Random matrix theory of polarized light scattering in disordered media

Abstract

In this work we present a method for generating random matrices describing electromagnetic scattering from disordered media containing dielectric particles with prescribed single particle scattering characteristics. Resulting scattering matrices automatically satisfy the physical constraints of unitarity, reciprocity and time reversal, whilst also incorporating the polarization properties of electromagnetic waves and scattering anisotropy. Our technique therefore enables statistical study of a variety of polarization phenomena, including depolarization rates and polarization-dependent scattering by chiral particles. In this vein, we perform numerical simulations for media containing isotropic and chiral spherical particles of different sizes for thicknesses ranging from the single to multiple scattering regime and discuss our results, drawing comparisons to established theory.

Full text

Random matrix theory of polarized light scattering in disordered
media
Niall Byrnes ID and Matthew R. Foreman ID 1
Blackett Laboratory, Department of Physics, Imperial College London, Prince Consort Road,
London SW7 2AZ, United Kingdom
ARTICLE HISTORY
Compiled May 20, 2022
ABSTRACT
In this work we present a method for generating random matrices describing electro-
magnetic scattering from disordered media containing dielectric particles with pre-
scribed single particle scattering characteristics. Resulting scattering matrices auto-
matically satisfy the physical constraints of unitarity, reciprocity and time reversal,
whilst also incorporating the polarization properties of electromagnetic waves and
scattering anisotropy. Our technique therefore enables statistical study of a variety of
polarization phenomena, including depolarization rates and polarization-dependent
scattering by chiral particles. In this vein, we perform numerical simulations for
media containing isotropic and chiral spherical particles of different sizes for thick-
nesses ranging from the single to multiple scattering regime and discuss our results,
drawing comparisons to established theory.
1. Introduction
Complex, disordered media are ubiquitous in nature, from cosmic dust in the inter-
stellar medium to tissue in the brain [1,2]. When light interacts with such media,
multiple scattering can cause severe deterioration of the spatio-temporal structure of
the incident field through randomization of amplitude, phase and polarization state.
Multiple scattering therefore can heavily degrade optical information [3,4] posing sig-
nificant challenges in many scientific disciplines, including telecommunications, remote
sensing, astronomy, medical diagnostics and optical imaging [5–9]. A detailed under-
standing of the transport of light in complex systems is paramount to overcoming
limitations imposed by multiple scattering, therefore necessitating development of ac-
curate modelling tools.
Theoretically modeling multiple scattering of polarized light is notoriously diffi-
cult. While in principle the scattered field follows exactly from Maxwell’s equations,
numerous approximations are generally required to render the mathematics tractable
[10]. Numerical solutions of Maxwell’s equations have been performed for scattering
systems of limited size, typically with dimensions on the order of ten wavelengths,
using the T-matrix method, time domain simulations and the coupled dipole approx-
imation [11–13]. A popular alternative approach for modeling low-density scattering
media is the radiative transfer equation (and its vectorial counterpart), which predicts
1Corresponding author: matthew.foreman@imperial.ac.uk
arXiv:2205.09423v1 [physics.optics] 19 May 2022

the specific intensity (Stokes vector) at a far field measurement point [14]. The radia-
tive transfer equations are frequently solved numerically using Monte Carlo approaches
that trace rays, thought of as ‘photons’, through the scattering medium [15–17]. The
path of each photon is simulated as a random walk where scattering events occur at
random positions at which the photon wavevector is updated probabilistically using a
specified phase function. The polarization state of a photon after scattering can then be
updated using an amplitude scattering matrix that can be customized according to the
type of particle being modeled. One drawback of the Monte Carlo technique is speed;
while much faster than directly solving Maxwell’s equations, a large amount of com-
putation is required to estimate statistical quantities with high accuracy. Simulations
must also be repeated when photons are injected with different angles of incidence.
In addition, traditional Monte Carlo methods are unable to reproduce correlations
such as the memory effect, although more recent studies have begun to address this
problem [18,19].
Scattering matrices, and the closely related transmission, reflection and transfer
matrices, provide an alternative description of a scattering medium [20]. Practically,
the scattering matrix (or other related matrices) can be determined through sequen-
tial measurements using a spatial light modulator to control the different degrees of
freedom of an electric field [21–25]. Once known, the scattering matrix determines the
response of a medium to an arbitrary incident field and enables the design of inci-
dent wavefronts that, rather than being distorted by multiple scattering, are tightly
focused or strongly transmitted well beyond the ballistic regime [26–28]. In addition,
when viewed statistically, correlations between different matrix elements can embody
phenomena such as the optical memory effect [29–32] or transmission-reflection correla-
tions, which have been exploited for imaging when the transmitted field is inaccessible
[33–35].
For complex media, the scattering matrix can be treated as a random matrix
sampled from some suitable matrix ensemble. This matrix ensemble is the set of all
scattering matrices corresponding to all possible microscopic configurations of a sys-
tem with a given set of macroscopic properties, such as particle density, mean particle
size etc. It is well known that for any non-absorbing system, physically admissible
scattering matrices are constrained to be unitary due to energy conservation, with an
additional symmetry constraint imposed when reciprocity or time reversal symmetry
holds [36,37]. The earliest random matrix models for the scattering matrix, namely
the circular ensembles, are based on the use of a uniform (Haar measure) distribution
over the unitary group [38]. A more sophisticated random matrix model is captured
in the DMPK equation, which describes the statistical evolution of the singular values
of the transmission matrix for a random medium [39,40]. While sufficient for revealing
universal properties of disordered media, such as the existence of highly transmitting
open eigenchannels [39], these models are largely limited to purely isotropic scattering
media and contain no adjustable parameters for exploring the multitude of phenomena
exhibited by real systems. Moreover, random matrix models typically only consider
scalar waves and thus can not describe polarization dependent effects. Generalizations
of the DMPK equation have been proposed, but are typically expressed in terms of
correlations between the singular values and vectors of the transmission and reflections
matrices [41–43]. These variables are physically unintuitive and their statistical prop-
erties can be difficult to relate to those of the elements of the scattering matrix. Monte
Carlo transfer matrix simulations for disordered waveguides with non-isotropic scat-
tering have also been performed, but to our knowledge have also not yet incorporated
polarization effects, which are particularly important for optical scattering [44]. In this
2

work we address these limitations by presenting a method for numerically generating
optical scattering matrices for random media of arbitrary thicknesses, incorporating
the polarization properties of light. Our method requires the prescription of the single
scattering properties of the particles that constitute the random medium and uses
a matrix cascade approach to simulate the multiple scattering regime. We consider
sparse distributions of randomly positioned particles such that each scatterer is in the
far field of all other scatterers. Arbitrary fields are expressed using a discrete angular
spectrum of plane waves, which facilitates the description of non-planar wavefronts
and allows the theory to be expressed in terms of the scattering of plane waves, for
which the literature is abundant.
The content of this paper is organized as follows. In Section 2, we cover the back-
ground theory relevant to the model. We begin in Section 2.1 by defining the scatter-
ing matrix and deriving expressions for its elements in the single scattering regime.
In Section 2.2, we detail the statistical properties of the scattering matrix and derive
expressions for the mean, covariance matrix and pseudo-covariance matrix associated
with the scattering matrix elements. The issue of enforcing necessary matrix symme-
tries on randomly generated matrices is briefly discussed in Section 2.3. We present
numerical simulations of random media consisting of dielectric spheres in Section 3.
Specifically, our method is explained in Section 3.1, with results presented in Section
3.2. In particular, we present statistical data for the transmission eigenvalues as well
as the scattered intensity, DoP, retardance and diattenuation for different outgoing
plane wave directions. For all of our results, we discuss their physical interpretations,
drawing comparisons to established theory. We end with a summary and conclusion
of our work.
2. Theory
In this section we give a comprehensive description of the theoretical model used in
our simulations. We begin by setting out the problem we wish to study and derive
expressions for the scattering matrix elements in the single scattering regime. We then
discuss the statistical properties of the scattering matrix elements, which can be related
to the properties of the individual scatterers in the medium. Finally, we discuss how
the matrix symmetries imposed by energy conservation and reciprocity are enforced.
2.1. The scattering matrix
Consider a slab of thickness ∆L, bounded by the planes z=−∆L/2 and z= ∆L/2
and infinite in transverse extent. Suppose that the slab contains Ndielectric particles
distributed sparsely enough so that each particle is in the far field (defined rigorously
below) of all the others. We assume that the boundaries of the slab are non-reflective so
that scattering only occurs due to the presence of the particles within the slab. Suppose
that the slab is illuminated by a right-propagating plane wave (‘right’ henceforth
meaning in the positive zdirection) with wavevector ki= (kix, kiy, kiz )T(T denoting
the transpose operator) where kiz >0, |ki|=k= 2π/λ and λis the wavelength.
The complex representation of the electric field associated with the incident wave at
position ris given by
Ei(r) = E0eiki·r=Zδ(κ−κi)E0ei(κ·ρ+kzz)dkxdky,(1)
3

