Compiled (140.3490) Lasers, distributed feedback; (060.2420) Fibers,
polarization-maintaining;(060.3735) Fiber Bragg gratings.

# Low-power optical bistability in $\mathcal{PT}$-symmetric chirped Bragg
gratings with four-wave mixing

S. Sudhakar  Department of Electronics and Communication Engineering, Vel Tech
High Tech Dr. Rangarajan Dr. Sakunthala Engineering College, Chennai - 600
062, India  S. Vignesh Raja  Department of Electronics and Communication
Engineering, SRM Institute of Science and Technology, Ramapuram Campus,
Chennai - 600 087, India  A. Govindarajan  Centre for Nonlinear Dynamics,
School of Physics, Bharathidasan University, Tiruchirappalli - 620 024, India
K. Batri  Department of Electronics and Communication Engineering, PSNA
College of Engineering and Technology, Dindigul - 624 622,India  M. Lakshmanan
Centre for Nonlinear Dynamics, School of Physics, Bharathidasan University,
Tiruchirappalli - 620 024, India

###### Abstract

The central theme of this article is the analysis of the usefulness of
introducing four-wave mixing or modulation of Kerr nonlinearity in a
nonuniform grating structure with gain and loss. To do so, we propose an
inhomogeneous system in which the nonlinearity of the $\mathcal{PT}$-symmetric
grating is modulated. It is proven that the proposed scheme can be fruitful
only if the nonlinearity and its associated modulation terms are assumed to be
self-defocusing ones. In order to cut down the intensity required for the
steering, the sign of chirping is taken to be negative and the wavelength of
operating light is considered to be higher than the Bragg wavelength.
Alongside these settings, launching the light from the rear end of the device
dramatically reduces the critical intensity to a value of 0.015
(approximately) which must be the lowest switching intensity ever reported in
the context of nonlinear $\mathcal{PT}$-symmetric gratings. The ramp-like
stable states in the broken $\mathcal{PT}$-symmetric regime prevail even in
the company of modulation in the nonlinearity index for both of the light
incident directions.

###### doi:

http://dx.doi.org/10.1364/XX.XX.XXXXXX

††journal: josab

##  1 Introduction

Fiber Bragg gratings (FBGs) are one of the inevitable building blocks in the
modern-day architecture of an all-optical network [1, 2, 3]. In the pursuit of
realizing an integrable all-optical and low-power FBG switch, researchers have
come up with a lot of innovative design solutions [4, 5, 6, 7]. Among the wide
variety of schemes available, the idea of modulation of Kerr nonlinearity [8,
9, 10, 11, 12] stands out to be one of the important methodologies to produce
low-power bistable optical switches [13, 14, 15, 16]. The reason for the
disappearance of multistable states in these types of systems is that the
cubic nonlinearity (Kerr) is compensated by a high value of nonlinearity
averaged over the structure. The above kind of system was thoroughly
investigated by Pelinovsky _et al._ in the perspective of optical bistability
(OB) under two regimes, namely zero (balanced) and non-zero (non-balanced)
modulation of Kerr nonlinearity regimes [13]. The former requires a nonlinear
(NL) modulation index ($n_{K}$) to be kept at a non-zero value but the
magnitude of the average value of nonlinear RI ($n_{nl}$) is set to zero.
Under this regime, it is possible to achieve true-optical limiting [14]. True
optical limiters are those nonlinear devices which favor uniformly stable
operating regimes for all pertinent incident intensities and this mechanism is
called all-optical limiting [14, 13, 17]. But, the other regime (in which both
these coefficients are kept at non-zero values) is more interesting for the
reason that it brings an additional degree of freedom (in the form of the
parameter $n_{K}$) for the realization of low-power NL applications like
switching, memory and signal processing [13, 16]. Hence, we explore this
regime in this article in the presence of $\mathcal{PT}$-symmetry and chirping
nonuniformity.

