# Noise and Pulse Dynamics in Backward Stimulated Brillouin Scattering

Oscar A. Nieves1  Matthew D. Arnold 1  M. J. Steel2  Mikołaj K. Schmidt2 and
Christopher G. Poulton1  1 School of Mathematical and Physical Sciences,
University of Technology Sydney, 15 Broadway, Ultimo NSW 2007, Australia.
2 Macquarie University Research Centre for Quantum Engineering (MQCEQ), MQ
Photonics Research Centre, Department of Physics and Astronomy, Macquarie
University, NSW 2109, Australia<EMAIL_ADDRESS>
###### Abstract

We theoretically and numerically study the effects of thermal noise on pulses
in backwards Stimulated Brillouin Scattering (SBS). Using a combination of
stochastic calculus and numerical methods, we derive a theoretical model that
can be used to quantitatively predict noise measurements. We study how the
optical pulse configuration, including the input powers of the pump and Stokes
fields, pulse durations and interaction time, affects the noise in the output
Stokes field. We investigate the effects on the noise of the optical loss and
waveguide length, and we find that the signal-to-noise ratio can be
significantly improved, or reduced, for specific combinations of waveguide
properties and pulse parameters.

††journal: oe††articletype: Research Article

##  1 Introduction

Backward Stimulated Brillouin Scattering (SBS) is an opto-acoustic process
that results from the coherent interaction between two counter-propagating
optical fields and an acoustic wave inside an optical waveguide [1, 2, 3, 4,
5]. The coupling between these fields occurs either through electrostriction
or radiation pressure [6, 7], in which the electromagnetic fields induce very
small strains in the material, which in turn stimulate the generation of
coherent phonons via photoelasticity or via physical motion of the waveguide
boundaries [6, 7, 8]. This interaction has been used in various applications,
including narrow-band RF and optical signal filtering [9, 10], phase
conjugation and precision spectroscopy [1], novel laser sources [11, 12], and
in recent experiments in opto-acoustic pulse storage and memory [13]. An
intrinsic property of SBS is noise: the interaction between the pump and the
thermal phonon background of the waveguide generates spontaneous Stokes
photons, which combine with the input signal to produce a noisy output signal.
SBS noise was first studied by Boyd and Gaeta [14, 15] and Ferreira et al.
[16], focusing on optical fibers. These studies assumed that the optical and
acoustic fields are all weakly guided, and furthermore examined the effects of
noise in the quasi-continuous wave regime. Later work [17] extended the
distributed fluctuating source model from Boyd and Gaeta to include the effect
of optical loss, with the addition of an acousto-optic effective area using
scalar modal fields, and was again restricted to quasi-CW pulses and weak
guidance. More recent studies have analyzed the effects of thermal noise in
the context of SBS lasers and micro-ring resonators [18, 19, 20, 21]. A
dynamic, fully-vectorial treatment that can be applied to predict noise in
modern integrated SBS waveguide experiments [22, 2] is important for the
further development of SBS signal-processing applications.

In this paper, we derive a theoretical model for SBS noise for pulses in
integrated optical waveguides with high-contrast materials, using fully
vectorial modal fields. We focus on amplitude noise arising from the thermal
phonon field, because in backwards SBS experiments this process has been
observed to be significantly stronger than phase noise [23, 24]. We explore
the dependence of the noise on the waveguide and pulse properties. We find
that in the regime where the pump is undepleted by the SBS process, it is
possible to derive analytic results for the absolute level of noise, as well
as for the signal-to-noise ratio, even for time-varying pulses of arbitrary
shape. Since numerical computation of ensemble averages are expensive, an
analytic model that computes the SBS noise properties of pulses is extremely
useful for the design of SBS-active devices. We use this analytic model to
explore the impact of pulse and waveguide properties on the noise and optical
signal-to-noise ratio (OSNR), and discuss the implications of these results in
SBS experiments, where it is often advantageous to minimize the Stokes noise.

##  2 Theory and formalism

We consider backward SBS interactions in a waveguide of finite length $L$
oriented along the $z$-axis (see Fig. 1): a pump pulse with angular frequency
$\omega_{1}$ is injected into the waveguide at $z=0$ and propagates in the
positive $z$-direction, while a signal pulse is injected at $z=L$ and
propagates in the negative $z$-direction. The signal pulse has frequency
centred around the Brillouin Stokes frequency $\omega_{2}=\omega_{1}-\Omega$,
which is down-shifted from the pump by the Brillouin shift $\Omega$, and has a
spectral width that lies entirely within the Brillouin linewidth. When these
two pulses meet, the pump transfers energy to the signal field, resulting in
coherent amplification of the signal around the Brillouin frequency. At the
same time, as the pump moves through the waveguide, it interacts with the
thermal phonon field and generates spontaneous Stokes photons which also
propagate in the negative $z$-direction. This spontaneous Stokes field
combines with the coherent signal to form a noisy amplified output field
centered around the Stokes frequency.

Figure 1: Propagation of two Gaussian pulses along a waveguide consisting of
substrate (yellow) and core (light blue) dielectrics. $L$ denotes the total
length of the waveguide medium. The input fields in (a) overlap in (b),
resulting in an exchange of energy between the pump and Stokes through the
acoustic field, whereby the pump’s energy is depleted by the Stokes. The
spontaneous Stokes noise generated by the pump interacting with the thermal
phonon field combines with the coherent signal in (b), leading to a noisy
output field at the Stokes frequency in (c).

To compute the noise transferred to the signal from the thermal field, we
first solve the coupled equations of the pump $\mathbf{E}_{1}$, Stokes
$\mathbf{E}_{2}$ and acoustic $\mathbf{U}$ fields. We write these fields using
the ansatz

$\displaystyle\mathbf{E}_{1}$
$\displaystyle=\tilde{a}_{1}(z,t)\,\mathbf{e}_{1}(x,y)\,\mathrm{e}^{i(k_{1}z-\omega_{1}t)}+\text{c.c.}$
(1) $\displaystyle\mathbf{E}_{2}$
$\displaystyle=\tilde{a}_{2}(z,t)\,\mathbf{e}_{2}(x,y)\,\mathrm{e}^{i(k_{2}z-\omega_{2}t)}+\text{c.c.}$
(2) $\displaystyle\mathbf{U}$
$\displaystyle=\tilde{b}(z,t)\,\mathbf{u}(x,y)\,\mathrm{e}^{i(qz-\Omega
t)}+\text{c.c.},$ (3)

where $\tilde{a}_{1}$, $\tilde{a}_{2}$ and $\tilde{b}$ are dimensionless
envelope functions, $\mathbf{e}_{1}$ and $\mathbf{e}_{2}$ are the
electromagnetic mode fields of the waveguide (with units of V/m), $\mathbf{u}$
is the waveguide displacement mode field (with units of m), and c.c. denotes
the complex conjugate. The parameters $k_{1}>0$ and $k_{2}<0$ represent the
pump and Stokes wavenumbers respectively. The total electric field is
$\mathbf{E}=\mathbf{E}_{1}+\mathbf{E}_{2}$. For the acoustic field, $\Omega$
is the phonon angular frequency and $\Omega$ is the acoustic wavenumber. Using
this ansatz along with the phase-matching conditions $k_{1}=k_{2}+q$ and
$\omega_{1}=\omega_{2}+\Omega$, the governing equations for the envelope
fields are [7, 8]

