.19). Eq. (4.21) tell us that the volume V enclosed by surface

S under the in�uence of the incident �eld U (i) will generate a �eld which will extinguish U (i),

and create a wave with di�erent wave number and phase velocity, U�. This theory was �rst

derived by Ewald [75] in 1912 and by Oseen [76] in 1915 for molecular optics, in which the

response of the medium to the incident �eld is expressed in terms of elementary dipole �elds,

generated by the interaction of the incident wave with the individual molecules in the medium.

Later on Wolf and Pattanayak [77, 78] presented a generalization of the extinction theorem,

and performed a thorough study on its implications and meaning. A great deal of works have

been published on this subject ever since, and general references can be found in Refs. [11, 59],

for example.

Similarly, Eq.(4.11) represents the Helmholtz-Kirchho� integral equation for DPDWs:

U+(r) = U (i)(r) +1

4�

ZS

U+(r0)

@G(�0jr� r0j)@n

�G(�0jr� r0j)@U+(r0)

@n

!dS 0 ; (4.22)

where we once again have made use of Eq. (4.19). Let us consider the case depicted in Fig.

4.5, in which a generic aperture Q is in an in�nite black screen SC, in the plane z = 0. A source

is located at z < 0. When measuring on a plane z = z0 > 0, we must make use of Eq. (4.14).

In this case, the Kircho� approximation or the physical optics approximation to Eq. (4.22)


or Eq. (4.14), assumes the boundary values of the �eld U = U (i) and its normal derivative

@U=@n = @U (i)=@n on the screen SC as if they were determined from geometrical optics, thus

giving the Kircho� di�raction formula:

U�(r) = � 1

4�

ZQ

U (i)(r0)

@G(�0jr� r0j)@n

�G(�0jr� r0j)@U(i)(r0)

@n

!dS 0 : (4.23)

This formula has proven very accurate in electromagnetic theory, but its implications and

rigourous demonstration are non trivial. A detailed account on this subject and relevant refer-

ences can be found in Ref. [59].

Figure 4.5: Schematic representation of the Kircho� or physical optics approximation.

The relationships between surface and volume integrals shown in Eqs. (4.11)-(4.15) also lead

to representations already used in previous works. By means of Eq. (4.17), we can combine

Eq. (4.11) with Eq. (4.15), and Eq. (4.12) with Eq. (4.14), to establish the following coupled

equations:

� r 2 ~V

U+(r>) = U (i)(r>) +1

4�

ZVG(�0jr> � r0j)��2(r0)U�(r0)d3r0

+1

4�


0)

D1(r0)� rU�(r0)d3r0

� 1

4�

ZS

1� D0

D1(r0)jS

!G(�0jr> � r0j)@U

+(r0)

@n0dS 0 ; (4.24)

� r 2 V

U�(r<) = U (i)(r<) +1

4�

ZVG(�0jr< � r0j)��2(r0)U�(r0)d3r0


+1

4�


0)

D1(r0)� rU�(r0)d3r0

+1

4�

ZS

1� D1(r

0)jSD0

!G(�0jr< � r0j)@U

�(r0)

@n0dS 0 : (4.25)

In Eqs. (4.24) and (4.25), there appears a D0=D1(r)jS factor inside the surface integrals due

to the saltus conditions in @U=@n. We wish to emphasize here the importance of this factor

for situations in which there is contrast in D. This has already been accounted for by some

authors (see for example Refs. [53], [63], and [73, 79]), by solving the problem by a perturbation

method, and in Ref. [54] where they present a study on the importance of this term. This was

also taken into account in Ref. [80], and in studies regarding the di�usion of light in nematic

liquid crystals, (see Ref. [81] for example). Also, whenever dealing with homogeneous media

and applying the appropriate boundary conditions (see Ref. [82] for the spherical symmetry

case) one is taking into account this contrast factor. Eqs. (4.24) and (4.25) must be handled

with care since they are coupled equations. In the special case in which D1(r) = D0, we recover

a well known expression similar to that for the scattering of scalar waves:

U(r) = U (i)(r) +1

4�

ZVG(�0jr� r0j)��2(r0)U(r0)d3r0 ; (4.26)

since both Eq. (4.24) and (4.25) reduce to the same equation, both inside and outside volume

V . In order to reconstruct and characterize di�usive objects in di�usive media, many authors

commonly use approximations such as Rytov or Born [30, 83, 84, 85, 86] to Eqs. (4.24) and

(4.25). The theory of such approximations can be found in many text books, see Refs. [11] and

[59], for example.

4.1.1 Rayleigh scattering for di�usive waves

Let us consider the case in which the volume V in Eq. (4.24) is a small particle located at ro,

which is so small that the �eld inside can be considered constant and equal to the incident �eld.

In this case, the volume integral in Eq. (4.24) simply converts into the volume of the particle

V :

U (sc)(r) =��

�a0D0

� �a1D1

�+i!

c

�n1D1

� n0D0

��V

4�U (i)(ro)G(�0jr� roj)

+�D1 �D0

D1

�1

4�

ZS

@U (i)(r0)

@n0G(�0jr� r0j)dS 0 ; (4.27)

where ��2 has been written explicitly as in Eq. (4.2). In Eq. (4.27) we have not written

the surface intergal explicitly since it must be handled with care, and we shall therefore derive

it separately. Even though we may not consider @U=@n0 constant in the scatterer, we may


consider G constant in the surface integral:

ZS

@U (i)(r0)

@n0G(�0jr� r0j)dS 0 ' G(�0jr� roj)

ZSn0 � rU (i)(r0)dS 0 ;

taking into consideration that rU (i)(r) = urs�r@U(i)(r)=@r:

G(�0jr� roj)ZSn0 � rU (i)(r0)dS 0 = G(�0jr� roj) @U(r)

@r

��r=ro

ZSn0 � urs�rodS 0 ;

which, since in spherical coordinates1 dS 0 = r2 sin �d�d�, we obtain that:

G(�0jr� roj)ZS

@U (i)(r0)

@n0dS 0 = 0 :

Therefore, the scattered wave from a small volume V can be approximated to:

U (sc)(r) '��

�a0D0

� �a1D1

�+i!

c

�n1D1

� n0D0

��V

4�U (i)(ro)G(�0jr� roj) : (4.28)

Eq. (4.28) can be considered as the scattered wave from a Rayleigh particle. Rayleigh scatterers

are considered those particles which size is much smaller than the wavelength, and whose

response to the incident �eld can be considered as linear and elastic: U (sc) / U (inc). When

dealing with propagating scalar waves (PSWs) of wavenumber k0 = !n0=c it is given by:

U(sc)PSW (r) '

"�!

c

�2(n1 � n0)

2

#V

4�U (i)(ro)G(�0jr� roj) ; (4.29)

that is, the �eld of the PSW has a dependence of a power of two on the oscillation frequency.

As seen from Eq. (4.28), this does not occur for DPDWs. What is more, in the case in which

there is no contrast in D, i.e. D0 = D1 = D, Eq. (4.28) reduces to:

U (sc)(r) =��a0 � �a1

D0

�V

4�U (i)(ro)G(�0jr� roj) ; (4.30)

therefore establishing that the scattered wave from such a particle is independent on the os-

cillation frequency. Eqs. (4.28) and (4.30) can also be considered as the �rst order Born

approximation to the scattering from a small particle. A thorough study of the limits of valid-

ity of the Born approximation when dealing with small particles has been performed by Boas

and can thus be found in Refs. [30, 87, 88].

1The integral term becomes �2�r2R �0cos �d[cos �] = �r2 cos2 �

��0= 0.


4.1.2 Source Anisotropy

Although only isotropic sources will be employed throughout this work, it is convenient to

derive the scattering integral equations for such sources, since in practice most sources are

anisotropic. In this case, we recall that the expression for the �ux J inside the volume which

contains the source is (see Sec. 2.3.2):

J+(r) = �D0rU+(r) + JE(r) ;

where JE is as de�ned in Eq. (2.65) in page 24. In this case, the di�usion equation is given by

Eq. (2.67) in page 24, and Eqs. (4.24) and (4.25) must be changed accordingly since now the

boundary condition for the �ux continuty is:

�D0@U+(r)

@n

��S

+ n � JE(r)jS = �D1@U�(r)

@n

��S

; (4.31)

and therefore the relationship between �S(+) and �S(�) is:

�S(�)(r) = �S(+)(r) +ZS

1� D0

D(r0)jS

!G(�0jr� r0j)@U

+(r0)

@n0dS 0+Z

S

1

D1n0 � JE(r0)G(�0jr� r0j)dS 0 :

Subsituting Eq. (4.31) into Eqs. (4.11)-(4.15), and including the rJE term of Eq. (2.65) in

page 24 we obtain:

� r 2 ~V

U+(r>) = U(i)Q (r>) +

1

4�

ZVG(�0jr> � r0j)��2(r0)U�(r0)d3r0

+1

4�


0)

D1(r0)� rU�(r0)d3r0

� 1

4�

ZS

1� D0

D1(r0)jS

!G(�0jr> � r0j)@U

+(r0)

@n0dS 0�

1

4�

ZS

1

D1n0 � JE(r0)G(�0jr> � r0j)dS 0 ;

� r 2 V

U�(r<) = U(i)Q (r<) +

1

4�

ZVG(�0jr< � r0j)��2(r0)U�(r0)d3r0

+1

4�


0)

D1(r0)� rU�(r0)d3r0

+1

4�

ZS

1� D1(r

0)jSD0

!G(�0jr< � r0j)@U

�(r0)

@n0dS 0


� 1

4�

ZS

1

D0n0 � JE(r0)G(�0jr< � r0j)dS 0 ;

where U(i)Q (r) is in this case given by:

U(i)Q = U (i)(r) = � 1

4�

Z~V

S0(r0) +r � JE(r0)

D0G(�0jr� r0j)d3r0 :

In the special case in which D1(r) = D0, we obtain [compare with Eq. (4.26)]:

U(r) = U(i)Q (r)� 1

4�

ZS

n0 � JE(r0)D0

G(�0jr� r0j)dS 0

+1

4�

ZVG(�0jr� r0j)��2(r0)U(r0)d3r0 :

4.2 Relationship with the angular spectrum

Returning to the Helmholtz-Kircho� integral, Eq. (4.22), when the surface S(x; y) delimits a

plane interface at z = 0, we can rewrite it as:

U+(r) = U (i)(r) +1

4�

ZS(x;y)

U+(R0)

@G(�0jr�R0j)@z0

�G(�0jr�R0j)@U+(R0)

@z0

!dS 0 ; (4.32)

where, by writing G and U in terms of their angular spectrum2:

U+(r) = U (i)(r) +4�2

4�

ZS(x;y)

Z +1

�1

~U+(K; z0)

@ ~G(K; z � z0)

@z0

� ~G(K; z � z0)@ ~U+(K; z0)

@z0

!exp[iK �R0]dKdS 0 : (4.33)

If we use the relationship demonstrated in Sec. 3.2, Eq. (3.27):

@ ~U(K; z)

@z= iq(K) ~U(K; z) ;

and include it in Eq. (4.33),

U+(r) = U (i)(r) + �ZS(x;y)

Z +1

�1

~U+(K; z0)

@ ~G(K; z � z0)

@z0

� iq(K) ~U+(K; z0) ~G(K; z � z0)�exp[iK �R0]dKdS 0 ;

2In order to present the Fourier transform of a convolution we have made use of the formulaRvf(r0)g(r �

r0)dr0 = 4�2

R +1�1

~f(k)~g(k) exp[ik � r]dk.


and substitute the values derived in Sec. 3.2.2 of:

~G(K; z � z0) =i

2�q(K)exp[iq(K)(z � z0)] ; (4.34)

@ ~G(K; z � z0)

@z0=

1

2�exp[iq(K)(z � z0)] ; (4.35)

we obtain:

U+(r) = U (i)(r) + �ZS(x;y)

Z +1

�1

2

2�~U+(K; z0) exp[iq(K)(z � z0)] exp[iK �R]dKdS 0 ; (4.36)

which gives us:

U+(r) = U (i)(r) +Z +1

�1B(K) exp[iq(K)z] exp[iK �R]dK ; (4.37)

where B(K) = ~U+(K; z0), and therefore:

B(K) =1

4�2

ZS(x;y)

U+(r)jz=0 exp[�iK �R]dR :

If we go back to Ch. 3, Sec. 3.2� we shall see that Eq. (4.37) is precisely the angular spectrum

representation of U+(r) , where B(K) is the angular spectrum of the scattered wave. The surface

S(x; y) will de�ne the surface of interaction, which, in case of a plane, implies B(K) = A(K).

The cases in which S(x; y) represents a circular or rectangular aperture Q on a black plane

where shown in Sec. 3.4 of Ch. 3. These where derived by means of the Kircho� approximation

Eq. (4.23), which in terms of the angular spectrum can be written as:

BK(K) =1

4�2

ZQU (i)(r)jz=0 exp[�iK �R]dR :

Another relationship can be obtained if we rewrite Eq. (4.36) again in terms of @G=@z in real

space3:

U+(r) = U (i)(r) +1

2�

ZS(x;y)

U+(r0)@G(�0jr� r0j)

@z0dS 0 ; (4.38)

which is exactly the expression of the �rst Rayleigh-Sommerfeld integral formula encountered

in Sec. 3.2.2, Eq. (3.41). The second Rayleigh-Sommerfeld integral formula can be obtained if

the substitute for U instead of G in Eq. (4.36), i.e. by using U+ = �i=q(K)@U+=@z, which

gives:

U+(r) = U (i)(r)� 1

2�

ZS(x;y)

@U+(r0)

@z0G(�0jr� r0j)dS 0 : (4.39)

Of course, if we combine Eqs. (4.38) and (4.39) we once again recover Eq. (4.32). Again we

wish to recall that this was derived for a plane interface. Therefore, whenever the interface is

3exp[iq(K)z] = 2�@G=@z .


not plane and it is employed as an approximation, it is called the Rayleigh hypothesis (see Ref.

[59] for details on this approximation), which assumes that the angular spectrum representation

Eq. (4.37), is valid in regions occupied by the volume.

Figure 4.6: Volume V located between the planes z = 0 and z = L.

Now, let us assume we have the situation depicted in Fig. 4.6, with a volume V located

in�L � z � 0 and a source in z = zs > 0. As in Chapter 3, Eq. (3.24), the �eld measured at

any plane z > 0 can be written in terms of the angular spectrum as:

U+(R; z) = U (i)(R; z) +Z +1

�1A(K) exp[iK �R] exp[iq(K)z]dK : (4.40)

On expressing the terms in Eq. (4.24) by their Fourier transform at a certain plane z we

obtain4:

U+(R; z) = U (i)(R; z) +1

4�

ZV

Z +1

�1

~G(K; z � z0)4�2~��(K) exp[ik � r0]dKdr0

+1

4�

ZV

Z +1

�1

~G(K; z � z0)4�2~�rD(K) exp[ik � r0]dKdr0

� 1

4�

ZS

Z +1

�1

~G(K; z � z0)4�2~�S(K) exp[ik � r0]dKdS 0 : (4.41)

where5:

~��(K) =1

4�2

ZV��2(r0)U�(r0) exp[�ik � r0]dr0 ; (4.42)

4We once again have made use of the formulaRvf(r0)g(r� r

0)dr0 = 4�2R +1�1

~f(K)~g(K) exp[ik � r]dK.5We remind that k = (K; q); where q(K) =

p�20 �K2.


~�rD(K) =1

4�2

ZV

rD(r0)

D(r0)� rU�(r0) exp[�ik � r0]dr0 ; (4.43)

~�S(K) =1

4�2

Zr02S

1� D0

D1(r0)jS

!� @U

�(r0)

@n0exp[�ik � r0]dr0 ; (4.44)

represent the Fourier transform of the sources inside the volume. If we now introduce the

expression for ~G(K; z � z0) of Eq. (4.34), and write k � r = (K �R; q(K)z) in Eq. (4.41):

U+(R; z) = U (i)(R; z) +i

2

ZV

Z +1

�1

~��(K)

q(K)exp[iq(K)z] exp[iK �R0]dKdr0

+i

2

ZV

Z +1

�1

~�rD(K)

q(K)exp[iq(K)z] exp[iK �R0]dKdr0

� i

2

ZS

Z +1

�1

~�S(K)

q(K)exp[iq(K)z] exp[iK �R0]dKdS 0 : (4.45)

Comparing Eq. (4.45) with Eq. (4.40) we obtain:

A(K) =i

2

1

q(K)f~��(K) + ~�rD(K)� ~�S(K)g : (4.46)

In the simple case in which D1(r) = D0 as in Eq. (4.26), A(K) is expressed as:

A(K) =i

2q(K)

ZV��2(r0)U(r0) exp[�iq(K)z0] exp[�iK �R0]dR0dz0 : (4.47)

Eqs. (4.46) and (4.47) imply that A(K) contains all the information on the object. As men-

tioned in Ch. 1, one of the main objectives of the development and applications of DPDWs is

to locate, detect and characterize hidden objects. Therefore, Eqs. (4.46) or (4.47) are indeed

useful for this purpose. They both are the basis for performing DPDWs di�raction tomogra-

phy experiments. In Refs. [63, 64] for example, this formulation is employed, obtaining good

reconstructions of thin objects. In these references, an approximation to Eq. (4.47) is used, in

which the object is assumed thin, i.e. L ' 0, and that the value of the �eld inside the object

can be adequately described by the incident �eld:

A(K) ' i

2q(K)

ZS��2(R0)U (i)(R0; z0 = 0) exp[�iK �R0]dR0 :

In this way, the values of ��2 can be obtained from a measurement of U = Udata at z = z0, by

means of Eq. (4.40) as:

A(K) =�~Udata(K)� ~U (i)(K)

�exp[�iq(K)z0] ; (4.48)


and the values of ��2 are obtained by inverse transforming:

��2(R) ' �

U (i)(R; z = 0)

Z +1

�1

~Udata(K)� ~U (i)(K)~G(K; z0)

exp[iK �R]dK : (4.49)

It can easily be proven that Eq. (4.26) yields the same solution as Eq. (4.49) in the �rst Born

approximation. Eq. (4.48) is used in thechniques of back-propagation [89, 90], which will be

addressed in Ch. 5. There are two main drawbacks associated to Eq. (4.49): First, the factor

1= ~G(K; z0) fastly diverges for large values of z0 and K, and it therefore ampli�es the noise

existent at high frequencies. This e�ect was studied in Ref. [64], by using Eq. (4.49) for object

reconstruction, and in Ref. [90] as regards to spatial resolution. The second drawback is due

to experimental uncertainty of the exact value of the incident �eld U (i)(r), since the amplitude,

source location, and exact expression are unknown in most realistic cases.

When ��2 is R independent, i.e. ��2(r) = ��2(z), from Eq. (4.47) we obtain:

A(K) =i

2q(K)

Z 0

�L��2(z0) ~U(K; z0) exp[�iq(K)z0]dz0 : (4.50)

This expression is directly connected with the re�ection and transmission coe�cients (see Sec.

6.1).

4.3 Multiple Volumes of Scattering

In most practical cases, what we usually have is an outer di�usive medium, where the source

is placed, and an inner di�usive medium where the object to be located and characterize is

embedded. We shall therefore derive the scattering integral equations for such a medium, and

present the method for solving this system. Numerical results will be presented later on, in Ch.

6.

Let us take into consideration the con�guration shown in Fig. 4.7. It consists of an arbitrary

interface Sout that separates an otherwise in�nite di�usive medium Vout[r > Sout], characterized

by a linear, spatially uniform and isotropic di�usion coe�cient Dout, absorption coe�cient

�aout and wavenumber �out, from a closed di�usive medium Vin[r < Sout] with a linear, spatially

uniform and isotropic di�usion coe�cient Din, absorption coe�cient �ain , and wavenumber �in.

An object enclosed by the surface Sobj, with di�usion coe�cient Dobj, absorption coe�cient

�aobj , and wavenumber �obj is immersed in Vin. The refractive index is constant throughout

the media. A point source is placed in the outside medium r > Sout at point rs, its intensity

being sinusoidally modulated at frequency !. Considering these three di�usive media, imposing

average intensity continuity conditions and conservation of the total current density across their


Figure 4.7: Con�guration for the case of three scattering volumes.

common boundaries [see Eqs. (4.8) and (4.9)], we write the following saltus conditions:

hU (out)(r)

iSout

=hU (in)(r)

iSout

;hU (in)(r)

iSobj

=hU (obj)(r)

iSobj

;

Dout

"@U (out)(r)

@m

#Sout

= Din

"@U (in)(r)

@m

#Sout

;

Din

"@U (in)(r)

@n

#Sobj

= Dobj

"@U (obj)(r)

@n

#Sobj

: (4.51)

m being the outward normal vector of the surface pro�le Sout, and n the outward normal

vector of the surface of the object Sobj. Since we have three di�erent volumes with constant

parameters, it is more convenient to describe the problem by means of three di�erent Green's

functions, one for each scattering volume [see Eqs. (4.3) and (4.4)]:

. r 2 Vout

r2U (out)(r) + �2outU(out)(r) = �S0(r)

Dout;

r2G(�outjr� r0j) + �2outG(�outjr� r0j) = �4�Æ(r� r0) ; (4.52)

. r 2 Vin


r2U (in)(r) + �2inU(in)(r) = 0 ;

r2G(�injr� r0j) + �2inG(�injr� r0j) = �4�Æ(r� r0) ; (4.53)

. r 2 Vobj

r2U (obj)(r) + �2objU(obj)(r) = 0 ;

r2G(�objjr� r0j) + �2objG(�objjr� r0j) = �4�Æ(r� r0) : (4.54)

Proceeding in a similar way as in Section 4.1, but using the corresponding Green's function

for each homogeneous medium, Eqs. (4.52)-(4.54), we obtain the following set of coupled

integral equations:

. r0; r 2 Vout

U (out)(r) = U (i)(r) +1

4�

ZSout

(U (out)(r0)

@G(�outjr� r0j)@m0

�G(�outjr� r0j)@U(out)(r0)

@m0

)dS 0 ; (4.55)

. r 2 Vout; r 62 Vout

0 = U (i)(r)+

1

4�

ZSout

(U (out)(r0)

@G(�outjr� r0j)@m0

�G(�outjr� r0j)@U(out)(r0)

@m0

)dS 0 ; (4.56)

. r0; r 2 Vin

U (in)(r) = � 1

4�

ZSout

(U (in)(r0)

@G(�injr� r0j)@m0

�G(�injr� r0j)@U(in)(r0)

@m0

)dS 0

+1

4�

ZSobj

(U (in)(r0)

@G(�injr� r0j)@n0


@n0

)dS 0 ; (4.57)

. r0 2 Vin; r 62 Vin

0 = � 1

4�

ZSout

(U (in)(r0)

@G(�injr� r0j)@m0


@m0

)dS 0

+1

4�

ZSobj

(U(r0)

@G(�injr� r0j)@n0


@n0

)dS 0 ; (4.58)


. r0; r 2 Vobj

U (obj)(r) = � 1

4�

ZSobj

(U (obj)(r0)

@G(�obj jr� r0j)@n0

�G(�objjr� r0j)@U(obj)(r0)

@n0

)dS 0 ; (4.59)

. r0 2 Vobj; r 62 Vobj

0 = � 1

4�

ZSobj

(U (obj)(r0)

@G(�objjr� r0j)@n0

�G(�objjr� r0j)@U(obj)(r0)

@n0

)dS 0 : (4.60)

Depending on the symmetry of the problem, we must use as Green's functions G(��jr � r0j)those corresponding to the dimension of the system. In 2D these are given by the zeroth-order

Hankel function of the �rst kind: G(��jr � r0j) = �iH(1)0 (��jr � r0j) (see Ref.[58]), for which

U (i)(r � rs) = S0G(�outjr � rsj)=4�Dout represents in 2D a damped cylindrical wave with its

origin at rs, and in 3D, G(��jr � r0j) is given by: G(��jr� r0j) = exp[i�0jr� r0j]=jr� r0j, forwhich U (i)(r) represents a damped spherical wave, see Eq. (3.13) in Sec. 3.1.1.

It should also be remarked that by following arguments similar as before, the above equations

can be straightforwardly generalized to cases in which there are more than one scattering domain

(i.e. several objects) embedded in the volume Vin, even if the objects themselves are also rough

or of any arbitrary shape.

4.4 Solving coupled integral equations

Figure 4.8: Schematic representation of the case when s � r! Sout and so � r! Sobj.


The method for solving the system of coupled integral equations (4.55)-(4.60), consists of

applying the boundary conditions Eqs. (4.51), and making r tend to a surface point Sout

in Eqs. (4.55) and (4.57), and r tend to r ! Sobj in Eqs. (4.57) and (4.59) as shown in

Fig. 4.8. This method has been extensively used in scattering by electromagnetic waves from

arbitrary surfaces [74, 91, 92, 93, 94], and its accuracy and convergence has been repeatedly

demonstrated. It has also been applied to stress measurements, where it is called the boundary

element method (see Ref. [95]). In the electromagnetic environment it is usually denoted as

the extinction theorem method, name we shall use from now on.

After applying the boundary conditions Eqs. (4.51), the system is reduced to the following

coupled integral equations:

. r! Sout

US(s) = U (i)(s) +1

4�

ZSout

(US(s

0)@G(�outjs� s0j)

@m0�G(�outjr� r0j)@US(s

0)

@m0

)dS 0 ; (4.61)

US(s) = � 1

4�

ZSout

(US(s

0)@G(�injs� s0j)

@m0�G(�injs� s0j)Dout

Din

@US(s0)

@m0

)dS 0

+1

4�

ZSobj

(UO(s

0o)@G(�injs� so0j)

@n0�G(�injs� so0j)@UO(so

0)

@n0

)dS 0 ; (4.62)

. r! Sobj

UO(so) = � 1

4�

ZSout

(US(s

0)@G(�injs� s0j)

@m0�G(�injs� s0j)Dout

Din

@US(s0)

@m0

)dS 0

+1

4�

ZSobj

(UO(s

0o)@G(�injso � so0j)

@n0�G(�injso � so0j)@UO(so

0)

@n0

)dS 0 ; (4.63)

UO(so) = � 1

4�

ZSobj

(UO(so

0)@G(�obj jso � so0j)

@n0

�G(�objjso � s0oj)Din

Dobj

@UO(s0o)

@n0

)dS 0 ; (4.64)

where we have performed the change of variables s � r ! Sout, and so � r ! Sobj. As seen,

now the unknowns are US, @US=@m, and UO , @UO=@n which represent the average intensity

and its normal derivative at the interfaces Sout and Sobj, respectively:

U (out)��Sout

= U (in)��Sout

= US ;

@U (out)

@m

��Sout

=Din

Dout

@U (in)

@m

��Sout

=@US

@m;

U (in)��Sobj

= U (obj)��Sobj

= UO ;

@U (in)

@n

��Sobj

=Dobj

Din

@U (obj)

@n

��Sobj

=@UO

@n: (4.65)


To date, the most accurate way to solve Eqs. (4.61)-(4.64) is to numerically proccess them.

Thus, by means of a quadrature scheme [58], the integration is converted into a summation for

M discretization points for Sout which will be represented as SM , and N discretization points

for Sobj, represented as ON :

US(Si) = U (i)(Si) +MXj=1

US(Sj)@G(out)(Sj; Si)� @US(Sj)G

(out)(Sj; Si) ; (4.66)

US(Si) = �MXj=1

US(Sj)@G(in)(Sj; Si)� @US(Sj)

Dout

DinG(in)(Sj; Si)

+NXj=1

UO(Oj)@G(in)(Oj; Si)� @UO(Oj)G

(in)(Oj; Si) ; (4.67)

UO(Oi) = �MXj=1

US(Sj)@G(in)(Sj; Oi)� @US(Sj)

Dout

Din

G(in)(Sj; Oi)

+NXj=1

UO(Oj)@G(in)(Oj; Oi)� @UO(Oj)G

(in)(Oj; Oi) ; (4.68)

UO(Oi) = �NXj=1

UO(Oj)@G(obj)(Oj; Oi)� @UO(Oj)

Din

DobjG(obj)(Oj; Oi) ; (4.69)

where the elements G and @G have the following expressions in 2D and 3D (see Eqs. (3.14)-

(3.17) in Sec. 3.1.1):

. In 2D :

G(�)(p; q) =1

4�

Z q+�x=2

q��x=2�iH

(1)0 (��jp� x0j)dx0 =

1

4��

8>>><>>>:�iH

(1)0 (��jp� qj)�x p 6= q ;

I(2D)�� (�x) p = q ;

(4.70)

@G(�)(p; q) =1

4�

Z q+�x=2

q��x=2�i��n � up�x0H(1)

1 (��jp� x0j)dx0 =

1

4��

8>>><>>>:�i��n � up�qH(1)

1 (��jp� qj)�x p 6= q ;

12

p = q ;

(4.71)

. In 3D :

G(�)(p;q) =1

4�

Z qx+�x=2

qx��x=2

Z qy+�y=2

qy��y=2G(3D)(��jp� r0j)dx0dy0 =

1

4��

8>>><>>>:G(3D)(��jp� qj)�x�y p 6= q ;

I(3D)�� (�x;�y) p = q;

(4.72)


@G(�)(p;q) =1

4�

Z qx+�x=2

qx��x=2

Z qy+�y=2

qy��y=2n � rG(3D)(��jp� r0j)dx0dy0 =

1

4��

8>>><>>>:n � rG(3D)(��jp� q0j)�x�y p 6= q ;

12

p = q ;

(4.73)

where,

I(2D)�� (�x) =

Z +�x=2

��x=2�iH

(1)0 (��x)dx ; (4.74)

I(3D)�� (�x;�y) =

Z +�x=2

��x=2

Z +�y=2

��y=2

exp[i��px2 + y2]p

x2 + y2dxdy ; (4.75)

are the principal values of G(2D;3D), i.e. the value at the discontinuity r0 = r, or as de�ned in

Ref. [96], the values of the Green function in the source region. I(2D;3D) greatly depend on

the discretization cells, and on the accuracy of their value greatly depends the solution to the

system of equations (4.66)-(4.69). Their expressions are6 :

I(2D)�� (�x) = �iH

(1)0

��x

2e

�; (4.76)

I(3D)(�x;�y) ' 2� + 8 ln�tan

�3�

8

��i��x

2

�+ 4

�i��x

2

�2; (4.77)

where we have taken the discretization cell in 3D to be square, i.e. �x = �y.

Once we have the values of the elements G(�) and @G(�) we can write Eqs. (4.66)-(4.69) in

matrix form as:

[G]2(M+N)�2(M+N) � [U ]2(M+N)�1 = [Uinc]2(M+N)�1 ; (4.78)

where we have de�ned

[G] �

0BBBBBB@IM�M � @G

(out)M�M G

(out)M�M 0M�N 0M�N

IM�M + @G(in)M�M �Dout

DinG

(in)M�M �@G(in)

M�N G(in)M�N

@G(in)N�M �Dout

DinG(in)N�M IN�N � @G(in)

N�N G(in)N�N

0N�N 0N�N IN�N + @G(obj)N�N � Din

DobjG

(obj)N�N

1CCCCCCA ; (4.79)

[U ] �

0BBBBBB@US

@US

UO

@UO

1CCCCCCA ; [Uinc] �

0BBBBBB@U (i)

0

0

0

1CCCCCCA : (4.80)

In Eq. (4.79) IM�M and 0M�M represent the unitary and zero matrix of order M .Then, by

6The derivation of these expressions is quite cumbersome, and we can see that I(3D) is an approximation,and as such should be handled with care. The derivation of the principal values @I(2D) of @G(2D) can be foundin Ref. [91].


inverting matrix [G], we obtain the values of the �eld and the normal derivative at surface Sout

and Sobj:

[U ] = [G]�1[Uinc] :

Once the values of the �eld and the normal derivative are found, we can build the total �eld in

any point in space by means of Eqs. (4.55), (4.57) and (4.59) as:

. r 2 Vout ;

U (out)(r) = U (i)(r) +MXj=1

US(Sj)@G(out)(Sj; r)� @US(Sj)G

(out)(Sj; r) ;

. r 2 Vin ;

U (in)(r) = �MXj=1

US(Sj)@G(in)(Sj; r)� @US(Sj)

Dout

DinG(in)(Sj; r)

+NXj=1

UO(Oj)@G(in)(Oj; r)� @UO(Oj)G

(in)(Oj; r) ;

. r 2 Vobj ;

U (obj)(r) = �NXj=1

UO(Oj)@G(obj)(Oj; r)� @UO(Oj)

Din

Dobj

G(obj)(Oj; r) :

Chapter 5

Spatial resolution of Di�use Photon

Density Waves

In Chapter 3 we have de�ned the propagation of DPDWs, and the expressions for the impulse

response and transfer function have been derived. However, the important issue of resolution

was not addressed, not withstanding its connection with wave propagation. A robust criterion

for spatial resolution is of great help, as it enables us not only to predict the limits of spatial

resolution that a particular system can deliver, but in the future, to calibrate and compare

di�erent clinical imaging systems which use di�usive photons, in terms of their resolving power.

In this Chapter, we shall derive the expressions for the limits of spatial resolution by directly

starting from the transfer function concept presented in Ch. 3. We will discuss a limiting case

of spatial resolution, namely, the electrostatic limit, and perform a comparison between di�use

photon density waves (DPDWs) and propagating scalar waves (PSWs). Later on, we test the

predictions from the resolution limit with rigorous numerical results for two di�usive cylinders.