Physically, this modulation of Kerr nonlinearity term would have originated
because of the complex four-wave mixing (FWM) process happening inside the
fiber because of the fundamental interactions among different frequency
components [18]. The literature provides a clear scientific explanation or
practical validation in the form of experiments for the origin of the FWM in
physically realizable gratings [19, 20, 21, 22, 23, 24]. Nonlinear processes
inside an optical fiber involving modulation of one or more system parameters
are generally referred as parametric processes and in the present work the
Kerr nonlinearity parameter is modulated. For such a parametric process, to
get a build up in any fiber based device, phase matching is an essential
ingredient. The FWM arises as a result of the nonlinear response of bound
electrons of a material to the incident optical field [25]. The interaction
between these frequency components generate a new frequency component at the
output end. Remarkably, the signal generated by the FWM process also occurs
within the photonic bandgap (PBG) (range of wavelengths in which the
transmission of incident optical light is inhibited [1]) of the grating. As a
result, they can be generated at desired frequencies simply by manipulating
the system and the signal parameters [25, 18].

Presently, the notion of $\mathcal{PT}$-symmetry is reckoned as one of the
research areas to be subjected to an intense investigation by the scientific
community (See Refs. [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36] and
therein). This notion would have remained as an unrealistic subject in real
physical settings forever if not for the fact that the fundamental postulates
of $\mathcal{PT}$-symmetry can be translated from quantum mechanics to optics
[37, 26, 38]. As far as FBGs are concerned, $\mathcal{PT}$-symmetric
structures reveal some atypical light propagation characteristics such as
direction-dependent (linear) [39, 29, 40, 41, 42, 43, 44], nonreciprocal
(nonlinear) [45, 46, 47], and unidirectional [39] wave transport provided that
the mathematical condition $n(z)=n^{*}(-z)$ for the refractive index is
fulfilled by the structure via a precise equilibrium between gain and loss
regions. Also, it is proven that different kinds of nonlinearity including
Kerr modulation could essentially suppress the instability in certain
$\mathcal{PT}$\- symmetric systems with equal amount of gain and loss [48,
49].

In the context of uniform $\mathcal{PT}$-symmetric FBGs (PTFBGs), quite a good
number of nonlinear models have been proposed so far to realize all-optical
switches. To cite a few, time domain and time-independent models of PTFBG with
Kerr nonlinearity [50], a combination of third-order nonlinearity with
dispersive and saturable gain and loss profiles [17, 51, 52], PTFBGs with
modulated Kerr nonlinearity [16], and higher-order nonlinearities [45] are
some of the important works in the literature, as of now. Subsequently, three
of the present authors with Mahalingam came up with a theoretical modeling of
a low power switch involving a nonlinear PTFBG with a chirping nonuniformity
[46].

Chirping is a kind of nonuniformity that is intentionally introduced into a
grating in order to vary the spatial frequency of the device (rather than
keeping it constant as in the case of uniform gratings) when the laser light
is launched into it. As mentioned above, the variations in chirping can either
increase (positively chirped) or decrease (negatively chirped) the spatial
frequency along the propagation direction ($z$) (See Refs. [53, 54, 55, 56,
57] and therein). In this article, we investigate the optical bistability
features exhibited by a chirped PTFBG (CPTFBG) whose nonlinear RI is modulated
over the propagation distance, and from here onwards we denote the
corresponding term by $n_{K}$ or call it as modulated Kerr nonlinearity term.
In the literature, one can find that the authors have considered positive
(self-focusing) and negative values (self-defocusing) for both the cubic
nonlinearity ($\gamma_{nl}$) and modulation of Kerr nonlinearity
($\gamma_{K}$) parameters. Our motivation is to find the best regime among the
different operating domains of the system that leads to a low-power switching
as discussed in Sec. 33.1. Even though the same study (without introducing the
detuning and chirping) has been carried out by Kommisorrova _et al._ [16], the
present work is different from their work due to the following considerations:
The current study includes the chirping nonuniformity. Due to the presence of
chirping, different combinations of chirping ($C$), detuning ($\delta$), Kerr
nonlinearity ($\gamma_{nl}$), and modulation of Kerr nonlinearity
($\gamma_{K}$) parameters can give rise to different kinds of OB curves and
these sign combinations (positive or negative) are likely to have a key role
in dictating the spectral span and power required to switch between the on and
off states of a bistable curve which was not reported in detail in Ref [16].
The dynamics of a nonuniform PTFBG with its nonlinearity term modulated in the
broken regime needs to be depicted in an elaborate fashion and so an
individual section (Sec. 33.2) is dedicated to this study. The presence of
perturbation in the form of modulated Kerr nonlinearity casts doubt upon the
nonreciprocal and low power switching dynamics under the reversal of direction
of light incidence and hence a separate numerical investigation is required
(from this perspective) and the results are shown in Sec. 4.