$\displaystyle\frac{\partial a_{1}}{\partial z}+\frac{1}{v}\frac{\partial
a_{1}}{\partial t}+\frac{1}{2}\alpha a_{1}$
$\displaystyle=i\omega_{1}Q_{1}a_{2}b^{*},$ (4) $\displaystyle\frac{\partial
a_{2}}{\partial z}-\frac{1}{v}\frac{\partial a_{2}}{\partial
t}-\frac{1}{2}\alpha a_{2}$ $\displaystyle=i\omega_{2}Q_{2}a_{1}b,$ (5)
$\displaystyle\frac{\partial b}{\partial z}+\frac{1}{v_{a}}\frac{\partial
b}{\partial t}+\frac{1}{2}\alpha_{\text{ac}}b$ $\displaystyle=i\Omega
Q_{a}a_{1}^{*}a_{2}+\sqrt{\sigma}R(z,t).$ (6)

In these equations, the new envelope functions $a_{1}$, $a_{2}$ and $b$ are
related to the dimensionless envelope functions via the modal powers
$\mathcal{P}_{j}$ via $a_{1,2}=\sqrt{\mathcal{P}_{1,2}}\tilde{a}_{1,2}$ and
$b=\sqrt{\mathcal{P}_{a}}\tilde{b}$, and related to the measured optical and
acoustic powers $P_{1,2}$ and $P_{a}$ (in Watts) via $|a_{1,2}|^{2}=P_{1,2}$
and $|b|^{2}=P_{a}$. The boundary conditions for the pump and signal fields
are applied by specifying the input values $a_{1}(0,t)$ and $a_{2}(L,t)$
respectively. These boundary conditions can be deterministic, as is the case
in our calculations, or stochastic if laser noise is incorporated into the
model. The analytic results presented in this paper are valid for both cases.

The waveguide has optical loss $\alpha$ and acoustic attenuation
$\alpha_{\text{ac}}$ (in units of m-1), with optical group velocity $v$ and
acoustic group velocity $v_{a}$. The coefficients $Q$ represent the coupling
strength of the SBS interaction, and can be calculated from the optical and
acoustic modes of the waveguide [25]. From conservation of energy, we have
$Q_{2}=Q_{1}^{*}$ and $Q_{a}=Q_{1}$ [7].

The function $R(z,t)$ represents a complex-valued space-time white noise with
zero mean $\left\langle R(z,t)\right\rangle=0$ and auto-correlation function
$\left\langle
R(z,t)R^{*}(z^{\prime},t^{\prime})\right\rangle=\delta(z-z^{\prime})\delta(t-t^{\prime})$,
where $\delta$ is the Dirac-delta function. This type of delta correlation
arises in the standard treatment of noise in optical fibers [26], and arises
from the effective continuum of phonon modes supported by the waveguide and
coupled through dissipation to the coherent excitation of the elastic field.
The strength of the noise $\sigma$ gives the conversion of coherent elastic
energy in the medium into thermal phonons, in the absence of any nonlinear
optical effects [14], and is given by (see Appendix A)

$\sigma=k_{\text{B}}T\alpha_{\text{ac}},$ (7)

where $k_{B}T$ is the thermal energy per phonon mode. This quantity is
directly proportional to the acoustic loss in the medium $\alpha_{\text{ac}}$
[27] in accordance with the fluctuation-dissipation theorem [28].

###  2.1 Noise under the undepleted pump approximation

An analytic solution to equations (4)$-$(6) exists under the undepleted pump
approximation, which is valid for the case in which the signal power $P_{2}$
remains much smaller than the pump power $P_{1}$. We also assume that the
depletion of the pump by the thermal field is negligible compared to optical
loss [16]. The equation for the pump envelope is therefore

$\frac{\partial a_{1}}{\partial z}+\frac{1}{v}\frac{\partial a_{1}}{\partial
t}+\frac{\alpha}{2}a_{1}=0,$ (8)

which, for an input field $a_{p}(t)=\sqrt{P_{\text{pump}}^{\text{in}}(t)}$ at
$z=0$ has the solution

$a_{1}(z,t)=a_{p}\left(t-\frac{z}{v}\right)\mathrm{e}^{-\alpha z/2},$ (9)

by the method of characteristics. Next, we simplify (6) by letting the term
$\partial_{z}b\rightarrow 0$, because for experimentally relevant time scales
the propagation speed of the acoustic pulses is much smaller than for optical
pulses [4]. This leads to the equation for the acoustic field

$\frac{1}{v_{a}}\frac{\partial b}{\partial
t}+\frac{1}{2}\alpha_{\text{ac}}b=i\Omega
Q_{a}a_{1}^{*}a_{2}+\sqrt{\sigma}R(z,t),$ (10)

which can be integrated in time formally as

$b(z,t)=iv_{a}\Omega
Q_{a}\int_{-\infty}^{t}\mathrm{e}^{\frac{1}{2}\Gamma(s-t)}a_{1}^{*}(z,s)a_{2}(z,s)\,\text{d}s+D(z,t),$
(11)

where $\Gamma=v_{a}\alpha_{\text{ac}}$ is the decay rate of the acoustic
phonons, which is related to the Brillouin linewidth $\Delta\nu_{B}$ by
$\Gamma=\Delta\nu_{B}/2\pi$, and the time-integrated noise function $D(z,t)$
is defined as

$D(z,t)=v_{a}\sqrt{\sigma}\int_{-\infty}^{t}\mathrm{e}^{\frac{1}{2}\Gamma(s-t)}R(z,s)\,\text{d}s.$
(12)

According to the properties of normally distributed random processes, the
integral of a normal random process is also a normal random process [29, 30].
This means that $D(z,t)$ has zero mean $\left\langle D(z,t)\right\rangle=0$
and auto-correlation function

$\left\langle
D(z,t)D^{*}(z^{\prime},t^{\prime})\right\rangle=\frac{v_{a}\sigma}{\alpha_{\text{ac}}}\delta(z-z^{\prime})\exp\left\\{-\frac{1}{2}\Gamma\left|t-t^{\prime}\right|\right\\}.$
(13)

At this stage, we can use a separation of time-scales in the integral in (11),
since $1/\Gamma$ is large compared to the characteristic timescales of the
optical envelope fields, and this allows us to use the Markov approximation
$a_{1}^{*}(z,s)a_{2}(z,s)\approx
a_{1}^{*}(z,t)a_{2}(z,t)\mathrm{e}^{-i\Delta(s-t)}$ [21], where $\Delta$ is a
detuning parameter relative to the center of the Brillouin gain profile
written as $\Delta=\omega_{2}-\omega_{1}+\Omega$. Next, pulling
$a_{1}^{*}(z,t)a_{2}(z,t)$ outside of the first integral in (11) yields the
approximate expression for the acoustic envelope field

