July 15, 2008 / Vol. 33, No. 14 / OPTICS LETTERS 1581

Simulation of polarization-sensitiveoptical coherence tomography images

by a Monte Carlo method

Igor Meglinski,1,* Mikhail Kirillin,2 Vladimir Kuzmin,3 and Risto Myllylä2

1Cranfield Health, Cranfield University, Cranfield, MK43 0AL, UK2Optoelectronics and Measurement Techniques Laboratory, University of Oulu, P.O. Box 4500, 90014, Oulu, Finland

3St. Petersburg Institute of Commerce and Economics, 194021 St. Petersburg, Russia*Corresponding author: i.meglinski@cranfield.ac.uk

Received April 28, 2008; accepted May 27, 2008;posted June 16, 2008 (Doc. ID 95558); published July 11, 2008

We introduce a new Monte Carlo (MC) method for simulating optical coherence tomography (OCT) images ofcomplex multilayered turbid scattering media. We demonstrate, for the first time of our knowledge, the useof a MC technique to imitate two-dimensional polarization-sensitive OCT images with nonplanar bound-aries of layers in the medium like a human skin. The simulation of polarized low-coherent optical radiationis based on the vector approach generalized from the iterative procedure of the solution of Bethe–Saltpeterequation. The performances of the developed method are demonstrated both for conventional andpolarization-sensitive OCT modalities. © 2008 Optical Society of America

OCIS codes: 110.4500, 290.0290, 290.5855.

Since Wilson and Adam [1] first introduced a MonteCarlo (MC) method into the field of laser-tissue inter-action, this technique has been used extensively in anumber of studies of photon migration for diverse op-tical diagnostic applications [2–5], including opticalcoherence tomography (OCT) [6–8]. Recently a MCtechnique has been widely used to simulate coherentphenomena of multiple scattering [9–11] and thechanges of polarization of optical radiation scatteredwithin the biological media [12–14]. Typically, withinthe MC algorithms the state of the polarization andits evolution during the propagation is described inframework of Stokes–Mueller or Jones formalism[12–14].

In this Letter, we introduce a new MC techniquethat is able to imitate two-dimensional (2D)polarization-sensitive OCT images of complex multi-layered turbid scattering media avoiding implemen-tation of Stokes–Mueller and/or Jones formalism.The concept of the approach is based on the develop-ment of a unified MC program as a natural extensionfor the most popular standard MC code [2].

Following this we consider propagation of photonpackets along each possible trajectory within the me-dium. The free photon path between two successivescattering events is governed by Poisson distribution:f�li�=�s exp�−�sli�, and defined as li=−�s

−1 ln �, where� is the probability that the mean free path l is noless than li, arbitrary � value is chosen in the [0,1] in-terval using a random number generator, and �s isthe scattering coefficient: �s= l−1. A new direction ofthe photon packet after each scattering event is de-termined by the Henyey–Greenstein scattering phasefunction [2], originally developed to approximate Miescattering of light from particles with size compa-rable with the wavelength of the incident light. Thisexpression is characterized by a so-called anisotropyfactor g, which is equal to the mean cosine of the

scattering angle � g=cos � (g� �0,1�, with g=0 corre-

0146-9592/08/141581-3/$15.00 ©

sponding to the isotropic scattering). A considerationof light scattering within an absorbing medium ��sl�1� requires a proportional reduction of the statisti-cal weight W of each photon packet according its tra-jectory [15]: W=W0 exp�−�i=1

N �ali�, where W0 is theinitial photon packet weight, �a is the absorption co-efficient of the medium, and N is the number of ex-perienced scattering events.

The photon packets fitting the given detection cri-teria [8], including size, numerical aperture, and de-tector position are taken into account. Completingthe tracing of the photon trajectories for a large num-ber of photons Nph (typically Nph�107), the OCT sig-nal is calculated as a convolution of distribution ofthe detected photons over their optical path lengthsL=�i=1

N li with the envelope of coherence function [16]:

I�z� = 2�j=1

Nph

�WrWs�Lj� exp�− z − Lj

lcoh2� , �1�

where Wr and Ws are the total weights of the detectedphoton packets from the reference arm and scatter-ing medium, respectively; lcoh is the coherence lengthof probing laser radiation; and z is the depth.

Using a combination of the MC technique and theiteration procedure of the solution of Bethe–Salpeterequation [10], it has been shown that simulation ofthe optical path of a photon packet undergoing Nscattering events directly corresponds to the Nth or-der ladder diagram contribution. In this correspon-dence we generalized the above mentioned MC tech-nique for the direct simulation evolution of thepolarization vector for each photon packet. The polar-ization is described in terms of the polarization vec-tor P� undergoing a sequence of transformations aftereach scattering event [17], and all the photon packetsare weighted in accordance with the polarization

state. In Rayleigh and Rayleigh–Gans approxima-

2008 Optical Society of America

1582 OPTICS LETTERS / Vol. 33, No. 14 / July 15, 2008

tions the polarization vector of the scattered wave P� iis transformed upon the ith scattering such that [17]

P� i = − e�i�e�i · P� i−1� = �I − e�i � e�i�P� i−1, �2�

where e�i is the unit vector aligned along the trajec-tory element of a photon packet after the ith scatter-ing event. Thus, for simulating the electromagneticfield transfer we trace the transformation of the po-larization vector of incident field P0 under the actionof the chain of diadic operators:

P = �I − k−2ks � ks��i=2

N

�I − k−2kjj−1 � kjj−1�P0. �3�

The final stage of the parallel and perpendicular com-ponents of vector P determines the polarized and de-polarized components of scattering radiation. In sucha manner, we consider the propagation of polarizedand depolarized components of the electromagneticfield along the trajectory defined for the intensitypropagation (i.e., by scalar approach). However, andit should be pointed out here, based on the opticaltheorem [18] the scalar approach gives [19,20]

k04 G�k� i,k� s�d�s =

1

l, �4�

whereas for the electromagnetic field in the limit ofweak scattering [19,20]

k04 G�k� i,k� s�d�s =

1

l

2

1 + cos2 �. �5�

Here, G�k� i ,k� s� is the scattering phase function; k� i andk� s are the wave vectors for incident and scattered ofcomplex-conjugated fields, respectively; and k0=2� /�, where � is the wavelength.