##  2 Model

Figure 1: Schematic of a nonlinear $\mathcal{PT}$ -symmetric chirped FBG. The
system satisfies $\mathcal{PT}$-symmetric condition in each unit cell
featuring alternate regions of gain and loss as indicated in red and green,
respectively. The nonlinear interaction of the incident field inside the
device is self-defocusing type.  Figure 2: Illustration of a typical OB curve
showing various stable, unstable states, and switching intensities.

Consider a PTFBG of length $L$ whose grating period ($\Lambda$) is chirped
[$\Lambda(z)$] along the direction of propagation of light ($z$) as shown in
Fig. 1. Modern-day fiber optic experiments use high-power laser sources and so
invoking the nonlinearity inside the fiber will not be a tedious task. For our
theoretical investigation, we consider a high-power laser source with input
intensity $P_{0}$ that induces a third-order nonlinearity ($n_{nl}$) inside
the CPTFBG. Unlike highly nonlinear chalcogenide fibers, conventional silica
fibers support only cubic nonlinearity when a continuous wave laser is
launched into it. It is assumed that the layered and distributed feedback
structure accounts for the modulation of nonlinear RI coefficient ($n_{k}$) in
addition to the modulation of the linear RI terms ($n_{1R}$ and $n_{1I}$). As
the input field is launched into the system, the beam expands as it propagates
inside the system due to the self-defocusing effect and the nonlinear
interaction is pictorially depicted in the system model shown in Fig. 1. The
RI distribution of the system that describes the combined effects of RI of the
background material ($n_{0}$), modulation of real ($n_{1R}$) and imaginary
($n_{1I}$) parts of RI, third-order nonlinear coefficient ($n_{nl}$),
modulation of the third-order nonlinear coefficient ($n_{K}$) is given by

$\displaystyle n(z)=n_{0}+n_{1R}cos(2\pi z/\Lambda)+in_{1I}sin(2\pi
z/\Lambda)+n_{nl}|E|^{2}$ $\displaystyle+n_{K}|E|^{2}cos(2\pi z/\Lambda)$ (1)

The optical field ($E(z)$) launched into the CPTFBG can be described in
mathematical terms as the sum of the envelope of forward ($E_{+}$) and
backward propagating fields ($E_{-}$) as

$\displaystyle E(z)=E_{+}\exp(ikz)+E_{-}\exp(-ikz)$ (2)

As we substitute Eq. (2) in Eq. (1), we can readily arrive at the coupled-mode
equations (CMEs) that characterize the wave propagation in the proposed system
provided that a slowly varying envelope approximation is applied during the
mathematical manipulations. The final form of the CMEs can be written as [14,
13, 16]

$\displaystyle\begin{aligned}
\color[rgb]{0,0,1}+&i\left(\frac{dE_{+}}{d\zeta}\right)+\bar{\delta}(\zeta)+(\kappa+g)E_{-}+\gamma_{nl}\left(\left|E_{+}\right|^{2}+2\left|E_{-}\right|^{2}\right)E_{+}\\\
&+\gamma_{K}\left(\left|E_{-}\right|^{2}+2\left|E_{+}\right|^{2}\right)E_{-}+\gamma_{K}E_{+}^{2}E_{-}^{*}=0\\\
-&i\left(\frac{dE_{-}}{d\zeta}\right)+\bar{\delta}(\zeta)+(\kappa-g)E_{+}+\gamma_{nl}\left(\left|E_{-}\right|^{2}+2\left|E_{+}\right|^{2}\right)E_{-}\\\
&+\gamma_{K}\left(\left|E_{+}\right|^{2}+2\left|E_{-}\right|^{2}\right)E_{+}+\gamma_{K}E_{-}^{2}E_{+}^{*}=0\end{aligned}$
(3)

The mathematical notations used in Eq. (3) are as follows: $\zeta$-normalized
propagation coordinate, $\kappa$-coupling coefficient which describes the
amount of power coupled between the transmitted and reflected signals,
$g$-gain and loss term which accounts for the effect of
$\mathcal{PT}$-symmetry, $\gamma_{nl}$-cubic nonlinearity, $\gamma_{K}$\-
modulation of the Kerr nonlinearity which is averaged over the layered
structure, and $\bar{\delta}$\- modified detuning parameter which accounts for
the effect of chirping as well as detuning. The mathematical forms of these
parameters are described below,