$b(z,t)\approx i\frac{v_{a}\Omega
Q_{a}}{\Gamma/2-i\Delta}a_{1}^{*}a_{2}+D(z,t).$ (14)

Now we can eliminate the elastic field by substituting (14) into (5), which
yields

$\frac{\partial a_{2}}{\partial z}-\frac{1}{v}\frac{\partial a_{2}}{\partial
t}+\frac{1}{2}\left[g(\Delta)P_{1}(z,t)-\alpha\right]a_{2}=i\omega_{2}Q_{2}a_{1}(z,t)D(z,t),$
(15)

where $g(\Delta)$ is the usual SBS gain parameter in (m-1W-1)

$g(\Delta)=\frac{2v_{a}\omega_{2}\Omega|Q_{2}|^{2}}{\Gamma/2-i\Delta}=g_{0}\frac{\Gamma/2}{\Gamma/2-i\Delta},$
(16)

with $g_{0}=g(0)$. Equation (15) is then solved by making a variable
transformation to the frame of reference that is co-propagating with the
Stokes field [31], as shown in Appendix B. This leads to the solution of the
Stokes envelope field

$a_{2}(z,t)=a_{2}(L,t)G_{N}(z,L,t)\\\
-i\omega_{2}Q_{2}\int_{z}^{L}G_{N}(z,z^{\prime},t)a_{1}\left(z^{\prime},\
t+\frac{z-z^{\prime}}{v}\right)D\left(z^{\prime},\
t+\frac{z-z^{\prime}}{v}\right)\text{d}z^{\prime},$ (17)

where the net gain function is expressed as

$G_{N}(z,z^{\prime},t)=\exp\left\\{\frac{1}{2}\int_{z}^{z^{\prime}}\left[g(\Delta)P_{1}\left(\eta,\
t+\frac{z-\eta}{v}\right)-\alpha\right]\text{d}\eta\right\\},$ (18)

for $z<z^{\prime}$. The modulus squared $|G_{N}(z,z^{\prime},t)|^{2}$ of this
function gives the net cumulative gain in the Stokes power between $z$ and
$z^{\prime}$, taking into account the effect of optical losses. As the
distance between $z$ and $z^{\prime}$ increases, both the amplification from
the pump and the optical loss increase. However, for any time $t$ in which the
pump pulse $P_{1}(z,t)$ lies outside the domain $[z,z^{\prime}]$, optical loss
dominates over the SBS amplification.

We compute the total, noisy, Stokes power by considering the Stokes envelope
as a sum of two contributions: a coherent signal field, which is the Stokes in
the absence of any thermal noise, and a spontaneous Stokes field which
contains all the contributions from the background thermal noise. This is
expressed by separating the two terms in Eq. (17) as
$a_{2}(z,t)=a_{2}^{\text{sig}}(z,t)+a_{2}^{\text{spo}}(z,t)$ where

$a_{2}^{\text{sig}}(z,t)=a_{2}(L,t)G_{N}(z,L,t),$ (19)
$a_{2}^{\text{spo}}(z,t)=-i\omega_{2}Q_{2}\int_{z}^{L}G_{N}(z,z^{\prime},t)a_{1}\left(z^{\prime},\
t+\frac{z-z^{\prime}}{v}\right)D\left(z^{\prime},\
t+\frac{z-z^{\prime}}{v}\right)\text{d}z^{\prime}.$ (20)

The ensemble-average of the total Stokes power is then

$\left\langle P_{2}(z,t)\right\rangle=\left\langle
P_{2}^{\text{sig}}(z,t)\right\rangle+\left\langle
P_{2}^{\text{spo}}(z,t)\right\rangle=\left\langle\left|a_{2}^{\text{sig}}(z,t)\right|^{2}\right\rangle+\left\langle\left|a_{2}^{\text{spo}}(z,t)\right|^{2}\right\rangle,$
(21)

where the relation $\left\langle
a_{2}^{\text{sig}}a_{2}^{\text{spo}}\right\rangle=0$ holds since the input
laser noise in $a_{2}^{\text{sig}}$ and thermal noise in $a_{2}^{\text{spo}}$
are statistically independent from each other. The average signal power is
computed by taking the ensemble average of the input field defined by the
noisy boundary condition $a_{2}(L,t)$

$\left\langle
P_{2}^{\text{sig}}(z,t)\right\rangle=\left\langle\left|a_{2}(L,t)\right|^{2}\left|G_{N}(z,L,t)\right|^{2}\right\rangle.$
(22)

Here $G_{N}$ is taken inside the ensemble average because it is dependent on
the pump $P_{1}$, which may contain input laser noise. The spontaneous Stokes
power is

$P_{2}^{\text{spo}}(z,t)=\frac{\alpha_{\text{ac}}}{4}\frac{\omega_{2}}{\Omega}g_{0}\int_{z}^{L}G_{N}(z,z^{\prime},t)\,a_{1}\left(z^{\prime},\
t+\frac{z-z^{\prime}}{v}\right)D\left(z^{\prime},\
t+\frac{z-z^{\prime}}{v}\right)\text{d}z^{\prime}\\\
\times\int_{z}^{L}G_{N}^{*}(z,z^{\prime\prime},t)\,a_{1}^{*}\left(z^{\prime\prime},\
t+\frac{z-z^{\prime\prime}}{v}\right)D^{*}\left(z^{\prime\prime},\
t+\frac{z-z^{\prime\prime}}{v}\right)\text{d}z^{\prime\prime}.$ (23)

This product of integrals can be cast as a double-integral by applying the
stochastic Fubini theorem [32]. A sufficient condition for this to hold is
that the integrand in (20) be square-integrable and finite [33]. This
condition holds since $D(z,t)$ reaches a steady-state after a short time $t$
[34] and both $a_{1}$ and $G_{N}$ are bounded functions [30]. The spontaneous
power becomes

$P_{2}^{\text{spo}}(z,t)=\frac{\alpha_{\text{ac}}}{4}\frac{\omega_{2}}{\Omega}g_{0}\int_{z}^{L}\int_{z}^{L}G_{N}(z,z^{\prime},t)G_{N}^{*}(z,z^{\prime\prime},t)\times\\\
a_{1}\left(z^{\prime},\
t+\frac{z-z^{\prime}}{v}\right)a_{1}^{*}\left(z^{\prime\prime},\
t+\frac{z-z^{\prime\prime}}{v}\right)D\left(z^{\prime},\
t+\frac{z-z^{\prime}}{v}\right)D^{*}\left(z^{\prime\prime},\
t+\frac{z-z^{\prime\prime}}{v}\right)\text{d}z^{\prime\prime}\text{d}z^{\prime}.$
(24)

