Symmetry constraints for vector scattering and transfer matrices containing evanescent components: Energy conservation, reciprocity, and time reversal

Abstract

In this paper we study the scattering and transfer matrices for electric fields defined with respect to an angular spectrum of plane waves. For these matrices, we derive the constraints that are enforced by conservation of energy, reciprocity, and time reversal symmetry. Notably, we examine the general case of vector fields in three dimensions and allow for evanescent field components. Moreover, we consider fields described by both continuous and discrete angular spectra, the latter being more relevant to practical applications, such as optical scattering experiments. We compare our results to better-known constraints, such as the unitarity of the scattering matrix for far-field modes, and show that previous results follow from our framework as special cases. Finally, we demonstrate our results numerically with a simple example of wave propagation at a planar glass-air interface, including the effects of total internal reflection. Our formalism makes minimal assumptions about the nature of the scattering medium and is thus applicable to a wide range of scattering problems.

Full text

PHYSICAL REVIEW RESEARCH 3, 013129 (2021)
Symmetry constraints for vector scattering and transfer matrices containing evanescent
components: Energy conservation, reciprocity, and time reversal
Niall Byrnes and Matthew R. Foreman *
Blackett Laboratory, Department of Physics, Imperial College London, Prince Consort Road, London SW7 2AZ, United Kingdom
(Received 30 November 2020; accepted 29 January 2021; published 10 February 2021)
In this paper we study the scattering and transfer matrices for electric fields defined with respect to an angular
spectrum of plane waves. For these matrices, we derive the constraints that are enforced by conservation of
energy, reciprocity, and time reversal symmetry. Notably, we examine the general case of vector fields in
three dimensions and allow for evanescent field components. Moreover, we consider fields described by both
continuous and discrete angular spectra, the latter being more relevant to practical applications, such as optical
scattering experiments. We compare our results to better-known constraints, such as the unitarity of the scattering
matrix for far-field modes, and show that previous results follow from our framework as special cases. Finally, we
demonstrate our results numerically with a simple example of wave propagation at a planar glass-air interface,
including the effects of total internal reflection. Our formalism makes minimal assumptions about the nature of
the scattering medium and is thus applicable to a wide range of scattering problems.
DOI: 10.1103/PhysRevResearch.3.013129
I. INTRODUCTION
Scattering and transfer matrices are important mathemati-
cal tools that have been applied in a variety of fields, including
the transport of electrons in wires [1,2], telecommunications
[3], acoustics [4,5], and photonic crystals [6,7]. Within both
formalisms, a scattering object or medium is viewed as a
“black box” and the scattering and transfer matrices describe
the coupling between different modes in the surrounding
regions.
In optics, the scattering matrix, which describes the reflec-
tion and transmission of electromagnetic modes by a system,
is particularly powerful in the study of disordered media for
which the underlying scattering potential is practically un-
knowable, but for which the reflected and transmitted fields
are experimentally measurable [8]. In many modern scattering
experiments, the degrees of freedom of an electromagnetic
field are explored through wave-front shaping using spatial
light modulators, which has facilitated experimental measure-
ments of scattering matrices of complex media for both scalar
and vectorial light [9–11]. Knowledge of these matrices and
their statistical properties has revealed the existence of highly
transmitting “open eigenchannels,” even for optically thick
systems [12–14], and has greatly enhanced the prospect of
imaging through multiple scattering media, such as biological
tissue, through careful wave-front control and the exploitation
of scattered field correlations, such as the memory effect
*Corresponding author: matthew.foreman@imperial.ac.uk
Published by the American Physical Society under the terms of the
Creative Commons Attribution 4.0 International license. Further
distribution of this work must maintain attribution to the author(s)
and the published article’s title, journal citation, and DOI.
[15–21]. Scattering matrices have also been employed in
theoretical studies of random laser modes [22], information
transfer through random media [23], coherent backscattering
[24], and Anderson localization [25].
Alternatively, the transfer matrix describes coupling be-
tween modes on either side of a scattering medium, provided
that the geometry of the system permits a meaningful iden-
tification of two opposite sides. Examples of such systems
include optical waveguides [26–28] and stratified media con-
sisting of a series of contiguous slabs [29,30], such as
superlattices [31,32] and multilayer thin films [33,34]. The
most important property of the transfer matrix is that for a
layered system the overall transfer matrix can be computed
by taking the correctly ordered matrix product of the transfer
matrices of each separate layer [35]. The corresponding com-
position law for scattering matrices is more mathematically
complex, but can be advantageous in certain circumstances,
such as in mitigating numerical divergences in large systems
with exponentially growing waves [36]. This property makes
transfer matrices particularly well suited to numerical simula-
tions [37–39] and theoretical studies, such as in photonic band
structures [40] and mesoscopic scattering [41,42].
Traditionally, scattering and transfer matrices are defined
only for modes that propagate to the far field. The exact values
of the elements of these matrices depend strongly on the
type of system being considered and may vary significantly
from one example to another. Nevertheless, under rather gen-
eral conditions, both matrices can be shown to obey certain
mathematical constraints valid for large classes of scattering
media. For example, it has long been known that a system
that conserves energy, i.e., does not absorb or generate light,
regardless of its microscopic configuration, must possess a
scattering matrix that is unitary [43]. More recently, tech-
niques such as scanning near-field optical microscopy have
enabled studies in which the scattering of evanescent fields
2643-1564/2021/3(1)/013129(13) 013129-1 Published by the American Physical Society