$\displaystyle\bar{\delta}(z)=\delta+C(\zeta-L/2)/L^{2},\quad\delta=2\pi
n_{0}\left(\cfrac{1}{\lambda_{0}}-\cfrac{1}{\lambda_{b}}\right)$
$\displaystyle\lambda_{b}=2n_{0}\Lambda,\quad\kappa=\pi
n_{1R}/\lambda_{0},\quad g=\pi n_{1I}/\lambda_{0},$
$\displaystyle\gamma_{nl}=2\pi
n_{nl}/\lambda_{0}\quad\text{and}\quad\gamma_{K}=2\pi n_{K}/\lambda_{0}$ (4)

Here $C$, $\delta$, $\lambda_{b}$, and $\lambda_{0}$ indicate the normalized
chirping, detuning parameter, Bragg and operating wavelengths, respectively.
Note that Eq. (3) holds good when the light incidence direction is left. If
the light incidence direction is right, then the coupled mode equations are
modified as

$\displaystyle\begin{aligned}
+&i\left(\frac{dE_{+}}{d\zeta}\right)+\bar{\delta}(\zeta)+(\kappa-g)E_{-}+\gamma_{nl}\left(\left|E_{+}\right|^{2}+2\left|E_{-}\right|^{2}\right)E_{+}\\\
&+\gamma_{K}\left(\left|E_{-}\right|^{2}+2\left|E_{+}\right|^{2}\right)E_{-}+\gamma_{K}E_{+}^{2}E_{-}^{*}=0\\\
-&i\left(\frac{dE_{-}}{d\zeta}\right)+\bar{\delta}(\zeta)+(\kappa+g)E_{+}+\gamma_{nl}\left(\left|E_{-}\right|^{2}+2\left|E_{+}\right|^{2}\right)E_{-}\\\
&+\gamma_{K}\left(\left|E_{+}\right|^{2}+2\left|E_{-}\right|^{2}\right)E_{+}+\gamma_{K}E_{-}^{2}E_{+}^{*}=0\end{aligned}$
(5)

It is assumed that the normalized length and coupling coefficient are fixed
(throughout the paper) as $L=2$ and $\kappa=4$ (unless specified). Similarly,
the value of gain and loss coefficient is kept at $g=3.75,5,4$ for the
unbroken, broken $\mathcal{PT}$-symmetric regimes, and unitary transmission
point, respectively. The transmitted field emanating at the other end of the
fiber will have a measurable intensity and it is taken to be $P_{1}(L)$. Then,
the OB curves that describe the nonlinear input-output characteristics of the
system can be obtained by plotting the input intensity ($P_{0}$) against the
transmitted intensity($P_{1}(L)$) under the variation of different control
parameters. Implicit Runge-Kutta fourth-order method is one of the efficient
numerical schemes used by many researchers working with nonlinear FBGs and so
we adhere to the same. The boundary conditions which we have chosen are

$\displaystyle E_{+}(0)=E_{0},\text{and}\quad E_{-}(L)=0$ (6)

##  3 Light launching direction: Left

Before we present our results, we would like to recap some of the basic
concepts related to the optical bistability phenomenon. Formally, it is
defined as a nonlinear physical process in which two output states are
generated for a given input state provided that a sufficient feedback is given
to the system. In the absence of any one of these requirements (nonlinearity
and feedback), OB will not occur. As we tune the input intensity of the laser,
the system will behave linearly at first till a threshold power value, say
$P_{up}^{th}$, is reached as shown in Fig. 2. The corresponding linear curve
between $0<P_{0}<P_{up}^{th}$ represents the first stable state (AB). Once the
intensity crosses this threshold value, then it will undergo a sudden jump (to
C) in the magnitude of the output intensity $P_{1}(L)$. This action is
generally referred to as the up-switching phenomenon. Post the occurrence of
this switching action, it remains to behave linearly (again) to any variations
in the input power and this is taken to be the second stable state (CD). Once
again the switching can occur in the perspective of highly nonlinear glasses
like chalcogenide and hence it is named as optical multistability phenomenon
which is not the subject of interest in this article. If one intends to return
to the original state (second stable state to first), then the input intensity
can be slowly reduced. But due to the inherent property of the device, the
down-switching (jump from second to the first stable state) will occur at a
different power level $P_{down}^{th}$ far away from point B, say E. Between
these two stable states, an unstable state (BE) will appear in the numerical
output.

