# Correlated Nonreciprocity around Conjugate Exceptional Points

Arnab Laha<EMAIL_ADDRESS>Institute of Spintronics and Quantum
Information, Faculty of Physics, Adam Mickiewicz University, 61-614 Poznań,
Poland  Department of Physics, Indian Institute of Technology Delhi, New
Delhi-110016, India  Adam Miranowicz  Institute of Spintronics and Quantum
Information, Faculty of Physics, Adam Mickiewicz University, 61-614 Poznań,
Poland  R. K. Varshney  Department of Physics, Indian Institute of Technology
Delhi, New Delhi-110016, India  Somnath Ghosh  Department of Physics, Indian
Institute of Technology Jodhpur, Rajasthan-342037, India

###### Abstract

The occurrence of exceptional points (EPs) is a fascinating non-Hermitian
feature of open systems. A level-repulsion phenomenon between two complex
states of an open system can be realized by positioning an EP and its time-
reversal ($\mathcal{T}$) conjugate pair in the underlying parameter space.
Here, we report the fascinating nonreciprocal response of such two conjugate
EPs by using a dual-mode planar waveguide system having two
$\mathcal{T}$-symmetric active variants concerning the transverse gain-loss
profiles. We specifically reveal a comprehensive all-optical scheme to achieve
correlative nonreciprocal light dynamics by using the reverse chirality of two
dynamically encircled conjugate EPs in the presence of local nonlinearity. A
specific nonreciprocal correlation between two designed
$\mathcal{T}$-symmetric waveguide variants is established in terms of their
unidirectional transfer of light with a precise selection of modes. Here, the
unconventional reverse chiral properties of two conjugate EPs allow the
nonreciprocal transmission of two selective modes in the opposite directions
of the underlying waveguide variants. An explicit dependence of the
nonlinearity level on a significant enhancement of the nonreciprocity in terms
of an isolation ratio is explored by investigating the effects of both local
Kerr-type and saturable nonlinearities (considered separately). The physical
insights and implications of harnessing the properties of conjugate EPs in
nonlinear optical systems can enable the growth and development of a versatile
platform for building nonreciprocal components and devices.

##  I Introduction

The synergy of non-Hermitian quantum physics and photonics has been revealing
a novel and promising direction for building a range of photonics components
and devices Midya  _et al._ (2018). An extensive study on the perturbation
theory in quantum mechanics once revealed the occurrence of exceptional point
(EP) singularities as an explicit mathematical feature of non-Hermitian or
open systems Kato (1966). EPs usually appear as topological defects in the
system’s parameter space, affecting the eigenspace dimensionality, which
results in the simultaneous coalescence of at least two coupled eigenvalues
and the associated eigenstates Kato (1966); Heiss (2000); Berry (2004); Heiss
(2012); Eleuch and Rotter (2016). The parity-time ($\mathcal{PT}$)-symmetric
systems (a special class of non-Hermitian system with real eigenvalues) Bender
and Boettcher (1998); Bender (2007) encounter an EP at a spontaneous
transition from real (exact-$\mathcal{PT}$-phase) to complex
(broken-$\mathcal{PT}$-phase) eigenvalues Guo  _et al._ (2009); Rüter  _et
al._ (2010); El-Ganainy  _et al._ (2018); Özdemir  _et al._ (2019). Recently,
the engineering of ubiquitous non-Hermitian components (e.g. loss and gain) in
photonic systems has revealed such EP-like mathematical objects as a powerful
tool to manipulate and detect the energy-states of light El-Ganainy  _et al._
(2018); Miri and Alù (2019); Özdemir  _et al._ (2019); Bergholtz  _et al._
(2021); Wang  _et al._ (2023); Yan  _et al._ (2023); Parto  _et al._ (2021). A
controlled variation of the system’s parameters in the vicinity of EPs can
immensely boost a versatile range of quantum-photonic technologies in the
context of, e.g., asymmetric energy transfer Xu  _et al._ (2016), programmable
state-switching Arkhipov  _et al._ (2023); Laha  _et al._ (2021), phonon
lasing Zhang  _et al._ (2018), coherent perfect absorption Wang  _et al._
(2021), slow-light engineering Goldzak  _et al._ (2018), enhanced energy
harvesting Fernández-Alcázar  _et al._ (2021), parametric instability
Zyablovsky  _et al._ (2016) and highly-precise sensing Chen  _et al._ (2017);
Wiersig (2020).

The concept of the occurrence of conjugate EPs has recently been introduced
based on the complex parameter dependence of a non-Hermitian Hamiltonian Laha
_et al._ (2022). This can be described by considering a generic two-level
(without loss of generality for higher-order situations) non-Hermitian
Hamiltonian $\mathcal{H}(\lambda)$, which depends on a complex parameter
$\lambda=\lambda^{\text{R}}+i\lambda^{\text{I}}$. The associated eigenvalues
$\mathcal{E}_{1,2}(\lambda)$ and the eigenvectors $\Psi_{1,2}(\lambda)$ would
be analytical functions in the complex-$\lambda$ plane except at a singularity
$\lambda=\lambda_{s}$, known as an EP. Concerning the imaginary part of the
dependent parameter $\lambda$ (i.e., $\lambda^{\text{I}}$), the considerations
of $\lambda^{\text{I}}<0$ and $\lambda^{\text{I}}>0$ ideally define two
complementary variants of $\mathcal{H}(\lambda)$. Such two complementary
systems can be correlated based on time-reversal ($\mathcal{T}$)-symmetry.
Here, two variants of $\mathcal{H}(\lambda)$ under $\mathcal{T}$-symmetry
separately host two EPs in the complex $\lambda$-plane at
$\lambda_{s}=\lambda_{s}^{\text{R}}+i\lambda_{s}^{\text{I}}$ and
$\lambda_{s}^{*}=\lambda_{s}^{\text{R}}-i\lambda_{s}^{\text{I}}$ (say, EP and
its conjugate EP*, respectively), which are in the complex conjugate relation.
Such two correlated EPs in two $\mathcal{T}$-symmetric complementary systems
can be called as conjugate EPs.

Unconventional light guidance mechanism based on the chirality of EPs has
extensively been studied, where a sufficiently slow length-dependent gain-loss
dynamics along a closed 2D loop around an EP can steer the adiabatic and
nonadiabatic conversions of modes Gilary  _et al._ (2013); Milburn  _et al._
(2015). Here, even though the adiabaticity is maintained in the sense of the
exchange of eigenvalues for a quasistatic gain-loss variation Dembowski  _et
al._ (2004), the associated eigenmodes fail to meet adiabaticity while
propagating along the length, which results in the conversion of all the modes
into different particular dominating modes, based on the device chirality (in
terms of direction of light propagation) Doppler  _et al._ (2016); Zhang and
Chan (2019); Laha  _et al._ (2018); Dey  _et al._ (2020); Laha  _et al._
(2020). Such a chirality-based asymmetric transfer of modes has recently been
explored to reveal a distinct reverse-chiral behavior of a pair of conjugate
EPs, while dynamically encircling them in two $\mathcal{T}$-symmetric active
variants of a waveguide-based optical system Laha  _et al._ (2022).