where κ= (kx, ky)Tand ρ= (x, y)Tare the transverse wavevector and transverse
position vector, kz= (k2−k2
x−k2
y)1/2and δis the Dirac delta function. The vector
E0is constant and characterizes the polarization state of the incident wave.
Suppose now that the slab thickness ∆Lis sufficiently small so that the total
scattered field can be assumed to be composed of only single scattering contributions
from each particle. If the centre of the p’th particle is located at position rp, its single
scattering contribution to the total field Epin the far field (i.e. k|r−rp|  1) is given
by
Ep(r) = eik|r−rp|
ik|r−rp|Ap(∆r,ui)E0eiki·rp,(2)
where ∆r= (r−rp)/|r−rp|and ui=ki/k are unit vectors [45]. The 3 ×3 matrix
Ap(∆r,ui), which depends on the shape, size, orientation and morphology of the
scatterer, describes the transformation of the polarization state of the incident field to
that of the scattered field in the far field observation direction ∆r. Eq. (2) admits an
angular spectrum representation, which is given by
Ep(r) = ZAp(u,ui)ei(ki−k)·rp
2πkkz
E0ei(κ·ρ+kzz)dkxdky,(3)
where now r= (ρ, z)T,k= (κ, kz)Tand u=k/k [46]. Since rlies in far field of the
scatterer, the domain of integration in Eq. (3) is restricted to the set of all wavevectors
for which |κ|< k, i.e. homogeneous plane waves. Considering now the total electric
field on the planar boundaries of the scattering medium, we find the expressions
E(ρ,∆L/2) = Zhδ(κ−κi)I3+
N
X
p=1
At
p(κ,κi)
2πkkz
ei(ki−k)·rpiE0ei(κ·ρ+kz∆L/2)dkxdky,
(4)
E(ρ,−∆L/2) = ZN
X
p=1
Ar
p(κ,κi)
2πkkz
ei(ki−
e
k)·rpE0ei(κ·ρ−kz(−∆L/2))dkxdky
+E0ei(κi·ρ+kiz(−∆L/2)) ,
(5)
where Inis the n×nidentity matrix and we have defined At
p(κ,κi) = Ap(u,ui)
and Ar
p(κ,κi) = Ap(e
u,ui). We use a tilde to denote a wavevector with negative z
component, i.e. if u= (ux, uy, uz)Twith uz= (|u|2−u2
x−u2
y)1/2>0, then e
u=
(ux, uy,−uz)T. Assuming that the planar boundaries also lie in the far field of every
particle within the scattering medium, evanescent wave contributions to the integrals
in Eqs. (4) and (5) can also be neglected.
In Eqs. (4) and (5), the matrices At
pand Ar
pare continuous functions of transverse
wavevector. In reality, however, it is only possible to simulate the scattered field up
to some minimal resolution. We hence construct discrete counterparts to Eqs. (4) and
(5) by replacing the integrals with sums over a finite set of wavevectors. We define the
set K={−κNk,...,−κ2,−κ1,0,κ1,κ2,...,κNk},which consists of Nktransverse
wavevectors (henceforth referred to as ‘modes’) together with their additive inverses
and the two component zero vector 0= (0,0)T, which corresponds to the wavevector
4

(0,0, k)T. For each mode κi∈K, we also define an associated weight wi, where
Piwi=πk2, so that for any function fwe have the cubature scheme
Zf(κ)dkxdky≈X
i
f(κi)wi.(6)
Naturally, increasing the number of modes improves the accuracy of Eq. (6), albeit at
the expense of an increase in computation. Many different choices of modes and weights
are possible in principle, and the optimal choice of cubature scheme may depend non-
trivially on the forms of At
pand Ar
p. In this work, we used modes distributed on a
Cartesian grid in kspace, each having an equal weight given by wi=w=πk2/(2Nk+1)
for all i. Finally, we note that it is necessary to choose modes in inverse pairs to fully
exploit scattering reciprocity [37].
Given a cubature scheme defined as above, Eqs. (4) and (5) can be discretized to
E(ρ,∆L/2) =
Nk
X
j=−Nk
t(κj,κi)E0ei(κj·ρ+kjz ∆L/2)w, (7)
E(ρ,−∆L/2) = E0ei(κi·ρ−kiz∆L/2) +
Nk
X
j=−Nk
r(κj,κi)E0ei(κj·ρ+kjz ∆L/2)w, (8)
where
t(κj,κi) = δij
wI3+1
2πkkj z
N
X
p=1
At
p(κj,κi)ei(ki−kj)·rp,(9)
r(κj,κi) = 1
2πkkj z
N
X
p=1
Ar
p(κj,κi)ei(ki−
e
kj)·rp(10)
are transmission and reflection matrices. Note that we have replaced the differential
product dkxdkywith wand the delta function δ(κ−κi) with the normalized Kronecker
delta δij/w. For the transverse wavevectors, we use integer subscripts where negative
values correspond to modes listed in Kwith a negative sign, e.g. κ−1=−κ1, and
κ0=0.
As there are no sources in the planes z=−∆L/2 and z= ∆L/2, it follows from
the Maxwell equation ∇· E= 0 that only four of the nine elements of the transmission
and reflection matrices are independent [10]. These matrices may therefore be reduced
to 2 ×2 matrices, which is facilitated by introducing the standard spherical polar
coordinates basis vectors
ek(κ, kz) = k
k,eφ(κ, kz) = ˆ
z×ek
|ˆ
z×ek|,eθ(κ, kz) = eφ×ek
|eφ×ek|.(11)
For the special cases ek=±ˆ
z, we set eφ=±ˆ
y. We define the reduced 2×2 transmission
and reflection matrices to be t(j,i)and r(j,i)whose elements are defined by
t(j,i)mn =eT
m(κj, kjz )t(κj,κi)en(κi, kiz ),(12)
r(j,i)mn =eT
m(κj,−kjz )r(κj,κi)en(κi, kiz ),(13)
5

where mand nstand for either θor φ. Finally, for mathematical convenience (see Ref.
[37] for more details), we normalize the transmission and reflection matrices to ¯
t(j,i)
and ¯
r(j,i), which are given by
¯
t(j,i)=rkjz
kiz
t(j,i)w=δijI2+Cji
N
X
p=1
At
p(j,i)ei(ki−kj)·rp,(14)
¯
r(j,i)=rkjz
kiz
r(j,i)w=Cji
N
X
p=1
Ar
p(j,i)ei(ki−
e
kj)·rp,(15)
where Cji =w/(2πkpkj z kiz ) and At
p(j,i)and Ar
p(j,i)are defined analogously to t(j,i)
and r(j,i).
The indices iand j, which label the matrices ¯
t(j,i)and ¯
r(j,i), span from −Nkto
Nk, meaning there are a total of (2Nk+ 1)2transmission and reflection matrices. We
may form an overall transmission and reflection matrix by concatenating 2 ×2 blocks
¯
t(j,i)and ¯
r(j,i)for all pairs of incoming and outgoing modes taken from the set K.
Specifically, we define ¯
t(and ¯
ranalogously) to be the block matrix
¯
t=






















¯
t(−Nk,−Nk)· · · ¯
t(−Nk,−1) ¯
t(−Nk,0) ¯
t(−Nk,1) · · · ¯
t(−Nk,Nk)
.
.
.....
.
..
.
..
.
.....
.
.
¯
t(−1,−Nk)· · · ¯
t(−1,−1) ¯
t(−1,0) ¯
t(−1,1) · · · ¯
t(−1,Nk)
¯
t(0,−Nk)· · · ¯
t(0,−1) ¯
t(0,0) ¯
t(0,1) · · · ¯
t(0,Nk)
¯
t(1,−Nk)· · · ¯
t(1,−1) ¯
t(1,0) ¯
t(1,1) · · · ¯
t(1,Nk)
.
.
.....
.
..
.
..
.
.....
.
.
¯
t(Nk,−Nk)· · · ¯
t(Nk,−1) ¯
t(Nk,0) ¯
t(Nk,1) · · · ¯
t(Nk,Nk)
.(16)
The block ¯
t(3,−2), for example, describes transmission through the medium from mode
−κ2to mode κ3, i.e. from the incident right-propagating plane wave with wavevector
k−2= (−k2x,−k2y, k2z)Tto that with wavevector k3= (k3x, k3y, k3z)T. It is impor-
tant to remember that in reflection, each outgoing plane wave component propagates
to the left and has a wavevector with a negative zcomponent. The corresponding
block of the reflection matrix ¯
r(3,−2) therefore describes the scattering from the same
incident plane wave component to the left-propagating plane wave with wavevector
e
k3= (k3x, k3y,−k3z)T.
Analogous expressions to those presented thus far can be derived for a left-
propagating plane wave incident upon the right side of the scattering medium, yielding
an additional pair of transmission and reflection matrices ¯
t0and ¯
r0. Together, the ma-
trices ¯
t,¯
r,¯
t0and ¯
r0form the normalized scattering matrix ¯
S, which is given by
¯
S=¯
r¯
t0
¯
t¯
r0.(17)
Put simply, the scattering matrix fully describes how waves incident upon the medium
scatter into modes that propagate away from the medium, up to the resolution afforded
by the mode discretization.
6

