# Polar magneto-optic Kerr and Faraday effects in finite periodic
$\mathcal{P}\mathcal{T}$-symmetric systems

Antonio Perez-Garrido<EMAIL_ADDRESS>Departamento de Física Aplicada,
Universidad Politécnica de Cartagena, E-30202 Murcia, Spain  Peng Guo
<EMAIL_ADDRESS>Department of Physics and Engineering, California State
University, Bakersfield, CA 93311, USA  Vladimir Gasparian
<EMAIL_ADDRESS>Department of Physics and Engineering, California State
University, Bakersfield, CA 93311, USA  Esther Jódar
<EMAIL_ADDRESS>Departamento de Física Aplicada, Universidad
Politécnica de Cartagena, E-30202 Murcia, Spain

###### Abstract

We discuss the anomalous behavior of the Faraday (transmission) and polar Kerr
(reflection) rotation angles of the propagating light, in finite periodic
parity-time ($\mathcal{P}\mathcal{T}$) symmetric structures, consisting of $N$
cells. The unit cell potential is two complex $\delta$-potentials placed on
both boundaries of the ordinary dielectric slab. It is shown that, for a given
set of parameters describing the system, a phase transition-like anomalous
behavior of Faraday and Kerr rotation angles in a parity-time symmetric
systems can take place. In the anomalous phase the value of one of the Faraday
and Kerr rotation angles can become negative, and both angles suffer from
spectral singularities and give a strong enhancement near the singularities.
We also shown that the real part of the complex angle of KR, $\theta^{R}_{1}$,
is always equal to the $\theta^{T}_{1}$ of FR, no matter what phase the system
is in due to the symmetry constraints. The imaginary part of KR angles
$\theta^{R^{r/l}}_{2}$ are related to the $\theta^{T}_{2}$ of FR by parity-
time symmetry. Calculations based on the approach of the generalized
nonperturbative characteristic determinant, which is valid for a layered
system with randomly distributed delta potentials, show that the Faraday and
Kerr rotation spectrum in such structures has several resonant peaks. Some of
them coincide with transmission peaks, providing simultaneous large Faraday
and Kerr rotations enhanced by an order one or two of magnitude. We provide a
recipe for funding a one-to-one relation in between KR and FR.

##  I Introduction

The study of the magneto-optic effects (Faraday rotation (FR) and Kerr
rotation (KR)), has played an important role in the development both of
electromagnetic theory and atomic physics. The magneto-optical materials that
exhibiting FR and KR are essential for optical communication technology
Yoshino (2005); Zamani et al. (2011); Birch (1982), optical amplifiers Corzo
et al. (2012); Stubkjaer (2000), and photonic crystals Wang et al. (2008,
2009). In addition to this important application, the KR is also an extremely
accurate and versatile research tool and can be used to determine quantities
as varied as anisotropy constants, exchange-coupling strengths and Curie
temperatures (see, e.g., McGee et al. (1993).)

In polar or magneto-optical Kerr effect, the magnetization of the system is in
the plane of incidence and perpendicular to the reflecting surface. Reflection
can produce several effects, including 1) rotation the direction of light
polarization, 2) introducing ellipticity into the reflected beam, and 3)
changing by the intensity of the reflected beam.

FR is similar to KR in terms of rotation and ellipticity and has a wide range
of applications in various fields of modern physics, such as measuring
magnetic field in astronomy Longair (2011) and construction of optical
isolators for fiber-optic telecommunication systems Furdyna (1988), as well as
the design of optical circulators used in the development of microwave
integrated circuits. Berger (2003); Turner and Stolen (1981); Firby et al.
(2018).

Note, that large Faraday and Kerr rotations are needed for all the
applications mentioned. However, the standard method, based on increasing the
sample size or applying a strong external magnetic field, is currently
ineffective due to the small size of systems in which the de Broglie
wavelength is compatible with the size of quantum devices. In other words,
thin film materials exhibiting a large FR angle should be desirable for
promote progress optical integrated circuits.

A large enhancement of the FR and as well as a change in the sign of the FR
can be obtained by incorporating several nanoparticles and their composites in
nanomaterials, see e.g. Refs. Uchida et al. (2011); Andrei and Mayergoyz
(2003); Gevorkian and Gasparian (2014). A phase transition-like anomalous
behavior of Faraday rotation angles in a simple parity-time
$\mathcal{P}\mathcal{T}$-symmetric model of a regular dielectric slab was
reported recently in Ref.Gasparian et al. (2022a). In anomalous phase, the
value of one of Faraday rotation angles turns negative, and both angles suffer
spectral singularities and yield strong enhancement near singularities.

As for the enhancing of the KR, in which we are interested too, it is mainly
related to spin-orbit coupling strength Oppeneer et al. (1992), to
interference effects Chen et al. (1990) and as well as by the plasma resonance
of the free carriers of magnetic materials Feil and Haas (1987). As it was
mentioned in Ref. Loughran et al. (2018), with addition of a gold nano-disc to
the periodic magnetic system, yields a strong wavelength-dependent enhancement
of the KR. Generally, the enhancement factor is expected to be less than three
even for materials with high refractive index $\approx 2$, such as
semiconductors with zero extinction coefficients in the near or mid infrared
range (like tellurium or aluminum gallium arsenide).

In this paper we aim to present a complete and quantitative theoretical
description of the Faraday and Kerr complex rotations for an arbitrary one
dimensional finite periodic $\mathcal{P}\mathcal{T}$-symmetric system,
consisting from (2N+1) cells for some simple cases we give simple closed form
expressions, describing the FR and KR.

We illustrate that the Faraday and Kerr rotation angles of the polarized light
traveling through a $\mathcal{P}\mathcal{T}$-symmetric periodic structure
display phase transition-like anomalous behaviors.

In one phase (normal phase), the FR and KR angles behave normally as in
regular passive system with a positive permittivity, and stay positive all the
time as expected. In the second anomalous phase, the angle of FR and KR angles
may change the sign and turn into negative. In addition, spectral
singularities arise in the second anomalous phase, where the angles FR and KR
increase strongly. In this sense, $\mathcal{P}\mathcal{T}$-systems seem to be
a good candidate for constructing fast tunable and switchable polarization
rotational ultrathin magneto-optical devices in a wide frequency range with a
giant FR and KR rotations. Despite that the obtained results are, in general,
only suitable for numerical analysis. However, in some simple cases
approximate expressions can be derived and a qualitative discussion is
possible.

The paper is organized as follows. In Sec. II the complex Faraday and Kerr
effects are introduced and discussed for $\mathcal{P}\mathcal{T}$-symmetric
unit cell with two complex $\delta$-potentials. We will assume that the
strengths of two Dirac $\delta$ functions $Z_{1}$ and $Z_{2}$ are arbitrary
complex numbers. The periodic system with $2N+1$ cells is is discussed in Sec.
III. Followed by the discussions and summary in Sec. IV.

##  II General theory of Faraday and Kerr effects in
$\mathcal{P}\mathcal{T}$-symmetric dielectric slabs

Figure 1: Schematic of a one-dimesnionl $\mathcal{P}\mathcal{T}$-symmetric
photonic heterostructure, consisting of $2N+1$ arbitrary number of slabs that
are $\mathcal{P}\mathcal{T}$ -symmetric about $x_{0}=0$, that is
$\epsilon(x)=\epsilon^{*}(-x)$. Each slab of the photonic heterostructure, has
two balanced complex tiny slabs placed at both ends of a real dielectric slab.
The green slab indicates the loss and the red slab indicates the gain region.

In this section, before discussing in detail the Faraday and Kerr effects in a
simple unit cell—an ordinary dielectric slab with two complex
$\delta$-potentials located at both boundaries (the unit cell located
symmetrically about $x_{0}=0$ in Fig. 1), we present some details of the
rotation angle calculation for an arbitrary one-dimensional dielectric
permittivity profile $\epsilon(x)$. Later, we will impose the condition
$\epsilon(x)=\epsilon^{*}(-x)$, which guarantees the system is
$\mathcal{P}\mathcal{T}$-symmetric, that its eigenstates are real-valued
solutions. In such a $\mathcal{P}\mathcal{T}$-symmetric dielectric system with
a finite spatial extension in $x$ direction (see Fig. 1), the permittivity of
the system (as well as the single slab) has a balanced gain and loss.

Assume a linearly polarized electromagnetic plane wave with angular frequency
$\omega$ enters the system from the left at normal incidence propagating along
the $x$ direction. The polarization direction of electric field of incident
wave is taken as the z-axis: $\bm{E}_{0}(x)=e^{ik_{0}x}\hat{z}$, where
$k_{0}=\frac{\omega}{c}\sqrt{\epsilon_{0}}$ stands for the wave vector and
$\epsilon_{0}$ denotes the dielectric constant of vacuum. A weak magnetic
field $\bm{B}$, which preserves the linearity of Maxwell’s equations, is
applied in the $x$-direction and is confined into the system, see Fig.1. The
scattering of incident wave by the system is described by Schrödinger-like
equations, see e.g. Refs. Lofy et al. (2020); Gasparian et al. (1995),

$\left[\frac{d^{2}}{dx^{2}}+\frac{\omega^{2}\epsilon_{\pm}(x)}{c^{2}}\right]E_{\pm}(x)=0,$
(1)

where $E_{\pm}=E_{y}\pm iE_{z}$ are circularly polarized electric fields. The
$\epsilon_{\pm}(x)$ is defined,

$\epsilon_{\pm}(x)=\begin{cases}\epsilon(x)\pm g,\ \ &\ \
x\in[-\frac{L}{2}-N(L+L_{0}),\frac{L}{2}+N(L+L_{0})],\\\ \epsilon_{0},\ \ &\ \
\mbox{otherwise},\end{cases}$ (2)

where $(L,L_{0},2N+1)$ stand for spatial extent of a unit cell, the spatial
separation of neighbouring two cells and number of cells, see Fig. 1. The $g$
is the gyrotropic vector along the magnetic-field direction. The external
magnetic field $\bm{B}$ is included into the gyrotropic vector $g$ to make the
calculations valid for the cases of both external magnetic fields and magneto-
optic materials.

