# Control of wave scattering for robust coherent transmission in a disordered
medium

Zhun-Yong Ong<EMAIL_ADDRESS>Institute of High Performance
Computing (IHPC), Agency for Science, Technology and Research (A*STAR), 1
Fusionopolis Way, #16-16 Connexis, Singapore 138632, Republic of Singapore

(August 28, 2024)

###### Abstract

The spatial structure of the inhomogeneity in a disordered medium determines
how waves scatter and propagate in it. We present a theoretical model of how
the Fourier components of the disorder control wave scattering in a two-
dimensional disordered medium, by analyzing the disordered Green’s function
for scalar waves. By selecting a set of Fourier components with the
appropriate wave vectors, we can enhance or suppress wave scattering to filter
out unwanted waves and allow the robust coherent transmission of waves at
specific angles and wavelengths through the disordered medium. Based on this
principle, we propose an approach for creating selective transparency, band
gaps and anisotropy in disordered media. This approach is validated by direct
numerical simulations of coherent wave transmission over a wide range of
incident angles and frequencies and can be experimentally realized in
disordered photonic crystals. Our approach, which requires neither nontrivial
topological wave properties nor a non-Hermitian medium, creates opportunities
for exploring a broad range of wave phenomena in disordered systems.

##  I Introduction

Understanding how wave scattering in a disordered medium depends on the
spatial structure of its inhomogeneity is important for progress in
fundamental topics such as Anderson localization (Lagendijk  _et al._ , 2009;
Modugno, 2010; Gredeskul  _et al._ , 2012; Segev  _et al._ , 2013; Mafi  _et
al._ , 2019) and other transport phenomena (Sheng, 2006) as well as for a wide
range of applications in photonics, (Wiersma, 2013; Yu  _et al._ , 2021) wave
shaping, (Cao  _et al._ , 2022) imaging, (Bertolotti and Katz, 2022; Gigan,
2022) acoustics (Maynard, 2001), matter waves. (Billy  _et al._ , 2008) and
even signal filtering. (Zangeneh-Nejad and Fleury, 2020) A remarkable insight
into this relationship is found in Ref. (Kim and Heller, 2022) which shows how
the electron wave interaction with the _Berry potential_ can result in sharp
Bragg-like scattering in spite of the absence of periodicity, with the
outgoing waves distributed at specific angles in a manner akin to powder x-ray
diffraction. Although this phenomenon is reported for electron waves in the
Schroedinger equation, its underlying mechanism is not quantum mechanical and
instead depends on the orientation of the incident wave vector $\bm{k}$ with
respect to the Fourier (plane-wave or $\bm{q}$) components of the
disorder,111In our paper, we use the letters $\bm{k}$ and $\bm{q}$ to denote
the wave vectors of the plane-wave states and disorder Fourier components,
respectively.  as represented by the Berry potential (Kim and Heller, 2022)

$U(\bm{r};\\{\phi_{j}\\})=\frac{A}{\sqrt{N}}\sum_{j=1}^{N}\cos(\bm{p}_{j}\cdot\bm{r}+\phi_{j})\
,$ (1)

where $\bm{r}$ is the position vector in two dimensions, $A$ is a constant
having the dimension of energy, $N$ is the number of Fourier components,
222For convenience, we use the term ‘components’ to label the plane-wave
components of the disorder. We reserve the term ‘state’ for the plane-wave
solutions to the Helmholtz wave equation.  and $\bm{p}_{j}$ and $\phi_{j}$
denote the wave vector and phase of the $j$-th component, respectively, with
$|\bm{p}_{j}|=q_{\text{c}}$. 333Equation (1) also includes the contribution of
the component at $-\bm{p}_{j}$ for $j=1,\ldots,N$. The set of phases
$\\{\phi_{j}\\}$ uniquely determines the spatial configuration of $U$, with
each $\phi_{j}$ taking a value between $0$ and $2\pi$. Equation (1) describes
a two-dimensional (2D) random potential constructed from a superposition of
plane waves of equal wave number $q_{c}$ and distributed over all propagation
angles. Because the Fourier components of the Berry potential are localized on
a circular manifold of radius $q_{\text{c}}$, electron waves with
$|\bm{k}|<q_{\text{c}}/2$ undergo minimal scattering and thus propagate
without attenuation.

The finding suggests that the distribution of the Fourier components of the
spatial disorder in a medium, as described by its reciprocal-space ($\bm{q}$)
spectrum, has profound bearing for scattering and coherent wave transmission.
This principle has been utilized in partially disordered media, (Yu  _et al._
, 2021; Vynck  _et al._ , ) in which the spatial distribution of particulate
scatterers, characterized by the structure factor $S(\bm{q})$, can be
correlated to generate transparency for a range of low-frequency waves.
(Leseur  _et al._ , 2016) In _stealthy hyperuniform_ (SH) systems in
particular, (Torquato, 2018, 2016) where $S(\bm{q})=0$ for $0<|\bm{q}|\leq
q_{\text{SH}}$ and $q_{\text{SH}}$ is the length scale of the absence of long-
wavelength density fluctuations, this results in optical transparency for
incident waves with wave numbers in the range of $|\bm{k}|<q_{\text{SH}}/2$,
(Leseur  _et al._ , 2016) a finding closely related to that of Ref. (Kim and
Heller, 2022). Indeed, one may interpret Eq. (1) as an analog of SH-type
disorder for the Schroedinger equation, with the Berry potential behaving as a
random scalar field with stealth hyperuniformity. (Torquato, 2016) Similar
conditions for wave transparency in other SH systems have also been found by
Kim and Torquato. (Kim and Torquato, 2020; Torquato and Kim, 2021)

A common conceptual thread that runs through the earlier articles on wave
propagation in SH systems and Ref. (Kim and Heller, 2022) is that multiple
wave scattering is suppressed when long-wavelength fluctuations are absent
from the disorder configuration. In a classical disordered medium such as an
inhomogeneous dielectric material with a position-dependent permittivity
$\epsilon(\bm{r})$, (Frank and Lubatsch, 2011; Sheremet  _et al._ , 2020) the
control of scattering is realized by modulating the Fourier components of
$\epsilon(\bm{r})$ such that its reciprocal-space spectrum conforms to a
particular distribution. Because multiple scattering underlies the wave
transport phenomena of diffusion and Anderson localization, (Sheng, 2006) this
control of scattering can potentially allow us to engineer the spectrum of
waves transmitted through the disordered medium. We remark here that there is
a slight difference between the complementary approaches of Ref. (Kim and
Heller, 2022) and existing work on SH systems. (Torquato, 2018, 2016) The
former proposes the reciprocal-space loci of the Fourier components of the
disorder while the latter determines where the Fourier components should be
excluded.

In our paper, we propose a systematic approach to engineer disorder for the
selective suppression of scattering to allow the robust transmission of waves
through the disordered medium for isotropic and selective transparency with
orientation-dependent frequency-space windows. Although it is known that SH
systems can act as isotropic low-pass filters, (Leseur  _et al._ , 2016;
Torquato, 2018; Torquato and Kim, 2021) we go beyond the current state of the
art through the introduction of a more elaborate substructure in the disorder
spectrum, perhaps foreshadowed by the notion of directional hyperuniformity in
Ref. (Torquato, 2016), and show, in the context of a scalar wave model, how
disorder can be more precisely engineered to create mid-band gaps and
orientation-dependent frequency-space transmission windows. We discuss its
theoretical basis by analyzing how the Fourier-space distribution of the
disorder affects wave scattering within the framework of the perturbative
expansion of the disordered Green’s function and the incident wave function. A
connection is also made with wave transparency in SH systems. Our analysis
sheds light on the relationship between the disorder components and the on-
shell scattering contributions to the disordered Green’s function, and
predicts which plane waves are suppressed by disorder scattering. We use the
insights from the analysis to discuss the conditions for isotropic and
selective transparency. For validation, we compute the coherent transmission
coefficient $t(\bm{k})$ (Anderson, 1985) for a wide range of incident angles
and frequencies, using the Atomistic Green’s Function (AGF) method adapted
from Refs. (Ong, 2018a, b), and obtain excellent agreement between the theory
and simulation results. We demonstrate with an 2D example of how the disorder
components can be combined to suppress wave scattering and to enable robust
coherent transmission for certain plane-wave states at specific incident
angles and frequencies. Possible experimental realizations are also suggested.

##  II Theory of disorder scattering of scalar waves

###  II.1 Scalar wave model

To discuss the scattering of a harmonic wave $\psi(\bm{r})$ of angular
frequency $\omega$ at position $\bm{r}$ in a disordered medium, we use the
Helmholtz wave equation

$\left[\nabla^{2}+\left(\frac{\omega}{c}\right)^{2}\epsilon(\bm{r})\right]\psi(\bm{r})=0$
(2)

where $\nabla^{2}$ and $c$ denote the 2D Laplace operator and wave speed,
respectively. Equation (2) has been used to model transverse magnetic (TM)
wave scattering in dielectric materials. (Davy  _et al._ , 2015; Froufe-Pérez
_et al._ , 2017; Brandstötter  _et al._ , 2019) For the purpose of this paper,
we interpret the wave function $\psi(\bm{r})$ as the out-of-plane electric
field component in a 2D dielectric medium (Davy  _et al._ , 2015; Froufe-Pérez
_et al._ , 2017; Brandstötter  _et al._ , 2019) although our results can be
generalized to non-photonic systems. The static disorder is described by the
permittivity function

$\epsilon(\bm{r})=\epsilon_{0}[1+f(\bm{r})]\ ,$

where $\epsilon_{0}=1$ denotes the permittivity of the disorder-free medium;
$f(\bm{r})$, which denotes the _disorder function_ corresponding to the
position-dependent fluctuations of the permittivity, can be written as a sum
of $N$ Fourier components like in Eq. (1), i.e.,