We shall then consider the issue of noise present in the measurements. This is of basic interest

to understanding the evolution of the spatial resolving power on propagation. Finally, the

procedure of back-propagation will be studied in terms of the spatial resolution. We shall

present numerical examples on how to achieve greater spatial resolution.

5.1 Spatial Resolution

Before going on, it is convenient to de�ne the term spatial resolution, in order to avoid miscon-

ceptions. As we de�ne spatial resolution, it is not related to the characterization limit of an

object, i.e. the limits of recovery of its optical parameters by inverse scattering. The de�nition

of spatial resolution to be used is in its standard meaning [11], namely, the ability to separate

two point objects, or �ne details, from measurements at a certain distance from the scattering

object. The criterion we shall use to de�ne the limit of spatial resolution is the well known

Rayleigh criterion, and as such it must be borne in mind that it does not measure the absolute

73

CHAPTER 5. SPATIAL RESOLUTION OF DIFFUSE PHOTON DENSITY WAVES 74

limit of resolution, but it yields a criterion to predict whether or not two objects will be resolved

at a certain measurement plane. The limit of spatial resolution is given by the full width at

half maximum (FWHM) of the transfer function ~F :

~F (K; z) = exp[iq(K)z] ; (5.1)

which therefore states that any spatial frequency higher than that given by the FWHM will

not be detected. The FWHM of both the transfer function F and the impulse response H:

H(R�R; z) =Z +1

�1

~F (K; z) exp[iK �R]dK ; (5.2)

can be evaluated analytically.

5.1.1 Propagating Scalar Waves

In the case of PSWs, since for evanescent components we have:

q(K) = �qjKj2 � �20 ;

the FWHM, which we shall de�ne as �jKj, of the transfer function is obtained from the

condition:

j ~F (K; z)j = exp[�qjKj2 � �20z] =

1

2:

On taking logarithms on both sides, one obtains:

jKj2 � �20 =

ln 2

z

!2

:

Therefore, the value of jKj that yields the value ~F = 1=2, is:

jKj1=2 =vuut�20 +

ln 2

z

!2

:

Since what we are looking for is the FWHM �jKj , it follows that �jKj = 2jKj1=2 so that the

limit of spatial frequency is given by:

�jKj = 2

vuut�20 +

ln 2

z

!2

: (5.3)

Now, making use of the relationship between the FWHM of a function (H in our case) and

that of its Fourier transform ( ~F ):

�d =2�

�jKj ;


�d being the FWHM of H, we obtain from Eq. (5.3):

�d

�0=

1

2

241 + ln 2

2�z=�0

!235�1=2 ; (5.4)

where �d=�0 is the spatial resolution limit in units of the wavelength. When z increases, we

see in Eq. (5.4) that the spatial resolution limit tends to:

limz��0

�d =�02; (5.5)

namely, we retrieve the well known Rayleigh limit, for z � �0, of optical imaging. In frequency

space, Eq. (5.3) shows that as z increases,

limz��0

�jKj = 2�0 :

5.1.2 Di�use Photon Density Waves

In the case of DPDWs, the expression for q is:

q(K) =q�20 � jKj2 =

q�2<e � �2=m � jKj2 + 2i�<e�=m ;

where �0 = �<e+ i�=m, which after some algebra can be written in terms of q(K) = q<e+ iq=m

as:

q<e(K) =1p2

��j�0j4 � 2(�2<e � �2=m)jKj2 + jKj4

�1=2+ �2<e � �2=m � jKj2

�1=2; (5.6)

q=m(K) =1p2

��j�0j4 � 2(�2<e � �2=m)jKj2 + jKj4

�1=2 � �2<e + �2=m + jKj2�1=2

: (5.7)

If can be seen that when jKj = 0 we recover q(K = 0) = �0. Since the maximum of ~F decays as

z increases, we must de�ne the FWHM by normalizing ~F to its value at K = 0. Therefore the

value of jKj1=2 that brings the modulus of the normalized transfer function to its half maximum

is given by:j ~F (K; z)j

j ~F (K = 0; z)j = exp[�(q=m(K)� �=m)z] =1

2;

so that, proceeding as with the PSWs, the FWHM of ~F is obtained from the equation:

�j�0j4 � 2(�2<e � �2=m)jKj2 + jKj4

�1=2 � �2<e + �2=m + jKj2 = 2

�=m +

ln 2

z

!2

:


After some tedious algebra, the value for �jKj for DPDWs is:

�jKj = 2

vuut �=m +ln2

z

!2

+ �2<e � �2=m ��2=m�

2<e

(�=m + ln 2=z)2; (5.8)

and using the fact that �0 = 2�=�0 + i=L! (see Ch. 3, Sec. 3.1), the corresponding FWHM of

H is:

�d

�0=

1

2

24 �02�L!

+ln 2

2�z=�0

!2

�

�02�L!

!2

+ 1� 1 +

ln 2

z=L!

!�235�1=2 : (5.9)

As z increases, z � �0, we see in Eq. (5.8) that

limz��0

�jKj = 0 :

Also, Eq. (5.9) shows that �d=�0 has no upper limit:

limz��0

�d = �

szL!

2 ln 2; (5.10)

and tends to in�nity as z1=2, monotonically worsening the resolution. Hence, in contrast with

the case of PSWs, all components of spatial frequencies K always propagate into z > 0, even

though there exists attenuation of the signal in the whole K range due to di�usion.

Figure 5.1: Spatial resolution limit �d in cm as z increases, for the following cases: DPDWsin breast tissue (�a = 0:035cm�1, �0s = 15cm�1): ! = 0 [solid curve]; f = !=2� = 100MHz,�0 = 13:38cm [dotted curve]; f = 200MHz, � = 7:53cm [dashed curve]; f = 300MHz,�0 = 5:60cm [dotted-dashed curve]. PSWs: f = 0 [u]; �0 = 13:38cm [�]; �0 = 7:53cm [Æ].n = 1:333 in all cases.

This is seen in Fig. 5.1, which shows the values of the spatial resolution limit both for


DPDWs and PSWs versus the observation distance z. However, since in practical cases the

values of z are small (of the order of a few cm), we �nd in Fig. 5.1 that in many cases the loss

of resolution as z increases is less critical for DPDWs than for PSWs (compare in Fig. 5.1 the

cases with � = 7:53cm for PSWs and DPDWs in the interval [0; 3cm], for example).

The analysis presented here can be used to discuss experimental data, as those reported in

Refs. [63] and [86]. In Fig. 2 of Ref. [63], the authors characterize two di�usive objects with a

relative diameter of' 0:1�0, 3:26cm apart, embedded in a 0.75% Intralipid solution, illuminated

by a modulated source of ! = 100MHz (in this case �0 ' 15:12cm and L! ' 7:7cm). If we

take a look at our Fig. 5.1, approximating the di�usive parameters of the Intralipid solution

to those of breast tissues, we see that two objects 3:26cm apart can be resolved as long as

measuring at distances z � 2cm. This is what is observed in Fig. 2 of Ref. [63], where the

measurements are performed at a distance of 2cm and are therefore within the limit of spatial

resolution discussed above.

Figure 5.2: DPDWs spatial resolution limit �d in cm as we increment z, for the following cases:�a = 0:1cm�1, f = 50MHz [�]; �a = 0:1cm�1, f = 1GHz [Æ]; �a = 1:0cm�1, f = 50MHz[solid line]; �a = 1:0cm�1, f = 1GHz [dashed line]. In all cases �0s = 6:0cm�1, n = 1:333.

As another example, we can take Figs. 3(c)-(f) of Ref. [86] where the authors present four

images of two absorbing spheres with di�erent background parameters and di�erent modulation

frequencies. The spheres had a diameter of 1cm, and even though their separation is not

mentioned in Ref. [86], from the image it can be seen that it is approximately �d � 2:5cm.

The distance of the plane of measurement from the objects is also not mentioned, but from

Fig. 1 of Ref. [86] we can assume it is �z ' 3cm. The parameters used for the background

medium presented in Figs. 3(c) and 3(d) of Ref. [86] were �a = 0:1cm�1, �0s = 6:0cm�1, and

in Figs. 3(e) and 3(f) it was �a = 1:0cm�1, �0s = 6:0cm�1. The cases in Fig. 3(c) and 3(e)

correspond to f = 50MHz, and in Figs. 3(d) and 3(f) correspond to f = 1GHz. These cases


were introduced in Eq. (5.9) and are plotted in Fig.5.2. In Ref. [86] the authors observed

that for the �a = 0:1cm�1 case, there was an increase in resolution for higher modulation

frequencies. This is exactly what we observed in Fig. 5.2, were we see that for �z � 3cm, there

is an increase of resolution of more than one centimeter, changing from a non-resolved image at

f = 50MHz , to a resolved image at f = 1GHz (if we take the separation between the spheres

to be �d � 2:5cm). In the �a = 1:0cm�1 case, the authors in Ref. [86] observed that there

was no increase in resolution with the modulation frequency. This also can be observed in our

Fig. 5.2, were we see that the case �a = 1:0cm�1 does not present any appreciable di�erence

between f = 50MHz and f = 1GHz. The reason put forward in Ref. [86] was that in order

for the modulation frequency to change appreciable the value of �0, we need �av � !, where

v = c=n. This fact was discussed earlier in Ch. 3, Sec. 3.1.

5.2 The Electrostatic Limit

Since DPDWs are damped waves, the detection of the wave�eld U(R; z) is usually done in

the near �eld, i.e., at sub-wavelength distance from the source object (considered either as a

primary source or as a scattering object). In this range, if all distances involved are much

smaller than the wavelength, retardation e�ects can be neglected. This property is well-known,

for example, in near-�eld optics [65, 66, 67]. When retardation e�ects are neglected, one is in

the domain of the electrostatic limit .

At a given frequency !, this electrostatic limit is obtained when �0 !1. Then, for PSWs,

the limit of resolution within the electrostatic limit �de can be obtained from Eq. (5.4):

�de = lim�0!1

�d =�

ln 2z : (5.11)

On the other hand, for DPDWs we �nd from Eq. (5.9) that the limit of resolution in the

electrostatic limit is:

�de = lim�0!1

�d = �

24 1

2�Ld+

ln 2

2�z

!2

��

1

2�Ld

�235�1=2 ; (5.12)

where Ld =qD=�a is the di�usion length as de�ned in Sec. 2.3.2, i.e. Ld = L!=0. It is

interesting to note from Eqs. (5.11) and (5.12) that in the electrostatic limit, resolution does

not depend on the background medium for PSWs, whereas in the case of DPDWs, resolution

still depends on the background medium, through the di�usion length Ld. In the case in which

the absorption coe�cient is negligible (i.e. �a ' 0), the expression �d for DPDWs does not

depend on the background medium, and we obtain �de � z, as previously found by Ref. [97]

in the time-domain.

The transition to the electrostatic limit when z decreases can be clearly seen in Fig. 5.1,


Figure 5.3: DPDWs spatial resolution limit �d in cm as we increment z, in DC regime (! = 0),for the following cases: breast parameters �a = 0:035cm�1, �0s = 15cm�1[solid line]; abdomenparameters �a = 0:09cm�1, �0s = 9:5cm�1 [dotted line]; back parameters �a = 0:09cm�1,�0s = 10:5cm�1 [short-dashed line]; white matter �a = 0:22cm�1, �0s = 9:1cm�1 [long-dashedline]; grey matter �a = 0:27cm�1, �0s = 20:6cm�1 [dotted-dashed line]. In all cases n = 1:333.

for both DPDWs and PSWs. As seen, at low z, one does no longer increase the resolution by

lowering the incident wavelength, so that the behavior is similar to that observed with constant

illumination, i.e., DC regime (! = 0). However, it is important to note that in the case of

DPDWs, the �electrostatic region� extends to higher values of z, and therefore is a good approx-

imation even at modulation frequencies of the order of ! = 100MHz. We also observe that,

for modulation frequencies lower than 100MHz, no increase in spatial resolution is obtained

in AC, and therefore it is cheaper, and experimentally simpler, to perform measurements in

DC, if one is only interested in the location of the objects, namely, in obtaining an image. For

modulation frequencies higher than 100MHz we �nd that, even though the spatial resolution

limit increases very quickly for such frequencies, the decay length L! is considerable smaller

and, therefore, the attenuation is much stronger, thus making detection at practical distances

di�cult.

An important consequence of the existence of this electrostatic limit is that measurements in

DC (i.e. at ! = 0) performed within the domain of validity of this limit, give the same spatial

resolution as measurements in AC. To illustrate this point, we show in Fig. 5.3 the values of

the spatial resolution limit �d as z increases in several human tissues [98], corresponding to

a DC illumination. These curves give the spatial resolution that can be reached at a given

observation distance z, in each situation. If we once again refer to the situation depicted in

Fig. 2 of Ref. [63], we see that at a distance of 2cm, it is possible to resolve two objects 3:26cm

apart by measuring in DC (see the solid curve in Fig. 5.3). Thus, we infer that in the case


of Ref. [63], measurements in DC would have led to the same spatial resolution. Referring to

the situation depicted in Figs. 3(e) and 3(f) in Ref. [86], the resolution corresponding to these

images (solid and dashed lines in Fig. 5.2) presents no di�erences with the DC case, whatever

the measurement distance. This is to be expected from the results put forward in Sec. 3.1,

where the dependence of the wave number for DPDWs was studied versus the modulation

frequency.

5.3 Numerical Results

In order to illustrate the discussion of the previous section and to check the resolution criteria

Eqs. (5.8) and (5.9), we now present rigorous numerical results on scattering of DPDWs. Since

the theory of the scattering integral equations was already put forward in Ch. 4, we shall here

simply present the equations corresponding to N di�usive cylinders, later on to consider the

simple case of two cylinders, i.e. N = 2.

5.3.1 Scattering Integral equations for N bodies

Figure 5.4: Scattering geometry of N di�usive objects embedded in an otherwise in�nite homo-geneous medium.

Let us assume the con�guration depicted in Fig. 5.4, where there are N di�usive objects


of volumes Vj respectively, bounded by their corresponding surfaces Sj and of parameters �j,

Dj and �aj. These N objects are embedded in an otherwise homogeneous in�nite di�usive

medium of parameters �0, D0 and �a0. By following the steps presented in Ch. 4, we obtain

the following integral equations:

U (out)(r) = U (i)(r) +1

4�

NXj=1

ZSj

24U (obj)j (r0)

@G(�0jr� r0j)@n0j

�G(�0jr� r0j)@U

(obj)j (r0)

@n0j

35 dS 0j ;U

(obj)j (r) = � 1

4�

ZSj

24U (obj)j (r0)

@G(�j jr� r0j)@n0j

�G(�jjr� r0j)@U

(obj)j (r0)

@n0j

35 dS 0j ;where U (out) represents the total outside �eld, U (i) the incident �eld, U

(obj)j the �eld inside the

jth object, and nj the surface normal unit of object j. By imposing the boundary conditions

corresponding to index matched media:

U (out)jSj = U(obj)j jSj ;

�D0nj � rU (out)jSj = �Djnj � rU (obj)jSj ;

we obtain the following set of coupled equations:

Um(rm) = U (i)(rm) +1

4�

NXj=1

ZSj

"Uj(r

0)@G(�0jrm � r0j)

@n0j�G(�0jrm � r0j)@Uj(r

0)

#dS 0j ;

Uj(rj) = � 1

4�

ZSj

"U

(obj)j (r0)

@G(�j jrj � r0j)@n0j

�G(�jjrj � r0j)@U (obj)j (r0)

#dS 0j ;

where now the unknowns are Uj and @Uj, and rm represents the limiting point r approaching

Sm.The solution can be obtained exactly as shown in Sec. 4.4, on inversion of a N�(2M�2M)

matrix, where M is the number of discretization points for each surface Sj.

5.3.2 Numerical Results for Two Cylinders

Once we have put forward the system of equations to solve for N bodies, we shall consider

the case in which the geometry under consideration is two-dimensional, as depicted in Fig.

5.5. It consists of two di�usive in�nite cylinders (the hidden objects), with axis along OY ,

both with radius R and separated by a distance d. The cylinders are embedded in an in�nite,

homogeneous, di�usive medium. Constant refractive index n = 1:333 is supposed throughout

all the media. A point source with modulation frequency ! is located at rsource, and the

detection is performed in a plane z = zd. We shall rigorously solve this system by means of the

extinction theorem method, presented in Sec. 4.4, which allows to deal with multiple bouncing

contributions between objects. In the following, we shall be interested in the scattered DPDW,


Figure 5.5: Scattering geometry corresponding to two di�usive cylinders embedded in an other-wise in�nite di�usive medium.

de�ned by:

U (sc)(r) = U (out)(r)� U (i)(r) ;

where U (out) represents the total DPDW upon interaction with the objects, and U (i) corresponds

to the incident DPDW, namely, that created by the point source in the absence of the two

objects.

Following experimental procedures (see for example [10, 82]), we have considered a point

source emitting light at a wavelength of 780nm, with a modulation frequency ! = 200MHz.

The parameters chosen for the background medium correspond to breast tissue, with �a =

0:035cm�1 and �0s = 15cm�1. As for the cylinders, we have used the parameters of a breast

tumor, �a = 0:24cm�1 and �0s = 10cm�1. In all cases, the refractive index in the media is

n = 1:333. In order to reach numerical convergence, due to the small sizes of the cylinders

under study, we have used a discretization ds = 0:004cm for the surface of the cylinders.

Fig. 5.6 shows the amplitude of the scattered DPDW, jU (sc)j, at di�erent detection planes,

and for di�erent cylinder distances, when the two di�usive cylinders have a radius R = 0:1cm.

As expected, the further apart the cylinders are from each other, the better they are resolved,

and as we locate the detection plane further away, this resolution power diminishes. In order

to compare these results with the conclusions of the previous section, we refer again to Fig.

5.1, for breast tissue illuminated with a modulation frequency ! = 200MHz (dashed line). In

the range of cylinder distances represented in Fig. 5.6, i.e. for d from 1cm to 2:5cm, Fig. 5.1

indicates that, in order to keep within the spatial resolution limit, we must place the detection

plane in a range between 0:5cm and 1:5cm. This is precisely what is observed in Fig. 5.6. If we


Figure 5.6: Scattered amplitude corresponding to two di�usive cylinders of R = 0:1cmwith breast tumor parameters �a1 = 0:24cm�1, �0s1 = 10cm�1, embedded in breast tissue�a0 = :035cm�1, �0s0 = 15cm�1, with the source located at rsource = (0; 2:0cm) with modulationfrequency ! = 200MHz, separated by distances: d = 1cm (a), d = 1:5cm (b), d = 2:0cm (c)and d = 2:5cm (d), for the following z detector distances: zd = 0:2cm [solid line], zd = 0:4cm[dotted line], zd = 0:6cm [dashed line], zd = 0:8cm [long-dashed line], and zd = 1:0cm [dot-dashed line]. In all cases n = 1:333.

look at Fig. 5.6(a), for the case of zd = 1cm, we �nd that for a separation distance d = 1cm,

the objects are not spatially resolved. In this ideal noiseless situation, the cylinders start being

resolved at a detection plane at a distance zd = 0:4cm [dotted line in Fig. 5.6(a)].

Once the data are above the spatial resolution limit, it is convenient to de�ne another

quantity that allows to discriminate the image signal from a certain noise level, present in

the data, i.e. the contrast. It is also useful, in order to compare with previous de�nitions of

contrast, to introduce �rst a noise free contrast (nf), which we express in percentage % as:

Cnf (%) =jU (sc)

maxj � jU (sc)minj

jU (sc)maxj+ jU (sc)

minj� 100 : (5.13)

In Eq. (5.13), jU (sc)j is the noise-free scattered amplitude, and jU (sc)minj is the minimumvalue of

jU (sc)j between maxima jU (sc)maxj. In Fig. 5.7, we plot this noise-free contrast Cnf for two cylinders,

in the same situation as in Fig. 5.6, versus the detection distance zd. The modulation frequency


Figure 5.7: Values of the noise-free contrast Cnf(%) as we vary the detector plane distance zdwith the con�guration and parameters used in Fig. 5.6 for the following cylinder distances:d = 1cm [solid line], d = 1:5cm [dotted line], d = 2:0cm [dashed line] and d = 2:5cm [longdashed line]. In all cases n = 1:333.

is 200MHz. Results for di�erent separation distances d of the two cylinders are shown. These

curves give the basis to deal with more realistic data, that is, with noise. Calculations for

noise-free scattering data from cylinders with smaller radii R are not presented here because

the resulting contrast curves are similar to those of Fig. 5.7. This is because, as far as noise is

not taken into consideration, the main e�ect of reducing the size of the scattering object is a

decrease in the amplitude of the scattered wave, but has no e�ect in the resolution limit. The

reason for this has been put forward previously in Sec. 4.1.1, Eq. (4.27) where we see that for

small volumes, the Born approximation is quite accurate and the particles can be considered

Rayleigh scatterers. Therefore, as seen in Eq. (4.27), the scattered wave from these particles

has a linear dependence with the volume V , i.e. U (sc)(r) / V G(�0jr� roj)U (i)(ro). However,

in order to derive useful consequences when Fig. 5.7 is applied to actual (noisy) experimental

data, a contrast threshold must be introduced in Eq. (5.13) below which no signal can be

discriminated from the noise background.

5.4 E�ects of Noise on Resolution

Let Unoise be an image containing additive noise in amplitude N and in phase �. We shall

express it in terms of the DPDW U (sc) scattered by the objects as:

Unoise(R; z) = jU (sc)(R; z)j [1 +N(R)] exp (i[�(R; z) + �(R)]) ; (5.14)


where � is the phase of the scattered DPDW. The random variables N(R) and �(R) are

Gaussian distributed with correlation lengths TN and T�:

hN(R)N(R + �)i = �N exp[�T 2N=�

2] ; (5.15)

h�(R)�(R+ �)i = �� exp[�T 2� =�

2] ; (5.16)

where h�i indicates average. We shall consider TN = T� = 0, i.e. white noise. The root mean

square �N , and �� are de�ned as:

�N =qhN2(R)i � hN(R)i2 ; (5.17)

�� =qh�2(R)i � h�(R)i2 ; (5.18)

respectively. We shall consider the variables N and � with zero mean, and therefore:

�N =qhN2(R)i ; �� =

qh�2(R)i :

As de�ned in Eq. (5.14), the values of the random variables N and � are in the ranges:

N(R) 2 [�1; 1] ;�(R) 2 [��; �] :

When the noise present in an image is modelled as in Eq. (5.14), the noise-to-signal ratio �(%),

in percentage, is introduced as:

�(%) = �N � 100 : (5.19)

Then, we shall de�ne the contrast C(%) in the presence of noise as:

C(%) = Cnf(%)� �(%) :

With this de�nition, and referring again to Fig. 5.7, � is the aforementioned contrast threshold.

That is, resolving two objects requires placing the detector at such a distance z that the contrast

remains above �. This is shown in Fig. 5.8 where the same results plotted in Fig. 5.7 are

presented with � = 10%. As shown in this �gure, once the largest detection distance, z, that

yields a given resolution limit �d has been derived from Fig. 5.1, if for example we have a data

uncertainty � = 10%, the details which can be resolved with resolution �d can be found from

Fig. 5.8.

5.4.1 Filtering out the Noise

One wishes to �get rid� of as much noise as possible, with minimum loss of information. As we

will demonstrate, this is not a di�cult chore in the case of DPDWs since the Fourier spectra


Figure 5.8: Values of the noise contrast C(%) in the presence of a noise-signal ratio � = 10%.Parameters as those de�ned in Fig. 5.6.

of these waves is centered at the low frequencies. We shall use one of the most simple noise

�lters with the numerical data, so as to demonstrate how e�ciently we can remove noise from

the images with minimum loss of information.

Let us assume an image with noise Unoise(R; z) measured at z = zd, as shown in Eq. (5.14).

As seen from this equation, in order to distinguish the real value of U (sc) from Unoise we must

know the value of N and � at each point R. From Eq. (5.14) we infer that there is no possible

way in real space to extract the value of U (sc) from Unoise: On the other hand, if we take into

consideration that, since N and � both have zero correlation TN = T� = 0, then their spatial

frequency will be very high. In that case, their Fourier spectra will concentrate at high values

of the frequency. Therefore, in order to extract the value of U (sc) from Unoise, it is convenient

to work in Fourier space, i.e. with the angular spectrum representation (see Sec. 3.2). The

Fourier transform of our �noisy� image is:

~Unoise(K; zd) =1

4�2

Z +1

�1Unoise(R; zd) exp[�iK �R]dR :

Now, since the noise contribution will gather at high frequencies, we will introduce a �lter

N (K) which will practically leave untouched the low frequencies, and convert to zero these

high frequency components. One very commonly used is the Hanning �lter, de�ned as:

N (K) =

8><>:h12+ 1

2cos

�Kx

Kcutx��i��12+ 1

2cos

�Ky

Kcuty��

K < Kcut ;

0 K � Kcut ;


where Kcut = (Kcutx ; Kcut

y ) represents the cuto� frequency. This parameter must be modi�ed in

order to �nd the value which eliminates the largest possible amount of noise, while leaving the

value of U (sc) untouched. One of the advantages of the Hanning �lter is that it smoothly tends

to zero, thus avoiding problems such as aliasing (see Ref. [99], for further insight on noise and

image formation). There are of course many other �lters, each with its own advantages and

inconveniences, see Ref. [99], for example. Once we have de�ned the noise �lter to apply, the

�ltered spectrum ~Ufilt is obtained from:

~Ufilt(K) = ~Unoise(K)N (K) :

Therefore, the estimation of the scattered signal from the noisy data Unoise, with its high

frequency components �ltered out is given by:

Ufilt(R; zd) =Z +Kcut

�Kcut

~Unoise(K; zd)N (K) exp[iK �R]dK : (5.20)

As already discussed in Sec. 3.4, due to high damping and low re�ectivity, DPDWs are

not subjected to strong interference processes, as PSWs are. Therefore, the scattering and

di�raction patterns of DPDWs do not present high frequency interference fringes. This means

that the Fourier spectrum of DPDWs is mostly concentrated at frequenciesK � Kcut, therefore,

the �ltering neither substantially alters the image, nor appreciably reduces resolution. This can

be resumed as:~Ufilt(K) ' ~U (sc)(K) :

Then, after �ltering, one can estimate that:

Cfilt(%) ' Cnf (%) ;

Cfilt being the contrast of jUfiltj. This can be seen in Fig. 5.9 where we plot the values of~Unoise(K), ~U (sc)(K; z), and ~Ufilt(K; z), for z = 1cm, and Kcut = 12cm�1. These quantities

are computed for data from two di�usive cylinders with the same parameters as in Fig. 5.6,

radius R = 0:1cm, separated a distance d = 2:5cm, with the detection plane at zd = 1:0cm.

A numerical noise has been added to the scattered �eld, as described by Eq. (5.14). The

noise amplitude N(R) has a r.m.s. �N = 0:1 and therefore has a ratio � = 10%. The phase

noise has �� = 10Æ. A comparison of Fig. 5.9(b) and 5.9(c), con�rms that this �ltering does

not appreciably remove information in the spectrum. We believe that this should be the case

in most practical situations with DPDWs. On the other hand, the corresponding scattering

amplitudes in real space, both prior and after �ltering, are shown in Fig. 5.10.

As regards the simulation of Fig. 5.10, notice that, according to Fig. 5.8, the contrast that

we can expect in data taken at z = 1cm [Fig. 5.10(a)] and z = 1:5cm [Fig. 5.10(b)], for a

cylinder separation distance d = 2:5cm, is about 20% and 10%, respectively. In Fig. 5.10(a)


Figure 5.9: Values of (a) ~Unoise(K; z = 1cm), (b) ~U (sc)(K; z = 1cm) and (c) ~Ufilt(K; z = 1cm)for a detector plane distance zd = 1cm for the con�guration and parameters of Fig. 5.6, witha cylinder separation distance of d = 2:5cm. N (K) is a Hanning �lter with Kcut = 12cm�1.Noise parameters: � = 10%, and �� = 10Æ. In all cases n = 1:333.

we have � = 10% and therefore we still have a 10% of signal contrast above the threshold for

z = 1cm. However, in Fig. 5.10(b) we have considered a rather extreme situation in which

� = 30%, which in the z = 1:5cm case places the contrast under the threshold. Even so,

once the image is �ltered, we �nd that in the z = 1cm case [Fig. 5.10(c)], which is above the

threshold, the �ltered image is very close to the noise-free image. Surprisingly, this occurs also

for the case z = 1:5cm [Fig. 5.10(d)], which is under the threshold for both values of �, i.e.

� = 10% and � = 30%. That is, even in very disfavorable signal detections, we �nd no e�ective

threshold for the �ltered image amplitude.


Figure 5.10: Normalized scattered amplitude with the con�guration and parameters of Fig. 5.6,with a cylinder separation distance of d = 2:5cm in the cases: (a) Unoiseat zd = 1cm with noiseparameters � = 10%, and �� = 10Æ. (b) Unoiseat zd = 1:5cm with noise parameters � = 30%,and �� = 10Æ. (c) Ufilt with Kcut = 12cm�1 for the image in (a) [solid line]; U (sc) at zd = 1cm[dotted line]; Ufilt with Kcut = 12cm�1 an image at zd = 1cm with noise parameters � = 30%,and �� = 10Æ[�]. (d) Ufilt with Kcut = 12cm�1 an image at zd = 1:5cm with noise � = 10%,and �� = 10Æ[solid line]; U (sc) at zd = 1:5cm [dotted line]; Ufilt with Kcut = 12cm�1 the imageobtained in (b) [�]. In all cases n = 1:333.

5.5 Back-propagation

In Ch. 3 we already studied the propagation of DPDWs, and how the �eld that leaves an

object at z = 0 is modi�ed as it travels deeper into the medium until it is detected at z = zd.

The function that modi�es the �eld on propagation is the transfer function ~F (K; z � z0) =

exp[iq(K)(z � z0)], and their relationship is:

~U(K; zd) = ~U(K; z = 0) ~F (K; zd) :

In Sec. 4.2 we already mentioned the e�ect of back-propagating the �eld measured at plane

zd. This basically consists of inverting the e�ect that the transfer function has on the original


Figure 5.11: Regions (represented by ) which are not accessible by back-propagation.

�eld, and therefore from a �eld measured at zd we can reconstruct the emerging �eld at z = 0

by [89, 90]:

~U(K; zd ! 0) =~U(K; zd)~F (K; zd)

; (5.21)

where we have denoted ~U(K; zd ! 0) as the �eld obtained by back-propagating from z = zd up

to z = 0. It must always be taken into consideration that the angular spectrum representation,

and Eq. (5.21) are only valid outside the scattering object. We may change the plane of

detection and back-propagate so as to adequately reconstruct the object, but there will in most

cases exist regions as shown in Fig. 5.11 where we will not be able to back-propagate. This

is the basic idea of optical tomography: by means of several di�erent planes of measurement

around the object, back-propagate each plane to reconstruct the �eld at the object's surface.

Therefore, we shall back-propagate only up to the object surface. We shall consider the back-

propagation of several �elds, namely those with noise ~Unoise(K; zd ! 0) and those noise-free~U (sc)(K; zd ! 0). As it will be shown, although the value of ~U(K; zd ! 0) will indeed be similar

to ~U (sc), it is by no means the same �eld. The reason for this can be seen in Eq. (5.21), which

shows the value at zd divided by ~F , i.e. multiplied by ~F�1:

~U(K; zd ! 0) = ~U(K; zd) exp[�iq<e(K)zd] exp[q=m(K)zd] ; (5.22)

where we have written q(K) = q<e + iq=m. From Eq. (5.22) we see that ~U(K; zd) is multiplied

by a positive exponential exp[q=m(K)zd]. Now, if we go back to Eq. (5.7) in Sec. 5.1.2, we see

that for relatively high values of K the values of q=m are:

q=m(K) ' jKj :


Therefore, when back-propagating from z = zd to z = 0 we are multiplying the amplitude

measured at zd by:

j ~U(K; zd ! 0)j ' j ~U(K; zd)j exp[jKjzd] ;

and all the high frequencies are thus exponentially increased. This function diverges for rel-

atively high frequencies and hence we must establish a cuto� frequency. Most importantly,

we see that this backward �ltering does not a�ect all frequencies equally but that it leaves

untouched the value K = 0 and exponentially increases for higher values of K. This means

that even the slightest noise contribution is greatly ampli�ed. As an example, consider we

have an image at zd = 1cm as in Fig. 5.9(a), which we wish to back-propagate to z = 0. A

noise contribution of 1% present at a low frequency such as K = 4cm�1 [see Fig. 5.9(a)], is

ampli�ed into a contribution of � 55%! Even so, we shall demonstrate that the most relevant

information of U (sc)(R; z = 0) is contained at even lower frequencies, were the inverse �ltering

does not cause such a tremendous e�ect, and we shall see that a great deal of information can

still be recovered by back-propagating the �eld. In summary, we have the same limitations as

those presented in Sec. 5.4, but with the addition of the exp[jKjzd] factor, which implies that

at high values of zd only values very near to K = 0 may be back-propagated.