NIALL BYRNES AND MATTHEW R. FOREMAN PHYSICAL REVIEW RESEARCH 3, 013129 (2021)
play a critical role, such as in single molecule near-field
imaging [44], scattering from plasmonic nanoantennas [45],
particle tracking [46], and near-field speckle imaging [47].
Such studies have motivated the introduction of an extended
version of the scattering matrix that is capable of describing
scattering to and from evanescent field components. In a no-
table paper by Carminati et al., the mathematical constraints
obeyed by the extended scattering matrix were explored for
scalar waves under the conditions of conservation of energy,
reciprocity, and time reversal symmetry [48]. The correspond-
ing scattering matrix constraints due to reciprocity for vector
evanescent waves has also been considered separately [49].
Matrix constraints such as those mentioned place limits
on the set of all physically possible scattering and transfer
matrices, and hence can serve as useful guides in determining
whether a given experimental or simulated matrix satisfies
the corresponding physical law. In addition, these constraints
may also be useful in designing matrix-based models and
simulations for scattering in complex media. Furthermore,
random matrix theory, in which the set of constraints satis-
fied by a matrix is generally the only assumption made, has
proven to be very fruitful at uncovering universal properties of
random scattering media [23,43,50]. We therefore believe that
an accurate knowledge of the constraints satisfied by both the
scattering and transfer matrices is important for future studies
of scattering problems.
Compared to the scattering matrix, theoretical analysis of
the transfer matrix seems to have received less attention in
the optics literature. Moreover, while previous works have
explored matrix constraints pertinent to a continuous decom-
position of an electric field containing an infinite set of modes,
such as in a continuous angular spectrum decomposition
[51,52], the corresponding constraints satisfied by the scat-
tering and transfer matrices defined with respect to a finite set
of modes are equally important, particularly for experiments
and simulations in which only a finite description is physically
possible. The purpose of this paper is therefore to present
a self-contained derivation of the set of constraints imposed
upon the scattering and transfer matrices by conservation
of energy, reciprocity, and time reversal symmetry. Notably,
our treatment takes full account of the vector nature of light
and allows for evanescent components. A treatment of vector
fields is essential in optics, as scattering typically gives rise to
polarization mixing and depolarization, neither of which can
be described within a scalar wave formalism. We describe ar-
bitrary fields using a vector angular spectrum decomposition,
first over a continuous range encompassing all possible wave
vectors and then for a discrete angular spectrum containing
a finite set of modes. The angular spectrum decomposition is
particularly useful as it is able to conveniently discriminate
between propagating and evanescent modes. This work builds
upon previous results and, for the sake of literary continuity,
we adopt similar notation and follow a similar format to that
of Ref. [48].
In Sec. II, we define the scattering and transfer matrices for
a continuous angular spectrum of an electric field and derive
the constraints imposed by the aforementioned conditions. In
Sec. III, we introduce the scattering and transfer matrices for
a discrete angular spectrum and derive the associated con-
straints from the continuous case. We then compare our results
FIG. 1. Geometry of the scattering problem. The integrating sur-
face ∂Vused in deriving the energy conservation and reciprocity
constraints is also depicted. Dashed lines denote the planar sections
of ∂Vand dotted-dashed curves denote the spherical section of
radius R.
to previously reported results and show that the latter follow
as special cases. We end Sec. III by giving a simple numerical
example of wave propagation at a glass-air planar interface,
including the effects of total internal reflection. Finally, in
Sec. IV, we summarize and conclude our work.
II. THE SCATTERING AND TRANSFER MATRICES FOR A
CONTINUOUS ANGULAR SPECTRUM
A. Preliminaries
We consider the scattering problem depicted in Fig. 1.
A dielectric scattering medium is situated within the region
−lzl. We denote by R−and R+the regions −L<z<
−land l<z<Lsurrounding the scattering medium, which
are assumed to be dielectric with constant permittivity 1.
All sources are contained in the regions z<−Land z>L,
and may produce both propagating and evanescent incident
fields. We assume that the scattering medium is linear and
all fields are monochromatic with angular frequency ω.The
scattering medium can be described by a spatially inhomoge-
neous, complex-valued permittivity function (r,ω), where
r=(x,y,z)Tis the position vector and the superscript T is
used to denote the transpose of a vector or matrix. We also
assume that both the scattering and background media are
nonmagnetic and have magnetic permeabilities equal to the
vacuum permeability μ0. We denote by E(r) and H(r)the
complex phasor representations of the real electric and mag-
netic fields with a suppressed time factor of exp(−iωt).
The frequency-domain Maxwell equations for the entire
region −L<z<Lare given by [53]
∇·E(r)=0,(1)
∇·H(r)=0,(2)
∇×E(r)=iωμ0H(r),(3)
∇×H(r)=−iω (r,ω)E(r).(4)
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SYMMETRY CONSTRAINTS FOR VECTOR SCATTERING … PHYSICAL REVIEW RESEARCH 3, 013129 (2021)
From Eqs. (3) and (4), the vector wave equation for the electric
field can be shown to be
∇×∇×E(r)−k2E(r)=0,(5)
where k2=ω2μ0(r,ω). In the regions R−and R+,wemay
write the electric field as a sum of plane-wave components
using the well-known angular spectrum representation [54]
E±(r)=a±(κ)ei(κ·ρ+kzz)dκ
+b±(κ)ei(κ·ρ−kzz)dκ,(6)
where the plus or minus sign is chosen according to the
region under consideration and dκ=dkxdky.InEq.(6)we
have introduced the transverse position vector ρ=(x,y)Tand
transverse wave vector κ=(kx,ky)T. For each plane wave, the
zcomponent of the associated wave vector kz=kz(κ)isgiven
by
kz(κ)=k2−|κ|2if |κ|2k2,
i|κ|2−k2if |κ|2>k2.(7)
The vector a±(κ) denotes the amplitude of the right-traveling
plane wave with wave vector k=(kx,ky,kz)T. Similarly, the
vector b±(κ) denotes the amplitude of the left-traveling plane
wave with wave vector 
k=(kx,ky,−kz)T.InEq.(6) and all
integrals that follow, the domain of integration is assumed to
be from −∞to ∞for all integration variables unless specified
otherwise. The corresponding angular spectrum representa-
tion for the magnetic field can be obtained by taking the curl of
Eq. (6) and using Eq. (3). It is also necessary that the electric
field satisfies the divergence condition in Eq. (1). By taking
the divergence of Eq. (6) and using Eq. (1), we find that
a±(κ)·k=0,(8)
b±(κ)·
k=0,(9)
for all κ.
In R−and R+,k2is a constant since k2=n2k2
0=
n2(2π/λ0)2, where λ0is the wavelength in vacuum, n=
√1/0is the refractive index, and 0is the vacuum per-
mittivity. Clearly when |κ|2k2,kzis real and the plane
waves are homogeneous, or propagating. When |κ|2>k2,
kzis imaginary and the plane waves are inhomogeneous,
or evanescent. For convenience, we define the sets pand
e, where p={κ:|κ|2k2}is the set of all transverse
wave vectors corresponding to propagating plane waves and
e={κ:|κ|2>k2}is the set of all transverse wave vectors
corresponding to evanescent plane waves.
For linear scattering, we may relate the amplitudes of
plane waves traveling towards and away from the scattering
medium using the scattering matrix S. Specifically, if we de-
fine the column vectors I(κ)=[a−(κ),b+(κ)]Tand O(κ)=
[b−(κ),a+(κ)]T, then the continuous scattering matrix is de-
fined to be the matrix that satisfies
O(κ)=S(κ,κ)I(κ)dκ.(10)
For a given κand κ, the scattering matrix S(κ,κ)isa6×6
matrix of complex entries, but it is useful to write it as a 2 ×2
block matrix in the form
S(κ,κ)=r(κ,κ)t(κ,κ)
t(κ,κ)r(κ,κ),(11)
where r,r,t, and tare 3 ×3 matrix generalizations of
transmission and reflection coefficients.
An alternative description of the scattering problem is
possible using the transfer matrix M, which relates the am-
plitudes of plane waves on the left- and right-hand side
of the medium. Letting L(κ)=[a−(κ),b−(κ)]Tand R(κ)=
[a+(κ),b+(κ)]T, the continuous transfer matrix is defined to
be the matrix that satisfies
R(κ)=M(κ,κ)L(κ)dκ.(12)
As with the scattering matrix, it is useful to write the transfer
matrix as a 2 ×2 block matrix in the form
M(κ,κ)=α(κ,κ)β(κ,κ)
γ(κ,κ)δ(κ,κ).(13)
Unlike the scattering matrix, however, the submatrices α,β,
γ, and δ, do not have such an obvious physical interpretation.
B. Conservation of energy
We now consider the constraints placed upon the continu-
ous scattering and transfer matrices by energy conservation. In
order to enforce conservation of energy, we consider the time-
averaged Poynting vector associated with the fields, which is
given by S(r)=1
2Re[E(r)×H∗(r)]. The average net rate
Wat which electromagnetic energy flows out of any closed
surface Ais given by [54]
W=AS·ˆ
ndA,(14)
where ˆ
nis the outward unit normal vector to the surface. If
energy is not absorbed within the volume enclosed by the
surface, then conservation of energy demands that W=0
and the integral in Eq. (14) must vanish. We shall henceforth
assume that this is the case.
Consider now the surface ∂Vshown in Fig. 1bounding
the volume V. The surface consists of several parts: two