The relation $\left\langle a_{1}(z,t)D(z,t)\right\rangle=\langle
a_{1}(z,t)\rangle\langle D(z,t)\rangle=0$ holds since the laser noise in
$a_{1}$ and the thermal noise in $D$ are not coupled to each other, so they
are statistically independent. Therefore, the ensemble-averaged spontaneous
Stokes is

$\left\langle
P_{2}^{\text{spo}}(z,t)\right\rangle=\frac{k_{B}T\Gamma}{4}\frac{\omega_{2}}{\Omega}g_{0}\int_{z}^{L}\left\langle
P_{1}\left(z^{\prime},\
t+\frac{z-z^{\prime}}{v}\right)\left|G_{N}(z,z^{\prime},t)\right|^{2}\right\rangle\text{d}z^{\prime}.$
(25)

Equation (25) is one of the main results in this paper. It gives the ensemble-
averaged spontaneous Stokes power, which is proportional to the pump power
$P_{1}(z,t)$ multiplied by the net gain over the length of the waveguide. As
the temperature increases, more spontaneous phonons are generated, leading to
more collisions of phonons with pump photons. The annihilation of these
photons then creates spontaneous Stokes photons, which amplifies the Stokes
noise in the medium. The integration from $z$ to $L$ indicates that as the
pump propagates through the medium, there is an accumulation of spontaneous
Stokes photons propagating in the opposite direction.

Once this spontaneous power has been calculated, a useful measure to quantify
the integrity of the Stokes signal detected at the waveguide output is the
OSNR. This is computed by taking the ratio of the average coherent signal
power to spontaneous noise power over a specified bandwidth [35]. In time-
domain, the OSNR may be calculated over an arbitrary but suitably long time
period $T_{0}$ using the time averaged powers

$\text{OSNR}=\frac{\int_{0}^{T_{0}}\left\langle
P_{2}^{\text{sig}}(0,t)\right\rangle\text{d}t}{\int_{0}^{T_{0}}\left\langle
P_{2}^{\text{spo}}(0,t)\right\rangle\text{d}t}.$ (26)

In the following calculations, using a Gaussian pulse we choose $T_{0}$ as
four times the intensity full width at half maximum (FWHM).

##  3 Results for a chalcogenide SBS chip

Figure 2 shows the ensemble-averaged signal power and spontaneous Stokes power
computed using Eqs (22) and (25). These calculations assume a dielectric
waveguide at temperature 300 K of length 23.7 cm and SBS gain coefficient
$g_{0}=423$ m-1W-1, which corresponds to the SBS chip structure used in [36]
at zero detuning ($\Delta=0$). We consider a Gaussian pump pulse with peak
power 110 mW and FWHM of 200 ps, and a Gaussian signal pulse with peak power
100 nW and FWHM of 200 ps. The optical pump frequency is fixed at 200 THz,
while the acoustic frequency is set to 7.6 GHz, with a Brillouin linewidth of
$\Omega/2\pi=30$ MHz. The linear optical loss is 0.05 dB/cm. Equations (4)–(6)
are solved numerically with a split-step integration scheme that calculates
the drift of the pulses first, and then uses an Euler-Mayurama step for
computing the noisy nonlinear interaction between the pump and Stokes.

Figure 2: Numerical simulation of the Stokes noise inside a 23.7 cm long
waveguide. (a) Temporal evolution of the signal and spontaneous Stokes powers
across the waveguide for a single run of the numerical method (red). The
dashed line represents the ensemble-averaged expression for the total Stokes
power being the sum of Eqs (22) and (25). (b) Several independent runs of the
Stokes field (gray), with the numerical ensemble average (blue) and the
analytic ensemble average (black), at $t=1.8$ ns.

For these simulations, the numerical grid consists of 600 points in $z$ and
900 points in $t$. These values were chosen, using convergence tests in the
absence of noise, to ensure an error in the gain of under $1\%$. We see in
Fig. 2(a) that as the pump propagates through the waveguide, a noisy Stokes
field is generated that trails the leading edge of the pump pukse. This field
combines with the input signal, with the resulting noisy Stokes field further
amplified by the pump. Figure 2(b) shows the total Stokes power as a function
of $z$ at $t=1.8$ ns, where the dashed line represents the analytic solution
to (25) added to the coherent Stokes power in (22), and the blue line is a
numerical ensemble average over 20 independent runs of the numerical method.

Figure 3: Time-averaged Stokes noise power, coherent signal power and OSNR for
a dielectric waveguide of length $L=23.7$ cm. Plots (a)$-$(c) are for
$\alpha=0.01$ dB/cm, while plots (d)$-$(f) are for $\alpha=1.0$ dB/cm. The
signal pulse is a Gaussian with FWHM of 200 ps and peak power of 100 nW. The
pump consists of a rectangular pulse with constant energy, ranging from
$P_{\text{p0}}=5$ mW ($\tau_{p}=10$ ns) to $P_{\text{p0}}=5$ W ($\tau_{p}=10$
ps). The vertical dashed lines mark the point at which the interaction time
between pump and signal becomes equal to the transit time of the signal pulse.

We now use (25) to study the effect on the noise of the tunable properties of
the pump such as peak power and pulse duration, and how this may interact with
fixed parameters such as waveguide length and optical loss. We consider a
Gaussian signal pulse with FWHM of 200 ps, and a rectangular pump pulse of
width $\tau_{p}$ that varies between 10 ps$-$10 ns. The total pulse energy of
the input pump field is kept constant, while the peak power $P_{\text{p0}}$
varies with the pulse width, ranging from $P_{\text{p0}}=5$ mW ($\tau_{p}=10$
ns) to $P_{\text{p0}}=5$ W ($\tau_{p}=10$ ps). In all the calculations, the
output Stokes field at $z=0$ is averaged over a time period of 800 ps, which
is four times the signal FWHM and contains more than 99.7% of the pulse’s
energy. The results for the spontaneous Stokes power, signal power and OSNR
are shown in Fig. 3. We observe in Fig. 3(a) and (b) that the Stokes noise and
signal powers remain constant for pump widths $\tau_{p}\lesssim 3$ ns before
decreasing. In this low optical loss case, the near-constant values of the
Stokes noise and signal power can be attributed to the constant energy in the
pump pulse, as shown in Fig. 4: for short pulse-lengths, the signal and the
pump interact entirely within the waveguide. As the pump pulse becomes longer
than the waveguide and its peak power decreases, the Stokes field is able to
interact with less of its energy inside the medium, resulting in less
amplification of the Stokes. The signal and spontaneous fields fall off at
different rates with respect to $\tau_{p}$ because the signal is proportional
to the net gain squared $|G_{N}|^{2}$ via (22), while the spontaneous Stokes
power is proportional to the pump power $P_{1}$ multiplied by $|G_{N}|^{2}$
via (25). The gain $|G_{N}|^{2}$ decreases with larger $\tau_{p}$ because more
of the pump pulse lies outside of the interaction region, and at the same time
$P_{1}$ decreases while its peak power is lowered for longer $\tau_{p}$,
causing the spontaneous Stokes to decrease more rapidly than the signal. This
results in an OSNR figure which increases with $\tau_{p}$, as shown in Fig.
3(c).

