Helicity flip of the backscattered circular polarized light

Helicity flip of the backscattered circular polarized light

Vladimir Kuzmina and Igor Meglinskib,c

a Department of Mathematics, St. Petersburg Institute of Commerce and Economics,194021 St. Petersburg, Russia

b Cranfield Health, Cranfield University, Cranfield, MK43 0AL, Bedfordshire, UKc Jack Dodd Centre for Quantum Technology, Department of Physics,

University of Otago, New Zealand

ABSTRACT

We study coherent and non-coherent backscattering of circularly polarized light from turbid media. We find that the signof helicity of circular polarized light does not change for a medium of point-like scatterers and can change significantly forthe medium with high anisotropy of scatterers. The helicity flip is observed when the light scattering is described in termsof the Henyey-Greenstein scattering phase function. The angular dependence of the sum of coherent and non-coherentparts of backscattering also exhibits a helicity flip.

Keywords: Backscattering, circularly polarized light, Monte Carlo, helicity flip

1. INTRODUCTION

Earlier there has been shown that multiple scattering of circularly polarized light exhibits peculiar features. 1–5 It has beenshown that the polarization of backscattered light survives more scattering events than the direction of its propagation; ithas also been pointed out that the helicity of backscattered light depends noticeably on the size of scatterers. 5–8

The backscattered circular polarized light is expected to consist mainly of the cross-polarized component resultingfrom the specular reflection. However, it turns to be true only for the scatterers, which size is smaller than wavelength ofincident radiation; for the media with larger scatterers the opposite situation happens. 6, 7 This phenomenon, known as thepolarization memory, was explained by specific features of the Mie tensor phase function.

Kim and Moscoso6 numerically solved the vector radiative transfer equation for a circular polarized plane-wave pulse;it has been concluded that for a medium with large scattering particles the helicity is determined by the successive near-forward scattering events strongly depending on the angular characteristic of the Mie scattering. In 7 the backscattering ofcircular polarized pulses has been studied for spatially separated source-detector geometry; the helicity flip has been alsoascribed to the Mie scattering.

In this letter we study numerically the backscattering of circularly polarized light from a turbid tissue-like medium,where the scattering is described by the Henyey-Greenstein scattering phase function. We find that the cross-polarizedcomponent continues to exceed the co-polarized one independently of scattering anisotropy, in case of wide incident beam,as could be expected for specular reflection. We find, however, that for the spatially separated incident and backscatteredbeams the co-polarized component is dominant for a medium with high scattering anisotropy of scatterers. Similar resultshave been obtained in7 for Mie scattering.

2. METHOD DESCRIPTION

Recently, we have developed9,10 a numerical technique for a direct simulation of propagation of electromagnetic fieldavoiding the Mueller-Stokes formalism. In framework of the technique we trace the polarization of electric field along astochastic trajectory built for a scalar field. Calculating the n-th order contribution, we find four components of the Stokesvector, but only at the end of a trajectory, after final event of scattering. This approach has been comprehensively validatedby comparing the results of simulation with the exact theoretical solution. 9, 10 Here, based on this technique we considerthe backscattering of linear and circular polarized light for the typical experimental probe geometry, when the source and

Correspondence: kuzmin [email protected] or [email protected]

Biomedical Applications of Light Scattering IV, edited by Adam P. Wax, Vadim Backman, Proc. of SPIE Vol. 7573, 75730Z · © 2010 SPIE · CCC code: 1605-7422/10/$18 · doi: 10.1117/12.841193

Proc. of SPIE Vol. 7573 75730Z-1

detector on the surface of the medium are spatially separated from each other. In case of circular polarization for smallparticles the cross-polarized component is shown to exceed the co-polarized component; however for large particles atdistances between incidence and scattering sites exceeding transport length the co-polarized component becomes dominant;the important feature is that such a helicity flip occurs only for a contribution of several lower scattering orders. The novelfeature is that this phenomenon appears to be valid also for the Heney-Greenstein phase function. It means that such achange of dominant component with the transition from small to large scatterers, found in, 7 can be attributed rather to theform of pulse of incident radiation than to the scattering phase function.