planar sections at z=z−and z=z+, and a spherical section
of radius Rin between the two planar sections. In the limit
R→∞, the two planar sections expand to infinite planes
and the energy flow through the spherical section becomes
negligible. The total energy flow through the planes can be
written as
W++W−=z=z+S·ˆ
zdρ−z=z−S·ˆ
zdρ=0,(15)
where dρ=dxdy. The vector Scan be expressed in terms of
the angular spectra of the electric and magnetic fields defined
previously. Doing so and evaluating the integrals in Eq. (15)
yields
p
kz|a−|2−|a+|2−(|b−|2−|b+|2)dκ
+2ie
kzIm[a−·b−∗ −a+·b+∗]dκ=0,(16)
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NIALL BYRNES AND MATTHEW R. FOREMAN PHYSICAL REVIEW RESEARCH 3, 013129 (2021)
where the dependence of each term on κhas been temporarily
omitted for brevity.
To derive relations pertaining to the scattering matrix, we
notice that Eq. (16) can be recast in the form
p
kz[|I(κ)|2−|O(κ)|2]dκ
+e
kz[I(κ)·O∗(κ)−I∗(κ)·O(κ)]dκ=0.(17)
Introducing the scattering matrix into Eq. (17)usingEq.(10)
and simplifying the resultant expressions, we ultimately arrive
at p
kzS†(κ,κ)S(κ,κ)dκ
=⎧
⎪
⎪
⎨
⎪
⎪
⎩
k
zδ(κ−κ)I6if κ∈p,κ ∈p,
k
zS†(κ,κ)ifκ∈p,κ ∈e,
−k
zS(κ,κ)ifκ∈e,κ ∈p,
k
zS†(κ,κ)−k
zS(κ,κ)ifκ∈e,κ ∈e,
(18)
where δis the Dirac delta function, the superscript † denotes
the conjugate transpose, and we use Into denote the n×n
identity matrix. These so-called extended unitarity relations
are hence an expression of conservation of energy and gen-
eralize the better known unitarity condition on the scattering
matrix so as to include evanescent wave components [48].
The transfer matrix can also be shown to obey a similar set
of equations to Eq. (18). Returning to Eq. (16), we note that
the integrands can be written in the alternate form
p
kz[L†(κ)z
3L(κ)−R†(κ)z
3R(κ)]dκ
−e
ikz[L†(κ)y
3L(κ)−R†(κ)y
3R(κ)]dκ=0,(19)
where we have introduced the generalized Pauli matrices
z
n=InOn
On−In,y
n=iOn−In
InOn,(20)
where Ondenotes the n×nzero matrix. Similarly to before,
we introduce the transfer matrix into Eq. (19)usingEq.(12),
which yields
p
kzM†(κ,κ)z
3M(κ,κ)dκ
−e
ikzM†(κ,κ)y
3M(κ,κ)dκ
=⎧
⎨
⎩
k
zδ(κ−κ)z
3if κ∈p,κ ∈p,
−ik
zδ(κ−κ)y
3if κ∈e,κ ∈e,
O6otherwise.
(21)
Equation (21) therefore represents the constraint imposed
upon the transfer matrix by energy conservation.
C. Reciprocity
In this section, we consider the constraints imposed upon
the scattering and transfer matrices due to the reciprocity
principle. Roughly speaking, reciprocity describes a relation
between scattering matrices that are related by an exchange
of input and output modes. For a more detailed review, see
Ref. [55].
Consider again the volume Vshown in Fig. 1and its
bounding surface ∂V.LetE1and E2be two arbitrary fields
that satisfy Maxwell’s equations. Applying the vector analog
of Green’s second identity to these two fields gives [56]
V
(E1·∇×∇×E2−E2·∇×∇×E1)dV
=∂V
(E2×∇×E1−E1×∇×E2)·ˆ
ndA.(22)
Since both fields satisfy Eq. (5)inV, it can be shown that the
integral on the left-hand side of Eq. (22) vanishes. Further-
more, by writing the electric fields in the far field as sums of
incoming and outgoing spherical waves (see, e.g., Ref. [53]),
it can be shown that the integral on the right-hand side of
Eq. (22) over the spherical section of ∂Vvanishes in the
limit R→∞. Therefore, we conclude that the integral on the
right-hand side of Eq. (22) over the infinite planes z=z−and
z=z+is equal to zero. Expressing E1and E2using angular
spectrum representations, we arrive at the equation Iz−=Iz+,
where
Iz±=kz[a±
1(κ)·b±
2(−κ)+a±
1(−κ)·b±
2(κ)
−a±
2(κ)·b±
1(−κ)−a±
2(−κ)·b±
1(κ)]dκ.(23)
Written in terms of the vectors Iand O, the equation Iz−=Iz+
becomes
kz[I1(κ)·O2(−κ)−I2(κ)·O1(−κ)]dκ=0.(24)
By now introducing the scattering matrix into Eq. (24)using
Eq. (10), we are able to derive the reciprocity relation for the
scattering matrix
kz(κ)S(κ,κ)=kz(κ)ST(−κ,−κ),(25)
which is valid for all pairs of transverse wave vectors κand
κ. Reciprocity relations for the constituent transmission and
reflection matrices can be obtained by considering the block
form of the scattering matrix as in Eq. (11). Comparing blocks
on either side of Eq. (25), we obtain
kz(κ)r(κ,κ)=kz(κ)rT(−κ,−κ),(26)
kz(κ)r(κ,κ)=kz(κ)rT(−κ,−κ),(27)
kz(κ)t(κ,κ)=kz(κ)tT(−κ,−κ),(28)
which are consistent with those previously reported [49].
Alternatively, writing Eq. (23) in terms of the vectors Land
R, we find
kzLT
1(κ)y
3L2(−κ)−RT
1(κ)y
3R2(−κ)dκ=0,(29)
which, after introducing the transfer matrix using Eq. (12),
gives
kzMT(κ,κ)y
3M(−κ,κ)dκ=k
zδ(κ+κ)y
3.(30)
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SYMMETRY CONSTRAINTS FOR VECTOR SCATTERING … PHYSICAL REVIEW RESEARCH 3, 013129 (2021)
Equation (30) is thus the reciprocity relation for the transfer
matrix.
D. Time reversal symmetry
Let us now consider time reversal symmetry, which de-
scribes an invariance under the transformation t→−t.Let
E1be an arbitrary, real electric field and let E2be its time re-
versed counterpart, i.e., E2(r,t)=E1(r,−t). It can be shown
that the Fourier transforms of the fields satisfy E2(r,ω)=
E∗
1(r,ω)[57]. Making use of this property and comparing the
angular spectrums of the two fields, we find
a±
2(κ)=b±∗
1(−κ),(31)
b±
2(κ)=a±∗
1(−κ),(32)
for κ∈p, and
a±
2(κ)=a±∗
1(−κ),(33)
b±
2(κ)=b±∗
1(−κ),(34)
for κ∈e. In terms of the scattering matrix, a system is
time reversal invariant if both E1and E2satisfy Eq. (10)for
the same scattering matrix S. Combining this property with
Eqs. (31)–(34), we ultimately find
p
S(κ,κ)S∗(−κ,κ)dκ
=⎧
⎪
⎨
⎪
⎩
δ(κ+κ)I6if κ∈p,κ ∈p,
−S(κ,−κ)ifκ∈p,κ ∈e,
S∗(−κ,κ)ifκ∈e,κ ∈p,
−S(κ,−κ)+S∗(−κ,κ)ifκ∈e,κ ∈e,
(35)
which is therefore an expression of time reversal symmetry
for the scattering matrix.
If we instead require that both fields satisfy Eq. (12)forthe
same transfer matrix, we find
M(κ,κ)=⎧
⎪
⎨
⎪
⎩
x
3M∗(−κ,−κ)x
3if κ∈p,κ ∈p,
x
3M∗(−κ,−κ)ifκ∈p,κ ∈e,
M∗(−κ,−κ)x
3if κ∈e,κ ∈p,
M∗(−κ,−κ)ifκ∈e,κ ∈e,
(36)
where
x
n=OnIn
InOn.(37)
Equation (36) is therefore the time reversal symmetry con-
straint for the transfer matrix.
To conclude this section, we note briefly that for the
scattering matrix the time reversal symmetry equations also
follow from the conservation of energy and reciprocity equa-
tions, i.e., Eqs. (18) and (25)implyEq.(35). Similarly, for
the transfer matrix, the reciprocity relation can be derived
from conservation of energy and time reversal symmetry, i.e.,
Eqs. (21) and (36)implyEq.(30). We therefore conclude
that, in accordance with Ref. [48], for a system that conserves
energy, reciprocity and time reversal symmetry are equivalent.
III. THE SCATTERING AND TRANSFER MATRICES
FOR A DISCRETE ANGULAR SPECTRUM
The scattering and transfer matrices we have considered
so far are, in principle, defined for all pairs of conceivable
transverse wave vectors κand κ, which form a continuous
spectrum and are infinite in number. In both experiments
and numerical simulations, however, fields cannot be resolved
with infinite precision and must be described using some finite
set of modes [58]. Beyond merely being a practical limitation,
the number of independent modes a system can support in
reality must be finite due to the wave nature of light and is
often constrained by the geometry of the scattering system.
This is particularly relevant for waveguides, such as optical
fibers, where the number of modes is finite and is determined
by the fiber’s radius and refractive indices [59]. Even for
waves in free space, however, diffraction places a lower limit
on the resolution to which a field can be discretely sampled
[60]. Sampling beyond this limit would result in a set of modes
that would not be independent and would therefore not yield
additional information.
In this section, we shall consider a pixelwise discretization
of the continuous spectrum of transverse wave vectors. This
will allow us to define scattering and transfer matrices that
describe coupling between plane waves in a discrete angular
spectrum. We shall then derive the constraints that must be
satisfied by these matrices.
A. Definitions
Let us consider a finite set of plane waves indexed by their
transverse wave vectors. We first define two sets, Kpand Ke,
of transverse wave vectors for propagating and evanescent
waves, respectively. We form Kpby choosing Nptransverse
wave vectors κp
icorresponding to propagating waves together
with their additive inverses −κp
iand possibly the zero vector.
As we shall demonstrate, it is necessary to include modes in
inverse pairs to fully explore the effects of reciprocity and time
reversal symmetry. Similarly, we construct Keby taking Ne
transverse wave vectors κe
icorresponding to evanescent waves
and their additive inverses −κe
i.Thus,wehave
Kp=−κp
Np,...,−κp
2,−κp
1,0,κp
1,κp
2,...,κp
Np,(38)
Ke=−κe
Ne,...,−κe
2,−κe
1,κe
1,κe
2,...,κe
Ne,(39)
which, with the inclusion of the zero vector in Kp, contain
2Np+1 and 2Neelements, respectively. The set of all wave
vectors Kis then the union of these sets, i.e., K=Kp∪Ke.A
sample of several modes is shown in Fig. 2. Note that while
in Fig. 2we have, for simplicity, distributed the modes at
points on a rectangular lattice in kspace, the choice of other
geometries, such as a hexagonal lattice, may have practical ad-
vantages [61]. Note also that although kspace is unbounded,
there is a practical upper limit to the size of |κe
i|.Thisis
because when |κe
i|is sufficiently large, its corresponding
wave amplitude, even at positions very close to the scattering
medium, will have decayed to the point of being practically
unmeasurable.
As a result of Eqs. (8) and (9), the amplitude associated
with each plane wave has only two degrees of freedom.
013129-5