Figure 4: Snapshots of a rectangular pump pulse with varying widths and peak
powers for an optical loss of $\alpha=0.01$ dB/cm. The input pump energy is
kept constant. Each row represents a snapshot of the fields inside the
waveguide at fixed time $t$, for varying pump pulse widths and the
corresponding signal and spontaneous Stokes powers. Figure 5: Snapshots of a
rectangular pump pulse with varying widths and peak powers for an optical loss
of $\alpha=0.5$ dB/cm. The input pump energy is kept constant. Each row
represents a snapshot of the fields inside the waveguide at fixed time $t$,
for varying pump pulse widths and the corresponding signal and spontaneous
Stokes powers.

For the high optical loss case shown in Fig. 3(d)$-$(f), we observe a maximum
in both the Stokes noise and signal powers for a critical value of $\tau_{p}$,
before decreasing with longer pump lengths as in the lower loss case. This
results in a minimum in the OSNR; this minimum occurs because the optical loss
is dominant in the regions inside the waveguide that are not covered by the
pump (see Fig. 5). As the pump becomes wider, the pump coverage inside the
waveguide increases, compensating for the losses incurred at the uncovered
regions. This results in higher amplification of the Stokes despite the peak
pump power decreasing with larger $\tau_{p}$. However, this increased
amplification vanishes once the pump pulse becomes longer than the waveguide,
in which case the Stokes power decreases in a similar manner to the low-loss
case.

Figure 6: Contour plot of the Stokes noise power (a) and the OSNR (b) as
functions of the rectangular pump width $\tau_{p}$ and optical loss $\alpha$,
for constant energy pump pulses. Plots (c) and (d) are for Gassuan pump pulses
of varying FWHM $\tau_{p}$. The black curve denotes where the maximum in the
noise power occurs for each loss $\alpha$. The horizontal dashed line denotes
when the interaction time of the pump and Stokes is equal to the transit time
of the signal pulse. The cross indicates the value of $\alpha$ below which the
maximum peak in the noise power vanishes.

The emergence of the OSNR minimum in Fig. 3(f) is a function of the optical
loss $\alpha$, the pump pulse width $\tau_{p}$ and the waveguide length $L$.
This is shown as a contour plot in Fig. 6; we see that for smaller losses
there is no maximum peak in the spontaneous noise power, and therefore no
minimum in the OSNR. Below this value the Stokes power and OSNR follow the
behavior of Fig. 3(a)$-$(c), staying approximately constant for short pump
pulses, then decreasing and increasing respectively for higher values of
$\tau_{p}$. However, once the loss reaches a critical value, the noise power
develops a maximum at approximately the point in which the interaction time
$\tau_{i}$ between pump and signal is equal to the transit time $L/v$. This
corresponds to a minimum in the OSNR, because making $\tau_{p}$ longer than
the transit time results in a reduced level of noise, since the spontaneous
Stokes decreases more rapidly than the amplified signal. The value
$\tau_{\text{max}}$ that maximizes the noise corresponds to the pump-length
when the interaction time $\tau_{i}$ satisfies $\tau_{i}=L/v$ and can be
computed as follows: first, we compute an effective pulse-width
$\tau_{\text{eff}}$ using the equation

$\int_{0}^{\tau_{\text{eff}}/2}P_{1}(0,t)\text{d}t=\frac{1}{2}\tilde{f}\int_{-\infty}^{\infty}P_{1}(0,t)\text{d}t,$
(27)

where $0\leq\tilde{f}\leq 1$ represents the fraction of the total energy in
the pump that is contained within $\tau_{\text{eff}}$. In our calculations, we
set $\tilde{f}=0.95$. The interaction time between the pulses is
$\tau_{i}=\tau_{\text{eff}}/2$ because the signal and pump counter-propagate
at the same speed, so the time of overlap is halved. This gives an
approximation of the value $\tau_{p}$ for which the maximum amount of pump
energy is able to interact with the signal, before it no longer fits inside
the structure.

This noise behavior is not restricted to rectangular pump pulses. In Fig. 6(c)
and (d), we observe the same behavior when using a Gaussian pump pulse, and
again the same maximum in the Stokes noise can be observed. Here, some of the
pump’s energy is more localized near the central peak, while the long tails of
the pulse contain a very small portion of the energy. This limits the region
inside the waveguide in which the amplification from the pump is able to
compensate for the optical losses, and results in a maximum Stokes noise peak
that occurs at a smaller $\tau_{p}$ compared to the rectangular pump case.

Figure 7: Stokes noise power (a) and OSNR (b) for a rectangular pump pulse
using a constant peak pump power.

Finally, we compute the effects of the noise and OSNR in the case where the
pump pulse has a fixed peak power. Both the spontaneous Stokes noise and
signal power increase with $\tau_{p}$, reaching a steady state for $\tau_{p}$
longer than the transit time of the waveguide. This results in the OSNR
decreasing until reaching a constant value for larger $\tau_{p}$, as shown in
Fig. 6(b). This occurs due to the pump pulse covering more of the waveguide as
$\tau_{p}$ gets larger: initially there is more energy injected into the SBS
medium, but as soon as the pump pulse becomes comparable in size to the
waveguide; the amount of pump energy that interacts with the Stokes field
reaches a constant value. This then leads to a steady-state for both the
signal and spontaneous Stokes powers.

##  4 Validity of the analytic model

We now estimate the region of validity of our analytic model, by considering
an SBS chalcogenide chip with length 23.7 cm, with a Gaussian pump pulse of
$P_{\text{p0}}=300$ mW and fixed optical loss of $\alpha=0.3$ dB/cm. The
energy in both the pump and signal input fields is given by the expressions
$E_{1}=\int_{-\infty}^{\infty}P_{1}(0,t)\text{d}t$ and $\quad
E_{2}=\int_{-\infty}^{\infty}P_{2}(L,t)\text{d}t$ respectively. For each ratio
of energies $E_{2}/E_{1}$ where $E_{2}\leq E_{1}$ we compute the relative
error in the OSNR using both the analytic model in (25) and the numerical
solver for Eqs (4)$-$(6).

In Fig. 8(a) we see that the ONSR error remains below 2% when the signal
energy relative to the pump is small, but increases suddenly for larger signal
powers. This occurs because of pump depletion, which was neglected in Eq. 25
in deriving the analytic model: as the pump becomes depleted, some of its
energy gets transferred to the spontaneous Stokes as well as to the signal,
enhancing the coupling between the fields. This results in some of the
spontaneous Stokes energy feeding back into the pump, making the
multiplicative noise more dominant than the additive noise in Eq. (25). This
then leads to the sudden increase in the OSNR error observed in Fig. 8(a). The
analyic model is therefore expected to yield accurate results provided the
pump remains undepleted.

