1MstocsabgcsFnpmi(flmsbtvtwpdla

et

1030 J. Opt. Soc. Am. A/Vol. 25, No. 5 /May 2008 Nee et al.

Scattering polarization by anisotropicbiomolecules

Tsu-Wei Nee,1,* Soe-Mie F. Nee,1 De-Ming Yang,2 and Yu-Shan Huang1

1Institute of Biophotonics, National Yang Ming University, Taipei 11221, Taiwan, China2Department of Medical Research and Education, Taipei Veterans General Hospital, Taipei 11217, Taiwan, China

*Corresponding author: twnee.ym.edu.tw or twnee@aol.com

Received January 8, 2008; accepted February 22, 2008;posted March 4, 2008 (Doc. ID 91283); published April 15, 2008

The full polarization properties of anisotropic biomolecule optical scattering are investigated theoretically. Byusing a simple ellipsoid model of a single biomolecule, the scattering fields and Mueller matrices are derivedfrom fundamental electromagnetism theory. The energy of scattered photons is not necessarily equal to that ofthe incident laser beam. This theory can be generally applied to the experiments of fluorescence, Raman scat-tering, and second-harmonic generation. Fitting of a single tetramethylrhodamine-labeled lipid molecule’s an-isotropic imaging experiment is demonstrated. This theory has provided a fundamental simulation analysistool of understanding and developing the optical polarimetric sensing science and technology of the anisotropicbiomolecules and biomedium. The medium dielectric constant of the model ellipsoid provides a theoretic back-ground for correlating the optical polarization properties of a biomolecule to its microscopic electronicstructure. © 2008 Optical Society of America

OCIS codes: 290.5855, 290.5840, 290.5825, 290.1350, 170.2520, 170.3880.

mteatjrcs

2Osldid

sutss[

Iifita

. INTRODUCTIONolecular imaging technology of biomedical materials

uch as cells, protein, tissue, etc., is of current major in-erest in biophotonics [1]. Polarization is a basic propertyf light. Optical polarization sensing technology was re-ently extended to biomedical applications. Polarization-ensitive optical coherence tomography (OCT) can revealnisotropic properties of biological tissues [2–4]. It haseen shown that abnormal tissue has a stronger birefrin-ence than that of the normal tissue [5]. Polarization-ontrolled optical and second-harmonic-generation micro-copes can determine 3-D molecular orientation [6].luorescence polarization has also been used for biotech-ology applications [7,8]. Single-molecule fluorescenceolarization imaging of an anisotropic tetra-ethylrhodamine-labeled (TMR-DPPE) lipid molecule

n a 1-palmitoyl-2-oleoyl-sn-glycerol-3-phosphocholinePOPC) membrane and eYFP (enhanced yellow-uorescent protein) has been demonstrated [9,10]. Single-olecular orientation was determined by liquid-crystal

patial light modulators with a polarization-controlledeam [11]. The linear and nonlinear full polarization op-ical properties of an anisotropic object, material or de-ice, can be completely described by a 4�4 Mueller ma-rix [12–17]. This Mueller matrix approach has not beenidely applied to the optical polarization research of bio-hotonics. In this paper, a Mueller matrix model theory iseveloped for the simulation and analysis of the full po-arization optical properties of anisotropic biomoleculesnd cells.The polarizability of an anisotropic ellipsoid model mol-

cule is presented in the next section. In Sections 3 and 4,he scattered fields of a single ellipsoid biomolecule or cell

1084-7529/08/051030-9/$15.00 © 2

odel due to the incident electromagnetic wave are inves-igated. The Mueller matrix is derived from fundamentallectromagnetic theory. In Sections 5 and 6, numerical ex-mples showing the single-molecular scattering responseo incident light of different polarizations are reported. Toustify this theory, fitting of existing single-molecule fluo-escence polarization imaging data is reported. The dis-ussion and conclusion are given in Sections 7 and 8, re-pectively.