Moreover, the reciprocity of such a chiral light guidance process can be
broken by introducing nonreciprocal elements, where the occurrence of an EP
can considerably enhance nonreciprocity Thomas  _et al._ (2016); Choi  _et
al._ (2017). Nonreciprocal devices, such as isolators and circulators, allow
only one-way light transmission with an asymmetric scattering matrix, which is
indispensable to minimize unwanted back-reflection and multi-path interference
in photonic circuits Caloz  _et al._ (2018). However, the common magneto-
optical approaches (such as a Faraday rotator), mainly applied for bulky free-
space devices, are usually inefficient in enabling a sufficient nonreciprocity
for photonic circuits. Hence, there are growing demands to achieve high
nonreciprocity on the chip-scale footprint, where the chiral response of an EP
in nonlinear media can play a crucial role in meeting such demands. Recently,
an EP-induced mode-selective isolation scheme has been revealed, where local
nonlinearity has served as an efficient tool to enable all-optical
nonreciprocity without using any magneto-optical effect Laha  _et al._ (2020).
In this context, the chiral response of two conjugate EPs in nonlinear media
could have immense potential in developing correlative nonreciprocal devices
with highly precise mode manipulation. Moreover, the recently developed non-
Hermitian formalism of Liouvillian super operators Minganti  _et al._ (2019,
2020) can also be exploited for the quantum implementation of our waveguide-
based classical analysis to explore the correlated features of conjugate
quantum EPs.

In this article, we comprehensively report the correlated nonreciprocal
response of two $\mathcal{T}$-symmetric active variants of a gain-loss
assisted dual-mode planar waveguide, operating near two conjugate EPs. Here,
all-optical nonreciprocity is achieved with the introduction of local
nonlinearity. We investigate the hosting of conjugate EPs in complementary
gain-loss parameter planes based on Riemann surface connections associated
with two quasi-guided modes. Besides establishing the reverse-chiral response
concerning the asymmetric mode conversion process driven by dynamical
parametric variation in the vicinity of two conjugate EPs, we exclusively
investigate the asymmetric nonreciprocal waveguidence mechanism in the context
of all-photonic isolation through two $\mathcal{T}$-symmetric waveguide
variants. Here, a correlation in the nonreciprocal transmission of selective
modes with an enhanced isolation ratio (say, IR) through two complementary
waveguides is established. Moreover, a comparative study on the individual
effect of local Kerr-type nonlinearity and saturable nonlinearity is reported
by showing the possibility of enhancing the IR significantly.

##  II Results and Discussion

###  II.1 Designing two time-symmetric active waveguide variants

We design a 2D planar step-index optical waveguide having the geometrical
dimensions $w=20\lambda/\pi$ (width) and $l=l_{m}\times 10^{3}$ (length) with
$l_{m}=7.5\lambda/\pi$ (i.e., both the dimensions are considered in the unit
of wavelength $\lambda$), where we set $\lambda=2\pi$ corresponding to a
normalized wavenumber $k=1$ (in a dimensionless unit). The designed waveguide
with a glass based core ($n_{\text{co}}=1.5$), surrounded by a silica based
cladding ($n_{\text{clad}}=1.46$), is distributed in the $xz$-plane, where
$x\in[-w/2,w/2]$ and $z\in[0,l]$ are the transverse and propagation axes,
respectively. The real (background) refractive index profile is considered as

$\text{Re}[n(x)]=\left\\{\begin{array}[]{l}n_{\text{co}}\\\
n_{\text{clad}}\end{array}\begin{array}[]{l}\,:-w/6\leq x\leq w/6,\\\
\,:w/6\leq|x|\leq w/2.\end{array}\right.$ (1)

Based on the chosen dimensional parameters and Re$n(x)$\- profile, the
designed waveguide supports only two scalar modes: the fundamental mode
$\Psi_{\text{F}}$ and the first higher order mode $\Psi_{\text{H}}$ (scalar
modal analysis is valid in the presence of a small index difference between
the core and cladding; $\Delta n=0.04$).

Figure 1: (a) Schematic design of WG${}_{\text{A}}$ and WG${}_{\text{T}}$
($\mathcal{T}$-symmetric) based on the framework of a gain-loss assisted
planar waveguide. Two arrows indicate their opposite propagation directions.
Circular plus and minus signs in different segments are associated with the
positive (loss) and negative (gain) imaginary indices as in (2). (b)
Transverse background refractive index profile, i.e., Re[$n(x)$], (dotted
black line; corresponding to the left vertical axis) along with normalized
intensity profiles of two supported modes $\Psi_{\text{F}}$ and
$\Psi_{\text{H}}$ (corresponding to the right vertical axis). (c) Transverse
gain-loss distributions, i.e., Im[$n(x)$], for two $\mathcal{T}$-symmetric
variants WG${}_{\text{A}}$ (solid green line) and WG${}_{\text{T}}$ (solid
black line) for $\gamma=0.01$ and $\tau=2$. Figure 2: (a) Connections between
the Riemann surfaces associated with $\beta_{\text{F}}$ and
$\beta_{\text{H}}$, while varying the control parameters $\gamma$ and $\tau$,
simultaneously. (a.1) and (a.2) show the distributions of Re[$\beta$] and
Im[$\beta$], respectively. Dotted red and blue curves represent the
trajectories of $\beta_{\text{F}}$ and $\beta_{\text{H}}$ for a chosen
$\tau=3.1607$, which reveal the encounter of two conjugate EPs based on the
coalescence and bifurcations in Re[$\beta$] and Im[$\beta$] at $\gamma=\pm
8.1\times 10^{-3}$. Dotted blue squares separate the regions for
WG${}_{\text{A}}$ and WG${}_{\text{T}}$. (b) Variation of
$\left\langle\Psi_{\text{F}}|\Psi_{\text{H}}\right\rangle$ with respect to
$\gamma$ (when $\tau=3.1607$), which shows the coalescence of
$\Psi_{\text{F}}$ and $\Psi_{\text{H}}$ via
$\left\langle\Psi_{\text{F}}|\Psi_{\text{H}}\right\rangle=1$ at both EP and
EP*. (b) Parametric encirclement of two conjugate EPs in the
($\gamma$,$\tau$)-plane following (3) (shown in the ground surfaces) and
associated transfer process of $\beta_{\text{F}}$ and $\beta_{\text{H}}$ from
their respective surfaces.

Now, we enable non-Hermiticity via the introduction of an unbalanced gain-loss
profile [i.e., the imaginary part of $n(x)$] in the designed passive
waveguide, which results in the coupling between two quasiguided modes
$\Psi_{\text{F}}$ and $\Psi_{\text{H}}$. We can control such coupling with the
modulation of a gain-loss profile in a 2D parameter space characterized by the
gain-loss coefficient $\gamma$ and a loss-to-gain ratio $\tau$. Using this
waveguide framework, we consider two complementary active variants connected
via $\mathcal{T}$-symmetric Im[$n(x)$] profiles given by

$\text{Im}[n(x)]=\left\\{\begin{array}[]{l}-\,i\gamma\\\ +\,i\tau\gamma\\\
+\,i\gamma\end{array}\right.\begin{array}[]{c}|\\\ \leavevmode\hbox
to13.16pt{\vbox to13.16pt{\pgfpicture\makeatletter\hbox{\hskip
6.58061pt\lower-6.58061pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.38062pt}{0.0pt}\pgfsys@curveto{6.38062pt}{3.52396pt}{3.52396pt}{6.38062pt}{0.0pt}{6.38062pt}\pgfsys@curveto{-3.52396pt}{6.38062pt}{-6.38062pt}{3.52396pt}{-6.38062pt}{0.0pt}\pgfsys@curveto{-6.38062pt}{-3.52396pt}{-3.52396pt}{-6.38062pt}{0.0pt}{-6.38062pt}\pgfsys@curveto{3.52396pt}{-6.38062pt}{6.38062pt}{-3.52396pt}{6.38062pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.61111pt}{-3.41666pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\mathcal{T}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\\\
|\end{array}\begin{array}[]{l}+\,i\gamma\\\ -\,i\tau\gamma\\\
-\,i\gamma\end{array}\begin{array}[]{l}\,:-w/6\leq x\leq 0,\\\ \,:0\leq x\leq
w/6,\\\ \,:w/6\leq|x|\leq w/2.\end{array}$ (2)