NIALL BYRNES AND MATTHEW R. FOREMAN PHYSICAL REVIEW RESEARCH 3, 013129 (2021)
FIG. 2. Transverse wave-vector modes distributed on a rectan-
gular lattice in kspace. The circle |κ|2=k2defines the boundary
between propagating and evanescent modes. A selection of modes
and their additive inverses have been highlighted in red.
Therefore, in order to remove extraneous information and
simplify the ensuing mathematics, it is useful to introduce the
vectors
ek(κ,kz)=k
k=1
k⎛
⎝
kx
ky
kz
⎞
⎠,(40)
eφ(κ,kz)=(ˆ
z×ek)
|ˆ
z×ek|=1
|κ|⎛
⎝−ky
kx
0
⎞
⎠,(41)
eθ(κ,kz)=(eφ×ek)
|eφ×ek|=1
k|κ|⎛
⎝
kxkz
kykz
−k2
x−k2
y
⎞
⎠.(42)
Note that when κ∈p,kzis real and the vectors ek,eφ,
and eθare the standard unit basis vectors in spherical polar
coordinates. When κ∈e,ekand eθbecome complex vectors
and the latter may be interpreted using a complex polar angle.
The vectors eφand eθarealsoclassicallyreferredtoass
and pmodes in polarization theory [54], but we have chosen
to reserve the letter pin this work for “propagating.” By
definition, it follows that
eφ(κ,kz)·k=eθ(κ,kz)·k=0,(43)
eφ(κ,−kz)·
k=eθ(κ,−kz)·
k=0,(44)
where 
kis defined as in Eq. (9). This means we may express
a±(κ) and b±(κ)inEq.(6) in terms of their θand φcompo-
nents. Explicitly, we have
a±(κ)=a±
θ(κ)eθ(κ,kz)+a±
φ(κ)eφ(κ,kz),(45)
b±(κ)=b±
θ(κ)eθ(κ,−kz)+b±
φ(κ)eφ(κ,−kz).(46)
It is now possible to reduce the matrices r,r,t,t,α,β,
γ, and δin Eqs. (11) and (13)to2×2 matrices that couple
the θand φcomponents of a±(κ) and b±(κ). As an example,
consider r(κi,κj), where κiand κjare any two vectors taken
from K. This matrix describes the reflection at the left-hand
side of the medium of an incident wave with wave vector
(κj,kzj)Tand amplitude a−(κj) to a final wave with wave
vector (κi,−kzi )Tand amplitude b−(κi), where kzj =kz(κj).
We define the 2 ×2 reduced reflection matrix for the pair of
modes κiand κjas the matrix r(i,j), where
r(i,j)=r(i,j)θθ r(i,j)θφ
r(i,j)φθ r(i,j)φφ(47)
and
r(i,j)mn =eT
m(κi,−kzi )r(κi,κj)en(κj,kzj),(48)
where mand nmay be chosen to be either θor φ. Note that
eT
k(κi,−kzi )r(κi,κj)=0T(49)
for all κiand κj, but, since a−(κj) has no kcomponent,
r(κi,κj)ek(κj,kzj) is undefined. We therefore assign
r(κi,κj)ek(κj,kzj)=0,(50)
which justifies the construction of the reduced reflection ma-
trix as only the four components in Eq. (47) are unconstrained.
The other submatrices of Sand Mcan be treated similarly and
a brief summary is given in the Appendix.
We can now construct the discrete scattering matrix for
our finite set of modes by considering a discretized version
of Eq. (10). To achieve this, we partition kspace into a series
of regions, each of which is centered on a mode in the set K.
We denote the area of these regions by κ, which, for conve-
nience, we assume is the same for each region. If, for example,
a rectangular partitioning is used, then κ =kxky, where
kxand kyare the distances in kspace between adjacent
modes. We assume that the choice of modes and k-space
partitioning are such that the scattering and transfer matrices
are approximately constant over each region.
Let κi∈Kbe any transverse wave vector. We may replace
the integral in Eq. (10) with a sum and write
⎛
⎜
⎜
⎝
b−
θ(κi)
b−
φ(κi)
a+
θ(κi)
a+
φ(κi)
⎞
⎟
⎟
⎠=
κj∈Kr(i,j)t
(i,j)
t(i,j)r
(i,j)⎛
⎜
⎜
⎝
a−
θ(κj)
a−
φ(κj)
b+
θ(κj)
b+
φ(κj)
⎞
⎟
⎟
⎠κ, (51)
where κjin the sum ranges over all modes in K.Bynow
letting κivary over the set K, we obtain a system of equations,
each of which have the same form as Eq. (51), which can be
combined into a single matrix equation. To facilitate this, we
first introduce the notation
u±
q=u±
θ−κq
Nq,u±
φ−κq
Nq,...,u±
θκq
Nq,u±
φκq
Nq,(52)
where udenotes either aor band qis either por e. Regardless
of the choice of q, we order the transverse wave vectors within
ufrom left to right in the same way as they are presented in
Eqs. (38) and (39). Using the notation in Eq. (52), we define
the four vectors
ci=(a−
p,b+
p,a−
e,b+
e)T,(53)
013129-6

