Geometric depolarization in patterns formed bybackscattered light
David Lacoste
Laboratoire de Physico-Chimie Théorique, Ecole Supérieure de Physique et de Chimie Industrielles, 10 Rue de Vauquelin,F-75321 Paris Cedex 05, France
Vincent Rossetto
Laboratoire de Physique et Modélisation des Milieux Condensés, Maison des Magistères, B.P. 166, 38042 Grenoble Cedex 9, France
Franck Jaillon and Hervé Saint-Jalmes
Laboratoire de Résonance Magnetique Nucléaire, Méthodologie et Instrumentation en Biophysique, Unité Mixte de Recherche CentreNational de la Recherche Scientifique 5012, Domaine de la Doua— CPE—3, Rue Victor Grignard, 69616 Villeurbanne, France
The transport of light through human tissues is oneof the most promising techniques to detect breastcancer, for instance, in a noninvasive way. For medi-cal imaging applications, it is important to extractthe information contained not only in the intensitybut also in the polarization of backscattered light.This extraction is not easy in general because ofthe complexity of vector-wave multiple scattering.In this Letter we study a simple experiment, inwhich polarized light is backscattered from a diffusemedium. In these conditions a fourfold symmetrypattern can be observed between crossed polarizersthat was f irst interpreted qualitatively by Dogariu andAsakura.1 Recently more quantitative approacheswere developed by use of Mueller matrices.2,3 In thisLetter we propose an alternate approach, which isquite simple to implement because it is not basedon a vector radiative-transfer method as generallyused in the literature. Instead our approach is basedon the notion of geometric phase, which was intro-duced by Berry4 in his interpretation of experimentsshowing optical activity in a helically wound opticalfiber.5 Berry’s geometric phase in these referencesis the phase acquired by light when its directionof propagation is slowly changed on a sphere of di-rections (i.e., in momentum space). This geometricphase is equal to the solid angle on the sphere ofwave-vector directions. A different geometric phase,called Pancharatnam’s phase is the phase acquiredby paraxial polarized light wave when its polariza-tion undergoes some transformation on the Poincarésphere. That geometric phase is equal to half thesolid angle on the Poincaré sphere.6 So far the ap-plications of geometric phases to polarized light havebeen limited to situations in which light is traveling ina homogeneous medium. In this Letter we use only
the first geometric phase, Berry’s phase, and applyit to multiple light scattering in a random medium.Before presenting our application of Berry’s phase,which follows closely and extends the recent Ref. 7, wediscuss the cross-shaped patterns, using the standardStokes formalism to make a connection with previouswork.2,3
We assume that linearly polarized light is incidentupon a medium and that the direction of the incidentbeam is normal. The intensity and polarization ofthe backscattered light are completely characterizedby Stokes parameters �I , Q,U , V �. Transformationsof the Stokes parameters are represented by 4 3 4Mueller matrices. Scattering matrix S is such amatrix and contains contributions from all ordersof scattering.3 The dependence of the outgoingStokes parameters as a function of azimuthal anglef measured about the incident beam direction canbe obtained by a product of the appropriate Muellermatrices. We find that the outgoing intensities inthe backscattered direction and between parallel(perpendicular) polarizers are3
I� �14
�2S11 1 S33 2 S22�
214
�S22 1 S33�cos 4f , (1)
Ik �14
�2S11 2 S33 1 S22� 2 S12
3 cos 2f 114
�S22 1 S33�cos 4f , (2)
corresponding to Stokes parameters I � I� 1 Ik andQ � Ik 2 I�. Equations (1) and (2) are valid for
any distribution of randomly oriented particles witha symmetry plane. Note that Eq. (1) implies thatthe cross-polarized pattern has a fourfold symme-try, whereas Eq. (2) implies an additional twofoldsymmetry in the copolarized pattern because of theterm proportional to S12. In the particular casethat is satisfied in multiple light scattering,8 whenS � S11 diag�1, C, C, D�, with C � S22�S11 andD � S44�S11, Eqs. (1) and (2) take the simple form
I� �12
I0�1 2 C cos 4f� , (3)
Ik �12
I0�1 1 C cos 4f� , (4)
corresponding to outgoing Stokes parameters I � I0 �S11 and Q � CI0 cos 4f. Note that a cross is expectednow in both polarization channels and that C measuresthe contrast of this pattern.