On back-propagating, we shall use a Hanning �lter, Eq. (5.20) so as to avoid ampli�cation

of noise at high frequencies. In Fig. 5.12 we show the back-propagated amplitudes from the

detection planes z = 1cm and z = 1:5cm, onto the plane z = 0:2cm. This back-propagation

is performed for all cases shown in Fig. 5.10, i.e., noise-free and �ltered images. As quoted

above, the noise parameters for data shown in Figs. 5.12(c) and 5.12(d) are �� = 10Æ, � = 10%,

and �� = 10Æ, � = 30%, respectively. As shown in Fig. 5.12, and as already mentioned

before, since the di�raction patterns from the scattered waves at z = 1cm and z = 1:5cm

do not present appreciable interference fringes, the back-propagation in these cases basically

constitutes an increase in contrast. Once again, we can see in Fig. 5.12 that the back-propagated

�ltered images corresponding to � = 30% are approximately the same as those corresponding

to � = 10%. Also, since the image directly taken at z = 0:2cm does not have a high frequency

contribution, the back-propagated image is a very good approximation to this image [see Figs.

5.10(a) and 5.10(b)]. The reason for this is that the frequency cut for the �lter does not have

to be very high in order to retrieve information from the reconstructed wave. The asymmetry

of the back-propagated noisy images with respect to x = 0 is due to residual noise in the

�ltered images at the detection z plane [cf. Fig. 5.10(c) and 5.10(d)]. We wish to emphasize

that we have not undertaken any additional processing of these data. In practice, however,

an averaging over several image recordings must be made in order to �lter out this e�ect,

together with a standard apodization procedure on these images, prior to the back-propagation

operation. This considerably improves the results shown in Figs. 5.12(c) and 5.12(d). These

apodization procedures will be employed with experimental data, shown in Sec. 10.1.1.


Figure 5.12: Normalized scattered amplitude back-propagated onto z = 0:2cm with con�gurationand the parameters used in Fig. 5.9 for: (a) ~U (sc)(K; 1:0cm! 0:2cm) with Kcut = 8cm�1[solidline]; ~U (sc)(K; zd = 0:2cm) [dotted line]. (b) ~U (sc)(K; 1:5cm! 0:2cm) with Kcut = 8cm�1 [solidline]; ~U (sc)(K; zd = 0:2cm) [dotted line]. (c) ~Unoise(K; 1:0cm! 0:2cm) with Kcut = 7cm�1, andnoise parameters �� = 10Æ, � = 10% [solid line]; ~Unoise(K; 1:0cm! 0:2cm) with Kcut = 7cm�1

, and noise parameters �� = 10Æ, � = 30% [dotted line]. (d) ~Unoise(K; 1:5cm ! 0:2cm) withKcut = 6cm�1, and noise parameters �� = 10Æ, � = 10% [solid line]; ~Unoise(K; 1:5cm! 0:2cm)with Kcut = 5cm�1, and noise parameters �� = 10Æ, � = 30% [dotted line]. In all casesn = 1:333.

Chapter 6

Index matched di�usive/di�usive

interfaces

So far, we have studied the propagation of di�use density wave (DPDWs) in in�nite media, and

have considered di�raction by apertures. However, other basic properties have not yet been

established. These pertain to the re�ection and refraction coe�cients of DPDW plane wave

components at the interface between two di�usive media[100], or at a di�usive/non-di�usive

interface[101], in analogy with the Fresnel coe�cients for light. Like in optical characterization

of materials, the use of these coe�cients will be of great importance for directly determining

the optical parameters of turbid media. In addition, a detailed knowledge of the re�ectivity of

interfaces between di�usive media is the key to understanding the multiple scattering strength

[102] of objects hidden in these media and hence to establish e�ective inverse scattering tech-

niques to characterize them. So far, there is a very scarce bibliography [10, 100] on these

recently establishd coe�cients [100] and very little progress and experimental veri�cation of

these coe�cients exists, since only the equivalent to Snell's law has been presented for DPDWs

[10]. More insight on the experimental veri�cation of these coe�cients will be presented in Ch.

10.

In this Chapter, we establish these re�ection and transmission coe�cients for the case of

index matched interfaces. We shall discuss special consequences such as the appearance of a

zero re�ectivity oscillating frequency, which bears some formal analogy with Brewster modes

for TM-polarized electromagnetic waves [11], and the appearance of frequency independent

re�ection and transmission coe�cients. By means of these coe�cients, we shall study the

e�ect that the zero re�ectivity frequency has on the re�ected wave from a di�usive/di�usive

interface. Finally, a detailed study of the detection of an object buried in rough di�usive media

is presented.

93

CHAPTER 6. INDEX MATCHED DIFFUSIVE/DIFFUSIVE INTERFACES 94

Figure 6.1: Two semi-in�nite di�usive media separated by a plane interface at z = 0, whereU (i), U (r) and U (t) represent the incident, re�ected and transmitted DPDW, respectively.

6.1 Re�ection and Transmission coe�cients

Let two semi-in�nite media be separated by a plane interface at z = 0. A point source is placed

in the upper medium, of certain di�usion parameters �a0, D0, and n0, which characterize the

complex wavenumber �0 =q��a0=D0 + i!n0=cD0. The lower medium, has di�usion param-

eters �a1 , D1, and n1, which characterize the complex wavenumber �1. In this Chapter we

will assume n0 = n1, and thus we will consider no index mismatch e�ects. The point source

is located at a distance zd from the z = 0 plane, and will be modulated at frequency !, thus

generating an incident DPDW represented by U (i). At any plane z = constant, with z > 0, we

can express this incident wave U (i)(r) by its angular spectrum representation of plane waves

(see Sec. 3.2):

U (i)(R; z) =Z +1

�1A(i)(K) exp[iK �R+ iq(K)jzd � zj]dK ;

where R = (x; y), and we have taken the origin at z = 0. The decomposition of wave-vector

k = (K; q(K)) is shown in Fig. 6.2.

We shall now study, in the same manner as in classical optics (see [11], for example), how

one of these spectral components interacts with a plane interface, i.e. how each incident plane

wave A(i)(K) undergoes refraction and re�ection at an interface.

By studying one spectral component at a time, we shall then de�ne, on interaction with a

plane interface (see Fig. 6.1):


Y

Z

X

kq(K)

K

Figure 6.2: Decomposition of wave-vector k onto the XY plane [K], and the Z � axis [q(K)].

� Incident Wave:

~U (i)(K; z) = A(i)0 (K) exp[iqi(K)jzj] exp[iKi �R] ; z � 0 ; (6.1)

� Re�ected Wave:

~U (r)(K; z) = A(r)(K) exp[iqr(K)jzj] exp[iKr �R] ; z � 0 ; (6.2)

� Transmitted Wave:

~U (t)(K; z) = A(t)(K) exp[iqt(K)jzj] exp[iKt �R] ; z � 0 ; (6.3)

where we have used the superscripts i; r; t to design the incident, re�ected, and transmitted

wave, respectively, and:

A(i)0 (K) = A(i)(K) exp[iqi(K)zd] ;

A(i), A(i)0 being the angular spectrum of the source at its origin (i.e. z = zd), and at z = 0,

respectively. Therefore, the total wave in the upper and lower media are de�ned respectively

as:

~U0(K; z) = ~U (i)(K; z) + ~U (r)(K; z) ; z � 0 ;

~U1(K; z) = ~U (t)(K; z) ; z � 0 :

Now we will impose the saltus conditions at the boundary, for index matched media (see Eqs.

(4.9)-(4.8), in page 53):

~U (i)(K; z)��z=0

+ ~U (r)(K; z)��z=0

= ~U (t)(K; z)��z=0

; (6.4)


D0

@ ~U (i)(K; z)

@z

��z=0

+@ ~U (r)(K; z)

@z

��z=0

!= D1

@ ~U (t)(K; z)

@z

��z=0

: (6.5)

A clear analogy of Eqs. (6.4) and (6.5) with the continuity conditions for the magnetic compo-

nent of electromagnetic TM-waves between two media of permitivities �0 and �1, can be found

when n0 = n1, if we substitute D0;1 by 1=�0;1.

From Eqs. (6.4) and (6.5) and expressions (6.1)-(6.3) we obtain the following set of equa-

tions:

A(i)0 (K) +A(r)(K) = A(t)(K) ;

D0

�qi(K)A(i)

0 (K) + qr(K)A(r)(K)�= D1qt(K)A(t) ;

Ki �R = Kr �R = Kt �R :

After some basic algebra, this set of equations reduces to:

A(r)(K) =D0q0(K)�D1q1(K)

D0q0(K) +D1q1(K)A(i)

0 (K) ; (6.6)

A(t)(K) =2D0q0(K)

D0q0(K) +D1q1(K)A(i)

0 (K) ; (6.7)

jKij = jKrj = jKtj ; (6.8)

where we have used the fact that qr = �qi since they have opposite propagation directions (see

Fig. 6.1), and we have made the change of notation qr = �qi = q0, and qt = q1. Eq. (6.8)

is the general expression of Snell's law, since we can express jKij as �0 sin[�i] (see Sec. 3.2),

taking into consideration that, since �0 is complex, we obtain that �i is a complex angle, in

order to verify q20 +K2 = �20. The complex re�ection and transmission coe�cients (R and T ,respectively), in the case of index matched interfaces are therefore expressed as:

R(K) =D0

q�20 � jKj2 �D1

q�21 � jKj2

D0

q�20 � jKj2 +D1

q�21 � jKj2

; (6.9)

T (K) =2D0

q�20 � jKj2

D0

q�20 � jKj2 +D1

q�21 � jKj2

; (6.10)

where we have introduced the expression for q0;1 =q�20;1 � jKj2. Snell's law reduces to:

�1 sin �t = �0 sin �i : (6.11)

If in Eq. (6.11) one assumes that both media have the same refractive index n and the same


absorption coe�cient �a, then we obtain the result put forward in Ref. [10]:

j sin �tj =pD1pD0

j sin �ij : (6.12)

The conservation of total �ux density, Eq.(6.5), can be rewritten in terms of the incident,

re�ected and transmitted �ux density as:

J(i) � n + J(r) � n = J(t) � n ;

which, after substituting for the J's, according to Eqs. (6.9) and (6.10), and considering that

J(i) and J(r) point in opposite directions, yields the relationship:

R(K) +D1q1(K)

D0q0(K)T (K) = 1 ; (6.13)

which is general for DPDWs. R can therefore be seen as the re�ectivity �ux, and D1q1=D0q0Tas the transmittivity �ux. Once again, both coe�cients are complex, and the sum of their

moduli is not unity (as in the electromagnetic case). In the index matched case, we also obtain

from Eq. (6.4) the relationship:

T (K) = R(K) + 1 ;

which, as we shall see in Ch. 7, is not true for index mismatched interfaces.

6.1.1 Wave Scattered from a Plane Interface

The re�ected wave, in the angular spectrum representation is:

U (r)(R; z) =Z +1

�1A(r)(K) exp[iq0(K)z] exp[iK �R]dK :

If we use the relationship implicit in Eq. (6.7):

A(r)(K) = R(K) ~U (i)(K; z)��z=0

;

we can represent the re�ected wave as:

U (r)(R; z) =Z +1

�1R(K) ~U (i)(K; z)

��z=0

exp[iq0(K)z] exp[iK �R]dK : (6.14)

The same can be done for the transmitted wave, by using:

A(t)(K) = T (K) ~U (i)(K; z)��z=0

;


and therefore1:

U (t)(R; z) =Z +1

�1T (K) ~U (i)(K; z)

��z=0

exp[�iq1(K)z] exp[iK �R]dK : (6.15)

6.1.2 Zero re�ectivity frequency

Figure 6.3: Values of the zero re�ection frequency normalized to j�0j =q�a0=D0, in the ! = 0

case, for di�erent D0=D1 ratios versus �a0=�a1.

From Eqs. (6.9) and (6.10), we see that, since �0 is complex, there exists no value of jKj thatmakes q0(K) zero, and therefore, total internal re�ection is not possible for DPDWs. However,

zero re�ection can be achieved at certain ratios D1=D0 of the di�usion coe�cients. Equating to

zero the numerator of Eq.(6.9) when n0 = n1 and ! = 0, one obtains the corresponding value

Kzero:

Kzero =

sD1�a1 �D0�a0

D20 �D2

1

: (6.16)

The values of Kzero versus di�erent media parameters are shown in Fig. 6.3. As will be

shown, the values of Kzero which cause a great modi�cation of the re�ected wave are those

1As we did in Sec. 5.5, if we wish to back-propagate the transmitted wave up to the source, once the interfaceis reached we must change the q(K) component, i.e. back-propagate with q1(K) up to z = 0, divide by T , andthen back-propagate with q0(K) up to z = zd.


Kzero � 2cm�1. Since Kzero is real, the condition that the di�usive parameters must hold is:

D1�a1 �D0�a0D2

0 �D21

> 0 :

The condition (6.16) for zero re�ectivity is qualitatively analogous to that of existence of Brew-

ster modes [11], as manifested on performing a change �a0;1=D0;1 ! ��20;1 and D0;1 ! 1=�0;1.

Figure 6.4: Values of the re�ection coe�cient R versus the frequency jKj near the zero re�ectionfrequency, with ! = 0, D1 = 4=5D0 = 0:03333cm and �a1 = 0:025cm�1 (see Fig. 6.3), for thefollowing cases: �a0 = �a1 = 0:0250cm�1 [solid line]; �a0 = 0:0200cm�1 , Kzero = 0:0 [dashedline]; �a0 = 0:0175cm�1, Kzero = 0:408cm�1 [?]; �a0 = 0:0150cm�1, Kzero = 0:577cm�1 [�];�a0 = 0:0125cm�1, Kzero = 0:707cm�1 [Æ]; �a0 = 0:0100cm�1, Kzero = 0:816cm�1 [4]. In allcases n0 = n1.

In Fig. 6.4 we show the re�ectivities, for the cases near the zero re�ection frequency. As

seen, they are observed to be low. This is a general characteristic of DPDWs at interfaces

between di�erent di�usive biological media. This, together with the existence of high damping,

underlines the low contribution of multiple scattering from hidden objects upon DPDWs [52,

102] so far observed. As seen in Fig. 6.4, for values of K < Kzero, we have negative values

of R. This implies that the re�ected wave has a �-phase change with respect to the incident

wave, which therefore interferes destructively with the incident wave, thus concluding that

the lower medium is absorbing. Even so, as seen in Fig. 6.4, for values of K > Kzero, we

obtain that the lower medium is re�ecting for high frequency values. However, as shown by

the case �a0 = �a1 (solid line in Fig. 6.4), this only occurs in the cases in which Kzero exists,

and in the other cases R does not contain a change of sign. This must always be taken into

consideration, speci�cally in the cases in which we have an incident wave with a large frequency

range. Since the angular spectrum of DPDWs is concentrated at low jKj due to high damping

[90], the Kzero e�ect is strongest when D1�a1 = D0�a0, which corresponds to Kzero = 0;

then R(K) (see Fig. 6.4) constitutes a high pass �lter. Hence, by changing the value of �a0


for example, one can "tune" the K-components to be �ltered. Note that this presents two

important di�erences with electromagnetic TM-waves: �rst, the values of R(K) below the

Brewster K-vector are signi�cantly lower than those at larger values of K, and second, the

values of Kzero for electromagnetic waves have a lower limit at 2��0=p2, while that of DPDWs

is Kzero = 0, except when the magnetic permeabilities �(m) are greater than unity. In this case

a "tuning" of the Brewster angle can be achieved with electromagnetic TM-waves, and values

of Kzero < (2��0=�(m))=

p2, or even Kzero = 0, can be reached by varying the quotient �=�(m)

[103, 104]. As can be seen from Eq. (6.9), Kzero does not exist for ! 6= 0. Even so, a minimum

Figure 6.5: Amplitude and phase of R in the case D0 = 0:041666cm, D1 = 0:03333cm, �a0 =0:015cm�1, �a1 = 0:025cm�1, n0 = n1 = 1:333, (see Fig. 6.4) for the cases: ! = 0, i.e. DC(solid line); f = !=2� = 50MHz (dashed line); f = !=2� = 100MHz (?); f = !=2� =200MHz ( �); f = !=2� = 500MHz (Æ).

of the re�ection coe�cient amplitude is observed at the Kzero position, as shown in Fig. 6.5.

As seen in Fig. 6.5, this minimum gradually disappears and shifts towards higher values of K

as we increase the modulation frequency !.

Figure 6.6: Same as Fig. 6.4 but for the transmission coe�cient T .

In Fig. 6.6, we plot the values of the transmission coe�cient, which are very near 1. This


is expected since for the index matched case T = R+ 1, and, as shown in Fig. 6.4, R has low

values. In this �gure it can be clearly seen that what accounts for the re�ection and transmission

coe�cients is not the actual value of the parameters, but their relative values. In Figs. 6.4

and 6.6 we see that as the medium which contains the incident wave becomes less absorbing,

the interface has a lower transmission value, and a more negative re�ection coe�cient. This

implies that the amplitude of the re�ected (scattered) wave grows, but that the total wave

(U (i) + U (r)) in the upper medium diminishes. In general, we therefore state that the lower

medium is absorbing, since the total wave in the upper medium diminishes, and so does the

transmitted wave, the key fact being the change in the �a0=�a1 ratio.

The e�ect that thisK-�ltering has on the re�ection of a spherical incident wave (i.e., a point

source in three-dimensions) is shown in Fig. 6.7, where the re�ected amplitude is plotted versus

the detector xy-position, at a certain plane z = z0. As seen, near the condition in which Kzero

is real, the re�ected wavefront is shaped in such a manner that its shape resembles re�ection

from a complex geometry rather than from a plane. Also, we have observed that for values of

Kzero � 0:8cm�1 or higher, the K-�ltering has little e�ect on the re�ected wave. To further see

Figure 6.7: Re�ected amplitude [see Eq. (6.14)] at z0 = 1cm, with ! = 0, n0 = n1, D0 =0:041666cm, D1 = 0:03333cm, �a1 = 0:025cm�1, for the cases: (a) �a0 = 0:0150cm�1, Kzero =0:577cm1; (b) �a0 = 0:0175cm�1, Kzero = 0:408cm�1.

the e�ect that Kzero has even more clearly, we plot a section of Fig. 6.7 at y = 0, for di�erent

values of �a0.

6.1.3 Frequency independent coe�cients

Another interesting result in the index matched case is derived when ! = 0, and the relationship

between the di�usive parameters is such that:

D0

D1=

�a0�a1

= ;


Figure 6.8: Re�ected amplitude [see Eq. (6.14)] at z0 = 1cm, with ! = 0, n0 = n1, D0 =0:041666cm, D1 = 0:03333cm, �a1 = 0:025cm�1, for the cases: �a0 = �a1 = 0:0250cm�1 [solidline]; �a0 = 0:0200cm�1 , Kzero = 0:0 [dashed line]; �a0 = 0:0175cm�1, Kzero = 0:408cm�1

[?]; �a0 = 0:0150cm�1, Kzero = 0:577cm�1 [�]; �a0 = 0:0125cm�1, Kzero = 0:707cm�1 [Æ];�a0 = 0:0100cm�1, Kzero = 0:816cm�1 [4]. In all cases n0 = n1.

with real. Then �0 = �1, both media being optically di�erent. In this case, the expressions

for the re�ection and transmission coe�cients become:

R(K) = R = � 1

+ 1; (6.17)

T (K) = T =2

+ 1; (6.18)

that is, we obtain frequency independent re�ection and transmission coe�cients. The corre-

sponding expressions for the re�ectivity and transmittivity will then be ( � 1)=( + 1) and

2=( + 1), respectively. This means that all frequency components incident on the interface

are equally re�ected and transmitted, independently of their propagation direction, and do not

su�er refraction, (since �0 = �1, implies �t = �i and �r = ��i). The interface then re�ects like

a di�usive mirror that does not re�ect 100%, namely as if the lower medium were a partially

absorbing perfect conductor (R = constant < 1). Also, if is replaced by 1= , we obtain a

180o phase change in the re�ected wave, the amplitude remaining unchanged. The transmitted

wave would then travel inside the lower medium as if there were no interface, but with a lower

amplitude reduced by the factor T , since it would then have the same phase as the incident

wave. This frequency independent coe�cient e�ect has been studied for electromagnetic waves,

and have a great variety of applications [103, 104, 105]. A direct analogy with Eqs. (6.17) and

(6.18) for electromagnetic waves when �0=�(m)0 = �1=�

(m)1 , can be found in Ref. [105].


6.2 Detection of buried objects

Figure 6.9: Con�guration for a di�usive object embedded in a rough di�usive medium, withinan otherwise in�nite homogeneous di�usive medium.

In order to study the propagation of di�usive waves in inhomogeneous media, and how do

the re�ection and transmission coe�cients put forward in Sec. 6.1 in�uence the incident wave

in complex geometries, we have taken into consideration a two dimensional con�guration as

shown in Fig. 6.9. It consists of a random rough cylinder with axis along OY , with a boundary

with random pro�le �(�) Gaussian distributed:

�(�) = [R +D(�)] � ur ;

with mean R = h�(�)i, where h�i denotes the ensemble average, and ur is the radial unit vector,

and Gaussian covariance:

< �(�)�(� + �) >= �2 exp[��2=T 2� ] ;

T� being the correlation length in radians, and � denoting the root mean square height (r.m.s.):

� =qh�2(�)i �R2 :

We shall de�ne T and � in cm units, i.e. T = T�=R. This outer rough cylinder separates an

otherwise in�nite di�usive medium Vout[r > �(�)], characterized by a linear, spatially uniform

and isotropic di�usion coe�cient Dout, and absorption coe�cient �aout, from a closed di�usive


medium Vin[r < �(�)] with a linear, spatially uniform and isotropic di�usion coe�cient Din,

and absorption coe�cient �ain . A cylinder of radius Robj, di�usion coe�cient Dobj, absorption

coe�cient �aobj , and axis along OY is immersed in Vin. A point source is placed in the outside

medium r > �(�) at point rs, its intensity being sinusoidally modulated at frequency !. With

this con�guration, we shall use directly the equations presented in Sec. 4.3, for multiple volumes

of scattering.

The set of equations (4.55)-(4.60) provide the solution to the scattering of a di�usive wave

at any point in space, which we shall write as: U(r) = U (i)(r) + U (sc)(r), where U (i) and U (sc)

are the incident and scattered DPDWs, respectively. We shall study separately the relative

amplitude jU(r)j=jU (i)(r)j and the total phase �(r):

U(r) = jU(r)j exp[i�(r)] :

In order to obtain the numerical solution to this set of equations we have employed the numerical

procedure fully described in Sec. 4.4 and Refs. [52, 74, 91].

6.2.1 Numerical Results

Figure 6.10: Di�erent geometries corresponding to two values of the correlation length T : T =0:5cm (�rst column) and T = 1:0cm (second column), for three di�erent r.m.s. heights �:� = 0:05cm (�rst row), � = 0:1cm (second row) and � = 0:2cm (third row). In all cases themean radius R is 5cm.


We have considered a source with a modulation frequency ! = 200MHz. The parameters

chosen for the three media Vout, Vin and Vobj, are: Medium outside, typical intralipid values,

�aout = 0:02cm�1 and Dout = 0:0333cm. Medium inside, brain tissue, as de�ned in [4] i.e.

�ain = 0:035cm�1 and Din = 0:019004cm. Finally, we have used the parameters of a brain

hematoma [98], for the cylinder, �aobj = 1:0cm�1 and Dobj = 0:019608cm. In all cases the

speed of light in the media was v = 2:25 � 1010cm=s, i.e. nout = nin = nobj = 1:333. We

shall �rst study the e�ect that a rough interface has on the relative amplitude and total phase,

without the presence of the object, by generating di�erent rough pro�les with two correlation

distances, T = 0:5cm and T = 1:0cm and several r.m.s. heights: � = 0:05cm; 0:1cm; and

0:2cm. Afterwards, we shall perform a study with a rough surface with statistics T = 0:5cm

and � = 0:2cm, in the presence of an object. We shall consider two object sizes, namely

Robj = 1cm and Robj = 0:2cm and place them at di�erent positions in order to see how the

relative amplitude and total phase are a�ected. When studying small objects, we must always

take into consideration that, for the di�usion approximation to remain valid, light must travel

at least a few mean free paths. As stated in Ref. [106], on comparing with a Monte Carlo

result, sizes smaller than, or equal to, the mean free path start causing deviations. In our case,

this would occur when Robj � 1=�0s, Robj being the radius. We have kept within this limit, and

our objects are of sizes at least of the order of twice the mean free path. To reach numerical

convergence, we have used a discretization dS = 0:025cm for a rough cylinder of mean radius

R = 5cm, consisting of 1250 points. 250 points have been employed to de�ne an R = 1cm

object, and 50 points for an R = 0:2cm object, both with dS = 0:025cm. The surface roughness

did not noticeably change the discretization distance between points of the rough cylinder. The

numerical calculations for the Robj = 1cm object took 53 minutes on a PC at 200MHz, 128Mb

RAM, this time being reduced to 35 minutes for the Robj = 0:2cm object.

6.2.2 Rough Interface

In order to study the e�ect that roughness has on the relative phase and amplitude, we have

simulated the geometries shown in Fig. 6.10, that correspond to the following statistics: T =

0:5cm and T = 1:0cm (in degrees, this would be T = 5:73o and T = 11:46o:) for three r.m.s

heights: � = 0:05cm; 0:1cm and 0:2cm. For future reference we show in Fig. 6.11 the relative

amplitude (left), and the total phase (right), of a di�usive wave scattered by an empty (i.e. with

no immersed object) smooth cylinder of radius 5cm. The refractive e�ect of the cylinder surface

on the wave, as it di�racts on its surface is clearly noticeable, specially in its phase. The relative

amplitude jU j=jU (i)j is plotted in Fig. 6.12 in four of the six geometries of Fig. 6.10, where

the lines denote constant relative amplitude contours. We do not plot the case of � = 0:05cm

because it presents no observable deviations from the � = 0:1cm case. Fig. 6.12 demonstrates

that the pro�les with low � produce small di�erences in the scattered amplitude on comparison

with smooth cylinders (see Fig. 6.11), as expected. As the roughness � increases, we �nd


Figure 6.11: Relative amplitude jU j=jU (i)j (left �gure), and total phase � (right �gure), fora smooth cylinder of radius 5cm. Source position rs = (x = 0; z = 6cm). �aout = 0:02cm�1,Dout = 0:0333cm, �aout = 0:035cm�1, Din = 0:019004cm. The gray scale indicates 180ochanges.

Figure 6.12: Relative amplitude, jU j=jU (i)j, for the following rough cylinder parameters: (a)T = 0:5cm and � = 0:1cm, (b) T = 0:5cm and � = 0:2cm, (c) T = 1:0cm and � = 0:1cm, and(d) T = 1:0cm and � = 0:2cm. Parameters as in Fig. 6.11.

greater perturbations in the scattered amplitude, specially in the transmission region, within

180o � 45o. The reason for this can be seen in Fig. 6.11(right), for example, where we observe

that the wave generated by the source is mostly refracted, and we do not �nd an appreciable

contribution in re�ection. Therefore, interference between the incident and scattered wave

outside the cylinder is not observed, since practically all the incident energy is transmitted,

both in the smooth (Fig. 6.11) and the rough cases (Fig. 6.12). Once inside the cylinder, we

still do not �nd any interference between the backward (re�ected from the inner boundary) and


forward scattered waves. All this is due to the small re�ectivity of the interfaces as put forward

in Sec. 6.1.2. Hence, if one takes into consideration a rough interface, since on the one hand the

wave is greatly attenuated on propagation (see Ch. 3), and on the other hand multiple re�ection

processes are negligible2, wave interaction with an interface results in a distortion due to simple

refraction. This result constitutes an advantage for modeling the scattering from rough di�usive

objects without taking much into account their surface roughness. However, it indicates, on the

other hand, that information about the roughness may be hard to extract on re�ection from

the scattering of di�usive waves. In summary, we �nd that, in transmission which is where

objects are more easily detected, dealing with high values of �, the relative amplitude of the

wave scattered from the rough cylinder can present great di�erences when compared to that

from a smooth cylinder, in some cases even being 20 times greater in the forward scattering

direction (180o). It is also interesting to note from Fig. 6.12, that diminishing the correlation

length T has a small e�ect on the relative amplitude. This stems from the fact that we are

dealing with wavelengths about �0 � 7cm, and, since multiple scattering processes are almost

negligible on the surface, increasing the corrugation slope by a factor of 2, does no further

perturb the DPDW.

6.2.3 Rough Interface in the Presence of an Object

We now study the e�ect that an embedded object has on the propagation of the di�usive wave

when it is placed at di�erent positions inside a di�usive medium with either a smooth or rough

boundary. To do so, we shall address two object sizes, namely Robj = 1cm and Robj = 0:2cm

and we will place it inside either a smooth or rough cylinder. The roughness parameters being

T = 0:5cm and � = 0:2cm [see Fig. 6.10]. We shall consider the object in one of the four

basic positions: near the top, and hence near the source, in the center: (x; y) = (0; 0), at the

bottom, and at the side of the cylinder center, respectively. Except when the object is placed

at (0; 0), the center of the object will always be at a radial distance of 3cm from this origin of

coordinates. We shall �rst study the e�ect of the object immersed in a smooth cylinder, and

then consider the case in which the cylinder is rough.

In Fig. 6.13 we plot the total phase of the wave scattered from a smooth cylinder containing

an object of radius 1cm or 0:2cm, located at either (x; z) = (0; 0), (0; 3cm), (0;�3cm), or

(3cm; 0). We see, on comparison with Fig. 6.11, that the wavefront is distorted in transmission

by the presence of the object, but it manifests no signi�cant object contribution in re�ection,

as expected. We also observe that the closer the object is to the source, the greater is the

distortion su�ered by the wave. This is expected, considering that the �eld is highly damped

on propagation, and hence only a small percentage of the incident wave reaches the object

when this is situated far away from the source. It is interesting to see, for example in Fig.

6.13(d), how the wave distorted by a small object looses much information of its presence as

2This will be clearly demonstrated in Ch. 9, in relation to re�ection from di�usive slabs.


Figure 6.13: Total Phase �, for the following object sizes and positions: Radius Robj = 1cm:(a) (x; z) = (0; 0), (c) (x; z) = (0; 3cm), (e) (x; z) = (0;�3cm), and (g) (x; z) = (3cm; 0).Radius Robj = 0:2cm: (b) (x; z) = (0; 0), (d) (x; z) = (0; 3cm), (f) (x; z) = (0;�3cm), and (h)(x; z) = (3cm; 0). In all cases a smooth cylinder of radius R = 5cm was used. rs = (x = 0; z =6cm). Parameters as in Fig. 6.11.

it travels deeper into the medium. We shall see, however, that still the presence of this object

can be detected in several cases. After a few centimeters, we �nd that the distorted wave is

approximately the same as in the case without object, [compare with Fig. 6.11(right)].

In order to see where the presence of the object has a greater contribution to the wavefront,

we represent in Fig. 6.14 the ratio of the amplitudes between the case of an object present and


Figure 6.14: Relative contribution of the object, i.e. jU (object)j=jU (no�object)j for the followingobject sizes and positions: Radius Robj = 1cm: (a) (x; z) = (0; 0), (c) (x; z) = (0; 3cm), (e)(x; z) = (0;�3cm), and (g) (x; z) = (3cm; 0). Radius Robj = 0:2cm: (b) (x; z) = (0; 0), (d)(x; z) = (0; 3cm), (f) (x; z) = (0;�3cm), and (h) (x; z) = (3cm; 0). In all cases a smoothcylinder of radius R = 5cm was used. Parameters as in Fig. 6.11.

the case of object absence, i.e. jU (object)j=jU (no�object)j. These plots show how this ratio varies

in propagation. As before, we see that the nearer the object is to the source, the greater its

contribution is to the scattered wave. We can also observe from these �gures that there are no

interference processes between the waves from the object and from the cylinder, due to the wave

damping and the low re�ectivity of the surfaces. This can be understood from the Sec. 6.2.2:

in the same manner as multiple scattering processes due to the cylinder are negligible, so are

multiple scattering processes between the object and the cylinder. Also, we observe that even

an object of radius 0:2cm yields a noticeable contribution in transmission. The actual value in

percentage of the contribution of the object to the scattered amplitude is quantitatively better

seen in the next �gures.


Figure 6.15: Total Phase �, for a rough cylinder of parameters: T = 0:5cm, � = 0:2cm,R = 5cm, for the following object sizes and positions: Radius Robj = 1cm: (a) (x; z) = (0; 0),(c) (x; z) = (0; 3cm), (e) (x; z) = (0;�3cm), and (g) (x; z) = (3cm; 0). Radius Robj = 0:2cm:(b) (x; z) = (0; 0), (d) (x; z) = (0; 3cm), (f) (x; z) = (0;�3cm), and (h) (x; z) = (3cm; 0).Parameters as in Fig. 6.11.

We now proceed with an embedded object, but now the cylinder boundary is rough. Results

are presented for the total phase in Fig. 6.15. On comparing Figs.6.13 and 6.15, one sees that

both �gures are remarkably similar in the region between the source and the object, however,

they present noted di�erences in the transmission region, (i.e., below the object in these �gures).


This again suggests that there is no signi�cant wave bouncing by multiple scattering between

the object and the rough surface due to the low re�ectivity of these interfaces. We also carried

out the study of the relative amplitude, compared to the case with no object present, like in

Fig. 6.14, i.e. jU (object)j=jU (no�object)j, but for a rough surface cylinder. The results are not

presented here, however, since they exhibit no signi�cant di�erences with respect to the case of

a smooth cylinder. That is, although the rough cylinder presents deviations in transmission on

comparison to the smooth cylinder, since there is no major contribution from multiple scattering

events between the object and the cylinder, in both cases, rough and smooth, the contribution

from the cylinder surface can be subtracted if one can estimate it beforehand. In this way,

the contribution of the object can be isolated with a very small loss of information on it. It is

because of this that the magnitude jU (object)j=jU (no�object)j is so similar both in the smooth and

rough cases (see Fig. 6.14): what is left from this ratio is approximately the wave scattered by

the object, normalized to the incident �eld.