. POLARIZABILITY OF AN ELLIPSOIDur model treats an anisotropic molecule as a uniaxially

ymmetric ellipsoid of dielectric constant �. As such an el-ipsoid is exposed to incident light, an electric dipole is in-uced and the dipole reradiates. The principal polarizabil-ty and induced dipole moment for a biomolecule will beerived in this section.Let the principal radii of an ellipsoid be b along the

ymmetric axis and a along the transverse axes. The vol-me of the ellipsoid is V=4�a2b /3. If an externally elec-ric field Eo is present, the induced charges on the ellip-oid surface will generate an induced electric field E1uch that the macroscopic field E inside the ellipsoid is18]

E = Eo + E1. �1�

f the incident field is along one of the principal axes, thenduced field is in the direction opposite to the incidenteld. Otherwise, the induced field may be off aligned withhe incident field. Let us consider the case for the fieldlong a principal axis, called the z axis:

E = − 4�q P = − 4�� E ,

1z z z z z

008 Optical Society of America

Hcfaaeg

F�p

d

Eptdp

T

wu

fm

3AFfckCttcapa

mvtiifid

Hsta�As

Fo

Fq

Nee et al. Vol. 25, No. 5 /May 2008/J. Opt. Soc. Am. A 1031

�z = qz�� − 1�/4�. �2�

ere Pz is the polarization along the z axis, �z is the sus-eptibility along the z axis, and qz is the depolarizationactor for the principal z axis of the ellipsoid. Let the zxis be the symmetric axis of the ellipsoid, the x and yxes be the transverse axes, and the aspect ratio of thellipsoid be u=b /a; the depolarization factors are theniven as [19,20]

qz =1

1 − u2�1 −u cos−1 u

�1 − u2 � for u � 1,

=1

u2 − 1�u cosh−1 u

�u2 − 1− 1� for u � 1,

qx = qy = �1 − qz�/2. �3�

igure 1 shows the dependence of qz and qx on u; qz1/3 for an oblate ellipsoid with u�1 and qz�1/3 for a

rolate ellipsoid with u�1.From Eqs. (1) and (2), we get the relation between the

ipole moment pz and the incident field as

pz = VPz =1

3

a2b�� − 1�

1 + qz�� − 1�Eoz. �4�

quations (2) and (4) apply equally well to the other tworincipal axes. The principal polarizability �j in the direc-ion of the jth principal axis is defined as the ratio of theipole moment pj to the incident field Eoj for that princi-al direction. It is given by

�j =1

3

a2b�� − 1�

1 + qj�� − 1�, j = x,y,z. �5�

he polarizability ratio � is determined as

� =�z

�x=

1 + qx�� − 1�

1 + qz�� − 1�, �6�

here �= ��+1� /2 for qz=0 or u1 and �=1/� for qz=1 or=0. The polarizability ratios � as a function of u for dif-

ig. 1. Incident and scattering beams and the model ellipsoidrientations in the 3-D coordinate system (x ,y ,z axes).

erent � are shown in Fig. 2. The polarization effect isore obvious for larger dielectric constants.

. VECTOR SCATTERING BY ANNISOTROPIC MOLECULE

or a monochromatic electromagnetic wave of angularrequency incident along direction ki onto a scatteringenter at the coordinate origin, the incident wave vector is

i= ki /c. As shown in Fig. 1, we choose the laboratoryartesian coordinates such that the plane of incidence is

he plane containing the z axis and ki. The incident elec-ric field can be decomposed into two components: the somponent that is perpendicular to the plane of incidence,nd the p component that is in the plane of incidence anderpendicular to ki. The incident electromagnetic field Eit a point x� of the scattering medium is

Ei�x�,t� = Eio exp�iki · x� − it�,

Eio = siEis + piEip, �7�

si = ki � z/�ki � z�, pi = si � ki. �8�

The scattering of visible to near-infrared light by bio-olecules or cells of sizes of the order of 10–100 nm is in-

estigated. Because particle sizes are small compared tohe wavelength �400–1000 nm� of incident light, the fieldnside the particle is nearly homogeneous and the scatter-ng is basically a dipole scattering. The electromagneticeld scattered by a dipole p at x� to an observer at x inirection ks is [21]

E�x,t� = − exp�iksr − ist�ks � �ks � p�exp�− iks · x��/r.

�9�

ere ks=s /c, x= ksr, ks= ksks, r is the distance of the ob-erver from the origin, and s is the frequency of the scat-ered field. Note that s is not necessarily equal to . Ex-mples are (i) fluorescence [9,10] and Raman scatterings�� and (ii) second-harmonic generation �s�� [17].s shown in Fig. 1, the scattered field and the associatedand p polarizations are

E�x,t� = Esc exp�iks · x − ist�,

ig. 2. Depolarization factors of a uniaxial ellipsoid: �=1.1896,, q versus u�=b /a�.