When the reflection within the boundaries is important, the outgoing
transmitted/reflected wave is generally elliptically polarized even without
absorption, where the major axis of the ellipse is rotated with respect to the
original direction of polarization and the maximum FR (KR) angle does not
necessarily coincide with angular frequencies $\omega$ of light at which zero
ellipticity can be measured. The real part of the rotation angle describes the
change of polarization in linearly polarized light. The imaginary part
describes the ellipticity of transmitted or reflected light. Once we know the
scattering matrix elements $r_{\pm}(\omega)$ and $t_{\pm}(\omega)$ of the one-
dimensional light propagation problem, e.g. the reflection and transmission
amplitudes with an incoming propagating wave from left are defined by

$E_{\pm}(x)\rightarrow\begin{cases}\pm
i\left[e^{ik_{0}x}+r_{\pm}(\omega)e^{-ik_{0}[x+L+2N(L+L_{0})]}\right],&\quad
x\rightarrow-\infty,\\\ \pm it_{\pm}(\omega)e^{ik_{0}[x-L-2N(L+L_{0})]},&\quad
x\rightarrow+\infty.\end{cases}$ (3)

The two characteristic rotational parameters of transmitted light (magneto-
optical measurements of complex Faraday angle) can be written as a complex
form as (see, e.g., Refs. Gasparian et al. (1995); Lofy et al. (2020))

$\theta^{T}_{1}=\frac{\psi^{T}_{+}-\psi^{T}_{-}}{2},\ \ \ \
\theta^{T}_{2}=\frac{1}{4}\ln\frac{T_{+}}{T_{-}},$ (4)

where $T_{\pm}$ and $\psi^{T}_{\pm}$ are the transmission coefficients and
phase of transmission amplitudes, $t_{\pm}=\sqrt{T_{\pm}}e^{i\psi^{T}_{\pm}}$,
of transmitted electric fields. For weak magnetic field ($g\ll 1$), the
perturbation expansion in terms of weak magnetic field can be applied. The
leading order contribution can be obtained by expanding $\psi_{\pm}$ and
$T\pm$ around the refractive index of the slab in the absence of the magnetic
field B:

$\theta^{T}_{1}=\frac{g}{2n}\frac{\partial\psi^{T}}{\partial n},\ \ \ \
\theta^{T}_{2}=\frac{g}{4n}\frac{\partial\ln{T}}{\partial n},$ (5)

where $n=\sqrt{\epsilon}$ is the refractive index of the slab. The Kerr
rotation complex angles are defined in a similar way as in Eq.(4). In the weak
magnetic field, the leading order expressions can be written in the form

$\theta^{R}_{1}=\frac{g}{2n}\frac{\partial\psi^{R}}{\partial n},\ \ \ \
\theta^{R}_{2}=\frac{g}{4n}\frac{\partial\ln{R}}{\partial n},$ (6)

where $R$ and $\psi^{R}$ are the reflection coefficients and phase of
reflection amplitudes in the absence of magnetic field B:
$r(\omega)=\sqrt{R}e^{i\psi^{R}}$.

We remark that FR and KR angles are not all independent due to the constraints
of $\mathcal{P}\mathcal{T}$ symmetry. As mentioned in Ref. Guo and Gasparian
(2022a), the parametrization of scattering matrix only requires three
independent real functions in a $\mathcal{P}\mathcal{T}$-symmetric system: one
inelasticity, $\eta\in[1,\infty]$, and two phaseshifts, $\delta_{1,2}$. In
terms of $\eta$ and $\delta_{1,2}$, the reflection and transmission amplitudes
are given by

$t=t^{r}=t^{l}=\eta\frac{e^{2i\delta_{1}}+e^{2i\delta_{2}}}{2},\ \ \ \
r^{r/l}=\eta\frac{e^{2i\delta_{1}}-e^{2i\delta_{2}}}{2}\pm
i\sqrt{\eta^{2}-1}e^{i(\delta_{1}+\delta_{2})},$ (7)

where subscript $(r/l)$ are used to label amplitudes corresponding to two
independent boundary conditions: right ($e^{ik_{0}x}$) and left
($e^{-ik_{0}x}$) propagating incoming waves respectively. Therefore we find
relations:

$\sqrt{T}=\eta\cos(\delta_{1}-\delta_{2}),\ \ \ \
\psi^{T}=\delta_{1}+\delta_{2},\ \ \ \
\sqrt{R^{r/l}}=\left|\eta\sin(\delta_{1}-\delta_{2})\pm\sqrt{\eta^{2}-1}\right|,\
\ \ \ \psi^{R}=\psi^{T}+\frac{\pi}{2},$ (8)

and the pseudounitary conservation relations take place (see, e.g.
Refs.Mostafazadeh (2014); Ahmed (2011); Ge et al. (2012)):

$|T-1|=\sqrt{R^{l}R^{r}}.$ (9)

The FR and KR angles are given by

$\theta^{T}_{1}=\theta^{R}_{1}=\frac{g}{2n}\frac{\partial(\delta_{1}+\delta_{2})}{\partial
n},\ \ \ \ \theta^{T}_{2}=\frac{g}{2n}\frac{\partial}{\partial
n}\ln\left[\eta\cos(\delta_{1}-\delta_{2})\right],\ \ \ \
\theta^{R^{r/l}}_{2}=\frac{g}{2n}\frac{\partial}{\partial
n}\ln\left|\eta\sin(\delta_{1}-\delta_{2})\pm\sqrt{\eta^{2}-1}\right|.$ (10)

The $\theta^{R^{r/l}}_{2}$ and $\theta^{T}_{2}$ are hence related by

$\frac{\theta^{R^{r}}_{2}+\theta^{R^{l}}_{2}}{2}=\frac{T}{T-1}\theta^{T}_{2}.$
(11)

We thus conclude that only three FR and KR angles are independent due to the
symmetry constraints. The special case of zero inelasticity ($\eta=0$) thus
represents the results for real spatially symmetric dielectric systems with
$\epsilon(x)=\epsilon(-x)$ and $Im[\epsilon(x)]=0$, hence

$\theta^{T}_{2}\stackrel{{\scriptstyle Im[\epsilon(x)]\rightarrow
0}}{{\rightarrow}}\frac{g}{2n}\frac{\partial}{\partial
n}\ln\cos(\delta_{1}-\delta_{2}),\ \ \ \
\theta^{R^{r/l}}_{2}\stackrel{{\scriptstyle Im[\epsilon(x)]\rightarrow
0}}{{\rightarrow}}\frac{g}{2n}\frac{\partial}{\partial
n}\ln\left|\sin(\delta_{1}-\delta_{2})\right|.$ (12)

##  III Unit cell: two complex $\delta$-potentials placed on both boundaries
of the ordinary dielectric slab

We first present some main results of FR and KR for a unit cell in this
section, all the technical details can be found in Appendix A. The properties
of the spectral singularities are also discussed in current section, and we
draw attention to the parameter ranges where a phase-like transition can take
place for both Faraday and Kerr effects. A simple
$\mathcal{P}\mathcal{T}$-symmetric model for a unit cell is adopted in this
work: two complex $\delta$-potential are placed at both ends of the dielectric
slab,

$\epsilon(x)=\epsilon+Z_{1}\delta(x+\frac{L}{2})+Z_{2}\delta(x-\frac{L}{2}),\
\ \ \ Z_{1}=V_{1}+iV_{2},\ \ \ \ Z_{2}=Z_{1}^{*},$ (13)

where $L$ denotes the spatial extent of unit cell of dielectric slab and
$\epsilon>0$ is positive and real permittivity of slab. The transmission
$t_{0}(\omega)$ and reflection $r_{0}(\omega)$ amplitudes for the unit cell
can be obtained rather straightforwardly by matching boundary condition method
or using explicit form of characteristic determinant $D_{2}$ in Eq.(43).

First of all, inserting Eq.(40) in Eq.(42) and also using (38) it is easy to
see that $t_{0}(\omega)$, phase $\psi^{T}$ and transmission coefficient
$T_{0}$ for a unit cell are respectively given by

$t_{0}(\omega)=\sqrt{T_{0}}e^{i\psi^{T}}=\frac{\csc\left(\frac{\omega
n}{c}L\right)}{{\it\mathcal{R}}(\omega)-i{\it\mathcal{I}}(\omega)},\ \ \ \
\psi^{T}=\tan^{-1}\left[\frac{\mathcal{I}(\omega)}{\mathcal{R}(\omega)}\right],\
\ \ \ T_{0}=\frac{\csc^{2}\left(\frac{\omega
n}{c}L\right)}{{\it\mathcal{R}}^{2}(\omega)+{\it\mathcal{I}}^{2}(\omega)},$
(14)

where

${\it\mathcal{R}}(\omega)=\cot\left(\frac{\omega n}{c}L\right)-\frac{\omega
V_{1}}{cn},\ \ \ \ {\it\mathcal{I}}(\omega)=\frac{\omega
V_{1}}{cn_{0}}\cot{\left(\frac{\omega
n}{c}L\right)}+\frac{1}{2}(\frac{n}{n_{0}}+\frac{n_{0}}{n})-\frac{\omega^{2}}{2c^{2}n_{0}n}\left(V^{2}_{1}+V^{2}_{2}\right).$
(15)

The $n=\sqrt{\epsilon}$ and $n_{0}=\sqrt{\epsilon_{0}}$ denote the refractive
index of the dielectric slab and vacuum respectively. We remark that
unphysical units are adopted throughout the rest of presentation: the length
of slab $L$ is used to sent up the physical scale, $V_{1,2}$ and
$\epsilon=n^{2}$ hence carry the dimensions of $1/L$ and $1/L^{2}$
respectively. The $\omega/c$ is a dimensionless quantity.