##  5 Conclusion

In this paper, we have developed a general mathematical model for solving the
dynamical SBS equations in the presence of thermal noise. We present a simple
analytic model in the undepleted pump regime whereby the amplitude noise in
the Stokes field can be computed for pulses of arbitrary shape and size, which
is useful in Brillouin signal processing. The analytic results show that in
the case of a constant energy pump field and lossy media, the OSNR has a
minimum peak near the region in which the interaction time of the pulses
matches the transit time of the signal in the waveguide. This occurs as the
spontaneous Stokes noise increases towards a maximum value, which results from
longer pump pulses compensating for the linear optical losses in the medium.
Once the pump pulse becomes longer than the waveguide, the loss dominates
again as less pump energy fits inside the SBS medium. This behaviour of the
OSNR is mediated by the pump energy, pump shape, waveguide length and optical
loss, making it important to choose the right parameter combination to
maximize the OSNR in a specific device.

Figure 8: Plots for the signal percentage error in the OSNR (a) and signal
gain (b). All the plots are represented as functions of the input signal
energy as a percentage of the input pump energy, over a range of signal powers
between 1 nW$-$1 mW relative to a 300 mW input pump at a fixed optical
attenuation of 0.3 dB/cm. The dashed vertical lines represent the point at
which the near-constant gain begins to decrease, indicating that the pump is
being depleted by the Stokes field.

##  Appendix A Derivation of the noise strength

In this appendix, we find an expression for the noise strength parameter
$\sigma$ in Eq. (7) by considering the acoustic envelope field $b_{j}$ at a
set of discrete positions along the waveguide $z_{j}$ separated by a small
width $\Delta z$, such that $b_{j}=b(z_{j})$, and assuming $\partial_{z}b$ is
negligible due to the small propagation length of the phonons [4]. The thermal
noise field in each slice is $\tilde{R}_{j}$, and we assume it to be
statistically independent between any two distinct slices. Additionally, we
assume a weak opto-acoustic coupling so that the $b_{j}$ field is mainly
dominated by the thermal noise. This leads to the following equation for the
acoustic envelope field in the $j$th slice in the waveguide

$\frac{\mathrm{d}b_{j}}{\mathrm{d}t}+\frac{1}{2}\Gamma
b_{j}=v_{a}\sqrt{\sigma_{j}}\tilde{R}_{j}(t),$ (28)

where $\sigma_{j}=\sigma/\Delta z$ and $\Gamma=v_{a}\alpha_{\text{ac}}$. The
complex-valued noise $\tilde{R}_{j}(t)$ is written as

$\tilde{R}_{j}(t)=\frac{\tilde{R}_{j}^{(1)}(t)+i\tilde{R}_{j}^{(2)}(t)}{\sqrt{2}},$
(29)

with the properties

$\langle\tilde{R}_{j}(t)\rangle=0,\quad\langle\tilde{R}_{j}^{(1)}(t)\tilde{R}_{j}^{(2)}(t)\rangle=0,\quad\langle\tilde{R}_{j}(t)\tilde{R}_{j}^{*}(t^{\prime})\rangle=\delta(t-t^{\prime}).$
(30)

Equation (28) has the solution

$b_{j}(t)=b_{j}(0)\mathrm{e}^{-\frac{1}{2}\Gamma
t}+v_{a}\sqrt{\sigma_{j}}\int_{0}^{t}\mathrm{e}^{-\frac{1}{2}\Gamma(t-t^{\prime})}\tilde{R}_{j}(t^{\prime})\,\mathrm{d}t^{\prime},$
(31)

and thus the mean squared amplitude can be found via

$\langle|b_{j}(t)|^{2}\rangle=|b_{j}(0)|^{2}\mathrm{e}^{-\Gamma
t}+v_{a}^{2}\sigma_{j}\int_{0}^{t}\int_{0}^{t^{\prime}}\mathrm{e}^{-\frac{1}{2}\Gamma(2t-t^{\prime}-t^{\prime\prime})}\langle\tilde{R}_{j}(t^{\prime})\tilde{R}_{j}^{*}(t^{\prime\prime})\rangle\,\mathrm{d}t^{\prime\prime}\mathrm{d}t^{\prime},$
(32)

and using
$\langle\tilde{R}_{j}(t)\tilde{R}_{j}^{*}(t^{\prime})\rangle=\delta(t-t^{\prime})$
one obtains

$\displaystyle\langle|b_{j}(t)|^{2}\rangle$
$\displaystyle=|b_{j}(0)|^{2}\mathrm{e}^{-\Gamma
t}+v_{a}^{2}\sigma_{j}\int_{0}^{t}\int_{0}^{t^{\prime}}\mathrm{e}^{-\frac{1}{2}\Gamma(2t-t^{\prime}-t^{\prime\prime})}\delta(t^{\prime}-t^{\prime\prime})\,\mathrm{d}t^{\prime\prime}\mathrm{d}t^{\prime}$
$\displaystyle=|b_{j}(0)|^{2}\mathrm{e}^{-\Gamma
t}+v_{a}^{2}\sigma_{j}\int_{0}^{t}\mathrm{e}^{-\Gamma(t-t^{\prime})}\,\mathrm{d}t^{\prime}$
$\displaystyle=|b_{j}(0)|^{2}\mathrm{e}^{-\Gamma
t}+\frac{v_{a}\sigma_{j}}{\alpha_{\text{ac}}}\left(1-\mathrm{e}^{-\Gamma
t}\right),$

which in the limit $t\rightarrow\infty$ yields the steady-state expression

$\langle|b_{j}(t)|^{2}\rangle_{\infty}=\frac{v_{a}\sigma_{j}}{\alpha_{\text{ac}}}.$
(33)

This quantity represents the average acoustic power, and thus we can then
relate this to the thermal energy per slice $\Delta z$ through the
equipartition theorem [28], by dividing both sides by the acoustic speed
$v_{a}$

$\frac{1}{v_{a}}\langle|b_{j}(t)|^{2}\rangle_{\infty}=\frac{\sigma_{j}}{\alpha_{\text{ac}}}=\frac{k_{B}T}{\Delta
z},$ (34)

which then yields

$\sigma_{j}=\frac{k_{B}T\alpha_{\text{ac}}}{\Delta z}.$ (35)

Then, we obtain the value for $\sigma$ based on the slice thickness $\Delta z$

$\sigma=\sigma_{j}\Delta z=k_{B}T\alpha_{\text{ac}}.$ (36)

##  Appendix B Full derivation of the solution to the Stokes field in the
undepleted pump approximation

In this Appendix we show the complete derivation of the analytic solution for
the Stokes envelope field in (17). Starting with Eq. (15) and by using the
transformation of variables $\xi=z$ and $\tau=t+z/v$, one moves into a frame
of reference which moves along with the Stokes field. The partial derivatives
in the new coordinate system $(\xi,\tau)$ are thus