$f(\bm{r})=\alpha\sqrt{\frac{2}{N}}\sum_{j=1}^{N}\cos(\bm{p}_{j}\cdot\bm{r}+\phi_{j})\
,$ (3)

with the normalization constraint
$\lim_{\Omega\rightarrow\infty}\frac{1}{\Omega}\int_{\Omega}d\bm{r}|f(\bm{r})|^{2}=\alpha^{2}$
where $\Omega$ is the area of integration. The dimensionless constant $\alpha$
is the root-mean-square value of $f(\bm{r})$ and determines the relative
disorder strength in Eq. (2) as well as the ‘coupling constant’ in the Dyson
expansion which in our calculations, we set as $\alpha=0.1$. Instead of a
discrete sum, Eq. (3) can also be expressed as an integral
$f(\bm{r})=\alpha\int
d\bm{q}\rho(\bm{q})\exp[i(\bm{q}\cdot\bm{r}+\phi_{\bm{q}})]$ where
$\rho(\bm{q})$ is the ‘density of states’. The Fourier transform of
$f(\bm{r})$ is given by
$\mathcal{F}(\bm{q})=\frac{1}{h^{2}}\int_{\Omega}d\bm{r}e^{-i\bm{q}\cdot\bm{r}}f(\bm{r})=\alpha(\frac{2\pi}{h})^{2}\sum_{j=1}^{N}\sqrt{\frac{2}{N}}[e^{i\phi_{j}}\delta(\bm{q}-\bm{p}_{j})+e^{-i\phi_{j}}\delta(\bm{q}+\bm{p}_{j})]$
where $h$ is the 2D unit cell spacing. To facilitate our discussion of
scattering, we rewrite Eq. (2) as (Frank and Lubatsch, 2011; Sheremet  _et
al._ , 2020)

$\left[\nabla^{2}+\kappa_{0}(\omega)^{2}+V(\omega,\bm{r})\right]\psi(\bm{r})=0\
,$ (4)

where $\kappa_{0}(\omega)=\omega\sqrt{\epsilon_{0}}/c$ is the frequency-
dependent wave number for a plane-wave state in the disorder-free medium and
$V(\omega,\bm{r})=\kappa_{0}(\omega)^{2}f(\bm{r})$ denotes the perturbation
term which scales linearly with $\alpha$. In the absence of disorder where
$V(\omega,\bm{r})=0$, Eq. (4) admits the plane-wave solution
$\phi_{0}(\bm{r})=\frac{1}{\sqrt{\Omega}}e^{i\bm{k}\cdot\bm{r}}$ which
describes a state with the wave vector $\bm{k}$, such that
$|\bm{k}|=\kappa_{0}$.

###  II.2 Dyson expansion of the disordered Green’s function

To elucidate the relationship between wave scattering and disorder, we analyze
the retarded Green’s function $G^{+}(\omega,\bm{r},\bm{r}^{\prime})$, which
describes the $\omega$-dependent response at point $\bm{r}$ from a source at
$\bm{r}^{\prime}$ in the disordered medium (Economou, 1983; Sheng, 2006) and
is used in perturbative expansion of the wave function $\psi(\bm{r})$.
Although some of the material in the following discussion on the Dyson
expansion can be found in textbooks, we still discuss it in order to clarify
the role of the Fourier components of $V(\omega,\bm{r})$ in scattering and
wave transparency. Before we proceed, we give a qualitative bird’s eye view of
our approach to determining wave transparency. In our analysis, we discuss how
$V(\omega,\bm{r})$ affects the perturbative corrections to the Green’s
function and the incoming plane-wave state, which are related through the
Lippmann-Schwinger equation, and identify the condition for wave transparency
under which these corrections vanish. For simplicity, we describe these
corrections in terms of their Fourier transforms $\mathcal{V}(\bm{q})$ with
respect to the reciprocal-space coordinates $\bm{k}$ and $\bm{q}$. This sheds
light on the role of $\mathcal{V}(\bm{q})$ in the integrals for the
perturbative corrections. We exploit the chain structure of these integrals to
identify the $\mathcal{V}(\bm{q})$ components that suppress the perturbative
corrections when the former is set to zero. In addition, we show that when we
set $\mathcal{V}(\bm{q})=0$ for these components, the higher-order
perturbative corrections beyond the Born approximation are also suppressed.

We remark that in our treatment of the disordered Green’s function $G^{+}$, we
do not rely on any kind of configuration averaging and instead rely on the
analysis of wave scattering for each disorder configuration corresponding to a
unique set of phases $\\{\phi_{j}\\}$ from Eq. (3). This greatly simplifies
our analysis because it eliminates the need to evaluate the configuration-
averaged products of $V$ in the higher-order perturbative corrections to
$G^{+}$. (Sheng, 2006) Without configuration averaging, the integrals for each
order in the perturbative expansion have a chain structure that can be
exploited. This chain structure allows us to show that if that disorder
configuration satisfies the condition $\mathcal{V}(\bm{q})=0$ for an
identified set of $\bm{q}$ vectors such that the lowest-order correction to
$G^{+}$ vanishes, then all the higher-order corrections also vanish and we can
determine the condition for wave transparency.

To quantify the $O(\alpha^{n})$ perturbative scattering correction to the
incident $\bm{k}$ plane-wave state, we introduce the function
$\mathcal{P}_{n}(\bm{k},\bm{r})$, which is also used in the perturbative
expansion of $G^{+}$, i.e.,

$\psi(\bm{r})=\phi_{0}(\bm{r})[1+\mathcal{P}_{1}(\bm{k},\bm{r})+\mathcal{P}_{2}(\bm{k},\bm{r})+\ldots]\
,$ (5)

and analyze its relationship to $V(\omega,\bm{r})$. The connection between
$\psi(\bm{r})$ and $\phi_{0}(\bm{r})$ can be realized through the Lippmann-
Schwinger equation, (Economou, 1983)

$\psi(\bm{r})=\phi_{0}(\bm{r})+\int
d\bm{r}^{\prime}G_{0}^{+}(\omega,\bm{r},\bm{r}^{\prime})V(\omega,\bm{r}^{\prime})\psi(\bm{r}^{\prime})$

which we can rewrite as

$\psi(\bm{r})=\phi_{0}(\bm{r})+\int
d\bm{r}^{\prime}G^{+}(\omega,\bm{r},\bm{r}^{\prime})V(\omega,\bm{r}^{\prime})\phi_{0}(\bm{r}^{\prime})$
(6)

where the second term on the right represents the correction to the
unperturbed state after the perturbation $V$ is switched on. The significance
of $\mathcal{P}_{n}(\bm{k},\bm{r})$ is that it depends on the scattering
strength of the plane wave by the disorder $V$. Hence, the relation
$\mathcal{P}_{n}(\bm{k},\bm{r})=0$ is important for determining the condition
for wave transparency.

In the absence of disorder ($\alpha=0$), we define the retarded Green’s
function $G_{0}^{+}$ (Economou, 1983; Sheng, 2006) using the equation
$\left[\nabla^{2}+\kappa_{0}(\omega)^{2}\right]G_{0}^{+}(\omega,\bm{r},\bm{r}^{\prime})=\delta(\bm{r}-\bm{r}^{\prime})$,
where $\delta(\bm{r}-\bm{r}^{\prime})$ denotes the Dirac delta function. 444In
two dimensions,
$G_{0}^{+}(\omega,\bm{r},\bm{r}^{\prime})=-\frac{i}{4}H_{0}^{(1)}(\kappa_{0}|\bm{r}-\bm{r}^{\prime}|)$
where $H_{0}^{(1)}$ denotes the zeroth-order Hankel function of the first
kind. (Economou, 1983; Sheng, 2006) When disorder is present, the retarded
Green’s function $G^{+}(\omega,\bm{r},\bm{r}^{\prime})$ is defined by the
equation
$\left[\nabla^{2}+\kappa_{0}(\omega)^{2}+V(\omega,\bm{r})\right]G^{+}(\omega,\bm{r},\bm{r}^{\prime})=\delta(\bm{r}-\bm{r}^{\prime})$.
Unlike $G_{0}^{+}$, $G^{+}$ has no closed form but is formally related to
$G_{0}^{+}$ through the Dyson equation
$G^{+}(\omega,\bm{r},\bm{r}^{\prime})=G_{0}^{+}(\omega,\bm{r},\bm{r}^{\prime})+\int
d\bm{r}_{1}G_{0}^{+}(\omega,\bm{r},\bm{r}_{1})V(\omega,\bm{r}_{1})G^{+}(\omega,\bm{r}_{1},\bm{r}^{\prime})$,
(Economou, 1983; Sheng, 2006) which we can expand as a power series

$\displaystyle G^{+}(\omega,\bm{r},\bm{r}^{\prime})$
$\displaystyle=\sum_{n=0}^{\infty}G_{n}^{+}(\omega,\bm{r},\bm{r}^{\prime})$
(7)

where $G_{n}^{+}$ corresponds to the $O(\alpha^{n})$ correction to $G^{+}$ and
can be expressed as a convolution of $G_{0}^{+}$ and $V$.

In the following discussion, we drop $\omega$ from the arguments of
$G_{n}^{+}$ and $V$ for the sake of brevity. Assuming that the disorder is
sufficiently weak for the series expansion to be valid, we have for $n=1$

$G_{1}^{+}(\bm{r},\bm{r}^{\prime})=\int
d\bm{r}_{1}G_{0}^{+}(\bm{r},\bm{r}_{1})V(\bm{r}_{1})G_{0}^{+}(\bm{r}_{1},\bm{r}^{\prime})\
,$ (8)

and for $n>1$, in general,

$\displaystyle G_{n}^{+}(\bm{r},\bm{r}^{\prime})$ $\displaystyle=\int
d\bm{r}_{1}\ldots\int d\bm{r}_{n}G_{0}^{+}(\bm{r},\bm{r}_{1})V(\bm{r}_{1})$
$\displaystyle\times G_{0}^{+}(\bm{r}_{1},\bm{r}_{2})\ldots
V(\bm{r}_{n})G_{0}^{+}(\bm{r}_{n},\bm{r}^{\prime})\ .$ (9)

We note that for $n\geq 1$, the integral in Eq. (9) has the chain arrangement

$G_{n}^{+}(\bm{r},\bm{r}^{\prime})=\int
d\bm{r}_{1}G_{0}^{+}(\bm{r},\bm{r}_{1})V(\bm{r}_{1})G_{n-1}^{+}(\bm{r}_{1},\bm{r}^{\prime})\
.$ (10)

The expression for $G_{n}^{+}$ depends on $G_{n-1}^{+}$ which in turns depends
on $G_{n-2}^{+}$ and so on. Therefore, if we can prove that
$G_{1}^{+}(\bm{r},\bm{r}^{\prime})$, the lowest-order correction to
$G^{+}(\bm{r},\bm{r}^{\prime})$, vanishes for any $\bm{r}$ and
$\bm{r}^{\prime}$ when $V(\bm{r})$ has the right disorder configuration, then
all the higher-order $G_{n}^{+}(\bm{r},\bm{r}^{\prime})$ terms in Eq. (7) also
vanish by induction. For the isotropic transparency condition, this reduces
the problem to a matter of determining the $V(\bm{r})$ configuration for which
$G_{1}^{+}(\bm{r},\bm{r}^{\prime})=0$.