SYMMETRY CONSTRAINTS FOR VECTOR SCATTERING … PHYSICAL REVIEW RESEARCH 3, 013129 (2021)
co=(b−
p,a+
p,b−
e,a+
e)T,(54)
cl=(a−
p,b−
p,a−
e,b−
e)T,(55)
cr=(a+
p,b+
p,a+
e,b+
e)T,(56)
with which we define the discrete scattering matrix Sas the
matrix satisfying
co=Sci.(57)
Given our ordering of the modes, we may partition the matrix
Sin the block form
S=Spp Spe
Sep See =⎛
⎜
⎜
⎝
rpp t
pp rpe t
pe
tpp r
pp tpe r
pe
rep t
ep ree t
ee
tep r
ep tee r
ee
⎞
⎟
⎟
⎠,(58)
where
Sqq=rqqt
qq
tqqr
qq(59)
and qand qare either por e. For each submatrix, the right
subscript denotes the type of incident mode (i.e., propagating
or evanescent) and the left subscript denotes the type of out-
going mode. For example, rpe describes the reflection at the
left-hand side of the system of incoming evanescent modes
to outgoing propagating modes. It is formed by concatenating
2×2 reduced reflection matrices of the form in Eq. (47). All
other submatrices of Scan be understood in an analogous
manner.
Similarly, we define the discrete transfer matrix Mto be
the matrix that satisfies
cr=Mcl.(60)
This matrix can also be partitioned in an analogous way to the
discrete scattering matrix. Explicitly,
M=Mpp Mpe
Mep Mee =⎛
⎜
⎜
⎝
αpp βpp αpe βpe
γpp δpp γpe δpe
αep βep αee βee
γep δep γee δee
⎞
⎟
⎟
⎠.(61)
In order to simplify the equations in the following section,
it is useful to normalize the scattering and transfer matrices.
We use a bar to indicate the normalized version of a matrix
and define the normalized continuous scattering and transfer
matrices by
¯
S(κ,κ)=√|kz(κ)|
√|kz(κ)|S(κ,κ)κ, (62)
¯
M(κ,κ)=√|kz(κ)|
√|kz(κ)|M(κ,κ)κ. (63)
Physically, this normalization is equivalent to redefining the
angular spectrum of Eq. (6) so that each plane wave delivers
an equal energy flux per unit cross-sectional area perpen-
dicular to the zdirection. Explicitly, this means making the
replacement eik·r→eik·r/√|kz|for each plane wave. The
√|kz|factors in Eqs. (62) and (63) then rescale the amplitudes
of each plane wave by the appropriate amount to preserve the
original energy flow. The corresponding normalized discrete
scattering and transfer matrices are given by
¯
S=η1
2Sη−1
2κ, (64)
¯
M=η1
2Mη−1
2κ, (65)
where
η±1
2=diagη±1
2
p,η±1
2
p,η±1
2
e,η±1
2
e,(66)
η±1
2
p=diagkz−κp
Np±1
2I2,...,kzκp
Np±1
2I2,(67)
η±1
2
e=diagkz−κe
Ne±1
2I2,...,kzκe
Ne±1
2I2.(68)
In the case that κ is different for different k-space regions,
we can instead incorporate its different values into the matri-
ces η±1
2.
To end this section, we note that it is possible to convert
between the discrete scattering and transfer matrices. It can
be shown from Eqs. (57) and (60) that
r=−δ−1γ,α=t−r(t)−1r,(69)
r=βδ−1,β=r(t)−1,(70)
t=α−βδ−1γ,γ=−(t)−1r,(71)
t=δ−1,δ=(t)−1,(72)
where, for example,
r=rpp rpe
rep ree ,(73)
and the other matrices are defined analogously. These equa-
tions also hold for the normalized scattering and transfer
matrices.
B. Conservation of energy
We now aim to derive the constraints imposed upon ¯
S
and ¯
Mby energy conservation. We begin by discretizing the
conservation of energy equation for the continuous scatter-
ing matrix, namely Eq. (18). As before, we assume that the
continuous scattering matrix is constant over each k-space
region. The Dirac delta function δ(κ−κ), whose integral
is by definition unity, can be replaced by the normalized
Kronecker delta δij/κ. Furthermore, by using the normal-
ized scattering matrix, it is possible to remove all kzterms
from the equation. Note that a factor of iis introduced when-
ever a kzfactor corresponds to an evanescent wave. Replacing
the integral with a sum and rewriting κ,κ, and κ as κl,κi,
and κj, respectively, we obtain