z x

F

m

Htaleett

c

Bs

4SFsoa

F

Tb

dkt�s�eJ

1032 J. Opt. Soc. Am. A/Vol. 25, No. 5 /May 2008 Nee et al.

Esc = �sEsp + ssEss, �10�

ss = ks � z/�ks � z�, ps = ss � ks. �11�

rom Eqs. (9) and (10), we have

Esp = ks2ps · p exp�− iks · x��/r,

Ess = ks2ss · p exp�− iks · x��/r. �12�

The vector representation of the induced dipole mo-ent is

p = � · Eio exp�iki · x��, �13�

� = xd�xxd + yd�yyd + zd�zzd. �14�

ere �z, �x, and �y are given by Eq. (5), and �xd , yd , zd� ishe set of the unit vectors along the three principal axes ofn ellipsoid, as shown in Fig. 1. For a single scattering el-ipsoid at the origin, exp�iki ·x���1�exp�−iks ·x��, thexponential factors can be neglected. If a complex mol-cule is composed of several ellipsoids at different posi-ions, this factor introduces a phase difference betweenhe ellipsoids.

The Jones matrix relates the scattering field to the in-ident field by

Esc = �Esp

Ess� = �Jpp Jps

Jsp Jss��Eip

Eis� = J · Ei. �15�

y using Eqs. (11)–(15) that relate the incident field to thecattering field, the Jones matrix is represented by

J = jks2 exp�i�ki − ks� · x�/r,

j = �ps · � · pi ps · � · si

ss · � · pi ss · � · si� . �16�

. NORMAL INCIDENCE SCATTERING BY AIMPLE ELLIPSOIDor a single ellipsoid located at x�=0, J= jks

2 /r. We con-ider the plane of incidence as the zx plane and the anglef incidence is �i. The vectors for the incident polarizationre given by

ki = x sin �i − z cos �i,

pi = x cos �i + z sin �i,

si = − y. �17�

or the scattering direction, the directions are given by

ks = x sin �s cos s + y sin �s sin s + z cos �s,

ps = − x cos �s cos s − y cos �s sin s + z sin �s,

ss = x sin s − y cos s. �18�

he orientations of the ellipsoid principal axes are giveny

zd = x sin �d cos d + y sin �d sin d + z cos �d,

xd = x sin d − y cos d,

yd = x cos �d cos d + y cos �d sin d − z sin �d. �19�

For the application to microscope optics [9,10,22], theirection of incident light ki is along the −z axis, �i=0°,

ˆi=−z, si− y, and pi= x. Using the coordinate transforma-ion equations between the ellipsoid body coordinatexd , yd , zd� and the scattered coordinate �ps , ss , ks�, thecattered field �Esp ,Ess� is related to the incident fieldEip ,Eis� by the Jones matrix J as defined by Eq. (15). Thelements of Jones matrix J�r ,�s , s ,�d , d�, Jpp, Jps, Jsp,

, can be calculated using Eqs. (14)–(19) as

ss

J�r,�s, s,�d, d� =Jou�� − 1�

1 + qx�� − 1��− cos �s cos s − cos �s sin s sin �s

sin s − cos s 0 ��

1 + �� − 1�sin2 �d cos2 d �1 − ��sin2 �d sin d cos d

�� − 1�sin2 �d sin d cos d − 1 + �1 − ��sin2 �d sin2 d

�� − 1�sin �d cos �d cos d �1 − ��sin �d cos �d sin d� , �20a�

Jo = 4�2a3/3r�s2, �20b�

wd

p

Tmr

wttM

H+

5MPWticattea

=ra

�upilmp�cm

=c1swi

F1=T

F�

Fp

Nee et al. Vol. 25, No. 5 /May 2008/J. Opt. Soc. Am. A 1033

here �s is the wavelength of scattered light and r is theistance of the detector from the scattering center.If the Jones matrix is diagonal, then the ellipsometric

arameters � and � are defined by the ratio of jpp / jss as

tan � exp�i�� = jpp/jss. �21�

he relation between the complex fields and the Jonesatrix can be transformed to the relation between the

eal Stokes vector and the Mueller matrix [12].

Is

Qs

Us

Vs

��r,�s, s,�d, d� = M�r,�s, s,�d, d�Ii

Qi

Ui

Vi

� , �22�

here �Ii ,Qi ,Ui ,Vi� and �Is ,Qs ,Us ,Vs� are the Stokes vec-ors for the incident and scattered light beams, respec-ively. For a diagonal Jones matrix, the correspondingueller matrix is

M = R1 − cos 2� 0 0

− cos 2� 1 0 0

0 0 sin 2� cos � sin 2� sin �

0 0 − sin 2� sin � sin 2� cos �� .