The most appealing conclusion forwarded from these results is therefore that, due to the

highly damped nature of di�use density waves, and for the parameters that these waves involve,

the detectability of an object is not substantially in�uenced by the roughness of the medium

boundary enclosing it. The buried object can also be characterized if one knows a priori the

pro�le of the boundary (rough or smooth) of the medium enclosing it, (this information can be

inferred for instance from measurements without object), then one can isolate the contribution

of the object to the scattered wave.

In order to quantify the contribution of the object at di�erent positions for the case of the

rough cylinder in comparison with the case of the smooth cylinder, we plot in Fig. 6.16, the con-

tribution of the object to the total scattered amplitude (in percentage), i.e jU(sc�obj)j�jU(sc�no�obj)j

jU(sc�obj)j�

100 for di�erent object positions. This �gure represents a scan performed at a constant dis-

tance from either the rough or the smooth cylinder center, equal to the source distance. In

Fig. 6.16 we study how the presence of the object a�ects the scattered wave by placing an

object of radius 1cm and 0:2cm at distances of 0cm, and 3:5cm from the center. As seen in

Fig. 6.16, when the object is close to the surface, the contribution of the object can mount

up to 50%. This, of course, is taking into consideration an object of radius 1cm, at a distance

of 1:5cm from the surface. For the case of an object of radius 0:2cm, as expected, such small

dimensions contribute in such a small manner for most positions, that the object is hard to be

experimentally detected. Even so, this result indicates that such small objects can be actually

detected in re�ection when located at distances from the surface smaller than 1:5cm [see Fig.

6.16(d)].

In Fig. 6.17, we perform the same scan and represent the same magnitude as in Fig. 6.16

but for di�erent object positions, namely (x; y) = (3cm; 0) and (0;�3cm). In this �gure we see

that very small di�erences can be found when comparing the object located at the center [see

Fig. 6.17(a) and (c)], with positions deeper in the cylinder that encloses it [see Figs. 6.17(b)

and (d)]. It is interesting to see in Figs. 6.17(a) and (c), that the maximum contribution of


Figure 6.16: Relative contribution of the object, jU(sc�obj)j�jU(sc�no�obj)jjU(sc�obj)j

� 100, to the scattered

wave in percentage, for an object of radius Robj = 1cm placed at positions: (a) (x; z) = (0; 0),and (b) (x; z) = (0; 3:5cm), and for an object of radius Robj = 0:2cm placed at positions: (c)(x; z) = (0; 0), and (d) (x; z) = (0; 3:5cm). In all cases the scan was performed at a constantradial distance of 6cm, equal to the source position distance, rs = (x = 0; z = 6cm). Performedin the case of a rough cylinder of statistics T = 0:5cm, � = 0:2cm and R = 5cm (solid line)and in the case of a smooth cylinder (dotted line). Parameters as in Fig. 6.11.

the object is not at the nearest detector position (i.e. � = 90o) but about � ' 120o. If we take

into consideration a point particle, and trace a straight line from the source through its center,

we see that the angle at which it cuts the surface of the cylinder is at � ' 120o for the object

at (0; 3cm). This is exactly the position of the maxima for the object of radius 0:2cm [see Fig.

6.17(c)].

To get information on the shape of the scattered wave, we perform a scan as in the previous

�gures, and represent in Fig.6.18 the scattered amplitude [Fig. 6.18(a) and (b)] and relative

phase [Fig. 6.18(c) and (d)] for a rough and a smooth cylinder, respectively. In these �gures we

plot the curves corresponding to: no object present, an object of radius 1cm at (0; 2:5cm) and

at (0; 3:5cm). We only plot the scattered amplitude and relative phase because they are the

quantities that present greater di�erences between the three cases. As seen, when representing

the data in this manner, no noticeable di�erence between the position at (0; 2:5cm) and the

con�guration with no object can be found in the scattered amplitude, however, the object at

(0; 3:5cm) has the e�ect of increasing this scattered amplitude, and is therefore detectable. On

the other hand, when the relative amplitude is evaluated, no noticeable contribution is found


Figure 6.17: Same as Fig. 6.16 but for object positions: (x; z) = (3cm; 0), and (x; z) =(0;�3cm).

in re�ection for any position of the object, and therefore it is not shown. Data corresponding

to the object of radius 0:2cm are not shown because they present no signi�cant departure

from those data without any object. The greater variations in the scattered phase appear at

those positions closest to the object, these di�erences being of the order of 30o, for the smooth

cylinder, and 15o for the rough cylinder. These results are not shown for the sake of brevity,

since we can �nd di�erences up to 45o when evaluating the relative phase in transmission in the

smooth cylinder case [see Fig. 6.18(c)]. Compared with the rough cylinder case [Fig. 6.18(d)],

we �nd that these di�erences are much smaller, and only 15o di�erences can be found.


Figure 6.18: (a) Scattered Amplitude for a rough cylinder with statistics T = 0:5cm, � = 0:2cmand � = 5cm. (b) Scattered Amplitude for a smooth cylinder of radius 5cm. (c) Relative Phase(� ��no�object) for a rough cylinder with statistics T = 0:5cm, � = 0:2cm and � = 5cm. (d)Relative Phase (��no�object) for a smooth cylinder of radius 5cm. In all cases performed forthe following positions of a Robj = 1cm object: No object [dot-dashed line], (x; z) = (0; 2:5cm)[dotted line and (1)], and (x; z) = (0; 3:5cm) [solid line and (2)]. Parameters and source-detector locations as in Fig. 6.16.

Chapter 7

Index mismatched di�usive/di�usive

interfaces

Figure 7.1: Interaction of a di�use photon density wave with an interface S (left), and thebehavior at a lower scale of the speci�c intensity incident at the interface, when n0 6= n1(right).

At di�usive/di�usive interfaces in the di�usion approximation context, one commonly con-

siders the refractive indices equal to each other and constant throughout both media. However,

when there exists refractive index mismatch, boundary conditions drastically change and are no

longer easy to impose. The reason for this can be understood as follows: in light propagation,

the average intensity su�ers both re�ection and refraction as shown in Sec. 6.1.1, but due to

the present index mismatch, each contribution of the speci�c intensity to the total average in-

tensity also su�ers re�ection and refraction (see Fig. 7.1), as put forward in Sec. 2.1.1. So far,

many works (see Refs. [107, 108] for example) have been reported dealing with the problem

of index mismatched boundaries, but these researches have mainly been concerned with the

non-di�usive/di�usive interface (see Ch. 8). We �nd therefore that a rigorous study on the

di�usive/di�usive interface in the case of index mismatch is needed, and although the complete

boundary conditions for this case are known [107, 108], to our knowledge very few results have

115

CHAPTER 7. INDEX MISMATCHED DIFFUSIVE/DIFFUSIVE INTERFACES 116

been published [109], these dealing with simple con�gurations (see for example [110] where a

cylinder is considered).

Following the theory employed for the derivation of the radiative transfer equation (RTE)

put forward in Ch. 2, we shall obtain the boundary conditions for a di�usive/di�usive interface

with index mismatch. By means of these boundary conditions, we present the expressions for

the re�ection and transmission coe�cients as done in Sec. 6.1. Later on, we derive and solve

the scattering integral equations corresponding to these boundary conditions, which we shall

test with Monte Carlo simulations for the case of a planar interface. Since the expression for

the boundary condition is quite cumbersome, we present an approximation to the complete

boundary conditions, and perform a study of its limits of validity. We �nally put forward the

e�ect of index mismatch at a random rough di�usive/di�usive interface, comparing the results

with those obtained with the approximation to the complete boundary condition and with those

obtained without taking into account the index mismatch.

7.1 Boundary Conditions for Index Mismatched media

To arrive to the expressions for the boundary conditions for a di�usive/di�usive interface, we

recall that the average intensity U(r) and total �ux density J(r) are related to the speci�c

intensity I(r; s) by (see Sec. 2.1):

U(r) =Z4�I(r; s)d ;

J(r) =Z4�I(r; s)sd :

It is important to state that when one measures photon densities in experiments, one must

take into consideration that the energy density1 and the average intensity are both related by

a factor that involves the speed of light c. This factor must be accounted for when di�erent

indexes throughout the media are dealt with.

Let an interface be at z = 0, separating an upper semi-in�nite homogeneous di�usive

medium (medium 0 of Fig. 7.1) of parameters D0, �a0, and n0, from a lower semi-in�nite

di�usive medium (medium 1 of Fig. 7.1) with parameters D1, �a1, and n1. In this case, both

media have di�erent refractive indices, i.e. n0 6= n1. Then, as derived in Sec. 2.1, the upward

and downward �uxes passing through a small area da with n as its unit normal are:

J+(r; n) =Z2�[1�R1!0(�i)] I1(r; si)si � ndi ; (7.1)

J�(r; n) =Z2�[1�R0!1(�i)] I0(r; si)si � ndi ; (7.2)

1We remind that the energy density u is proportional to the photon density �, u = h��, where h� is thephoton energy. Also, u = U=c.


where Rj!k represents the power re�ectivity on passing from medium j (with index nj) to

medium k (index nk), given by the corresponding Fresnel re�ection coe�cient (see Sec. 2.1.1).

We remind that the total �ux density Jn that traverses da is J � n = J+(r; n)� J�(r; n).

We shall de�ne the average intensity in the upper and lower media by U0 and U1, respectively,

and the total �ux density by J0 and J1, respectively. We also make use of the condition for the

di�usion approximation, Eq. (2.55) in Sec. 2.3.1:

I(r; s) ' 1

4�U(r) +

3

4�J(r) � s ; (7.3)

where we shall write J(r) = Jn(r)n + Jt(r)t, Jn and Jt being the normal and tangential

components of J, respectively. Introducing Eq. (7.3), into Eqs. (7.1) and (7.2), we obtain:

J+(r; n) =U1(r)

4�

Z2�[1�R1!0(�i)] si � ndi +

3J1n4�

Z2�[1�R1!0(�i)] (n � si) � (si � n)di ;

J�(r; n) =U0(r)

4�

Z2�[1� R0!1(�i)] si � ndi � 3J0n

4�

Z �

0[1�R0!1(�i)] (n � si) � (si � n)di ;

where there is no Jt contribution since (t � si) � (si � n) = 0. Substituting for d = d�d cos �

we obtain:

J+(r; n) =U1(r)

4�

Z �

0[1� R1!0(�i)] cos �i(2�d cos �i)+

3J1n4�

Z �

0[1�R1!0(�i)] cos

2 �i(2�d cos �i) ;

J�(r; n) =U0(r)

4�

Z �

0[1� R0!1(�i)] cos �i(2�d cos �i)�

3J0n4�

Z �

0[1�R0!1(�i)] cos

2 �i(2�d cos �i) ;

which can be rewritten as:

J+(r; n) =U1(r)

2

Z 1

0[1�R1!0(�i)]�id�i +

3J1n2

Z 1

0[1� R1!0(�i)]�

2id�i ; (7.4)

J�(r; n) =U0(r)

2

Z 1

0[1� R0!1(�i)]�id�i � 3J0n

2

Z 1

0[1� R0!1(�i)]�

2id�i : (7.5)

In order to derive Eqs. (7.4) and (7.5) we have introduced the notation �i = cos �i, where �i

is the angle of incidence. Therefore, since the total �ux Jn that traverses the interface is the

same when considered from medium 0 or 1, i.e. Jn = J0n = J1n, by means of Eqs. (7.4) and

(7.5) is:

J � n = J+ � J� =U1

2R1!0

U +J1n2R1!0

J � U0

2R0!1

U +J0n2R0!1

J ; (7.6)


where we have employed the notation:

Rj!kU =

Z 1

0[1� Rj!k(�)]�d� ; (7.7)

Rj!kJ = 3

Z 1

0[1�Rj!k(�)]�

2d� : (7.8)

The coe�cients given by Eqs. (7.7) and (7.8) have been analytically calculated in Ref. [108],

but we shall evaluate them numerically. Taking into consideration that the de�nition Jn is the

total �ux that traverses an interface, J0n and J1n must be the same quantity, i.e., J � n = J0n =

J1n = Jn. Using the fact that:

R0!1U =

n21n20R1!0

U ;

after grouping terms one obtains the boundary conditions for index mismatched di�usive/di�usive

interfaces [108, 109]:

U1(r)jS ��n1n0

�2U0(r)jS = CnJn(r) ; (7.9)

Jn = �D0@U0

@n

��S

= �D1@U1

@n

��S

; (7.10)

where:

Cn =2�R1!0

J �R0!1J

R1!0U

: (7.11)

Figure 7.2: Typical values of Cn [see Eq. (7.11)] versus n1 for the values: n0 = 1:0 [solid line];n0 = 1:333 [dashed line]; n0 = 1:5 [�]; n0 = 2:0 [Æ].

From a look at Eqs. (7.7) and (7.8), when n1 = n0 one getsRj!kU = 1=2, andRj!k

J = 1, thus

obtaining from Eq. (7.9) average intensity continuity (U0 = U1) as employed in the previous

chapters, since Cn(n0 = n1) = 0. Some typical values of Cn are shown in Fig. 7.2, where we


see that in the cases in which n1 < n0 we obtain values of Cn < 2:5. The most relevant case in

Fig. 7.2 corresponds to n0 = 1:333, since it is the most common value in biological media. As

shown in Ref. [111], 1:3 < n < 1:5 is the typical range of refractive indices in biological media,

and therefore the maximum expected value for n1 > n0 is Cn � 5. Notwithstanding, we must

consider that when scattering media, such as resins, are built, the value n of the refractive

index may be n > 1:5. As shown in Ch. 10, these resins are very useful for experimental

measurements as exit interfaces.

7.2 Re�ection and Transmission coe�cients

By means of the angular spectrum representation, we can derive the re�ection and transmission

coe�cients like in the n0 = n1 case shown in Sec. 6.1. In order to do so, we express the values

of U1 and U0 as:

~U0(K; z) = ~U (i)(K; z) + ~U (r)(K; z) ; z � 0 ;

~U1(K; z) = ~U (t)(K; z) ; z � 0 :

which, introduced into the boundary conditions, Eqs. (7.9) and (7.10), gives us:

n21n20

�~U (i)(K; z)

��z=0

+ ~U (r)(K; z)��z=0

��

CnD0

@ ~U (i)(K; z)

@z

��z=0

+@ ~U (r)(K; z)

@z

��z=0

!= ~U (t)(K; z)

��z=0

; (7.12)

D0

@ ~U (i)(K; z)

@z

��z=0

+@ ~U (r)(K; z)

@z

��z=0

!= D1

@ ~U (t)(K; z)

@z

��z=0

: (7.13)

which, introducing the expressions for the angular spectrum (see Eqs. (6.1)-(6.3) in Sec. 6.1):

n21n20

hA(i)

0 (K) +A(r)(K)i� CnD0

h�iq0(K)A(i)

0 (K) + iq0(K)A(r)(K)i= A(t)(K) ;

D0

hq0(K)A(i)

0 (K) + q0(K)A(r)(K)i= D1[q1A(t)(K)] ;

where A(i)0 = A(i) exp[iq0zs], as in Sec. 6.1, jKij = jKrj = jKtj, and the q's have their usual

meaning qj =q�2j �K2 (see Ch. 3). After some basic algebra, this set of equations reduces

to:

A(r)(K) =D0q0(K)n20 [1� iCnD1q1(K)]� n21D1q1(K)

D0q0(K)n20 [1� iCnD1q1(K)] + n21D1q1(K)A(i)

0 (K) ; (7.14)

A(t)(K) =2n21D0q0(K)

D0q0n20 [1� iCnD1q1(K)] + n21D1q1(K)A(i)

0 (K) : (7.15)


Therefore, the re�ection and transmission coe�cients for di�use photon density waves (DPDWs)

in a general manner (including index mismatch) are2:

R(K) =n20D0

q�20 � jKj2

�1� iCnD1

q�21 � jKj2

�� n21D1

q�21 � jKj2

n20D0

q�20 � jKj2

�1� iCnD1

q�21 � jKj2

�+ n21D1

q�21 � jKj2

; (7.16)

T (K) =2n21D0

q�20 � jKj2

n20D0

q�20 � jKj2

�1� iCnD1

q�21 � jKj2

�+ n21D1

q�21 � jKj2

: (7.17)

As seen, Eqs. (7.16) and (7.17) obey the conservation of total �ux, Eq. (6.13):

R(K) +D1q1(K)

D0q0(K)T (K) = 1 :

Figure 7.3: Re�ection coe�cient versus the frequency for D0 = 0:04166cm, D1 = 0:0333cm,�a0 = 0:0175cm�1, �a1 = 0:025cm�1, n0 = 1:333, for the n1 values (compare with Fig. 6.4):n1 = 1:2 [solid line]; n1 = 1:3 [dashed line]; n1 = n0 = 1:333 [�]; n1 = 1:4 [Æ]; n1 = 1:5 [4]. Inall cases ! = 0.

In order to study the e�ect that index mismatch has on R, in Fig. 7.3 we plot the values of

R versus the frequency. The cases displayed correspond to a zero re�ection frequency Kzero =

0:0408cm�1 at n0 = n1. As seen, index mismatch shifts the values of R but practically leaves

untouched its overall shape, and we observe that a Kzero value also exists for index mismatched

cases. This zero re�ection frequency is obtained from R(K) = 0, and can be approximated to3

2Care must be taken when the values of Cn are calculated, since the value of Cn changes if it is consideredon going from medium 1 onto medium 0 [see Eq. (7.11)].

3The exact expression is di�cult to obtain and presents no appreciable di�erences with the approximation.


(see Eq. (6.16) in Sec. 6.1.2):

Kzero 'vuutn41D1�a1 � n40D0�a0

n40D20 � n41D

21

: (7.18)

7.2.1 Characterization of di�usive media

We now illustrate how the characterization of di�usive media by means of Eqs. (7.16) and

(7.17) can be made in experiments. A frequency modulated point source is located in the

upper di�usive medium, of known �a0, D0 and n0, at a distance zs > 0 from the interface

z = 0. The re�ected (scattered) wave U (r)(R; zd) is determined from measurements at a certain

distance z = zd from this interface. By representing U (r)(R; zd) by its angular spectrum (see

Sec. 3.2), we obtain:

A(r)(K) exp[iq0(K)zd] =1

4�2

Z +1

�1U (r)(R; zd) exp[�iK �R]dR ;

and therefore,

R(K) U (i)(K; z)��z=0

exp[iq0(K)zd] = ~U (r)(K; zd) : (7.19)

If the angular spectrum of the incident wave is known at z = 0, then we can directly determine

the value of R from Eq. (7.19):

R(K) =~U (r)(K; zd)

U (i)(K; z)jz=0exp[�iq0(K)zd] : (7.20)

If there is an incident spherical wave, generated by a point source of strength S0 located at

z = zd (see Sec. 3.2.2), the expression for U (i)(K; z)��z=0

at z = 0 is:

U (i)(K; z)��z=0

=S0

4�D0

i

2�q0(K)exp[iq0(K)zs] ; (7.21)

and therefore, the expression for R would be:

R(K) =4�D0

S0

2�q0(K)

i~U (r)(K; zd) exp[�iq0(K)(zs + zd)] : (7.22)

Then, �a1, D1 and n1 are univocally determined by �tting them to the analytic expression, Eq.

(7.16) for R(K).~U (r)(K; zd) has been numerically found at zd = 3cm in a two-dimensional simulation.

Namely, the incident numerically generated �eld was a cylindrical wave in the xz-plane. The

estimated values retrieved for n1, �a1 and D1 were obtained from Eq. (7.20) employing a range

of K-values between 10 and 15 for noiseless simulations and 15 or more for noisy data. These

retrieved values are shown in Table 7.1. Estimations were performed both without noise and


Coe�. Upper Medium Lower Medium No Noise 10% NoiseD 0.041666 0.044444 0.04452 0.046�a 0.02 0.03 0.02993 0.0281n 1.33 1.30 1.3003 1.33D 0.041666 0.033333 0.03328 0.034�a 0.02 0.2 0.1998 0.215n 1.33 2.0 2.008 1.91

Table 7.1: Retrieved values of lower medium using Eq. ( 7.16). f = !=2� = 140MHz, zs =2cm, zd = 3cm.

with additive noise up to 10% in the amplitude data and with 10Æ in the phase, �nding that,

by using apodization smoothing procedures and �tting techniques, the error in the recovery of

each parameter was never over 10%. As seen from this table, the parameters are thus retrieved

with remarkable accuracy. It should also be remarked that the parameters �a, D, and n at both

sides of the interface can be also determined by using T (K) in addition to R(K):

T (K) =~U (t)(K; zd)

U (i)(K; z)jz=0exp[�iq1(K)zd] :

7.3 Integral Equations for index mismatched di�usive me-

dia

Figure 7.4: Two semi-in�nite homogeneous di�usive media separated by a rough surface withpro�le z = S(x). The source is located in medium 0 . The geometry is constant in the OY -axis.

We shall now address arbitrary surfaces. In order to study a generic interface, as regards the

two dimensional con�guration depicted in Fig. 7.4, let us consider a rough interface with pro�le

de�ned by z = S(x), with unit normal n, that separates a di�usive semi-in�nite homogeneous


medium of volume ~V (the upper medium in Fig. 7.4), with constant parameters �a0, D0 and

n0, from a di�usive semi-in�nite homogeneous medium of volume V (the lower medium in Fig.

7.4), characterized by constant parameters �a1, D1 and n1. We shall assume a point source

modulated with frequency !, located at rs, in medium 0. In this manner, the equations for the

di�usive wave function at the upper (de�ned by U0) and lower (de�ned by U1) are obtained

as in Ch. 4, on applying Green's theorem, to U(r) and the corresponding Green's function

G(r; r0):

� r > S(x)

U0(r) = U (i)(r) +1

4�

ZS(x)

(U0(r

0)@G(�0jr� r0j)

@n0�G(�0jr� r0j)@U0(r

0)

@n0

)dS 0 : (7.23)

� r < S(x)

U1(r) = � 1

4�

ZS(x)

(�n1n0

�2 @G(�1jr� r0j)@n0

U0(r0) �"

D0

D1

G(�1jr� r0j) + CnD0@G(�1jr� r0j)

@n0

#@U0(r

0)

@n0

)dS 0 : (7.24)

In Eqs. (7.23) and (7.24) are rigorously solved by means of the numerical method put forward

in Sec. 4.4. The volume integral over the source has been written as U (i):

U (i) =1

4�

Z~VG(�0jr� r0j)S0(r

0)

D0d3r :

7.4 Numerical results for index mismatched media

We shall consider two-dimensional con�gurations, as depicted in Fig. 7.4, namely, constant in

the y-direction, in which the point source is replaced by and in�nite line along the y-axis.

The �rst system addressed is a �at interface, S(x) = 0, for which we have considered a

DC source (! = 0). The parameters chosen for the media 0 (volume ~V ), and 1 (volume V ),

are: Medium 0, breast tissue, �a0 = 0:035cm�1 and �0s0 = 15cm�1; medium 1, breast tumor,

�a1 = 0:24cm�1 and �0s1 = 10cm�1. In all cases the refractive index in medium 0 was n0 = 1:333,

and n1 was varied 1:0 � n1 � 3:0. In order to reach numerical convergence, we have used a

discretization dS = 0:025cm for a two-dimensional �at surface of 20cm in length.

7.4.1 Comparison with Monte Carlo simulations

In order to asses the accuracy of the calculations based on Eqs. (7.23) and (7.24), we compared

with results obtained from Monte Carlo simulations. The description of the Monte Carlo

method for photon di�usion is well known (see for example [112, 113, 114]) and will not be


Figure 7.5: Percentage of photons that cover a distance lt assigned by Beer's law, found by MonteCarlo [histogram] and Eq. (7.25) [solid line]. �a = 0:02cm�1, �s = 10cm�1, W0 = 0:998. 30000photons launched.

described here in detail. A complete description can be found in Refs. [112, 115]4 . The Monte

Carlo process is as follows: A photon is launched at the source position, and the pathlength

travelled lt before it su�ers another scattering event will be assigned by the uniform random

variable � 2 [0; 1]. This expression is obtained from Beer's law:

I(lt) = I(0) exp[�(�a + �s)lt] ; (7.25)

where I(lt)=I(0) can be seen as the probability that a photon travels a distance lt, and therefore:

lt =1

�s + �aln(�) :

The results obtained from Monte Carlo, and Beer's law, Eq. (7.25) are shown in Fig. 7.5.

Every time the photon is scattered, its initial weight w = 1, decreases by a factor w = (W0)k,

where W0 = �s=(�a + �s) is the albedo and k is the number of scattering events. When the

weight is below a certain threshold, the photon extinguishes and another photon is launched

from the source position. At each scattering event, the exit polar angle � is given by the

4Although Monte Carlo simulations presented in this section were obtained from our own computer code, avery sophisticated Monte Carlo code for multi-layered di�usive media can be found in Ref. [116].


Henyey-Greenstein function (see Sec. 2.3.2), whose probability is given by:

cos � =

8><>:12g

�1 + g2 �

�1�g2

1�g+2g�

�2�; g 6= 0 ;

2�� 1 ; g = 0 ;

where g = hcos �i, and � has a new random value. The value of the exit azimuthal angle � is

given by a new value of � as:

� = 2�� :

Figure 7.6: Projection onto the XY plane of the trajectory of a single photon. Medium 0:�s0 = 75cm�1, g = 0:8, �a0 = 0:035cm�1, n1 = 1:333. Medium 1: �s1 = 50cm�1, g = 0:8,�a0 = 0:24cm�1, n1 = 1:0.

On encountering an interface, the angle of incidence is calculated, and the value of the

Fresnel re�ection coe�cient R(�i) is determined. Then, a new value of � is generated, the

photon being re�ected when R(�i) � �, and transmitted if R(�i) < �.

The quantity measured was the average intensity, after �nding the value of the photon

deposition due to absorption, as in Ref. [112]. The anisotropy factor g used in all simulations

was g = 0:8, and the phase function used was given by the Henyey-Greenstein formula. As

shown in Sec. 2.3.3, the di�usion approximation starts to break down at values of g ! 1.

However, the value of g that simulates realistic experiments of photon di�usion in biological

media is g � 0:8. This value was therefore used to test not only the validity of Eqs. (7.23)

and (7.24), but also to test the saltus condition Eq. (7.9). Lower values of g were also tested,

�nding no important di�erences. The medium parameters were those mentioned in the previous

paragraph. In all cases 40000 photons were launched, allowing a maximum of 20000 interactions

per photon. Since the results obtained from the integral equations where for a two dimensional


Figure 7.7: Comparison of the normalized quantity jr � rsjjU(r)j calculated by means of theintegral equations (solid line) and by Monte Carlo simulation (Æ) for the following parameters:Medium 0: Breast parameters, �s0 = 75cm�1, g = 0:8, �a0 = 0:035cm�1, n1 = 1:333. Medium1: Breast tumor parameters, �s1 = 50cm�1, g = 0:8, �a0 = 0:24cm�1, n1 = 1:0. 40000 photonslaunched and 20000 interactions permitted per photon in Monte Carlo. DC (! = 0) source atrs = (0; 1cm). Scan performed in the z-direction at x = 0.

con�guration, we integrate the calculated intensity in Monte Carlo along the y axis, the photon

injection source in Monte Carlo being a line at x = 0; z = 1:0.

In Figs. 7.7 and 7.8 we show the normalized value of jr� rsj � jU(r)j, where jU(r)j is thetotal amplitude of the di�use wave, expressed as the sum of the incident and scattered wave,

i.e. U(r) = U (i)(r) + U (sc)(r), both for the integral equation numerical results and for Monte

Carlo simulations. The detector scan was performed along the z-direction at the x = 0 plane.

In both cases we have chosen a rather extreme value of n1, i.e., n1 = 1:0 (Fig. 7.7) and n1 = 2:0

(Fig. 7.8). As seen from these �gures, we obtain excellent agreement between both methods.

Also, the results were as expected, i.e.: if n1 < n0 there is a higher average intensity in medium

0 directly at the surface boundary due to an increase of the total internal re�ection e�ect. The

opposite thing occurs for n1 > n0 as then the total internal re�ection is greater in medium 1.

This e�ect is due to the discontinuity in U across S given by Eq. (7.7).

In Fig. 7.9 we plot the total amplitude for scans along the x-direction performed at a

distance of z = 0:2cm and z = �0:2cm from the surface. Numerical results and Monte Carlo

simulations are shown for n1 = 1:0 and n1 = 3:0. As in the previous �gures, we obtain excellent

agreement between Monte Carlo simulations and numerical results.


Figure 7.8: Same as Fig. 7.7 but for n1 = 2:0.

7.5 Approximate Boundary Conditions

Since the application of the boundary conditions across interfaces of index mismatched media

in Eqs. (7.9) and (7.10) is in most cases complicated, we have studied an alternative which

would simplify these boundary conditions and aid in the use of analytical methods for index

mismatched interfaces. We propose to approximate the saltus conditions in Eqs. (7.9) and

(7.10) by:

U1(r)jS '�n1n0

�2U0(r)jS ; (7.26)

�D0@U0(r)

@n

��S

= �D1@U1(r)

@n

��S

: (7.27)

This approximation is valid as long as we can consider�n1n0

�2U0 � CnJn [cf. Eqs. (7.9) and

(7.11)]. In order to study the validity of such an approximation we will compare the average

intensities and the total �ux densities obtained from this approximation applied to Eqs. (7.23)

and (7.24), versus results from the complete conditions (7.9) and (7.10). With this aim we shall

study:

� The total upward and downward �ux density J+;�total that passes through the surface S,


Figure 7.9: Comparison of U(r) calculated by means of the integral equations at zdetect = 0:2cm(solid line) and zdetect = �0:2cm (dotted line), and by Monte Carlo simulation at zdetect = 0:2cm(�) and zdetect = �0:2cm (Æ) for two values of n1: (a) n0 = 1:333, n1 = 1:0, (b) n0 = 1:333,n1 = 3:0. Parameters as in Fig. 7.7. Scan performed in the x direction at a constant z distancefrom the surface.

normalized to the total incident average intensity at S:

jJ+;�totalj

jU (inc)total j

=

RS j jJ+;�(r)j dSRS jU (i)(r)j dS : (7.28)

� The total average intensity jU total0;1 j at the surface both at medium 0 and medium 1,

jU total0;1 j =

ZSjU0;1(r)j dS : (7.29)

The quantities (7.28) and (7.29) are calculated as the refractive index of medium 1 varies in

the interval, 1:0 � n1 � 3:0, for di�erent di�usive media (n0 = 1:333 in all cases). Even though

the range of values of the refractive index in biological media is actually much smaller, we have

chosen a larger range of variation so as to perform a more general study of the limits of validity

of Eqs. (7.32) and (7.32). Figs. 7.10 and 7.11 show the results. Fig. 7.10 shows how the

upward and downward �ux density increases up to a maximum value that corresponds to the


Figure 7.10: Values of jJ�totalj=jU inctotalj (solid line) and jJ+

totalj=jU inctotalj (dotted line) calculated

with the complete boundary conditions, compared to jJ�totalj=jU inctotalj (�) and jJ+

totalj=jU inctotalj (Æ)

calculated with the approximate boundary conditions. Media Parameters: (a) D0 = 2D1 =0:0666cm, �0s0 = 4:98cm�1, �0s1 = 10:0cm�1, �a0 = �a1 = 0:02cm�1. ! = 0. (b) D0 =D1=2 = 0:01665cm, �0s0 = 20:0cm�1, �0s1 = 10:0cm�1, �a0 = �a1 = 0:02cm�1. ! = 0. (c)D0 = D1=2 = 0:0333cm, �0s0 = 10:0cm�1, �0s1 = 4:98cm�1, �a0 = �a1 = 0:02cm�1. ! = 0.(d) Breast and breast tumor parameters, �0s0 = 15:0cm�1, �0s1 = 10:0cm�1, �a0 = 0:035cm�1,�a1 = 0:24cm�1. ! = 0. (e) Parameters used in Ref. [10], D0 = 0:0333, D1 = 0:1095,�0s0 = 10:0cm�1, �0s1 = 3:02cm�1, �a0 = �a1 = 0:02cm�1. ! = 200MHz. (f) Breast andbreast tumor parameters, �0s0 = 15:0cm�1, �0s1 = 10:0cm�1, �a0 = 0:035cm�1, �a1 = 0:24cm�1.! = 200MHz. In all cases rs = (0; 1cm), and n0 = 1:333.

index matched case (n1 = 1:333), and then decreases as n1 grows, always in such a way that the

total �ux density (pointing downward) increases, due to an increase in total internal re�ection.

We then see that the approximate boundary conditions yield very good results up to a value of

n1 approximately being n1 = 1:7, and a larger error is found for the approximated downward

�ux density. In Fig. 7.11 we see that, right at the boundary, the average intensity in medium

0 decreases as n1 increases, whereas, the average intensity in medium 1 increases, as expected.