The non-coherent ladder part of the correlation function of electric field E(r) at a distance r from scattering volume canbe presented as: ⟨

δE∗β2(r)δEβ1(r)

⟩= r−2S Lβ2 β1 α2 α1 (k f , ki) × E∗α2

Eα1 , (1)

where 〈...〉 denotes ensemble average, β2 and β1 are polarization indices of pair of complex-conjugated scattered fields, andα2 and α1 are polarizations of incident field E, S is the detection square, and k i and k f are the incident and scattered wavevectors. We consider the weak scattering limit, λ � l, where λ is the wavelength and l is the photon mean free path. Thefourth rank tensor L(k f , ki) describes the radiation transfer in a random medium and obeys the Bethe-Salpeter equation.Iterating the equation we present the tensor L(k f , ki) as the sum in scattering orders10

L(k f , ki) =∑

n≤m

L(n)(k f , ki) + ΔLm(k f , ki), (2)

where L(n)(k f , ki) is the contribution of the n-th scattering order, and Δ Lm(k f , ki) is the contribution of scattering ordershigher than the m-th one.

A normal incidence of optical radiation at the semi-infinite (z ≥ 0) medium is considered; x, y, z are the cartesian axes.If the incident light is linearly polarized along the x axis, the intensities of co-polarized and cross-polarized components ofscattered light are I‖ = Lxxxx and I⊥ = Lyyxx. Arguments ki and k f are omitted for brevity.

The circular cross-polarized backscattered component can be written as:

Icross =12

(Lxxxx + Lyyxx + Lyxyx − Lxyyx

), (3)

and co-polarized backscattered component:

Ico =12

(Lxxxx + Lyyxx − Lyxyx + Lxyyx

)(4)

The n-th order contribution L(n)(k f , ki) can be calculated as an average over sampling of N ph photon packets, or trajec-tories, given by a succession of n scattering events,10

L(n)(k f , ki) =1

Nph

Nph∑

i=1

W (i)n M′(k f , kn n−1, . . . , ki) ⊗ M(k f , kn n−1, . . . , ki) exp(−l−1z(i)

n / cos θ f ). (5)

Here W (i)n and z(i)

n are, respectively, the weight and the distance to the medium boundary of i-th photon after n scatteringevents. The tensor

M(k f , kn n−1, . . . , ki) = P(k f )∞∏

j=2

P(k j j−1)P(ki) (6)

presents the chain of projection operators P(k) = I − k−2k ⊗ k, transforming the polarization of the incident field alongthe trajectory containing n scattering events in points R 1,R2, . . . ,Rn; the wave vector k j j−1 = k(R j − R j−1)/

∣∣∣R j − R j−1

∣∣∣describes the wave propagating between two successive scattering events. Tensor M(k f , kn n−1, . . . , ki) acts upon field E,and M′(k f , kn n−1, . . . , ki) – upon E∗. The direct product P′(k j j−1) ⊗ P(k j j−1) multiplied by the Heney-Greenstein phasefunction describes the scattering matrix transforming pair of complex-conjugated fields at the j-th scattering event.

It should be pointed out that typically simulation of the electromagnetic field propagation is based on the Stokes vectortracing. We trace directly the electromagnetic field following the transformation of the initial polarization under action of


tensor M(k f , kn n−1, . . . , ki) calculated along a particular trajectory. In case of circular polarization six components of twoorthogonal vectors, shifted by phase in a quarter of wavelength, are traced along each trajectory.

The series in scattering orders is known to converge very slowly. It has been shown that scattered intensity 11 continuesto increase even for the upper cut m = 104 of scattering orders. We have studied numerically the intensity dependence onscattering orders and find an approximate decay I n ∼ n−α for the large n and α ≈ 1.5, for the scalar and electromagneticfield, independently of scattering anisotropy. Extrapolating this decay beyond m we obtain:

ΔLm(k f , ki) ≈ (α − 1)−1mL(m)(k f , ki). (7)

With this approximation the results obtained appear to be independent of m beginning with m = 10 3 for Rayleighscattering, g = 0, and with m = 104 for anisotropic scattering, g = 0.9. The mean cosine of scattering angle g = 〈cos θ〉characterizes the size of scatterers.

3. RESULTS AND DISCUSSION

Using the described approach we calculate the backscattering of polarized light from a turbid scattering medium. InTable 1 the polarized components of scattered radiation are presented for two media, with small, g = 0, and large, g = 0.9,anisotropy of scattering. The small g values correspond to Rayleigh scattering from a point-like particle system, whereasthe large g values, 1 − g � 1 corresponds to scattering from a medium with scatterers much larger than the wavelength.