To simplify the convolution in Eq. (8), we use the Fourier transforms
$\mathcal{G}_{0}^{+}(\bm{k})$ and $\mathcal{V}(\bm{q})$, defined by the
equations
$G_{0}^{+}(\bm{r},\bm{r}^{\prime})=h^{2}\int_{\text{BZ}}\frac{d\bm{k}}{(2\pi)^{2}}e^{i\bm{k}\cdot(\bm{r}-\bm{r}^{\prime})}\mathcal{G}_{0}^{+}(\bm{k})$
and
$V(\bm{r})=h^{2}\int_{\text{BZ}}\frac{d\bm{q}}{(2\pi)^{2}}e^{i\bm{q}\cdot\bm{r}}\mathcal{V}(\bm{q})$,
respectively, with the explicit form of $\mathcal{V}(\bm{q})$ as
$\mathcal{V}(\bm{q})=\kappa_{0}^{2}\mathcal{F}(\bm{q})$. It is well-known that
$\mathcal{V}(\bm{q})$ is proportional to the scattering amplitude in the Born
approximation. (Economou, 1983) From Eqs. (5) and (6), we obtain the
expression

$\mathcal{P}_{1}(\bm{k},\bm{r})=h^{4}\int\frac{d\bm{q}}{(2\pi)^{2}}e^{i\bm{q}\cdot\bm{r}}\mathcal{V}(\bm{q})\mathcal{G}_{0}^{+}(\bm{k}+\bm{q})$
(11)

which implies that we can rewrite $G_{1}^{+}$ from Eq. (8) as

$G_{1}^{+}(\bm{r},\bm{r}^{\prime})=h^{2}\int\frac{d\bm{k}}{(2\pi)^{2}}e^{i\bm{k}\cdot(\bm{r}-\bm{r}^{\prime})}\mathcal{G}_{0}^{+}(\bm{k})\mathcal{P}_{1}(\bm{k},\bm{r})\
,$ (12)

where $\mathcal{G}_{0}^{+}(\bm{k})=\lim_{\eta\rightarrow
0^{+}}\frac{1}{h^{2}}\frac{1}{(\kappa_{0}+i\eta)^{2}-|\bm{k}|^{2}}$. The pole
in $\mathcal{G}_{0}^{+}(\bm{k})$ at $|\bm{k}|=\kappa_{0}$ (Sheng, 2006) allows
us to simplify Eq. (12) as an angular integral, i.e.,

$G_{1}^{+}(\bm{r},\bm{r}^{\prime})=\frac{i}{8\pi}\int_{0}^{2\pi}d\theta\exp(i\kappa_{0}\cos\theta|\bm{r}-\bm{r}^{\prime}|)\mathcal{P}_{1}(\kappa_{0}\bm{\hat{k}},\bm{r})\
,$ (13)

where $\theta$ is the angle of $\bm{k}$ with respect to
$\bm{r}-\bm{r}^{\prime}$ and $\bm{\hat{k}}$ denotes the unit vector parallel
to $\bm{k}$, and Eq. (13) implies that $G_{1}^{+}$ depends on the value of
$\mathcal{P}_{1}(\bm{k},\bm{r})$ over the kinematically constrained
$|\bm{k}|=\kappa_{0}$ ‘shell’. Similarly, we can express Eq. (9) as

$G_{n}^{+}(\bm{r},\bm{r}^{\prime})=h^{2}\int\frac{d\bm{k}}{(2\pi)^{2}}e^{i\bm{k}\cdot(\bm{r}-\bm{r}^{\prime})}\mathcal{G}_{0}^{+}(\bm{k})\mathcal{P}_{n}(\bm{k},\bm{r})\
,$ (14)

where

$\displaystyle\mathcal{P}_{n}(\bm{k},\bm{r})$
$\displaystyle=h^{4n}\int\frac{d\bm{q}_{1}}{(2\pi)^{2}}\ldots\int\frac{d\bm{q}_{n}}{(2\pi)^{2}}e^{i(\bm{q}_{1}+\ldots+\bm{q}_{n})\cdot\bm{r}}\mathcal{V}(\bm{q}_{1})$
$\displaystyle\times\mathcal{G}_{0}^{+}(\bm{k}+\bm{q}_{1})\ldots\mathcal{V}(\bm{q}_{n})\mathcal{G}_{0}^{+}(\bm{k}+\bm{q}_{1}+\ldots+\bm{q}_{n})\
,$ (15)

which we can rewrite as

$\displaystyle\mathcal{P}_{n}(\bm{k},\bm{r})$
$\displaystyle=h^{4}\int\frac{d\bm{q}_{1}}{(2\pi)^{2}}e^{i\bm{q}_{1}\cdot\bm{r}}\mathcal{V}(\bm{q}_{1})\mathcal{G}_{0}^{+}(\bm{k}+\bm{q}_{1})$
$\displaystyle\times\mathcal{P}_{n-1}(\bm{k}+\bm{q}_{1},\bm{r})\ ,$ (16)

with $\mathcal{P}_{0}=1$. The identity in Eq. (16) relates $\mathcal{P}_{n}$
to $\mathcal{P}_{n-1}$ for $n\geq 1$ and like in Eq. (10), has a chain
arrangement within the integrand in which $\mathcal{P}_{n}(\bm{k},\bm{r})$
depends on $\mathcal{P}_{n-1}(\bm{k}+\bm{q}_{1},\bm{r})$ over the same shell
where $|\bm{k}|=|\bm{k}+\bm{q}_{1}|=\kappa_{0}$. This will be useful for
understanding why all higher-order corrections to the $\bm{k}$ plane-wave
state are suppressed for $|\bm{k}|<q_{\text{c}}/2$ when the disorder is
described by Eq. (3).

###  II.3 On-shell scattering in perturbative expansion

The expression in Eq. (15) lends itself to an intuitive physical
interpretation when it is fully expanded in $\mathcal{G}_{0}^{+}$ and
$\mathcal{V}$. If we confine ourselves to the first-order (Born) approximation
or $\psi(\bm{r})\approx\phi_{0}(\bm{r})[1+\mathcal{P}_{1}(\bm{k},\bm{r})]$, we
can interpret $\mathcal{P}_{1}(\bm{k},\bm{r})$ from Eq. (11) as the
perturbative correction in $\psi(\bm{r})$ due to all the possible
$\bm{k}\stackrel{{\scriptstyle\mathcal{V}(\bm{q}_{1})}}{{\longrightarrow}}\bm{k}+\bm{q}_{1}$
scattering processes in which $\mathcal{V}(\bm{q}_{1})$ determines the
strength of the scattering process and $\bm{k}$ and $\bm{k}+\bm{q}_{1}$ are
the wave vectors of the incident and outgoing plane-wave states, respectively.
Similarly, we can interpret
$\mathcal{P}_{2}(\bm{k},\bm{r})=h^{8}\int\frac{d\bm{q}_{1}}{(2\pi)^{2}}\int\frac{d\bm{q}_{2}}{(2\pi)^{2}}e^{i(\bm{q}_{1}+\bm{q}_{2})\cdot\bm{r}}\mathcal{V}(\bm{q}_{1})\mathcal{G}_{0}^{+}(\bm{k}+\bm{q}_{1})\mathcal{V}(\bm{q}_{2})\mathcal{G}_{0}^{+}(\bm{k}+\bm{q}_{1}+\bm{q}_{2})$
as the next-order perturbative correction in $\psi(\bm{r})$ due to all the
possible
$\bm{k}\stackrel{{\scriptstyle\mathcal{V}(\bm{q}_{1})}}{{\longrightarrow}}\bm{k}+\bm{q}_{1}\stackrel{{\scriptstyle\mathcal{V}(\bm{q}_{2})}}{{\longrightarrow}}\bm{k}+\bm{q}_{1}+\bm{q}_{2}$
scattering processes, and likewise for the remaining
$\mathcal{P}_{n}(\bm{k},\bm{r})$ terms where
$\bm{k}\stackrel{{\scriptstyle\mathcal{V}(\bm{q}_{1})}}{{\longrightarrow}}\bm{k}+\bm{q}_{1}\stackrel{{\scriptstyle\mathcal{V}(\bm{q}_{2})}}{{\longrightarrow}}\ldots\stackrel{{\scriptstyle\mathcal{V}(\bm{q}_{n})}}{{\longrightarrow}}\bm{k}+\bm{q}_{1}+\ldots+\bm{q}_{n}$.
Because $\mathcal{G}_{0}^{+}(\bm{k})$ has poles at $|\bm{k}|=\kappa_{0}$,
(Sheng, 2006) the presence of the $\mathcal{G}_{0}^{+}(\bm{k})$,
$\mathcal{G}_{0}^{+}(\bm{k}+\bm{q}_{1})$, $\ldots$,
$\mathcal{G}_{0}^{+}(\bm{k}+\bm{q}_{1}+\ldots+\bm{q}_{n})$ terms in the
integrand of Eq. (15) implies that the contributions to
$\mathcal{P}_{n}(\bm{k},\bm{r})$ for $n\geq 1$ are maximized when the wave
vectors of the virtual states ($\bm{k}$, $\bm{k}+\bm{q}_{1}$, $\ldots$,
$\bm{k}+\bm{q}_{1}+\ldots+\bm{q}_{n}$) are kinematically restricted to the
circular frequency shell of radius $\kappa_{0}$, i.e.,

$|\bm{k}|=|\bm{k}+\bm{q}_{1}|=\ldots=|\bm{k}+\bm{q}_{1}+\ldots+\bm{q}_{n}|=\kappa_{0}\
.$ (17)

The singularity of $\mathcal{G}_{0}^{+}$ implies that the integral associated
with $\mathcal{P}_{n}(\bm{k},\bm{r})$ in Eq. (15) is limited to scattering
between these on-shell states. The kinematic constraint in Eq. (17) also
implies that the arbitrarily large momentum transfers
($\Delta\bm{k}=\bm{q}_{1}+\ldots+\bm{q}_{n}$) from multiple scattering are not
possible.