Next the reflection amplitude $r^{r/l}_{0}(\omega)$ to the left/right of an
individual cell can be obtained conveniently from the following relation
related to the derivative of the transmission amplitude $t_{0}(Z_{1},Z_{2})$
with respect to $Z_{1}/Z_{2}$ located on the left/right border of the slab,
see Ref. Gasparian et al. (2005):

$r^{r}_{0}(\omega)=-{i}\frac{cn_{0}}{\omega}{\frac{\partial\ln
t_{0}(\omega)}{\partial Z_{1}}}-1,\ \ \ \
r^{l}_{0}(\omega)=-{i}\frac{cn_{0}}{\omega}{\frac{\partial\ln
t_{0}(\omega)}{\partial Z_{2}}}-1.$ (16)

Hence we find

$r^{r/l}_{0}(\omega)=\sqrt{R_{0}^{r/l}}e^{i\psi^{R}}=i\frac{Q^{r/l}(\omega)}{\mathcal{R}(\omega)-i\mathcal{I}(\omega)},\
\ \ \
\psi^{R}=\tan^{-1}\left[\frac{\mathcal{I}(\omega)}{\mathcal{R}(\omega)}\right]+\frac{\pi}{2},\
\ \ \
R^{r/l}_{0}=\frac{\left[Q^{r/l}(\omega)\right]^{2}}{{\it\mathcal{R}}^{2}(\omega)+{\it\mathcal{I}}^{2}(\omega)},$
(17)

where

$Q^{r/l}(\omega)=\frac{\omega V_{1}}{cn_{0}}\cot\left(\frac{\omega
n}{c}L\right)+\frac{1}{2}\bigg{(}\frac{n}{n_{0}}-\frac{n_{0}}{n}\bigg{)}\pm\frac{\omega
V_{2}}{cn}-\frac{\omega^{2}}{2c^{2}n_{0}n}\left(V^{2}_{1}+V^{2}_{2}\right).$
(18)

Note, that in case of $n_{0}$ = $n$ we recover the result of a reflection
amplitude from a simple diatomic system, discussed in Ref. Guo et al. (2022).
It is easy to verify that the phase of the reflection amplitude indeed
coincides with the phase of the transmission amplitude as previously
discussed. Later, in the next subsections we used these expressions to
illustrate a number of quite general features of Faraday and Kerr rotations in
$\mathcal{P}\mathcal{T}$-symmetric periodic systems.

A simple inspection of the Eq.(14), show that replacing $\omega$ with
$-\omega$ does not affect $t_{0}(\omega)$, which means that the transmission
is equal for the left-to-right and right-to-left scattering, that is
$t_{0}^{l}(\omega)=t^{r}_{0}({-\omega})\equiv t_{0}(\omega)$. The situation is
somewhat more complicated in the case of the reflection amplitude in Eq.(17).
Simultaneous sign change of both $\omega$ and $V_{2}$ is required to satisfy
the condition $r_{0}^{l}(-\omega,-V_{2})=r^{r}_{0}(\omega,V_{2})$. There in
fact are indeed the general properties of $\mathcal{P}\mathcal{T}$ systems,
see e.g. Eq.(B33) in Ref. Guo and Gasparian (2022a).

(a)

(b)

(c)

(d)

Figure 2: Faraday and Kerr rotation angles $\theta^{T}_{1,2}$ and
$\theta^{R^{r}}_{1,2}$ versus $\frac{\omega}{c}$ are shown for two values of
$\varphi_{V}=0.1\pi$ and $0.4\pi$ for $\mathcal{P}\mathcal{T}$-symmetric unite
cell: (a) $\theta^{T}_{1}=\theta^{R}_{1}$ for $\varphi_{V}=0.1\pi$; (b)
$\theta^{T}_{1}=\theta^{R}_{1}$ for $\varphi_{V}=0.4\pi$; (c)
$\theta^{R^{r}}_{2}$ (solid red) vs. $\theta^{T}_{2}$ (solid black) for
$\varphi_{V}=0.1\pi$; (d) $\theta^{R^{r}}_{2}$ (solid red) vs.
$\theta^{T}_{2}$ (solid black) for $\varphi_{V}=0.4\pi$. The rest of
parameters are taken as: $|V|=1$, $L=0.8$, $n=\sqrt{2}$ and $n_{0}=1$.

###  III.1 Spectral singularities in a unit cell

We now turn to a closer investigation of the spectral singularities for FR and
KR angles. Spectral singularities are spectral points belonging to non-
Hermitian Hamiltonian operators with $\mathcal{P}\mathcal{T}$-symmetry,
characterized by real energies. At these energies, the reflection and
transmission coefficients tend to infinity, i.e., they correspond to
resonances having zero width. Interesting to note that a slight imbalance
between gain and loss regions, can change the shape of the transition from
zero width to the symmetric shape of the ”bell curve” (for more details see
Ref. Gasparian et al. (2022a)).

For our model and for FR and KR rotational effects, spectral singularities
arise when both conditions, $\mathcal{R}(\omega)=0$ and
$\mathcal{I}(\omega)=0$, are satisfied simultaneously, see Eq.(15). By solving
Eq.(15) for $\cot(kL)$ and $\omega$ one obtains straightforwardly

$\bigg{(}\frac{\omega_{cr}|V|}{c}\bigg{)}^{2}\cos{2\varphi_{V}}+n^{2}+n^{2}_{0}=0,\
\ \ \ |V|=\sqrt{V_{1}^{2}+V_{2}^{2}},$ (19)

where
$\tan(\varphi_{V})=\frac{\mathcal{I}(\omega_{cr})}{\mathcal{R}(\omega_{cr})}$.
The condition necessary for the existence of a solution of the spectral
singularities exist only when the transmission phase is in the range
$\varphi_{V}\in[\pi/4,\pi/2]$. Hence the critical value of $\omega_{cr}$ is
defined as

$\omega_{cr}=\frac{c}{|V|}\frac{\sqrt{n^{2}+n^{2}_{0}}}{\sqrt{|\cos{2\varphi_{V}}|}},$
(20)

provided that

$\cot\bigg{(}\frac{\omega_{cr}}{c}nL\bigg{)}=\frac{\sqrt{n^{2}+n_{0}^{2}}}{n}\frac{\cos{\varphi_{V}}}{\sqrt{|\cos{2\varphi_{V}}|}}.$
(21)

For a fixed $|V|$, as follows from Eq.(20) the solutions of spectral
singularities can only be found in a finite range:
$\varphi_{V}\in[\frac{\pi}{4},\varphi_{c}]$, where $\varphi_{c}$ stands for
upper bound of range. Hence as $\varphi_{V}$ approaches lower bound of range
at $\frac{\pi}{4}$, the spectral singularity solution occurs at large
frequency: $\omega\rightarrow\infty$. When $\varphi_{V}$ is increased, the
solution of spectral singularity moves toward lower frequencies. As
$\varphi_{V}$ approaches the upper bound of range at $\varphi_{c}$, the
spectral singularity solution thus reaches its lowest value. The graphical
illustration of the distribution of spectral singularities can be found in
Fig.2 in Ref. Gasparian et al. (2022a).

###  III.2 Faraday and Kerr rotation: transmitted and reflected light

A phase transition-like anomalous behavior and properties of Faraday rotation
angles in a simple $\mathcal{P}\mathcal{T}$-symmetric model with two complex
$\delta$-potential placed at both boundaries of a regular dielectric slab was
most recently reported in Ref.Gasparian et al. (2022a). Let us recall the
essential features of the FR and then focus our attention on the KR effect. In
a $\mathcal{P}\mathcal{T}$-symmetric systems a phase transition-like anomalous
behavior of Faraday rotation angle take place. In this phase, one of Faraday
rotation angles turns negative, and both angles yield strong enhancement near
spectral singularities.

As the consequence of $\mathcal{P}\mathcal{T}$ symmetry constraint, the phase
of reflected amplitude $\psi^{R}$ from left coincides with the phase of the
transmission amplitude $\psi^{T}$, see Eq.(8). Hence the real part of the
complex angle of KR, $\theta^{R}_{1}$, is always equal to the $\theta^{T}_{1}$
of FR, no matter what phase the system is in. In this sense, the situation is
similar to the passive symmetric system, where is always
$\theta^{R}_{1}=\theta^{T}_{1}$. It is interesting to note that Eq.(15) is
invariant under the symmetry transform: $V_{2}\rightarrow-V_{2}$. This is a
manifestation of the fact that the phase of reflected amplitude $\psi^{R}$ and
as well as the Kerr rotation angle for the right incident light preserve the
same behaviour, although the strengths of the right and left
$\delta$-potentials on the boundaries are not equal to each other (more
precisely, they are complex conjugate to each other). The mentioned asymmetry
should lead to different left-to-right and right-to-left reflection amplitudes
(see, e.g., Garmon et al. (2015)) and does not affect physical quantities
$\theta^{R}_{1}$ and $\theta^{T}_{1}$, which are related to the phase
accumulated during the process of reflection and transmission and as well as
to the density of states. However, this asymmetry will affect $\theta^{R}_{2}$
and $\theta^{T}_{2}$, and they will no longer be equal to each other, see Fig.
2(c) and Fig. 2(d). This is consistent with the general statement that the
Faraday and Kerr rotation profiles are very different from the corresponding
curves describing ellipticities. In addition, symmetry constraint also yields
the wavelength dependence of Faraday and Kerr ellipticity $\theta^{T}_{2}$ and
$\theta^{R^{r/l}}_{2}$ shown in Eq.(11).

Here we would like to add a few more brief comments to emphasize that upon
closer look at Fig.2 reveals some details of the similarities between curves
that are relevant to our further discussion.

Firstly, the Faraday (Kerr) rotation local maximum/minimum (see Fig. 2)
coincide with the local peak on the ellipticity curves with some accuracy. The
ellipticity, at that frequencies, approaches zero non-linearly, becomes zero
(linearly polarized light), and then the resulting polarization reverses its
original direction.

Secondly, ellipticity (imaginary part the spectra) for $\theta^{T}_{2}$ and
$\theta^{R}_{2}$ depend little on frequency and are close to zero in almost
the entire frequency range, except for some regions associated with the
maximum/minimum or spectral singularities of the Faraday rotation and Kerr
rotation.