2.2. Statistics of the scattering matrix elements
The expressions we have derived for the scattering matrix elements in Eqs. (14) and
(15) are deterministic: if the locations and properties of all the scatterers are known,
then in principle one can calculate the elements of ¯
S. In practice, however, the precise
locations of every scatterer within the slab may be unknown and may vary consider-
ably from one complex medium to another. It is therefore useful to think of ¯
Sas a
random matrix. Observing Eqs. (14) and (15), we see that the ‘randomness’ arises from
two physical sources: the positions of the scatterers, which contribute to the complex
exponential terms, and the morphological properties of the scatterers, i.e. shape, size,
orientation etc., which contribute to the matrix factors At
pand Ar
p.
Observing Eqs. (14) and (15), with the exception of the diagonal elements of
the transmission matrix (i=j), for which the argument of the complex exponential
is always 0, the expressions for the transmission and reflection matrix elements are
essentially random phasor sums. Under rather general conditions, such expressions are
known to be asymptotically Gaussian random variables as N→ ∞ [47]. For this to
hold, we require the assumption that a scatterer’s morphology is statistically indepen-
dent of its position, which we shall take to be the case. We may therefore reasonably
suppose that each of the matrix elements is marginally Gaussian distributed. It does
not automatically follow that the the elements of ¯
Sfollow a multivariate Gaussian
distribution, but we shall nevertheless assume that this is the case. The statistics of
a complex multivariate Gaussian distribution are fully described by three parameters:
the mean, covariance matrix and pseudo-covariance matrix, expressions for which we
shall now derive [48].
Starting from Eq. (14), we see that the mean value of ¯
t(j,i)is given by
h¯
t(j,i)i=δijI2+NCj ihAt
(j,i)ihei(ki−kj)·ri,(18)
where we have used the independence of scatterer position and morphology. We have
also assumed that each particle’s At
pmatrix is identically distributed, which allows
us to drop the psubscript. In order to compute the hexp[i(ki−kj)·r]iterm, it is
first necessary to specify a probability distribution function for the particle position
r. We suppose that the particles are distributed uniformly in the slab so that the
single particle distribution function is given by p(r)=1/V , where Vis the volume of
the slab (momentarily taken to be finite). This assumption is reasonable given that
each particle is in the far field of the others [49]. Since the slab is infinite in transverse
extent, both Nand Vare in fact infinite. We assume, however, that the particle density
n=N/V is finite and take the limit N, V → ∞, holding nconstant. Therefore, we
have
Nhei(ki−kj)·ri → nZ∆L/2
−∆L/2Z∞
−∞ Z∞
−∞
ei(ki−kj)·rdxdydz
= (2π)2n∆Lsinc (kiz −kjz )∆L
2δ(kix −kjx)δ(kiy −kjy ),
(19)
where sinc(x) = sin(x)/x. Replacing the delta functions in Eq. (19) with normalized
Kronecker delta symbols [δ(kix −kjx)δ(kiy −kjy)→δij /w], Eq. (18) ultimately becomes
h¯
t(j,i)i=δij I2+2πn∆L
kkiz
hAt
(j,i)i!.(20)
7

It is evident from Eq. (20) that the mean values of the transmission matrix elements are
only non-zero for blocks lying on the diagonal of ¯
t, which describe forward scattering.
The mean values of the reflection matrix elements can be calculated similarly. Starting
from Eq. (15), we arrive at the analogous result
h¯
r(j,i)i=δij
2πn∆L
kkiz
sinc kiz∆LhAr
(j,i)i.(21)
These values are also only non-zero for blocks lying on the diagonal of ¯
r. These blocks
correspond to reflections of plane waves whose wavevectors transform according to
k→e
k, i.e. scattering in the ‘specular reflection’ direction. The sinc function in Eq.
(21) is due to the randomness in zposition of the particles, which imparts a random
phase onto each singly scattered component of the total field [50].
Computing the covariances of the scattering matrix elements requires finding cor-
relations of the form h¯
t(j,i)ba¯
t∗
(v,u)dci, where i, j, u and vrefer to transverse wavevectors
(taken from K) and a, b, c and drefer to polarization states (θor φ). Assuming for
simplicity that we are not considering diagonal blocks of ¯
t(i.e. i6=j, u 6=v), we have
h¯
t(j,i)ba¯
t∗
(v,u)dci=CjiCvu
N
X
p,q=1
hAt
p(j,i)baAt∗
q(v,u)dcihei[(ki−kj)·rp−(ku−kv)·rq]i.(22)
The sum in Eq. (22) can be separated into two types of terms: those for which p=q
and those for which p6=q. Assuming that the particles in the medium are statistically
independent in all senses, the terms for which p6=qdecouple and, in the limit N→ ∞,
the right hand side of Eq. (22) reduces to the product h¯
t(j,i)abih¯
t∗
(v,u)cdi. In handling
the terms for which p=q, the average of the complex exponential can be dealt with
as in Eq. (19). Setting η=ki−kj−ku+kv, we find
Nheiη·ri= (2π)2n∆Lsinc ηz
∆L
2δ(ηx)δ(ηy).(23)
The right hand side of Eq. (23) is non-zero when ηx=ηy= 0, i.e.
kix −kjx =kux −kvx , kiy −kjy =kuy −kvy .(24)
This condition is precisely that of the memory effect, which manifests here as a cor-
relation between certain pairs of transmission matrix blocks [49]. Incorporating this
result into Eq. (22), we find that
h¯
t(j,i)ba¯
t∗
(v,u)dci−h¯
t(j,i)baih¯
t∗
(v,u)dci
=δRCijuv hAt
(j,i)baAt∗
(v,u)dcisinc ∆L
2(kiz −kjz −kuz +kvz ),(25)
where Cijuv =wn∆L/(k2pkizkjz kuz kvz) and δR= 1 when Eq. (24) is satisfied and 0
otherwise. The superscript Rhere stands for ‘regular’ correlations (to be contrasted
with ‘pseudo’ correlations shortly). An analogous result holds for h¯r(j,i)ba¯r∗
(v,u)dci −
h¯r(j,i)baih¯r∗
(v,u)dci, which can be found in Appendix A.
Calculating the correlation in Eq. (25) requires knowledge of the scattered field
due to a single particle, which is described by the elements of the matrices At
pand
8

Ar
p. It is worth noting, however, that these correlations can be equivalently described
by ensemble averaged Mueller matrices for the slab. Transformations between Mueller
matrix elements and field correlations are well documented in the literature (see for
example Ref. [51]). While both formalisms are informationally equivalent, reformulat-
ing the theory presented here in terms of Mueller matrices may be preferable in some
circumstances. For example, decompositions of the Mueller matrix are well known
and allow one to express a Mueller matrix in terms of simpler matrices that corre-
spond to familiar optical elements, such as a diattenuator, retarder and depolarizer
[52]. For the purpose of modelling a random medium, it may be simpler to begin
with a custom Mueller matrix with desired scattering characteristics, which can then
be translated into the corresponding field correlations. Moreover, the Mueller matrix
is relatively easy to determine experimentally as it can be calculated from intensity
measurements, without requiring interferometric techniques. In cases where the form
of a Mueller matrix is known, but analytic expressions for At
pand Ar
pare not, the
Mueller matrix still allows for the extraction of covariances that can used in numerical
simulations.
In addition to regular correlations as in Eq. (22), it is also necessary to consider
‘pseudo’ correlations, i.e. correlations of the form h¯
t(j,i)ba¯
t(v,u)dciwithout complex con-
jugation of the second term. These can be calculated in a similar manner to the regular
correlations, yielding the pseudo-covariance
h¯
t(j,i)ba¯
t(v,u)dci−h¯
t(j,i)baih¯
t(v,u)dci
=δPCijuv hAt
(j,i)baAt
(v,u)dcisinc ∆L
2(kiz −kjz +kuz −kvz ),(26)
where δP= 1 when
kix −kjx =−(kux −kvx ), kiy −kjy =−(kuy −kvy ).(27)
and 0 otherwise. It is worth nothing that pseudo-correlations do not influence the
statistics of any individual, non-diagonal 2 ×2 block within the transmission matrix,
which can be seen by noting that δP= 0 for i=uand j=v(i6=j). Given that
non-diagonal blocks also have 0 mean, it follows that every element within a non-
diagonal block of the transmission matrix is a circularly symmetric complex random
variable. The joint statistics of all of the elements of the transmission matrix, however,
do not obey circular symmetry, owing to the presence of pairs of modes for which
δP6= 0. For example, consider the pair of blocks ¯
t(j,i)and ¯
t(i,j), which are related
by swapping the incident and outgoing plane wave directions. Referring to Eq. (14),
the complex exponential terms for these blocks are given by exp[i(ki−kj)·rp] and
exp[i(kj−ki)·rp] = exp[−i(ki−kj)·rp] respectively. Thus, regardless of the distribution
of the particles within the medium, the complex exponential terms associated with
¯
t(i,j)are always the complex conjugates of those associated with ¯
t(j,i). This manifests
as a non-zero pseudo-correlation between the elements of the matrices ¯
t(j,i)and ¯
t(i,j),
for which it is simple to show that δP= 1. Analogously, pseudo-covariances can be
found for the other blocks of the scattering matrix, a summary of which is given in
Appendix A.
Finally, we note that correlations (both regular and pseudo) between elements
of different blocks of the scattering matrix, e.g. h¯
t(j,i)ba¯r∗
(v,u)dci, can be computed in
an identical fashion to those presented. For simplicity, however, we neglect these so
9