Such two $\mathcal{T}$-symmetric waveguide variants, say WG${}_{\text{A}}$ and
WG${}_{\text{T}}$ are shown in Fig. 1(a). Figure 1(b) shows the profile of
Re$[n(x)]$ [dotted black line; given by (1)] of the background framework along
with the normalized intensity profile of two supported modes $\Psi_{\text{F}}$
and $\Psi_{\text{H}}$, whereas Fig. 1(c) shows the profiles of Im$[n(x)]$ of
two active variants WG${}_{\text{A}}$ and WG${}_{\text{T}}$ (represented by
green and black lines, respectively). As per the constraints of
$\mathcal{T}$-symmetry, i.e.,
$\mathcal{T}:\left\\{i,t,x\right\\}\rightarrow\left\\{-i,-t,x\right\\}$ ($i$
is the imaginary quantity; $t$ and $x$ are the time and space coordinates,
respectively), WG${}_{\text{A}}$ and WG${}_{\text{T}}$ host exactly two
complex conjugate profiles of $n(x)$ with respect to the transverse axis [as
can be understood from (2) and Fig. 1(c)]. Here, we have to consider two
opposite propagation directions for WG${}_{\text{A}}$ and WG${}_{\text{T}}$ to
maintain $\mathcal{T}$-symmetric equivalence based on the quantum-optical
analogy $t\equiv z$.

###  II.2 Riemann surface connections: Hosting conjugate EPs

To host the pair of conjugate EPs, we study the interaction between two
coupled eigenvalues associated with $\Psi_{\text{F}}$ and $\Psi_{\text{H}}$,
while varying the control parameters $\gamma$ and $\tau$, simultaneously,
within chosen ranges. In this context, an analytical treatment toward hosting
conjugate EPs using a non-Hermitian Hamiltonian, which is analogous to our
proposed waveguide-based system, is discussed in detail in an appendix. Here,
the complex propagation constants ($\beta$-values), i.e., $\beta_{\text{F}}$
and $\beta_{\text{H}}$ (associated with $\Psi_{\text{F}}$ and
$\Psi_{\text{H}}$, respectively) are considered as the system eigenvalues,
which are calculated by computing the solutions of the 1D scalar wave equation
$\left[\partial_{x}^{2}+k^{2}n^{2}(x)-\beta^{2}\right]\psi(x)=0$. We identify
the connections between the Riemann sheets associated with coupled
$\beta_{\text{F}}$ and $\beta_{\text{H}}$ in Fig. 2(a) [with the distributions
of Re($\beta$) and Im($\beta$) as shown in Figs. 2(a.1) and 2(a.2)], where the
formation of a pair of conjugate EPs is clearly evident. Dotted red and blue
curves show the trajectories of $\beta_{\text{F}}$ and $\beta_{\text{H}}$
concerning a continuous variation of $\gamma$, when we particularly choose
$\tau=3.1607$. Here, we can observe a simultaneous bifurcation and a
coalescence of the associated Re($\beta$) and Im($\beta$) values at
$\gamma=-8.1\times 10^{-3}$, as in Figs. 2(a.1) and 2(a.2), respectively. In
contrary, a simultaneous coalescence and bifurcation of the associated
Re($\beta$) and Im($\beta$) values can be observed at $\gamma=8.1\times
10^{-3}$. Hence, two different circumstances corresponding to $\gamma<0$ and
$\gamma>0$ for a specific $\tau$ refers to perfect complex conjugate
situations (as the parameters $\gamma$ and $\tau$ are associated with
Im[$n(x)$], i.e., gain-loss), which can ideally be observed in two active
variants WG${}_{\text{A}}$ and WG${}_{\text{T}}$. The associated
characteristics of $\beta_{\text{F}}$ and $\beta_{\text{H}}$ refer to the
encounter of two conjugate EPs at ($\pm 8.1\times 10^{-3},3.1607$) (say, an EP
and its conjugate EP* for WG${}_{\text{A}}$ and WG${}_{\text{T}}$,
respectively) in the respective ($\gamma,\tau$)-planes. Topological
dissimilarities in ARC-type interactions between $\beta_{\text{F}}$ and
$\beta_{\text{H}}$ can clearly be observed alongside these conjugate EPs. The
coalescence of the eigenmodes ($\Psi_{\text{F}}$ and $\Psi_{\text{H}}$) at
both the conjugate EPs can be understood from the variation of
$\left\langle\Psi_{\text{F}}|\Psi_{\text{H}}\right\rangle$ with $\gamma$ at a
fixed $\tau=3.1607$, where
$\left\langle\Psi_{\text{F}}|\Psi_{\text{H}}\right\rangle=1$ only at EP and
EP*, as shown in Fig. 2(b).

The effect of parametric encirclement of the embedded conjugate EPs in terms
of chiral branch-point features is investigated in Fig. 2(c). Here, we
consider two parametric loops in the 2D ($\gamma,\tau$)-plane according to the
equations

$\gamma(\varphi)=\gamma_{c}\sin\left(\dfrac{\varphi}{2}\right)\quad\text{and}\quad\tau(\varphi)=\tau_{c}+r\sin(\varphi),$
(3)

which leads to a closed and simultaneous variation of gain and loss around the
EP and EP*. A slow variation of $\varphi\in[0,2\pi]$ governs the stroboscopic
encirclements based on the characteristic parameters $\gamma_{c}$, $\tau_{c}$,
and $r\in(0,1]$, where the conjugate EPs would be inside the parametric loop
only for $|\gamma_{c}|>|\gamma_{\text{EP}}|$ ($\gamma_{\text{EP}}=8.1\times
10^{-3}$; $\gamma$-value at the location of the EP). Here, the variations
$\varphi:0\rightarrow 2\pi$ and $\varphi:0\leftarrow 2\pi$ enable a clockwise
(CW) and a counter-clockwise (CCW) gain-loss variation around the EP for
$\gamma_{c}>0$, and vice-versa around the EP* for $\gamma_{c}<0$. Such two
parametric loops are shown in the ground surfaces of both Figs. 2(c.1) and
2(c.2) (for $r=0.3$, $\gamma_{c}=\pm 1.5\times 10^{-2}$. and
$\tau_{c}=3.1607$), where the associated trajectories of coupled
$\beta_{\text{F}}$ and $\beta_{\text{H}}$ are shown on their respective
Riemann surfaces. Here, we observe that $\beta_{\text{F}}$ and
$\beta_{\text{H}}$ are swapping their identities from their respective
surfaces [concerning both Re($\beta$) and Im($\beta$), as can be seen in Figs.
2(c.1) and 2(c.2), respectively], and exchange their initial positions upon
the completion of encirclement schemes. Such switching between complex
$\beta_{\text{F}}$ and $\beta_{\text{H}}$ around both EP and EP* justify their
branch-point behavior.