The questions discussed above can be straightforwardly generalized for the
periodic $\mathcal{P}\mathcal{T}$-symmetric system. This will be done in the
next section. We will show that the anomalous effect, similar to a phase
transition, occurred more often due to the complex structure of the
transmission and reflection amplitudes.

###  III.3  Limiting cases

The phase transition-like behavior of $\theta^{T}_{1}$ for two limiting cases
$(|V|\rightarrow\infty$ and $V_{1}\rightarrow 0$) was discussed in Ref.
Gasparian et al. (2022a). It was shown that in the case of
$|V|\rightarrow\infty$ the sign of $\theta^{T}_{1}$ is completely determined
by $\varphi_{V}$. As for the case $V_{1}=0$
($\varphi_{V}\rightarrow\frac{\pi}{2}$), then again for the given parameters
of the problem the anomalous negative behavior of $\theta^{T}_{1}$ is
illustrated analytically. The latter case, that is a
$\mathcal{P}\mathcal{T}$-symmetric optical lattice with a purely imaginary
scattering potential has been discussed in detail in a number of
investigations both theoretically and experimentally, see, e.g. Refs. Garmon
et al. (2015) and references therein.

####  III.3.1  $|V|\rightarrow\infty$

The situation is slightly different for Kerr rotation. In the same limiting
case $|V|\to\infty$, given that $\frac{n\omega L}{c}\neq\pi l$, we can show
that $\theta^{R}_{2}\propto\frac{1}{|V|^{2}}$, hence the ellipticity is almost
zero for all frequencies excluding $\frac{n\omega L}{c}=\pi l$ where
$l\in\mathbb{Z}$ and the reflected light remains linearly polarized. At the
discrete values of $\omega/c=\frac{\pi l}{nL}$ that yield the location of the
resonance poles, $\theta^{R}_{2}$ display sharp peak with narrow resonance
width. It reflects the fact that the reflected light is again linearly
polarized but rotated 90 degrees from the initial direction.

####  III.3.2  $V_{1}\rightarrow 0$

Bound state solutions of the Schrödinger equation for a
$\mathcal{P}\mathcal{T}$-symmetric potential with Dirac delta functions was
study in Ref.Uncu and Demiralp (2006). In Ref.Gasparian et al. (2022a) it was
pointed out that despite the fact that the expression for $\theta^{T}_{1}$ is
valid for the case $V_{1}\rightarrow 0$, it can still explain not only the
sign change of $\theta^{T}_{1}$ ($\theta^{R}_{1}$) in Fig.2 (a) where
$V_{1}\neq 0$, but also explain existence first local maximum. It is clear
that further features of the $\theta^{T}_{1}$ ($\theta^{R}_{1}$) in Fig. 3(a)
near the frequencies of spectral singularities, is related to the behavior of
$T(\omega)$.

Figure 3: Plot of Faraday $\theta^{T}_{2}$ (solid red) and Kerr
$\theta^{R^{r}}_{2}$ (solid black) rotation angles are shown for $|V|=1$,
$\varphi_{V}=\frac{\pi}{2}$, $L=0.8$, $n=\sqrt{2}$ and $n_{0}=1$ for
$\mathcal{P}\mathcal{T}$-symmetric unite cell, respectively. The resonant
frequencies at $\frac{\omega_{l}n}{c}L=\pi l,\ l=1,2,\cdots$ are plotted as
dashed pink vertical lines.

As for the imaginary portion of Kerr effect $\theta^{R^{r/l}}_{2}$, it is
straightforward to show that in the same limit of $V_{1}\rightarrow 0$ the
$\theta^{R^{r/l}}_{2}$ reads

$\theta^{R^{r/l}}_{2}\stackrel{{\scriptstyle V_{1}\rightarrow
0}}{{\rightarrow}}\frac{g}{2nQ(\omega)}\left[\frac{1}{2}\bigg{(}\frac{1}{n_{0}}+\frac{n_{0}}{n^{2}}\bigg{)}\mp\frac{\omega}{cn^{2}}V_{2}\bigg{(}1\mp\frac{\omega}{2cn_{0}}V_{2}\bigg{)}+\frac{R^{r/l}_{0}}{Q^{r/l}(\omega)}\bigg{(}\frac{\omega
L}{c}\frac{\cot(\frac{n\omega}{c}L)}{\sin^{2}(\frac{n\omega}{c}L)}-\frac{n^{2}-(n^{2}_{0}-\frac{\omega^{2}}{c^{2}}V^{2}_{2})^{2}}{4n^{2}_{0}n^{3}}\bigg{)}\right],$
(22)

where the reflection coefficient $R^{r/l}_{0}$ is given by Eq.(14) and
$Q^{r/l}(\omega)$ is defined by Eq.(18). The dependence of the imaginary part
of the Kerr rotation $\theta^{R^{r}}_{2}$ (solid black line) on
$\frac{\omega}{c}$ for $V_{1}\rightarrow 0$ is illustrated in Fig. 3. A number
of basic features of $\theta^{R}_{2}$ can be observed even in this simplest
case of $V_{1}\rightarrow 0$. One of the key features is the single resonant
peak that show up clearly when $R_{0}\rightarrow\infty$, see Eq.(22). As
mentioned above the resonance frequencies are spectral singularities when both
conditions, ${\it\mathcal{R}}(\omega)=0$ and ${\it\mathcal{I}}(\omega)=0$, are
satisfied simultaneously. In the particular case of $V_{1}\rightarrow 0$ there
is only one $\omega_{cr}$ that can be directly calculated from Eq.(20) by
putting $\varphi_{V}=\frac{\pi}{2}$:
$\omega_{cr}=\frac{c\sqrt{n^{2}+n^{2}_{0}}}{V_{2}}$. The second condition
$\cot\bigg{(}\frac{\omega_{cr}}{c}nL\bigg{)}=0$ can be satisfied by choosing
the appropriate value of length is $L=0.8$ (the system parameters are:
$n_{0}=c=V_{2}=1$, $n=\sqrt{2}$ and $\omega_{cr}=\sqrt{3}$). Other maxima or
minima in the Kerr rotation, located near the resonant frequencies, are
associated with multiple reflections from the boundaries and are located at
$\frac{\omega_{l}n}{c}L=\pi l,\ l=1,2,\cdots$ (see vertical pink lines in
Fig.3).

Repeating similar calculations leading to Eq.(22), we arrive at an explicit
expression for ellipticity $\theta^{T}_{2}$ for Faraday rotation for this
simplest case with a purely imaginary potential:

$\theta^{T}_{2}\stackrel{{\scriptstyle V_{1}\rightarrow
0}}{{\rightarrow}}\frac{g}{2n}{\cot(kL)}\frac{\omega}{c}L\left[T_{0}\left(1-\frac{{\sin^{2}(kL)}}{\cot(kL)}\frac{c}{\omega
L}\frac{n^{2}-(n^{2}_{0}-\frac{\omega^{2}}{c^{2}}V^{2}_{2})^{2}}{4n^{2}_{0}n^{3}}\right)-1\right].$
(23)

We observe that the smoothed maxima and minima that appeared around the zeros
of $\sin{kL}$ at $\frac{\omega_{l}n}{c}L=\pi l,\ l=1,2,\cdots$ coincides with
maxima and minima of $\theta^{R}_{2}$ and associated with multiple reflections
from the boundaries, see e.g. vertical yellow lines in Fig.3. Secondly, the
large value of $\theta^{T}_{2}$ at $3\pi/2$ is related to the frequency of the
spectral singularity $\omega_{cr}=\frac{c\sqrt{n^{2}+n^{2}_{0}}}{V_{2}}$,
where $T_{0}\rightarrow\infty$.

The physical background of the relatively simple mathematical structure of the
Faraday rotation angle $\theta^{T}_{1}$ on the frequency of comparison Kerr
ellipticity ($\theta^{R^{r/l}}_{2}$ is that in the first case the rotation
maximum is direct proportional to the optical anisotropy (for example, the
larger ( $n_{+}-n_{-}$ ), the larger is $\theta^{T}_{1}$. However, the
maximisation of $\theta^{R^{r/l}}_{2}$ is, not so straightforward, since
anisotropy indices are mixed (see, e.g, Ref. Němec et al. (2018) and and
references therein).

##  IV Periodic system with $2N+1$ cells

It is known that when the wave propagation through a medium is described by a
differential equation of second order, the expression for the total
transmission from the finite periodic system for any waves (sound and
electromagnetic) depends on the unit cell transmission, the Bloch phase and
the total number of cells. As an example of collective interference effect,
let us mention the intensity distribution from $N$ slits (diffraction due to
$N$-slits), as well as the formula that describes the Landauer’s resistance of
a one-dimensional chain of periodically spaced $N$ random scatterers. In both
cases, the similarity of the results is obvious. However, the physics behind
these results is completely different both in spirit and in details. In
analogy to Hermitian Hamiltonian, one can expect interference effect holds
also for a non-Hermitian Hamiltonian system. In this sense it is a natural
result for a $\mathcal{P}\mathcal{T}$-symmetric system that a somewhat similar
formula for transmission and reflection amplitudes appears, for example, in
Refs. Achilleos et al. (2017); Kalozoumis et al. (2018); Lin et al. (2011a);
Guo et al. (2022). The infinite periodic $\mathcal{P}\mathcal{T}$-symmetric
structures, because of unusual properties, including the band structure, Bloch
oscillations, unidirectional propagation and enhanced sensitivity, are of
special interest and are presently the subject of intensive ongoing research
(see e.g., Refs. Bender et al. (1999); Shin (2004); Musslimani et al. (2008);
Midya et al. (2010) and references therein). However, the case of scattering
in a finite periodic systems composed of an arbitrary number of cells/scatters
has been less investigated, despite that any open quantum systems generally
consist of a finite system coupled with an infinite environment.

In many studies, to describe quantitatively, both amplification and absorption
in periodic $\mathcal{P}\mathcal{T}$-symmetric systems, the transfer matrix
method is used. The latter, can be reduced to the evolution of the product of
transfer matrices of complex, but identical unit cells, and using the
classical Chebyshev identity get the final result.