that each of the blocks of the scattering matrix, now assumed to be uncorrelated,
can be generated independently. The effects of these additional correlations will be
investigated in future works.
2.3. Matrix symmetries and random matrix generation
In addition to the correlations discussed in the previous section, additional relation-
ships exist between the elements of the scattering matrix due to fundamental physical
laws. Provided that there is no absorption or gain within the slab and that the scatter-
ing medium satisfies the reciprocity principle, the scattering matrix is constrained to
be unitary (S†S=I) and to possess certain lines of symmetries about which some of its
elements are equal [37]. These constraints must be satisfied in order for the scattering
matrix to represent a physically admissible scattering medium. In order to generate
a random scattering matrix that automatically satisfies these symmetry constraints,
it is first necessary to identity a set of independent parameters that fully capture
the degrees of freedom of the matrix. Once these parameters have been determined,
the matrix elements can be uniquely determined from the constraints. Importantly,
the set of independent parameters must be chosen so that their statistics can be re-
lated to the physical properties of the scattering medium. While it is straightforward
to accommodate the reciprocity constraint, unitarity, which manifests as a large sys-
tem of quadratic equations, is far less trivial to satisfy. A common strategy employed
in theoretical studies is the generalized polar decomposition, which parametrizes the
scattering matrix in terms of the singular values and vectors of its transmission and
reflection matrix blocks [39]. The connection between these parameters and the raw el-
ements of the scattering matrix, however, is non-trivial and unintuitive. Furthermore,
the singular vectors still comprise unitary matrices, and thus the problem of how to
randomly sample a unitary matrix with given statistics remains.
Instead of directly generating a random unitary matrix, an alternative strategy is
to first generate a non-unitary scattering matrix S0with desired statistical properties
and to then find a unitary matrix Sthat closely approximates S0. Naturally, the
resulting unitary matrix Sfrom this procedure will not possess the same statistical
properties as those prescribed for S0. Provided that the matrix Sis sufficiently ‘close’
to S0(in the sense that ||S0−S|| is small for some choice of matrix norm), however,
this issue becomes unimportant. Given any arbitrary matrix S0, it is well known that
the closest unitary approximation Sof S0is given by the unitary matrix that appears
in the polar decomposition of S0[53].
Using the results of Section 2.2,¯
t,¯
rand ¯
r0can be generated using a multivariate
Gaussian distribution. For diagonal blocks of ¯
t, as there is no phase variation in Eq.
(14), the matrix elements are non-random and we can instead use the result for the
mean transmission matrix in Eq. (20) as a fixed, non-random value. If reciprocity
holds, it is unnecessary to generate ¯
t0as it can always be calculated from ¯
t(see Ref.
[37]). Furthermore, reciprocity of ¯
rand ¯
r0is automatically enforced by a subset of the
correlations in Section 2.2. Given ¯
t,¯
r,¯
t0and ¯
r0, which form the non-unitary scattering
matrix S0, we then take the unitary part of the polar decomposition of S0to arrive at
a unitary scattering matrix S. Note that in light of, for example, Eq. (25), the squared
magnitudes of the elements of ¯
S0are proportional to the thickness ∆L. In the limit
∆L→0, it is clear that ¯
t,¯
t0→Iand ¯
r,¯
r0→O. The unitary approximation Salso
improves in accuracy as ∆Ldecreases, satisfying lim∆L→0||S0−S|| = 0.
Given the assumption of single scattering, we may only directly generate scatter-
10

ing matrices for thin slabs. Matrices for slabs of arbitrary thickness, however, can be
found by cascading many independent realizations of thin slabs. This is easily achieved
using transfer matrices, which possess the useful property that the transfer matrix for
a system composed of two contiguous slabs is given by the correctly-ordered product
of the transfer matrices of the individual slabs [39]. Scattering matrices can also be
cascaded, but the calculation is more complex (see Appendix B). An additional com-
plication however is that the statistical results in Section 2.2 assume that the slab is
centred at z= 0. This led to the emergence of the sinc factors in the expressions for
the covariances and pseudo-covariances associated with the matrix elements. If instead
the slab were centred at an arbitrary position z=L0, these factors would be different.
By performing a change of coordinates, we find that
¯
SL0= ΛL0
±¯
S0ΛL0
±,¯
ML0= ΛL0
∓¯
M0ΛL0
±,(28)
where ¯
S0and ¯
SL0are scattering matrices for the same physical medium, but located
with centers at z= 0 and L0respectively. The matrices ¯
M0and ¯
ML0are the corre-
sponding transfer matrices. The matrices ΛL0
±and ΛL0
∓are diagonal matrices containing
complex phasor terms, more details of which can be found in Appendix C. Thus, in
order to generate a random matrix describing a scattering medium centred at z=L0,
we can first generate ¯
S0, whose statistics are given by the results of Section 2.2, and
then compute ¯
SL0using Eq. (28).
Consider now the special case of a series of slabs, all of equal thickness ∆L,
positioned contiguously in the zdirection so as to constitute a single, continuous
medium. In this case, in order to find the scattering or transfer matrix for the overall
medium, it can be shown that it is sufficient to take the product of transfer matrices
of the form ¯
M= Λ∆L
±¯
M0, where ¯
M0can be randomly generated using the statistics
in Section 2.2 and the method outlined in this section. Taking the product of Nsuch
transfer matrices yields a transfer matrix for a scattering medium of thickness N∆L.
More details can be found in Appendix C. Finally, if necessary, scattering at the
boundaries of the slab can also be incorporated into the matrix cascade by including
additional scattering or transfer matrices that capture the surface effects at either
interface.
3. Numerical simulations
In this section we discuss numerical simulations of scattering matrices, performed for
random media containing different types of particles. We first outline our simulation
method and then present some results with discussion.
3.1. Method
Before generating random scattering matrices, it is first necessary to choose a set K
of transverse wavevectors and associated weights. Since we need only consider ho-
mogeneous plane waves, the set of all possible transverse wavevectors in k-space is
the interior of the circle |κ|2=k2
x+k2
y=k2. In a real scattering experiment, the
number of independent modes can be extremely large, on the order of millions per
square millimetre of illuminated surface area [28]. In our simulations, however, there
is a practical upper limit to the number of modes that can be used, as large scatter-
ing matrices quickly become unwieldy and computationally intensive. We distributed
11

modes on a Cartesian grid in kspace, including the origin and with lattice spacing
given by ∆kx= ∆ky= 0.1715k, rejecting modes lying on lattice points for which
k2
x+k2
y> k2. This spacing was chosen arbitrarily so that the set Kcontains a total
of 101 modes, which, given the block structure of ¯
S, means our scattering matrices
were of size 404 ×404. Of course, as the boundary of kspace is a circle, the interior
cannot be fully tessellated by a Cartesian grid and modes close to the boundary have
associated weights not given by the product ∆kx∆ky. To ensure that the weight for
each mode was equal and that the weights were correctly normalized, we decided to
give each mode the weight w=πk2/101. This value differs slightly to ∆kx∆ky, but
this discrepancy decreases as the number of modes increases.
We simulated two types of scattering media: one containing spherical, optically
inactive particles and one containing chiral particles exhibiting circular birefringence.
In either case, the single particle scattering properties are known theoretically (see,
for example, Ref. [45]). It was convenient to specify the matrix At/r/r0
(j,i)as a product of
rotation matrices and a 2 ×2 scattering matrix defined with respect to the scattering
plane. Details of this calculation can be found in Appendix D. We chose the wavelength
λ= 500 nm and considered isotropic spheres of three different sizes, namely x= 1,2
and 4, where x=ka is the dimensionless size parameter and ais the particle radius. For
each particle size, we used the same relative refractive index m= 1.2 and calculated
the At/r/r0
(j,i)factors using Mie theory. In addition, we performed simulations for chiral
spheres of two different size parameters, x= 1 and 4. For both size parameters, we
chose a mean relative refractive index ¯m= 1.2 and circular birefringence ∆m= 0.044
so that ml= ¯m+∆mand mr= ¯m−∆mwere the relative refractive indices experienced
by incident left and right handed circular polarization respectively. This birefringence
is such that left handed circularly polarized light is more strongly forward scattered
than right handed circularly polarized light.
For a given type of particle, the volume density nand slab thickness ∆Lthat
appear in the expressions for the mean and covariances in Section 2.2 are free param-
eters, not immediately constrained by any other variables. It is important, however,
that these parameters are chosen in a way that does not violate any of the basic as-
sumptions made in our model. To ensure that this was the case, we identified three
conditions that must be simultaneously satisfied. Firstly, we require kd 1, where
d= (1/n)1/3is a measure of the average spacing between the particles in the medium.
This condition ensures that the particles are all in the far field of each other. Secondly,
we require l/∆L1, where lis the mean free path of medium, given by the standard
formula l= (nσ)−1, where σis the scattering cross section [45]. This condition ensures
that the single scattering approximation holds. Since this second condition requires
that the slab thickness ∆Lis small, we identified a third condition ∆L/2a > 1 that
ensures that the slab is thick enough to contain the particles.
Instead of specifying ndirectly, it was simpler to start with a particle volume
fraction φand calculate the density via n=φ/Vp, where Vpis the volume of a single
particle. For all simulations we chose the value φ= 0.01. In specifying ∆L, a problem
we encountered was that, given the appearance of 1/kiz factors in, for example, Eq.
(20), the numerical values of the means and covariances can become large for graz-
ing incidence modes. In effect, these modes ‘see’ a larger thickness for the scattering
medium. To overcome this problem, we set a threshold value δ1 and demanded
that the elements of the mean transmission matrix were smaller than δfor all incident
modes (i.e. all blocks on the diagonal of ¯
t). Specifically, for all i, we solved the equation
δ= 2πn∆Lsmax,i/(kkiz ) for ∆L, where smax,i is the largest singular value of hAt
(i,i)i
12