###  II.4 Condition for wave transparency

If a plane-wave state with wave vector $\bm{k}$ propagates through the
disordered medium without attenuation, it means that the medium is transparent
and $\psi(\bm{r})=\phi_{0}(\bm{r})$ because of the absence of scattering,
i.e., $\mathcal{P}_{n}(\bm{k},\bm{r})=0$ for $n\geq 1$. To determine the
condition for the absence of scattering, we analyze the structure of the
integrand in $\mathcal{P}_{n}(\bm{k},\bm{r})$ from Eq. (15) and find the
configuration of the disorder $V(\bm{r})$, in the form of the
$\mathcal{V}(\bm{q})=0$ spectrum, that is compatible with
$\mathcal{P}_{n}(\bm{k},\bm{r})=0$.

####  II.4.1 Isotropic transparency

In the case of _isotropic_ transparency like in SH systems, we have to find
the $\mathcal{V}(\bm{q})$ spectrum compatible with
$\mathcal{P}_{n}(\bm{k},\bm{r})=0$ for _all_ $\bm{k}$’s that satisfy
$|\bm{k}|=\kappa_{0}$ and Eq. (14) implies the more stringent condition
$G^{+}(\bm{r},\bm{r}^{\prime})\approx G_{0}^{+}(\bm{r},\bm{r}^{\prime})$. This
problem is however greatly simplified given the chain structure of Eq. (10)
which suggests that we need only to determine the $\mathcal{V}(\bm{q})$
spectrum corresponding to $G_{1}^{+}(\bm{r},\bm{r}^{\prime})=0$ because Eq.
(10) implies that $G_{n}^{+}(\bm{r},\bm{r}^{\prime})=0$ for $n>1$.

The structure of the angular integral in Eq. (13) means that
$G_{1}^{+}(\bm{r},\bm{r}^{\prime})$ depends on
$\mathcal{P}_{1}(\bm{k},\bm{r})$ distributed over all $\bm{k}$’s that satisfy
$|\bm{k}|=\kappa_{0}$. In this angular integral, the value of
$\mathcal{P}_{1}(\bm{k},\bm{r})$ on the shell depends on the magnitude of
$\mathcal{V}(\bm{q})$ for $|\bm{q}|<2\kappa_{0}$, which we physically
interpret as a bottleneck limiting the availability of the on-shell
$\bm{k}\rightarrow\bm{k}+\bm{q}$ scattering phase space for
$|\bm{k}|=|\bm{k}+\bm{q}|=\kappa_{0}$ . Hence, if we wish to make the
approximation $G_{1}^{+}(\bm{r},\bm{r}^{\prime})\approx 0$, we should set
$\mathcal{V}(\bm{q})=0$ for $|\bm{q}|<2\kappa_{0}$ to suppress this on-shell
scattering contribution to $G_{1}^{+}(\bm{r},\bm{r}^{\prime})$. This is
similar to the condition $S(\bm{q})=0$ for $|\bm{q}|<q_{\text{SH}}$ in SH
systems which are also transparent for incoming plane-wave states that satisfy
$|\bm{k}|<q_{\text{SH}}/2$. (Leseur  _et al._ , 2016; Kim and Torquato, 2020;
Torquato and Kim, 2021) In our case, we may regard $|\mathcal{V}(\bm{q})|^{2}$
as the analog of $S(\bm{q})$ for $|\bm{q}|>0$ and like in SH systems, define a
finite ‘exclusion zone’ centered around the origin in reciprocal space for the
nonzero $\mathcal{V}(\bm{q})$ components.

Therefore, if we set $V(\bm{r})$ from Eq. (4) to be proportional to
$U(\bm{r};\\{\phi_{j}\\})$ from Eq. (1) such that the Fourier components in
$f(\bm{r})$ are only non-zero when $|\bm{q}|=q_{\text{c}}$, then we can
approximate $G^{+}(\omega,\bm{r},\bm{r}^{\prime})\approx
G_{0}^{+}(\omega,\bm{r},\bm{r}^{\prime})$ for $\omega<\omega_{\text{c}}$ where
$\omega_{c}=\frac{cq_{\text{c}}}{2\sqrt{\epsilon_{0}}}$ is the cutoff
frequency. This implies that any incident plane wave with frequency
$\omega<\omega_{\text{c}}$, or equivalently with wave number
$|\bm{k}|<q_{\text{c}}/2$, can propagate through the disordered region with
near total transparency, consistent with the principal finding of Ref. (Kim
and Heller, 2022). Similar conditions for the cutoff frequency in SH systems
have also been derived in Ref. (Leseur  _et al._ , 2016) and also by Torquato
and Kim (Kim and Torquato, 2020; Torquato and Kim, 2021). Figure 1 shows the
geometrical interpretation for the on-shell scattering contribution in
$\mathcal{P}_{1}(\bm{k},\bm{r})$. We draw an Ewald circle of all the possible
$|\bm{k}|=\kappa_{0}$ states and another circle containing the disorder modes.
If $\omega>\omega_{\text{c}}$, then the Ewald circle is large enough for the
two circles to intersect and disorder scattering is allowed. If
$\omega<\omega_{\text{c}}$, no disorder scattering is allowed.

Figure 1: The black Ewald circle is the shell containing the loci of all
possible $\bm{k}$ states such that $|\bm{k}|=\kappa_{0}$. The green circle of
radius $q_{\text{c}}$ is centered at $\bm{k}$ and contains the loci of all the
possible $\bm{q}$ disorder Fourier components in the Berry potential. (a) If
$\kappa_{0}>q_{\text{c}}/2$, then the on-shell
$\bm{k}\rightarrow\bm{k}+\bm{q}$ scattering process is allowed because the two
circles intersect. (b) If $\kappa_{0}<q_{\text{c}}/2$, then the two circles do
not intersect and there is no on-shell scattering.

####  II.4.2 Selective transparency

Beyond isotropic transparency, we can also fine-tune $\mathcal{V}(\bm{q})$ to
generate wave transparency for a set of $\bm{k}$ plane-wave states that is
more selective than the $0<\omega<\omega_{\text{c}}$ frequency band in
isotropic transparency. Instead of setting $\mathcal{V}(\bm{q})=0$ in the
entire $|\bm{q}|<q_{\text{c}}$ neighborhood, we limit the
$\mathcal{V}(\bm{q})=0$ condition to a subset of the Fourier components in the
$|\bm{q}|<q_{\text{c}}$ region to generate a smaller window of wave
transparency containing the selected $\bm{k}$ plane-wave states. In other
words, we permit some non-zero Fourier components in the ‘exclusion zone’ to
interact (scatter) with the incoming plane-wave states that are outside of
this window. To do this, we need to determine the condition for
$\mathcal{P}_{n}(\bm{k},\bm{r})=0$ for the $\bm{k}$ states in this window. We
first show how this is determined for $\mathcal{P}_{1}(\bm{k},\bm{r})$ in Eq.
(11), which is associated with the
$\bm{k}\stackrel{{\scriptstyle\mathcal{V}(\bm{q}_{1})}}{{\longrightarrow}}\bm{k}+\bm{q}_{1}$
scattering processes of an individual $\bm{k}$ state, as the procedure can be
generalized to higher-order $\mathcal{P}_{n}(\bm{k},\bm{r})$ terms because the
integrand for $\mathcal{P}_{n}(\bm{k},\bm{r})$ in Eq. (16) also contains the
$\mathcal{V}(\bm{q})\mathcal{G}_{0}^{+}(\bm{k}+\bm{q})$ term which determines
the loci of $\bm{q}$ for the $\mathcal{V}(\bm{q})$ contributing to the
integral.

Given the presence of the
$\mathcal{V}(\bm{q})\mathcal{G}_{0}^{+}(\bm{k}+\bm{q})$ term in the integrand,
which we associate with the $\bm{k}\rightarrow\bm{k}+\bm{q}$ scattering
process shown in Fig. 1(a), we minimize the integral in Eq. (11) and hence
$\mathcal{P}_{1}(\bm{k},\bm{r})$ by setting $\mathcal{V}(\bm{q})=0$ for all
possible values of $\bm{q}$ that satisfy $|\bm{k}+\bm{q}|=\kappa_{0}$. These
loci of $\bm{q}$ values for $\mathcal{V}(\bm{q})=0$ comprise a circle of
radius $\kappa_{0}$ centered at $\bm{q}=-\bm{k}$ in reciprocal space and
contain all the Fourier components that can interact with that particular
$\bm{k}$ state. For each $\bm{k}$ state, we have one circle. We note that this
circle is contained within the $|\bm{q}|<2\kappa_{0}$ neighborhood. Hence, to
generate transparency for a window of $\bm{k}$ states, we set the
$\mathcal{V}(\bm{q})=0$ condition over the reciprocal-space region defined by
the superposition of these circles. If this transparency window includes all
the $\bm{k}$ states with $|\bm{k}|<q_{\text{c}}/2$ like in SH systems, then,
as expected, we have to impose the $\mathcal{V}(\bm{q})=0$ condition over the
entire $|\bm{q}|<q_{\text{c}}$ neighborhood. Otherwise, the
$\mathcal{V}(\bm{q})=0$ condition has to be applied to only a subregion of the
$|\bm{q}|<q_{\text{c}}$ neighborhood. This procedure also minimizes the
higher-order $\mathcal{P}_{n}(\bm{k},\bm{r})$ terms, which are associated with
the
$\bm{k}\stackrel{{\scriptstyle\mathcal{V}(\bm{q}_{1})}}{{\longrightarrow}}\bm{k}+\bm{q}_{1}\stackrel{{\scriptstyle\mathcal{V}(\bm{q}_{2})}}{{\longrightarrow}}\ldots\stackrel{{\scriptstyle\mathcal{V}(\bm{q}_{n})}}{{\longrightarrow}}\bm{k}+\bm{q}_{1}+\ldots+\bm{q}_{n}$
process for multiple scattering, because it suppresses the
$\bm{k}\stackrel{{\scriptstyle\mathcal{V}(\bm{q}_{1})}}{{\longrightarrow}}\bm{k}+\bm{q}_{1}$
part of the scattering process.