$\displaystyle\frac{\partial}{\partial t}$
$\displaystyle=\frac{\partial\xi}{\partial
t}\frac{\partial}{\partial\xi}+\frac{\partial\tau}{\partial
t}\frac{\partial}{\partial\tau}=\frac{\partial}{\partial\tau},$ (37)
$\displaystyle\frac{\partial}{\partial z}$
$\displaystyle=\frac{\partial\xi}{\partial
z}\frac{\partial}{\partial\xi}+\frac{\partial\tau}{\partial
z}\frac{\partial}{\partial\tau}=\frac{\partial}{\partial\xi}+\frac{1}{v}\frac{\partial}{\partial\tau}.$
(38)

Upon substituting these derivatives into Eq. (5) we obtain

$\frac{da_{2}}{d\xi}+\frac{1}{2}\left[g(\Delta)P_{1}(\xi,\tau-\xi/v)-\alpha\right]a_{2}=i\omega_{2}Q_{2}a_{1}(\xi,\tau-\xi/v)D(\xi\tau-\xi/v),$
(39)

Identifying the integrating factor

$I(\xi,\tau)=\exp\left\\{-\frac{1}{2}\int_{\xi}^{L}\left[g(\Delta)P_{1}\left(\xi^{\prime},\tau-\frac{\xi^{\prime}}{v}\right)-\alpha\right]\text{d}\xi^{\prime}\right\\},$
(40)

leads to the equation

$\int_{\xi}^{L}\frac{d}{d\xi^{\prime}}\left[a_{2}(\xi^{\prime},\tau)I(\xi^{\prime},\tau)\right]\text{d}\xi^{\prime}=i\omega_{2}Q_{2}\int_{\xi}^{L}I(\xi^{\prime},\tau)a_{1}\left(\xi^{\prime},\
\tau-\frac{\xi^{\prime}}{v}\right)D\left(\xi^{\prime},\
\tau-\frac{\xi^{\prime}}{v}\right)\text{d}\xi^{\prime},$ (41)

The left-hand side of Eq. (39) is
$a_{2}(L,\tau)I(L,\tau)-a_{2}(\xi,\tau)I(\xi,\tau)$. Then, using
$a_{2}(L,\tau)$ as the input boundary condition for the Stokes field, and
introducing the cumulative net gain function

$\tilde{G}_{N}(\xi,\xi^{\prime},\tau)=\exp\left\\{\frac{1}{2}\int_{\xi}^{\xi^{\prime}}\left[g(\Delta)P_{1}\left(\xi^{\prime\prime},\tau-\frac{\xi^{\prime\prime}}{v}\right)-\alpha\right]\text{d}\xi^{\prime\prime}\right\\}.$
(42)

we write the solution for the Stokes envelope field at the point $\xi$ as

$a_{2}(\xi,\tau)=a_{2}(L,\tau)\tilde{G}_{N}(\xi,L,\tau)-i\omega_{2}Q_{2}\int_{\xi}^{L}\tilde{G}_{N}(\xi,\xi^{\prime},\tau)a_{1}\left(\xi^{\prime},\
\tau-\frac{\xi^{\prime}}{v}\right)D\left(\xi^{\prime},\
\tau-\frac{\xi^{\prime}}{v}\right)\text{d}\xi^{\prime}.$ (43)

Finally, we switch back to the coordinate system $(z,t)$ by replacing $\tau$
in the above equation by $\tau=t+z/v$, yielding Eq. (17).

## Disclosures

The authors declare no conflicts of interest.

## Acknowledgments

The authors acknowledge funding from the Australian Research Council (ARC)
(Discovery Projects DP160101691, DP200101893), the Macquarie University
Research Fellowship Scheme (MQRF0001036) and the UTS Australian Government
Research Training Program Scholarship (00099F). Part of the numerical
calculations were performed on the UTS Interactive High Performance Computing
(iHPC) facility.