κl∈Kp
¯
S†(κl,κi)¯
S(κl,κj)
=⎧
⎪
⎪
⎨
⎪
⎪
⎩
δijI6if κi∈Kp,κj∈Kp,
i¯
S†(κj,κi)ifκi∈Kp,κj∈Ke,
−i¯
S(κi,κj)ifκi∈Ke,κj∈Kp,
i¯
S†(κj,κi)−i¯
S(κi,κj)ifκi∈Ke,κj∈Ke.
(74)
013129-7

NIALL BYRNES AND MATTHEW R. FOREMAN PHYSICAL REVIEW RESEARCH 3, 013129 (2021)
We can consider each of the four cases in Eq. (74) sep-
arately. In each case, we may form four matrix equations
by considering different submatrix blocks individually. Since
there are a lot of equations and the algebra involved is rather
lengthy and repetitive, we shall only present a single example
here. Suppose κi,κj∈Kpand we consider the first case of
Eq. (74). Equating the top-left blocks of the matrices on either
side, we obtain

κl∈Kp
[¯
r†(κl,κi)¯
r(κl,κj)+¯
t†(κl,κi)¯
t(κl,κj)] =δijI3.(75)
We may further extract four equations from Eq. (75)by
premultiplying and postmultiplying both sides by different
combinations of eθand eφ. We also make use of the fact
that, since ek(κ,±kz), eθ(κ,±kz), and eφ(κ,±kz)forman
orthonormal basis of C3,wehave[62]
ekeT
k+eθeT
θ+eφeT
φ=I3.(76)
This means that multiplying any matrix by the combination of
vectors on the left-hand side of Eq. (76) will leave the matrix
unchanged. Inserting this combination of vectors in between
the matrix products in Eq. (75), premultiplying the equation
by eT
θ(κi,kzi ) and postmultiplying the equation by eθ(κj,kzj)
yields

κl∈KpeT
θ(κi,kzi )¯
r†(κl,κi)eθ(κl,−kzl )eT
θ(κl,−kzl )+eφ(κl,−kzl )eT
φ(κl,−kzl )r(κl,κj)eθ(κj,kzj)
+eT
θ(κi,kzi )¯
t†(κl,κi)eθ(κl,kzl )eT
θ(κl,kzl )+eφ(κl,kzl )eT
φ(κl,kzl )¯
t(κl,κj)eθ(κj,kzj)=δij.(77)
Note that all terms involving either ekor eT
kvanish, as pre-
viously discussed. The left-hand side of Eq. (77) consists of
eight bilinear forms, each of which can be simplified sepa-
rately. For example, the first part of the first term in the sum
in Eq. (77) can be shown to be
eT
θ(κi,kzi )¯
r†(κl,κi)eθ(κl,−kzl )=¯r∗
(l,i)θθ,(78)
where we use the fact that eθ=e∗
θfor propagating waves.
Simplifying all terms in Eq. (77) in a similar way gives

κl∈Kp¯r∗
(l,i)θθ ¯r(l,j)θθ +¯r∗
(l,i)φθ ¯r(l,j)φθ
+¯
t∗
(l,i)θθ ¯
t(l,j)θθ +¯
t∗
(l,i)φθ ¯
t(l,j)φθ =δij.(79)
Repeating this process for all combinations of eθand eφ
and all subblocks in Eq. (74) yields a large system of equa-
tions that can be shown to be equivalent to the single matrix
equation
¯
S†
pp ¯
Spp =I4Np+2,(80)
which is the classic result that a scattering matrix for a system
that conserves energy and only considers propagating modes
is unitary.
The other three cases for κiand κjcan be treated similarly.
After a lot of algebra, we obtain the equations
¯
S†
pp ¯
Spe =i¯
S†
ep,(81)
¯
S†
pe ¯
Spe =i(¯
S†
ee −¯
See ).(82)
By introducing the matrices Ip=diag(I4Np+2,O4Ne) and Ie=
diag(O4Np+2,I4Ne), we can combine Eqs. (80)–(82) into the
single equation
¯
S†Ip¯
S=Ip+i(¯
S†Ie−Ie¯
S),(83)
which is the most general form of the conservation of en-
ergy constraint for the scattering matrix, incorporating both
vector properties of the electric field as well as evanescent
components.
We note that our results here are consistent with those
in Ref. [63], in which energy conservation is examined in
detail in the context of a generalized optical theorem. A
simple example of some of the peculiarities of evanescent
waves can be seen by considering Eq. (82). Suppose that
the field incident upon the scattering medium consists of a
single evanescent component that decays towards the medium
and is polarized parallel to one of the basis states (e.g., eφ).
Examining the on-diagonal elements of the matrices on either
side of Eq. (82) then reveals that the total energy radiated away
from the medium by propagating waves is intimately tied to
the reflected evanescent wave whose transverse wave vector
is equal to that of the incident wave, but which decays away
from the medium (in particular, the imaginary part of its scat-
tering amplitude). If, due to scattering, there are any outgoing
propagating waves, then this reflected evanescent wave must
also be present. Conversely, if this reflected evanescent wave
is not present, then there cannot be any outgoing propagating
waves, neither in reflection nor transmission.
To derive the conservation of energy constraint for the
discrete transfer matrix, we begin by discretizing Eq. (21)to
obtain

κl∈Kp
¯
M†(κl,κi)z
3¯
M(κl,κj)+
κl∈Ke
¯
M†(κl,κi)y
3¯
M(κl,κj)
=⎧
⎨
⎩
δijz
3if κi∈Kp,κj∈Kp,
δijy
3if κi∈Ke,κj∈Ke,
O6otherwise.
(84)
Performing similar steps to those demonstrated for the scatter-
ing matrix, we find, again after a lot of algebra, the equations
¯
M†
ppz
4Np+2¯
Mpp +¯
M†
epy
4Ne
¯
Mep =z
4Np+2,(85)
¯
M†
ppz
4Np+2¯
Mpe +¯
M†
epy
4Ne
¯
Mee =O,(86)
¯
M†
pez
4Np+2¯
Mpe +¯
M†
eey
4Ne
¯
Mee =y
4Ne,(87)
where the zero matrix in Eq. (86)isofsize(4Np+2) ×
4Ne. Equations (85)–(87) can be combined into the single
013129-8