This result can be reached from another perspective by considering the self-
energy, which describes the frequency shift and inverse lifetime caused by the
disorder scattering, as defined by
$\Sigma(\omega,\bm{k})=\mathcal{G}_{0}^{+}(\bm{k})^{-1}-\langle\mathcal{G}(\bm{k})\rangle^{-1}$
where $\langle\mathcal{G}(\bm{k})\rangle$ is the configuration-averaged
disordered Green’s function in reciprocal space. To the lowest non-zero
$O(\alpha^{2})$ approximation, we can write (Mahan, 2000)

$\displaystyle\Sigma(\omega,\bm{k})$
$\displaystyle\approx\int\frac{d\bm{q}}{(2\pi)^{2}}\langle\mathcal{V}(\bm{q})\mathcal{V}(-\bm{q})\rangle\mathcal{G}_{0}^{+}(\bm{k}+\bm{q})$
$\displaystyle\approx\int\frac{d\bm{q}}{(2\pi)^{2}}|\mathcal{V}(\bm{q})|^{2}\mathcal{G}_{0}^{+}(\bm{k}+\bm{q})$
(18)

since $\mathcal{V}(-\bm{q})=\mathcal{V}(\bm{q})^{\dagger}$. Given the
singularity in $\mathcal{G}_{0}^{+}(\bm{k}+\bm{q})$ and the linear dispersion
$\omega(\bm{k})=c|\bm{k}|$, the $\bm{q}$ integral for $\Sigma(\omega,\bm{k})$
is effectively taken over a circular shell of radius $\kappa_{0}$ centered at
$\bm{q}=-\bm{k}$ in reciprocal space, like in the integral for
$\mathcal{P}_{1}(\bm{k},\bm{r})$ in Eq. (11). Therefore, if we set
$\mathcal{V}(\bm{q})=0$ on this shell and use the on-shell approximation
$|\bm{k}|=\kappa_{0}$, then we have $\Sigma(\omega,\bm{k})\approx 0$ which we
interpret as the absence of scattering for the $\bm{k}$ plane-wave state.

##  III Coherent transmission simulation and analysis

###  III.1 Simulation setup

To validate our analysis, we compute the _coherent_ transmission coefficient
$t(\bm{k})$, (Anderson, 1985) which determines the proportion of the wave flux
passing through the disordered region without blurring, for a range of
$\bm{k}$ states. If $|t(\bm{k})|^{2}=1$, the disordered medium is completely
transparent. We approximate Eq. (2) using a 2D square lattice (Sheng, 2006),
in which the second derivatives are replaced by finite differences, i.e.,
$\nabla^{2}\psi(\bm{r})\approx\frac{1}{a^{2}}[-4\psi(\bm{r})+\psi(\bm{r}+a\hat{\bm{x}})+\psi(\bm{r}-a\hat{\bm{x}})-\psi(\bm{r}+a\hat{\bm{y}})+\psi(\bm{r}-a\hat{\bm{y}})]$
where $a$ is the 2D lattice constant determined by the length scale of the
scattering problem. The resulting eigenvalue equation
$\frac{1}{a^{2}}[4\psi(\bm{r})-\psi(\bm{r}+a\hat{\bm{x}})-\psi(\bm{r}-a\hat{\bm{x}})-\psi(\bm{r}+a\hat{\bm{y}})-\psi(\bm{r}-a\hat{\bm{y}})]=\omega^{2}\frac{\epsilon(\bm{r})}{c^{2}}\psi(\bm{r})$
can be written in the matrix form $\bm{K}\Psi=\omega^{2}\bm{M}\Psi$, where
$\bm{K}$ is the finite-difference matrix, $\Psi$ is a column vector with
$\psi(\bm{r})$ as its vector elements, and $\bm{M}$ is a diagonal matrix with
$\frac{\epsilon(\bm{r})}{c^{2}}$ as its diagonal elements. This formulation
sets us up for the direct scattering amplitude calculations using the AGF
method, (Ong, 2018a, b) a technique developed to study phonon scattering. (Ong
_et al._ , 2020; Ong, 2021) In the disorder-free lattice where
$\epsilon(\bm{r})=\epsilon_{0}$, the dispersion relationship is given by
(Sheng, 2006)
$\omega(\bm{k})^{2}=\frac{4c^{2}}{a^{2}}[\sin^{2}(\frac{1}{2}k_{x}a)+\sin^{2}(\frac{1}{2}k_{y}a)]$.
In the continuum ($a\rightarrow 0$) limit, we obtain $|\bm{k}|=\kappa_{0}$ and
recover the linear $\omega(\bm{k})=c|\bm{k}|$ relationship. (Sheng, 2006)

Our AGF simulation setup is shown in Fig. 2. A disordered region of width $W$
and length $L$ is sandwiched between the semi-infinite and disorder-free left
and right leads where $f(\bm{r})=0$. We fix $W=200a$ and let $L$ vary. We use
the dimensionless variable $\kappa_{0}a$ to represent the frequency. For each
$\bm{k}$ plane wave, we compute $|t(\bm{k})|^{2}=|S(\bm{k},\bm{k})|^{2}$,
where $S(\bm{k}^{\prime},\bm{k})$ is the scattering amplitude between the
incoming state $\Psi_{\text{in}}(\bm{k})$ and the outgoing state
$\Psi_{\text{out}}(\bm{k}^{\prime})$ at frequency $\omega$. At each frequency
step, we determine all the kinematically allowed $\bm{k}$ states and compute
their $S(\bm{k}^{\prime},\bm{k})$ and $|t(\bm{k})|^{2}$ values. For
consistency with our analysis based on the continuum limit, we restrict the
range of $\bm{k}$ in our AGF calculations to $|\bm{k}|\apprle 1/a$ or
$\kappa_{0}a\leq 1$.

Figure 2: Schematic of the simulation setup for the AGF calculations. The
disordered region ($0\leq x\leq L$) is confined between the two semi-infinite
disorder-free leads on the left ($x<0$) and right ($x>L$) where $f(\bm{r})=0$.
In the transverse $y$ direction, we impose periodic boundary conditions.

###  III.2 Simulation results

####  III.2.1 Isotropic wave transparency

Figure 3(a) shows an instance of $f(\bm{r})$ in which its Fourier components
are distributed uniformly over a circular manifold (or ‘ring’) with a radius
of $|\bm{q}|=q_{\text{c}}=0.5/a$, as shown in the inset of Fig. 3(a). The
corresponding $|t(\bm{k})|^{2}$ spectrum for $L=5W$ is shown in Fig. 3(b) for
the frequency range $0<\kappa_{0}a\leq 1$. We observe that for
$\kappa_{0}>q_{\text{c}}/2$, $|t(\bm{k})|^{2}\approx 0$ for almost every state
as a result of multiple scattering by the disorder. This reduction in
$|t(\bm{k})|^{2}$ for $\kappa_{0}>q_{\text{c}}/2$ is a result of the
disordered domain being much larger than the mean free paths of the plane-wave
states with $\kappa_{0}>q_{\text{c}}/2$. At $\kappa_{0}=q_{\text{c}}/2$, there
is a sharp transition or cutoff frequency, below which $|t(\bm{k})|^{2}\approx
1$ or near-perfect transparency for every state regardless of the angle of
incidence. Nonetheless, we observe some speckling in the $|t(\bm{k})|^{2}$
spectrum above the cutoff frequency due to the random phase in the disorder
modes. If we distribute the disorder components uniformly over a band of rings
($0.5/a\leq|\bm{q}|\leq 1/a$), as shown in the inset of Fig. 3(c), instead of
a single ring, then the speckling is smoothed out, as shown in Fig. 3(d),
because of the greater range of $\mathcal{V}(\bm{q})$ components available for
on-shell scattering.

Figure 3: The disorder function $f(\bm{r})$ for disorder modes at (a)
$|\bm{q}|=0.5/a$ and (b) $0.5/a\leq|\bm{q}|\leq 1/a$. The value of $f(\bm{r})$
is indicated by the color bar in the bottom right corner. The bottom left
corner insets show the loci of the disorder modes in $\bm{q}$ space. The
corresponding coherent transmission $|t(\bm{k})|^{2}$ spectra for $L=5W$ are
shown in (c) and (d), respectively, with each $\bm{k}$ plane wave represented
by a colored dot indicating its numerical value according to the color bar. To
guide the eye, we draw the dimensionless frequency-dependent $\kappa_{0}a$
contour lines at intervals of $0.025$ (gray lines) and $0.25$ (black lines).

####  III.2.2 Orientation and frequency-dependent transparency window

The smoothing of the speckling in Fig. 3(d) implies that the the effects of
the disorder Fourier components are additive, i.e., by selectively including
more Fourier components in $f(\bm{r})$ we can introduce more scattering
pathways to modify the $|t(\bm{k})|^{2}$ spectrum by filtering out unwanted
$\bm{k}$ states. We go beyond the SH systems by introducing an anisotropic
substructure in the distribution of the Fourier components. We elaborate on
this idea with the example in Fig. .4. Figure 4(a) shows the anisotropic
$f(\bm{r})$ distribution ($f=f_{\text{inner}}+f_{\text{outer}}$) obtained from
combining two disorder distributions $f_{\text{inner}}(\bm{r})$ and
$f_{\text{outer}}(\bm{r})$, shown in Figs. 4(b) and (c), respectively. Each of
the disorder distributions corresponds to a set of distinctive scattering
pathways. The highly anisotropic $f_{\text{inner}}$ comprises Fourier
components distributed over two circular pockets with radius of $0.2/a$ and
centered away from the origin at $q_{x}=\pm 0.3/a$ unlike the examples in Fig.
3. On the other hand, $f_{\text{outer}}$, which describes SH-like disorder, is
isotropic and comprises Fourier components that are centered at the origin and
distributed over a band of rings with radii varying between $1/a$ and $1.5/a$.