Therefore, the extra multiplicative factor

� =2

1 + cos2 ��6�

should be taken into account at every scatteringevent. Finaly, the expressions for the intensities ofthe copolarized �I�� and cross-polarized �I�� compo-nents valid for Henyey–Greenstein phase functionare

I���� = �j=1

Nph

Wj�NjP�j

2 ,

I���� = �j=1

Nph

Wj�NjP�j

2 , �7�

where N is the number of scattering acts for the jthphoton packet and � denotes the dependence on anyvariable of interest such as depth, time, number ofscattering events, etc.

This MC algorithm has been validated against anaccepted analytic solution for a semi-infinite medium[11], and cross validated for a slab geometry with an

absorbing inclusion. The results are compared with

the exact solutions for pointlike Rayleigh scatteringparticles. For normal incidence and exact back-scattering detection the generalized Milne solutionfor the electromagnetic field gives the ratio of polar-ized and depolarized intensities I� /I��1.92 [19]. Nu-merical stimulation suggests that I� /I��1.93 withall significant digits. The generalized Milne solutiongives the depolarization ratio DP=I�−I� /I�+I�=0.31[19], while the results of our simulation suggestDP�0.326. The close value DP=0.33 was obtainedin [9].

The performances of this extended computationalapproach for simulating 2D conventional OCT andpolarization-sensitive OCT images are shown in Fig.1 for the skin model with six nonplanar layers [4,16](see Fig. 1a). The optical properties of the layers are�s: 35, 5, 10, 10, 7, 12 �mm−1�; �a: 0.02, 0.015, 0.02,0.1, 0.7, 0.2 �mm−1�; g: 0.9, 0.95, 0.85, 0.9, 0.87, 0.95;and refractive index n: 1.54, 1.34, 1.4, 1.39, 1.4, 1.39,respectively.

The source-detector parameters are chosen in ac-cordance with the parameters of the OCT system de-scribed in [21]: wavelength �=900 nm, lcoh=15 �m,and numerical aperture 0.2. To simulate OCT imagesthe sequential A-scans are simulated with the de-fined 20 �m transversal scanning step.

The results of simulation of 2D OCT andpolarization-sensitive OCT images for both copolar-

Fig. 1. (Color online) Schematic presentation of a skinmodel used in the simulation. a: 1, upper stratum corneum(average thickness 0.02 mm); 2, lower stratum corneum�0.18 mm�; 3, epidermis �0.2 mm�; 4, upper blood net der-mis �0.2 mm�; 5, reticular dermis �0.8 mm�; 6, deep bloodnet dermis �0.6 mm�. 2D simulated images for b, conven-tion OCT and c, copolarized and d, cross-polarized OCT mo-

dalities, respectively.

July 15, 2008 / Vol. 33, No. 14 / OPTICS LETTERS 1583

ized and cross-polarized modes are presented in Figs.1b–1d. The OCT image obtained in copolarized mode(see Fig. 1c) is similar to the OCT image obtained forthe convention OCT (Fig. 1b), whereas the intensityof the image obtained in cross-polarized mode (seeFig. 1d) is much lower compared with the co-polarized mode (Fig. 1c). This is due to the origin ofthe OCT signal formed mostly by so-called “snake” orlow-scattering-order photons [22]. Therefore, contri-bution of the photon packets depolarized owing tomultiple scattering in the formation of the final OCTimage is relatively small in upper layers and signifi-cantly rises for deep layers.

Thus, based on the generalization of iterative pro-cedure of the solution of Bethe–Saltpeter equation, anew vector-based MC approach has been developed.For the first time to our knowledge using a MC tech-nique we have imitated 2D OCT and polarization-sensitive OCT images of a skin model with nonplanarlayers’ boundaries. The performances of the devel-oped method are shown both for conventional andpolarization-sensitive OCT modalities. The currentapproach allows considerably simplified simulation ofpolarized low-coherent optical radiation propagationwithin turbid scattering media avoiding implementa-tion of Stokes–Mueller and/or Jones formalism, withthe substantial reducing of the computational re-sources. In a similar manner this approach can be ap-plied for circular polarization allowing one to avoid acumbersome calculation of the Mueller matrix. Thetechnique can provide detailed information on polar-ization propagation within/outside the medium, scor-ing various physical quantities simultaneously. Thetechnique will find a number of straightforward ap-plications related to the noninvasive diagnostics ofvarious disperse random media, including biologicaltissues, polymers, liquid crystals, and others.

The authors acknowledge the support of UK Bio-technology and Biological Sciences Research Council(project BBS/B/04242) and Royal Society, NATO(project PST.CLG.979652), GETA graduate school,

and Tauno Tönning Foundation, Finland.

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