###  3.1 Unbroken $\mathcal{PT}$-symmetric regime

It should be recalled that when the magnitude of the coupling coefficient
($\kappa$) dominates the magnitude of the gain and loss parameter ($g$), then
the system is said to be operating in the unbroken $\mathcal{PT}$-symmetric
regime.

####  3.1.1 Type of Nonlinearity: Self-focusing ($\gamma_{nl},\gamma_{K}>0$)

Figure 3: OB in a CPTFBG operating in the unbroken $\mathcal{PT}$-symmetric
regime with modulation of the Kerr nonlinearity ($\gamma_{K}$) and self-
focusing cubic nonlinearity ($\gamma=1$) for the left light incidence. (a)
Role of $\gamma_{K}$ at $\delta=C=0$. (b) Effect of variations in the chirping
parameter ($C$) at $\delta=0$ and $\gamma_{K}=0.05$. The role of self-focusing
nonlinearity ($\gamma_{nl}>0$) on the switching intensities is depicted in
(c).

Since different combinations of signs among the chirping, detuning, nonlinear
parameter, and nonlinear modulation term are feasible, it is important to find
the regime under which the system gives rise to optical steering at low
powers. To do so, we fix the value of the nonlinearity parameter to be
$\gamma_{nl}=1$ in Fig. 3 which means that the nonlinearity is of self-
focusing type. From Fig. 3(a), one can infer that any increase in the value of
the modulation of Kerr nonlinearity parameter ($\gamma_{K}$) will increase the
power required ($P_{up}^{th}$ and $P_{down}^{th}$) to switch between the
stable states and the width of hysteresis starts to grow with an increase in
the value of $\gamma_{K}$ as shown in Fig. 3(a). This clearly explains that
the combination of self-focusing nonlinearity along with self-focusing
nonlinear modulation term is not the preferable regime for realizing all-
optical switches. In our previous work, we have already demonstrated that any
sign mismatch between the type of nonlinearity and chirping parameter will
result in switching at higher powers [46]. This conclusion holds good in the
presence of modulation of Kerr nonlinearity term ($\gamma_{K}$) also as shown
in Fig. 3(b). In other words, the combination of self-focusing nonlinearity,
modulation of nonlinear parameters, and positive chirping can reduce the
switching power considerably, which will be proven in the next subsection. The
role of nonlinearity parameter on the switching is depicted in Fig. 3(c) and
we infer that with increase in the value of $\gamma_{nl}$ the intensity
required to switch between the states decreases. We would like to note that
the value of the length of the device ($L$) and the coupling coefficient
($\kappa$) should be chosen carefully so that the feedback to the system is
sufficient without which the optical bistability phenomenon cannot occur [45].
Through a series of numerical experiments we found that if the value of $L$ is
too large ($L>5$) and the coupling coefficient ($\kappa$) is comparatively
less than $L$ [$L>>k$], the system produces undesirable bistable states which
indicate the onset of instability. The instability can be managed by suitably
increasing the value of the nonlinear coefficient ($\gamma_{nl}$) and keeping
the value of gain and loss parameter ($g$) closer to $k$. This once again
proves that tuning the value of the nonlinearity and the value of gain and
loss can essentially suppress the instability [48, 49].

####  3.1.2 Type of Nonlinearity: Self-defocusing ($\gamma_{nl},\gamma_{K}<0$)

Figure 4: Plots illustrating the optical bistability dynamics under self-
defocusing nonliear regime ($\gamma=-1$) at $\delta=C=0$. (a) Role of
modulation of the Kerr nonlinearity parameter ($\gamma_{K}$) on switching. (b)
Effect of variations in the cubic nonlinearity ($\gamma_{nl}$) at
$\gamma_{K}=-0.05$ on switching. Illustration of variation in the switching
intensities with respect to (c) the modulation of Kerr nonlinearity
($\gamma_{K}$) and (d) nonlinearity ($\gamma_{nl}$) parameters.