To understand how combining disorder distributions affects the scattering
pathways, we plot in Figs. 4(d) to (f) the $\bm{k}$ plane-wave states that are
unaffected by on-shell scattering for $f(\bm{r})$, $f_{\text{inner}}(\bm{r})$,
and $f_{\text{outer}}(\bm{r})$, respectively. At each given $\kappa_{0}$, we
compute all the possible $\bm{k}\rightarrow\bm{k}+\bm{q}$ on-shell scattering
transitions for each $\bm{k}$ state. If none of the $\bm{q}$’s lie in the
region occupied by the Fourier components, then that $\bm{k}$ state is
considered unaffected by on-shell scattering. The distribution of unscattered
$\bm{k}$ states for $f_{\text{inner}}$ in Fig. 4(e) shows a distinctive
anisotropy. At more oblique angles of incidence, the $\bm{k}$ states are more
likely to be scattered. The low-frequency plane waves with $\kappa_{0}a<0.05$
are however unaffected by on-shell scattering because the disorder component
closest to the origin is at $\bm{q}=(\pm 0.1/a,0)$. In the
$0.05\leq\kappa_{0}a\leq 0.25$ frequency range, there is a transmission
gap555We define the transmission gap as the frequency range in which plane
waves cannot propagate through the disordered medium. because of the absence
of unscattered plane waves. The upper bound of this gap is determined by the
position of the disorder mode furthest from the origin at $\bm{q}=(\pm
0.5/a,0)$ and can be adjusted by changing the radius and position of the
circular pockets in $f_{\text{inner}}$. At higher frequencies
($\kappa_{0}a>0.25$), a range of unscattered $\bm{k}$ states exists for more
acute incident angles because there are no disorder modes available in
$f_{\text{inner}}$ to scatter the plane wave at higher frequencies for small
incident angles. For $f_{\text{outer}}$, the distribution of unscattered
$\bm{k}$ states in Fig. 4(f) show a sharp cutoff at $\kappa_{0}a<0.5$ because
the disorder Fourier components are located in the $1/a\leq|\bm{q}|\leq 1.5/a$
band.

Hence, for $f$, Fig. 4(d) shows the distribution of unscattered $\bm{k}$
states, which is equal to the intersection of the unscattered $\bm{k}$ states
in Figs. 4(e) and (f). Figures 4(g) to (i) show the $|t(\bm{k})|^{2}$ spectra
for $L=20W$ where a larger $L$ is chosen to magnify the effects of multiple
scattering. We see a close correspondence between Figs. 4(d) and (g), with the
$|t(\bm{k})|^{2}$ values highest for the unscattered $\bm{k}$ states,
validating our theory of how disorder Fourier components affect wave
scattering and transport. The $|t(\bm{k})|^{2}$ spectrum for $f$ can also be
approximated by taking the product of the $|t(\bm{k})|^{2}$’s for
$f_{\text{inner}}$ and $f_{\text{outer}}$ from Figs. 4(h) and (i). Figure 4(g)
shows that by combining the Fourier components of $f_{\text{inner}}$ and
$f_{\text{outer}}$ to filter out the unwanted $\bm{k}$ states, we are able to
create transmission gaps and a window of high transparency [Fig. 4(g)] in the
$|t(\bm{k})|^{2}$ spectrum, in which incoming plane waves can be robustly
transmitted through the disordered medium without blurring. The correspondence
between Figs. 4(d) and (g) is however not perfect because of wave reflection
at the boundary between the leads and the disordered region. Nevertheless, the
example from Fig. 4 shows that we can combine groups of disorder Fourier
components to engineer specific wave transport properties such as the
transparency window; we use $f_{\text{in}}$ to create anisotropy and a
transmission gap and $f_{\text{out}}$to create a low-pass filter. Furthermore,
the orientation, size and position of the transparency window can be modified
by changing the loci of the circular pockets and ring in $f$. In Figs. 6 and
7, we also show how the $|t(\bm{k})|^{2}$ spectrum changes when we modify
$f_{\text{inner}}$ by changing the orientation of the circular pockets.

Figure 4: Plot of the disorder function for (a) $f(\bm{r})$, (b)
$f_{\text{inner}}(\bm{r})$, and (c) $f_{\text{outer}}(\bm{r})$. The value of
$f(\bm{r})$ is indicated by the color bar in the bottom right corner. The
bottom left corner insets show the loci of the Fourier components in $\bm{q}$
space. The corresponding distributions of the scattered (black dots) and
unscattered (red dots) $\bm{k}$ plane-wave states are shown in (d) to (f)
while the $|t(\bm{k})|^{2}$ spectra for $L=20W$ are shown in (g) to (i) with
the numerical values represented by the color bar. The white arrow in (g)
points to the window of transparency for $f(\bm{r})$.

For greater clarity, we plot in Fig. 5 the coherent transmission
$|t(\bm{k})|^{2}$ spectrum from Fig. 4(g) for incoming plane-wave states at
normal incidence ($k_{y}=0$). We observe two transmission bands – one at low
frequencies ($\kappa_{0}a<0.05$) and the other in the ‘window’
($0.25<\kappa_{0}a<0.5$). The two transmission bands are separated by a ‘mid
gap’ ($0.05\leq\kappa_{0}a\leq 0.25$) that originates from scattering by the
Fourier components associated with $f_{\text{inner}}$ while the ‘window’ is
bounded from above by a ‘top gap’ ($\kappa_{0}a\geq 0.5$) originating from
scattering by the Fourier components associated with $f_{\text{outer}}$.

Figure 5: Coherent transmission $|t(\bm{k})|^{2}$ spectrum (yellow shaded area
under the curve) for the disorder function $f(\bm{r})$ as a function of
frequency $\kappa_{0}a$ at normal incidence ($k_{y}=0$) with the label
‘Transmit’. The spectrum is derived from Fig. 4(g). The $|t(\bm{k})|^{2}$
spectra for $f_{\text{inner}}$ and $f_{\text{outer}}$ are labeled ‘Inner’ (red
solid line) and ‘Outer’ (dashed blue line), respectively, with the
corresponding transmission gaps labeled ‘Mid gap’ (red shaded rectangle) and
‘Top gap’ (blue shaded rectangle). The transmission window in the
$0.25<\kappa_{0}a<0.5$ range is indicated by the double-headed arrow
(‘Window’).

###  III.3 Possible experimental realization

This phenomenon can be simulated for TM waves in a 2D photonic crystal (PC)
with tunable site disorder. A prototype would be a 2D square lattice of
cylindrical dielectric rods (Istrate and Sargent, 2006; Busch  _et al._ ,
2007) with a radius of $R$ and nearest-neighbor distance of $h$. To realize
the _spatial structure_ of the disorder described by $f(\bm{r})$, we can let
the cylinder radius be site dependent, i.e., $R(\bm{r})=R_{0}[1+f(\bm{r})]$,
where $R_{0}$ is the average radius of the rods in the photonic crystal, so
that the perturbation is proportional to $f(\bm{r})$. The coherent
transmission spectrum through a disordered PC sandwiched between two disorder-
free PC leads as in Fig. 2 should be similar to those in Fig. 3.
Alternatively, the coherent transmission spectrum should be observed in a
disordered PC with a position-dependent refractive index. (Schwartz  _et al._
, 2007)

Another possible approach for experimental realization would be to allow the
amplitude of the nonzero Fourier components of $f(\bm{r})$ to vary while we
fix the loci of the zero-amplitude Fourier components in $\bm{q}$ space. For
instance, in the example in Section III.2.2, the zero-amplitude Fourier
components would be confined to the region surrounding the two circular
pockets and bounded by the ring as shown in the inset of Fig. 4(a). This
provides us with more flexibility in the design of disordered media with
selective transparency and such disorder configurations can be realized
through collective-coordinate optimization. (Uche  _et al._ , 2004; Torquato,
2018)

##  IV Summary

We have elucidated the role of disorder, in terms of its Fourier components,
in wave scattering in a 2D disordered medium. We show how the disorder
configuration, as determined by $\mathcal{V}(\bm{q})$, can be engineered for
isotropic and selective wave transparency. Using numerical simulations, we
demonstrate an approach where, by choosing the appropriate combination of
Fourier components, wave scattering can be selectively suppressed and the
transport properties of the material can be engineered to create transmission
gaps and allow specific incident waves to be robustly and coherently
transmitted in orientation-dependent frequency windows. This Fourier
components-based approach, which requires neither nontrivial topological wave
properties nor a non-Hermitian medium, can be generalized to tailor the
transport properties of disordered media for fundamental investigations of
disordered systems and applications in acoustics, imaging, photonics, and
structural health monitoring.

###### Acknowledgements.

I thank Donghwan Kim and Eric Heller of Harvard University for clarifying some
concepts in their paper. (Kim and Heller, 2022) I thank Salvatore Torquato of
Princeton University for drawing my attention to the phenomenon of wave
transparency in stealthy hyperuniform systems. I acknowledge support for this
work by A*STAR, Singapore with funding from the Polymer Matrix Composites
Program (SERC Grant No. A19C9a004).

##  Appendix A Coherent transmission spectra for other combinations of
$f_{\text{inner}}$ and $f_{\text{outer}}$

We plot in Figs. 6 and 7 the simulation data for different $f_{\text{inner}}$.
The data in Fig. 6 are for an $f_{\text{inner}}$ that is like the one in Fig.
4 but rotated by 45 degrees with respect to the $x$ axis. The coherent
transmission spectra for $f$ and $f_{\text{inner}}$ are also rotated by 45
degrees with respect to the $x$ axis. Hence, we do not observe the window of
transparency unless the angle of incidence is close to 45 degrees. Similarly,
the data in Fig. 7 are for an $f_{\text{inner}}$ that is rotated by 90
degrees. Hence, the window of transparency can only be observed at very
oblique angles of incidence close to 0 degree.