Let us now discuss the origin of the depolariza-tion of polarized light. For Rayleigh scattering, the(linear) polarization vector after scattering, E0, isE0 � k0 3 �E 3 k0�, in terms of the polarization vectorbefore scattering E and the scattered wave vector k0.This implies that E evolves by parallel transport inthe limit of small scattering angles and diffuses onthe sphere of wave-vector directions until the memoryof the polarization has been lost. This depolarizationhas a characteristic length lp equal to 2.8l, where lis the elastic mean free path of the light.9 As theanisotropy in the scattering increases, lp approachesthe transport mean free path l�.10,11 Here we assumeforward-peaked scattering because it applies to manybiological tissues, and because in this case there is aclear analogy between Berry’s geometrical phase inoptics and the twist and writhe of polymers.7 Thehypothesis of forward-peaked scattering nicely satis-fies the requirement for the Berry’s phase of a slowvariation of the circuit in momentum space6 andcorresponds to a special recently investigated limit ofthe radiative-transfer equation.12
Let us consider a path of light, which we assume to benormally incident on a semi-infinite random medium.Following Ref. 4, we express polarization vector E ina basis of two vectors �n, b� normal to tangent vectoru (if the path is regular enough, the Frénet frame is apossible choice), as shown in Fig. 1:
E�t� � c1�t�n�t� 1 c2�t�b�t� , (5)
where t is a parameter that goes from 0 to s alongthe path. Let us call f the angle between E and n att � 0, so c2�0��c1�0� � tan f. Since the polarizationevolves by parallel transport, �c1 � tc2 and �c2 � 2tc1,where t denotes the torsion on the trajectory, as foundmany years ago by Rytov.13 In the backscattering ge-ometry, n�t � s� �2n�t � 0� and b�t � s� � b�t � 0�;therefore we find that the polarization vector at theend of the path is
E�t � s� �2 cos�f 1 V�s��n�s�
1 sin�f 1 V�s��b�s� , (6)
where V�s� is a geometrical phase equal to the oppo-site of the integral of the torsion between t � 0 andt � s modulo 4p. In the analogy between a path oflight and a semif lexible polymer, the twist of the pathis zero for light (it would be nonzero only in a chiralmedium), and the writhing angle is precisely V. Thiswrithe is a real value since the path is open, and thatreal value is equal to the algebraic area of a randomwalk on a unit sphere, with the constraint that thepath goes from the north pole to the south pole in thebackscattering geometry. From Eq. (6), we find thatthe output intensity after the light has gone throughan analyzer crossed with respect to the direction of theincident polarization is proportional to sin2�2f 1 V�.Because the medium is random, this intensity must beaveraged with respect to all paths:
I��R� �Z
P 0�s,R�ds�sin2�2f 1 V�s��� , (7)
where P 0�s, R� is the distribution of the path lengthfor a given distance to the incident beam R and �. . .�denotes the average over paths of length s. Usingthe identity 2 sin2�2f 1 V� � 1 2 cos�4f�cos�2V� 1
sin�4f�sin�2V� and the fact that �sin�2V�� � 0, becausethe distribution of V is even, we write Eq. (7) in theform of Eq. (3) with I0�R� �
RP 0�s, R�ds and
C�R� �1
I0�R�
ZP 0�s,R�ds�cos�2V�s��� . (8)
The factor cos�2V� in Eq. (8) means that the contrastresults from grouping pairs of paths of opposite geo-metrical phases, and the sum over s means that thephases of any other paths are uncorrelated. Inter-estingly, a similar randomization of the phase occursin the theory of magnetoconductance of Andersoninsulators.14
To evaluate the distributions of V for fixed s,P �s, V� shown in Fig. 2, we use a Monte Carlo algo-rithm originally developed for semif lexible polymers.Random paths are generated with an exponentialdistribution of path length with a characteristic stepequal to l. The incident photons are normal to theinterface, but when light is exiting the medium all
Fig. 1. Representation of a typical path in a semi-infiniterandom medium in backscattering. The Frénet frame con-sists of tangent u, normal n, and binormal b vectors. Rdenotes the distance between end points, f is the initialangle between polarization vector E and normal n, and Vis the geometric phase.
Fig. 2. Distribution of geometric phase V for different val-ues of path length s and in the inset variance of the distri-bution as a function of s�l�.
Fig. 3. Contrast as a function of R: the crosses were ob-tained from Monte Carlo simulations by use of Eq. (8),and the squares are experimental values, obtained froman analysis of Stokes parameter Q.
the outgoing angles of the emergent photons are ac-cepted. The paths can be generated for an arbitraryratio of l��l. We calculated the geometric phase byclosing the paths on the momentum sphere with ageodesic.7 Because of this closure the distributionof V for short paths, s ,, l�, is peaked at zero, asis also found for planar random walks (Levy’s law).For long paths, s .. l�, the distribution of V widensuntil the polarization is completely lost. In thisregime the distribution P �s, V� should be Gaussianaccording to the central limit theorem. Indeed, wehave confirmed this point by numerically evaluatingthe moment of order four of the distribution. Fur-thermore, the variance of the distribution, whichwas quadratic for s ,, l�, becomes linear for s .. l�,as seen from the inset of Fig. 2. This means thatP �s, V� �
plp�ps exp�2V2lp�s�, which implies that
�cos 2V�s�� � exp�2s�lp�. In Fig. 3, we show thecorresponding curve for the contrast of the patterncalculated from Eq. (8), together with experimen-tal points, which we obtained by averaging Stokes
parameter Q of an image along two perpendiculardirections, thereby suppressing a possible contributionin cos�2f� present in Eq. (2). In the experiment acolloidal suspension of latex particles of negligibleabsorption (diameter 0.5 mm, wavelength l � 670 nm)was used, and the sample was �8.8l� thick. Thevalue of anisotropy parameter g in the simulation waschosen to match the experimental value g 0.82. Inthis figure one can see that the contrast decreasesexponentially as a function of distance R with acharacteristic distance of the order of lp l�, whichagrees with both theory and experiments.10,11 In thecentral region of the pattern, low-order scattering isdominant, as was confirmed numerically. This couldexplain the discrepancy between experiments andsimulations in this region, since our model only treatslow-order scattering events in an approximative way.
To conclude, we have developed a simple theo-retical framework to extend the idea of Berry’stopological phase to the backscattering of light in amultiple scattering medium. The randomization ofthe geometric phases is the process that leads todepolarization, which is most clearly seen when thescattering is peaked in the forward direction. Wehave substantiated our theory with experiments. Wehope that our work will motivate further studies on therole of geometric phases in the transport properties ofpolarization in random media.
We acknowledge many stimulating discussionswith T. Maggs, M. Cloitre, F. Monti, C. Boccara, andB. A. van Tiggelen. D. Lacoste’s e-mail address [email protected].
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