In the following, we present the amplitudes of transmission and reflection
form the left and right sides of the incident wave based on the characteristic
determinant approach, the technical details are given in Appendix A. The
latter, in principle, is compatible with the transfer matrix method and is
convenient for both numerical and analytical calculations.

###  IV.1 Amplitudes of transmission and reflection form left and right

We now turn to a closer investigation of the Faraday and Kerr rotations for
various parameter ranges of our $\mathcal{P}\mathcal{T}$-periodic symmetric
system that consists of $2N+1$ cells, see Fig.1. Following Refs. Gasparian et
al. (1997); Guo et al. (2022) and also see Appendix A, a generic expressions
for the transmission and left/right reflection amplitudes for the
$\mathcal{P}\mathcal{T}$ can be presented as:

$\displaystyle t(\omega)$
$\displaystyle=\frac{e^{-ik_{0}L_{0}}}{\cos(\beta(2N+1)\Lambda)+iIm\left[\frac{e^{-ik_{0}L_{0}}}{t_{0}(\omega)}\right]\frac{\sin(\beta(2N+1)\Lambda)}{\sin(\beta\Lambda)}},$
(24)

where $k_{0}=n_{0}\frac{\omega}{c}$ and $k=n\frac{\omega}{c}$ are the wave
vectors in the respective medium. The quasi-momentum $\beta$ is the Bloch wave
vector of the infinite periodic system with unit cell length or spatial
periodicity $\Lambda=L_{0}+L$:

$\cos({\beta\Lambda})\equiv{\it
Re}\left[\frac{e^{-ik_{0}L_{0}}}{t_{0}(\omega)}\right]=\sin(kL)\left[\cos(k_{0}L_{0})\mathcal{R}(\omega)-\sin(k_{0}L_{0})\mathcal{I}(\omega)\right].$
(25)

please confirm two equations in blue. The left/right reflection amplitude can
be written in the form Guo et al. (2022); Gasparian et al. (1997)

$\displaystyle\frac{r^{(r/l)}(\omega)}{t(\omega)}$
$\displaystyle=\left[\frac{r^{(r/l)}_{0}(\omega)}{t_{0}(\omega)}\right]\frac{\sin(\beta(2N+1)\Lambda)}{\sin(\beta\Lambda)},$
(26)

where $t_{0}(\omega)$ and ${r^{(r/l)}_{0}(\omega)}$ are the transmission and
reflection amplitudes for a single cell ($N=0$) that are given in Eq.(14) and
Eq.(17) respectively.

An important feature of expressions (24) and (26) is that both contain factor
$\frac{\sin(\beta(2N+1)\Lambda)}{\sin(\beta\Lambda)}$ which naturally occur in
Hermitian one-dimensional finite periodic systems due to interference or
diffraction effects and reflects a combine effect of all $2N+1$ cell. The
appearance of this factor in non-Hermitian systems is highly non-trivial from
the view of the usual probability conservation property for Hermitian systems
(the reflection and transmission coefficients must sum to unit in either
classical or quantum mechanical regimes) or unitary scattering matrix theory.
However, in Refs. Achilleos et al. (2017); Guo et al. (2022) a simple closed
form expressions is obtained for the total transmission and reflection
(left/right) amplitudes from a lattice of $N$ cells. As pointed out in Refs.
Guo et al. (2022), the transmission and reflection amplitudes for a periodic
many scatters system are related to single cell amplitudes in a compact
fashion. This is intimately connected with the fact that the factorization of
short-range dynamics in a single cell and long-range collective effect of
periodic structure of entire system: the short-range interaction dynamics that
is described by single cell scattering amplitudes and the $\beta$ represents
the collective mode of entire lattice. The occurrence of factorization of
short-range dynamics and long-range collective mode has been known in both
condensed matter physics and nuclear/hadron physics. In the cases such as
particles interacting with short-range potential in a periodic box or trap,
where two physical scales, (1) the short-range particles dynamics and (2)
long-range geometric effect due to the periodic box or trap, are clearly
separated. The quantization conditions are given by a compact formula that is
known as Korringa–Kohn–Rostoker (KKR) method Korringa (1947); Kohn and
Rostoker (1954) in condensed matter physics, Lüscher formula Lüscher (1991) in
LCQD and Busch-Englert-Rzażewski-Wilkens (BERW) formula Busch et al. (1998) in
a harmonic oscillator trap in nuclear physics community. Other related useful
discussions can be found in e.g. Refs.Guo and Long (2022); Guo and Gasparian
(2021, 2022b); Guo (2021).

Above statement can also be demonstrated by the expression of transmission
coefficient $T=|t|^{2}$ for the finite system with $2N+1$ cells,

$\frac{1}{T}=1+{\frac{r_{0}^{r}}{t_{0}}}{\frac{{r_{0}^{l}}^{*}}{{t_{0}}^{*}}\frac{\sin^{2}(\beta(2N+1)\Lambda)}{\sin^{2}(\beta\Lambda)}}=1+\bigg{(}\frac{1}{T_{0}}-1\bigg{)}\frac{\sin^{2}(\beta(2N+1)\Lambda)}{{\sin^{2}(\beta\Lambda)}}.$
(27)

In addition, Equation (27) shows that there are two distinct cases for which
an incident wave is totally transmitted, i.e. $T=1$. This implies perfect
resonant transmission with no losses and no gain, regardless of the complex
nature of the coupling constants.

The first case occurs when there is no reflected wave from any individual cell
and this matches the condition when the product of
$\frac{r_{0}^{r}}{t_{0}}\frac{{r_{0}^{l}}^{*}}{{t_{0}}^{*}}$ in Eq. (27) is
zero (or $T_{0}=1$). This would lead to the unidirectional propagation
discussed in several studies on $\mathcal{P}\mathcal{T}$-symmetric systems,
see, e.g, Refs. Lin et al. (2011b); Feng et al. (2013); Longhi (2011); Longhi
and Della Valle (2013). This phenomenon is also referred as the effect of
exceptional points (EPs) that separate the broken and unbroken
$\mathcal{P}\mathcal{T}$-symmetric phases, see e.g. Refs. El-Ganainy et al.
(2018); Bender et al. (2019); Miri and Alù (2019); Özdemir et al. (2019).

In the second case $\sin(\beta(2N+1)\Lambda)/\sin(\beta\Lambda)=0$. It
corresponds to constructive interference between path reflected from different
unit cells at

${\beta\Lambda}={\frac{\pi l}{2N+1}},\qquad|l|=1,\cdots,N.$ (28)

In both cases mentioned, we have a perfect transmission, that is, $T=1$. As a
consequence, the product of two the reflection coefficients on the left and
right should disappear according to the formula (9). In the case, when one of
reflections reach zero while the other remains non-zero, so-called
unidirectional transparency can occur when we have an ideal non-reflective
transmission in one direction but not in the other. The experimental
demonstration of a unidirectional reflectionless at optical parity-time
metamaterial at optical frequencies is reported in Ref. Feng et al. (2013). An
outlook on the potential directions and applications of controlling sound in
non-hermitian acoustic systems can be found in Ref.Gu et al. (2021).

###  IV.2 Spectral singularities in periodic system with $2N+1$ cells

To illustrate the influence of the two factors mentioned above, as well as the
role of spectral singularities on the formation of Faraday and Kerr rotations
and their shapes let us note that (i) the spectral singularities arise when
both conditions, $\it Re(\omega)=0$ and $Im(\omega)=0$, are satisfied
simultaneously (ii) the location of these poles can be found by solving
$1/t(k)=0$. Based on Eq.(24), there are two types of solutions, as was
mentioned above:

(i) Type I singularities are given by solutions of
$\frac{1}{t_{0}(\omega)}=0$. Hence $\cos(\beta\Lambda)=0$ and
$\frac{1}{t(\omega)}=0$ are both automatically satisfied:

$\beta\Lambda=\pi l+\frac{\pi}{2},\ \ \ \ l\in\mathbb{Z}.$ (29)

The type I singularities are originated from a single cell ($N=0$), and shared
by the entire system of $2N+1$ cells. The type I solutions are independent of
number of cells and the size of system. The detailed discussion about type I
singularities can be found in Sec.III.1.

(ii) type II singularities depend on the size of the system and are given by
two conditions,

$\cos\left(\beta(2N+1)\Lambda\right)=0,\ \ \ \
Im\left[\frac{e^{-ik_{0}L_{0}}}{t_{0}(k)}\right]=0.$ (30)

Hence $\beta\Lambda=\frac{\pi(l+\frac{1}{2})}{2N+1}$ where $l\in\mathbb{Z}$,
above two conditions are given explicitly by

$\displaystyle\sin\left(\frac{n\omega_{cr}}{c}L\right)\left[\cos\left(\frac{n_{0}\omega_{cr}}{c}L_{0}\right)\mathcal{R}(\omega_{cr})-\sin\left(\frac{n_{0}\omega_{cr}}{c}L_{0}\right)\mathcal{I}(\omega_{cr})\right]$
$\displaystyle=\cos\frac{\pi(l+\frac{1}{2})}{2N+1},$
$\displaystyle\cos\left(\frac{n_{0}\omega_{cr}}{c}L_{0}\right)\mathcal{I}(\omega_{cr})+\sin\left(\frac{n_{0}\omega_{cr}}{c}L_{0}\right)\mathcal{R}(\omega_{cr})$
$\displaystyle=0.$ (31)

At the limit of $V_{1}\rightarrow 0$, two conditions are reduced to

$\frac{V^{2}_{2}\omega^{2}_{\omega_{cr}}}{2c^{2}nn_{0}}-\tan\left(\frac{n_{0}\omega_{cr}L_{0}}{c}\right)\cot\left(\frac{n\omega_{cr}L}{c}\right)=\frac{1}{2}\left(\frac{n_{0}}{n}+\frac{n}{n_{0}}\right),\
\ \
\cos\frac{\pi(l+\frac{1}{2})}{2N+1}=\frac{\cos\left(\frac{n\omega_{cr}L}{c}\right)}{\cos\left(\frac{n_{0}\omega_{cr}L_{0}}{c}\right)}$
(32)