Figure 6: Plot of the disorder function for (a) $f(\bm{r})$, (b)
$f_{\text{inner}}(\bm{r})$, and (c) $f_{\text{outer}}(\bm{r})$ with
$f_{\text{inner}}$ rotated by 45 degrees. The value of $f(\bm{r})$ is
indicated by the color bar in the bottom right corner. The bottom left corner
insets show the loci of the disorder modes in $\bm{q}$ space. The
corresponding distributions of the scattered (black dots) and unscattered (red
dots) $\bm{k}$ plane-wave states are shown in (d) to (f) while the
$|t(\bm{k})|^{2}$ spectra for $L=20W$ are shown in (g) to (i) with the
numerical values represented by the color bar. The white arrow in (g) points
to the window of transparency for $f(\bm{r})$. Figure 7: Plot of the disorder
function for (a) $f(\bm{r})$, (b) $f_{\text{inner}}(\bm{r})$, and (c)
$f_{\text{outer}}(\bm{r})$ with $f_{\text{inner}}$ rotated by 90 degrees. The
value of $f(\bm{r})$ is indicated by the color bar in the bottom right corner.
The bottom left corner insets show the loci of the disorder modes in $\bm{q}$
space. The corresponding distributions of the scattered (black dots) and
unscattered (red dots) $\bm{k}$ plane-wave states are shown in (d) to (f)
while the $|t(\bm{k})|^{2}$ spectra for $L=20W$ are shown in (g) to (i) with
the numerical values represented by the color bar. The white arrows in (g)
point to the window of transparency for $f(\bm{r})$.

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# Investigating the image lag of a scientific CMOS sensor in X-ray detection

Qinyu Wu  Zhixing Ling<EMAIL_ADDRESS>Chen Zhang  Quan Zhou  Xinyang
Wang  Weimin Yuan  Shuang-Nan Zhang

###### Abstract

In recent years, scientific CMOS (sCMOS) sensors have been vigorously
developed and have outperformed CCDs in several aspects: higher readout frame
rate, higher radiation tolerance, and higher working temperature. For silicon
image sensors, image lag will occur when the charges of an event are not fully
transferred inside pixels. It can degrade the image quality for optical
imaging, and deteriorate the energy resolution for X-ray spectroscopy. In this
work, the image lag of a sCMOS sensor is studied. To measure the image lag
under low-light illumination, we constructed a new method to extract the image
lag from X-ray photons. The image lag of a customized X-ray sCMOS sensor
GSENSE1516BSI is measured, and its influence on X-ray performance is
evaluated. The result shows that the image lag of this sensor exists only in
the immediately subsequent frame and is always less than $0.05\%$ for
different incident photon energies and under different experimental
conditions. The residual charge is smaller than $0.5\ \rm{e^{-}}$ with the
highest incident photon charge around $8\ \rm{ke^{-}}$. Compared to the
readout noise level around $3\ \rm{e^{-}}$, the image lag of this sensor is
too small to have a significant impact on the imaging quality and the energy
resolution. The image lag shows a positive correlation with the incident
photon energy and a negative correlation with the temperature. However, it has
no dependence on the gain setting and the integration time. These relations
can be explained qualitatively by the non-ideal potential structure inside the
pixels. This method can also be applied to the study of image lag for other
kinds of imaging sensors.

###### keywords:

CMOS detector , Image lag , X-ray detector

††journal: NIM A

[inst1]organization=National Astronomical Observatories, Chinese Academy of
Sciences,addressline=20A Datun Road, Chaoyang District, city=Beijing,
postcode=100101, country=China [inst2]organization=School of Astronomy and
Space Science, University of Chinese Academy of Sciences,addressline=19A
Yuquan Road, Shĳingshan District, city=Beijing, postcode=100049, country=China
[inst3]organization=Gpixel Inc.,addressline=No. 588 Yingkou Road, Economical
and Technological Development Zone, city=Changchun, postcode=130033,
country=China [inst4]organization=Institute of High Energy Physics, Chinese
Academy of Sciences,addressline=19B Yuquan Road, Shĳingshan District,
city=Beijing, postcode=100049, country=China

##  1 Introduction

Silicon image sensors, including charge-coupled devices (CCDs) and
Complementary Metal Oxide Semiconductor (CMOS) sensors, have been widely used
for optical and X-ray applications in the fields of medical imaging,
industrial radiography, nuclear physics and astronomical research. For
examples, since the 1990s, CCDs have been used as soft X-ray imaging and
spectroscopy detectors in many synchrotron light sources [1, 2, 3], and in
space X-ray astronomical missions, such as ASCA [4], Chandra [5] and XMM-
Newton [6]. In recent years, scientific CMOS (sCMOS) sensors have progressed
rapidly and have outperformed CCDs in several aspects: high readout frame
rate, high radiation tolerance, and relaxed working temperature requirement,
making the sCMOS sensor a favorable choice in both ground experiments and
space missions. In August 2022, the performance of a customized 6 cm $\times$
6 cm large-format X-ray sCMOS sensor has been successfully verified in orbit
by the Lobster Eye Imager for Astronomy (LEIA) onboard the SATech-01 satellite
[7, 8]. In the future, missions such as Einstein Probe (EP) [9, 10] and
THESEUS [11] will use sCMOS sensors as focal plane detectors.

As a new type of imaging sensors, the performance of sCMOS sensors has been
studied less extensively than that of mature CCDs. Since 2015, We have tested
several on-shelf sCMOS sensors and evaluated their X-ray performance and
capabilities of X-ray imaging and spectroscopy [12, 13, 14]. Crosstalk was
also found in a standard GSENSE400BSI sensor [15]. In this work, we will focus
on the image lag of the sCMOS sensor. CCD and CMOS sensors can detect the
input signal (including optical, UV, and X-ray photons and charged particles)
by converting it to photon-electrons, and collect them into the nearby pixels.
If the photon-generated electrons are not completely transferred and collected
in one frame, image lag will occur as the residual electrons are transferred
in the subsequent frames. Image lag is important for silicon image sensors and
is closely related to the pixel’s physical structure. For optical
applications, severe image lag will leave a ghost image after an exposure to a
bright target. For X-ray applications, image lag can deteriorate the energy
resolution. Many works have been done to reduce image lag by optimizing the
pixel design [16, 17, 18, 19, 20, 21]. The image lag of a silicon image sensor
can be measured with an optical flash of a LED [18, 19, 20], and the lag is
typically less than 1$\%$ for modern CCD and CMOS sensors. However, for some
specialized sensors, such as aluminum-coated sCMOS sensors designed for X-ray
applications [13], the optical method cannot be applied and new methods have
to be adopted. X-ray photons can produce single-pixel events on a sCMOS sensor
and make it possible to assess the image lag performance on the pixel level.

Cooperating with Gpixel Inc., our lab has customized a 6 cm $\times$ 6 cm
large-format sCMOS sensor, which is available in 2019. This sCMOS sensor,
named GSENSE1516BSI [22], is optimized for soft X-ray detection. It has an
array of $4096\times 4096$ pixels with a pixel size of $\rm{15\ \mu m}$. The
epitaxial layer is $\rm{10\ \mu m}$ thick, and it is fully depleted. The dark
current of this sensor is lower than 0.02 $\rm{e^{-}}$/pixel/s and the readout
noise is lower than 5 $\rm{e^{-}}$ at $\rm{-30^{\circ}\\!C}$. The nominal
energy resolution is about 180 eV at 5.9 keV. We have demonstrated [23] that,
by making pixel-level gain correction, its energy resolution can be improved
to 124.6 eV at 4.5 keV and 140.7 eV at 6.4 keV at room temperature. A level of
extra noise of $\sim 7\ \rm{e^{-}}$ is needed to explain the difference from
the theoretical limit, and image lag is a candidate source of the noise. To
investigate the image lag of this X-ray sCMOS sensor and assess its effect on
the energy resolution, we developed a new method using X-ray measurement and
carried out a series of experiments.

The experimental setup is introduced in Section 2; the measurement method is
described in Section 3; the results about the image lag are shown and
discussed in Section 4; and the conclusions are summarized in Section 5.

##  2 Experimental Setup

The experimental device for image lag measurement is shown in Fig. 1. An
Oxford Jupiter 5000 X-ray tube with a Ti anode is placed at the top left side
of the setup. At the center, a Ti plate target is installed to produce
secondary X-rays and to reduce the initial X-ray flux. When the X-ray tube is
powered on, an X-ray continuum spectrum with emission lines from the plate is
produced. These X-rays are then recorded by a camera at the bottom. The camera
is made of a GSENSE1516BSI sCMOS sensor, readout electronics, and temperature
control structures [24]. The camera has two working modes: the image mode, in
which raw images are read out and saved into a disk directly; and the event
mode, in which X-ray events are extracted from raw images in real time and
only such events are saved into the disk. The event mode can greatly reduce
the bandwidth needed for high-speed exposures, especially at the highest frame
rate of 20 Hz.

In this study, the X-ray tube is operated under a current of 0.12 mA and a
high-voltage of 30kV. Therefore, X-rays close to 30 keV can be seen in the
final spectrum, as shown in Fig. 2. However, at this high energy, the quantum
efficiency of the sensor is pretty low [25, 26], and only a small number of
the incident photons can be captured by the sensor. Characteristic lines of
not only Ti, but also Cr, Fe and Ni can be found in the spectrum, which come
from the stainless steel of the surrounding structures of the device. The
countrate is controlled to be about 20000 counts per frame to make the pixel
occupancy of each frame less than 0.2%. In a vacuum of around $10^{-3}$ Pa, a
series of experiments were carried out with different gain settings and
different integration times, and at different temperatures, ranging from
$\rm{-30^{\circ}\\!C}$ to $\rm{20^{\circ}\\!C}$

Under each of the experiment conditions, dark exposures without X-ray
illumination are first taken to obtain the raw background images. The
experiments are then conducted with the X-ray tube powered on. For the
following data reduction, the continuous raw images captured by the sCMOS
sensor have to be stored, so we make the camera run in the image mode.

---
Figure 1:  The experimental setup. The center Ti plate target in this setup,
illuminated by the X-ray tube at left side, can reduce the initial X-ray flux
and produce secondary X-ray photons. These photons are then recorded by the
sCMOS sensor in the camera at the bottom.
---
Figure 2:  A typical spectrum obtained by the sCMOS sensor. X-rays close to 30
keV can be seen. Characteristic lines of not only Ti, but also Cr, Fe and Ni
can be found in the spectrum, which come from the stainless steel of the
surrounding structures of the device.