Table 1. The backscattered intensities of scalar and electromagnetic fields, presented in units of energy density (4π)−2∣∣∣E(in)∣∣∣2, and polar-

ization ratios, calculated numerically, and theoretical data.

g I(scalar) I‖ + I⊥ I⊥/I‖ Ico/Icross

g = 0, theory 4.228 4.588 0.517 0.617g = 0 4.228 4.615 0.523 0.618g = 0.9 4.55 4.40 0.81 0.95

For the point-like particle system the achieved results agree well with the values obtained theoretically by Milne’s exactsolution and its generalization12, 13 (see Table 14.5.1 in14).

As is seen the polarized component I‖ exceeds the depolarized component I⊥, whereas the cross-polarized componentIcross exceeds the co-polarized component I co, as it could be expected for a specular reflection, for any scattering anisotropy.The residual polarization is larger for Rayleigh scattering due to greater share of lower scattering orders, than for themedium with higher anisotropy of scattering particles. The sign of helicity remains unchanged for any value of anisotropy.The data presented in Table 1 are intensities collected from the infinite area, whereas the effect of helicity flip has beenobserved for the spatially separated incident and backscattered beams. 7 We have also simulated the backscattering for sucha geometry.

In Figs. 1 and 2 the polarized components are shown as function of distance d between incidence and backscatteringpoints, in the units of transport length l∗ = l(1 − g)−1. We assume that the transversal size of incident and backscatteredthin plane wave beams is smaller than the transport length. In Fig. 1 there are presented the linear and circular polarizedcomponents, related to the total non-coherent part of backscattered intensity, I = I ‖+ I⊥, for Rayleigh scattering, g = 0. Thepolarized component I‖ is larger than the depolarized one I⊥, and the cross-polarized component I cross remains dominantover Ico as it could be expected for a specular reflection, at any distances. With d increasing the light becomes moredepolarized due to smaller weight of lower scattering orders. In Fig.2 the relative components of backscattered light arepresented for a system with high scattering anisotropy, g = 0.9. For linear polarization the picture is qualitatively the sameas for the system with Rayleigh scattering. However for circular polarized light at d of order of the transport length thereoccurs the helicity flip. Let the incident light be right-hand circular polarized; the backscattered light, being dominantly leftcircular polarized at d < l∗, becomes the right circular polarized at d > l∗; at distances of order of two transport length theright circular polarized component exceeds approximately two times the left circular polarized component. Thus, for theright circular polarized incident light the backscattered light is dominated by the left polarized component in case of smallscatterers at any distances while in case of large scatterers and large incidence-detector distances it is dominated by the


Figure 1. The backscattered intensity of linear (left) and circular (right) polarized light as function of source-detector separation d, forisotropic scattering, g = 0. Diamonds represent polarized I‖/I (left) and cross-polarized Icross/I (right) components; circles show thedepolarized I⊥/I and co-polarized Ico/I components, left and right, respectively.

Figure 2. The backscattered intensity for linear (left) and circular (right) polarized light as function of source-detector separation d, forhighly anisotropic scattering medium, g = 0.9. As in Fig. 1 diamonds represent polarized I‖/I (left) and cross-polarized Icross/I (right)components; circles show the depolarized I⊥/I and co-polarized Ico/I ones, left and right, respectively.


Figure 3. Angular dependence of the backscattered polarized light for Rayleigh g = 0 (left) and anisotropic g = 0.9 (right) scattering:rhombus, circles, triangles and squares represent I‖ + ICBS

‖ , I⊥ + ICBS⊥ , Icross + ICBS

cross, and Ico + ICBSco , respectively; same units, as in Table 1

right polarized component. The important feature is that such a helicity flip occurs for a finite number of scattering eventsm of order of several dozens in case g = 0.9; for larger values of m the difference between the co- and cross- polarizedcomponents becomes of order of statistical error at distances d exceeding transport length l ∗.

We perform also simulations of coherent backscattering. The coherent backscattering (CBS) is known to be presentedas a sum of cyclic diagrams C(k f , ki). A cyclic diagram can be obtained from the ladder one, performing a permutationof polarization indices as well as wave vectors k∗f ↔ k∗i of complex conjugated field.15 In particular, for strictly backwardscattering, k f = −ki, we can write Cβ2β1α2α1 (−ki, ki) = Lα2β1β2α1 (−ki, ki) less the single scattering contribution. Thus, for thelinear polarization the co- and cross- polarized components of CBS are readily expressed through the ladder, non-coherentcomponents: ICBS

‖ = Lxxxx − I(single) and ICBS⊥ = Lxyyx. Similarly, for the circular cross- and co- polarized components wehave

ICBScross =

12

(Lxxxx − Lyyxx + Lyxyx + Lxyyx

)− I(single) (8)

ICBSco =

12

(Lxxxx + Lyyxx − Lyxyx + Lxyyx

). (9)

In general case, for backscattered light inclined at the angle θ along the x axis, the factor cos(k(x(i)

n − x(i)0 ) sin θ

)should be

inserted into the sum (5).