�23�

ere R is the reflectance or transmittance, R= �Jpp2

Jss2� /2.

. NUMERICAL EXAMPLE. I: SINGLE-OLECULE ANISOTROPY SCATTERING

OLARIZATION PROPERTYe consider in the configuration of a microscope’s optics

hat the z axis is normal to the sample plane and that thencident light direction is ki=−z, [9,10,22]. The scatteringenter is a single ellipsoid molecule with principal radii and b, volume V�=4�a2b /3�, and dielectric constant �. Forhe backscattering (ks= z, �s=0°, s=180°) by a dipole onhe zx plane � d=0° �, the Jones matrix of Eq. (20) and thellipsometric parameters �� ,�� of Eq. (21) can be evalu-ted as

ig. 3. Polarization ratio � versus u�=b /a�. Curves for �=1.1,.1896, and 1.3 are shown for comparison. The data of �1.1896 and u=3 and 1 used in the experimental fitting (seeable 1) are marked.

J��d� =Jou�� − 1�

1 + qx�� − 1��1 + �� − 1�sin2 �d 0

0 − 1� ,

� = 180 ° , � = tan−1�1 + �� − 1�sin2 �d. �24�

For a spherically symmetric molecule, u=1, qx=qz1/3 (Fig. 2) and �=1 (Fig. 3), �=tan−1 1=45° (Fig. 4). Itepresents an isotropic scattering case in which the p-nd s-polarization scattering are the same [12].For the direction of the symmetric axis along the x axis,

d=90°, and we have �=tan−1 �, where � as a function ofcan be obtained from Eqs. (3) and (6). The ellipsometric

arameter � as a function of u for �=1.1, 1.1896, and 1.3s calculated and shown in Fig. 3. There is greater p po-arization than s polarization for ��45°. Since the sym-

etric axis is along the x axis that is defined also for polarization, � is �45° for an oblate ellipsoid and �45° for a prolate ellipsoid, in agreement with the con-

ept that a dipole moment is induced along the longer di-ension of an object.For a prolate ellipsoid with u=3 on the zx plane � d

0° � and with different orientations ��d=0° –90° �, wealculate � as functions of �d using Eq. (24) for �=1.1,.1896, and 1.3. The computed ���d� for different � arehown in Fig. 5. Polarization is the largest at �d=90°hen the ellipsoid has the largest cross section as viewed

n the backscattering direction.

ig. 4. For �=1.1, 1.1896, and 1.3, the ellipsometric parameterversus u�=b /a� curves are shown.

ig. 5. For u=3 and �=1.1896, 1.1, and 1.3, the ellipsometricarameter � versus � curves are shown.

d

6METmTaoatoeteoal1t

cmtoicla

t�mrmt

HiT

FT

M

w

plp

dl0tt

Fs

1034 J. Opt. Soc. Am. A/Vol. 25, No. 5 /May 2008 Nee et al.

. NUMERICAL EXAMPLE. II: SINGLE-OLECULE ANISOTROPY IMAGING

XPERIMENThe polarization image of an individual TMR-DPPE lipidolecule on a phospholipid membrane was measured [9].he sample was illuminated on an inverted microscope byn incident laser of 514 nm wavelength with an intensityf 5 kW/cm2. The fluorescence images of s and p wave im-ging were detected using a Wollaston prism polarizer inhe backward direction path toward the CCD imager. Thebserved s and p-signals of the single TMR-DPPE mol-cule were different due to the anisotropic molecule struc-ure. As shown in Fig. 2 of [9], if the incident laser is lin-arly polarized in the x direction (parallel to the long axisf the attached molecule), then the s and p signals are 130nd 285 counts, respectively (ratio 0.456). If the incidentaser is circularly polarized, then the s and p signals are36 and 149 counts, respectively (ratio 0.913). We shall fithese data using the theory developed in Sections 3 and 4.

As shown in Fig. 6, we propose a molecule model as aombination of two ellipsoids to imitate the TMR-DPPEolecule shown in Fig. 1 of [9]. The aspect ratios u of the

wo ellipsoids are u1=3 and u2=1. The dielectric constantf this molecule is assumed to be �=1.1896. The depolar-zation factors qx, qz and the polarizability ratio � are cal-ulated from Eqs. (3) and (6), respectively. The results areisted in Table 1 and marked as solid squares in Fig. 3. Wessume that the symmetric axis of the ellipsoid is along

ig. 6. Two-ellipsoid model to imitate the TMR-DPPE moleculehown in Fig. 1 of [9].

he x axis; i.e., the molecular orientation parameters are�d1 , d1�= ��d2 , d2�= �90° ,0° �. A laser is incident nor-ally �ki=−z� on the molecule. In the backscattering di-

ection, ks= z, ss= y, and ps= x. The Jones matrix for theolecule is the sum of the individual Jones matrices for

he two ellipsoids,

J�r,�s, s,�d1, d1,�d2, d2� = J1�r,�s, s,�d1, d1�

+ J2�r,�s, s,�d2, d2�.