If we compare Figs. 7.11(b) and 7.11(c), we see that both are very similar to each other. The

values of D0 and D1 in Fig. 7.11(a) are di�erent to those in Fig. 7.11(b), but D0=D1 = 0:5

in both. Thus, the expression jU total0;1 j=jU inc

totalj does not depend on the di�usion coe�cients

themselves but on their quotient D0=D1. Also, looking at �gures 7.11(d) and 7.11(f), we see

that the relative value of jU total0;1 j=jU inc

totalj does not change with the modulation frequency. This


Figure 7.11: Values of jU total0 j=jU total

n1=1:333j (solid line) and jU total

1 j=jU totaln1=1:333

j (dotted line)calculated with the complete boundary conditions, compared to U total

0 =U totaln1=1:333 (�) and

U total1 =U total

n1=1:333(Æ) calculated with the approximate boundary conditions. Parameters as in

Fig. 7.10.

is expected, since the values of Rj!kU;J only depend on the refractive indexes and do not depend

on !. Even so, we must take into consideration that the only expression for the boundary

conditions that is not frequency dependent when the quotient jU total0;1 j=jU inc

totalj is considered, isthe approximate boundary condition (7.32). For the average intensity we again see that the

approximate boundary conditions (7.32)-(7.32), yield good results up to a value of n1 = 1:7.

In order to study how the approximate boundary conditions (7.32)-(7.32) deviate from the

complete boundary conditions (7.9) and (7.10), in Fig. 7.11 we plot the value of jU total1 j=jU total

0 jversus n1, i.e:

jU total1 j

jU total0 j =

�n1n0

�2� CnD0

��@U0

@n

��S

=jU total0 j

!: (7.30)

The approximate boundary conditions are not valid when this quantity deviates from (n1=n0)2,

i.e. when the condition�n1n0

�2U0 � CnJn does not hold. As seen from Fig. 7.12, this starts

occurring at values of n1 of the order of n1 ' 1:7.

To gain understanding on the error committed when the approximate boundary conditions

are used, or, equivalently, when the index mismatch is not taken into consideration, in Figs.

7.13 and 7.14 we plot the error committed when the values of U total0 and U total

1 are estimated


Figure 7.12: Values of jU total1 j=jU total

0 j calculated with the complete boundary conditions forthe media parameters: D0 = 2D1 = 0:0666cm, �0s0 = 4:98cm�1, �0s1 = 10:0cm�1, �a0 =�a1 = 0:02cm�1, ! = 0 (Æ); D0 = D1=2 = 0:01665cm, �0s0 = 20:0cm�1, �0s1 = 10:0cm�1,�a0 = �a1 = 0:02cm�1, ! = 0 (u); D0 = D1=2 = 0:0333cm, �0s0 = 10:0cm�1, �0s1 = 4:98cm�1,�a0 = �a1 = 0:02cm�1, ! = 0, (?); breast and breast tumor parameters, �0s0 = 15:0cm�1,�0s1 = 10:0cm�1, �a0 = 0:035cm�1, �a1 = 0:24cm�1, ! = 0 (solid line); parameters used in Ref.[10], D0 = 0:0333, D1 = 0:1095, �0s0 = 10:0cm�1, �0s1 = 3:02cm�1, �a0 = �a1 = 0:02cm�1,! = 200MHz (dotted line). breast and breast tumor parameters, �0s0 = 15:0cm�1, �0s1 =10:0cm�1, �a0 = 0:035cm�1, �a1 = 0:24cm�1, ! = 200MHz (�). In all cases rs = (0; 1cm),and n0 = 1:333.

by means of these approximate boundary conditions. This error is in medium i, (i = 0; 1):

Errori(%) =jU total

i japprox � jU totali jcomplete

jU totali jcomplete

� 100 : (7.31)

If we take the range of refractive indexes: 1:3 < n1 < 1:5 as typical for biological tissues

[111], we observe from Fig. 7.13 that when the approximate boundary conditions are used, the

error committed in the worst of the cases under study is never above 3%. On the other hand, if

we do not take into consideration the refractive index mismatch, this error can be of the order

of 15% (see Fig. 7.14). If we go to more extreme cases of refractive index mismatch, n1 = 2:0

for example, we �nd that not taking the index mismatch into consideration can lead to an error

of even 100%. However, we see that on using the approximate boundary conditions this error

is not higher than 20%.


Figure 7.13: Error committed in % [see expression (7.31)] when using the approximate boundaryconditions to �nd out the values of: (a) jU total

0 j; (b) jU total1 j. Parameters as in Fig. 7.12.

7.5.1 Approximate re�ection and transmission coe�cients

Established their limits of validity, we can proceed in a similar manner as in Sec. 7.2, and

derive the re�ection and transmission coe�cients from the approximate boundary conditions:

~U (i)(K; z)��z=0

+ ~U (t)(K; z)��z=0

'�n1n0

�2~U (t)(K; z)

��z=0

;

D0

@ ~U (i)(K; z)

@z

��z=0

+@ ~U (r)(K; z)

@z

��z=0

!= D1

@ ~U (t)(K; z)

@z

��z=0

;

which therefore result as:

R(K) ' n20D0

q�20 � jKj2 � n21D1

q�21 � jKj2

n20D0

q�20 � jKj2 + n21D1

q�21 � jKj2

; (7.32)

T (K) ' 2n21D0

q�20 � jKj2

n20D0

q�20 � jKj2 + n21D1

q�21 � jKj2

: (7.33)


Figure 7.14: Error committed in % (see expression (7.31)) when considering index matchedboundary conditions to �nd out the values of: (a) jU total

0 j; (b) jU total1 j. Parameters as in Fig.

7.13.

On comparing with the complete expressions (7.16) and (7.17) one sees that from the approxi-

mate expressions, Eqs. (7.32) and (7.33), the parameters are more easily retrieved. From Eq.

(7.32) we obtain the expression for the zero re�ection frequency Kzero presented in Eq. (7.18).

7.5.2 Approximate surface integrals

Introducing the approximate boundary conditions into Eqs. (7.23) and (7.24) we obtain:

� r > S(x)

U0(r) = U (i)(r) +1

4�

ZS(x)

(U0(r

0)@G(�0jr� r0j)

@n�G(�0jr� r0j)@U0(r

0)

@n0

)dS 0 :

� r < S(x)

U1(r) ' � 1

4�

ZS(x)

(�n1n0

�2 @G(�1jr� r0j)@n0

U0(r0)� D0

D1G(�1jr� r0j)@U0(r

0)

@n0

)dS 0 :


By following arguments such as those in Sec. 4.1, the relationship between the surface integrals

when considered from above (�0) and below (�1) S(x) when using the approximate boundary

conditions are:

�1(r) ' �0(r) +ZS(x)

�1� D0

D1

�G(�0jr� r0j)@U0(r

0)

@n0dS 0+Z

S(x)

n20n21� 1

!U0(r

0)@G(�0jr� r0j)

@n0dS 0 ;

and U0 and U1 can be thus written as:

� r > S(x)

U0(r) = U (i)(r) +1

4��0(r) ;

r < S(x)

U1(r) ' � 1

4��0(r)�

ZS(x)

�1� D0

D1

�G(�0jr� r0j)@U0(r

0)

@n0dS 0�Z

S(x)

n20n21� 1

!U0(r

0)@G(�0jr� r0j)

@n0dS 0 :

A particular instance is that in which D0 = D1, but we have n0 6= n1. In this case, we may

write the integral equations in terms of the volume contributions, as in Eqs. (4.24) and (4.25)

in Sec. 4.1:

� r > S(x)

U0(r) ' U (i)(r) +1

4�

ZVG(�0jr� r0j)��2(r0)U1(r

0)d3r0 ;

� 1

4�

ZS

n20n21� 1

!U0(r

0)@G(�0jr� r0j)

@n0dS 0 ; (7.34)

� r < S(x)

U1(r) ' U (i)(r) +1

4�

ZVG(�0jr� r0j)��2(r0)U1(r

0)d3r0 ;

+1

4�

ZS

n21n20� 1

!U1(r

0)@G(�0jr� r0j)

@n0dS 0 : (7.35)

Eqs. (7.34) and (7.35) are very useful for characterizing the refractive index of a di�usive object

by means of an approximation such as that of Born or Rytov.


Figure 7.15: Rough surface pro�les S(x) with Gaussian statistics: T = 0:5cm and � = 0:1cm(solid line); T = 1:0cm and � = 0:2cm (dotted line). Notice the di�erence in scale of the x andy axes.

7.6 Rough Di�usive/Di�usive Interfaces

Let us now consider a randomly rough di�usive/di�usive interface z = S(x) with zero mean,

hS(x)i = 0, homogeneous Gaussian statistics, and Gaussian correlation function hS(x)S(x+ �)i =�2 exp[��2=T 2], of correlation length T and root mean square height � =

qhS(x)2i. We address

the two dimensional con�guration depicted in Fig. 7.4. We shall study two simulations, each

corresponding to a rough surface as shown in Fig. 7.15, with statistics T = 0:5cm, � = 0:1cm,

and T = 1:0cm, � = 0:2cm, respectively. No averaging on surface realizations is done, i.e., only

one particular surface (see Fig. 7.15) is studied at a time. The generation of rough surfaces

with a given statistics as well as the electromagnetic scattering from them can be seen in more

detail in Refs. [74, 91, 93, 117]. Since we do no longer address a plane interface, we must

inquire about the limit of validity of Eqs. (7.7) and (7.8), and in consequence Eq. (7.9), when

numerical calculations are performed, due to the discretization of the rough surface. This limit

is studied in Section 7.6.1. In order to reach numerical convergence, we have used a discretiza-

tion dS for a 20cm surface, dS = 0:025cm, and, as stated in Section 7.6.1, in order to keep

working within the limits of validity of the boundary conditions (7.9) and (7.10), we must deal

with local radii of curvature R higher than R = 0:15cm. Also, one may wonder about the

validity of the di�usion approximation inside the asperities of the rough interface. However, as

stated in Ref. [106], the di�usion approximation starts causing departures from correct results

for media with sizes of the order of the transport-scattering mean free path ltr, where ltr = 1=�0s(see Sec. 2.2). In the cases considered in this work ltr � 1mm. Therefore, with rough surfaces

of parameters T and � as considered above, we always keep within both limits.


Figure 7.16: Normalized Scattered Amplitude jU (sc)(r)j=jU (sc)n1=n0jx=0 at zdetect = 0:5cm found by

using the complete boundary conditions for n1 = n0 = 1:333 (solid line), n0 = 1:333, n1 = 1:0(dotted line), and n0 = 1:333, n1 = 1:5 (dashed line), and by using the approximate boundaryconditions for n0 = 1:333, n1 = 1:0 (�), and n0 = 1:333, n1 = 1:5 (Æ). Scattering geometries:(a) Plane interface. (b) Rough surface with Gaussian statistics T = 0:5cm, � = 0:1cm. (c)Rough surface with Gaussian statistics T = 1:0cm, � = 0:2cm. In all cases rs = (0; 1cm) andf = !=2� = 200MHz.


In Fig. 7.16 we plot the scattered amplitude jU (sc)(r)j, i.e U (sc) = U � U (i), normalized to

the peak value of the scattered amplitude jU (sc)n1 jx=0, in the case in which n0 = n1 = 1:333,

i.e., jU (sc)j=jU (sc)n1=n0

jx=0, and in Fig. 7.17 we show the corresponding scattered phase � (where

U (sc) = jU (sc)j exp[i�]). These quantities are drawn both for a plane interface and for the two

rough surfaces considered above and depicted in Fig. 7.15. The detector scans along the x-

direction at z = 0:5cm with the source located at rs = (0; 1:0cm). The small di�erence between

the shapes of those curves corresponding to the �at interface and those obtained from the rough

surfaces, is due to the z ordinate di�erence from the source point rs to the surface, rather than

to the surface roughness, (see in Fig. 7.15 how the value of S(x) just under the source, i.e. at

x = 0, is di�erent for each rough surface). This is due to the highly absorptive character of

di�use photon density waves, which concentrates the contribution to the scattered wave within

the interface area closest to the source. Nevertheless, due to the similarity of the wave scattered

from the plane and that from rough surfaces we, once again, reach the conclusion put forward

in Ref. [52], and in Ch. 6, namely, that due to absorption of di�use photon density waves,

the multiple scattering contribution produced by the surface roughness is very small for these

waves. As seen from �gures 7.16 and 7.17, even for rough surfaces, the approximate boundary

conditions yield very good results on comparison with those obtained from conditions (7.9) and

(7.10). On the other hand, it is important to state that a refractive index mismatch not only

changes the peak value of the curves, but also their shape.

7.6.1 Discretization of Curved Boundaries

Numerically solving the integral equations (7.23) and (7.24) require to discretize the surface

boundary. In the case of a curved boundary, the limit of validity of the saltus condition

expressed in Eq. (7.9) should be studied, since this expression is originally derived for a locally

plane interface. In order to do so, the polar angle �, de�ning a surface with curvature radius

R, is discretized by elements Æ, as shown in Fig. 7.18(a). This discretization depends on the

limiting value of dS = RÆ needed for numerical convergence. As seen from Fig. 7.18(b), in

order to apply Eqs. (7.1) and (7.2) to �nd the values of the backward and forward �uxes, one

must change the limits of the integrals:

J+(r) =Z 2�

0d�Z �

2��

0I(r; s)s � nd cos � ;

J�(r) =Z 2�

0d�Z �

�2+�

I(r; s)s � nd cos � ;

where � = Æ=2, as shown in Fig. 7.18(b). Therefore, the expression of Rj!kU;J in this case is:

Rj!kU =

Z 1

sin Æ=2[1� Rj!k(�)]�d� ;


Figure 7.17: Same as Fig. 7.16 but for the scattered Phase �(r), of U (sc) = jU (sc)j exp[i�], atzdetect = 0:5cm .

Rj!kJ = 3

Z 1

sin Æ=2[1� Rj!k(�)]�

2d� :

We have done a comparison of these equations with those corresponding to a plane surface. We

have found that for practical values of the refractive index n1 (we used values between 1.0 and


Figure 7.18: (a) Discretized curved surface studied to �nd the minimum value of Æ required forthe expressions of Rj!k

U;J to remain valid when a numerical calculation is performed by samplingof a surface with curvature radius R by elements dS. n is the surface normal. (b) Detail of(a), where � = Æ=2. J�;+ represent the downward and upward �ux density, respectively.

3.0), when Æ � 10o, the error committed by using the expression of Rj!kU;J for a plane interface

is never higher than 0:5%. Considering that in the cases addressed in this work dS = 0:025cm

in order to reach numerical convergence, we obtain that the minimum allowed curvature radius

is R ' 0:15cm. If we wanted to numerically study surfaces with a smaller curvature radius, we

should diminish the value of dS accordingly, and, therefore, we should increment the number

of sampling points at the surface so as to make locally plane each surface element dS. The

number of sampling points for a surface of area S is S=dS. Even so, we must always bear in

mind that the di�usion approximation starts causing deviations from correct values at sizes of

the order of 1=�0s [106] (in this work 1=�0s � 0:1cm).

Chapter 8

Di�usive/non-di�usive interfaces

In Ch. 7, the re�ection and transmission coe�cients have been established for interfaces between

di�usive media [100]. However, in experimental con�gurations, measurements are usually taken

from an outer non-di�usive medium. Therefore, the coe�cients for the DPDW plane wave

components at di�usive/non-di�usive interfaces must be known. Such coe�cients will be of

importance for straightforwardly determining the optical parameters of turbid media, the value

of the so-called extrapolated distance1 [107, 108], and the shape and location of the photon

source. As mentioned in Ch. 6, detailed knowledge of the re�ectivity of the interfaces between

di�usive media, and therefore their multiple scattering contribution, is also the key to put

forward e�ective inverse scattering techniques. In a similar context, much research is motivated

by the ability of optical radiation to diagnose brain tumors. However, a problem arises in

studying complex systems such as the brain, in which not all regions di�use light. To solve

this situation, many works have studied the way to correctly modelling di�usive/non-di�usive

interfaces [34, 45, 46, 107, 108, 118, 119, 120, 121, 122, 123, 124], but these researches have only

considered the scattered light being detected from outside the di�usive medium. Cases in which

there are inner non-di�usive regions, are those in which new boundary conditions, which model

the interaction of the di�use light with non-di�usive media, are needed [125, 126, 127, 128, 129].

The di�usion approximation does not hold near non-di�usive interfaces, and up to date there

exists no closed correct expression for such boundary conditions. Even so, in a non-di�usive

region, embedded in a di�usive medium, (for instance, an air bubble in milk), the di�usion

approximation, in combination with the radiative transfer equation (RTE) (see Ch. 2) in a

non-di�usive medium, have been shown to yield correct results [125, 126, 129, 130]. Namely,

light propagation can still be accurately modelled by the di�usion approximation when light

incident on the non-di�usive region is already di�use. The brain is an example of such a system,

as it is mostly a scatterer except for those regions �lled with cerebro-spinal �uid. The limit

of validity of such an approximation will depend both on the ratio between the di�usive and

non-di�usive volumes, and on the refractive index.

1The extrapolated distance Le appears when dealing with the boundary conditions at di�usive/non-di�usiveinterfaces. Its expression is derived in Sec. 8.1, and is given by Le = �U=rU at the boundary.

140

CHAPTER 8. DIFFUSIVE/NON-DIFFUSIVE INTERFACES 141

Figure 8.1: X-ray images of a human head (left, top view; right, side view) following Ref. [131],where the brain can clearly be distinguished. The clear areas indicate the regions �lled with anon-di�usive substance.

In this Chapter, we derive the boundary conditions at plane di�usive/non-di�usive inter-

faces, and we establish the corresponding re�ection and transmission coe�cients. We shall

discuss their e�ect on the DPDW wavefront, as well as special consequences such as the appear-

ance of a zero re�ectivity oscillating frequency, analogous to that occurring in di�usive/di�usive

interfaces [100] (see Sec. 6.1.2). We derive results from numerical computations. Further, we

put forward a method to characterize the di�usive medium, and to determine the extrapolated

distance Le and the location of the photon source. This is done from transmission measurements

containing up to 10% noise, added to the signal in the non-di�usive medium, and constitutes

a direct, novel and systematic procedure. Then, following previous works in which advances in

this subject were made [125, 126], we present a compact expression of the boundary conditions

for di�usive media with inner non-di�usive regions[130]. In order to obtain the expression for

these boundary conditions, we will make use of the radiative transfer equation (see Ch. 2)

[9]. Further, the scattering integral equations are put forward, which include the interaction of

the incident wavefront of di�use light with a non-di�usive region by means of these boundary

conditions. These equations will be rigorously solved by means of the extinction theorem (ET)

integral method presented in Sec. 4.4 [52, 74, 91], which when applied both above and below

the interface, yields the non-local boundary values for both the average intensity and its normal

derivative. We compare results derived from the ET with those obtained with the �nite element

model (FEM) [70, 71, 72], which takes into account the non-di�usive medium by approximating

the boundary conditions. Both methods are then contrasted with Monte Carlo (MC) simula-

tions, and the advantages of using either procedure are discussed accordingly. Results are put

forward for two distinct situations: the case of a cylindrical non-di�usive region embedded in

a di�usive cylinder, and that of a cylindrical non-di�usive gap embedded in a di�usive cylin-


der. The limit of validity of the boundary conditions established in this work is discussed by

considering the in�uence of the non-di�usive volume versus that of the di�usive volume. Once

the limits of validity of the novel boundary conditions are established, the e�ect of roughness

in the di�usive/non-di�usive boundary is studied. Finally, the scattered wave from a di�usive

and from a non-di�usive object, both separated by a certain distance, is studied.

8.1 Boundary conditions for plane interfaces

Figure 8.2: Interaction of a di�use photon density wave with an interface S (left) between adi�usive and a non-di�usive medium, and what occurs at a lower scale with the speci�c intensityincident at the interface, in the case n0 6= n1 (right). Note the direction of the unit normal n.

Let z = 0 be a plane interface which separates an upper di�usive medium (medium 0) of

parameters D0, �a0, and n0, from a lower non-di�usive medium (medium 1) of refractive index

n1 (see Fig. 8.2). A modulated point source at frequency ! is located at z = zs > 0 in the

di�usive medium. As shown in Sec. 2.1, the average intensity U and the total �ux density Jn

are de�ned as:

U(r) =Z4�I(r; s)d ; (8.1)

Jn(r) = J(r) � n =Z4�I(r; s)s � nd ; (8.2)

and the outward J+ and inward2 J� �uxes traversing the interface are de�ned, respectively as:

J+(r) =Z2�

�1� jR0!1(�i)j2

�I0(r; si)si � ndi ; (8.3)

J�(r) =Z2�

�1� jR1!0(�i)j2

�I1(r; si)si � ndi ; (8.4)

2When dealing with non-di�usive interfaces, we shall denote the �uxes that traverse and interface as outward(i.e. from the di�usive into the non-di�usive medium), and inward �uxes, instead of upward and downward asin Sec. 2.1.


where I0 and I1 represent the speci�c intensities incident on S from the di�usive and non-

di�usive media, respectively, Ri!j(�i) is the re�ection coe�cient, and n is the surface unit

normal, which points into the non-di�usive medium. Since the source is located in the di�usive

medium, we assume that all the light exiting the di�usive medium, will not reenter it. This

is true when no sources are located in the outer non-di�usive medium, the surface boundary

being plane. This, in terms of the �ux is described as:

J�(r) = 0 ;

Jn(r) = J+(r) ; (8.5)

As shown in Sec. 7.1, on introducing the expression for the di�usion approximation (see Sec.

2.3.1):

I(r; s) =U(r)

4�+

3

4�J(r) � s ; (8.6)

into Eqs. (8.3) we obtain:

J+(r) = RUU0(r)

2

��z=0

+RJJn(r)

2r 2 S ; (8.7)

where RU and RJ have the expressions and meaning presented in Sec. 7.1. Eq. (8.7) by means

of Eq. (8.5) reduces to:

U0(r)jz=0 = �Jn(r) ; (8.8)

where,

� =2� RJ

RU; (8.9)

is the coe�cient which takes into account the refractive index mismatch between the di�usive

and non-di�usive medium. It has been denoted by � to distinguish it from Cn (see Sec. 7.1)

which takes into account the index mismatch between two di�usive media. Eq. (8.8) is usually

referred to the zero �ux or partial �ux boundary condition [46, 107, 108], and by means of

Fick's law, J(r) = �DrU can be rewritten as:

U0(r)jz=0 = ��D0@U0(r)

@n

��z=0

; (8.10)

where we have taken the surface unit normal n = uz pointing outward, i.e. into the non-

di�usive medium (when the surface normal is considered pointing inward, � in Eq. (8.10) must

be replaced by ��). Using the fact that D0 = ltr=3, where ltr is the transport mean free path,

Eq. (8.10) in the case of isotropic sources (JE = 0) can be expressed as:

U0(r)jz=0 = ��3ltr

@U0(r)

@n

��z=0

: (8.11)


The quantity ltr�=3 is usually called the extrapolated distance [108]:

Le =�

3ltr ;

which, in the index matched case (n0 = n1), reduces to:

Le =2

3ltr : (8.12)

It must be stated that these values are exact within the di�usion approximation, and therefore

constitute an approximation to the complete physical problem. There are many works dealing

with this boundary condition (see Ref. [108] and references therein), and constantly more

accurate values of Le which can be employed in Eq. (8.11) appear in literature [108, 118, 120].

We can get some idea of how approximate Eq. (8.8) is by comparing it with an exact boundary

condition for a semi-in�nite slab containing isotropic scatterers. This is called theMilne problem

[13]. For the n0 = n1 case, it gives a value Le ' 0:7104ltr [9]. Compared to the value obtained

from the di�usion approximation [see Eq. (8.12)], Le = 0:6666ltr, it gives us an idea of the

approximation taken. These values change further the more absorbing the di�usive medium is,

i.e. as the albedoW0 departs from unity (a complete study on the values of Le for di�erent values

of W0 can be found in Ref. [108]).The reason why the quantity Le is named the extrapolated

distance is that one of the solutions to Eq. (8.11) is:

U0(r)jz=0 � exp[�Lez] ; (8.13)

and therefore Eq. (8.11) is approximated by:

U0(r)jz=Le = 0 : (8.14)

This approximation is the most commonly used, since the method of images can then be

employed [11], considerably simplifying the calculations. However, since Eq. (8.11) tends

asymptotically to Eq. (8.13), Eq. (8.14) is a good approximation for solutions deeply inside the

di�usive medium, and therefore it deviates from the already approximated expression (8.8) when

U0 is evaluated near the interface3. In any case, we shall always employ the exact expression,

Eq. (8.8), within the di�usion approximation. The only situation in which Eq. (8.14) is exact

is when the non-di�usive medium is a perfect absorber, i.e. a black material, in which case

Le = 0.

In the presence of source anisotropy, we must make use of the expression for the �ux J

which takes anisotropy into account (see Eq. (2.64) in Sec. 2.3.2):

J = �D0rU0(r) + JE(r) ; (8.15)

3which, on the other hand, is where usually measurements are taken.


where JE was de�ned as:

JE(r) = D0

Z4��(r; s)sd ;

being � the power radiated by the source per unit volume in direction s. In this case Eq. (8.10),

by means of Eq. (8.15), can be rewritten in general terms as (see Sec. 9-2 of Ref. [9]):

U0(r)jz=0 = �� D0

@U0(r)

@n

��z=0

� JE � njz=0!: (8.16)

8.2 Re�ection and transmission coe�cients

Now that the boundary conditions for di�usive/non-di�usive interfaces have been presented, we

may proceed as in Sections 6.1 and 7.2 to derive the re�ection and transmission coe�cients at

these interfaces. In a con�guration as depicted in Fig. 8.2, where the light source is located at

zs > 0, the total wave is represented by U0 = U (i) +U (r), and U1 = U (t) in the upper and lower

media, respectively. The expression for U (t) is only valid at the interface, since propagation

into the non-di�usive medium cannot be described by the di�usion approximation [130] (this

fact will be studied in Sec. 8.3). In order to �nd the relationship between U (i), U (r), and U (t)

across the interface, and hence the expressions for the re�ection and transmission coe�cients

Rnd and Tnd, we must apply the saltus condition at z = 0, Eq. (8.10). Writing Jn and U in

their angular spectrum representation (see Sec. 3.2) substituting into Eq. (8.10), we obtain:

~U (i)(K; z = 0) + ~U (r)(K; z = 0) = � ~Jn(K; z = 0) ; (8.17)

~Jn(K; z = 0) = �D0

hiq(K) ~U (i)(K; z = 0)� iq(K) ~U (r)(K; z = 0)

i; (8.18)

where Eq. (8.18) represents Fick's law, and takes into account the opposite propagation direc-

tions of U (i) and U (r). After some simple algebra Eqs. (8.17) and (8.18) reduce to:

A(r)(K) = Rnd(K)A(i)0 (K) ; (8.19)

Jn(K; z = 0) =1

�Tnd(K)A(i)

0 (K) ; (8.20)

where,

Rnd(K) =i�D0q(K) + 1

i�D0q(K)� 1; (8.21)

Tnd(K) =2i�D0q(K)

i�D0q(K)� 1; (8.22)

Rnd and Tnd being the frequency dependent re�ection and transmission coe�cients for di�usive/non-

di�usive interfaces, respectively. Both Rnd and Tnd are complex, and the sum of their moduli


is not unity, but they hold the relationship 4 Tnd(K) = Rnd(K) + 1. As seen from Eq. (8.20),

we have expressed the transmission coe�cient in terms of the total density �ux that traverses

the interface. The reason for this is that when measurements are performed from an outer non-

di�usive medium, usually the measured quantity is Jn, through Lambert's cosine law [11, 130],

and not U (t) (this fact will be shown later in Sec. 8.3). In any case, the value of U (t) can be

directly obtained from Eq. (8.10), and therefore:

~U (t)(K; z = 0) = Tnd(K) ~U (i)(K; z = 0) : (8.23)

8.2.1 Black interface

As a particular case of di�usive/non-di�usive interface, we consider that in which the lower

medium is perfectly absorbing, i.e. a black slab. The boundary condition is obtained by

substituting Le = 0 in Eq. (8.14):

U0(R; z)jz=0 = U (i)(R; z)��z=0

+ U (r)(R; z)��z=0

= 0 ; (8.24)

which results in the following re�ection and transmission coe�cients for di�usive/black inter-

faces:

Rblack(K) = �1 ; (8.25)

Tblack(K) = 0 : (8.26)

This relationship is what is expected from Eq. (8.24) since absorption is simply scattering with

a � dephase, i.e. U (r)(R; z = 0) = U (i)(R; z = 0) exp[i�], and in the case of a perfect absorber,

with the property jU (r)j = jU (i)j. Eqs. (8.25) and (8.26) are equivalent to introducing � = 0

into Tnd and Rnd, [see Eq. (8.10)].

8.2.2 Zero re�ection frequency

Exactly as for di�usive/di�usive interfaces (see Secs. 6.1.2 and 7.2) Rnd given by Eq. (8.21)

can be zero whenever �0 is a pure complex number, which occurs when there is a continuous

DC source, i.e. ! = 0. The corresponding value Kzero then is:

Kzero =

s�1

�D0

�2� �a0

D0: (8.27)

Typical values of the re�ectivities Rnd are shown in Fig. 8.3, and are observed to be rather

low. As also shown in Ch. 6, this is a general characteristic of di�use photon density waves

(DPDWs) at interfaces between biological media, and it also applies to di�usive/non-di�usive

4This relationship was also found in Sec. 6.1, for index matched di�usive/di�usive interfaces.


Figure 8.3: Amplitude of the re�ection coe�cient jRnd(K)j, and the transmission coe�cientjTnd(K)j with ! = 0, for the cases: D0 = 0:06666cm, �a0 = 0:2, n0 = n1 = 1:3333, Kzero =7:3cm�1, (jRndj [solid line], jTndj [�]); D0 = 0:06666cm, �a0 = 0:2, n0 = 1:45, n1 = 1:0,Kzero ' 1:5cm�1, (jRndj [dotted line], jTndj [Æ]); D0 = 0:03333cm, �a0 = 0:017, n0 = 1:45,n1 = 1:0, Kzero ' 4:5cm�1, (jRndj [dot-dashed line], jTndj [u]).

interfaces. The transmitivities, also exhibited in Fig. 8.3, are therefore quite high, and in

the case in which measurements are made from the non-di�usive outer medium constitutes an

advantage. Hence, this fact, together with the existence of high damping, states that there will

be a very low contribution of multiple scattering between any object embedded in the di�usive

medium and the di�usive/non-di�usive interface. As seen in Fig. 8.3, the values of Kzero range

between zero and very high values. Since the angular spectrum of DPDWs is concentrated at

lowK due to high damping [90] (see Ch. 5), Fig. 8.3 shows important variations in the di�erent

plane wave component contributions versusK, both for the re�ected and transmitted waves. As

seen with the simulated speci�c cases, when the value of Kzero is above 2cm�1, no appreciable

e�ect is seen in the scattered wave. Also, as shown in Fig. 8.3, the transmission coe�cient,

even in the range in which Kzero exists, shows no signi�cant changes, and therefore its e�ect in

transmission is di�cult to appreciate (a similar thing occurs in the di�usive/di�usive case, as

shown in Ref. [100] and Sec. 6.1.2). This e�ect is strongest when Kzero < 2cm�1. In order that

this occurs, values of � � 5 are needed, so as to maintain �a0 and D0 within physical values

and the di�usion approximation. In the index matched case, in which � = 2, in order to reach

values of Kzero < 2cm�1, D0 and �a0 must have values which are in the limit of the di�usion


approximation. If we consider the case in which n0 = 1:45, n1 = 1:0, D0 = 0:0666cm�1, and

�a0 = 0:2, we obtain a very interesting situation (see Fig. 8.3), since then R(K) constitutes

a high pass �lter5. Hence, as shown in Sec. 6.1.2, by changing the value of �a0, for example,

then one can "tune" the K-components to be �ltered, the corresponding value of Kzero also

changing. Kzero does not exist when ! 6= 0, since its expression is then complex. Even so,

Figure 8.4: Re�ected amplitude (a) and Transmitted amplitude (b) at z0 = 1cm, with ! = 0,D0 = 0:066666cm, �a0 = 0:2cm�1, n0 = 1:45, n1 = 1:0, Kzero ' 1:5cm�1 for the cases:zs = 0:5cm (solid lines); zs = 1:0cm (dotted lines). Note how the zero re�ection a�ects there�ected wave but does not noticeable a�ect the transmitted wave. See Fig. 8.3 for the re�ectioncoe�cients.

a dip can be appreciated at the corresponding value of Kzero for the ! = 0 case in a similar

manner than for di�usive/di�usive interfaces presented in Sec. 6.1.2.

The e�ect that thisK-�ltering has on the re�ection of a spherical incident wave (i.e., a point

source in three-dimensions) is shown in Fig. 8.4(a), where the re�ected amplitude is plotted

versus the detector x-position, at a certain plane z = z0, y = 0. As seen, near the condition

in which Kzero < 2cm�1 is real , the re�ected wavefront is shaped in such a manner that its

measured images appear to resemble re�ection from complex geometries rather than from a

plane. As shown in Fig. 8.4(b), there is no signi�cant distortion on the transmitted wave. At

this point it must also be stated, that in order to obtain re�ected images such as those plotted

5These values can easily be obtained in some resins, as will be shown in Ch. 10.


in Fig. 8.4(a), the incident angular spectra should have some high frequency components, such

that Kzero can have some e�ect on them. High frequency components are obtained whenever

the source is close to the interface (zs = 0:5cm; 1:0cm in the cases presented here). As the

source moves further away, the angular spectrum starts to concentrate in the low frequency

values (i.e. jKj ' 1cm�1) and the zero re�ection coe�cient has no e�ect.