Table 1. Summary of the physical parameters used in simulations.
Input Calculated Parameters Physical Checks
x m ∆m n/µm−3∆L/µml/µma/nm d/µmkd ∆L/2a l/∆L
1 1.2 0 4.737 1.177 311.57 79.58 0.595 7.48 7.34 264.7
2 1.2 0 0.592 1.126 88.08 159.15 1.191 14.96 3.53 78.2
4 1.2 0 0.074 1.173 35.87 318.31 2.382 29.93 1.84 30.6
1 1.2 0.044 4.737 0.969 311.57 79.58 0.595 7.48 6.09 321.7
4 1.2 0.044 0.074 0.969 35.87 318.31 2.382 29.93 1.52 37.0
and took ∆Lto be the minimum of all these values. We found that using a threshold
value δ= 0.1 gave values of ∆Lthat satisfied our conditions. A summary of all the
simulation parameters is given in Table 1, where each row corresponds to a different
parameter set. For chiral particles, the presented mean free path is that calculated
from Mie theory for an isotropic sphere with the same size parameter.
For each parameter set we generated the matrices ¯
t,¯
rand ¯
r0using a multivariate
Gaussian distribution, calculating ¯
t0from ¯
tas previously discussed. For each matrix
¯
S0we computed the unitary approximation ¯
Sas described in Section 2.3 and its
associated transfer matrix ¯
M. To properly account for propagation along the zaxis
when cascading multiple slabs, we then pre-multiplied each of these transfer matrices
by the constant matrix Λ∆L
±. In total, we randomly generated pools of 104transfer
matrices for each parameter set for slabs with thicknesses as shown in Table 1.
In order to access the multiple scattering regime, it is necessary to cascade at least
∼l/∆Ltransfer matrices, which, as can be seen, can be on the order of 102matrices.
Additionally, in order to compute good statistics, it is necessary to have a large number
of scattering matrices at any given thickness. Consequently, in total, a large number
of random matrices are required to generate data for random media with thicknesses
beyond a mean free path. To alleviate this computational burden, we first decided upon
a thickness step size (0.5lin our simulations) and calculated a secondary pool of 104
transfer matrices by cascading random selections of transfer matrices from the initial
matrix pool so that each resulting transfer matrix corresponded to a random medium
of thickness equal to the step size. In generating this secondary pool, some matrices
from the initial matrix pool are reused, which may introduce unwanted statistical
correlations between members of the secondary pool. Given that the number of possible
permutations in performing the matrix products is far greater than any realistic size
for the secondary pool, however, we found this issue to be unimportant. Finally, we
used an additional set of 104transfer matrices for actual data collection. For this
final set of transfer matrices, we progressed through media of increasing thicknesses
in steps of 0.5lto a final thickness of 30l, collecting data at each step. Progressing to
the next thickness is performed by multiplying each matrix in our final collection with
a randomly selected matrix from the secondary pool. Therefore, after the secondary
pool has been generated, no further random matrices are required.
When continuing to multiplying transfer matrices together, the elements tend to
diverge, as the set of transfer matrices is not a compact group [44]. Therefore, after a
certain point, it is necessary to convert all matrices used in the calculations into their
corresponding scattering matrices. While slower to cascade, unitarity of the scattering
matrices means they do not suffer from the same numerical problem.
13

Figure 1. (a) Mean transmission as a function of thickness for size parameters x= 1,2 and 4. Fitting curves
are of the form hτi= (1 + L/αl)−1, where αwas calculated from the data points. (b) Probability density
functions of transmission eigenvalues for thicknesses L/l = 1,5 and 30 for size parameter x= 2.
3.2. Model validation
In the following section we present a variety of statistical data calculated from our
simulations for thicknesses Lranging from the single to multiple scattering regimes. As
we have access to the entire scattering matrix, in addition to analyzing more familiar
characteristics of the scattered field in individual modes, such as the intensity and
DoP, we may also calculate parameters that are functions of larger sections of ¯
S,
such as correlations between different blocks or the transmission eigenvalues. In all of
the following data, averages were computed over all 104realizations of the scattering
matrix for each thickness.
3.2.1. Isotropic spheres
The following results are for optically inactive spheres whose parameters are given
in the first three rows of Table 1.
3.2.1.1. Transmission eigenvalues
Figure 1(a) shows the mean transmission eigenvalue hτi=htr(¯
t†¯
t)i/N, where tr
denotes the trace operator and Nis the size of the transmission matrix, as a function
of medium thickness. When all incident light is transmitted, regardless of incident
mode or polarization state, hτi= 1, whereas hτi= 0 when no light is transmitted. By
conservation of energy, a decrease in hτimust be compensated for by an increase in
the mean reflection eigenvalue hρi= 1 − hτi=htr(¯
r†¯
r)i/N. The main characteristics
of Figure 1are that hτidecreases monotonically with increasing medium thickness, as
is known to occur for isotropic systems [20], and that the rate of decrease is smaller
for larger size parameters. The dependence on particle size can be explained by single
particle scattering anisotropy: larger particles preferentially scatter light in the forward
direction, which results in a smaller decay rate for hτi. In Ref. [54], it was found
that in a quasi-one dimensional system with isotropic scattering, to lowest order, the
14

Figure 2. Mean intensity as a function of thickness for size parameters (a) x= 1 and (b) x= 2. The
intensity is shown in four different outgoing modes: forward transmission (FT), oblique transmission (OT),
oblique backscattering (OB) and direct backscattering (DB). A visual aid is provided in (a).
mean transmission eigenvalue decays as hτi= (1 + L/l)−1. We found that our curves
were reasonably well fit by functions of the form hτi= (1 + L/αl)−1, where αis
a fitting parameter given by 4.02, 13.25 and 37.51 for x= 1,2 and 4 respectively.
Physically, αl can be interpreted as a length scale over which the random medium
scatters isotropically, similar to the transport mean free path l∗=l/(1 −g), where g
is the anisotropy factor [55]. We found however that our value for αwas larger than
1/(1 −g). To explain this, we note that the expression 1/(1 −g) only accounts for
randomization of direction, whereas αalso incorporates isotropization of polarization
state.
Figure 1(b) shows the probability density function for the transmission eigenval-
ues of scattering matrices at thicknesses L/l = 1,5 and 30 for size parameter x= 2.
The distribution transitions from being highly peaked at τ= 1 for small thicknesses
to highly peaked at τ= 0 for large thicknesses. Notably, even for the largest thickness
L/l = 30, there still exist channels for which τ= 1. These open eigenchannels are
well known and have been studied extensively, both theoretically and experimentally,
particularly for scalar waves [20,28]. In our simulations however, these eigenchannels
also have a specific polarization structure. In order to construct such an eigenchannel
experimentally, such as in a wavefront shaping experiment, it would be necessary to
control both the relative intensity and polarization state of each plane wave compo-
nent of the incident field. Considering the eigenchannel with largest transmission, we
found that altering the polarization state of any individual plane wave component
while keeping its relative intensity constant resulted in a decrease of the total trans-
mitted intensity. Careful control of the incident polarization state may therefore lead
to enhanced transmission over the case of scalar waves. We found similar behaviour
for x= 1 and 4, but the rate at which the distribution evolves with thickness is greater
for x= 1 and smaller for x= 4, as expected due to scattering anisotropy.
3.2.1.2. Scattered intensity
Figures 2(a) and (b) show the mean plane wave intensity hIiin several outgoing
15