�25�

ere J1 and J2 are calculated using Eq. (20) and the an-sotropic parameters � for u=3 and 1, respectively (seeable 1).

J1�r,�s, s,�d1, d1� = 4.58Jo�1 0

0 − 0.9411� ,

J2�r,�s, s,�d2, d2� = 1.47Jo�1 0

0 − 1� . �26�

J�r,�s, s,�d1, d1,�d2, d2� = 6.05Jo�1 0

0 − 0.9554� . �27�

or this Jones matrix, we have �=180° and �=46.31°.he Mueller matrix M for the above J is

�r,�s, s,�d1, d1,�d2, d2�

= R1 0.0456 0 0

0.0456 1 0 0

0 0 − 0.9990 0

0 0 0 − 0.9990� , �28�

here R=35 Jo2.

We consider the following two cases for which the ex-erimental data were reported [9]: one for incident circu-arly polarized light, and the other for incident linearlyolarized light:(i) For an incident circularly polarized light, the inci-

ent Stokes vector is Ii (1, 0, 0, 1). The Stokes vector foright scattered toward the detector is RIi (1, 0.046, 0,.999). The Stokes vectors for the scattered light passinghrough a linear analyzer oriented in the x and y direc-ions are, respectively,

I0

Q0

U0

V0

� =RIi

2 1 1 0 0

1 1 0 0

0 0 0 0

0 0 0 0�

1

0.046

0

0.999� =

RIi

2 1.046

1.046

0

0� ,

�29a�

Table 1. Depolarization Factors and PolarizabilityRatios of Ellipsoid Molecules with �=1.1896

u �b /a� qx qz �=�x /�z

3 0.4456 0.1087 1.06261 0.3333 0.3333 1.0000

ToTt

s

S(cliRwgp

7Ti

H=cpt

siwa−tm

FpsplsgleFetrslttl

s

Nee et al. Vol. 25, No. 5 /May 2008/J. Opt. Soc. Am. A 1035

I90

Q90

U90

V90

� =RIi

2 1 − 1 0 0

− 1 1 0 0

0 0 0 0

0 0 0 0�

1

0.046

0

0.999� =

RIi

2 0.954

− 0.954

0

0� .

�29b�

he intensities at the detector for the analyzer at the tworthogonal directions are I0=0.523 RIi and I90=0.477 RIi.he ratio is I90/I0=0.477/0.523=0.912, in agreement withhe experimental data of [9], 136/149=0.912.

(ii) If the incident laser is partially x polarized, we as-ume an incident Stokes vector of I �1,0.334,0,0�. The

i

spdi

wshsbTatimWtt7del

g

Iceolt

tokes vector for light scattered toward the detector is RIi1.015, 0.379, 0, 0). Using a calculation similar to that inase (i), we obtain the intensity of light passing through ainear analyzer in the x direction as I0=0.697 RIi and thentensity for the analyzer in the y direction as I90=0.318Ii. The ratio is I90/I0=0.318/0.697=0.456, in agreementith the experimental data of [9], 130/285=0.456. Theseood data-fitting results have shown that our theory is ex-erimentally justified.

. DISCUSSIONhe Mueller matrix M�r ,�s , s ,�d , d� of a general scatter-

ng system has the following form [23–25]:

M = R1 − P cos 2� 0 0

− P cos 2� 1 − 2Dv 0 0

0 0 P sin 2� cos � P sin 2� sin �

0 0 − P sin 2� sin � P sin 2� cos �� . �30�

ere P is the polarization, and the depolarization is D1−P=DU+DV; DU and DV are the co-polarized andross-polarized depolarization, respectively. The five inde-endent parameters in Eq. (30) describe the full polariza-ion properties of a scattering system [23,25].