8.2.3 Characterization of Di�usive Media

We next illustrate how the characterization of di�usive media by means of Eq. (8.22) should

be made in experiments when measuring from the outer non-di�usive medium, in a similar

manner as that presented in Sec. 7.2.1. As before, a frequency modulated point source is

located inside the di�usive medium, of unknown �a0, D0 and n0, at a distance zs > 0 from the

interface z = 0. Since U (t) = �Jn, the transmitted wave U (t)(R; z = 0) into the outer space

is determined from measurements of Jn at this interface. On inverse-Fourier transforming

U (t)(R; z = 0), ~U (t)(K; z = 0) is obtained. By means of Eq. (8.23), using the angular spectrum

of the wave emitted by a point source [61] (see Sec. 3.2.2),

A(i)(K) =S0i

2�q(K)exp[iq(K)zs] ;

we get:

Tnd(K) =2�

S0i

~U (t)(K; z = 0)q(K)

exp[iq(K)zs]: (8.28)

�a0, D0, and � univocally determined by �tting them by means of Eq. (8.22) to the analytic

expression, Eq. (8.28), for Tnd(K). Since Tnd depends on n0 only through �, once � is found,

n0 can be estimated (see Ref. [107, 108]). If the only unknown is �, it can be directly obtained

from:

� =1

iD0

0@ d~Jn(K)� exp[i(q(K)� �0)zs]d~Jn(K)q(K)� exp[i(q(K)� �0)zs]�0

1A ; (8.29)

whered~Jn(K) is the value of ~J(K) normalized to ~J(K = 0).

~U (t)(K; z = 0) was numerically found at z = 0 in a three-dimensional simulation, in which

the incident �eld was a point source at zs > 0. In this illustration, we chose a DC source,

i.e. ! = 0, in order to compare with the experimental setup used in Ch. 10, in which a

CCD camera is used. We employ the experimental CCD setup parameters used in Ch. 10,

such as high resolution pixels � 0:03cm, and images of 300 � 500 points. The parameters

used for the di�usive medium correspond to typical resin phantom values of �a0 = 0:2cm�1m,

D0 = 0:0333cm, and n0 = 1:45. In the DC regime, not all parameters �a0, D0 and � can be

simultaneously retrieved in the presence of noise [132], although they can be retrieved for very

low noise values < 0:1%. Two di�erent cases are presented: one in which the values of zs and

� are assumed to be known a priori, and another in which we assume the values of D0 and


�a0 known, and we aim to retrieve zs and �. In both cases estimations were performed with

additive noise up to 10% in the amplitude data. We have also tested situations in which a

modulated source (i.e. ! 6= 0) is used and all parameters can be retrieved at once with errors

lower than � 10%, even with noise values of up to � 5%.

Coe�. Real Value 2% Noise 10% Noise�a0 0.0200 0.0206 0.0236D0 0.0333 0.0343 0.0386� 6.559 6.539 6.764

Table 8.1: Real and retrieved values by transmission measurements using Eq. (8.22) and (8.28)with ! = 0, zs = 2:0cm.

In the �rst case, the retrieved values of �a0 and D0 retrieved are shown in Table 8.1. The

error in the recovery of each parameter, as shown in this table, was never over 15%. Typical

values of noise in CCD images are within the range of � 2%. Thus, as seen from this table,

the parameters are retrieved with remarkable accuracy.

In the second case, the values of zs and � were retrieved, assuming D0 and �a0 known.

Estimated values for � are also shown in Table 8.1. The values for zs are not shown since, even

in the presence of 10% noise, they were retrieved with less than 0:2% error. As observed in this

case, this method can be e�ectively used to determine the value of the extrapolated distance

Le [107, 108], Eq. (8.12). In general, we have obtained faithful reconstructions with errors in

the a priori parameters �a0, D0, and zs up to 0:5%, whereas the error allowed in � can be up

to � 10%. This is also an advantage since usually the value of � is never well known.

8.3 Boundary conditions at arbitrary interfaces

Let us consider the situation depicted in Fig. 8.5, where a rough non-di�usive region of volume

V , bounded by the surface S, with absorption coe�cient �a1 and refractive index n1, is embed-

ded in an otherwise homogeneous di�usive medium of volume ~V , characterized by an absorption

coe�cient �a0, refractive index n0, and di�usion coe�cient D0. A modulated point source at

frequency ! is located at r = rs and generates a di�use photon density wave (DPDW) for the

average intensity U(r). Eqs. (8.4) and (8.3) give us the expressions for the inward and outward

�ux at surface S, the former in terms of the speci�c intensity inside the non-di�usive medium,

the latter in terms of the average intensity and density �ux inside the di�usive medium. What

is needed at this point is a boundary condition for the light wave that matches both media. The

most simple approximation, which will be used throughout this Chapter, is to consider the light

isotropically radiated from the di�usive medium (see Fig. 8.6). This involves the assumption

that the speci�c intensity, radiated from the surface S into the non-di�usive medium, will not

have an angular dependence. Then, since the total �ux that emerges from a point in the surface


Figure 8.5: Scattering Geometry of a non-di�usive region of volume V and surface S embeddedin a di�usive medium.

Figure 8.6: Scheme for the speci�c intensities inside the di�usive medium and emerging into thenon-di�usive medium. Note the angular dependence to �rst order of I0 in the di�usive medium,whereas in the non-di�usive medium I1 is angle independent. The light distribution radiated intothe non-di�usive medium, i.e. I1(r), gives rise to Lambert's power law: dp(r) = I1(r) cos �dS.

r is J+, the speci�c intensity inside the non-di�usive medium I1 must be expressed as:

I1(r; s) = I1(r) = �J+(r) ; (8.30)


where � is a constant. Since the total �ux radiated into the non-di�usive medium must be J+,

by means of Eq. (8.3) we obtain � = 1=�. Therefore, the power radiated into the non-di�usive

medium from a certain point r of the surface element dS will be (see Eq. (2.9) of Sec. 2.1):

dp(r; �) = I1(r) cos �dSd =J+(r)

�cos �dSd : (8.31)

Eq. (8.31) is Lambert's cosine law [11] and is frequently used for describing light emerging from

a strongly scattering medium, the surface boundary S acting as a secondary source (see Fig.

8.6). The amount of power incident on the elementary area dS 2 S located at point r, and

Figure 8.7: Scheme for the solid angle relationship between dS = ndS and dS0 = n0dS 0.

emitted from the elementary area dS0 2 S located at r0 (see Fig. 8.7) will be (see Sec. 2.2.1):

dpi(r; �) = I1(r0; ur�r0) exp

��a1 + i

!n1c

�jr� r0j

�cos �dSd ; (8.32)

where cos � = n � ur0�r, and comes from the solution to the RTE in free space (see Sec. 2.2.1):

I1(r; ur�r0) = I1(r0; ur�r0) exp

��a1 + i

!n1c

�jr� r0j

�;

where I1 represents the speci�c intensity inside the non-di�usive volume, and �a1 is the ab-

sorption coe�cient of the non-di�usive medium. Note that we consider S to be the union of

all surfaces contributing to the �ux at r which in general may have a complex topology; see

[126] for further discussion of this topic. The total power transmitted at r, taking all possible

incidence angles into consideration, is (cf. Eq. (2.14) in Sec. 2.1.1):

dP t(r) =Z(2�)+

�1� jR1!0(�)j2

�dpi(r; �) ;


dP t(r) =Z(2�)+

�1� jR1!0(�)j2

�I1(r

0; ur�r0) exp��a1 + i

!n1c

�jr� r0j

�cos �dSd :

By using the fact that (cf. Fig. 8.7):

d =dS0 � ur�r0jr� r0j2 ;

and (see Eq. (2.16) of Sec. 2.1.1):

J�(r) =dP t(r)

dS;

the total �ux density transmitted through every point r of S is written in terms of the speci�c

intensity radiated by the complete surface as:

J�(r) =ZSI1(r

0; ur�r0)G(r� r0)dS 0 r 2 S ; (8.33)

where we have de�ned:

G(r� r0) =�1� jR1!0(�)j2

� exp h��a1 + i!n1c

�jr� r0j

ijr� r0j2 V(r� r0) cos �0 cos � ; (8.34)

cos � = n � (r0 � r)

jr0 � rj ; cos �0 = n0 � (r� r0)

jr0 � rj :

Figure 8.8: Example of the visibility factor V(r� r0) for an arbitrary surface. The lines denotethe surface points visible from one surface position.

In Eq. (8.34) V(r � r0) is a visibility factor, which is either unity if both points r and r0

can be joined by a straight line without intersecting an interface (i.e., when they are visible

to each other), or zero when such a straight line does not exist (see Fig. 8.8). From now on,

and for the sake of simplicity, this factor will be implicitly assumed and therefore omitted. An

expression, similar to that of Eq. (8.33) for the inward density �ux, can be found by means of


Eq. (8.1) for the average intensity U1 inside the non-di�usive volume:

U1(r) =ZSI1(r; ur0�r)

n0 � ur�r0jr� r0j2 dS

0 r 2 V ; (8.35)

which can be rewritten as:

U1(r) =ZSI1(r

0; ur0�r)�(r� r0)dS 0 r 2 V ; (8.36)

where �(r� r0) is:

�(r� r0) = exph��a1 + i!n1

c

�jr� r0j

ijr� r0j2 cos �0 : (8.37)

It must be stated, that even though Eqs. (8.33) and (8.36) will be used within the di�usion

approximation, their expressions are general for the RTE. Substituting Eq. (8.30) into Eq.

(8.33) we obtain the following expression for the total downward �ux:

J�(r) =1

�

ZSJ+(r0)G(r� r0)dS 0 r 2 S : (8.38)

Figure 8.9: Contribution of the inner �uxes at r0 to the total �ux at a point r.

The total �ux going into the di�usive medium is therefore a superposition of all outward

going �uxes at the surface S that are transmitted at point r (see Fig. 8.9). Therefore, by means

of Eq. (8.7) the total normal �ux Jn = J+ � J� is:

Jn(r) = RUU0(r)

2+RJ

Jn(r)

2�

1

�

ZS

"RU

U0(r0)

2+RJ

Jn(r0)

2

#G(r� r0)dS 0 r 2 S :


Grouping terms we then obtain that:

U0(r) = �Jn(r) +1

�

ZS

�U0(r

0) +RJ

RU

Jn(r0)�G(r� r0)dS 0 r 2 S ; (8.39)

where � = (2�RJ)=RU , as shown in Eq. (8.8). Eq. (8.39) is the non-local boundary condition

that we were seeking for light propagation in di�usive media with non-di�usive regions. It must

be stated that this boundary condition cannot take into account multiple re�ections inside the

non-di�usive region when dealing with index mismatched media, and therefore is only valid

for small index mismatch. In the case of a plane interface (i.e. an open surface), there is

only outgoing �ux and we recover the expression for the boundary condition in the di�usion

approximation [9, 108] derived in Sec. 8.1, U0(r)jS = �Jn(r).

The average intensity inside the non-di�usive medium, Eq. (8.36), within this di�usion

approximation can be rewritten with the aid of Eq. (8.30) as:

U1(r) =1

�

ZS

"RU

U0(r0)

2+RJ

Jn(r0)

2

#�(r� r0)dS 0 r 2 V: (8.40)

At this point it must be remarked that Eqs. (8.39) and (8.40) have been derived for a three

dimensional (3D) con�guration. The expressions for these equations in two dimensions (2D)

are derived in Sec. 8.3.1, and are those that will be employed in our numerical simulations.

Whenever dealing with non 3D con�gurations, great care must be taken with the RTE, since

as mentioned in Ch. 2, conversion from 3D to 2D is non-trivial. A �rst order approximation

to Eq. (8.39) was used in Refs. [125] and [126] to deal with non-di�usive regions in di�usive

media, for the index matched case. This approximation consisted of applying the boundary

condition Eq. (8.10) and including a secondary source at the interface given by:

U(sec)0 (r) =

1

�

ZSJ (0)n (r0)G(r� r0)dS 0 r 2 S ; (8.41)

where J (0)n denotes the zeroth order approximation to the total �ux, i.e. J (0)

n = U=�. The main

assumption that this approximation makes is that the total normal �ux is equal to the total

outward �ux, i.e:

J (0)n (r) = J+(r) = RU

U0(r)

2+RJ

J (0)n (r)

2r 2 S ; (8.42)

which gives the boundary condition �J (0)n = U . This approximation is expected to break down

when the total inward �ux is not small, but as shown in Refs. [125] and [126], it yields accurate

results with biological parameters and will be employed in Section 8.4.1 with the FEM model.

Also, as mentioned before, whenever dealing with great di�erences in the refractive indices,

both Eqs. (8.39) and (8.42) are expected to break down, since they cannot account for multiple

re�ections at inside the non-di�usive medium. That is, both Eqs. (8.39) and (8.42) deal with


transmitted light, and do not consider the re�ected part, which would give rise to non-linear

equations.

8.3.1 Boundary conditions in two dimensions

The expressions for the boundary conditions in two dimensions (2D) are quite useful since the

great majority of numerical simulations are performed in 2D due to their lower computational

cost. Let us assume that in Fig. 8.7 the con�guration is constant in the z direction. We write

r = (R; z), the unit normal n(r) = N(R), and the unit area as dS = dAdz. Since in this case

U and J are z independent, we can rewrite Eq. (8.40) as:

U1(R) =1

�

ZAJ+(R0)�2D(R�R0)dA0 ; (8.43)

�2D(R�R0) =Z +1

�1�(r� r0)dz0 : (8.44)

In Eq. (8.37), cos �0 can be rewritten as cos �0 = cos�0 cos �, where cos �0 = uR�R0 � N0. By

performing the change of variable: cos � = R=pR2 + z2, Eq. (8.44) can be expressed as:

�2D(R�R0) =uR0�R � NjR0 �Rj

Z �=2

��=2exp

"��a1 + i

!n1c

� jR�R0jcos �

#cos �d� : (8.45)

Proceeding as in Sec. 8.3, we obtain the following expression for the boundary condition Eq.

(8.39):

U0(R) = �Jn(R) +1

�

ZA

�U0(R

0) +RJ

RUJn(R

0)�G2D(R�R0)dA0 R 2 A ; (8.46)

G2D(R�R0) =Z +1

�1G(r� r0)dz0 (8.47)

Performing the same substitutions as in Eq. (8.45), we obtain:

G2D(R�R0) =(uR0�R � N)(uR�R0 � N0)

jR�R0j �Z �=2

��=2

�1� jR1!0(�

(i))j2�exp

"��a1 + i

!n1c

� jR�R0jcos �

#cos2 �d� ; (8.48)

where cos �(i) = cos � cos�. In the case in which there is no index mismatch, n0 = n1, the

expression (8.48) for G2D can be approximated to:

G2D(R�R0) ' �

2

cos� cos�0

jR�R0j exp��a1 + i

!n1c

�jR�R0j

�: (8.49)

Eq. (8.49) is exact when ! = 0, such as in the cases addressed in Refs. [52, 125, 126], and in

the instances presented in this Chapter.


8.4 Scattering Integral Equations

Figure 8.10: Scattering geometry consisting of a rough non-di�usive cylinder with statisticalparameters � and T , embedded in a di�usive cylinder of radius Rcyl.

Let us consider the two-dimensional situation depicted in Fig. 8.10. It consists of a rough

cylinder with axis alongOZ, embedded in an otherwise homogeneous di�usive cylinder of radius

Rcyl = 2:5cm, surrounded by air. The rough non-di�usive region has a Gaussian distributed

random pro�le given by:

�(�) = [R +D(�)]ur ; (8.50)

with mean R = h�(�)i, where h� � �i denotes the ensemble average, D(�) represents the randomfunction with hD(�)i = 0, and ur is the radial unit vector. The Gaussian covariance is:

h�(�)�(� + �)i = �2 exp(��2=T 2� ) ; (8.51)

where T� is the correlation length in radians, and � is the root mean square (r.m.s.) height:

� =qh�2(�)i �R2 : (8.52)

We shall de�ne the correlation length T in cm units as T = T�R. A point source with constant

illumination (! = 0) is located at rs � (r = 2:45cm; � = 0o) inside the di�usive medium. The

rough non-di�usive region of volume V , bounded by the surface S, has absorption coe�cient

�a1 and refractive index n1. The outer di�usive cylinder of volume ~V is characterized by an

absorption coe�cient �a0, refractive index n0, and di�usion coe�cient D0. A DC point source

at r = rs generates a di�use photon density wave (DPDW) for the average intensity U(r).

Proceeding as in Ch. 4 [52, 130], by applying Green's theorem [58], the total �eld inside the


di�usive medium is given by:

U0(r) = U (i)(r)� 1

4��Rcyl(r)�

1

4�

ZS

U0(r

0)@G(�0jr� r0j)


0)

@n

!dS 0 r 2 ~V ; (8.53)

where U (i), represents the incident DPDW in its usual sense, and G(�0jr� r0j) is the Green'sfunction in 2D for the di�usive medium, presented in Sec. 3.1.1. In Eq. (8.53) we have de�ned

n as the S surface unit normal pointing inward (see Fig. 8.10), and �Rcyl represents the surface

contribution of the outer surface:

�Rcyl(r) =ZRcyl

U0(r0)

@G(�0jr� r0j)

@m+

1

�cylD0G(�0jr� r0j)

!dR0 ; (8.54)

where m is the outer surface unit normal pointing outward, i.e. m = ur. In order to reach Eq.

(8.54) we have applied the zero �ux boundary condition (8.8), since there is no incoming �ux

into the outer cylinder:

U(r)jRcyl = ��cylD0@U(r)

@m

��Rcyl

; (8.55)

where �cyl = (2 � RcylJ )=Rcyl

U , accounts for the Fresnel re�ection coe�cients at the boundary

and depend on n0 and nout = 1:0. In the presence of source anisotropy, Eq. (8.15) must be

employed to relate U and @U=@n, and Eqs. (8.53), (8.54) and (8.55) changed accordingly (see

Sec. 4.1.2, where this was considered for di�usive/di�usive interfaces).

In order to construct the total wave�eld at any point within the di�usive medium, r 2 ~V ,

we must �nd the values of the sources at the boundary, i.e. the values of U(r 2 S) and

@U(r 2 S)=@n at the rough interface, as well as the value of U(r 2 Rcyl) at the outer cylinder.

To this aim, we need to apply the boundary condition between the di�usive and non-di�usive

media, Eq. (8.46) in the 2D case, which in the index matched case for ! = 0 is:

U0(R) = 2Jn(R) +1

�

ZS[U0(R

0) + 2Jn(R0)]G2D

0 (R�R0)dS 0 r 2 S ; (8.56)

where,

G2D0 (R�R0) =

�

2

cos� cos�0

jR�R0j exp [��a1jR�R0j] :

In Eq. (8.56), � = (2 � RJ)=RU = 2, where RclrJ = 1 and Rclr

U = 1=2 for the index matched

case.

Therefore, Eqs. (8.53) and (8.56), and Fick's law, Jn = �D0@U0=@n, constitute a linear

system of equations which will be rigorously solved numerically, by means of the extinction

theorem integral method [52, 74, 91] presented in Sec. 4.4. It should be stated that Eqs. (8.53)

and (8.56) can be generalized to any number of di�usive and non-di�usive regions, in two or


three dimensions, and can be easily adapted to an AC regime, i.e. ! 6= 0, by using Eq. (8.47)

or (8.34). Also, in the case of source anisotropy Eq. (8.15) must be used to relate @U=@n and

Jn, instead of Fick's law. Once the values of U1 and Jn are found, U1 and U0 can be univocally

determined at any point inside both the di�usive and the non-di�usive medium by means of

Eqs. (8.53) and (8.40), respectively.

8.4.1 Smooth boundaries

Figure 8.11: Cases considered: (a) Non-di�usive cylinder of radius Rin embedded in a di�u-sive cylinder of radius Rout. (b) Non-di�usive gap of inner radius Rin and outer radius Rgap

embedded in a di�usive cylinder of radius Rout.

In order to assess the validity of the boundary condition Eq. (8.56), we will �rst study the

2D con�gurations depicted in Fig. 8.11 and contrast the results obtained from the ET method

with the �nite element method (FEM) and Monte Carlo (MC) calculations. We shall �rst

consider smooth boundaries, i.e. � = 0, T =1, the geometries consisting on an outer cylinder

of �xed radius Rout = 2:5cm, �lled with a di�usive medium of parameters �0s0 = 20cm�1 and

�a0 = 0:1cm�1, is surrounded by air. We shall consider two cases: one in which a non-di�usive

cylinder of radius Rin is included, [Fig. 8.11(a)]; and another in which there is a non-di�usive

gap of outer radius Rgap and inner radius Rin between two di�usive regions, [Fig. 8.11(b)]. In

both cases the absorption coe�cient for the non-di�usive region is �a1 = 0:05cm�1. The speed

of light is constant throughout the media, with refractive index n0 = n1 = 1:4. A continuous

isotropic point source (! = 0) is placed at one transport mean free path ltr = 3D1 ' 0:05cm

from the outer surface within the di�usive medium (i.e., rsource = (Rout � ltr) ' 2:45cm). In

order to quantify the limits of validity of the boundary condition we shall consider several

object sizes. For small objects, we must always take into consideration that, for the di�usion

approximation to remain valid, light must travel at least a few mean free paths. As shown in

Ref.[106], sizes smaller than, or equal to, the mean free path, start causing deviations.

In order to reach numerical convergence with the ET method, all surfaces have a discretiza-

tion dS = 0:025cm. The numerical calculations took 3 minutes for Fig. 8.11(a) and 9 minutes

for Fig. 8.11(b) on a 200MHz PC with a 128Mb RAM. When using the FEM, the mesh had


between 949 nodes, 1614 elements (Rin = 2cm) and 2323 nodes, 4456 elements (Rin = 0:25cm)

for the case Fig. 8.11(a) and ' 2000 nodes and ' 4000 linear elements for the case in Fig.

8.11(b) (exact numbers were di�erent for each position of the gap). Computation times were

5 � 20 seconds on a 450MHz Pentium II. The FEM has been extensively reported in many

cases [70, 71, 72], and with this particular con�guration in Ref. [126]. Therefore we shall not

explain it here. In order to assess the accuracy of the calculations performed with the ET and

FEM, we compared with results from MC simulations. The description of the MC method

for photon di�usion is well known (see for example [112, 113]) and was brie�y described in

Sec. 7.4.1. The program is the same used in Ref. [126]. 107 photons were launched when

performing MC simulations. In all three methods the detector scanning was performed along

the surface of the outer cylinder, i.e. at rd = 2:5cm, varying the angular distance from the

source. Once again, we would like to state that when applying the ET model, we will make use

of the complete boundary condition, Eq. (8.39), whereas in the FEM model the approximate

boundary condition, Eqs. (8.41) and (8.42), will be employed. It must also be remarked that

in the absence of non-di�usive regions both FEM and ET yield results which are similar to a

high degree of accuracy. Thus, any deviation between both methods must be due solely to the

di�erence in the boundary conditions applied.

Fig. 8.12 shows the results for the MC, FEM and ET calculations in the case represented

in Fig. 8.11(a). Rin has been varied from Rin = 0:25cm to Rin = 2:0cm in 0:25cm steps. This

implies that the volume ratio of di�usive medium to total volume varies from Vr = 36% up

to Vr = 100%. As shown in Fig. 8.12, both FEM and ET models accurately account for the

photon transport in all cases, noting that ET is slightly more accurate. The reason for this

is twofold: �rst of all, ET is an exact method, whereas FEM assumes an approximation on

@U=@r. Secondly, the FEM uses the approximated boundary condition Eqs. (8.41) and (8.42),

while ET is able to include the complete boundary condition Eq. (8.39). As expected, results

have a greater deviation from MC for greater values of Rin, i.e. for lower values of Vr. As a

general rule, we may say that for values of Vr > 75% we obtain quite accurate results with both

FEM and ET. Even so, results obtained for the case with Rin = 2:0cm, i.e. Vr = 36%, are quite

impressive if we take into account that then 64% of the total volume is non-di�usive. Also, if

we look at the case with Rin = 0:25cm, we see that there is a greater deviation than expected.

The reason for this is mainly that such a small non-di�usive volume stands in the limit of

length scales within which the di�usion approximation works. That is, within the di�usion

approximation context, in the same manner as di�usive regions with sizes of the order of the

transport mean free path ltr can barely be considered as actually di�usive [106], non-di�usive

objects with sizes of the order of ltr embedded in di�usive media, can barely be consider as

actually non-di�usive. Statistically, this would mean that in a particular region of the di�usive

object, we have a lower concentration of scatterers, which, since the di�usion approximation

always deals with averages, has a very low contribution. Thus, following these considerations,

the results put forward by the FEM and ET methods for the case Rin = 0:25cm show lower


Figure 8.12: Average intensity measured on the outer surface Rout = 2:5cm versus de-tector angle separation, for the con�guration depicted in Fig. 8.11(a). Values of Rin =0:25cm; 0:5cm; 0:75cm; 1:0cm; 1:25cm; 1:5cm; 1:75 and 2:0cm. Results presented for simula-tions performed with MC (solid line), FEM (Æ) and ET (�). In all cases a DC source (! = 0)was located at rs = 2:45cm at � = 0. �a1 = 0:05cm�1, �a0 = 0:1cm�1, �s0 = 20cm�1, g = 0.

intensities (i.e., the non-di�usive object is less visible) than those for the MC.

In Fig. 8.13 we show the results for the MC, FEM and ET calculations in the case repre-

sented in Fig. 8.11(b). Now we have considered two values of Rin, namely Rin = 1:0cm [Fig.

8.13(a)], and Rin = 1:5cm [Fig. 8.13(b)]. For each value of Rin we computed �ve values of

Rgap which conferred the gap widths W = Rgap�Rin = 0:1cm; 0:3cm, 0:5cm, 0:7cm and 0:8cm

respectively. In all cases, we were always above a volume ratio Vr > 75%, namely, between

Vr = 94:7% (W = 0:8cm) and Vr = 99:84% (W = 0:1cm), so that a deviation due to low values

of the volume ratio is not expected. As seen in Fig. 8.13, both the FEM and ET show excel-

lent agreement with MC. In this con�guration, we once again reach more accurate results with

the ET for high values of W , namely W = 3mm ; 5mm; 7mm; and 8mm, for the reason stated

above. Also, a greater deviation of both FEM and ET occurs for smaller values of the gap width

W , namely W = 0:1cm, but in these cases FEM yields better results than ET. The reason for

this is that this width is of the order of ltr, and therefore the arguments that were applied to

the case Rin = 0:25cm of Fig. 8.12 can now be applied here. That is, when dealing with widths

in the order of ltr, the di�usion equation fails and therefore the boundary conditions do not

yield correct results. Also, since FEM uses the approximate boundary condition, it appears


Figure 8.13: Average intensity measured on the outer surface Rout = 2:5cm versus detectorangle separation, for the con�guration depicted in Fig. 8.11(b). Values of (a) Rin = 1:5cm,Rgap = 1:6cm (W = 0:1cm), 1:8cm (W = 0:3cm), 2:0cm (W = 0:5cm), 2:2cm (W = 0:7cm)and 2:3cm (W = 0:8cm). (b) Rin = 1:0cm, Rgap = 1:1cm (W = 0:1cm), 1:3cm (W = 0:3cm), 1:5cm (W = 0:5cm), 1:7cm (W = 0:7cm) and 1:8cm (W = 0:8cm). Results presented forsimulations performed with MC (solid line), FEM (Æ) and ET (�). Parameters as in Fig. 8.12.

that in cases of low values of W < 3mm, the approximate boundary condition works better

than the complete boundary condition. Also, on comparing the cases W = 0:5cm; 0:7cm; and

0:8cm from Figs. 8.13(a) and 8.13(b), we �nd that the depth of the gap buried in the di�usive

medium, i.e. the value of Rin, does not change the accuracy of the results. We would also like

to draw the atention to the curve corresponding to W = 0:8cm, in Fig. 8.13(a), where we �nd a

sharp intensity decrease6 that is also predicted by ET. This decrease in intensity appears when

the inner di�usive volume, r � Rin, is in the line of sight between source and detector, and its

magnitude is greater the closer the outer radius, Rgap , is from the source.

It is important to remark that what has been presented here corresponds to an isotropic

point source. Whenever dealing with more realistic sources, as would be a laser source impinging

the di�usive object from outside, the light entering the di�usive region is non-isotropic at least

within a few mean free paths, in which case the integral equations must be modi�ed accordingly

by using Eq. (8.15). In any case, if the �rst non-di�use interface, i.e. Rgap, is of the order

6This e�ect was also found in Ref. [125], where it is referred to as a �kink�.


of a few mean free paths ltr near to the source (typically � 2ltr), the boundary condition

Eq. (8.39) is expected to break down, since now light incident at the interface will no longer

be well described by the di�usion approximation, Eq. (8.6) [see Fig. 8.6]. However, in the

cases presented here, the values of Rgap are such that they are located at least 4ltr from the

source, and as shown in Fig. 8.13(a) the di�usion approximation still accurately describes light

transport.

8.4.2 Rough surfaces

Figure 8.14: Cases considered: (a) Non-di�usive rough cylinder of radius Rin embedded in adi�usive cylinder of radius Rout. (b) Non-di�usive rough gap of inner radius Rin and outerradius Rgap embedded in a di�usive cylinder of radius Rout.

Now that we have established the accuracy of the boundary condition, Eq. (8.39), in order

to quantify the e�ect of roughness on the total �eld U(r), we shall study the perturbation that

di�erent values of � and T cause in the scattered DPDW, U (sc)(r) = U(r) � U (i)(r) ; where

U (i)(r) is the incident �eld, given in Eq. (8.53). The di�erent con�gurations shown in Fig.

8.14 will be studied (see Sec. 8.4.1 for their comparison with MC and FEM): one in which we

have a non-di�usive rough cylinder for the case R = h�(�)i = 1:0cm; and another in which

we have a rough non-di�usive gap of inner radius Rin = h�in(�)i = 1:0cm and outer radius

Rout = h�out(�)i = 1:5cm (see Fig. 8.10). In both cases, the media are characterized by the

parameter values used in Sec. 8.4.1 for the smooth pro�le case.

In order to contrast the e�ect that roughness has on U (sc), we plot in Fig. 8.15 in logarithmic

scale the values of U (sc) throughout the di�usive cylinder in the cases: no rough cylinder present,

Fig. 8.15(a); smooth cylinder (� = 0; T =1), Fig. 8.15(c); and smooth gap (� = 0; T =1),

Fig. 8.15(c). As seen in this �gure, a very interesting e�ect occurs due to the presence of the

non-di�usive region. We see that there are some areas inside the di�usive cylinder in which

the value of U (sc) ' 0. These values of r correspond to the cases in which the distance traveled

by the incident wave inside the di�usive medium is equal to the distance traveled by the total

�eld U , that has traveled through the non-di�usive volume. In terms of photon statistics, this


Figure 8.15: Scattered wave U (sc) represented in logarithmic scale for the cases: (a) No non-di�usive cylinder present. (b) A non-di�usive cylinder of R = 1:0cm. (c) A non-di�usive gapof Rin = 1:0cm, Rout = 1:5cm. In all cases Rcyl = 2:5cm. Parameters as in Fig. 8.12.

condition is met when both optical path-lengths are equivalent, i.e.:

DL(i)(rs � r)

E= hL(rs � rv)i+ hL(rv � r)i ; (8.57)

where L represents the optical pathlength, rv is r 2 V , rs is the position of the source, and

h� � �i denotes average over all possible path-lengths. As the radius of the non-di�usive region

increases, this equivalent optical pathlength distance starts closer to the source, as in Fig.

8.15(c), since Rout = 1:5cm for the gap. We shall see that these optical path-lengths represented

by Eq. (8.57) are greatly modi�ed by the presence of roughness in the interface.

Fig. 8.16 shows the amplitude of the scattered wave jU (sc)(r)j in logarithmic scale, for

di�erent values of � and T , namely: � = 0:02cm; 0:05cm, and 0:10cm; T = 0:02cm; 0:05cm;

and 0:1cm, therefore dealing with roughness in the millimeter scale. In order to quantify how

rough is a con�guration compared to another, we shall use the following empirical formula for

a rough cylinder of statistical parameters �; T :

Roughness =�=R

T=Rexp[�=R] ; (8.58)

where, as de�ned in Section 8.4, R = h�(�)i, and � and T are in cm units. It can be seen that

in the case in which R!1, i.e. a plane surface, we recover from Eq. (8.58) the expression for

the slope of a rough surface [74, 117], �=T . By using the criterion of Eq. (8.58), the increase

in roughness for the nine cases in Fig. 8.16 is observed as:

Roughness �

0BBB@1:02 0:41 0:20

2:63 1:05 0:53

5:42 2:17 1:08

1CCCA : (8.59)

Using the values in Eq. (8.59), we see in Fig. 8.16 that, as we increase the roughness, these

equivalent optical pathlength distances start concentrating in the � = � direction, and �nally,


Figure 8.16: Scattered wave U (sc) represented in logarithmic scale for di�erent statistical pa-rameters � and T for the case of a non-di�usive cylinder of R = 1:0cm embedded in a di�usivecylinder Rcyl = 2:5cm. Parameters as in Fig. 8.12.

at the value of � = 0:10cm, T = 0:02cm, they disappear. The situation in which these optical

path-lengths disappear is very interesting, since, as seen on comparison with Fig. 8.16(a), the

overall shape of the scattered wave looks like if the non-di�usive region were absent. This e�ect

can be better understood if we compare the measured value of U at the interface r 2 Rcyl, for

di�erent values of the roughness, Eq. (8.58). Since this e�ect could be obscured by the fact

that we are working with random pro�les, we have performed the same calculations, by using

a cosine modulated surface pro�le of amplitude h and period d:

�(�) = R + h cos�2�

d��: (8.60)

In this case, h; d are proportional to �; T [133, 134] and therefore the corresponding values

of Eq. (8.58) will be proportional. In any case, their relationship is non-trivial [133, 134]. In

Fig. 8.17 we plot the values of the total average intensity U(r) for di�erent h and d for a

pro�le generated with Eq. (8.60), in the case R = 1:0cm. As seen, as the roughness given by

Eq. (8.58) increases, the signal calculated at Rcyl tends to that produced in the absence of the


Figure 8.17: Total average intensity U measured at Rcyl, for the case of a non-di�usiverough cylinder with cosine pro�le, with the following amplitude h and period d: Smooth(h = 0; d = 1) [solid line]; (h = 0:05cm; d = 0:32cm)[Æ]; (h = 0:05cm; d = 0:16cm) [�];(h = 0:05cm; d = 0:1cm) [4]; Absence of the non-di�usive object [dotted line]. In all casesRcyl = 2:5cm. Parameters as in Fig. 8.12.

non-di�usive volume. From the curves presented in Fig. 8.17, we also infer that if roughness

is not taken into consideration, these curves could be mistaken for non-di�usive volumes of

high absorption coe�cient �a1, or for smaller non-di�usive volumes. We should state that from

random rough pro�les as in Eq. (8.50), whenever calculating intensities from di�erent pro�les

with the same statistical parameters � and T , we always obtain similar results.