###  IV.3 Large $N$ limit

As number of cells is increased, all FR and KR angles demonstrate fast
oscillating behavior due to $\sin(\beta(2N+1)\Lambda)$ and
$\cos(\beta(2N+1)\Lambda)$ functions in transmission and reflection
amplitudes. These behaviors are very similar to what happens for tunneling
time of a particle through layers of periodic
$\mathcal{P}\mathcal{T}$-symmetric barriers that is discussed in Ref. Guo et
al. (2022). For the large $N$ systems, we can introduce the FR and KR angles
per unit cell

$\displaystyle\widehat{\theta}^{T/R}(\omega)=\frac{\theta^{T/R}(\omega)}{(2N+1)\Lambda}.$
(33)

The $N\rightarrow\infty$ limit may be approached by adding a small imaginary
part to $\beta$: $\beta\rightarrow\beta+i\epsilon$, where
$\epsilon\gg\frac{1}{(2L+1)\Lambda}$. As discussed in Ref. Guo et al. (2022),
adding a small imaginary part to $\beta$ is justified by considering the
averaged FR and KR angles per unit cell, which ultimately smooth out the fast
oscillating behavior of FR and KR angles. Using asymptotic behavior of

$\sec(\beta(2N+1)\Lambda)\propto 2e^{i\beta(2N+1)\Lambda},\ \ \ \
\tan(\beta(2N+1)\Lambda)\propto 1,$ (34)

we find

$\frac{1}{(2N+1)\Lambda}\ln t(\omega)\stackrel{{\scriptstyle
N\rightarrow\infty}}{{\rightarrow}}i\beta,\ \ \ \ \frac{1}{(2N+1)\Lambda}\ln
r(\omega)\stackrel{{\scriptstyle N\rightarrow\infty}}{{\rightarrow}}0.$ (35)

Therefore, as $N\rightarrow\infty$, FR and KR angles per unit cell approach

$\widehat{\theta}^{T}_{1}\stackrel{{\scriptstyle
N\rightarrow\infty}}{{\rightarrow}}-\frac{g}{2n}\frac{\partial
Re[\beta]}{\partial n},\ \ \ \ \widehat{\theta}^{T}_{2}\stackrel{{\scriptstyle
N\rightarrow\infty}}{{\rightarrow}}\frac{g}{2n}\frac{\partial
Im[\beta]}{\partial n},\ \ \ \
\widehat{\theta}^{R^{r/l}}_{1,2}\stackrel{{\scriptstyle
N\rightarrow\infty}}{{\rightarrow}}0.$ (36)

Also noted that at large $N$ limit,

$\sin(\beta(2N+1)\Lambda)\propto-\frac{1}{2i}e^{-i\beta(2N+1)\Lambda}\stackrel{{\scriptstyle
N\rightarrow\infty}}{{\rightarrow}}\infty,$ (37)

using Eq.(27), one can show that transmission coefficient therefore approaches
zero: $T\stackrel{{\scriptstyle N\rightarrow\infty}}{{\rightarrow}}0$. The
relation between $\widehat{\theta}^{T}_{2}$ and
$\widehat{\theta}^{R^{r/l}}_{2}$ given in Eq.(11) hence is still valid as
$N\rightarrow\infty$.

(a)

(b)

(c)

(d)

Figure 4:  Plot of FR and KR angles with $N=2$ vs. large $N$ limit result:
$i\frac{g}{2n}\frac{d\beta}{dn}$ (solid purples): (a)
$\widehat{\theta}^{T}_{1}=\widehat{\theta}^{R}_{1}$ (solid black) vs.
$\frac{g}{2n}\frac{dRe[\beta]}{dn}$ (solid purple); (b)
$\widehat{\theta}^{T}_{2}$ (solid black) vs.
$-\frac{g}{2n}\frac{dIm[\beta]}{dn}$ (solid purple); (c)
$\widehat{\theta}^{R^{r}}_{2}$ (solid black) vs.
$\widehat{\theta}^{R^{l}}_{2}$ (dashed red); (d) Band structure plot in
unbroken phase. The parameters are taken as: $V_{1}=0.31$, $V_{2}=0.95$,
$L=0.2$, $L_{0}=0.8$, $n=2$ and $n_{0}=1$

(a)

(b)

(c)

(d)

Figure 5:  Plot of FR and KR angles with $N=2$ vs. large $N$ limit result:
$i\frac{g}{2n}\frac{d\beta}{dn}$ (solid purples): (a)
$\widehat{\theta}^{T}_{1}=\widehat{\theta}^{R}_{1}$ (solid black) vs.
$\frac{g}{2n}\frac{dRe[\beta]}{dn}$ (solid purple); (b)
$\widehat{\theta}^{T}_{2}$ (solid black) vs.
$-\frac{g}{2n}\frac{dIm[\beta]}{dn}$ (solid purple); (c)
$\widehat{\theta}^{R^{r}}_{2}$ (solid black) vs.
$\widehat{\theta}^{R^{l}}_{2}$ (dashed red); (d) Band structure plot in broken
phase. The parameters are taken as: $V_{1}=0.31$, $V_{2}=0.95$, $L=0.2$,
$L_{0}=0.8$, $n=2$ and $n_{0}=1$.

The examples of FR and KR angles per unit cell for
a$\mathcal{P}\mathcal{T}$-symmetric finite system with five cells are shown in
Fig. 4 and Fig. 5, compared with the large $N$ limit results. As we can see in
Fig. 4 and Fig. 5, the $\theta^{T}_{1}$ and $\theta^{T}_{2}$ angles
oscillating around the large $N$ limit results. Even for the small size
system, we can see clearly that the band structure of infinite periodic system
is already showing up. The oscillating KR angles are consistent with zero at
large $N$ limit. In addition, in broken $\mathcal{P}\mathcal{T}$-symmetric
phase in Fig. 5, EPs can be visualized even for a small size system, where two
neighbouring bands merge and the $\mathcal{P}\mathcal{T}$ becomes totally
transparent: both $\theta^{T}_{1}$ and $\theta^{T}_{2}$ approach zero.

For a real refractive index profile, the sign of $\theta^{T}_{1}$ is always
positive due to the fact that $\theta^{T}_{1}$ is closely related to the
density of states. However, in $\mathcal{P}\mathcal{T}$-symmetric systems,
$\theta^{T}_{1}$ is now associated with a generalized density of states, which
can be either positive or negative, see discussion in Ref.Guo et al. (2022);
Guo and Gasparian (2022a). In this sense the negative spike(s) in Fig. 4 and
Fig. 5 around the some frequencies provide the formal justification of the
existence of such negative states. Turning negative of $\theta^{T}_{1}$ is
closely related to the motion of poles across the real axis moving from
unphysical sheet (the second Riemann sheet) into physical sheet (the first
Riemann sheet), for more details see Refs.Guo et al. (2022); Gasparian et al.
(2022b). Since $\theta^{T}_{1}$ ($\theta^{R}_{1}$) is assumed to be related to
the density of states, it is natural that it is practically zero in all
forbidden bands and takes a giant leap to a very large number at the end of
each band.

##  V Discussion and summary

In summary, we studied the anomalous behavior of the Faraday (transmission)
and polar Kerr (reflection) rotation angles of the propagating light, in
finite periodic parity-time ($\mathcal{P}\mathcal{T}$) symmetric structures,
containing $2N+1$ cells.

We have obtained closed form expressions for FR and KR angles for a single
cell consisting of two complex $\delta$-potentials placed on both boundaries
of the ordinary dielectric slab. It is shown that, for a given set of
parameters describing the system, a phase transition-like anomalous behavior
of Faraday and Kerr rotation angles in a parity-time symmetric systems can
take place. In the anomalous phase the value of one of the Faraday and Kerr
rotation angles can become negative, and both angles suffer from spectral
singularities and give a strong enhancement near the singularities. It is
shown that due to symmetry constraints, the real part of the complex angle of
KR, $\theta^{R}_{1}$, is always equal to the $\theta^{T}_{1}$ of FR, no matter
what phase the system is in. The imaginary part of KR angles
$\theta^{R^{r/l}}_{2}$ are also related to the $\theta^{T}_{2}$ of FR by
parity-time symmetry.

We find that, in the limit of weak scattering, the Kerr and Faraday rotation
angles increase linearly with the length of the system. In this approximation
the effects of multiple reflections within the layers are not significant. We
have also shown, based on the modified Kramers-Kronig relations, that only the
three angles FR and KR are completely independent.

###### Acknowledgements.

P.G. and V.G. acknowledge support from the Department of Physics and
Engineering, California State University, Bakersfield, CA. V.G., A.P-G. and
E.J. would like to thank UPCT for partial financial support through the
concession of ”Maria Zambrano ayudas para la recualificación del sistema
universitario español 2021-2023” financed by Spanish Ministry of Universities
with financial funds ”Next Generation” of the EU.

##  Appendix A Determinant Approach

This section is devoted to more mathematical interest. We combine two non-
perturbative approaches, that sufficiently completely describe of photon
(electron) behaviour in a random potential to study the energy spectrum and
scattering matrix elements in the $\mathcal{P}\mathcal{T}$ system without
actually determining the photon eigenfunctions.

In both approaches, the Green’s function was calculated exactly for two
different models. In the first model, we are dealing with the sum of
$\delta$-potentials distributed randomly with an arbitrary strength. The
second model was used to calculate the passage of a free particle through a
layered system, which is characterized by random parameters of the layers.

A convenient formalism to study one dimensional scattering systems satisfying
the stationary Schrödinger equation or the Helmholtz equation relevant to
optical Bragg grating is developed in Ref Aronov et al. (1991); Gasparyan
(1989). The approach allows one to express the transmission and reflection
amplitudes of a wave propagating in a one-dimensional random layered structure
through the characteristic determinant $D_{N}$ ($N$ is the number of the
boundaries), which depends on the amplitudes of reflection of a single scatter
only. The transmission amplitude $t_{N}$ of waves through the systems can be
presented in the form

$t_{N}=\frac{e^{ik|x_{N}-x_{1}|}}{D^{0}_{N}},$ (38)

where the characteristic determinant $D^{0}_{N}$ reduces to a recursive
equation that is convenient for both numerical and analytical approaches.