SYMMETRY CONSTRAINTS FOR VECTOR SCATTERING … PHYSICAL REVIEW RESEARCH 3, 013129 (2021)
equation
¯
M†¯
M=,(88)
where =diag(z
4Np+2,y
4Ne). Equation (88) therefore rep-
resents conservation of energy for the discrete transfer matrix.
C. Reciprocity/time reversal symmetry
As noted previously, when conservation of energy holds,
reciprocity and time reversal symmetry are equivalent. In this
section we shall therefore henceforth refer to both conditions
as “reciprocity.” We begin by deriving the reciprocity con-
straint for the discrete scattering matrix. Note that for the
continuous scattering matrix the reciprocity relation Eq. (25)
is considerably simpler than the time reversal symmetry rela-
tion Eq. (35), particularly as it does not involve an integral.
We therefore begin our derivation with Eq. (25).
In terms of the normalized scattering matrix, the reci-
procity constraint of Eq. (25) has two different cases:
¯
S(κi,κj)=¯
ST(−κj,−κi) if either κi,κj∈Kpor κi,κj∈Ke,
and ¯
S(κi,κj)=i¯
ST(−κj,−κi) otherwise. Let us consider first
the case where κi,κj∈Kp. As in the previous section, we
again consider each submatrix of ¯
Sseparately. Comparing
top-left blocks, we have
¯
r(κi,κj)=¯
r(−κj,−κi).(89)
If we premultiply Eq. (89)byeT
θ(κi,−kzi ) and postmultiply by
eθ(κj,kzj), the left-hand side becomes r(i,j)θθ and the right-
hand side, after a bit of manipulation, becomes r(−j,−i)θθ,
where here the subscript −jrefers to the mode with transverse
wave vector −κj. Repeating this for all four combinations of
eθand eφ, we obtain the relations
¯r(i,j)θθ =¯r(−j,−i)θθ ,¯r(i,j)θφ =−¯r(−j,−i)φθ ,(90)
¯r(i,j)φθ =−¯r(−j,−i)θφ,¯r(i,j)φφ =¯r(−j,−i)φφ.(91)
Equations (90) and (91) are equivalent to the single matrix
relation
¯
r(i,j)=¯
rR
(−j,−i),(92)
where we have introduced the reciprocal operator R, which we
define such that if [A]mn is the (m,n) element of the matrix A,
then
[AR]mn =[A]nm (−1)m+n.(93)
This particular symmetry of the reflection matrix is a well-
known result in scattering theory for polarized light and is
sometimes referred to as the backscattering theorem [64]. The
operator R has also been discussed previously in the context
of reciprocal Jones matrices [65].
By now carefully considering the structure of the matrix
¯
rpp,itfollowsfromEq.(92) that
¯
rpp =σp¯
rR
ppσp,(94)
where
σp=⎛
⎝
OI
2
...
I2O
⎞
⎠(95)
is the matrix containing 2Np+1 copies of the 2 ×2 identity
matrix on its antidiagonal and zeros elsewhere. The effect
of multiplying a matrix on either side by σpis to reflect the
positions of all 2 ×2 submatrices horizontally and vertically
about the central rows and columns of the matrix, but to leave
the submatrices themselves unchanged. This is necessary so
that Eq. (94) correctly equates submatrices of rpp that are
related by an inversion of transverse wave vectors. Similarly,
by considering the other submatrices of ¯
S, we find
¯
tpp =σp¯
tR
ppσp,(96)
¯
r
pp =σp¯
rR
ppσp,(97)
which can be combined into the single equation
¯
Spp =σpp ¯
SR
ppσpp,(98)
where σpp =diag(σp,σp). If we similarly introduce σeas the
matrix containing 2Necopies of I2on its antidiagonal and
zeros elsewhere, and σee =diag(σe,σe), we can further derive
¯
Spe =σppi¯
SR
epσee,(99)
¯
See =σee ¯
SR
eeσee .(100)
Finally, Eqs. (98)–(100) can be combined into the single
equation
¯
S=ω∗¯
SRω,(101)
where ω=diag(σpp,iσee ). Equation (101) is the reciprocity
constraint for the scattering matrix, which generalizes the
well-known equation S=ST.
To derive the reciprocity constraint for the transfer matrix,
we begin with the time reversal invariance constraint Eq. (36),
which is notably simpler than the reciprocity constraint in
Eq. (30), and perform analogous steps to those used in the
derivation for the scattering matrix. We eventually arrive at
the equations
¯
Mpp =σpp ¯
M†R
ppσpp,¯
Mpe =σpp ¯
M†R
pe σee,(102)
¯
Mep =σee ¯
M†R
ep σpp,¯
Mee =σee ¯
M†R
ppσee ,(103)
where
σpp =x
4Np+2σpp =Oσp
σpO.(104)
Equations (102) and (103) can be combined into the single
equation
¯
M=ω¯
M†Rω,(105)
where ω=diag(σpp,σee ). Equation (105) is hence the reci-
procity constraint for the discrete transfer matrix.
Equations (83), (88), (101), and (105) constitute the main
results of our work and must be satisfied for any system that
conserves energy and is reciprocal/time reversal invariant.
We note that all four of these equations are algebraic and
are therefore much simpler to work with than the integral
constraints for the continuous scattering and transfer matrices.
013129-9

NIALL BYRNES AND MATTHEW R. FOREMAN PHYSICAL REVIEW RESEARCH 3, 013129 (2021)
D. Comparison with previous results
Here, we briefly compare our results with those that already
exist in the literature. In particular, we show that the more
commonly presented equations follow from our results as
special cases. From Refs. [8,41,43,66] and others we see that,
for a system that conserves energy and and is reciprocal/time
reversal invariant, the matrices ¯
Sand ¯
Mobey the equations
¯
S¯
S†=I,(106)
¯
S=¯
ST,(107)
¯
M†z¯
M=z,(108)
x¯
Mx=¯
M∗,(109)
where I,x, and zare of the appropriate size.
There are three important differences between the matrices
in Eqs. (106)–(109) and those of our results. First, the scat-
tering and transfer matrices in Eqs. (106)–(109) are defined
only for propagating modes in the far field. Therefore, for the
scattering matrix, it is sufficient to only consider ¯
Spp.Aspre-
viously noted, Eq. (80) is identical to the unitarity condition
of Eq. (106). For the transfer matrix, since in the far field we
have ¯
Mep =O, we see that Eq. (85) reduces to the same form
as of Eq. (108).
The second important difference is that Eqs. (106)–(109)
are defined only for scalar waves. A scalar wave formalism
is appropriate when there is no change in polarization state
induced by the scattering medium. For this to be the case
within our formalism, it is necessary for each 2 ×2 submatrix
within ¯
Sand ¯
Mto have zero off-diagonal elements. Mathe-
matically, this means that [¯
S]mn =[¯
M]mn =0when m+nis
an odd number. By now inspecting Eq. (93) we see that, if this
is the case, then the reciprocal operator is identical to a regular
matrix transpose and in all of our previous equations we may
make the notational transformation R →T.
The final difference is that Eqs. (106)–(109) are defined
in a quasi-one-dimensional geometry for which each mode
has a wave vector with a unique zcomponent. Consequently,
there are no two modes whose wave vectors have the same
zcomponents, but different transverse components. More
specifically, this means that there is no distinction between
the two modes with wave vectors (κ,kz)Tand (−κ,kz)T.Ifwe
were to enforce this constraint within our formalism, all trans-
verse wave vectors in the sets defined in Eqs. (38) and (39)
containing a minus sign would become extraneous and could
be removed. Moreover, the use of the matrix σpto correctly
associate submatrices of ¯
Sand ¯
Mthat describe scattering
between modes with inverse transverse wave vectors would no
longer be necessary and σpwould be replaced by the identity
matrix of the appropriate size. Making this change, combined
with the previous change regarding the reciprocal operator,
transforms Eq. (98) into Eq. (107). Finally, we see that un-
der these changes we also have σpp →x,¯
M†R →¯
M∗, and
hence Eq. (102) becomes identical to Eq. (109), completing
the comparison.
E. Numerical example: Glass-air interface
In this section we demonstrate the validity of our results
with a numerical example. We consider a planar glass-air
boundary with glass on the left and air on the right. We
choose ng=1.5 and na=1.0 to be the refractive indices of
the glass and air, respectively. In the glass, we consider three
right-traveling modes with wave vectors k1,k2and k3whose
zcomponents are kz1/k0=1.41, kz2/k0=0.90, and kz3/k0=
0.73i, where each numerical value is given to two decimal
places. In the air, we consider the set of refracted wave vectors
k
1,k
2, and k
3, whose zcomponents k
z1/k0=0.87, k
z2/k0=
0.66i, and k
z3/k0=1.33ifollow from those of k1,k2, and k3
by Snell’s law. We identify the “scattering medium” in this
example with the planar boundary and note that our choice
of wave vectors incorporates different types of scattering. For
example, the mode with wave vector k1is partially transmitted
and reflected, but the mode with wave vector k2is totally inter-
nally reflected. We describe left-traveling modes in the glass
medium using the reflected wave vectors 
k1,
k2, and 
k3and,
similarly, we describe left-traveling modes in the air using the
wave vectors 
k
1,
k
2, and 
k
3. For each wave-vector mode, we
consider both sand ppolarizations (we shall use sand pto
refer to polarization states in this section alone) with the usual
definitions for a planar interface and write k1sto refer to an
s-polarized wave with wave vector k1and similarly for k1p.
We calculate the scattering matrix elementwise using the
standard Fresnel equations (see, e.g., Ref. [54]). We then
normalize each element of the scattering matrix according to
Eq. (62), which gives
(110)
013129-10