###  II.3 Dynamically encircled conjugate EPs: Asymmetric transfer of modes in
a linear medium

Here, we consider the length dependence (analogous to the time dependence) on
the encirclements of the conjugate EPs to study the correlative propagation
dynamics of light (modes) through two $\mathcal{T}$-symmetric waveguide
variants. Figure 3(a) shows the chosen parametric loops for WG${}_{\text{A}}$
and WG${}_{\text{T}}$ [to encircle EP and EP*; exactly the same loops, as can
be seen in the ground surfaces of Fig. 2(c)]. We map the associated gain-loss
distribution along the length ($z$-axis) of respective waveguides. Here, the
reversal of the time axis ($t\rightarrow-t$) under the constraint of
$\mathcal{T}$-symmetry allows us to consider mapping obligatorily in opposite
directions (i.e., $z\rightarrow-z$ as $t\equiv z$) for WG${}_{\text{A}}$ and
WG${}_{\text{T}}$. Hence, we distribute the gain-loss profile from $z=0$ to
$z=l$ based on the encirclement of EP (EP*) governed by $\varphi:0\rightarrow
2\pi$ ($\varphi:2\pi\rightarrow 0$) for WG${}_{\text{A}}$ (WG${}_{\text{T}}$).
Such a $z$-dependent gain-loss distribution can be implemented by
reconsidering (3) as a function of $z$ as

$\gamma(z)=\gamma_{c}\sin\left(\dfrac{\pi
z}{l}\right)\quad\text{and}\quad\tau(z)=\tau_{c}+r\sin\left(\dfrac{2\pi
z}{l}\right)$ (4)

Figure 3(b) shows two complex conjugate 2D Im($n$)-profiles [governed by (4)]
to encircle the EP and EP* dynamically. Here, the CW and CCW directions of
encirclements are realized through one complete pass of light in the forward
direction ($z:0\rightarrow l$) and backward direction ($z:l\rightarrow 0$),
respectively for WG${}_{\text{A}}$, and vice-versa for WG${}_{\text{T}}$.

Now, we implement the scalar beam propagation method to investigate the
individual light transmission through WG${}_{\text{A}}$ and WG${}_{\text{T}}$.
Under the paraxial and adiabatic (which corresponds to a sufficiently slow
variation of gain-loss along $z$-direction) approximation, we implement the
scalar beam propagation equation, i.e.,

$-2ik\partial_{z}\Psi(x,z)=\left[\partial_{x}^{2}+k^{2}\Delta
n^{2}(x,z)\right]\Psi(x,z),$ (5)

associated with both $\Psi_{\text{F}}$ and $\Psi_{\text{H}}$ with extremely
fine split-step computation [with $\Delta n^{2}(x,z)\equiv
n^{2}(x,z)-n_{\text{clad}}^{2}$]. Figure 4 shows the resultant propagation
characteristics, while considering the dynamical encirclements around the EP
and EP* in WG${}_{\text{A}}$ and WG${}_{\text{T}}$, respectively. Here, we
initially verify the linear response (i.e., without any nonlinearity) in the
context of an asymmetric transfer between the modes, which occurs due to the
failure of the adiabatic approximation led by a dynamically encircled EP,
despite the associated omnipresent $\beta$-switching process. Here, the EP (or
EP*) itself acts as a source of chirality, which mainly steers the response of
the underlying system in the context of a direction-dependent transfer of
modes. Such an unconventional modal dynamics can be observed for both
WG${}_{\text{A}}$ and WG${}_{\text{T}}$, as shown in Figs. 4(a) and 4(b),
respectively.

Figure 3: (a) Parametric loops to encircle an EP and its conjugate EP* in the
($\gamma$,$\tau$)-plane [following (3)]. (b) Associated dynamical variation of
gain-loss profiles, i.e., two complex conjugate active potentials (separated
via a transparent plane), experienced by two $\mathcal{T}$-symmetric waveguide
variants WG${}_{\text{A}}$ and WG${}_{\text{T}}$.

To enable a dynamical encirclement of the EP in the CW direction, we consider
the propagation of light from $z=0$ to $z=l$ (forward direction) in
WG${}_{\text{A}}$. We can observe the corresponding dynamics of
$\Psi_{\text{F}}$ and $\Psi_{\text{H}}$ in the upper panel of Fig. 4(a), where
$\Psi_{\text{F}}$ is converted into $\Psi_{\text{H}}$, following the adiabatic
expectation. However, $\Psi_{\text{H}}$ violets the system adiabaticity, i.e.,
it becomes restructured and remains as $\Psi_{\text{H}}$. Thus a light signal
launched at $z=0$ of WG${}_{\text{A}}$ is converted into a dominating
$\Psi_{\text{H}}$ at $z=l$. The lower panel of Fig. 4(a) shows the modal
transitions, while considering light propagation in the backward direction
($z:l\rightarrow 0$; associated with the CCW encirclement process). Here,
$\Psi_{\text{F}}$ dominates at the output $z=0$ with the asymmetric
conversions
$\left\\{\Psi_{\text{F}},\Psi_{\text{H}}\right\\}\rightarrow\Psi_{\text{F}}$,
where only $\Psi_{\text{H}}$ maintains the adiabatic expectations (unlike the
case for the CW encirclement process). Thus during the dynamical encirclement
of an EP, the system partially maintains the adiabaticity, which however turns
into a fascinating chiral light dynamics, where irrespective of the excited
modes at the input, the device delivers two different dominating modes in the
opposite directions.

Such a violation in the system adiabaticity around an EP, can be predicted
with the associated nonadiabatic correction terms
($\mathbb{N}_{\text{F}\rightarrow\text{H}}$ and
$\mathbb{N}_{\text{H}\rightarrow\text{F}}$ for the adiabatic expectations
$\Psi_{\text{F}}\rightarrow\Psi_{\text{H}}$ and
$\Psi_{\text{H}}\rightarrow\Psi_{\text{F}}$, respectively) from the adiabatic
theorem Gilary  _et al._ (2013). These corrections mainly rely on the
accumulated relative-gain ($\Delta\gamma^{\text{ad}}_{\text{F,H}}$) factor
during the transition of modes as (generalized with a quantum-optical analogy
under the operating condition)

$\mathbb{N}_{\text{F\\{H\\}}\rightarrow\text{H\\{F\\}}}\propto-\\{+\\}\exp\displaystyle\int_{0}^{l}\Delta\gamma^{\text{ad}}_{\text{F,H}}(\gamma,\tau)dz.$
(6)

Here, $\Delta\gamma^{\text{ad}}_{\text{F,H}}$ can be estimated from the
relative difference between the average loss ($\gamma^{\text{m}}$) accrued by
the individual modes. The adiabatic trajectories of Im($\beta$)-values [as
shown in Fig. 2(c.2)] for $\Psi_{\text{F}}$ and $\Psi_{\text{H}}$ gives the
associated $\gamma^{\text{m}}$ with
$\oint\\{\text{Im}(\beta)/2\pi\\}d\varphi$.

Figure 4: (a) Propagation dynamics of $\Psi_{\text{F}}$ and $\Psi_{\text{H}}$
through WG${}_{\text{A}}$ (upper panel) from $z=0$ to $z=l$ (associated with
the CW dynamical EP-encirclement) followed by the asymmetric conversions
$\\{\Psi_{\text{F}},\Psi_{\text{H}}\\}\rightarrow\Psi_{\text{H}}$; (lower
panel) from $z=l$ to $z=0$ (associated with the CCW dynamical EP-encirclement)
followed by the asymmetric conversions
$\\{\Psi_{\text{F}},\Psi_{\text{H}}\\}\rightarrow\Psi_{\text{F}}$. (b) Similar
modal dynamics through WG${}_{\text{T}}$ (upper panel) for the CW dynamical
encirclement around the EP* with $z:l\rightarrow 0$, exhibiting the asymmetric
conversions $\\{\Psi_{\text{F}},\Psi_{\text{H}}\\}\rightarrow\Psi_{\text{F}}$;
(lower panel) for the CCW dynamical encirclement around the EP* with
$z:0\rightarrow l$, exhibiting the asymmetric conversions
$\\{\Psi_{\text{F}},\Psi_{\text{H}}\\}\rightarrow\Psi_{\text{H}}$. Intensities
are renormalized at each step of evolution along $z$ to show the inputs and
outputs clearly.