In order to have a better insight on the e�ect that roughness has on the measured signal,

we shall study the change of total transmitted intensity U trans versus roughness:

U trans =1

�

Z 3�2

�2

U(Rcyl;�)d� : (8.61)

The variation of U trans versus roughness is shown in Fig. 8.18 for di�erent values of R, and the

pro�le is as given in Eq. (8.50). In this �gure, it is clearly seen that as the roughness increases,

the total intensity U trans decreases. Whenever U trans reaches the value that would be obtained

without the non-di�usive volume (represented by a solid line), we obtain values for the scattered

wave like those presented in Fig. 8.16, for the case � = 0:1cm, T = 0:02cm. As expected, the

value of the roughness, Eq. (8.58), which reaches this condition depends on the average radius

R, needing higher roughness values the greater the non-di�usive volume is. Similar results are

shown in Fig. 8.19, but in the case in which we use the deterministic pro�le given by Eq. (8.60).

Then the same conclusions are reached, thus proving that the e�ect of roughness appears not

only for random pro�les but also for deterministic pro�les. It is interesting to see in Figs. 8.18


Figure 8.18: Total transmitted intensity U trans versus the roughness [see Eq. (8.58)] for a roughnon-di�usive cylinder with random rough pro�le and radii: R = 0:5cm [�]; R = 0:75cm [Æ];R = 1:0cm [4]; R = 1:25cm [?]; R = 1:25 [u]. U trans in the absence of the non-di�usivevolume is represented by a solid line. Parameters as in Fig. 8.12.

Figure 8.19: Same as Fig. 8.18 but in the case of a rough cylinder with cosine pro�le [see Eq.

(8.60)].


and 8.19, that in some cases the value of U trans obtained is smaller than that produced when

the non-di�usive volume is not present, i.e. we obtain a smaller output signal in the presence

of a highly rough non-di�usive volume, than in its absence. In these cases, if the existence of

the rough non-di�usive volume is not known a priori, the detected signal could be confused

with that corresponding to an absorbing di�usive object. Also, in these cases (see Fig. 8.19 for

the R = 0:5cm and R = 0:75cm cases), once this value is reached, the e�ect of roughness tends

to stabilize.

Figure 8.20: Scattered wave U (sc) represented in logarithmic scale for di�erent random pro�lesof statistical parameters � and T for the case of a non-di�usive gap of inner radius Rin = 1:0cmand outer radius Rout = 1:5cm embedded in a di�usive cylinder Rcyl = 2:5cm. Parameters asin Fig. 8.12.

In order to see that the conclusions reached for a rough non-di�usive cylinder are also

valid for a rough non-di�usive gap, Fig. 8.20 shows in logarithmic scale the scattered am-

plitude U (sc) for di�erent values of roughness, namely � = 0:02cm; 0:03cm; 0:05cm and T =

0:02cm; 0:05cm; 0:10cm, for the case Rin = 1:0cm, Rout = 1:5cm. As seen, the e�ect of rough-

ness is like in Fig. 8.16, the main di�erence being that the situation in which the wave�eld

is similar to that produced in the absence of the non-di�usive volume, is reached for a lower

value of the roughness, namely at � = 0:05cm, T = 0:02cm. Therefore, the conclusions reached


for the non-di�usive cylinder can be applied to the non-di�usive gap, the only di�erence now

being that in the case of gap presence, the e�ect of roughness is more critical. In the case

with the non-di�usive gap, low values of U trans are obtained at lower values of roughness, when

compared with those corresponding to the rough cylinder.

8.5 Non-di�usive volume versus di�usive object

Figure 8.21: Scattering con�guration for a non-di�usive rough cylinder and a di�usive cylinderseparated by a distance d from their centers, embedded in an otherwise in�nite homogeneousdi�usive medium.

Another important question that should be posed on non-di�usive volumes is how does their

presence a�ect scattering by di�usive objects, and how does this scattered intensity compare

for both objects. To clarify these questions, we have numerically calculated the scattered wave

from the con�guration depicted in Fig. 8.21. It consists of two objects separated a distance

d = 2cm, one of them is a di�usive cylinder of radiusR = 0:5cm with parameters �0sobj = 10cm�1

, �aobj = 0:2cm�1, and nobj = 1:4, the other being a non-di�usive rough cylinder with mean

radius h�(r)i = 0:5cm, and parameters �a1 = 0:05cm�1 and na1 = 1:4. Both are embedded in

an otherwise in�nite homogeneous medium of parameters �0s0 = 20cm�1, �a0 = 0:1cm�1, and

n0 = 1:4.

We shall study the scattered wave U (sc) from these objects, determined at a plane zd (see

Fig. 8.21). Two di�erent planes of measurement will be simulated, namely zd = 1:0cm, and

zd = �1:0cm. In all cases, a DC source will be located at rs = (xs; zs) = (0; 2:0cm). Fig.

8.22 shows the value of the scattered wave U (sc) at zd = 1:0cm for di�erent random rough

pro�les given by Eq. (8.50) for the boundary of the non-di�usive volume, namely, � = 0:02cm,


Figure 8.22: Scattered wave U (sc) measured at zd = 1:0cm for the case of a di�usive cylinderof radius R = 0:5cm placed at (0; 1:0cm) and a random rough non-di�usive cylinder of meanradius R = 0:5cm placed at (0;�1:0cm) with statistical parameters: (� = 0; T = 1) [solidline]; (� = 0:02cm; T = 0:10cm) [Æ]; (� = 0:02cm; T = 0:05cm) [�]; (� = 0:02; T = 0:02cm)[dotted line]. Both objects are embedded in an in�nite homogeneous di�usive medium (see Sec.8.5 for the optical parameters).

T = 0:02cm; 0:05cm; 0:10cm. As seen, there is no noticeable interaction between the di�usive

and the non-di�usive cylinders, whatever the roughness. Also, as shown in Fig. 8.22, as the

roughness increases, the value of U (sc) measured in the backward direction (i.e., at zd > 0)

also increases. This is due to the fact that the non-di�usive volume tends to isotropically

scatter the light when its roughness is high. This is clearly understood if we consider the case

zd = �1:0cm depicted in Fig. 8.23. We see that, as the roughness increases, the scattered wave

in the forward direction diminishes. Therefore, the scattered intensity in the backscattered

direction increases with roughness, whereas the scattered intensity in the forward direction

diminishes. It is interesting to note that for very rough surfaces at zd = �1cm (cf. Fig. 8.23),

the scattered pattern from the non-di�usive and the di�usive cylinders look very similar, and

thus one could confuse them. We should state that if one uses a modulated source, i.e. ! 6= 0,

the amplitudes still look very similar, but slight di�erences in phase are found.


Figure 8.23: Same as Fig. 8.22 but for determinations performed at zd = �1:0cm.

Chapter 9

Multi-layered di�usive media

Once the re�ection and transmission coe�cients have been established for di�usive/di�usive

(see Ch. 7) and di�usive/non-di�usive interfaces (see Ch. 8), we can use these coe�cients for

solving multiple layered media. This is of importance for straightforwardly determining the

optical parameters of realistic setups. Also, they give us the exact value of the Green function

in any of the layers, thus enabling us to improve the accuracy of hidden object reconstruction

within such systems. In many experimental con�gurations, the medium to be characterized is

usually bounded between two other media. As shown in Ch. 1, Fig. 1.3, a possible con�guration

for detecting breast cancer is to introduce the medium to characterize in a basin �lled with

intralipid, bounded with a resin. Hence, we will have a di�usive/non-di�usive interface at

the output. In most cases, in order to detect and characterize a hidden object within such

media, this system is approximated as a one-layered medium or as a semi-in�nite homogeneous

medium1. With the aid of the re�ection and transmission coe�cients, however, multiple layered

con�gurations in contact with non-di�usive media can be analytically solved in terms of these

coe�cients, thus speeding up the computation time.

In this Chapter, we solve any multiple layered system, without any approximation. For the

sake of clarity, we shall �rst address the simple system consisting of a slab, which is equivalent

to a three-layered medium, and we shall discuss how it compares with a dielectric slab, a side

consequence being that devising resonator interferometers[11] is not possible with di�use photon

density waves (DPDWs). This is so done for two reasons: �rst, the expression for a slab will be

very useful since in many cases, a stack of multiple layers can be approximated by a system of

slabs with no multiple re�ections between them; second, because a great deal of information,

such as the limiting depth within the layer at which one can detect or characterize an object,

can be recovered, or otherwise, the multiple re�ection contribution, can be extracted. Finally,

we discuss how the solution to an M -layered medium can be extrapolated to solving media

with smoothly varying parameters, and perform a study of how does a smooth varying pro�le

compare to a step boundary.

1These approximations will be compared in Ch. 10, in terms of their accuracy when retrieving the opticalparameters by means of experimental data.

172

CHAPTER 9. MULTI-LAYERED DIFFUSIVE MEDIA 173

9.1 Expression for a Slab

Figure 9.1: Con�guration for a slab, where three di�erent zones are distinguished, with di�erentdi�usive parameters: z � L, 0 � z � L, and z � 0.

Let us address the con�guration depicted in Fig. 9.1, namely, a slab of width L, located

at 0 < z < L: At z > L there is a semi-in�nite homogeneous medium of parameters D0, �a0,

n0, �0, with a source located at zs. At z < 0 we have a semi-in�nite homogeneous medium of

parameters D2, �a2, n2, �2. The slab has D1, �a1, n1 and �1 as parameters. We shall always

consider the normal n at the interfaces pointing in the �z direction, i.e. n = (0; 0;�1). Thereare two well known ways of solving this con�guration: the �rst consists of introducing the

boundary conditions at each of the interfaces and solving the linear system of equations. This

is the method that will be employed for solvingM -layered systems. The second method consists

of successively adding multiple re�ections and transmissions at the interfaces, and since it is

physically illustrating, we shall use it to derive the coe�cients for a slab.

The total wave in the three regions of Fig. 9.1 is in terms of the angular spectrum repre-

sentation (see Sec. 3.2):

A(i)0 (K) +A(K) exp[iq0(K)(z � L)] ; z > L ;

B(K) exp[iq1(K)(L� z)] + C(K) exp[iq1(K)z] ; L > z > 0 ;

D(K) exp[�iq2(K)z] ; L > z > 0 ;


Figure 9.2: Multiple re�ections present at the di�usive slab, and how they contribute to the totalre�ected and transmitted wave.

where A(i)0 (K) = A(i)(K) exp[iq0(K)(zs � L)], and the q's have their usual meaning:

q0(K) =q�20 � jKj2 ;

q1(K) =q�21 � jKj2 ;

q2(K) =q�22 � jKj2 :

Considering all multiple re�ections from the interfaces (see Fig. 9.2), the total re�ection and

transmission coe�cients for the slab are [11, 135]:

RSlab = R01 + T01R12 exp[2iq1L]T10+T01R12 exp[2iq1L]R10R12 exp[2iq1L]T10 + ::: ; (9.1)

T Slab = T01 exp[iq1L]T10 + T01R12 exp[2iq1L]R10 exp[iq1L]T10 + ::: ; (9.2)

where Rij and Tij are the re�ection and transmission coe�cients on going from medium i onto

medium j [100]. Eqs. (9.1) and (9.2) contain a geometric progression, and therefore A; B; C;and D are extracted as:

A =

"R01 +

T01 exp[2iq1L]R12T101�R10R12 exp[2iq1L]

#A(i)

0 ; C =T01 exp[iq1L]R12

1�R10R12 exp[2iq1L]A(i)

0 ; (9.3)

B =T01


0 ;D =T01 exp[iq1L]T12


0 :

From Eqs. (9.3) the re�ection and transmission coe�cients for a slab are de�ned as:

RSlab(K) = R01 +T01 exp[2iq1L]R12T101�R10R12 exp[2iq1L]

; (9.4)


T Slab(K) =T01 exp[iq1L]T12

1�R10R12 exp[2iq1L]: (9.5)

Figure 9.3: Amplitude of RSlab and T Slab at K = 0 for di�erent values of the slab width Lfor the cases: �0s1 = 10cm�1, ! = 0 [solid lines], �0s1 = 20cm�1, ! = 0 [�], �0s1 = 20cm�1,! = 200MHz [Æ]. In all cases �0s0 = �0s2 = 5cm�1, �a0 = �a1 = �a2 = 0:025cm�1, n0 = n1 =n2 = 1:333.

Eqs. (9.4) and (9.5) correspond to a dielectric slab [11]. For a gain medium, they represent

the equations for a Fabry-Perot laser cavity[135], and R10R12 exp[2iq1L] = 1 represents the

oscillation condition. In Eqs. (9.4) and (9.5), the denominator 1�R10R12 exp[2iq1L] takes into

account the multiple re�ections between the interfaces of the slab. Since in general for DPDWs

R10R12 exp[2iq1L]� 1, the geometric progression can be truncated to �rst order, i.e:

RSlab ' R01 + T01 exp[2iq1L]R12T10 ; (9.6)

T Slab ' T01 exp[iq1L]T12 (9.7)

These approximations are very accurate, even for low values of L. In Fig. 9.3 one sees

that for values of L greater than 3cm, the re�ection from the slab is simply R01. Therefore,

R10R12 exp[2iq1L] represents a way of determining the distance at which multiple re�ections

between the walls are important. As a rule of thumb, we can use the following criterion: as-

suming 0:1% noise in the measurements, the limiting value of exp[2iq1L] for absence of multiple

re�ection is 0:1% at best. This means L = � log[10�3]=2�0, for the case ! = 0; K = 0.


Therefore, the limiting value of L is approximately:

Llimit � 3

sD1

�a1= 3Ld1 ; (9.8)

where Ld1 is the di�usion length, de�ned in Sec. 2.3.2. For example, in Fig. 9.3, for �0s1 =

10cm�1 we have that Llimit � 3cm, and for �0s1 = 20cm�1, we obtain Llimit � 2:5cm.

Figure 9.4: Amplitude and Phase of RSlab and T Slab at K = 0 for di�erent values of themodulation frequency ! for the cases: �0s1 = 10cm�1,[RSlab(solid lines) and T Slab(dotted lines)],and �0s1 = 20cm�1[RSlab (�) and T Slab(Æ)]. In all cases �0s0 = �0s2 = 5cm�1, �a0 = �a1 = �a2 =0:025cm�1, n0 = n1 = n2 = 1:333.

For a dielectric slab, the condition R10R12 exp[2iq1L] � 1, conveys the existence of trans-

mission and re�ection resonant peaks. However, this requires values of R10 and R12 close to

unity. This is not the case for a di�usive slab. Also, we must take into consideration that q1

is always complex, and therefore exp[iq1L] represents a lossy medium as it decays with L. The

behavior of RSlab and T Slab versus the modulation frequency f = !=2� is shown in Fig. 9.4,

where we see that for f < 1MHz very little di�erence is observed with the DC (! = 0) case.

For high values of !, we must recall that the imaginary part of q1, and therefore the decay

of exp[2iq1L], grows as !1=2 (see Sec. 3.1). Nevertheless, in Fig. 9.4 we distinguish maxima

near f = 200MHz. These maxima correspond to maximum values of exp[2iq1L], and therefore

cannot be considered as resonant peaks as for the dielectric slab. We therefore conclude that


Fabry-Perot resonators are not possible for DPDWs.

Additional useful information contained in Eq. (9.4) pertains to the maximum distance

between the slab walls from which parameters from medium 2 can be extracted when re�ec-

tion measurements from medium 0 are performed, since all the information from medium 2 is

contained in R12 exp[2iq1L]. In this case, we can also use Eq. (9.8), thus concluding that on

re�ection measurements, it is not possible to characterize an object buried at a distance larger

than Llimit from the �rst interface. By contrast, as seen from Eq. (9.5), this does not occur in

transmission measurements, since information of all the media is present in the wave�eld.

9.2 Solving multiple layered media

Figure 9.5: Multiple layered con�guration of M slabs, where nin and nout are the refractiveindices of the input and output media, respectively, both being non-di�usive. In all cases, thenormal to each interface is considered to point from nin into nout, i.e. along the propagationdirection of the incident wave.

Let us address an M -layered system, as depicted in Fig. 9.5, where nin corresponds to

the refractive index of the non-scattering medium which bounds the medium containing the

source, and nout stands for the refractive index of the non-scattering medium where detection

is performed. An equivalent solution for multiple layered dielectric media can be found in Ref.

[11]. In any of the jth inner media, the total �eld for each frequency component is:

Uj = Aj exp[iqj(zj � z)] + Bj exp[iqj(z � zj+1)] ; zj � z � zj+1 ; (9.9)

where zj is the position of jth interface, i.e.:

zj =k=jXk=1

Lk ;

being Lk the width of medium k. On introducing the saltus conditions for each of the inter-

faces, taking into consideration that the �rst and last are di�usive/non-di�usive, we get to the

following set of equations: hMiM�M

�hXiM�1

=hYiM�1

; (9.10)


where:

M �

0BBBBBBBBBBBBBBBBBBB@

fin �ginE1 0 0 0 0 � � � 0 0

a12E1 b12 �1 �E2 0 0 � � � 0 0

q1D1E1 �q1D1 �q2D2 q2D2E2 0 0 � � � 0 0

0 0 a23E2 b23 �1 �E3 � � � 0 0

0 0 q2D2E2 �q2D2 �q3D3 q3D3E3 � � � 0 0

0 0 0 0...

.... . .

......

......

......

...... � � � �qMDM qMDMEM

0 0 0 0 0 0 � � � �goutEM fout

1CCCCCCCCCCCCCCCCCCCA

(9.11)

X �

0BBBBBBBBB@

A1

B1

...

AM

BM

1CCCCCCCCCA; Y �

0BBBBBBBBB@

gin ~U(i)(K; z = 0)

�a12 ~U (i)(K; z = 0)

0...

0

1CCCCCCCCCA; (9.12)

where:

Ej = exp[iqjLj] ; (9.13)

and:

ajk =

nknj

!2

+ iCjkDjqj ; (9.14)

bjk =

nknj

!2

� iCjkDjqj :

Eqs. (9.14) are the coe�cients for the boundary conditions between the jth and k = j + 1

di�usive media, which in the case nj = nk have the values ajk = bjk = 1 (see Sec.7.1 and Refs.

[100, 109]). For the input and output interfaces we have de�ned:

fin = i�inq1D1 � 1 ; gin = i�inq1D1 + 1 ; (9.15)

fout = i�outqMDM � 1 ; gout = i�outqMDM + 1 ; (9.16)

which when the input interface is black (�in = 0), yield fin = �1, gin = 1. Notice that

gx=fx = Rnd (see Eq. (8.21) in Sec. 8.2). As seen from Eq. (9.10), after solving the linear

system of equations we are left with an M � 1 matrix which represents the values of Aj, Bj foreach frequency component. That is, Eq. (9.10) must be solved for each frequency component.

Once these Aj, Bj coe�cients are found, the total �eld can be built inside any of the slabs by

Eq. (9.9).


9.3 Smoothly varying parameters

Figure 9.6: Values of the smooth varying parameters �a and �0s for the cases: step function[solid line]; variation width, w = 1:0cm [�]; variation width, w = 2:0cm [Æ]; variation width,w = 4:0cm [4]. In all cases, �a0 = 0:025cm�1, �0s0 = 5cm�1, �a1 = 0:075cm�1, �0s1 = 10cm�1,n0 = n1 = 1:333. Source modulated at f = !=2� = 200MHz, located in medium 0, atzs = �3:0cm. Detector in medium 1 at zd = 2:0cm.

By following the steps presented in Sec. 9.2, a system of length L with smoothly varying

parameters can be solved by means of Eqs. (9.10)-(9.12) , by dividing it intoM layers of width:

Lk =L

M;

with M su�ciently large so as to reach convergence. With this procedure we have performed

the following study: a plane abrupt interface between two di�usive media has been compared

to the cases in which this interface is de�ned by smoothly varying parameters. In this case, the

integral equations cannot be numerically solved by means of the method presented in Sec. 4.4,

since we also have volume integrals (see Sec. 4.1). Therefore, a study is needed which gives

us information on when a system with smoothly varying parameters can be approximated by

a step function.

In order to do so, we have studied the con�guration presented in Fig. 9.6. This con�guration

consists of two semi-in�nite di�usive media separated by an interface at z = 0: The medium

located at z < 0 (medium 0) has parameters �a0 = 0:025cm�1, �0s0 = 5:0cm�1, and n0 = 1:333,

and has a modulated point source at f = !=2� = 200MHz located at zs = �3cm. The

medium located at z > 0 (medium 1) has parameters �a1 = 0:075cm�1, �0s1 = 10cm�1, and

n1 = 1:333, and has a detector plane at zd = 2cm. In order to study the e�ect of smoothly


Figure 9.7: Ratio of the total average intensity between the smooth pro�le and the step function,for the amplitude (a), and phase (b), for the following variation widths: w = 1cm [solid line];w = 2cm [�]; w = 4cm [Æ]. Parameters as in Fig. 9.6.

varying parameters, we have generated the pro�les of �a and �0s following the function:

�a(z) = �a0 +(�a1 � �a0)

1 + exp[�12z=w] ;

�0s(z) = �0s0 +(�0s1 � �0s0)

1 + exp[�12z=w] ;

where w is the width of the variation. We have studied four di�erent widths, namely, w = 0,

w = 1cm, w = 2cm, and w = 4cm. As seen, these variations are much wider than what

would be expected within biological tissues. The results are shown in Fig. 9.7, for the ratio

of the average intensity for the smooth pro�le, U (smooth), and the average intensity for the

step pro�le, U (step). Results are shown for the amplitude, jU (smooth)j=jU (step)j, and the phase

�(smooth) � �(step), versus the detector position, performing a scan at zd = 2:0cm. As seen

in this �gure, even in the case in which w = 4cm, we obtain that the error committed if we

approximate the pro�le to a step function is only 5%. This error diminishes as we place the

detector plane zd further away. Therefore, we can infer from Fig. 9.7 that by approximating

the parameter pro�les of biological media to a step function we commit errors that are most

certainly lower than 2%.

Chapter 10

Experimental Results

In this Chapter we submit the theory to test with experimental data. In this way, we shall

characterize a two layered di�usive medium, from images taken with a CCD camera, in the DC

regime (! = 0). To this end, we shall employ the expressions for the re�ection and transmission

coe�cients in di�usive/di�usive interfaces put forward in Chapter 7, and those in di�usive/non-

di�usive interfaces presented in Chapter 3. We will employ the angular spectrum representation

presented in Chapter 8.

It should be stated that the experiment is performed in theDC regime [136] for the following

reasons: if characterization of the parameters is possible in this regime, it is clear that they can

also be obtained in the time-domain or in the AC regime. On the other hand, the experimental

setup for DC measurements is simpler.

10.1 Experimental Setup

Figure 10.1: Side and top view of the experimental con�guration.

In order to test the theory, we measured in a two-layer system composed of an intralipid

layer and a resin slab, as depicted in Fig. 10.1. The resin was polyester with a TiO2 suspension

[137] of length L1 = 2:17cm with optical parameters �0s1 = 10cm�1, �a1 = 0:017cm�1, and

181

CHAPTER 10. EXPERIMENTAL RESULTS 182

D1 [cm] �0s1 [cm�1] �a1 [cm

�1] n10.03333 10 0.017 1.5

Table 10.1: Parameters of the resin.

n1 = 1:5, shown in Table 10.1. The intralipid used was an Intralipid (Kabi Pharmacia, Clayton

Table 10.2: Di�erent parameters used for the intralipid. The value inside the boxes correspondsto the CCD exposure time, in seconds.

NC) emulsion, which consits of polydisperse suspension of fat particles ranging in diameter from

0:1�m to 1:1�m which served as the scattering background medium. This intralipid layer had

a length L0 = 4cm, and parameters �0s0; �a0; n0, for which three di�erent values of �0s0, where

used, namely: �(i=1;2;3)s0 = 5cm�1; 10cm�1, and 20cm�1: For each of these sets �

(i)s0 , �ve values

of �a0 where introduced, namely, �(j=1;::;5)a0 = 0:025cm�1; 0:050cm�1, 0:075cm�1, 0:1cm�1, and

0:125cm�1, constituting a total of �fteen di�erent (�(i)s0 ,�

(j)a0 ) pairs (see Table. 10.2). In all cases,

the refractive index of the intralipid was n0 = 1:333. In order to carry out the experiment, the

necessary amount of intralipid was mixed with water in order to obtain the desied value of �0s0.

Then, the value of �a0 was varied by introducing black India Ink (3080-4 KOH-I-NOOR Inc.

Bloomsbury NY 08804) into the mixture, thus performing a measurement for each (�(i)s0 ,�

(j)a0 )

pair.

As shown in Fig. 10.1, the system had side walls of black PVC. Measurements were per-

formed at the exit of the resin layer. This means that the con�guration had three inter-

faces, namely, black/intralipid, intralipid/resin, and resin/air surfaces. The basin had a height

h = 15cm, width w = 25cm, and total length L = 7:0cm (including the black PVC walls). In

spite of the �nite size of the basin, we will consider the system as being in�nite in the XY -

plane, and thus the expressions put forward in Chapters 7 and 8 will be employed. Illumination

was accomplished by an optical �ber of numerical aperture NA = 0:36, which emitted light of

constant intensity at � = 786nm with a laser diode power � 3mW , located at the basin rear

at (xs = 8:6cm; ys = 8:2cm; zs = 0). We shall model this source by a point source located at

rs = (xs; ys; zs = ltr), with constant illumination, i.e. ! = 0.

Measurements where taken with a liquid nitrogen cooled, 16bit CCD array (Princeton In-

struments), which had a resolution of 330� 1100 pixels which were 24 microns in linear dimen-


sion, each pixel representing an area dx� dy = 0:0287cm� 0:0287cm. The CCD was focused

by lenses on the exit surface of the resin, i.e., at z = L0 + L1 = 6:17cm.

10.1.1 Treating the CCD images

Figure 10.2: Unmodi�ed CCD image, displayed in log-scale, for the �0s = 5cm�1, �a =0:025cm�1 case.

Before performing the �t of the parameters, let us �rst take a look at the images taken. A

typical unprocessed CCD image of the recorded data set is as shown in Fig. 10.2, where the

average intensity received by the CCD is represented in log-scale. We see in this �gure, that the

basin extends in the x-direction from pixel number 200 up to pixel number 800, approximately.

A ruler was placed near the side of the basin in order to calculate pixel sizes, and can be seen

in Fig. 10.2, near the pixel number 800, approximately.

Figure 10.3: A slice of Fig. 10.2 in the x-direction (left) and y-direction (right). Averageintensity. Note the ruler placed near the pixel 800 (right).


On taking a slice of this unmodi�ed image in the x and y directions, we obtain the plots

shown in Fig. 10.3. As seen, noise is well under 1%. Before processing the image, we must �rst

subtract the background noise (see Fig. 10.4), i.e.:

U (data)(R) = U (raw)(R)� U (background)(R) :

Then, the data are apodized. In order to avoid aliasing caused by an abrupt step in the average

intensity, which translates into high frequency oscillations in Fourier space, this consists of

extending the tails of the function, so that they smoothly tend to zero.

Figure 10.4: Slices of a background image in the x-direction (left) and y-direction (right). In-tensity received.

In order to apodize the images, we proceed as follows:

Figure 10.5: Center of symmetry of the image (see Fig. 10.3 for the original image).

1. Since cylindrical symmetry is expected (from a point source and a plane layered con�gu-

ration), we must �rst �nd the center of symmetry Rc of the image.

2. Then we apodize the curve, by �tting the last 75 points of the tails to the function:

f(a; b; c) =

a+

b

jR�Rcj2!exp

"�jR�Rcj

c2

#;


Figure 10.6: Apodized function represented on top of the original data curve, showing no sig-ni�cant di�erences (compare with Fig. 10.3). In this case the slices in the x and y directionare equal.

for the values of a, b, and c. Once this function is found, the tails of the curve are extended

so that the resulting processed images are on the whole 1024x1024 pixels in size.

Figure 10.7: Final processed image in real space for the �0s0 = 5cm�1, �a0 = 0:025cm�1 case.Average intensity in log-scale versus pixel position. Note that now the center of symmetry Rc

is at the pixel (x; y) = (513; 513), and the image is 1024 � 1024 in size. Note that 1pixel '0:0287cm.

10.1.2 Preliminary results

Before proceeding to �t for the parameters of the intralipid, we will test some initial simple

theory with the data in real space. To do so, we shall make use of Eq. (2.71) from Sec. 2.3.2,


which states that the di�usion length of a medium, for ! = 0 is given by:

Ldj =

vuutDj

�aj;

and that in real space, the Green's function in 3D for ! = 0 is given by (see Sec. 3.1.1):

Gj(r� r0) =exp

��r

�ajDjjr� r0j

�4�Djjr� r0j2 :

Figure 10.8: Expected values from Eq. (10.1) [solid line] and experimental data [�] for theaverage intensity measured at the center of symmetry, Rc, versus the di�usion length values ofthe intralipid.

Therefore, we may expect the intensity measured at the center of symmetry Rc (see Fig.

10.5), at the detector plane zd = L0 + L1, i.e. r = (Rc; zd), to be something as:

U (data)(Rc; zd) � exp[�L0=Ld0]

L20D0

� exp[�L1=Ld1]

L21D1

; (10.1)

where Ld0 and Ld1 are the di�usion lengths of the intralipid and resin, respectively. The values

of U (data) at the center of symmetryRc versus the di�usion length of the intralipid Ld0 are shown

in Fig. 10.8. As seen, the average intensity follows the behavior predicted by the approximation

Eq. (10.1). The same results are shown in Fig. 10.9, but showing the behavior predicted by

Eq. (10.1) versus the parameter �a0 of the intralipid.


Figure 10.9: Expected values from Eq. (10.1) [solid line] and experimental data [�], for theintralipid case �0s0 = 5cm�1, versus �a0 .

10.2 Data Analysis

The two layer system shown in Fig. 10.1 can be rigorously solved numerically by using the

method presented in Sec. 9.2, with �in = 0, and �out ' 7:25, corresponding to nout =

1; n1 = 1:5. The three interfaces of the experimental set-up [see Fig. 10.1], two of which

are di�usive/non-di�usive, yield a linear system of four equations and four unknowns. How-

ever, an approximation to the system of Eqs. (9.10) can be introduced by using Eq. (9.8)

in Sec. 9.1, since we expect no contribution from multiple re�ections between the walls of

the intralipid, being L0 = 4cm. Also, the highest value of Llimit is Llimit � 5cm, for the

(�s0 = 5cm�1; �a0 = 0:025cm�1) case, taking into consideration that one of the interfaces

is black. Therefore, we shall derive the expression for a source near a black interface, and

approximate the experimental setup to a source impinging on a di�usive slab (see Sec. 9.1).

We will assume that the parameters of one of the media, namely the resin layer, are known,

and we must therefore �t the values of the intralipid. For the �tting functions, we shall use

three cases, namely, a two layered, a one-layered, and a semi-in�nite medium. For the three

expressions, we shall use zs = ltr = 3D0 = 1=�(i)s0 , which will vary for each �

(i=1;2;3)s0 set.

10.2.1 The incident �eld

In order to solve any multiple layered con�guration, we need an expression for the incident

�eld, i.e. for its angular spectrum representation. This was derived for a source located at zs

in Sec. 3.2.2, but it is here rewritten for convenience. In the angular spectrum representation,


the source wave in the intralipid is (see Fig. 10.1)1:

A(i)(K) =1

4�2

Z +1

�1U (i)(R; z = zs) exp[�iK �R]dR : (10.2)

If we approximate the photon source generated by the �ber by an isotropic point source

U (i)(R; z = zs) = S0 exp[i�0R]=(4�D0R) ; Eq. (10.2) reduces to (see Sec. 3.2.2, and Refs.