In Sections 5 and 6, single-molecule scattering is con-idered. The polarization of the scattering is P=1. Theres no depolarization effect. For more complicate systemsith more uncorrelated molecules, depolarization mayrise [23] such that P�1 and the depolarization D=1P�0. As an example, we consider the scattering by

hree uncorrelated ellipsoid molecules. The total Muelleratrix is

M�r,�s, s� = M1�r,�s, s,�d1, d1� + M2�r,�s, s,�d2, d2�

+ M3�r,�s, s,�d3, d3�. �31�

or molecules with u=2,3,4 and different �dj and �dj, theolarization properties P, �, and � are calculated andhown in Table 2. These results show the existence of de-olarization �P�1� for scattering by a system of uncorre-ated molecules (see Table 2). Similarly, scattering by aystem of many uncorrelated anisotropic molecules willive appreciable depolarization due to the random angu-ar distribution of anisotropic molecules. This is an inter-sting issue of tissue optics [26] for further investigation.or comparison, if the three ellipsoids belong to one mol-cule, we calculate the combined Jones matrix J similaro that of Eq. (25) instead of the Mueller matrix M. Theesults are shown in Table 3. The depolarization is verymall �1−P�10−4�. Due to the in-phase interference of el-ipsoids within a molecule, the R value for a system ofhree correlated ellipsoids can reach a maximum abouthree times generation that for a system of three uncorre-ated ellipsoids.

A more complicated biomolecule can be modeled as aystem of many ellipsoids. These ellipsoids are correlated,

ince they are parts of a single molecule. The opticalaths from different ellipsoids to an observer would beifferent such that the interference effect should be takennto account. The Jones matrix for such a molecule is

J�r,�s, s� = �j

Jj�r,�s, s,�dj, dj�exp�i�j�,

�j = �ki − ks� · xj�, �32�

here �j is the phase factor associated with the jth ellip-oid as specified by Eq. (16). Although each ellipsoid be-aves like a uniaxial medium, a molecular structure con-tructed by the combination of many ellipsoids willehave as a whole as a medium of any kind of symmetry.he Mueller matrix is calculated from this Jones matrix,nd then the Stokes vector of the scattering beam is ob-ained from this Mueller matrix. We now choose the casen Section 6 as an example. The fluorescence-labeled lipid

olecule is represented by two ellipsoids of u=3 and 1.e take �1=0 and �=�2−�1 as the phase difference of the

wo ellipsoids. For the case of incident circular polariza-ion, �, �, and I90/I0 as functions of � are shown in Figs.–9, (solid curves), respectively. The phase difference � isue mainly to the optical path difference between differ-nt ellipsoids. For a path difference of d�1 nm at a wave-ength of 514 nm, � is estimated to be 0.7°.

The electronic dielectric function of a molecule has theeneral form

��� = �r�� + i�i��. �33�

n principle, the complex � for a specific molecule can bealculated by the microscopic quantum theory of a mol-cule, which is beyond the scope of this paper. However,ur theory has provided a theoretical background corre-ating the optical polarization properties of a biomoleculeo its microscopic electronic structure. In the numerical

eonwa�1�7

8UtSpsomt

mcm[m=lcawm

McTtesci

ules 2 a

T

ules 2 a

1036 J. Opt. Soc. Am. A/Vol. 25, No. 5 /May 2008 Nee et al.

xamples, the anisotropic molecule is assumed to be madef a medium with a real dielectric constant �. If �i�� isonzero, then the polarizability and the dipole momentill be complex, and so the Jones matrix of scattering willlso be complex. The consequence is that the retardationwill have quite a wide range instead of having only 0° or

80° as for a transparent medium. For �i=0.5 and 1.0, thedependence of �, �, and I90/I0 are also shown in Figs.

–9, (dashed curves), respectively.

. CONCLUSIONsing a simple ellipsoid model of an anisotropic molecule,

he Jones matrix of scattering has been formulated inections 2 and 3. For scattering by a biomolecule, we ap-ly this theory to the microscope optics where only back-cattering is considered. The basic polarization propertiesf scattering by a single-ellipsoid-model molecule are for-ulated in Section 4, and the numerical evaluations of

hese properties are given in Section 5. For a two-ellipsoid

Table 2. Polarization Properties of a ScatteringMolecules of Differen

Molecule 2 Mole

u � u2 1.0440 4

�d d �d

90 0 9090 30 9090 90 9090 150 90

120 150 9030 0 6030 90 6030 45 6030 45 6030 −45 60

aMolecule 1 has u=3, �d=90°, d=0°, and �=1.0934. The parameters of Molec

able 3. Polarization Properties of a Scattering Sysof Different Sh

Molecule 2 Mole

u � u2 1.0653 4

�d d �d

90 0 9090 30 9090 90 9090 150 90120 150 9030 0 6030 90 6030 45 6030 45 6030 −45 60

aMolecule 1 has u=3, �d=90°, d=0°, and �=1.0934. The parameters of Molec

olecule model of b /a=3 and 1, the Mueller matrix wasalculated in Section 6 and applied to fit the single-olecule anisotropic fluorescence experimental data of