Here, the variant WG${}_{\text{A}}$ operating with a dynamically encircled EP
gives $\Delta\gamma^{\text{ad}}_{\text{F,H}}>0$ for the CW direction, whereas
$\Delta\gamma^{\text{ad}}_{\text{F,H}}<0$ for the CCW direction. These
particular relations result in the domination of the $\mathbb{N}$-factor
associated with the amplifying exponent of
$\Delta\gamma^{\text{ad}}_{\text{F,H}}$ over the overall adiabatic
expectations, whereas cooperation of the $\mathbb{N}$-factor corresponding to
the decaying exponent of $\Delta\gamma^{\text{ad}}_{\text{F,H}}$ with the
adiabatic expectations. Hence, the domination of
$\mathbb{N}_{\text{H}\rightarrow\text{F}}$ in the forward direction yields the
nonadiabatic transition of $\Psi_{\text{H}}(\rightarrow\Psi_{\text{H}})$,
whereas the cooperation of $\mathbb{N}_{\text{F}\rightarrow\text{H}}$ supports
the adiabatic conversion of $\Psi_{\text{F}}(\rightarrow\Psi_{\text{H}})$. On
the other hand, the domination of $\mathbb{N}_{\text{F}\rightarrow\text{H}}$
in the backward direction yields the nonadiabatic transition of
$\Psi_{\text{F}}(\rightarrow\Psi_{\text{F}})$, whereas the cooperation of
$\mathbb{N}_{\text{H}\rightarrow\text{F}}$ supports the adiabatic conversion
of $\Psi_{\text{H}}(\rightarrow\Psi_{\text{F}})$. The detailed analytical
predictions completely support our numerical beam-propagation results for
WG${}_{\text{A}}$, as shown in Fig. 4(a). From the dependence of the relative-
gain factor $\Delta\gamma^{\text{ad}}_{\text{F,H}}$ on the EP-induced
asymmetric mode conversions, one can generically conclude that the mode
transiting with a lower average loss ($\gamma^{\text{m}}$) follows the
adiabatic rules, whereas its coupled counterpart evolves nonadiabatically.

Now, if we consider the dynamical encirclement around EP*, then the concerned
waveguide variant WG${}_{\text{T}}$ exhibits reverse-chiral dynamics compared
to the chiral behavior of WG${}_{\text{A}}$, as can be seen in Fig. 4(b).
During the encirclement in the CW directions, $\Psi_{\text{F}}$ and
$\Psi_{\text{H}}$ transmit along the backward direction ($z:l\rightarrow 0$)
with $\Delta\gamma^{\text{ad}}_{\text{F,H}}<0$, which allows the nonadiabatic
transfer of $\Psi_{\text{F}}$ and the adiabatic transfer of $\Psi_{\text{H}}$
with the asymmetric conversions
$\left\\{\Psi_{\text{F}},\Psi_{\text{H}}\right\\}\rightarrow\Psi_{\text{F}}$
at $z=0$ [as shown in the upper panel of Fig. 4(b)]. In this case,
$\Psi_{\text{H}}$ evolves with a lower $\gamma^{m}$ and maintains the
adiabatic expectations. In contrary, the modal transmissions in the forward
direction ($z:0\rightarrow l$) of WG${}_{\text{T}}$ with a positive relative-
gain factor ($\Delta\gamma^{\text{ad}}_{\text{F,H}}>0$) yields the delivery of
the dominating $\Psi_{\text{F}}$ with the asymmetric conversions
$\left\\{\Psi_{\text{F}},\Psi_{\text{H}}\right\\}\rightarrow\Psi_{\text{H}}$,
while considering the encirclement in the CCW direction [as shown in the lower
panel of Fig. 4(b)]. Here, $\Psi_{\text{F}}$ evolves with a lower $\gamma^{m}$
and maintains the adiabatic expectations. Hence, based on the constraints of
the $\mathcal{T}$-symmetry, we exclusively demonstrate interesting opposite
chiral responses of two active variants designed on the same background
waveguide system, where the opposite encirclement directions around the EP and
EP* result in the delivery of modes of the same order.

###  II.4 Effect of nonlinearity on the asymmetric state-transfer process:
Enabling nonreciprocity around two conjugate EPs

The direction-dependent light transmission process with the asymmetric
transfer of modes in a dual-mode waveguide system (as described for two
variants) can be mimicked by a scattering matrix ($S$-matrix) given by

$[\Psi^{m}_{\text{op}}]_{4\times 1}=[S_{mn}]_{4\times
4}\left[\Psi^{n}_{\text{in}}\right]_{4\times 1}.$ (7)

Equation (7) describes the operation of an analogous four-port device via
$S$-matrix formalism, as shown in Fig. 5. Here the elements of $[S]$ can be
calculated as
$S_{mn}=\left\langle\Psi^{n}_{\text{in}}|\Psi^{m}_{\text{op}}\right\rangle$
with $\\{m,n\\}\in\\{1,2,3,4\\}$. We can safely consider the top-left and
bottom-right blocks of $[S_{mn}]$ as $2\times 2$ null matrices in order to
neglect the possible reflections from the same port of the designed waveguide.
Here, the forward ($T_{\text{F}}$) and backward ($T_{\text{B}}$) transmissions
can be estimated with $T_{\text{F}}=|\max\left(\text{B}_{bl}\right)|^{2}$ and
$T_{\text{F}}=|\max\left(\text{B}_{tr}\right)|^{2}$ (where, $\text{B}_{bl}$
and $\text{B}_{tr}$ represent the bottom-left and top-right blocks,
respectively). Now, it can be understood that if $[S]$ defines the scattering
matrix for WG${}_{\text{A}}$, then the analytical transpose form of $[S]$
would be associated with WG${}_{\text{T}}$ (the numerical values of matrix
elements would indeed be different for WG${}_{\text{A}}$ and WG${}_{\text{T}}$
due to the presence of two opposite gain-loss profile).

Figure 5: A schematic analogy between a 4-port optical device and our designed
dual-mode waveguide operating with a dynamically encircled EP or EP* in the
presence of nonlinearity. This analogy is essentially drawn to construct a
$4\times 4$ $S$-matrix [given by (7)] considering all the possible
transmissions.

In the linear regime, the chirality-drove asymmetric mode conversion process
in a particular waveguide variant follows Lorentz’s reciprocity with a
symmetric $S$-matrix, i.e., $[S]=[S]^{T}$. Now, the direction dependence on
the system’s response can bring up a special interest in achieving one-way
transmission, which is compulsory for designing nonreciprocal devices.
However, the presence of nonreciprocity obligatorily indicates the breakdown
of Lorentz’s reciprocity with an asymmetric $S$-matrix, i.e., $[S]\neq[S]^{T}$
Jalas  _et al._ (2013). In this context, unidirectional transmission with a
symmetric scattering matrix was reported in a photonic circuit Feng  _et al._
(2011), where isolation is not realizable Jalas  _et al._ (2013); Fan  _et
al._ (2012).