##  3 Measurement Method

Under each experimental condition, an averaged background image is calculated
from dark exposures. Then the X-ray exposure data is processed frame by frame.
For each frame (frame 0), we examine the preceding frame (frame m1) and the
two subsequent frames (frame 1 and frame 2), as shown in Fig. 3. After
subtracting the background image from the X-ray image, we extract all pixels
above a threshold of around 335 eV or 92 $\rm{e^{-}}$ in frame 0 to eliminate
false events caused by noise and keep those real events from incident X-ray
photons.

To obtain an unbiased image lag, we record the corresponding pixel values in
frame m1, frame 1 and frame 2. However, these pixel values may be affected by
another incident X-ray photon and cannot represent the real residual signal.
Therefore, an additional selection process is carried out on each over-
threshold pixel of frame 0: In frame m1, frame 1 and frame 2, if the values of
the pixel and its eight adjacent pixels are all below the threshold around 335
eV, which means that this pixel is not affected by photons in these three
frames, then the value difference of frame 1 and frame m1 is a reliable
measurement of the residual signal or the image lag in frame 1, and similarly
the difference between frame 2 and frame m1 is a reliable measurement for
frame 2. Finally, around 6% of these over-threshold pixels are discarded,
corresponding well to the occupancy around 0.2%. Fig. 3 gives two examples of
this selection process. For event 1 in the upper-left part of each panel,
there is no other incident photon in frame m1, frame 1 and frame 2. Therefore,
this event will pass the selection process. But for event 2 in the bottom part
of each panel, another incident photon is detected on an adjacent pixel in
frame 1, and therefore event 2 cannot be used for the image lag calculation.

For each selected pixel on each frame, we can get a single measurement value
of the image lag. We perform this selection and measurement for all over-
threshold pixels on all frames, so a large number of measurement results of
the image lag are obtained. The overall distribution of these image lags for
frame 1 and frame 2, respectively, are shown in Fig. 4. A Gaussian function is
used to fit the histograms, and the mean value $\mu$ of the Gaussian fit gives
the image lag level of the sensor. The direct average of the histogram gives a
similar result.

---
Figure 3:  Schematic diagram of the frames. Frame m1, 0, 1 and 2 are four
frames in a continuous exposure. The time between two frames is the
integration time. As examples, two over-threshold pixels are given in frame 0.
For event 1 in the upper-left part of each panel, there is no other incident
photon in frame m1, frame 1 and frame 2. Therefore, this event will pass the
selection process. But for event 2 in the bottom part of each panel, another
incident photon is detected on an adjacent pixel in frame 1, and therefore
event 2 cannot be used for the image lag calculation.
---
Figure 4:  The distribution of the measurement results of the image lag for
frame 1 and frame 2, respectively. The Gaussian fitting of the distributions
are also shown. Errors are given with a confidence level of 1 sigma. This
figure is for incident photons around 12 keV. The temperature is
$\rm{-30^{\circ}\\!C}$ and the gain register is 7.5. For frame 2, the second
subsequent frame of frame 0, the mean value $\mu$ is around 0, indicating that
there is no image lag in frame 2 under this experimental condition. For frame
1, the immediately subsequent frame of frame 0, $\mu$ gives a image lag of
around 0.89 eV, or 0.24 $\rm{e^{-}}$. The direct average of the histogram
gives a value of 0.26 $\rm{e^{-}}$, which is close to the Gaussian fitting
result.

##  4 Results and Discussion

Our results show that, under all experimental conditions adopted, the image
lag of frame 2, the second subsequent frame of a given frame, is statistically
consistent with zero. Fig. 4 gives such an example under a specific
experimental condition. We conclude that there is no measurable image lag in
the second subsequent frame.

However, for the immediately subsequent frame (frame 1), the image lag is
present clearly. We divide the data into several groups by the energy of the
incident photon to study the dependence of the image lag on the energy of
incident photons. Fig. 5 shows the image lags measured versus the energies of
incident photons. A positive correlation is clearly present,indicating that
the image lag is positively correlated to the number of electrons in the
original frame. However, as the photon energy increases, the percentage of the
residual charge among the total deposited charge decreases.

The effect of the gain setting on the image lag is also examined. The
conversion gain of this CMOS sensor can be tuned from about 6.5 eV/DN to
around 45 eV/DN, corresponding to a gain register value of 7.5 to 1. Fig. 5
also shows the energy-dependent image lags under different gain register
values. There is no obvious dependence of the image lag on the gain setting,
indicating that the image lag of this sensor does not come from the readout
circuits and the digital electronics.

---
Figure 5:  The image lag versus the incident photon energy, under different
gain settings. The conversion gain of this sCMOS sensor can be tuned from
about 6.5 eV/DN to around 45 eV/DN, corresponding to a gain register value of
7.5 to 1. The image lag increases as the incident energy increase, and it
shows no dependence on the gain setting. The integration time is 50 ms and the
temperature is kept at $\rm{-30^{\circ}\\!C}$.

The image lags are also measured at different temperatures. As shown in Fig.
6, the image lag decreases significantly as the temperature increases. The
positive correlation trend between the lag and the energy is kept the same at
these temperatures. Above $\rm{10^{\circ}\\!C}$, the image lag becomes small
enough and the statistical uncertainty starts to dominate the measurement;
therefore, the variation among the data points becomes obvious in a log scale
plot.

---
Figure 6:  The image lag versus the incident photon energy, under different
temperatures. The image lag decreases as the temperature increases. Above
$\rm{10^{\circ}\\!C}$, the image lag becomes small enough and the statistical
uncertainty starts to dominate the measurement; therefore, the variation among
the data points becomes obvious in a log scale plot. The gain register is set
to 7.5 and the integration time is 50 ms.

The image lags under different integration times, ranging from 50 ms to 5 s,
are shown in Fig. 7. Despite a change of the integration time by a factor of
ten, the image lag remains the same. This shows that the integration time of
the sensor has no measurable effect on the image lag and proves that the image
lag is not related to the charge collection process.

---
Figure 7:  The image lag versus the incident photon energy, under different
integration times. The image lag shows no dependence on the integration time.
The gain register is set to 7.5 and the temperature is kept at
$\rm{-30^{\circ}\\!C}$.

To summarize, the image lag is correlated positively with incident photon
energy and inversely with the temperature, but is independent of the gain
setting and integration time of the experiment. There have been extensive
discussions on the causes of image lag. They are usually thought to be related
to the pixel design of a sensor [16, 17, 18, 20]. Fig. 8 shows the typical
pixel design and its possible electric potential distribution of a
4-Transistor CMOS sensor. The pinned photodiode (PPD) is the place where
electrons are collected. After the collection process, the transfer gate (TG)
will be turned on and the electrons will flow down to the floating diffusion
(FD) node. Then the charge in FD will be converted to voltage signals and read
out. If the potential has a “barrier” or a “pocket” at the edge of the PPD and
the opened TG, as shown in Fig. 8, electrons can be blocked by the barrier or
trapped by the pocket, and cannot reach the FD during one charge transfer
process. In this case, the image lag is present. As the temperature increases,
the thermal energies of the electrons increase and fewer electrons will be
trapped by the small well. Therefore, the image lag is smaller at higher
temperatures. The gain settings and the charge integration process have
nothing to do with the charge transfer process, and therefore, have no effect
on the image lag. As the photon energy increases, more electrons will be
produced, and it is reasonable that more electrons will be blocked or trapped.
However, as the photon energy increases, the percentage of the trapped charge
among the total deposited charge decreases because the effective barrier or
pocket is weakened by the already trapped charges. These are consistent with
our experimental results.

---
Figure 8:  A typical pixel design of a 4T CMOS sensor (top) and its possible
electric potential distributions during the charge transfer process. The PPD
is the place where electrons are collected. After the collection process, the
TG will be turned on and electrons will flow down to the FD. Then the charge
in FD will be converted to voltage signals and read out. If the potential has
a “barrier” or a “pocket”, some electrons may be blocked or trapped, and image
lag will present.

The image lags of GSENSE1516BSI are always less than $0.05\%$ for different
incident photon energies and under experimental conditions. The residual
charge is smaller than $0.5\ \rm{e^{-}}$ with the highest incident energy
around 30 keV or $8\ \rm{ke^{-}}$. This value is much smaller than the readout
noise around $3\ \rm{e^{-}}$[23], indicating that the image lag of this sensor
is negligible and has no noticeable effect on the energy resolution.

In this work, the image lag of this sCMOS sensor is measured with X-rays. We
remark that this method can be applied to image lag measurements using other
means of radiation such as electrons and protons. We measured the image lag of
the GSENSE1516BSI sensor with cosmic rays in a normal environment, and the
results are similar to the results of X-rays, demonstrating the applicability
of our method.

##  5 Conclusions

Scientific CMOS sensors have been widely used in ground X-ray experiments and
space X-ray missions due to their excellent performance. As an important
factor, the image lag resulting from incomplete charge transfer in the pixels
can deteriorate the image quality and the energy resolution of sCMOS sensors.
In this work, we propose a new method to extract the image lag from X-ray
photons. This method is applied to an extensive study of the image lag for a
customized X-ray sCMOS sensor GSENSE1516BSI, for a range of incident energies
and under various experimental conditions. It is found that the image lag of
this sensor is always less than $0.05\%$, and the residual charge is smaller
than $0.5\ \rm{e^{-}}$ with the highest incident energy around $8\
\rm{ke^{-}}$. This low value indicates that the image lag of this sensor has
negligible effects on the imaging and energy resolution. The image lag
increases slightly with the increase of the incident energy and with the
decrease of the temperature. However, it shows no correlation with the gain
setting and the integration time of the sensor. These can be explained
qualitatively by the non-ideal potential scheme inside the pixel, lending
credence to the pixel-level origin of the image lag. In principle, this method
can be applied to not only X-ray radiation but other charged particles,
showing the generality of our method.

Funding: This work is supported by the National Natural Science Foundation of
China (grant No. 12173055) and the Chinese Academy of Sciences (grant Nos.
XDA15310000, XDA15310100, XDA15310300, XDA15052100).

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