We have concluded from Fig. 3(a) that operating the system with the self-
focusing nonlinear modulation parameter is not an ideal choice to reduce the
switching powers. For this reason, we look into the role of $\gamma_{K}$ under
the assumption that it is of a self-defocusing type ($\gamma_{K}=-1$). As we
tune the value of $\gamma_{K}$, we can observe that the power required to
switch between the stable states decreases as shown in Fig. 4(a). Comparing
Figs. 3(a) and 4(a), we arrive at a conclusion that the inclusion of
modulation of Kerr nonlinearity (FWM) term ($\gamma_{K}$) is beneficial only
if it is assumed to be of self-defocusing type. The thumb rule to realize an
all-optical switch with low switching powers is to use a material with a high
value of nonlinearity in magnitude and not in sign ($\gamma_{nl}$) [45] and
this is confirmed from Fig. 4(b). The negative sign before the nonlinearity
parameter indicates that the type of nonlinearity is self-defocusing. As we
decrease the value of the modulation of the Kerr nonlinearity ($\gamma_{K}$)
parameter, the value of $P_{up}^{th}$ reduces gradually as seen in Fig. 4(c).
One can infer from Fig. 4(d) that the bistable region gets narrowed as the
value of the nonlinear coefficient is increased (in magnitude) which confirms
the fact that more negative value of the nonlinear coefficient, lower will be
the intensity required to switch.

####  3.1.3 Role of chirping ($C$) and detuning ($\delta$) parameters for
$\gamma_{nl},\gamma_{K}<0$

Hereafter, both the Kerr nonlinearity and FWM terms are assumed to be self-
defocusing nonlinear ones (unless specified) throughout the paper, since such
a scheme results in low power steering. In this section, the Kerr and
modulated Kerr nonlinearity (FWM) coefficients are taken to be
$\gamma_{nl}=-1$ and $\gamma_{K}=-0.05$, for simplicity, and the role of
chirping nonuniformity ($C$) is investigated under the influence of the
detuning parameter. Figure 5(a) depicts the role of the negative detuning
parameter ($C<0$) on the bistable switching. Compared to the operation in the
absence of chirping (indicated by dashed-dotted lines), the inclusion of the
negative detuning parameter favors bistable switching at very low powers as
inferred from Fig. 5(a). Since all the laser sources exhibit a fine lasing
linewidth, the discrepancy between the Bragg ($\lambda_{b}$) and
operating($\lambda_{0}$) wavelengths must have a definite impact on the
nonlinear dynamics exhibited by the system, and the parameter which describes
the difference between the two wavelengths is designated as the detuning
parameter ($\delta$). The system is said to be negatively detuned if the
operating wavelength lies above the Bragg wavelength
($\lambda_{0}>\lambda_{b}$) and vice-versa. In the presence of a large value
of negative detuning parameter ($\delta<0$), the system supports low power
all-optical switching as depicted in Fig. 5(b). But it should be remembered
that detuning far away from Bragg wavelength or too much of reduced frequency
of the grating [as a result of the high value of negative detuning parameter
($C<0$)] will also reduce the effective feedback supplied to the system and so
the bistable states cannot occur. Before moving further, we would like to
introduce an important feature exhibited by chirped FBG structure, namely the
spectral span. This parameter refers to the range of operating wavelengths
over which we can observe a typical bistable phenomenon in the input-output
characteristics curve. Since chirping refers to the variation in the spatial
frequency of the grating along the propagation direction, any variation in the
chirping parameter could play a pivotal role in altering the range of
wavelengths over which OB occurs as depicted in Fig. 5(c). Alternatively, one
can tune the spectral range to the desired regime by judiciously manipulating
the detuning parameter as shown in Fig. 5(d).

Figure 5: OB in a CPTFBG operating in the unbroken $\mathcal{PT}$-symmetric
regime under left light incidence direction in the presence of self-defocusing
cubic nonlinearity ($\gamma_{nl}=-1$) and modulation of Kerr nonlinearity
($\gamma_{K}=-0.05$). (a) Role of negative chirping parameter ($C<0$) at
$\delta=0$. (b) Effect of varying the detuning parameter ($\delta$) on the OB
curve at $C=-1$. (c) and (d) portray the effects of continuous variation of
the chirping parameter ($C$) and detuning ($\delta$), respectively, on the
switching intensities ($P_{0}$).

###  3.2 Broken $\mathcal{PT}$-symmetric regime

Figure 6: Plots showing the same dynamics with the same system parameters as
in Fig. 5 except that it is operated in the broken $\mathcal{PT}$-symmetric
regime.