In order to break the reciprocity in EP-induced light dynamics, we exploit the
effect of local nonlinearity. We schematically represent our proposed scheme
in Fig. 5 with an operational analogy between one of the designed dual-mode
waveguide variants (hosting a dynamically encircled EP or EP*) with
nonlinearity and a 4-port isolator device. Here, we quantify a particular
nonlinearity level as $N_{l}=(\Delta n_{\text{NL}}/\Delta n)\times 100\%$
(with $\Delta n=0.04$; for the designed passive waveguide), where the
variation of $\Delta n_{\text{NL}}$ depends on the modal field-intensities
($I\equiv|\Psi|^{2}$) for a particular nonlinear coefficient ($n_{2}$). Here,
we initially study the effect of Kerr-type nonlinearities to achieve an
adequate level of nonreciprocity for both waveguide variants (in terms of an
isolation ratio, say, IR) with proper optimization. Then, we also explore the
effect of saturable nonlinearities to enhance the IR further and perform a
quantitative comparison. We consider the forms of two different types of
nonlinearities, viz.,

Kerr-type nonlinearity: $\displaystyle\Delta n_{\text{NL}}(x,z)$
$\displaystyle=n_{2}I,$ (8a) Saturable nonlinearity: $\displaystyle\Delta
n_{\text{NL}}(x,z)$ $\displaystyle=\dfrac{n_{2}I}{(1+I/I_{s})}.$ (8b)

$I_{s}$ in (8b) defines a saturating intensity. Here, the IRs under different
operating conditions are calculated from the forward and backward transmission
coefficients (i.e., $T_{\text{F}}$ and $T_{\text{B}}$) with

$\text{IR}=10\log_{10}\left[\dfrac{\max\left\\{T_{\text{F}},T_{\text{B}}\right\\}}{\min\left\\{T_{\text{F}},T_{\text{B}}\right\\}}\right].$
(9)

The operations of two $\mathcal{T}$-symmetric waveguide variants in terms of
nonlinearity-induced optical isolations are illustrated in Figs. 6 and 7.
Figure 6 shows prototype isolation schemes for both the variants with the one-
way transfer of selective modes at an optimized nonlinearity level ($N_{l}$),
where Fig. 7 illustrates how we optimize such a specific $N_{l}$.

Figure 6: (a) Nonreciprocal transition of modes with the asymmetric
conversions $\\{\Psi_{\text{F}},\Psi_{\text{H}}\\}\rightarrow\Psi_{\text{H}}$
through WG${}_{\text{A}}$ which is active in the forward direction
($z:0\rightarrow l$; associated with the CW dynamical EP encirclement
process). (b) Schematic nonreciprocal response of WG${}_{\text{A}}$ (which
allows light to pass in the forward direction, however, blocks in the backward
directions) along with outputs (O/P) at $z=l$ (for the allowed path
$z:0\rightarrow l$) and $z=0$ (for the blocked path $z:l\rightarrow 0$. (c)
Nonreciprocal transition of modes with the asymmetric conversions
$\\{\Psi_{\text{F}},\Psi_{\text{H}}\\}\rightarrow\Psi_{\text{F}}$ through
WG${}_{\text{T}}$ which is active in the backward direction ($z:l\rightarrow
0$; associated with the CW dynamical encirclement of the EP*). (d) Schematic
nonreciprocal response of WG${}_{\text{T}}$ (which allows light to pass in the
backward direction, however, blocks in the forward directions) along with
outputs (O/P) at $z=0$ (for the allowed path $z:l\rightarrow 0$) and $z=l$
(for the blocked path $z:0\rightarrow l$). For both WG${}_{\text{A}}$ and
WG${}_{\text{T}}$, the self-normalized outputs are shown for their active
directions, whereas relative outputs are shown in their blocked directions.

In Fig. 6(a), we show the one-way propagation of modes through
WG${}_{\text{A}}$ (which hosts a dynamically encircled EP) with Kerr-type
nonlinearity in the spatial index distribution. We judiciously optimize the
nonlinearity level at $N_{l}=6.75\%$. Here, we observe that the waveguide is
active for the encirclement in the CW direction, where both quasiguided modes
are fully transmitted from $z=0$ to $z=l$. Moreover, the adiabatic and
nonadiabatic relations [from (6)] for this specific encirclement condition
allow the asymmetric conversions
$\left\\{\Psi_{\text{F}},\Psi_{\text{H}}\right\\}\rightarrow\Psi_{\text{H}}$,
which results in delivery of the dominating $\Psi_{\text{H}}$ at $z=l$ of
WG${}_{\text{A}}$. Meanwhile, for the consideration of the dynamical EP
encirclement in the CCW direction, we also observe that almost no light is
transmitted from $z=l$ to $z=0$, which is shown in Fig. 6(b) via a relative
intensity difference. Figure 6(b) schematically shows the prototype isolation
scheme achieved using WG${}_{\text{A}}$ along with one of the output (O/P)
field intensities at both $z=l$ and $z=0$ (i.e., for the forward and backward
transmissions, respectively, with the inputs as already shown in Fig. 1(b); as
both modes are converted into a particular dominating mode for propagation in
a specific direction, we obtain almost similar output intensities at a
particular output-end, and hence we show only one of two output field-
intensities for each of the propagation directions). Here, the dotted blue
curve represents the normalized output field intensity ($\Psi_{\text{H}}$) at
$z=l$, while considering the forward propagation ($z:0\rightarrow l$).
However, during the backward propagation ($z:l\rightarrow 0$), the dotted red
curve shows the output field intensity ($\Psi_{\text{H}}$) at $z=0$, which is
relative with respect to the output at $z=l$ obtained during the forward
propagation (the relative output is considered to indicate the intensity
difference while considering the propagation in two opposite directions).
Here, output intensity at $z=0$ decreases almost 98.6% (during the backward
propagation) in comparison to the output at $z=l$ (during the forward
propagation). Two outputs at $z=0$ and $z=l$ perfectly imply the prototype
isolation scheme of WG${}_{\text{A}}$, which passes $\Psi_{\text{H}}$ in the
forward direction and blocks $\Psi_{\text{F}}$ in the backward direction. We
calculate the IR using (9), where a maximum of the IR of 18.6 dB is achieved.

In Fig. 6(c), we investigate a prototype isolation scheme based on
WG${}_{\text{T}}$ in the presence of Kerr-type nonlinearity with
$N_{l}=6.75\%$ (same as considered for WG${}_{\text{A}}$). Here, we observe
that the waveguide is surprisingly active in the backward direction
($z:l\rightarrow 0$) which is associated with the CW dynamical encirclement
scheme around the EP*. The waveguide passes the dominating $\Psi_{\text{F}}$
[based on the corresponding nonadiabatic correction factors from (6)] with the
asymmetric conversions
$\left\\{\Psi_{\text{F}},\Psi_{\text{H}}\right\\}\rightarrow\Psi_{\text{F}}$,
as can be observed via the associated beam propagation results. The light
becomes blocked in the forward direction, which is associated with the CCW
dynamical encirclement process around the EP*. The prototype isolation scheme
along with the output (O/P) field intensities for WG${}_{\text{T}}$ are shown
in Fig. 6(d). From the normalized output intensity at $z=0$
($\Psi_{\text{F}}$; during the backward propagation) and relative output
intensity at $z=l$ ($\Psi_{\text{H}}$; during the forward propagation;
relative with respect to the output $\Psi_{\text{F}}$ at $z=0$), it is clearly
evident that the intensity decreases $\approx 93.3\%$ during the forward
propagation through WG${}_{\text{T}}$. Hence, WG${}_{\text{T}}$ allows
$\Psi_{\text{F}}$ to pass in the backward direction, however, blocks
$\Psi_{\text{H}}$ in the forward direction, where we achieve a maximum of the
IR of 11.75 dB.

Hence, at a particular nonlinearity level, both $\mathcal{T}$-symmetric
waveguide variants behave as isolators, which allow the nonreciprocal
transmission of two different modes in opposite directions. For a particular
variant, a breakdown of the inversion symmetry in the length-depended gain-
loss variation occurs in two opposite directions, where the tailored
nonlinearity induces nonreciprocity. Hence, the intensity of the incoming
waves becomes completely attenuated in a particular direction, despite being
transmitted fully in the opposite direction. Here, a correlation between the
nonreciprocal transmissions to two different allowed modes in two waveguide
variants is dictated by the nonadiabatic corrections around EP and EP*.