[37, 61] for a detailed derivation):

A(i)(K) =S0

4�D0

i

2�q0(K); (10.3)

where S0 is the source strength. By means of Eq. (10.3), the expression for a point source at

any plane z inside the intralipid (i.e. 0 � z � L0, see Fig. 10.1) is:

~U (i)(K; z) =S0

4�D0

i

2�q0(K)exp[iq0(K)jz � zsj] : 0 � z � L0 : (10.4)

Now, considering that the source is close to a black PVC boundary, we may use the re�ection

coe�cient presented in Sec. 8.2.1, and �nd the incident �eld measured at a plane z � 0, due

to a point source located at distance zs from a black boundary as [cf. Eqs. (8.25) and (10.4)]:

~U (i)(K; z) =S0

4�D0

i

2�q0(K)(exp[iq0(K)jz � zsj]� exp[iq0(K)jz + zsj]) : (10.5)

For z > zs Eq. (10.5) can be rewritten as:

~U (i)(K; z) =S0

4�2D0

exp[iq0(K)z]

q0(K)sinh[iq0(K)zs] : (10.6)

We should state that the correct way to represent the source is to include an angular dependent,

or dipolar, term as in Ref. [23] (see Sec. 2.3.2). Nevertheless, since this e�ect decays at long

distances from the source, we will from now on use Eq. (10.5) to model the incident �eld.

10.2.2 Two layer expression

We shall assume an incident wave propagating from the intralipid, whose wavefunction is repre-

sented by Eq. (10.6), which interacts with a slab, i.e. the resin block (see Sec. 9.1). Therefore,

the expected average intensity at the exit of the resin, zd = L0 + L1 = 6:17cm is:

~U2layer(K; zd) = ~U (i)(K; L0)T SlabResin(K) ; (10.7)

1It should be remarked that, as stated in Ch. 3, this expression is only valid when we consider the systemin�nite in the XY plane. However, we will see that with dimensions such as those at hand, it is a goodapproximation.


Figure 10.10: Geometry for the two layer expression.

where ~U (i)(K; L0) is [see Eq. (10.6)]:

~U (i)(K; L0) =S0

4�2D0

exp[iq0(K)L0]

q0(K)sinh[iq0(K)zs] ;

T SlabResin is given by (see Sec. 9.1):

T SlabResin(K) =

T01(K) exp[iq1(K)L1]T1out(K)

1�R10(K)R1out(K) exp[2iq1(K)L1]; (10.8)

and we have introduced the coe�cients (see Secs. 7.2 and 8.2):

R10(K) =n21D1q1(K) (1� iC1!0

n D0q0(K))� n20D0q0(K)

n21D1q1(K) (1� iC1!0n D0q0(K)) + n20D0q0(K)

; (10.9)

T01(K) =2n21D0q0(K)

n20D0q0(K) (1� iC0!1n D1q1(K)) + n21D1q1(K)

; (10.10)

R1out(K) =i�outD1q1(K) + 1

i�outD1q1(K)� 1; (10.11)

T1out(K) =2i�outD1q1(K)

i�outD1q1(K)� 1: (10.12)

In the DC regime, ! = 0, the q's have the expressions:

q0(K) = i

s�a0D0

+ jKj2 ; (10.13)

q1(K) = i

s�a1D1

+ jKj2 ; (10.14)

since in this case �j =q��aj=Dj (see Sec. 3.1). We remark that Eq. (10.7) has been contrasted

with the exact solution obtained from the method presented in Sec. 9.2, �nding no noticeable

di�erences. In Eqs. (10.9) and (10.10), C1!0n and C0!1

n , are the coe�cients which take into


account the Fresnel re�ection at the di�usive/di�usive interface (see Sec. 7.1, Fig. 7.2), on

going from the resin to the intralipid, and from the intralipid to the resin, respectively.

Since in the resin case, the value of Llimit = 3qD1=�a1 = 3Ld1 (see Sec. 9.1), where Ld1

is the di�usion length of the resin, is Llimit ' 4cm, and we have L1 = 2cm, we shall use the

complete expression for transmission from a slab, Eq. (10.8). Even so, the di�erence from the

approximate expression presented in Sec. 9.1, T SlabResin ' T01 exp[iq1L1]T1out, is very small.

10.2.3 One layer expression

Figure 10.11: Geometry for the one layer expression.

In the same manner we derived in Sec. 10.2.2 the equation for a two-layer system, the

expression for the one-layer medium is obtained by considering that the parameters of the resin

(see Fig. 10.1) are the same as those of the intralipid, and therefore, the expected average

intensity at the exit of the system (see Fig. 10.11), zd = L0 + L1 = 6:17cm is:

~U1layer(K; zd) = ~U (i)(K; L0 + L1)T0out(K) ; (10.15)

where ~U (i)(K; L0 + L1) is [see Eq. (10.6)]:

~U (i)(K; L0) =S0

4�2D0

exp[iq0(K)(L0 + L1)]

q0(K)sinh[iq0(K)zs] ;

and T0out is given by [see Eq. (10.12)]:

T0out(K) =2i�0

outD0q0(K)

i�0outD0q0(K)� 1

: (10.16)

In this case, the value of the coe�cient which takes into account the refractive index mismatch,

�0out, has a value di�erent than the one in Sec. 10.2.2, since now it is de�ned from n0 = 1:333

to nout = 1, and therefore � ' 5. The expression for q0 is as given in Eq. (10.13).


10.2.4 Semi-in�nite expression

Figure 10.12: Geometry for the semi-in�nite expression.

Like we derived the expression for a two-layer and one-layer systems in Sections 10.2.2

and 10.2.3, respectively, the expression for a semi-in�nite medium is obtained by assuming

semi-in�nite the intralipid (see Fig. 10.12). Therefore, the expected average intensity at zd =

L0 + L1 = 6:17cm is:

~U(K; zd) = ~U (i)(K; L0 + L1) =S0

4�2D0

exp[iq0(K)(L0 + L1)]

q0(K)sinh[iq0(K)zs] : (10.17)

10.2.5 Fitting the data

Figure 10.13: Fourier spectrum corresponding to �s0 = 5cm�1, �a0 = 0:025cm�1. As seen, athigh frequencies we have noise contribution. Therefore we shall cut our data at Kcut, namelyat Kcut = 1:2cm�1 (see Sec. 5.4).


In order to �t for �(j=1;:::;5)a0 and �

(i=1;2;3)s0 , we �rst perform the 2D Fourier transform, ~U

(i;j)data(K),

of the CCD measurements, U(i;j)data(R):

~U(i;j)data(K) =

1

4�2

Z +1

�1

Z +1

�1U

(i;j)data(R) exp[�iK �R]dxdy ;

corresponding to the (�(i)s0 ; �

(j)a0 ) pair. In order to do so, we may take advantage of the cylindrical

symmetry and perform a Hankel transform of U(i;j)data(R), thus speeding up the computation time

by a power of two (see Sec. 3.2.2):

~U(i;j)data(K) =

1

2�

Z +1

�1U

(i;j)data(R)J0(KR)RdR ;

where R = jRj and K = jKj. We shall use a cuto� frequency Kcut = jKcutj = 1:2cm�1

to separate noise from the data. Then, we normalize all (�(i=1;2;3)s0 ; �

(j=1;:::;5)a0 ) sets to the value

Figure 10.14: Fitting procedure for �a0 and �0s0.

corresponding to (�(i=1)s0 = 5cm�1; �

(j=1)a0 = 0:025cm�1) atK = 0, �nally �tting for both �

(i=1;2;3)s0

and �(j=1;:::;5)a0 (see Fig. 10.14 for the �tting process). In this way, we make the assumption that

we only have one reference measurement for which the values �0s0 and �a0 are known. In all

cases, the only restriction that we impose is that each set �(j=1;:::5)a0 is associated to the same

value of �(i)s0 . The �tting procedure employed is the steepest descent method, in which we shall


always use �a0 = 0:1cm�1, D0 = 0:1cm, i.e. �s0 = 3:33cm�1, as initial values2. The function to

minimize is:

f(�(i)s0 ; �(j=1;:::;5)a0 ) =

XK

U (i;j=1;:::;5)theory (K)� U

(i;j=1;:::;5)data (K)

2 ; (10.18)

where the subindex �theory� stands for either �2layer�, �1layer�, or �semi � inf � [cf. Eqs.

(10.7) , (10.15), and (10.17)] depending on the expression used to �t, and U(i;j=1;:::;5)data represents

the Fourier data for the set (�(i)s0 ; �(j=1;:::;5)a0 ) normalized to the K = 0 value corresponding

to (�s0 = 5cm�1; �a0 = 0:025cm�1). In order to de�ne the accuracy of each expression for

retrieving the parameters, we de�ne the average error for the retrieval of �a0 as:

��a0 =1

5

5Xj=1

�founda0 (j)

�expecteda0 (j)� 1

!� 100 ; (10.19)

where �expecteda0 (j = 1:::5) = (0:025cm�1; :::; 0:125cm�1) are the expected values of �a0, and

�founda0 (j = 1:::5), are those �tted from Eqs. (10.7), (10.15), and (10.17).

10.3 Results and Discussion

Figure 10.15: Fitted values of U2slab(K) [�], U 1slab(K) [4], and U semi�inf(K) [?], comparedto the data values U data(K) [solid line] versus the frequency K, for the case �0s0 = 10cm�1,�a0 = 0:075cm�1. See Eqs. (10.7) and (10.15).

Fig. 10.15 shows the �tted curves of U 2layer, U 1layer, and U semi�inf versus the frequency

K, contrasted with the data set Udata for the case (�0s0 = 10cm�1; �a0 = 0:075cm�1). As

2We have found that the results are independent of the chosen initial values. This is a very important fact,since in practice one has no estimation of the actual parameters of the medium.


shown, the expressions U 2layer, U1layer, and U semi�inf yield very close �ts to the experimental

data U data, even at high values of the frequency K ' 1:2cm�1. However, as will be seen in

the reconstructions, the values of (�founds0 ; �founda0 ) which yield the curves in Fig. 10.15, when

using the one layer expression, Eq. (10.15), or the semi-in�nite expression, Eq. (10.17), are

far from the expected values. On the other hand, we obtain values very close to the expected

values when using the two layer expression, Eq. (10.7). The results obtained from the �tting

Figure 10.16: Fitted values using: The two-layer expression, Eq. (10.7), (a) �0s0 = 5cm�1; (b)�0s0 = 10cm�1; (c) �0s0 = 20cm�1. The one layer expression, Eq. (10.15), (d) �0s0 = 5cm�1; (e)�0s0 = 10cm�1; (f) �0s0 = 20cm�1. The semi-in�nite expression, Eq. (10.17), (g) �0s0 = 5cm�1;(h) �0s0 = 10cm�1; (i) �0s0 = 20cm�1.

procedure are presented in Fig. 10.16 and Table 10.3, which show that the values of both �(i)s0

and �(j)a0 are retrieved with remarkable accuracy with the two layer expression. On the other


hand, both the one layer and the semi-in�nite expressions yield acceptable reconstructed values

for �(j)a0 [see Figs. 10.16 (d)-(i)] but fail to retrieve �

(i)s0 (see Table 10.3, and Fig. 10.17). We

�expecteds0 �2layers0 ��2layera0 �1layers0 ��1layera0 �semi�infs0 ��semi�inf

a0

5:0 cm�1 4:1cm�1 33% 9:9cm�1 72% 2:4cm�1 31%10:0 cm�1 7:0cm�1 45% 4:7cm�1 60% 3:8cm�1 54%20:0 cm�1 16:8cm�1 7% 7:3cm�1 60% 6:9cm�1 50%

Table 10.3: Comparison of the �tted values obtained with Eqs. (10.7), (10.15), and (10.17):two-layered medium, one layered medium, and semi-in�nite medium, respectively. �theorys0 are

the �tted values for �s0, and ��theorya is the average error of retrieval of �a0, Eq. (10.19).

should draw attention to the values of ��a0 shown in Table 10.3. In this table, we show that the

average error of �a0 retrieval can amount up to 45%, for the two layer expression. If we look at

Fig. 10.16(b) which shows the retrieved values of �founda0 versus the expected values of �expecteda0 ,

we see that even in the case in which ��a0 = 45% (which corresponds to �s0 = 10cm�1), the

relative values of �(j)a0 are correct. That is, if the exact value of one of them is known, then

all the other lie correctly in place, yielding a very small value of ��a0. Also, as shown in Fig.

Figure 10.17: Fitted �0s0 values using: The two-layer expression, Eq. (10.7) [�]; The one layerexpression, Eq. (10.15) [4]; The semi-in�nite expression, Eq. (10.17) [?]. Refer to Table 10.3for the values of ��a0.

10.17, the retrieved values of �0s0 are all within a 30% error for the two layer expression, whereas

for the one layer and semi-in�nite expressions, they can mount up to 70%.


Figure 10.18: Values of the �tting function, Eq. (10.18), versus D0 and �a0 for the two layerexpression, Eq. (10.7), in the case �a0 = 0:025cm�1, �0s0 = 5cm�1. Note the existence of a�valley� which corresponds to the minimum values of Eq. (10.18).

10.4 Fitting independent measurements

We have shown in Sec. 10.3 how the media characterization can be performed by means of

the re�ection and transmission coe�cients for DPDWs. In order to �t for the parameters, we

imposed a single restriction: that each �(j)a0 set had the same value for �(i)s0 . This of course, limits

our space of solutions, not only yielding good �tted values, but also independence of initial guess

values. We shall now consider the case in which we have one independent measurement, from

which we do not know the value neither of �a0 nor �0s0, and have no other reference measurement

to contrast it with. As will be shown, in this case we �nd ourselves in a much wider space of

solutions, from which it will be di�cult to extract the values of both �a0 and �0s0, since the main

parameter now will be Ld0 =qD0=�a0. In order to get an idea of the space of solutions we are

dealing with, we plot in Fig. 10.18 the values of the �tting function, Eq. (10.18), for the two

layer expression, Eq. (10.7), versus the intralipid values of D and �a, for the �a0 = 0:025cm�1,

�0s0 = 5cm�1 case. As shown, there exists a �valley� of minimum values of Eq. (10.18). This

valley exactly corresponds to the values:

D(�a) = a+

sD0

�a0�a ;

a being a constant. In Fig. (10.19) we plot the minimum values of Eq. (10.18), versus �a0.

As shown, not all values of the D0, �a0 pairs near the �valley� give the same value for the

�tting function, Eq. (10.18), �nding an absolute minimum, and many relative minima. These

local minima are those that obstruct the steepest descent method from reaching the absolute

minimum, and will make our �tted values for �0s0 and �a0 dependent on the initial values.


Figure 10.19: Same as Fig. 10.18, but for the minimum values located at the �valley�.

Since in this case we have solutions which are dependent on the chosen initial values for

�a0 and �0s0, we shall characterize the intralipid by means of Eq. (10.7) by �tting the values

of �a0 and �0s0 to Eq. (10.18), but using initial values close to the expected ones. The main

purpose of this procedure is to prove that independent measurements can also be used to

characterize di�usive media, even in theDC regime. More powerful non-local �tting procedures,

independent of the initial value, require methods such as simulated annealing [138] (with the

computational cost it conveys).

Figure 10.20: Fitted values for �a0 and D0 from independent measurements, by using the twolayer expression. No restriction imposed on the �tting function 10.18. Initial values close tothe expected ones.

The results are shown in Fig. 10.20, where the values of �a0 and D0 are retrieved with

remarkable accuracy. However, it must be noted that the cases corresponding to high values

of Ld0, namely D0 = 0:01666cm, �a0 = 0:150cm�1, and D0 = 0:0333cm, �a0 = 0:150cm�1,

where not retrieved, yielding unrealistic �tted values. We must state that these �ts where also

performed using the one layer, Eq. (10.15), and semi-in�nite, Eq. (10.17), expressions but both

failed to recover the parameters in most of the cases.

Chapter 11

Conclusions

In this thesis, we have put forward a theory of di�use photon density waves (DPDWs) and have

studied their propagation and interaction with di�erent interfaces. This scheme was derived

from �rst principles, using radiative transfer as the starting equation, and the approximations

taken to derive the di�usion approximation, and their implications, were discussed. Two meth-

ods which rigorously solve the integral equations corresponding to the di�erent cases studied,

were presented, namely, one which makes use of the extinction and Helmholtz-Kirchho� the-

orems, and thus solves for the surface boundary values, and one that can solve any multiple

layered system by means of the re�ection and transmission coe�cients. As in most cases, the

main objective of the development of this theory was to predict and explain the great number

of experiments which use light to probe biological media, with special emphasis in the ability

to optically characterize such media. Focused on this last issue, the theory established was

successfully employed to characterize a di�usive medium from experimental data. Of special

importance is the fact that this characterization was done in the DC regime, i.e. ! = 0, there-

fore proving that within this regime, not only we do not loose resolution, as shown in Chapter

5, but in addition, optical characterization is possible. This means cheaper, portable, and sim-

pler experimental setups, that will be of importance for the development of these techniques in

medical applications.

In order to fully understand the conclusions to the work carried out in this thesis, we

shall present them in the same context as they were derived, namely, as regards propagation,

interaction, and characterization:

1. Propagation

(a) In Chapter 3 we studied the propagation of di�use photon density waves and derived

their spatial resolution limit. In most practical cases, the plane of measurement of

these waves is always located at a distance in the range of one wavelength from

the source or object. In this context, we found that the spatial resolution of the

DPDWs is in most cases better, than with propagating scalar waves (PSWs). Also of

198

CHAPTER 11. CONCLUSIONS 199

relevance is the fact that spatial resolution does not signi�cantly increase when the

modulation frequency ! grows, with the consequence that we may work in the DC

regime, ! = 0, without loss of spatial resolution. However, this leaves an important

question open which was later answered: Is the simultaneous characterization of the

di�usion and absorption coe�cient possible in the DC regime?

(b) In Chapter 4 we derived the di�raction formulas for DPDWs, �nding that there

is practically no interference between these waves from apertures. This fact was

further studied in the context of interaction between interfaces.

(c) We have studied in Chapter 5 the e�ect of noise present in the detected wave�eld, and

have seen that, due to high damping, DPDWs are concentrated at low frequencies,

and therefore high noise levels can be �ltered out with minimum loss of information.

This constitutes a great advantage for back-propagation procedures of the �eld up to

the object, thus increasing the resolution of the images, and is of major importance

when tomography with DPDWs si considered.

2. Interaction

(a) We have addressed two main interfaces with which the DPDWs interact, namely,

a di�usive/di�usive interface, and a di�usive/non-di�usive interface, for which the

re�ection coe�cients were derived in Chapters 6, 7, and 8. In particular we have

studied in Chapter 7 the e�ect of index mismatch between both media, �nding

that no matter how slight this index mismatch is, it must always be considered.

We have put forward an approximation to the boundary conditions between di�u-

sive/di�usive index mismatched interfaces, showing that, for the expected values in

biological media, it yields errors lower than 5%. The use of this approximation will

be of importance when characterizing buried objects by means of the Rytov or Born

approximation.

(b) We have seen in Chapters 6, 7, and 8, that multiple re�ections between interfaces

have a very small contribution, a fact which considerably simpli�es the development

of powerful inverse scattering techniques. These low re�ectivities were studied by

introducing the re�ection and transmission coe�cients.

(c) As regards the re�ection and transmission coe�cients, we have found the existence

of zero re�ection states, analogous to the Brewster modes in electromagnetic the-

ory, both at di�usive/di�usive and di�usive/non-di�usive interfaces. The existence

of these zero re�ection frequencies must be fully considered, since they drastically

change the shape of the re�ected DPDWs. In the future, these modes may be of

great importance to develop DPDWs lenses and �lters, in a manner such as that

presently employed for electromagnetic waves. Also, in di�usive/di�usive interfaces,


we have found that under certain conditions, the interface may act as a di�usive mir-

ror, equally re�ecting and transmitting all frequencies, and thus yielding frequency

independent re�ection and transmission coe�cients.

(d) Concerning more complex geometries, we studied in Chapter 8 the way to model,

within the di�usion approximation, the interaction of DPDWs with non-di�usive

regions embedded within a di�usive medium. This is of major importance for light

propagation studies in complex biological media such as the brain, where we �nd

areas �lled with a non-di�usive liquid, namely, the cerebro spinal �uid. We have

thus derived a compact expression for the boundary conditions for such geometries,

and have contrasted it with Monte Carlo calculations, �nding excellent agreement

between both. The e�ect of roughness in these non-di�usive volumes was studied,

�nding that knowledge of the actual rough pro�le is not necessary. However, the

statistics of the interface pro�le must be known to correctly predict the wavefront

shape of the scattered DPDW. We found that the overall e�ect of roughness of non-

di�usive interfaces was to decrease the transmitted average intensity, redistributing

the light almost isotropically.

3. Characterization

(a) We have found in Ch. 10 that characterization of multiple layered media can be

accomplished by means of the re�ection and transmission coe�cients for DPDWs.

We have experimentally performed such characterization in the DC regime, thus

proving that it is possible to characterize simultaneously both the di�usion and the

absorption coe�cients even when ! = 0.

(b) Having demonstrated the applicability of the re�ection and transmission coe�cients,

their use will be of great relevance for �nding more accurate Green's functions which

will increase the accuracy of the characterization of small di�usive objects embedded

in complex geometries. A practical instance of such media is the breast, when

measurements are performed in the compressed breast con�guration. In this case,

we deal with M layered di�usive media, and the objective is to detect, localize, and

characterize a strange object embedded in one of them. It is in these cases where

the use of the re�ection and transmission coe�cients will be of most importance.

11.1 Future Perspectives

The use of di�use light to detect, localize, and characterize lesions present in biological tissue

is a novel technique1, which in the last few years has been subject of intense activity. Re-1As seen in Ref. [3] the use of light as a diagnostic aid has been used since the mid-1800s. However, due

to the existing limitations, interest rapidly decreased because of the poor spatial resolution of the thechniquesemployed.


cently, experimental setups which employ near-infrared light are being developed with direct

clinical applications, such as breast cancer diagnosis [83]. Even so, the establishment of more

quantitative criteria between theory and controlled experiments is needed. Also, experiments

which study the e�ect of index mismatch in non-di�usive volumes within di�usive media, need

to be performed in order to develop accurate inverse methods for brain imaging [129].Theories

which go beyond the di�usion approximation, such as the Pn approximation to the radiative

transfer equation, need to be developed for an arbitrary number of volumes and geometries.

At the moment, codes which use high orders of the Pn can only be employed for simple ge-

ometries, such as planes and cylinders, and for a limited number of volumes. Also, another

issue of great importance still to be developed, is the connection between Maxwell's equations

in electromagnetics and the radiative transfer equation.

The clue factor for accurate characterization of inclusions, is the development of powerful

inverse methods. At the present moment, complex �nite element methods (FEM) [72, 139] are

being employed to �nd the solution to the inverse problem, with encouraging results. In order

to increase the accuracy of such procedures, a more rigorous treatment of the forward problem

is needed, by means of the theory presented in this work, or higher approximations to the RTE.

In the same context, scattering and di�raction tomography [63] must be developed based on

rigorous light propagation models.

As shown in Ch. 11, it is possible to recover the optical parameters of di�usive media from

one independent DC measurement. In order to do so, more powerful reconstruction methods

such as simulated annealing must be employed. Therefore, we �nd that at the same time a

more rigorous solution to the forward problem needs to be developed, in most cases a more

powerful (and thus, time consuming) reconstruction method must be applied. Since one of the

main objectives of optical imaging is to provide information in real time, the forward problem

solver must also be optimized [140]. Most importantly, the accuracy of the inversion methods

must be contrasted with controlled and in vivo experiments.

Appendix A

List of Symbols

Notation Description Units

d Unit solid angle sr

I(r; s) Speci�c Intensity Wcm�2sr�1

du(r) Energy density within d Jcm�3sr�1

u(r) Energy density Jcm�3

U(r) Average Intensity Wcm�2

J Total �ux density Wcm�2

J+;� Upward/downward �ux Wcm�2

Jn;t Total normal/tangential �ux Wcm�2

dp(r; s) Power in direction s W

dP (r) Total Power at r W

�(r; s) Speci�c intensity radiated by the medium Wcm�2sr�1

E(r) Total generated power density Wcm�3

Ea(r) Total absorbed power density Wcm�3

JE(r) Generated �ux density Wcm�2

rk;? Fresnel re�ection coe�cients -

tk;? Fresnel transmission coe�cients -

R Power re�ectance -

T Power transmittance -

R Di�usive/di�usive re�ection coe�. -

T Di�usive/di�usive transmission coe�. -

Rnd Di�usive/non-di�usive re�ection coe�. -

Tnd Di�usive/non-di�usive transmission coe�. -

RU;J Avg. intensity/�ux re�ectivity -

F (K; z) Transfer function -

H(R; z) Impulse response -

202

APPENDIX A. LIST OF SYMBOLS 203

Notation Description Units

p(s; s0) Phase function -

g Average cosine -

W0 Albedo -

�a;s Absorption/scattering cross-section cm2

�t Total macroscopic cross-section cm�1

�a;s Absorption/scattering coe�cient cm�1

�0s Reduced scattering coe�cient cm�1

D Di�usion coe�cient cm

lsc Scattering mean free path cm

ltr Transport mean free path cm

la Absorption length cm

Ld Di�usion length cm

L! Decay length cm

Le Extrapolated distance cm

Llimit Limiting distance between layers cm

�0 Wavelength of DPDW cm

Vp Phase velocity cms�1

� Complex wave number for DPDW cm�1

! Modulation frequency Hz

f = !=2� Modulation frequency Hz

k Wave vector cm�1

q Z-Projection of wave vector cm�1

K XY-Projection of wave vector cm�1

Appendix B

Summary of boundary conditions

B.1 Di�usive/di�usive interfaces

The general boundary conditions in the presence of index mismatch are:

U1(r)jS ��n1n0

�2U0(r)jS = CnJn(r) ;

Jn = �D0@U0

@n

��S

= �D1@U1

@n

��S

;

where: Cn =2�R1!0

J�R0!1

J

R1!0U

, and the unit surface normal n is de�ned pointing outwards from

volume S. In this case the integral equations reduce to:

� r > S(x)

U0(r) = U (i)(r) +1

4�

ZS(x)

(U0(r

0)@G(�0jr� r0j)


0)

@n0

)dS 0 :

� r < S(x)

U1(r) = � 1

4�

ZS(x)

(�n1n0

�2 @G(�1jr� r0j)@n0

U0(r0) �"

D0

D1G(�1jr� r0j) + CnD0

@G(�1jr� r0j)@n0

#@U0(r

0)

@n0

)dS 0 :

In this case, the re�ection and transmission coe�cients have the expression:

R(K) =n20D0

q�20 � jKj2

�1� iCnD1

q�21 � jKj2

�� n21D1

q�21 � jKj2

n20D0

q�20 � jKj2

�1� iCnD1

q�21 � jKj2

�+ n21D1

q�21 � jKj2

;

T (K) =2n21D0

q�20 � jKj2

n20D0

q�20 � jKj2

�1� iCnD1

q�21 � jKj2

�+ n21D1

q�21 � jKj2

;

204

APPENDIX B. SUMMARY OF BOUNDARY CONDITIONS 205

R(K) +D1

q�21 � jKj2

D0

q�20 � jKj2

T (K) = 1 :

B.1.1 Approximate boundary conditions

In the case in which 1:3 < n0 < 1:5; case the boundary conditions can be approximated to:

U1(r)jS '�n1n0

�2U0(r)jS ;

�D0@U0(r)

@n

��S

= �D1@U1(r)

@n

��S

;

for which the integral equations are expressed as:

� r > S(x)

U0(r) = U (i)(r) +1

4�

ZS(x)

(U0(r

0)@G(�0jr� r0j)


0)

@n0

)dS 0 :

� r < S(x)

U1(r) = � 1

4�

ZS(x)

(�n1n0

�2U0(r

0)@G(�1jr� r0j)

@n� D0


0)

@n0

)dS 0 :

The re�ection and transmission coe�cients can thus be rewritten as:

R(K) ' n20D0

q�20 � jKj2 � n21D1

q�21 � jKj2

n20D0

q�20 � jKj2 + n21D1

q�21 � jKj2

;

T (K) ' 2n21D0

q�20 � jKj2

n20D0

q�20 � jKj2 + n21D1

q�21 � jKj2

;

R(K) +D1

q�21 � jKj2

D0

q�20 � jKj2

T (K) = 1 :

B.1.2 Index matched conditions

In this case the boundary conditions recover the most commonly used expression:

U1(r)jS = U0(r)jS ;

�D0@U0(r)

@n

��S

= �D1@U1(r)

@n

��S

;

for which the integral equations are expressed as:


� r > S(x)

U0(r) = U (i)(r) +1

4�

ZS(x)

(U0(r

0)@G(�0jr� r0j)


0)

@n0

)dS 0 :

� r < S(x)

U1(r) = � 1

4�

ZS(x)

(U0(r

0)@G(�1jr� r0j)

@n� D0


0)

@n0

)dS 0 :

The re�ection and transmission coe�cients can thus be rewritten as:

R(K) =D0

q�20 � jKj2 �D1

q�21 � jKj2

D0

q�20 � jKj2 +D1

q�21 � jKj2

;

T (K) =2D0

q�20 � jKj2

D0

q�20 � jKj2 +D1

q�21 � jKj2

;

R(K) +D1

q�21 � jKj2

D0

q�20 � jKj2

T (K) = 1 ; R(K) + 1 = T (K) :

B.2 Di�usive/Non-di�usive interfaces

De�ning the average intensity inside the di�usive and non-di�usive volumes as U0 and U1,

respectively, U1 can in general be found from:

U1(r) =1

�

ZS

"R0!1

U

U0(r0)

2+R0!1

J

Jn(r0)

2

#�(r� r0)dS 0 r < S(x) ;

�(r� r0) = exph��a1 + i!n1

c

�jr� r0j

ijr� r0j2 cos �0 :

The corresponding re�ection and transmission coe�cients are de�ned as:

Rnd(K) =i�D0

q�20 � jKj2 + 1

i�D0

q�20 � jKj2 � 1

;

Tnd(K) =2i�D0

q�0 � jKj2

i�D0

q�0 � jKj2 � 1

;

Rnd(K) + 1 = Tnd(K) :


B.2.1 Convex or plane surfaces (no light re-entering)

The general boundary conditions in the presence of index mismatch are:

U0(r)jS = �Jn(r) ;

Jn = �D0@U0

@n

��S

;

where: � =2�R0!1

J

R0!1U

, the surface unit normal n pointing into the non-di�usive volume. In this

case the integral equations reduce to:

� r > S(x)

U0(r) = U (i)(r)� 1

4�

ZS(x)

U0(r0)

(@G(�0jr� r0j)

@n+

1

�D0

G(�0jr� r0j))dS 0 :

B.2.2 Non-di�usive within di�usive volumes

In this case the boundary conditions are expressed as:

U0(r) = �Jn(r) +1

�

ZS

�U0(r

0) +RJ

RU

Jn(r0)�G(r� r0)dS 0 r 2 S ;

where:

G(r� r0) =�1� jR1!0(�)j2

� exp h��a1 + i!n1c

�jr� r0j

ijr� r0j2 V(r� r0) cos �0 cos � ;

cos � = n � (r0 � r)

jr0 � rj ; cos �0 = n0 � (r� r0)

jr0 � rj ;

and the corresponding integral equation in the di�usive volume is:

� r > S(x)

U0(r) = U (i)(r) +1

4�

ZS(x)

(U0(r

0)@G(�0jr� r0j)

@n+

1

D0G(�0jr� r0j)Jn(r0)

)dS 0 :

List of Publications

Most of the work in this thesis can also be found at the following references:

[1] J. Ripoll and M. Nieto-Vesperinas, "Scattering Integral Equations for Di�usive

Waves. Detection of Objects Buried in Di�usive Media in the Presence of Rough Interfaces",

J. Opt. Soc. Am. A. 16, pp. 1453-1465 (1999).

[2] J. Ripoll, M. Nieto-Vesperinas, and R. Carminati, �Spatial resolution of di�use

photon density waves�, J. Opt. Soc. Am. A. 16, pp. 1466-1476 (1999).

[3] J. Ripoll and M. Nieto-Vesperinas, �Re�ection and transmission coe�cients for

di�use photon density waves�, Opt. Lett. 24, pp. 796-798 (1999).

[4] J. Ripoll and M. Nieto-Vesperinas, "Index Mismatch for Di�use Photon Density

Waves both at Flat and Rough Di�use-Di�use Interfaces", J. Opt. Soc. Am. A. 16 pp. 1947-

1957, (1999).

[5] J. Ripoll, S. R. Arridge, H. Dehghani, and M. Nieto-Vesperinas, �Boundary

conditions for light propagation in di�usive media with non-scattering regions�, J. Opt. Soc.

Am. A. 17 pp. 1671-1681 (2000).

[6] J. Ripoll, V. Ntziachristos, J. P. Culver, D. N. Pattanayak, A. G. Yodh,

and M. Nieto-Vesperinas, �Recovery of optical parameters in multiple layered di�usive

media: Theory and experiments�, to be published in J. Opt. Soc. Am. A. (2000).

[7] J. Ripoll, S. R. Arridge, and M. Nieto-Vesperinas, �E�ect of roughness in non-

di�usive regions within di�usive media�, to be published in J. Opt. Soc. Am. A. (2000).

Other publications in refereed journals:

[8] J. Ripoll, L.E. Bausa, C. Terrile, J. Garcia Sole and F.Diaz, �Optical Spec-

troscopy of Nd3+ doped KGd(WO4)2 monocrystals� , J. of Lumin. 72 pp. 253-254 (1997).

208


[9] J.Ripoll, A. Madrazo and M.Nieto-Vesperinas, �Scattering of electromagnetic

waves from a body over a random rough surface� , Opt. Comm. 142, pp.173-178 (1997).

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