modes for a normally incident plane wave and size parameters x= 1 and 2. We focused
our attention on four different modes: the transmitted wave parallel to the incident
field (forward transmission, or FT); the transmitted wave for which κ/k ≈(3∆kx,0)T
(oblique transmission, or OT); the reflected wave for which κ/k = (3∆kx,0)T(oblique
backscattering, or OB) and the backscattered wave propagating in the opposite di-
rection to the incident field (direct backscattering, or DB). For each mode, hIiwas
calculated by taking the ensemble average vector norm of the first column of the ap-
propriate matrix block. Since the scatterers are isotropic, hIiis independent of incident
polarization state.
Observing FT in Figure 2(a), we see that hIidecays exponentially, but the decay
rate changes at around L/l ∼10, becoming smaller for large thicknesses. The initial
exponential decay is the well-known Beer-Lambert law, which is given by hIi=e−L/l
and is shown in the figure as a black line. For larger thicknesses, the change in decay
rate occurs due to light being scattered back into the forward direction (i.e. an increase
in the ‘incoherent’ intensity). The notable bend in the decay curve can therefore be
thought of as a transition to the multiple scattering regime. Before this transition
occurs, our data points are systematically larger than those predicted by the Beer
Lambert law, which we attribute to numerical inaccuracies stemming from our sim-
plistic cubature scheme.
Looking at OT in Figure 2(a), for small thicknesses we see that the intensity is
small and increases with thickness. In this regime, scattering is weak and intensity
increases as more light is scattered away from FT and into OT. For large thicknesses,
the intensity behaviour is similar to FT, settling on a limiting decay trajectory. The
behaviour in reflection is conjugate to that of transmission. In OB, the intensity is
initially small, but increases monotonically. The same behaviour is observed in DB, but
the intensity values are ∼1.8 times larger. This intensity enhancement is a signature
of the coherent backscattering effect, which emerges naturally from our simulations
from the enforcement of reciprocity in the scattering matrices. This enhancement is
less than ideal (a factor of 2) due to the non-zero size each mode occupies in k-space.
Figure 2(b) shows similar trends to Figure 2(a). The most notable differences are that
the reflected intensities increase at slower rates and the transmitted intensities decay
at a slower rate, both of which are also a result of scattering anisotropy.
3.2.1.3. Degree of polarization
In Figure 3, we show the DoP in the same four modes discussed in Section 3.2.1.2
for both a linearly and circularly polarized, normally incident plane wave. The DoP
can be found by calculating the ensemble average Mueller matrix for each mode, from
which the average scattered Stokes vector for different incident polarization states,
and thus the DoP, can be deduced. We emphasize that for any individual realization
of a scattering medium the scattered field is fully polarized. The DoP in this context
is therefore a measure of the distribution of scattered polarization states across the
ensemble of random media.
Figure 3(a) shows the DoP versus thickness in FT. As is evident from the graph,
the DoP decays more slowly for larger particles, regardless of the incident polarization
state. Furthermore, for x= 1, we see that linear polarization better preserves its DoP
over greater thicknesses than circular polarization, but the opposite is true for x= 2
and 4. This phenomenon, sometimes called the polarization memory effect, is well
understood and can be explained by scattering anisotropy [55,56]. A similar trend can
16

Figure 3. Degree of polarization as a function of thickness for incident linearly (×markers) and circularly (◦
markers) polarized light and size parameters x= 1 (blue), 2 (orange) and 4 (green) in (a) forward transmission
(FT), (b) oblique transmission (OT), (c) oblique backscattering (OB) and (d) direct backscattering (DB).
be observed in Figure 3(b), which shows the DoP in OT. The most notable difference
is that, particularly for x= 1, the DoP begins to decay immediately, as opposed to
at L/l ∼5 for FT. This is due to the presence of the incident field in FT and absence
thereof in OT.
The behaviour of the DoP in OB, as shown in Figure 3(c) is much more inter-
esting. The most obvious feature is that the DoP retains a residual, non-zero value
as L/l → ∞ for all particle sizes and polarization states. This residual DoP can be
explained by noting that in reflection, unlike transmission, a significant contribution
to the total field comes from low-order scattering sequences that occur close to the
medium’s surface [57]. Another striking feature is the non-monotonicity of the DoP for
circular polarization and size parameters x= 2,4 (and the absence of such behaviour
for x= 1). Specifically, the DoP can be seen to dip to a minimum value before in-
creasing again and settling on a limiting value. This occurs at L/l ∼0.5 for x= 2 and
at L/l ∼6.5 for x= 4. There is also a non-trivial dependence between the limiting
DoP value, size parameter and incident polarization state.
17

To explain some of these phenomena, we note that, roughly speaking, the re-
flected field is the sum of three types of contributions: low scattering order contri-
butions from scattering sequences occurring close to the medium’s surface (type I);
polarization-randomizing, high order scattering contributions from long, circuitous se-
quences deep within the medium (type II) and polarization-maintaining, high order
scattering contributions from long, largely forward-directed sequences deep within the
medium (type III). As type I contributions occur near the slab boundary, their overall
magnitude should be largely independent of thickness. The latter two contributions,
however, should increase in magnitude with thickness.
For x= 1, since large angle scattering is more probable than for x= 2 or 4, type
I contributions dominate the total backscattered field for all thicknesses. The DoP de-
cays relatively slowly as type II contributions, which give a polarization-randomizing
background, gradually increase with thickness. As scattering is relatively isotropic,
type III contributions are comparatively weak and thus less relevant. To verify this
claim, we observed distributions of scattered polarization states over the Poincar´e
sphere for different thicknesses. We found that for all thicknesses, these distribu-
tions remained concentrated at the polarization state that would result from a single
backscattering event, with an increasing isotropic background for larger thicknesses.
The situation is different for x= 2 and 4. Since larger particles scatter more
strongly in the forward direction, type I contributions, which require large angle scat-
tering events, are comparatively much weaker. For incident linearly polarized light,
type I and III contributions both tend to preserve the incident polarization state. Al-
though type I contributions are weaker for x= 4 than x= 2, type III contributions
are greater for x= 4 than x= 2. There is thus a non-trivial relationship between the
relative magnitudes of these contributions as particle size changes, the exact balance
of which dictates the non-monotonicity of the limiting value of the DoP for linear
polarization.
For x= 2 and 4, the situation is again different for incident circularly polarized
light. While type III contributions maintain incident helicity, type I contributions
result in a helicity flip. Therefore, in transitioning from small to large thicknesses,
the distribution of scattered states on the Poincar´e sphere must transition from being
highly focused at the helicity flipped pole (a single scattering, type I dominant regime)
to being relatively isotropic, but concentrated at the pole with the same helicity as
the incident field (a multiple scattering, type III dominant regime). Although both of
these extremes correspond to relatively large values for the DoP, in performing this
transition, there is an intermediate thickness at which the distribution of scattered
states on the Poincar´e sphere shows no preference for either pole, in which case the
DoP is small. It is precisely this thickness that corresponds to the dips in the DoP.
The dip is more obvious for x= 4 than x= 2 and occurs at a larger thickness
because photons are able to penetrate further into the medium for x= 4 before their
directions are randomized. This behaviour has been observed experimentally in oblique
backscattering from suspensions of polystyrene spheres [58].
As a final remark, we note that in Figure 3(d), which shows similar trends to
Figure 3(c), the DoP tends to values close to 1/3 for x= 2 and 4. This is the value
predicted for scattering matrices drawn from the circular orthogonal ensemble in the
direct backscattering direction [59]. For x= 1, the dominance of type I contributions to
the reflected field means that the phase function of the slab better resembles that of the
individual particles in the medium, which is not isotropic. This may explain why the
DoP for x= 1 deviates strongly from this value, particularly for incident circularly
polarized light. The assumption of isotropic scattering, which is necessary for the
18

Figure 4. Diattenuation and retardance histograms for size parameter x= 1 in forward transmission. (a)
shows a heatmap of probability distribution functions for diattenuation at different thicknesses. The dashed
contour close to the origin indicates a region in which the colors are saturated and the probability density is
greater than 3. (b) shows a selection of histograms corresponding to horizontal cross-sections of data in (a).
(c) and (d) show analogous data for retardance, with the dashed contour in (c) showing a region for which the
probability density is greater than 1.
circular ensemble to be an appropriate model, is better satisfied at large thicknesses
for x= 2 and 4, whose scattered fields are dominated by multiply scattered light.
3.2.1.4. Diattenuation and retardance
An additional pair of parameters that can be useful in assessing the polarimetric
properties of a scattering medium are diattenuation and retardance. As we have access
to the full scattering matrix, these can computed for any 2 ×2 block using the polar
decomposition [51]. Unlike the DoP, which is dependent on the incident polarization
state, diattenuation and retardance are computed from the entire 2×2 block. We note
that, as the scattering matrix is unitary, the diattenuation we compute is solely due
to scattering and not absorption (dichroism).
Figure 4(a) shows a heat map of probability density functions for diattenuation D
in FT at different thicknesses for x= 1. The values of the color bar are dimensionless
and represent probability density. The color bar values are accurate for thicknesses
beyond 10l, but are saturated for shorter thicknesses in a small region close to the
origin as outlined by the dashed contour. In this region, Dis strongly peaked close to
0, as a weakly scattering medium, which largely preserves the incident field, cannot be
strongly diattenuating. In Figure 4(b), density functions for a selection of thicknesses
as indicated by the horizontal dashed lanes in Figure 4(a) are shown more clearly. As
can be seen, the diattenuation density function transitions from being a delta function
19