Such an exclusive nonreciprocal transmission of selective modes mainly relies
on the interplay between dynamical gain-loss variation (active components) and
the tailored local nonlinearity in the spatial index distribution (passive
components). During the propagation of light around an EP in the presence of
nonlinearity, the complex $\beta$-values of the supported modes become
affected significantly. The EP-induced interactions are led by the variations
of both Re($\beta$) (modal confinement) and Im($\beta$) (decay rates), where
the incorporation of nonlinearity directly influences Re($\beta$). Now, the
mode confinement factors enhance with an increasing amount of nonlinearity,
which results in the simultaneous reduction of the associated decay rates.
Hence, the onset of nonlinearity modifies the gain-loss parameter space
concerning the location of the EP (or EP*), and accordingly the relative-gain
factor [$\Delta\gamma^{\text{ad}}_{\text{F,H}}$; associated with (6)] between
the interacting modes is affected significantly during the evolution of modes
following the dynamical EP-encirclement scheme. Based on such an interplay,
the relative intensity difference at two opposite output-ends varies for
different nonlinearity amounts, which can be understood from the variation of
the IR concerning the nonlinearity level ($N_{l}$), as shown in Fig. 7(a).

The IR initially increases with an increasing Kerr-type nonlinearity level and
takes a maximum value of 18.6 dB for WG${}_{\text{A}}$ and 11.75 dB for
WG${}_{\text{T}}$ at a certain threshold nonlinearity-level of $N_{l}=6.75\%$
[as shown in Fig. 7(a)]. Here, the difference in the IR for two waveguide
variants at a particular $N_{l}$ can be observed, which occurs due to a
different gain-loss profile (exactly opposite; based on
$\mathcal{T}$-symmetry) as can be seen in Fig. 3(b). The operation of
WG${}_{\text{A}}$ is mainly dominated by loss, whereas WG${}_{\text{T}}$
operates with an overall higher amount of gain. Hence, WG${}_{\text{A}}$ is
able to induce a comparably higher output intensity difference for the light
propagation in two opposite directions. An additional gain-amplification in
WG${}_{\text{T}}$ might reduce such intensity difference between two outputs,
which results in achieving a lower IR for WG${}_{\text{T}}$ in comparison to
WG${}_{\text{A}}$ at a particular $N_{l}$. However, we interestingly observe
that both waveguide variants achieve their highest IR at a specific
$N_{l}=6.75\%$, which affirms their operational correlation based on the
chiral behavior of two conjugate EPs. It is further noticeable that while
increasing $N_{l}$ more than 6.75%, the IR decreases gradually [as shown in
Fig. 7(a)] for both variants. Such a decrease of the IR after a certain
threshold is mainly due to the abrupt effect of nonlinearity on the
encirclement loop that affects the location of the EP significantly (i.e., the
EP might come closer to the boundary of the modified loop in the parameter
space due to a higher amount of nonlinearity). Here, judicious care should be
taken to optimize the $N_{l}$, as a higher nonlinearity after a certain limit
may exclude the EP from the parametric loop, for which the overall observation
might be intangible. However, there is a sufficient scope of scalability to
investigate the device operation for different amounts of nonlinearities
within a broad range. The characteristic curve shown in Fig. 7(a) defines the
process to choose an optimized nonlinearity amount, from where we set
$N_{l}=6.75\%$ to obtain the beam propagation results in Fig. 6.

Figure 7: Dependence of the isolation ratio (IR) on the local nonlinearity
level ($N_{l}$), while considering (a) Kerr-type nonlinearity and (b)
saturable nonlinearity, separately. Red square and blue circular markers show
such a variation of IR for WG${}_{\text{A}}$ and WG${}_{\text{T}}$,
respectively. The green arrows in both (a) and (b) indicate the largest values
of the IRs, as achieved at the same $N_{l}$.

Then, instead of local Kerr-type nonlinearity, we introduce saturable
nonlinearity in the spatial index distribution to investigate the
nonreciprocal transmission through WG${}_{\text{A}}$ and WG${}_{\text{T}}$.
The saturable nonlinearity is considered with a chosen saturating intensity
[$I_{s}$; as per (8b)] based on the materials of the background waveguide. For
Kerr-type nonlinearity, a nonlinear interaction of light in the optical medium
gradually increases with an increasing signal intensity, which might ensemble
instability in the output signals after a certain limit. In this context, the
consideration of the saturable intensity in the associated nonlinear
interactions can potentially stabilize the output signals, where we can
observe a higher intensity difference at two output ends for propagation in
the opposite directions. Hence, we optimize the saturable nonlinearity level
at 7.5% from the characteristic dependence of the IR on $N_{l}$, as shown in
Fig. 7(b). Here, we observe an exactly similar nonreciprocal response of both
WG${}_{\text{A}}$ and WG${}_{\text{T}}$, as we have seen for the choice of
Kerr-type nonlinearity in Fig. 6. WG${}_{\text{A}}$ allows the nonreciprocal
transmission of $\Psi_{\text{H}}$ in the forward direction, whereas isolates
$\Psi_{\text{F}}$ in the backward direction. The field intensity decreases
$\approx 99.96\%$ during the backward propagation, where we achieve a maximum
of the IR of 34.6 dB. On the other hand, we achieve a maximum of 18.6 dB IR
for WG${}_{\text{T}}$, which allows $\Psi_{\text{F}}$ to transmit along the
backward direction and isolates $\Psi_{\text{H}}$ in the forward direction
with almost 98.7% reduction of the signal intensity.

##  III Summary

In summary, a significant stride in understanding and utilizing the concept of
conjugate EPs has been made in the context of a correlative nonreciprocal
light transmission process. Besides hosting the dynamical encirclements of two
conjugate EPs in two $\mathcal{T}$-symmetric variants using the framework of a
planar gain-loss assisted waveguide, a comprehensive all-optical platform has
been developed based on the onset of nonlinearity along with the encirclement
scheme to achieve nonreciprocal transmission of selective modes with a
specific chiral correlation. We have established that two
$\mathcal{T}$-symmetric waveguide variants, hosting two conjugate EPs, are
characterized by their ability to behave as isolators enabling nonreciprocal
transmission of selective modes in opposite directions. Here, they allow
active transmission of two different dominant modes in opposite directions,
whereas block light from passing in their respective reverse directions. We
have investigated the effect of both Kerr-type and saturable nonlinearities on
achieving nonreciprocity, where we have observed that the introduction of
saturable nonlinearity can induce a comparably higher nonreciprocal effect. We
have achieved a huge isolation ratio, even up to 34.6 dB under a specific
operating condition. The intricate interplay of the dynamical gain-loss
parameter space around the conjugate EPs in the presence of different types of
nonlinearities has been discussed in detail to understand such unconventional
chiral light dynamics. The insights and implementations of our approach
harnessing the fascinating features of conjugate EPs in nonlinear optical
systems would unlock a new avenue with exciting possibilities for boosting the
development of various nonreciprocal components, such as optical isolators and
circulators, for integrated (on-chip) photonic applications in next-generation
communication networks and quantum information processing.

## acknowledgments

A.L. and A.M. acknowledge the financial support from the Maestro Grant (No.
DEC-2019/34/A/ST2/00081) of the Polish National Science Center (NCN). A.L.
also acknowledges the support from the National Postdoctoral Fellowship Grant
(No. PDF/2021/001322) of the Science and Engineering Research Board (SERB),
India.