9]. Excellent fitting results are obtained. This ellipsoidolecule is considered to have a dielectric constant of �1.1896. These data are consistent with the electronic po-

arizability of molecules. Section 7 discussed the prin-iples to treat the systems of molecules also with nonzerobsorption coefficients, of more complicated moleculesith a system of ellipsoids at different locations, and ofany uncorrelated molecules.The major conclusions are listed as follows: (1) Theueller matrix formalism of anisotropic biomolecule opti-

al scattering is developed using a simple ellipsoid model.he energy of scattered photons is not necessarily equal

o that of the incident laser beam. This theory can be gen-rally applied to the experiments of fluorescence, Ramancattering, and second-harmonic generation. (2) A suc-essful fitting of a single TMR-DPPE lipid molecule’s an-sotropic imaging experiment is demonstrated. (3) The de-

em Consisting of Three Uncorrelated Ellipsoidapes with �=1.1896a

Polarization

�

1.0723

d P � (deg)

0 1.000 46.7430 0.9999 46.1690 0.9984 44.42

−150 0.9999 46.18−180 0.9998 46.38

0 0.9998 46.18−90 0.9990 45.01−90 0.9990 45.15−45 0.9996 45.5945 0.9996 45.59

nd 3 are also given.

onsisting of Three Correlated Ellipsoid Moleculeswith �=1.1896a

Polarization

�

1.1083

d P � (deg)

0 1.0000 46.7430 1.0000 46.1690 1.0000 44.42

−150 1.0000 46.16−180 1.0000 46.38

0 1.0000 46.17−90 1.0000 45.00−90 1.0000 45.15−45 0.9999 45.5945 0.9999 45.59

nd 3 are also given.

Systt Sh

cule 3

tem Capes

cule 3

pis(tlps

ATM

UTd

R

1

1

1

1

1

1

1

1

11

Fw

Fw

Ft

Nee et al. Vol. 25, No. 5 /May 2008/J. Opt. Soc. Am. A 1037

olarization effect for a system of noncorrelated moleculess demonstrated by the numerical analysis of a scatteringystem with three anisotropic ellipsoid model molecules.4) This theory has provided a foundation for (a) simula-ion analysis of the polarization properties of a biomolecu-ar medium and (b) correlating the polarization opticalroperties of a biomolecule to its microscopic electronictructure.

CKNOWLEDGMENTShis research was partly supported by the National Yang-ing University Development of Top-Level International

ig. 7. Phase difference effect: � for the two-ellipsoid moleculeith u=3 and 1; �r=1.1896 and �i=0, 0.5, and 1.

ig. 8. Phase difference effect: � for the two-ellipsoid moleculeith u=3 and 1; �r=1.1896 and �i=0, 0.5, and 1.

ig. 9. Phase difference effect: I90/I0 (incident circular polariza-ion case) for the two-ellipsoid molecule with u=3 and 1.

niversity Program, Ministry of Education, Taiwan.su-Wei Nee is grateful to Jens Krüger for an informativeiscussion of the TMR-DPPE molecular structure.

EFERENCES1. Z. Chen, H. Ren, Z. Ding, Y. Zhao, J. Miao, and J. S.

Nelson, “Biomedical imaging: simultaneous imaging of insitu tissue structure, blood-flow velocity, standarddeviation, birefringence and Stokes vectors in human skin,”Opt. Photon. News, December 2002, p. 14.

2. C. E. Saxer, J. F. de Boer, B. H. Park, Y. Zhao, Z. Chen, andJ. S. Nelson, “High-speed fiber-based polarization-sensitiveoptical coherence tomography of in vivo human skin,” Opt.Lett. 25, 1355–1357 (2000).

3. S. Liao and L. V. Wang, “Two-dimensional depth-resolvedMueller matrix of biological tissue measured with double-beam polarization-sensitive optical coherence tomography,”Opt. Lett. 27, 101–103 (2002).

4. J. F. de Boer and T. E. Milner, “Review of polarizationsensitive optical coherence tomography and Stokes vectordetermination,” J. Biomed. Opt. 7, 359–371 (2002).