Figure 5. As per Figure 4, albeit for scatterers with size parameter x= 4 and direct backscattering (DB).
Dashed contours in (a) and (c) demarcate regions for which the heat map has been clipped for probability
densities ≥2 and ≥1 respectively.
p(D) = δ(D) at L= 0 to a limiting distribution given by p(D) = 3D2as L→ ∞. This
limiting distribution is precisely that predicted by a random 2 ×2 matrix of uncorre-
lated, complex Gaussian entries [59]. The transition of the diattenuation distribution is
therefore related to the decorrelation of the elements of the scattering matrix. Figures
4(c) and 4(d) show analogous data for retardance in FT. Qualitatively, the behaviour
is similar to diattenuation and the density function makes a similar transition from
p(R) = δ(R) to the limiting distribution p(R) = 2 sin2(R/2)/π, which is also that
predicted by a random Gaussian matrix. For small thicknesses, we found that the
distributions of the diattenuation and retardance vectors were concentrated at po-
larization states expected from single scattering theory. These distributions however
became isotropic over the Poincar´e sphere for large thicknesses, meaning that no par-
ticular polarization state is preferentially scattered on average in the large thickness
limit. For individual medium realizations, however, as diattenuation tends to be quite
large (hDi= 0.75), there will exist random polarization states that are transmitted
much more strongly than others.
Figure 5shows a similar set of plots to those of Figure 4, but for particle size
x= 4 and for DB. The main differences between Figures 4and 5are the behaviour of
retardance, the rates of evolution of the density functions and the limiting probability
density functions. As shown in Figures 5(a) and 5(b), owing to the absence of the
incident field, the diattenuation distribution tends to a limiting distribution (this time
given by p(D) = 2D) at a shorter thickness. In Figure 5(c), for small thicknesses, the
retardance is peaked close to R=π, which is the value expected by single particle
backscattering. The retardance distribution evolves to p(R) = sin(R/2)/2 at larger
20

Figure 6. Mean intensity as a function of thickness for size parameter x= 4 and incident (a) left and
(b) right circular polarization. The intensity is shown in four different outgoing modes: forward transmission
(FT), oblique transmission (OT), oblique backscattering (OB) and direct backscattering (DB). A visual aid is
provided in (a).
thicknesses, as can be seen in Figure 5(d). The fact that these limiting densities differ
to those in Figure 4is another peculiarity of the DB direction. Due to reciprocity,
additional correlations exist between the elements of the 2×2 block, even in the large
thickness limit. The previous results relevant to a matrix of uncorrelated Gaussian
entries therefore no longer apply. It has been shown, however, that these limiting
densities are in fact those predicted for diagonal blocks of a random matrix sampled
from the circular orthogonal ensemble [59].
3.2.2. Chiral spheres
The following results are for chiral spheres, whose parameter sets are given in
the final two rows of Table 1. For these particles, since the mean free path depends
on the incident polarization state, to better illustrate the polarization dependence of
the statistics of the scattered field we decided to normalize the medium thickness L
by the mean free path calculated for an optically inactive sphere with the same size
parameter.
3.2.2.1. Transmission and reflection
Figures 6(a) and 6(b) show the mean scattered intensity for chiral spheres with
size parameter x= 4 for incident left handed circularly polarized light (LHC) and
right handed circularly polarized light (RHC) respectively. While the overall trends
closely resemble those in Figure 2, there is now a clear polarization dependence. As was
the case with the isotropic spheres, much of the behaviour can be explained through
consideration of scattering anisotropy. LHC, which is more preferentially forward scat-
tered than RHC, decays slower in FT. For RHC, the mean intensity is correspondingly
larger in the backscattering directions. Similar behaviour was seen for size parameter
x= 1.
21

Figure 7. DoP as a function of thickness for incident linearly polarized light (LIN), left handed circularly
polarized light (LHC) and right handed circularly polarized light (RHC) for size parameters x= 1 (Hmarkers)
and 4 (Nmarkers) in outgoing modes (a) forward transmission (FT), (b) oblique transmission (OT), (c) oblique
backscattering (OB) and (d) direct backscattering (DB).
3.2.2.2. Degree of polarization
The DoP statistics for chiral spheres of size parameters x= 1 and 4 (indicated by
downward and upward pointing triangles respectively) are shown in Figure 7. We have
included three different incident polarization states: LHC, RHC and LIN, the last of
which refers to incident linearly polarized light. The trends we see are similar to those
for isotropic spheres in Figure 3, but with a few interesting differences. In Figure 7(a),
a dip in the DoP can now be seen in FT for RHC and x= 1. For isotropic spheres,
these dips in the DoP were only present in reflection for larger spheres. We can explain
this phenomenon however by invoking a similar argument to before. For RHC and a
thin medium, the distribution of scattered polarization states on the Poincar´e sphere
was sharply peaked at the pole corresponding to RHC. For large thicknesses, however,
we found that this distribution transitioned to one that was relatively isotropic, but
with a slight concentration towards the LHC pole due to the particle chirality. Thus, as
22

before, in transitioning between these two distributions, there exists an intermediate
thickness at which the DoP attains a minimum value. This does not occur for incident
LHC, as the initial distribution of scattered states is already concentrated at the
LHC pole and no such transition occurs as thickness increases. For incident LIN, the
distribution is initially focused at a point on the equator of the Poincar´e sphere and, in
transitioning towards a distribution focused at the LHC pole, there is no intermediate
thickness at which the distribution is isotropic across the entire sphere. Therefore, no
such dip in the DoP occurs. We note that we also expect a dip in the DoP to occur for
incident RHC and x= 4, but as the DoP decay rate is small for this size parameter,
the medium is not thick enough, even at 30l, for the dip to occur. In OT, as shown
in Figure 7(b), we see that the behaviour resembles FT in the same way that Figure
3(b) resembles Figure 3(a).
In Figure 7(c), we see that for x= 1 the DoP behaviour is similar to that of Figure
3(c), but note that the DoP decays more quickly for RHC than for LHC. All three
incident polarization states settle on similar limiting DoP values, with LHC and RHC
∼0.2 and LIN ∼0.17. For x= 4, dips in the DoP are again visible for RHC and LHC.
Unlike in Figure 7(a), these dips arise due to the flipping or preservation of helicity for
different scattered field contributions, as was the case in Figure 3(c). For LHC, which
scatters more anisotropically, this dip occurs at a larger thickness (L/l ∼27) than
for RHC (L/l ∼3). In Figure 7(d), we see that while linearly polarized light retains
a large DoP for large thicknesses irrespective of particle size, dips in the DoP occur
again for LHC and RHC and x= 4. For x= 1 and incident circularly polarized light,
we also see dips in the DoP, but the exact trends are unclear. For DoP on the order of
10−2a larger number of realizations than was used in this work is required for good
numerical convergence.
4. Conclusion
To conclude, we have presented a method for randomly generating scattering matri-
ces for sparse, complex media that incorporates the polarization properties of light,
scattering anisotropy and the physical constraints of unitarity and reciprocity. Fur-
thermore, we are able to model random media in the multiple scattering regime using
a matrix cascade, only requiring knowledge of the single scattering properties of the
particles contained within the medium.
We have validated our model by reproducing known behaviour for systems con-
sisting of randomly distributed spherical particles, such as the dependence of the rate
of depolarization on the incident polarization state. We have also shown that some of
the polarization statistics of our scattering matrices in the large thickness limit can be
related to those of random Gaussian matrices and diagonal blocks of matrices drawn
from the circular orthogonal ensemble. We have demonstrated the flexibility of our ap-
proach by considering the example of a medium containing chiral particles, for which
we found that the polarization properties of the scattered field depend on the helicity
of the incident polarization state. We were able to analyze the more intricate details
of the rate of decay of DoP by considering the evolution of scattered polarization state
distributions on the Poincar´e sphere, which is easily done in our framework given that
we have access to the entire scattering matrix. In addition to the data presented here,
other possible studies include analyzing the polarization properties of the transmission
eigenchannels and the polarization properties of correlations between different matrix
blocks, such as, for example, the memory effect. We reserve these topics for future
23