It is evident from Sec. 33.1 that under favorable conditions and in the
presence of sufficient feedback [the product of coupling coefficient
($\kappa$) and length of the grating ($L$)], the system will support low-power
steering provided that the signs of nonlinearity ($\gamma_{n}l$), detuning
($\delta$), chirping ($C$), and modulation of the Kerr nonlinearity
($\gamma_{K}$) terms are negative in the unbroken $\mathcal{PT}$-symmetric
regime. Mathematically, the condition $g>\kappa$ indicates that the PTFBG is
operated in the broken $\mathcal{PT}$-symmetric regime. The high value of
nonlinearity alongside the gain supplied to the system may lead to instability
under some specific values of system parameters [16]. But, if the excitation
source is a high power continuous wave laser and the system parameters are
kept at optimal values, it is possible to obtain stable states in the broken
$\mathcal{PT}$-symmetric regime of the nonlinear uniform [45] and nonuniform
[46] PTFBGs. This proves that a judicious selection of the operating
parameters is an inevitable component to avoid numerical instabilities. Exotic
light dynamics like the existence of ramp-like stable states were discovered
during our studies in the broken $\mathcal{PT}$-symmetric regime [45, 46]. The
question that arises here is that whether the ramp-like stable states can
continue to evolve in the company of perturbations in the form of modulation
of the Kerr nonlinearity ($\gamma_{K}$) and chirping ($C$). To answer the
query, we carry out a numerical experiment with $g=5$ as shown in Fig. 6. In
Fig. 6(a), a sharp increase in the output intensity with respect to the
variations in the input intensity is visible in the plots. Thus, the bistable
curves corresponding to several chirping values are well isolated from each
other by a predictable amount of input intensity. In addition to this feature,
one can also notice that there exists a larger overlap of the first bistable
state and a high magnitude of horizontal separation between the second stable
states corresponding to different spectral components as inferred from Fig.
6(b). It is already well known that these two functionalities offered by the
inhomogeneous grating is designated as spectral uniformity parameter. The
presence of these two significant features indicates that the spectral
uniformity offered by the proposed system is large enough to operate as
optical memory [55, 54, 56].The intensity at which switching happens can be
used as the control signal to switch between the binary states (On and Off) of
a flip-flop [46]. What is more surprising is that the degree of spectral
uniformity is appreciable in the presence of modulation in the nonlinear index
of the grating. From the continuous variation of the curves plotted in
Figs.6(c) and 6(d), we infer that there exists a large range of input powers
common to different spectral components before the steering action takes
place. The power separation between the different components tends to vary
according to the magnitude of chirping and detuning parameters. From these two
inferences, we can justify the fact that the broken $\mathcal{PT}$-symmetry
offers large spectral uniformity [46] and the degree of spectral uniformity
can be uniquely controlled by varying either the chirping or detuning or both.

##  4 Light launching direction: Right

Komissarova _et al._ explored the consequences of operating the nonlinear
PTFBG under right light incidence for the first time [16]. Following this
work, we discovered that it is possible to realize ultra low power all-optical
switches by reversing the direction of light incidence in a nonlinear uniform
and nonuniform PTFBG [45, 46]. We are interested in finding a novel solution
to reduce the switching intensity further. For this reason, we carry out
simulations with the same device parameters as used in Fig. 5 except that the
optical field is now launched into the PTFBG from the rear end.

###  4.1 Unbroken $\mathcal{PT}$-symmetric regime ($\kappa>g$)

The inset in Fig. 7(a) compares the variation in the input intensities
required to switch between the two stable states in the absence and presence
of the modulation in the nonlinear index profile. Surprisingly, the power
required is reduced more than _1.25 times_ (roughly) in the presence of self-
defocusing modulation of the Kerr nonlinearity term ($\gamma_{K}=-0.05$).
Further, by decreasing the spatial frequency ($C=-1$) along the propagation
direction $z$, the switching intensity can be reduced to half of its original
value which is supposed to be a dramatic decrement in the intensity required
for the steering to occur between the stable states as shown in Fig. 7(a). The
theory of local field enhancement for the right light incidence direction in a
grating with gain and loss should be recalled at this juncture to explain the
physical reason behind such a drastic decrease in the input intensities [29,
46]. In simple words, the location of field maxima inside the gain regions of
the PTFBG is favored in the presence of right light incidence, whereas it is
not feasible in a conventional system or PTFBG operating with left light
incidence. So far, the lowest value of nonreciprocal switching intensity
reported in the context of nonlinear PTFBGs is 0.02 [46]. We intend to reduce
the value further by operating the PTFBG in the longer wavelength regime in
the presence of chirping and modulation of the Kerr nonlinearity parameter as
shown in Fig. 7(b). The switch-up intensity keeps on decreasing with any
variation in the negative detuning parameter ($\delta<0$). For a special case
plotted in the inset of Fig. 7(b) with numerical values $C=-1$,
$\gamma_{K}=-0.05$, $\gamma_{nl}$ = -1.5 and $\delta=-0.5$, the switch-up
intensity value is measured to be 0.015 which is less than the lowest
switching intensity reported by us previously [46]. This once again proves
that the quest to devise low power switches with the aid of newly engineered
FBG is ever-expanding. As an end note to this particular section, we would
like to stress that with a careful selection of other design parameters and
the operating regime, the nonuniformities as well the perturbations in the
grating can be turned into constructive ones (rather than being detrimental)
in the context of realization of low power switches. It should be noted that
the variation in the spectral span is effected as a consequence of the
variation in the chirping and detuning parameters as shown in Figs. 7(c) and
7(d) under the condition that the light is launched from the rear end. This
proves that the span over which the OB curves occur is independent of the
light launching direction and it is dependent on chirping nonuniformity and
the signal detuning parameter.

