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an understanding of scattered radiation is deeply embedded within such scientific disciplines as astronomy , meteorology , climatology , and more recently biomedical imaging  @xcite . measurement of this radiation allows for the remote detection of object properties , through inspection of both the spectral , and spatial redistribution of energy . notably , the inclusion of polarization in this analysis serves to enrich the source of available information . an area of immense interest concerning such behavior , is the study of optical radiation scattering within biological media  @xcite . it was first realized that through appropriate filtering , polarized light could enable the selective imaging of either surface , or subsurface tissue layers  @xcite . since then , similar applications have helped to improve imaging modalities such as optical coherence tomography  @xcite , and light scattering spectroscopy  @xcite . in addition to the advancement of existing techniques , the recovery of physical properties solely based on the observed transformation of polarized light is the objective of much research  @xcite . interestingly , in media consisting of particles comparable to the wavelength in size ( mie regime ) , circular polarization can survive many more scattering events than linear polarization due to excessive forward scattering . linear states depolarize once the direction of the incident photon stream is eventually isotropized  @xcite . on the other hand , due to the preference of shallow angle scattering which generally preserves helicity , incident circularly or elliptically polarized states persist on a longer length scale . this effect is known as polarization memory @xcite . the extent of the polarization memory is strongly dependent on the size parameter  @xcite , and on the refractive index mismatch between the particles and background medium  @xcite . this offers an additional gauge by which to characterize medium properties , as measurements of the polarization state of multiply scattered light contain this information , even beyond the distance required to randomize photon direction , known as the transport length @xmath0 .    in this article , we address the dynamic relationship between circular polarization memory and the particles present in the scattering medium . recent studies have shown that the survival of linear polarization exceeds that of circular polarization in many biological tissues  @xcite . such behavior is characteristic of smaller rayleigh scattering particles , although these tissues exhibit anisotropic scattering typical of the mie regime . this has suggested an inability to predict depolarization rates in complex media based purely on bulk scattering properties such as the reduced scattering coefficient @xmath1 , the absorption coefficient @xmath2 , and the anisotropy @xmath3 factor  @xcite . here , we demonstrate a quantitative description of circular polarization memory in media consisting of a variety of independent scattering spheres , with total generality in particle size , and refractive index distribution . our method uses direct analysis of the single scattering behavior attained from mie theory , and a flexible integral approach towards characterizing the depolarization rate within complex media . this will allow for additional information to be gathered from diffusing light measurements , and clarify the connection between particle distribution , and the observed crossover between scattering regimes . mie theory offers an exact solution for scattering of a plane wave by a single spherical particle of any size . the relationship between the incoming , and scattered stokes vectors describing polarization can be expressed in terms of the scattering matrix  @xcite : @xmath4 where the elements of the scattering matrix are dependent on the scattering angle @xmath5 , and the stokes vectors are both defined with an orientation aligned to the scattering plane . for an incident right - circularly polarized state , with @xmath6 , @xmath7 , the stokes vector for the partial wave scattered in the @xmath5 direction , with @xmath8 azimuth is : @xmath9 which is identical for all @xmath8 due to the rotational invariance of the input stokes vector . the expression for the degree of circular polarization of this partial wave is : @xmath10 while the ensemble average of this quantity , denoted @xmath11 , over all scattering angles , weighted by the intensity of each partial wave is thus , @xmath12 after any arbitrary number of scattering events @xmath13 from identical independent spheres , the value of @xmath11 for all partial waves can be shown to follow an exponential decay  @xcite . here we will define the characteristic helicity survival parameter @xmath14 , which is a measure of the number of scattering events required to depolarize an incident circular state . the average degree of circular polarization as a function of scattering order @xmath13 is then : @xmath15 where @xmath14 can be found from the single scattering behavior calculated in eq . ( [ integral ] ) and the natural logarithm of eq . ( [ exponential ] ) with @xmath16 . this procedure can be carried out for a monodisperse system for any size parameter @xmath17 , and any refractive index ratio @xmath18 , where @xmath19 and @xmath20 are the refractive indices of the sphere , and background respectively , and @xmath21 is the particle radius . figure  [ depolarization ] shows the calculated survival parameter for a range of sphere sizes , and refractive indices . the value of @xmath14 can be seen to fluctuate on both large and small scales with respect to a change in the size parameter @xmath22 . for a medium consisting of a range of particle sizes , eq . ( [ exponential ] ) must be generalized to a distribution of exponentials : @xmath23 where @xmath24 is appropriately normalized such that , @xmath25 @xmath24 is the fraction of total intensity scattered , on average , by particles of size @xmath22 with corresponding helicity survival parameters given by @xmath26 . this function is directly proportional to the scattering coefficient of each particle size , @xmath27 . in other words , the resulting circular depolarization rate of a polydisperse system is dependent on not only the helicity survival parameters of the constituent particles , but also the relative scattering cross sections @xmath28 , and number densities @xmath29 . we then have : @xmath30 and finally , in terms of the volume fraction distribution @xmath31 : @xmath32 where @xmath33 is the single particle volume for spheres of size parameter @xmath22 . equation  ( [ integral ] ) integrates over the helicity survival parameters of all particles present in the medium , with each @xmath34 being calculated from the single scattering behavior . thus , with a database of values stored for @xmath26 , and @xmath28 , the effective helicity survival parameter for any polydisperse system of spheres defined by the distribution function @xmath24 can be found immediately , a key advantage of this approach . narrow polydispersion in scattering media effectively washes out the short scale fluctuations of particle size on the helicity survival @xmath14 . for example , media with particle number densities @xmath35 obeying a normal distribution with a small ( @xmath36 ) coefficient of variance in particle size are shown in fig . [ poly ] . here the ratio @xmath37 is used to show the extent of the circular polarization memory , where @xmath38 is the number of scattering events required to randomize photon direction _ i.e. _ the number of scattering events in one transport length @xmath0 . narrow polydisperse distributions shown in fig . [ poly ] are common in media such as solutions of polystyrene microspheres in water ( @xmath39 ) , where the coefficient of variance is typically @xmath40 . these solutions are widely used as simulating media , or  phantoms " , due to their well defined bulk scattering properties . = 8.5 cm    scattering from more complex materials such as biological tissue can include contributions from a wide array of particle sizes . disagreement between measurements of circularly polarized light scattered from tissue samples , and from matched phantoms has been reported  @xcite . this is likely due in part to the presence of both large mie scattering structures in tissue such as cell nuclei ( typically @xmath41 @xmath42 m ) , as well as smaller rayleigh - gans scatterers ( mitochondira , lysosomes _ etc . _ @xmath43 @xmath42 m )  @xcite . these phantom studies often use a monodisperse solution with particle size and concentration chosen to match the reduced scattering coefficient @xmath44 , and the scattering anisotropy @xmath3 , to another more complex medium . however this is an oversimplification , and fails to reproduce the depolarization behavior , as we shall demonstrate . to approximate scattering from cell nuclei with smaller interspersed cellular organelles , we have calculated @xmath14 for media where the volume fraction contributions from small scatterers , and large scatterers have been varied . here , @xmath31 is chosen to take a bimodal form , with one normally distributed contribution with size parameter @xmath45 , and another normally distributed contribution at @xmath46 . the fraction of the total volume of scattering material taken by the small scatterers is varied from @xmath47 ( nuclei only scattering ) to @xmath48 ( small organelle only scattering ) . the refractive index ratio used is @xmath49 which is typical of biological media with @xmath50 ( organelles ) , and @xmath51 ( cytoplasm )  @xcite . the results are shown in fig . [ cell ] . = 8.5 cm    it can be seen that the circular polarization memory rapidly deteriorates with the addition of small scatterers to the medium . for relative volume fractions of small particles above @xmath36 , the effectiveness of the medium to preserve multiply scattered circularly polarized light is significantly diminished . to emphasize the importance of adequately accounting for polydispersion in scattering media , we have also calculated the circular polarization memory ( @xmath37 ) for a set of media with only one dominant size . these points on the graph  ( fig . [ cell ] ) represent typical  monodisperse " optical phantoms with a @xmath36 size distribution about the mean diameter . in each case , these phantoms have been chosen to match the scattering anisotropy of the bimodal distribution used to model tissue . the circles represent phantoms consisting of particles with @xmath49 . the squares are calculated with @xmath39 , to represent polystyrene phantoms . the branching of these points is due to the multiple possibilities in particle size for certain anisotropy @xmath3 values , and the partial absence of data points expresses that the upper limit of scattering anisotropy for polystyrene microspheres is @xmath52 ( in water ) . it is clear