## Appendix

The occurrence of conjugate EPs in any physical system can be understood as a
mathematical problem by constructing an analogous $2\times 2$ non-Hermitian
Hamiltonian given by

$\mathcal{H}(\lambda)=\mathcal{H}_{0}+\lambda\mathcal{H}_{p}=\left(\begin{array}[]{cc}\beta_{1}&0\\\
0&\beta_{2}\end{array}\right)+\lambda\left(\begin{array}[]{cc}\kappa_{1}&\gamma_{1}\\\
\gamma_{2}&\kappa_{2}\end{array}\right).$ (A.1)

Here, a passive Hamiltonian $\mathcal{H}_{0}$, consisting of two passive
eigenvalues $\beta_{j}\,(j=1,2)$, is subjected by a perturbation
$\mathcal{H}_{p}$, which is dependent on some coupling parameters $\kappa_{j}$
and $\gamma_{j}\,(j=1,2)$ with a perturbation strength $\lambda$.

A trivial case can be considered with real-valued $\beta_{j}$, $\kappa_{j}$,
and $\lambda$ along with $\gamma_{j}=0$, for which the effective Hamiltonian
$\mathcal{H}$ behaves as a Hermitian system and possesses two distinct
eigenvalues:
$\mathcal{E}_{j}(\lambda)=\beta_{j}+\lambda\,\kappa_{j}\,(j=1,2)$. Here, a
conventional degeneracy occurs at
$\lambda=-(\beta_{1}-\beta_{2})/(\kappa_{1}-\kappa_{2})$. Now, to ensure the
system to be non-Hermitian, all the elements in $\mathcal{H}_{p}$ might be
chosen as non-zero with a complex $\lambda$, where $[H_{0},H_{p}]\neq 0$. The
operation of our designed dual mode waveguide based optical system can be
understood based on such a non-Hermitian Hamiltonian. Here, $\beta_{1}$ and
$\beta_{2}$ represent two real propagation constants. The complex $\lambda$
defines the overall non-Hermitian elements based on gain-loss parameters
$\kappa_{j}$ and $\gamma_{j}\,(j=1,2)$, where $\kappa_{j}$ can be appeared as
individual modal decay rates, whereas $\gamma_{j}$ can be considered as
introduced gain-loss elements.

The eigenvalues of $\mathcal{H}$ can generically be written as,

$\mathcal{E}_{1,2}(\lambda)=\frac{\beta_{1}+\beta_{2}+\lambda\left(\kappa_{1}+\kappa_{2}\right)}{2}\pm
R;$ (A.2)

where,

$\displaystyle R=$
$\displaystyle\left[\left(\frac{\beta_{1}-\beta_{2}}{2}\right)^{2}+\lambda^{2}\left\\{\left(\frac{\kappa_{1}-\kappa_{2}}{2}\right)^{2}+\gamma_{1}\gamma_{2}\right\\}\right.$
$\displaystyle\left.+\frac{\lambda}{2}\left(\beta_{1}-\beta_{2}\right)\left(\kappa_{1}-\kappa_{2}\right)\right]^{1/2}.$
(A.3)

Owing to the coupling invoked by finite $\gamma_{j}\,(j=1,2)$, two levels
$\mathcal{E}_{1}$ and $\mathcal{E}_{2}$ exhibit avoided resonance crossing
(ARC; i.e., two levels do not cross but avoid each other) type interactions
with a continuous variation of $\lambda$. While exhibiting ARCs, the two
levels coalesce at two critical values of $\lambda$, which represent a complex
conjugate pair of EPs in the complex $\lambda$-plane. These two singularities
can be obtained in the complex $\lambda$-plane by setting $R=0$, which are
given by

$\lambda_{s}^{\pm}=-\frac{(\beta_{1}-\beta_{2})}{(\kappa_{1}-\kappa_{2})\mp
2i\sqrt{\gamma_{1}\gamma_{2}}}$ (A.4)

The connection between two conjugate EPs can be understood by rewriting $R$ in
terms of $\lambda_{s}^{+}$ and $\lambda_{s}^{-}$ as

$R=\left[\left(\dfrac{\lambda-\lambda_{s}^{+}}{2}\right)\left\\{\left(\frac{\kappa_{1}-\kappa_{2}}{2}\right)^{2}+\gamma_{1}\gamma_{2}\right\\}\left(\dfrac{\lambda-\lambda_{s}^{-}}{2}\right)\right]^{1/2}.$
(A.5)

Hence, the coupled levels are specified by the value of
$\sqrt{\lambda-\lambda_{s}^{+}}$ and $\sqrt{\lambda-\lambda_{s}^{-}}$ on two
different Riemann surfaces. The critical eigenvalues at two conjugate EPs
(i.e., at $\lambda_{s}^{+}$ and $\lambda_{s}^{-}$) are given by

$\mathcal{E}_{s}(\lambda_{s}^{\pm})=\dfrac{(\kappa_{1}\beta_{2}-\kappa_{2}\beta_{1})\mp
i\sqrt{\gamma_{1}\gamma_{2}}(\beta_{1}+\beta_{2})}{(\kappa_{1}-\kappa_{2})\mp
2i\sqrt{\gamma_{1}\gamma_{2}}}$ (A.6)

Now, an EP is associated with the occurrence of only one independent
eigenvector, unlike two orthogonal eigenvectors at a trivial Hermitian
degeneracy. Thus, using the bi-orthogonal norm for a non-Hermitian
Hamiltonian, two right-hand eigenvectors at two conjugate EPs (one for each of
the EPs) can be written as (approximated up to a factor)

$\displaystyle|\Psi_{s}^{+}\rangle$
$\displaystyle=\left(\begin{array}[]{c}\dfrac{+i\gamma_{1}}{\sqrt{\gamma_{1}\gamma_{2}}}\\\
1\end{array}\right)\qquad\textnormal{for}\quad\lambda=\lambda_{s}^{+},$ (A.7a)
$\displaystyle|\Psi_{s}^{-}\rangle$
$\displaystyle=\left(\begin{array}[]{c}\dfrac{-i\gamma_{1}}{\sqrt{\gamma_{1}\gamma_{2}}}\\\
1\end{array}\right)\qquad\textnormal{for}\quad\lambda=\lambda_{s}^{-};$ (A.7b)

with the associated left-hand eigenvectors

$\displaystyle\langle\widetilde{\Psi}_{s}^{+}|$
$\displaystyle=\left(\begin{array}[]{cc}\dfrac{+i\gamma_{2}}{\sqrt{\gamma_{1}\gamma_{2}}}&1\end{array}\right)\qquad\textnormal{for}\quad\lambda=\lambda_{s}^{+},$
(A.8a) $\displaystyle\langle\widetilde{\Psi}_{s}^{-}|$
$\displaystyle=\left(\begin{array}[]{cc}\dfrac{-i\gamma_{2}}{\sqrt{\gamma_{1}\gamma_{2}}}&1\end{array}\right)\qquad\textnormal{for}\quad\lambda=\lambda_{s}^{-}.$
(A.8b)

From Eqs. (10) and (11), it is evident that

$\langle\widetilde{\Psi}_{s}^{+}|\Psi_{s}^{+}\rangle=0\qquad\textnormal{and}\qquad\langle\widetilde{\Psi}_{s}^{-}|\Psi_{s}^{-}\rangle=0,$
(A.9)

These conditions are referred to as the self-orthogonality that holds at both
the conjugate EPs. The existence of only one self-orthogonal eigenvector
reflects the fact that the Hamiltonian $\mathcal{H}(\lambda)$ becomes non-
diagonalizable for both $\lambda=\lambda_{s}^{+}$ or $\lambda_{s}^{-}$, i.e.,
at the two conjugate EPs.

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