SYMMETRY CONSTRAINTS FOR VECTOR SCATTERING … PHYSICAL REVIEW RESEARCH 3, 013129 (2021)
where we have indicated the incident and outgoing modes above and to the right of the matrix, respectively. We have
included horizontal and vertical lines within the matrix as a visual aid to map out the block structure of ¯
Sas in Eq. (58).
We verify numerically that ¯
Ssatisfies the conservation of energy constraint given by Eq. (83) with Ip=diag(I6,O6) and
Ie=diag(O6,I6).
In this particular example, the scattering matrix includes evanescent components, but contains no polarization mixing and
does not consider pairs of inverse transverse wave-vector modes. As per the discussion in the previous section, the reciprocity
constraint for the scattering matrix can therefore be simplified. Specifically, in Eq. (101) we may make the transformations
R→T and σpp,σee →I6, which yields the modified equation
¯
S=ω∗¯
STω,(111)
where, in this instance, ω=diag(I6,iI6). Equation (111) is also satisfied by our scattering matrix ¯
S.
We calculate the transfer matrix from the scattering matrix using Eqs. (69)–(72). This gives
(112)
where, as before, the horizontal and vertical lines show the
block structure of ¯
Mas in Eq. (61). Since there is an uneven
number of propagating and evanescent modes on either side of
the boundary, ¯
Mhas an irregular block structure and, notably,
the matrices ¯
Mpp and ¯
Mee are no longer square. It is therefore
necessary to modify the sizes of the submatrices of and
ωin Eqs. (88) and (105) to account for this asymmetry.
Furthermore, in addition to the transformations made for the
scattering matrix reciprocity equation, we also replace σpp →
x. Ultimately, we find that ¯
Mmust satisfy
¯
M†1¯
M=2,(113)
ω
1¯
M∗ω
2=¯
M,(114)
where
1=z
2O
Oy
4,2=z
4O
Oy
2,(115)
ω
1=x
2O
OI
8,ω
2=x
4O
OI
4.(116)
Despite these modifications, we note that Eqs. (113) and (114)
still have the same basic form as Eqs. (88) and (105). Numer-
ical calculations confirm that Eqs. (113) and (114) are indeed
satisfied.
IV. CONCLUSION
In this paper we have derived the general set of constraints
for the scattering and transfer matrices imposed by conserva-
tion of energy, reciprocity, and time reversal symmetry. Our
formalism considers the general case of vectorial light and
allows for fields containing evanescent components. We have
extended previously known results, such as in Eqs. (106)–
(109), and have introduced a more general set of constraints
for the discrete scattering and transfer matrices, which are
given in Eqs. (83), (88), (101), and (105). We have discussed
the differences between our results and Eqs. (106)–(109) and
have demonstrated that the latter can be regarded as special
cases of the former. In particular, Eqs. (106)–(109) follow
from our results when one is able to neglect evanescent
field components, polarization mixing, and a distinction be-
tween inverse transverse wave-vector modes. We have given
a demonstration of the validity of our results in the simple
example of the scattering of polarized light at a planar glass-
air interface, including the effect of total internal reflection. In
our formalism, we have made minimal assumptions regarding
the nature of the scattering medium and our results should
therefore be valid for a wide range of systems.
Given the popularity of matrix methods in scattering stud-
ies and the increasing ease with which it is possible to
experimentally determine scattering and transfer matrices, we
believe that our results will serve as useful guides for future
experimental research. We also believe that our results will
help define the limits for physically realizable scattering and
transfer matrices in matrix-based numerical simulations and
theoretical studies of optical scattering. Our results should be
particularly important when one is interested in a description
of a scattering problem that incorporates any combination of
evanescent modes, vectorial light, and reciprocity/time rever-
sal symmetry in a system of more than one dimension.
013129-11

NIALL BYRNES AND MATTHEW R. FOREMAN PHYSICAL REVIEW RESEARCH 3, 013129 (2021)
ACKNOWLEDGMENT
This work was funded by the Royal Society.
APPENDIX: REDUCED MATRICES
To complement Eq. (48) in the main text, we present here
a list of equations for the θand φcomponents of the reduced
versions of each of the matrices r,r,t,t,α,β,γ, and δ.In
each equation, the subscripts mand nrepresent either k,θ,or
φ.Wehave
r(i,j)mn =eT
m(κi,−kzi )r(κi,κj)en(κj,kzj),(A1)
r
(i,j)mn =eT
m(κi,kzi )r(κi,κj)en(κj,−kzj),(A2)
t(i,j)mn =eT
m(κi,kzi )t(κi,κj)en(κj,kzj),(A3)
t
(i,j)mn =eT
m(κi,−kzi )t(κi,κj)en(κj,−kzj),(A4)
α(i,j)mn =eT
m(κi,kzi )α(κi,κj)en(κj,kzj),(A5)
β(i,j)mn =eT
m(κi,kzi )β(κi,κj)en(κj,−kzj),(A6)
γ(i,j)mn =eT
m(κi,−kzi )γ(κi,κj)en(κj,kzj),(A7)
δ(i,j)mn =eT
m(κi,−kzi )δ(κi,κj)en(κj,−kzj).(A8)
As discussed in the main text, if A(κi,κj) refers to any of the
above eight matrices, then
eT
k(κi,±kzi )A(κi,κj)=0T,(A9)
A(κi,κj)ek(κj,±kzj)=0,(A10)
where the plus or minus sign is chosen according to the choice
of A.
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