Figure 7: (a) – (d) Plots showing the same dynamics with the same system
parameters as in Fig. 5 except that the light launching direction is right.
Inset in (a) shows the decrease in the switching intensity in the presence of
self-defocusing modulation of Kerr nonlinearity parameter ($\gamma_{K}$).
Inset in (b) depicts the novel switching dynamics occurring at a very low
power ($P_{0}\leavevmode\nobreak\ 0.015$)

###  4.2 Broken $\mathcal{PT}$-symmetric regime ($g>\kappa$)

Figure 8: Plots showing the same dynamics with the same system parameters as
in Fig. 7 except that the light launching direction is right.

In Sec. 33.2, we disclosed the net effect of variation in the modulation of
Kerr nonlinearity ($\gamma_{K}$), chirping ($C$), and detuning ($\delta$)
parameters on the ramp-like bistable states. Encouraged by this, we intend to
study the existence of ramp-like stable states at lower powers under the
adjustment in the light launching direction. The common features between Figs.
6(a) and 8 are as follows: First, there exists a range of input intensities
which is common to all the spectral components (before switching occurs).
Second, the onset of switching is preceded by the occurrence of a ramp-like
first stable state. Post the occurrence of switching, the second stable are
well separated by significant amount of output intensities in Fig. 8(a). This
proves that it is possible to achieve large spectral uniformity even at lower
powers, thanks to the concept of nonreciprocal switching. Figure 8(b)
delineates the shift in the ramp-like bistable curves to the lower intensity
side while operating in the longer wavelength regime. Operating the system in
the broken $\mathcal{PT}$-symmetric regime under right light direction helps
in altering the feature of spectral uniformity as shown in Figs. 8(c) and 8(d)
similar to the conclusions drawn from Figs. 6(c) and 6(d)

##  5 Conclusions

Introducing modulation in the nonlinear refractive index in the form of the
parameter $\gamma_{K}$ has proven to be a beneficial degree of freedom to
realize low optical switches provided that it is of self-defocusing type
alongside the self-defocusing cubic nonlinearity. The presence of gain and
loss brings in nonreciprocal switching, low power steering dynamics, bistable
curves with larger spectral nonuniformity. The sign of chirping and detuning
plays a pivotal role in controlling the intensities at which steering occurs
and the spectral span in which the OB phenomenon occurs. In the presence of
high nonlinearity and modulation of the nonlinear index, manipulations in the
nonuniformities like operation in the negative detuning regime accompanied by
a simultaneous decrement in the spatial frequency of the grating resulted in
ultra-low power steering. The value of the critical intensity under these
conditions is measured to be 0.015 (approximately) when the incident optical
field is impelled into the CPTFBG from the rear end. We hope that experimental
researchers can bring the above theoretical results into reality in the near
future.

## Acknowledgments

AG and ML acknowledge the support by DST-SERB for providing a Distinguished
Fellowship (Grant No. SB/DF/04/2017) to ML in which AG was a Visiting
Scientist. AG is now supported by the University Grants Commission (UGC),
Government of India, through a Dr. D. S. Kothari Postdoctoral Fellowship
(Grant No. F.4-2/2006 (BSR)/PH/19-20/0025).

## Disclosures

The authors declare no conflicts of interest.

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