studies.
The biggest limitation of our model is the currently achievable angular resolution
of the scattered field, as this directly influences the size of the scattering matrix, which,
when large, requires a lot of memory and computation time when a large number of
samples is required for the study of statistical quantities. Generation of individual
scattering matrices, however, is very fast, taking only seconds or minutes, depending
on the medium thickness and number of modes. We therefore envisage that our method
will serve as a complement to the already existing Monte Carlo techniques and may
prove advantageous in certain applications, particularly where correlations between
different matrix elements are of interest.
Appendix A. Covariances and pseudo-covariances of scattering matrix
elements
Table A1 contains a list of expressions for the covariances and pseudo-covariances of
the elements of the scattering matrix. Referring to the first column of Table A1, type
‘Regular’ refers to the covariance of the form h¯
B(j,i)ba ¯
B∗
(v,u)dci−h¯
B(j,i)baih ¯
B∗
(v,u)dci,
where ¯
Bdenotes an arbitrary block of the scattering matrix (i.e. one of ¯r,¯
t,¯
t0or
¯r0). Type ‘Pseudo’ refers to the pseudo-covariance of the form h¯
B(j,i)ba ¯
B∗
(v,u)dci −
h¯
B(j,i)baih ¯
B∗
(v,u)dci. All symbols are as defined in the main text.
Table A1. Summary of the regular and pseudo covariances of the elements of the scattering matrix.
Type Block ¯
BExpression
Regular ¯
t δRCijuvhAt
(j,i)baAt∗
(j,i)dcisinc( L
2(kiz −kjz −kuz +kvz ))
¯r δRCijuvhAr
(j,i)baAr∗
(j,i)dcisinc( L
2(kiz +kjz −kuz −kvz ))
¯
t0δRCijuv hAt0
(j,i)baAt0∗
(j,i)dcisinc( L
2(−kiz +kjz +kuz −kvz ))
¯
r0δRCijuv hAr0
(j,i)baAr0∗
(j,i)dcisinc( L
2(−kiz −kjz +kuz −kvz ))
Pseudo ¯
t δPCijuvhAt
(j,i)baAt
(j,i)dcisinc( L
2(kiz −kjz +kuz −kvz ))
¯r δPCijuvhAr
(j,i)baAr
(j,i)dcisinc( L
2(kiz +kjz +kuz +kvz ))
¯
t0δPCijuv hAt0
(j,i)baAt0
(j,i)dcisinc( L
2(−kiz +kjz −kuz +kvz ))
¯
r0δPCijuv hAr0
(j,i)baAr0
(j,i)dcisinc( L
2(−kiz −kjz −kuz −kvz ))
Appendix B. Composition law for scattering matrices
Suppose two slabs, S1and S2, with planar faces perpendicular to the z-axis are ar-
ranged such that S1is to the left of S2, i.e. z1< z2where z1and z2are the zcoordinates
of the centers of the slabs. If S1and S2have scattering matrices S1and S2where
S1=r1t0
1
t1r0
1and S2=r2t0
2
t2r0
2,(B1)
24

then the scattering matrix Sfor the overall system composed of S1and S2is given by
S=r t0
t r0=r1+t0
1r2Qt1(J2Nk+1 ⊗σz)(t2Qt1)T(J2Nk+1 ⊗σz)
t2Qt1r0
2+t2Qr0
1t0
2,(B2)
where Q= (I−r0
1r2)−1,⊗is the Kronecker product, σz= diag(1,−1) and Jnis the
n×nexchange matrix containing 1s on its anti-diagonal and 0s elsewhere.
Appendix C. Scattering and transfer matrices centred at arbitrary
positions
Suppose that a slab of thickness ∆Lis centred at L0so that the zcoordinate of
the position of any particular particle within the slab is confined to the interval
[L0−∆L/2, L0+ ∆L/2]. Inspecting Eq. (14), it can be seen that if such a slab has
transmission matrix block ¯
tL0
(j,i), then ¯
tL0
(j,i)=¯
t0
(j,i)exp[i(kiz −kjz )L0], where ¯
t0
(j,i)de-
scribes a medium identical to that described by ¯
tL0
(j,i), but for which the zcoordinate
of the position of each particle has been translated by L0so that each particle is now
confined to the interval [−∆L/2,∆L/2] centred at the origin. Consideration of the
block structure of ¯
tL0, which is the full transmission matrix for scattering medium
centred at z=L0, then leads to the equation ¯
tL0= ΛL0
−¯
t0ΛL0
+, where
ΛL0
+=


eik−NkzL0. . . 0
.
.
.....
.
.
0. . . eikNkzL0


⊗1 0
0 1(C1)
and ΛL0
−= (ΛL0
+)∗. On the right hand side of Eq. (C1), the arguments of the exponen-
tials in the first matrix run through all modes in the set Kin order. Similar reasoning
for the other blocks of the scattering matrix leads to Eq. (28) in the main text, where
ΛL0
±= diag(ΛL0
+,ΛL0
−) and ΛL0
∓= (ΛL0
±)∗.
Suppose now that a series of Nscattering layers are situated with centres located
at (from left to right) L1, L2,· · · LNand let ¯
MLi
idenote the transfer matrix for the
i’th layer. Using Eq. (28), the overall transfer matrix is given by
¯
M=¯
MLN
N. . . ¯
ML3
3¯
ML2
2¯
ML1
1
= ΛLN
∓¯
M0
NΛLN
±. . . ΛL3
∓¯
M0
3ΛL3
±ΛL2
∓¯
M0
2ΛL2
±ΛL1
∓¯
M0
1ΛL1
±
= ΛLN
∓¯
M0
N. . . ¯
M0
3ΛL3−L2
±¯
M0
2ΛL2−L1
±¯
M0
1ΛL1
±
,(C2)
where, as always, a superscript 0 denotes the corresponding transfer matrix when the
slab is centred at the origin. Deriving the final line of Eq. (C2) makes use of the
identity ΛL2
±ΛL1
∓= ΛL2−L1
±, which follows trivially from the definitions. In the special
case L1= 0 and Li+1 −Li= ∆Lfor 1 ≤i≤N−1, as would be the case for contiguous
slabs of equal thicknesses ∆L, Eq. (C2) can be written in the form
¯
M= ΛN∆L
∓
N
Y
i=1
Λ∆L
±¯
M0
i.(C3)
25

Therefore, a transfer matrix for a medium of thickness N∆Lcan be computed by
cascading Nmatrices of the form Λ∆L
±¯
M0, where ¯
M0can be randomly generated as
discussed in the main text. Note that the final matrix ΛN∆L
∓outside of the product in
Eq. (C3) imparts global phase terms onto each 2×2 block of ¯
M(and ¯
S) and therefore
does not alter any of the intensity or polarization statistics of the random matrix given
by the product.
Appendix D. Computation of single particle scattering matrices
Consider a particular pair of incident and outgoing plane waves with wavevectors ki
and kjrespectively. Let eki,eφi ,eθi ekj ,eφj and eθj be the associated spherical polar
vectors as defined as in Eq. (11). The vectors eki and ekj define the scattering plane,
whose unit normal vector is given by e⊥= (eki ×ekj )/|eki ×ekj |. We then define the
vectors eki=e⊥×eki and ekj=e⊥×ekj so that (eki,e⊥i,eki) and (ekj,e⊥j,ekj )
form right-handed triads. In the case that eki and ekj are parallel, we take eki=eθi,
ekj=eθj and e⊥i=e⊥j=eφi.
Consider now the incident wavevector kiand let us temporarily drop the subscript
i. the vectors eθ,eφ,ekand e⊥all lie in the same plane with unit normal vector given
by ek. In general, however, the vectors eθand eφwill not align with ekand e⊥. Let
θbe the angle between eθand ek, defined such that −π < θ < π, where θ > 0 if
(eθ×ek)/|eθ×ek|=ek(i.e. ekis an anti-clockwise rotation of eθabout ek) and θ < 0
if (eθ×ek)/|eθ×ek|=−ek(i.e. ekis a clockwise rotation of eθabout ek). See Figure
D1 for a graphical representation of these vectors, along with the electric field vector
E, which also lies in the same plane.
Given θ, the electric field vector, which can be written as E= (Eθ, Eφ)Twith
respect to the basis vectors eθand eφ, can be transformed to E= (Ek, E⊥)Twith
respect to ekand e⊥by
Ek
E⊥=R(θ)Eθ
Eφ,R(θ) = cos(θ) sin(θ)
−sin(θ) cos(θ).(D1)
Note that conventions for the directions of the unit vectors described here are not con-
sistent throughout the literature. For example, in Ref [45], the normal to the scattering
plane is taken to be e0
⊥=−e⊥. In this case, the electric field component perpendicular
to the scattering plane is given by E0
⊥=−E⊥. Following the convention used in Ref
[45], it can ultimately be shown that
Eθj
Eφj=R(−θj)σzS2S3
S4S1σzR(θi)Eθi
Eφi,(D2)
where S1,S2,S3and S4are scattering coefficients defined with respect to the scattering
plane and θiand θjare the angles between eθi,ekiand eθj ,ekjrespectively, following
the sign convention as discussed. Finally, the matrix At/r
(j,i)is given by the product of
the five matrices in Eq. (D2).
26

Figure D1. Vectors used in scattering calculations. The vectors E,eθ,eφ,ekand e⊥all lie in the plane
perpendicular to ek. The angle θis positive in the diagram.
Acknowledgements
This work was funded by the Royal Society (grant numbers RGF\R1\180052,
UF150335 and URF\R\211029).
Disclosure statement
The authors report there are no competing interests to declare.
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