5. C. C. Wu, Y. M. Wang, L. S. Lu, C. W. Sun, C. W. Lu, M. T.Tsai, and C. C. Yang, “Optical birefringence of thehyperlipidemic rat liver with polarization-sensitive opticalcoherence tomography,” J. Biomed. Opt. 12, 64022 (2007).

6. M. Hashimoto, R. Kanamaru, K. Yoshiki, T. Araki, and N.Hashimoto, “Second-harmonic microscope with polarizationmode converter,” Presented at the Ninth InternationalConference on Optics Within Life Science (OWLS9),National Yang-Ming University, Taipei, Taiwan, November26–29, 2006, Paper O4-7.

7. C. L. Berger, J. S. Craik, D. R. Trentham, J. E. T. Corrie,and Y. E. Goldman, “Fluorescence polarization of skeletalmuscle fibers labeled with rhodamine isomers on themyosin heavy chain,” Biophys. J. 71, 3330–3343 (1966).

8. P. Wu, M. Brasseur, and U. Schindler, “Measurement ofspecific protease activity utilizing fluorescencepolarization,” Anal. Biochem. 247, 83–88 (1997).

9. G. S. Harms, M. Sonnleitner, G. S. Schutz, H. J. Gruber,and T. Schmidt, “Single-molecule anisotropy imaging,”Biophys. J. 77, 2864–2870 (1999).

0. G. S. Harms, L. Cognet, P. H. M. Lommerse, G. A. Blab, H.Kahr, R. Gamsjager, H. P. Spanink, N. M. Soldatov, C.Romanin, and T. Schmidt, “Single-molecule imaging of L-type Ca2+ channels in live cells,” Biophys. J. 81, 2639–2646(2001).

1. M. Hashimoto, K. Yamada, and T. Araki, “Proposition ofsingle molecular orientation determination usingpolarization controlled beam by liquid crystal spatial lightmodulators,” Opt. Rev. 12, 37–41 (2005).

2. S.-M. F. Nee, “Polarization measurement,” in TheMeasurement, Instrumentation and Sensors Handbook, J.G. Webster, ed. (CRC Press and IEEE Press, 1999), pp.60.1–60.24.

3. T.-W. Nee and S.-M. F. Nee, “Infrared polarizationsignatures for targets,” Proc. SPIE 2469, 231–241 (1995).

4. T. W. Nee, S. F. Nee, and E. J. Bevan, “Infrared polarizationsignatures of a target for enhanced discrimination,” inProceedings of the IRIS Specialty Group on Targets,Backgrounds and Discrimination (IRIA–IRIS, 1996), Vol.IV, pp. 349–368.

5. T.-W. Nee and S.-M. F. Nee, “Polarization of holographicgrating diffraction. I. General theory,” J. Opt. Soc. Am. A21, 523–531 (2004).

6. T.-W. Nee, S.-M. F. Nee, M. Kleinschmit, and S. Shahriar,“Polarization of holographic grating diffraction. II.Experiment,” J. Opt. Soc. Am. A 21, 532–539 (2004).

7. T.-W. Nee, “Second harmonic diffraction from holographicvolume grating,” J. Opt. Soc. Am. A 23, 2510–2518 (2006).

8. C. Kittel, Solid State Physics (Wiley, 1976), pp. 404–405.9. L. Davis, Jr. and J. L. Greenstein, “The polarization of

2

22

2

2

2

1038 J. Opt. Soc. Am. A/Vol. 25, No. 5 /May 2008 Nee et al.

starlight by aligned dust grains,” Astrophys. J. 114,206–240 (1951).

0. S.-M. F. Nee, “Ellipsometric analysis for surface roughnessand texture,” Appl. Opt. 27, 2819–2831 (1988).

1. J. D. Jackson, Classical Electrodynamics (Wiley, 1962).2. V. Prasad, D. Semwogerere, and E. R. Weeks, “Confocal

microscopy of colloids,” J. Phys.: Condens. Matter 19,113102 (2007).

3. S. F. Nee, “Polarization of specular reflection and near- 2

specular scattering by a rough surface,” Appl. Opt. 35,3570–3582 (1996).

4. S.-M. F. Nee, “Depolarization and retardation of abirefringent slab,” J. Opt. Soc. Am. A 17, 2067–2073 (2000).

5. S.-M. F. Nee and T.-W. Nee, “Principal Mueller matrix ofreflection and scattering measured for a one-dimensionalrough surface,” Opt. Eng. (Bellingham) 41, 994–1001(2002).

6. V. V. Tuchin, Tissue Optics (SPIE, 2007).