from the figure that a significant overestimation is made of circular polarization memory when such phantoms are used , even if they are chosen such that @xmath53 , @xmath3 , and hence @xmath1 are equivalent . the shortcomings of monodisperse phantoms evident here are in direct agreement with the previously mentioned studies  @xcite . it is important to note that the ability of small scatterers to reduce the polarization memory of larger constituent particles is heavily dependent on the relative scattering cross section @xmath54 . in the example situation shown in fig . [ cell ] , @xmath55 spheres were chosen because they are more effective at reducing circular polarization memory than smaller ( @xmath56 ) spheres , which have intrinsically lower values of @xmath14 . this is due to the trade off between faster depolarization , but lower scattering efficiency as size is reduced . this demonstrates that depolarization of incident circular states within polydisperse media is an intricate process dependent on the specific size and refractive indices of each contributing particle . equation  ( [ integral ] ) can be generalized further to include variations in particle refractive index mismatch @xmath57 : @xmath58 where @xmath59 now represents the fraction of total intensity scattered by particles for which @xmath60 . the effective helicity survival parameter of the polydisperse medium can once again be found by inverting eq . ( [ exponential ] ) after evaluating this integral . to further highlight the flexibility of the approach presented here , fig . [ heatmap ] shows the magnitude of circular polarization memory for a wide range of size parameters @xmath22 , and refractive index ratios @xmath57 . each point on the image represents @xmath37 for a @xmath36 particle size distribution about the mean size parameter @xmath22 . incident circularly polarized light is seen to be preserved most in media where both @xmath61 , and particle size is large . such properties are consistent with large cellular structures in tissue . a complex cellular model could be proposed by the integration over all types of organelles present , including vacuoles with @xmath62 . however , the closely packed nature of scattering centers in many biological tissues may cause discrepancies from such models when dependent scattering arises , as shown in ref . thus , care is required when generalizing these findings which are a result of the independent scattering approximation for spherical particles . in summary , we have presented here a method for characterizing circular polarization memory in polydisperse media , where the particle distribution has been demonstrated to intricately determine the decay rate of incident circularly polarized light : narrow polydispersion about one dominant size parameter has the effect of smoothing out the short scale fluctuations seen with theoretical solutions containing perfectly identical centers . this type of polydispersion is typical of phantom media such as polystyrene microspheres , and demonstrates that models using pure monodispersion can differ somewhat from realistic media where statistical variations in particle size are unavoidable ( see fig . [ poly ] ) . in complex media , where size distribution spans from the rayleigh regime to the mie regime , the extent of circular polarization memory is shown to be greatly impacted by only a small volume fraction of sub - wavelength scatterers . in our particular example ( fig . [ cell ] ) we showed that circular polarization decays within approximately two transport lengths ( @xmath63 ) for media consisting of over @xmath64 relative volume fraction of small @xmath65 particles . with the method put forward here , it is now possible to rapidly calculate the helicity survival parameter @xmath14 for a medium with any general form of polydispersion in both size parameter @xmath22 , and refractive index ratio @xmath57 . thus , recovery of this parameter from degree of polarization measurements of semi - diffuse light can provide an alternate means towards characterizing particle distribution . furthermore , although circular polarization memory is a manifestation of anisotropic scattering , we have demonstrated the important distinction that it can not be characterized solely by the scattering anisotropy @xmath3 parameter . we acknowledge support from the william evans fellowship provided by the university of otago for professor steven jacques , the dodd - walls center for photonic and quantum technologies , and the department of physics at the university of otago . w.  cai , _ et al . , _  analytical cumulant solution of the vector radiative transfer equation investigates backscattering of circularly polarized light from turbid media " , phys . e * 74 * , 056605 ( 2006 ) .
we investigate the survival of circularly polarized light in random scattering media . 
 the surprising persistence of this form of polarization has a known dependence on the size and refractive index of scattering particles , however a general description regarding polydisperse media is lacking . through analysis of mie theory , we present a means of calculating the magnitude of circular polarization memory in complex media , with total generality in the distribution of particle sizes and refractive indices . 
 quantification of this memory effect enables an alternate pathway towards recovering particle size distribution , based on measurements of diffusing circularly polarized light .