Table 2 presents the results of simulation of CBS enhancement, known as the ratio of the sum of coherent and non-coherent parts to the non-coherent component, for linear, H ‖,⊥ = (ICBS

‖,⊥ + I‖,⊥)/I‖,⊥, and circular, Hcross,co = (ICBScross,co +

Icross,co)/Icross,co, polarizations. The statistical error for number of photon packets 10 5 to 106 is less than one percent for

Table 2. The enhancement of coherent backscattering.

g HS calar H‖ H⊥ Hcross Hco

g = 0, theory 1.882 1.752 1.120 1.251 2g = 0 1.880 1.752 1.122 1.244 2g = 0.9 1.997 1.994 1.035 1.125 2

absolute values and about 0.001 for relative values; calculation of one plot takes about two hours. For the point-like particlesystem the results obtained agree remarkably with the theoretical data. 12, 14

The specific linear dependence on scattering angle in the narrow range kl ∗θ ≤ 1 is exhibited by the components Lxxxx

and Lyyxx only. Their difference turns to be a much wider Lorentzian than the triangle CBS peak. 13 It explains the features


of angular behavior of the polarized components of backscattering. For non-coherent contribution into backscatteringcomponent I‖ is larger than I⊥ and Icross is larger than Ico; therewith a gap between them decreases with the scattering angleincreasing. As for the CBS contribution the co-polarized component I CBS

co exhibits specific linear dependence on scatteringangle, and component I CBS

co does not.

Thus, for linear polarization the CBS enhancement is observed only in the polarized component I CBS‖ ; therewith the sum

of coherent and non-coherent contributions into the polarized components larger that that into depolarized one, I ‖ + ICBS‖ >

I⊥ + ICBS⊥ for any scattering angle. For circular polarization in case of Rayleigh scattering the sum of coherent and non-coherent cross-polarized components always exceeds the corresponding sum of co-polarized components due to largershare of the lower scattering orders. However in case of highly anisotropic scattering, 1 − g � 1, the co-polarizedcomponents, Ico + ICBS

co exceeds the sum of cross-polarized ones, Ico + ICBSco > Icross + ICBS

cross for strictly backward scatteringdue to smaller gap between non-coherent contributions into co- and cross- components. For larger scattering angles thepicture becomes ordinary, cross- polarized component exceeds the co-polarized one.

In Ref.8 the authors maintained that in case of exactly backward scattering the total co-polarized component exceedsthe cross-polarized one even for the Rayleigh scatterers. However this conclusion contradicts to the exact theoreticalresults12, 14 obtained earlier within a Milne-like approach; for ratio of co- and cross- components theory gives χ = (I co +

ICBSco )(Icross + ICBS

cross)−1 = 0.9865395, which can be obtained with data from Tables 1 and 2; we find χ = 0.994 in a fair

agreement with the theory We think that a slightly dominant co-polarized component was found in 8 due to non-zero sizeof scatterers.

In Fig 3 we present the angular dependence of the total backscattering polarized components for two scattering media,with g = 0 and g = 0.9, correspondingly. The co-polarized and circular cross- polarized components I CBS

‖ and ICBSco are

seen to exhibit the well-known linear dependence on scattering angle, in kl ∗θ units, in accordance with the diffusion theory.For the Rayleigh scattering the cross-polarized components exhibiting no CBS peak exceeds nevertheless the co-polarizedones; for a medium with high anisotropy of scattering particles, g = 0.9, the sign of helicity of backscattered light is seento change due to the coherent contribution decreasing with scattering angle.

4. SUMMARY

Thus, we have shown that there is no change of helicity sign with the scattering angle for a point-like particle system. Onthe contrary it is clearly observed for a system with larger scatterers.

Finally, we conclude that the helicity flip of the circular polarized light occurs in systems with different forms ofscattering, and can be observed in the tissue-like media where the scattering is typically described in terms of Henyey-Greenstein scattering phase function. We find that this phenomenon manifests itself in case of limited number of scatteringevents and, apparently, can be attributed to the pulse character of incident radiation rather than to the specific form ofscattering.

ACKNOWLEDGMENTS

The work is supported in part by NATO (project no. ESP.NR.CLG.982436).

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Helicity flip of the backscattered circular polarized light