This paper presents a generalization of the determinant approach to the case
of $\mathcal{P}\mathcal{T}$-symmetric (non-symmetric) systems consisting of
$(N-1)$ dielectric multilayers with two delta potentials in each. The detailed
and, in many respects, complete description and analysis of the Faraday and
Kerr effects in such a system discussed. Specifically, our investigations
focus on the periodic finite size diatomic $\mathcal{P}\mathcal{T}$-symmetric
model. We predict that for a given set of parameters describing the system the
Faraday and Kerr rotation angles show a non-trivial transition with a change
in sign. In the anomalous phase the value of one of the Faraday and Kerr
rotation angles can become negative, and both angles suffer from spectral
singularities and give a strong enhancement near the singularities.

Let us consider $(N-1)$ dielectric multilayer system labeled
$n={1,\cdots,N-1}$ between two semi-infinite media. The positions of the
boundaries of the $nth$ dielectric layer, characterized by the constant
$\epsilon_{n}$, are given by $x_{n}$ and $x_{n+1}$ respectively. The left and
right ends of the system are at $x=x_{{}_{N}}$ and $x=x_{1}$ with
$\epsilon_{0}=\epsilon_{N}$, respectively. We assume that a plain EMW wave is
incident from the left (with the dielectric permittivity $\epsilon_{0}$) onto
the boundary at $x=x_{1}$ and evaluate the amplitude of the reflected wave and
the wave propagating in the semi-infinite media for $x>x_{N}$, characterized
by $\epsilon_{N}$. In the further discussion we will assume, that the first
and last layers of the multilayer system make interfaces with the vacuum.

We also assume that we know the transmission $t_{n,n+1}$ and reflection
amplitudes (from the left $r_{n,n+1}$ and the right $r_{n+1,n}$) of the EMW
from a single $Z_{n}\delta(x-x_{n})$ scatter, located at the contact of two
semi-infinite media I and II at $x=x_{n}$. Using the results of the
transmission and reflection amplitudes for the single scatter, we will build
characteristic determinant $D_{N}$ for $N$ scatters and obtain the total
transmission $t_{N}$ and reflection amplitudes $r^{N}_{L}$ and $r^{N}_{R}$.
The transmission amplitudes from left and from right equal each other are
given by

$t_{n,n+1}=t_{n+1,n}\equiv\frac{2\sqrt{\frac{k_{n}}{k_{n-1}}}}{1+\frac{k_{n}}{k_{n-1}}-i\frac{\gamma}{k_{n-1}}Z_{n}},\
\ \ \ k_{n}=\frac{\omega}{c}n,\ \ \ \
\gamma\equiv\left(\frac{\omega}{c}\right)^{2}.$ (39)

Similarly,

$r_{n,n+1}=\frac{1-\frac{k_{n+1}}{k_{n}}+i\frac{\gamma}{k_{n}}Z_{n}}{1+\frac{k_{n+1}}{k_{n}}-i\frac{\gamma}{k_{n}}Z_{n}},\
\ \ \
r_{n+1,n}=\frac{\frac{k_{n+1}}{k_{n}}-1+i\frac{\gamma}{k_{n}}Z_{n}}{1+\frac{k_{n+1}}{k_{n}}-i\frac{\gamma}{k_{n}}Z_{n}}.$
(40)

We can easily verify by using Eq.(39) and Eq.(40) that the conservation law is
satisfied, provided that $Z_{n}$ is real:

$t_{n,n+1}t^{*}_{n,n+1}+r_{n,n+1}r^{*}_{n,n+1}=\frac{4\frac{k_{n}}{k_{n-1}}+(1-\frac{k_{n}}{k_{n-1}})^{2}+(\frac{\gamma}{k_{n-1}}Z_{n})^{2}}{(1+\frac{k_{n}}{k_{n-1}})^{2}+(\frac{\gamma}{k_{n-1}}Z_{n})^{2}}=1$
(41)

In the case of a complex value $Z_{n}$, the conservation law cannot hold,
since the system is not $\mathcal{P}\mathcal{T}$-symmetric and can be
described by only complex energy eigenvalue. Later, when we ”build” the
characteristic determinant $D_{N}$ for the entire system with $N$ complex
potentials, distributed arbitrary, we will return to the conservation law of
the system in more detail.

Assuming that we know the explicit expression for the amplitude of reflection
from a single-scattering delta potential, see Eq.(40), we now turn to a closer
investigation of the system with two complex potentials. Following Refs.Aronov
et al. (1991), we can present the determinant $D_{2}$ of two delta potentials
located at points $x_{1}$ and $x_{2}$ ($L$ = $x_{2}$\- $x_{1}$) on the left
and right boundaries of a dielectric slab surrounded by two semi-infinite
media with permittivities $\epsilon_{0}$ (left) and $\epsilon_{2}$ (right),
respectively. The dielectric slab itself is characterized by permittivity
$\epsilon_{1}$. The explicit form of $D_{2}$ is

$D^{0}_{2}=\frac{1}{(1+r_{21})(1+r_{32})}\det D_{2},$ (42)

where

$\displaystyle\det
D_{2}\equiv\begin{vmatrix}1&r_{23}e^{ik_{1}(x_{2}-x_{1})}\\\
r_{21}e^{ik_{1}(x_{2}-x_{1})}&1\end{vmatrix},$ (43)

and $r_{n,n+1}$ is given by Eq.(40) with the appropriate choice of $n$ and
$Z_{n}$. Let us add another boundary from the right, at the point $x_{3}$,
i.e. we consider a layered heterostructure consisting of two films with
permittivities $\epsilon_{1}$ and $\epsilon_{2}$, placed between two semi-
infinite media $\epsilon_{0}$ and $\epsilon_{3}$.

Next, adding another delta complex potential $Z_{3}$ at $x_{3}$ the new
$D_{3}$, which now is $3\times 3$ determinant, can be written as

$\displaystyle D^{0}_{3}=$
$\displaystyle\prod^{3}_{l=1}\frac{1}{(1+r_{l+1,l})}\det D_{3}$ (44)

where

$\displaystyle\det
D_{3}\equiv\begin{vmatrix}1&r_{23}e^{ik_{1}(x_{2}-x_{1})}&r_{34}e^{ik_{1}(x_{2}-x_{1})}e^{ik_{2}(x_{3}-x_{2})}\\\
r_{21}e^{ik_{1}(x_{2}-x_{1})}&1&r_{34}e^{ik_{2}(x_{3}-x_{2})}\\\
r_{21}e^{ik_{1}(x_{2}-x_{1})}e^{ik_{2}(x_{3}-x_{2})}&r_{32}e^{ik_{2}(x_{3}-x_{2})}&1\\\
\end{vmatrix}.$ (45)

By continuing adding new boundaries and complex potential $Z_{n}$ at the
points $x_{4}$,…, $x_{N}$, we will obtain an $N$-multilayer system, each layer
of which contains two delta potentials. This system will be characterized by
the product of $N$ by $N$ determinant $D_{N}$

$D^{0}_{N}=\prod^{N}_{l=1}\frac{1}{(1+r_{l+1,l})}\det D^{N}_{l,n},$ (46)

with the following matrix elements $D^{N}_{l,n}$:

$D^{N}_{l,n}=\left\\{\begin{array}[]{l
l}\delta_{ln}+(1-\delta_{ln})r_{l,l-1}e^{ik_{l}|x_{l}-x_{n}|},\ \ l\geq n,\\\
\delta_{ln}+(1-\delta_{ln})r_{l-1,l}e^{ik_{l}|x_{l}-x_{n}|},\ \ n\geq l.\\\
\end{array}\right.$ (47)

The characteristic determinant $D_{N}$ can be presented as a determinant of a
Teoplitz tridiagonal matrix that satisfies the following recurrence
relationship:

$D_{N}=A_{N}D_{N-1}-B_{N}D_{N-2},$

where $D_{N-1}$ ($D_{N-2}$) is the determinant equation (47) with the $Nth$
and also the $(N-1)th$ row and column omitted. The initial conditions for the
recurrence relations are $D_{0}=1$, $D_{-1}=0$, $D_{1}\equiv A_{1}=1$. The
coefficients $A_{N}$, $B_{N}$ can be obtained from the explicit form of
$D^{N}_{n,l}$ (see Eq. (47)). For $N>1$ we have

$A_{n}=1+\frac{r_{n,n+1}}{r_{n-1,n}}(1+r_{n-1,n}+r_{n,n-1})e^{2ik_{n}|x_{n}-x_{n-1}|}=1+B_{n}-r_{n,n-1}r_{n,n+1}e^{2ik_{n}|x_{n}-x_{n-1}|}$

and

$B_{n}=\frac{r_{n,n+1}}{r_{n-1,n}}(1+r_{n,n-1})(1+r_{n-1,n})e^{2ik_{n}|x_{n}-x_{n-1}|}$

In concluding, let us stress once more that Eqs.(46) and (47) may be viewed as
generalization of the characteristic determinant method that can be applied to
the Helmholtz (Shrödinger) equation with complex potentials, distributed
arbitrary and find scattering matrix elements without actually determining the
photon (electron) eigenfunctions.

## References

  * Yoshino (2005) T. Yoshino, Journal of the Optical Society of America B Optical Physics 22, 1856 (2005).
  * Zamani et al. (2011) M. Zamani, M. Ghanaatshoar, and H. Alisafaee, Journal of the Optical Society of America B Optical Physics 28, 2637 (2011).
  * Birch (1982) K. P. Birch, Optics Communications 43, 79 (1982).
  * Corzo et al. (2012) N. V. Corzo, A. M. Marino, K. M. Jones, and P. D. Lett, Phys. Rev. Lett. 109, 043602 (2012).
  * Stubkjaer (2000) K. E. Stubkjaer, IEEE Journal of Selected Topics in Quantum Electronics 6, 1428 (2000).
  * Wang et al. (2008) Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljačić, Phys. Rev. Lett. 100, 013905 (2008), eprint 0712.1776.
  * Wang et al. (2009) Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, Nature (London) 461, 772 (2009).
  * McGee et al. (1993) N. W. E. McGee, M. T. Johnson, J. J. de Vries, and J. aan de Stegge, Journal of Applied Physics 73, 3418 (1993).
  * Longair (2011) M. S. Longair, _High Energy Astrophysics_ (Cambridge University Press, 2011), 3rd ed.
  * Furdyna (1988) J. K. Furdyna, Journal of Applied Physics 64, R29 (1988), eprint https://doi.org/10.1063/1.341700, URL https://doi.org/10.1063/1.341700.
  * Berger (2003) U. Berger, in _Encyclopedia of Physical Science and Technology (Third Edition)_ , edited by R. A. Meyers (Academic Press, New York, 2003), pp. 777–798, third edition ed., ISBN 978-0-12-227410-7, URL https://www.sciencedirect.com/science/article/pii/B0122274105004452.
  * Turner and Stolen (1981) E. H. Turner and R. H. Stolen, Opt. Lett. 6, 322 (1981), URL http://opg.optica.org/ol/abstract.cfm?URI=ol-6-7-322.
  * Firby et al. (2018) C. J. Firby, P. Chang, A. S. Helmy, and A. Y. Elezzabi, J. Opt. Soc. Am. B 35, 1504 (2018), URL http://opg.optica.org/josab/abstract.cfm?URI=josab-35-7-1504.
  * Uchida et al. (2011) H. Uchida, Y. Mizutani, Y. Nakai, A. A. Fedyanin, and M. Inoue, Journal of Physics D: Applied Physics 44, 064014 (2011), URL https://doi.org/10.1088/0022-3727/44/6/064014.
  * Andrei and Mayergoyz (2003) P. Andrei and I. Mayergoyz, Journal of Applied Physics 94, 7163 (2003), eprint https://doi.org/10.1063/1.1625084, URL https://doi.org/10.1063/1.1625084.
  * Gevorkian and Gasparian (2014) Z. Gevorkian and V. Gasparian, Phys. Rev. A 89, 023830 (2014), URL https://link.aps.org/doi/10.1103/PhysRevA.89.023830.
  * Gasparian et al. (2022a) V. Gasparian, P. Guo, and E. Jódar, Physics Letters A 453, 128473 (2022a), eprint 2205.09871.
  * Oppeneer et al. (1992) P. M. Oppeneer, J. Sticht, T. Maurer, and J. Köbler, Zeitschrift fur Physik B Condensed Matter 88, 309 (1992).
  * Chen et al. (1990) L.-Y. Chen, W. A. McGahan, Z. S. Shan, D. J. Sellmyer, and J. A. Woollam, Journal of Applied Physics 67, 7547 (1990).
  * Feil and Haas (1987) H. Feil and C. Haas, Phys. Rev. Lett. 58, 65 (1987).
  * Loughran et al. (2018) T. H. J. Loughran, P. S. Keatley, E. Hendry, W. L. Barnes, and R. J. Hicken, Optics Express 26, 4738 (2018).
  * Lofy et al. (2020) J. Lofy, V. Gasparian, Z. Gevorkian, and E. Jódar, Reviews on Advanced Materials Science 59, 243 (2020).
  * Gasparian et al. (1995) V. Gasparian, M. Ortuño, J. Ruiz, and E. Cuevas, Phys. Rev. Lett. 75, 2312 (1995), URL https://link.aps.org/doi/10.1103/PhysRevLett.75.2312.
  * Guo and Gasparian (2022a) P. Guo and V. Gasparian, Phys. Rev. Research 4, 023083 (2022a), URL https://link.aps.org/doi/10.1103/PhysRevResearch.4.023083.
  * Mostafazadeh (2014) A. Mostafazadeh, Journal of Physics A: Mathematical and Theoretical 47, 505303 (2014), URL https://doi.org/10.1088/1751-8113/47/50/505303.
  * Ahmed (2011) Z. Ahmed, Journal of Physics A: Mathematical and Theoretical 45, 032004 (2011), URL https://doi.org/10.1088/1751-8113/45/3/032004.
  * Ge et al. (2012) L. Ge, Y. D. Chong, and A. D. Stone, Phys. Rev. A 85, 023802 (2012), URL https://link.aps.org/doi/10.1103/PhysRevA.85.023802.
  * Gasparian et al. (2005) V. Gasparian, B. Altshuler, and M. Ortuño, Phys. Rev. B 72, 195309 (2005).
  * Guo et al. (2022) P. Guo, V. Gasparian, E. Jódar, and C. Wisehart, arXiv e-prints arXiv:2208.13543 (2022), eprint 2208.13543.
  * Garmon et al. (2015) S. Garmon, M. Gianfreda, and N. Hatano, Phys. Rev. A 92, 022125 (2015), eprint 1505.04267.
  * Uncu and Demiralp (2006) H. Uncu and E. Demiralp, Physics Letters A 359, 190 (2006).
  * Němec et al. (2018) P. Němec, M. Fiebig, T. Kampfrath, and A. V. Kimel, Nature Physics 14, 229 (2018), eprint 1705.10600.
  * Achilleos et al. (2017) V. Achilleos, Y. Aurégan, and V. Pagneux, Phys. Rev. Lett. 119, 243904 (2017), eprint 1703.07428.
  * Kalozoumis et al. (2018) P. A. Kalozoumis, G. Theocharis, V. Achilleos, S. Félix, O. Richoux, and V. Pagneux, Phys. Rev. A 98, 023838 (2018), eprint 1712.08763.
  * Lin et al. (2011a) Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, Phys. Rev. Lett. 106, 213901 (2011a), eprint 1108.2493.
  * Bender et al. (1999) C. M. Bender, G. V. Dunne, and P. N. Meisinger, Physics Letters A 252, 272 (1999), eprint cond-mat/9810369.
  * Shin (2004) K. C. Shin, Journal of Physics A Mathematical General 37, 8287 (2004), eprint math-ph/0404015.
  * Musslimani et al. (2008) Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, Phys. Rev. Lett. 100, 030402 (2008).
  * Midya et al. (2010) B. Midya, B. Roy, and R. Roychoudhury, Physics Letters A 374, 2605 (2010), eprint 1004.3218.
  * Gasparian et al. (1997) V. Gasparian, U. Gummich, E. Jódar, J. Ruiz, and M. Ortuño, Physica B Condensed Matter 233, 72 (1997).
  * Korringa (1947) J. Korringa, Physica 13, 392 (1947), ISSN 0031-8914, URL https://www.sciencedirect.com/science/article/pii/003189144790013X.
  * Kohn and Rostoker (1954) W. Kohn and N. Rostoker, Phys. Rev. 94, 1111 (1954), URL https://link.aps.org/doi/10.1103/PhysRev.94.1111.
  * Lüscher (1991) M. Lüscher, Nucl. Phys. B354, 531 (1991).
  * Busch et al. (1998) T. Busch, B.-G. Englert, K. Rzażewski, and M. Wilkens, Found. Phys. 28, 549–559 (1998).
  * Guo and Long (2022) P. Guo and B. Long, Journal of Physics G: Nuclear and Particle Physics 49, 055104 (2022), URL https://doi.org/10.1088/1361-6471/ac59d5.
  * Guo and Gasparian (2021) P. Guo and V. Gasparian, Phys. Rev. D 103, 094520 (2021), URL https://link.aps.org/doi/10.1103/PhysRevD.103.094520.
  * Guo and Gasparian (2022b) P. Guo and V. Gasparian, Journal of Physics A: Mathematical and Theoretical 55, 265201 (2022b), URL https://doi.org/10.1088/1751-8121/ac7180.
  * Guo (2021) P. Guo, Phys. Rev. C 103, 064611 (2021), URL https://link.aps.org/doi/10.1103/PhysRevC.103.064611.
  * Lin et al. (2011b) Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, Phys. Rev. Lett. 106, 213901 (2011b), eprint 1108.2493.
  * Feng et al. (2013) L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, Nature Materials 12, 108 (2013).
  * Longhi (2011) S. Longhi, Journal of Physics A Mathematical General 44, 485302 (2011), eprint 1111.3448.
  * Longhi and Della Valle (2013) S. Longhi and G. Della Valle, Annals of Physics 334, 35 (2013), eprint 1306.0667.
  * El-Ganainy et al. (2018) R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, Nature Physics 14, 11 (2018), URL https://doi.org/10.1038/nphys4323.
  * Bender et al. (2019) C. M. Bender, P. E. Dorey, C. Dunning, A. Fring, D. W. Hook, H. F. Jones, S. Kuzhel, G. Lévai, and R. Tateo, _PT Symmetry_ (WORLD SCIENTIFIC (EUROPE), 2019), eprint https://www.worldscientific.com/doi/pdf/10.1142/q0178, URL https://www.worldscientific.com/doi/abs/10.1142/q0178.
  * Miri and Alù (2019) M.-A. Miri and A. Alù, Science 363, eaar7709 (2019), eprint https://www.science.org/doi/pdf/10.1126/science.aar7709, URL https://www.science.org/doi/abs/10.1126/science.aar7709.
  * Özdemir et al. (2019) Ş. K. Özdemir, S. Rotter, F. Nori, and L. Yang, Nature Materials 18, 783 (2019), URL https://doi.org/10.1038/s41563-019-0304-9.
  * Gu et al. (2021) Z. Gu, H. Gao, P.-C. Cao, T. Liu, X.-F. Zhu, and J. Zhu, Physical Review Applied 16, 057001 (2021).
  * Gasparian et al. (2022b) V. Gasparian, P. Guo, and E. Jódar, arXiv e-prints arXiv:2205.09871 (2022b), eprint 2205.09871.
  * Aronov et al. (1991) A. G. Aronov, V. M. Gasparian, and U. Gummich, Journal of Physics Condensed Matter 3, 3023 (1991).
  * Gasparyan (1989) V. M. Gasparyan, Soy. Phys. Solid State 31, 266 (1989). 3, 3023 (1989).