## References

  * [1] B. J. Eggleton, C. G. Poulton, P. T. Rakich, M. J. Steel, and G. Bahl, “Brillouin integrated photonics,” Nature Photonics 13, 664–677 (2019).
  * [2] R. Pant, D. Marpaung, I. V. Kabakova, B. Morrison, C. G. Poulton, and B. J. Eggleton, “On-chip stimulated Brillouin scattering for microwave signal processing and generation,” Laser Photonics Rev. 8, 653–666 (2014).
  * [3] L. Brillouin, “Diffusion de la lumière et des rayons X par un corps transparent homogène,” AnPh 9, 88–122 (1922).
  * [4] R. W. Boyd, _Nonlinear optics_ (Elsevier, 2003).
  * [5] A. Kobyakov, M. Sauer, and D. Chowdhury, “Stimulated Brillouin scattering in optical fibers,” Adv. Opt. Photonics 2, 1–59 (2010).
  * [6] P. T. Rakich, C. Reinke, R. Camacho, P. Davids, and Z. Wang, “Giant enhancement of stimulated Brillouin scattering in the subwavelength limit,” Phys. Rev. X 2, 011008 (2012).
  * [7] C. Wolff, M. J. Steel, B. J. Eggleton, and C. G. Poulton, “Stimulated Brillouin scattering in integrated photonic waveguides: Forces, scattering mechanisms, and coupled-mode analysis,” Phys. Rev. A 92, 013836 (2015).
  * [8] J. E. Sipe and M. J. Steel, “A Hamiltonian treatment of stimulated Brillouin scattering in nanoscale integrated waveguides,” New J. Phys. 18, 045004 (2016).
  * [9] A. Choudhary, Y. Liu, D. Marpaung, and B. J. Eggleton, “On-chip Brillouin filtering of RF and optical signals,” IEEE J. Sel. Top. Quantum Electron. 24, 1–11 (2018).
  * [10] H. Jiang, L. Yan, W. Pan, B. Luo, and X. Zou, “Ultra-high speed RF filtering switch based on stimulated Brillouin scattering,” Opt. Lett. 43, 279–282 (2018).
  * [11] B. Steinhausser, A. Brignon, E. Lallier, J.-P. Huignard, and P. Georges, “High energy, single-mode, narrow-linewidth fiber laser source using stimulated Brillouin scattering beam cleanup,” Opt. Express 15, 6464–6469 (2007).
  * [12] W. Loh, A. A. Green, F. N. Baynes, D. C. Cole, F. J. Quinlan, H. Lee, K. J. Vahala, S. B. Papp, and S. A. Diddams, “Dual-microcavity narrow-linewidth Brillouin laser,” Optica 2, 225–232 (2015).
  * [13] M. Merklein, B. Stiller, K. Vu, S. J. Madden, and B. J. Eggleton, “A chip-integrated coherent photonic-phononic memory,” Nat. Commun. 8, 1–7 (2017).
  * [14] R. W. Boyd, K. Rzazewski, and P. Narum, “Noise initiation of stimulated Brillouin scattering,” Phys. Rev. A 42, 5514 (1990).
  * [15] A. L. Gaeta and R. W. Boyd, “Stochastic dynamics of stimulated Brillouin scattering in an optical fiber,” Phys. Rev. A 44, 3205 (1991).
  * [16] M. Ferreira, J. Rocha, and J. Pinto, “Analysis of the gain and noise characteristics of fiber Brillouin amplifiers,” Opt. Quantum Electron. 26, 35–44 (1994).
  * [17] A. Kobyakov, S. Darmanyan, and D. Chowdhury, “Exact analytical treatment of noise initiation of SBS in the presence of loss,” Opt. Commun. 260, 46–49 (2006).
  * [18] W. Loh, S. B. Papp, and S. A. Diddams, “Noise and dynamics of stimulated Brillouin scattering microresonator lasers,” Phys. Rev. A 91, 053843 (2015).
  * [19] P. Kharel, R. Behunin, W. Renninger, and P. Rakich, “Noise and dynamics in forward Brillouin interactions,” Phys. Rev. A 93, 063806 (2016).
  * [20] R. O. Behunin, N. T. Otterstrom, P. T. Rakich, S. Gundavarapu, and D. J. Blumenthal, “Fundamental noise dynamics in cascaded-order Brillouin lasers,” Phys. Rev. A 98, 023832 (2018).
  * [21] N. T. Otterstrom, R. O. Behunin, E. A. Kittlaus, Z. Wang, and P. T. Rakich, “A silicon Brillouin laser,” Science 360, 1113–1116 (2018).
  * [22] W. Zhang and R. A. Minasian, “Widely tunable single-passband microwave photonic filter based on stimulated Brillouin scattering,” IEEE Photonics Technology Letters 23, 1775 (2011).
  * [23] M. Pelusi, T. Inoue, and S. Namiki, “Noise characterization of Brillouin amplified narrowband carriers for coherent communications,” in _2018 European Conference on Optical Communication (ECOC),_ (IEEE, 2018), pp. 1–3.
  * [24] A. Zarifi, M. Merklein, Y. Liu, A. Choudhary, B. J. Eggleton, and B. Corcoran, “Broadband sbs filter for optical carrier recovery applications in telecommunication systems,” in _2020 Conference on Lasers and Electro-Optics (CLEO),_ (IEEE, 2020), pp. 1–2.
  * [25] B. C. Sturmberg, K. B. Dossou, M. J. Smith, B. Morrison, C. G. Poulton, and M. J. Steel, “Finite element analysis of stimulated Brillouin scattering in integrated photonic waveguides,” J. Light. Technol. 37, 3791–3804 (2019).
  * [26] P. Drummond and J. F. Corney, “Quantum noise in optical fibers. I. Stochastic equations,” J. Opt. Soc. Am. B 18, 139–152 (2001).
  * [27] H. L. Pécseli, _Fluctuations in physical systems_ (Cambridge University Press, 2000).
  * [28] J. Lima and A. Plastino, “On the classical energy equipartition theorem,” Brazilian Journal of Physics 30, 176–180 (2000).
  * [29] R. Kubo, “Statistical-mechanical theory of irreversible processes. i. general theory and simple applications to magnetic and conduction problems,” Journal of the Physical Society of Japan 12, 570–586 (1957).
  * [30] K. Itô, “On a formula concerning stochastic differentials,” Nagoya Mathematical Journal 3, 55–65 (1951).
  * [31] P. Drábek and G. Holubová, _Elements of partial differential equations_ (Walter de Gruyter GmbH & Co KG, 2014).
  * [32] J. Van Neerven and M. C. Veraar, “On the stochastic Fubini theorem in infinite dimensions,” in _Stochastic Partial Differential Equations and Applications VII, Levico Terme, 2004, in: Lecture Notes Pure Appl. Math._ , (CRC Press, 2005), p. 323–336.
  * [33] M. Veraar, “The stochastic Fubini theorem revisited,” Stochastics 84, 543–551 (2012).
  * [34] R. Kubo, “The fluctuation-dissipation theorem,” Rep. Prog. Phys. 29, 255 (1966).
  * [35] G. P. Agrawal, _Fiber-optic communication systems_ (John Wiley & Sons, 2012), 4th ed.
  * [36] Y. Xie, A. Choudhary, Y. Liu, D. Marpaung, K. Vu, P. Ma, D.-Y. Choi, S. Madden, and B. J. Eggleton, “System-level performance of chip-based Brillouin microwave photonic bandpass filters,” J. Light. Technol. 37, 5246–5258 (2019).


# Gravastars in a Non-minimally Coupled Gravity with Electromagnetism

Özcan SERT<EMAIL_ADDRESS>Department of Physics, Faculty of Arts and
Sciences, Pamukkale University, 20070 Denizli, Türkiye  Muzaffer ADAK
<EMAIL_ADDRESS>Department of Physics, Faculty of Arts and Sciences,
Pamukkale University, 20070, Kınıklı, Denizli, Turkey

###### Abstract

In this paper we investigate the gravitational vacuum stars which called
gravastars in the non-minimally coupled models with electromagnetic and
gravitational fields. We consider two non-minimal models and find the
corresponding spherically symmetric exact solutions in the interior of the
star consisting of the dark energy condensate. Our models turn out to be
Einstein-Maxwell model at the outside of the star and the solutions become the
Reissner-Nordström solution. The physical quantities of these models are
continuous and non-singular in some range of parameters and the exterior
geometry continuously matches with the interior geometry at the surface. We
calculate the matter mass, the total gravitational mass, the electric charge
and redshift of the star for the two models. We notice that these quantities
except redshift are dependent of a subtle free parameter, $k$, of the model.
We also remark a wide redshift range from zero to infinity depending on one
free parameter, $\beta$, in the second model.

###### pacs:

Valid PACS appear here

##  I Introduction

Black holes are one of the most important predictions of Einstein’s theory,
which has been experimentally confirmed recently. Although black holes are
well known and analyzed as the exact solution of Einstein’s equation, they
have some properties that are not yet fully understood, such as event horizon,
singularity and information paradox. One of the alternative models that can
eliminate these complex properties of black holes is the gravastar
(gravitational vacuum star) proposed by Mazur and Mottola Mazur2001 ;
Mazur2004 ; Mazur20042 . Gravastar is a non-singular and horizon-less ultra
compact object, which has negative pressure in the interior, consisting of a
dark energy condensate. Since they are ultra compact objects, it is very
difficult to distinguish them from black hole observations. Although there is
not sufficiently observational evidence for the existence of gravastars, it is
important to investigate them as an alternative way to better understand black
holes. The increasing interest on the subject of gravastar has led researchers
to discuss and compare the physical properties of gravastars and black holes.
It has been shown that the energy flux emitted by the accretion disk surface
of gravastars is smaller than the energy flux emitted by black holes and then
the gravastars may have greater redshifts from black holes Harko2009 . Thus
the theoretical results might guide to observational researches in order for
distinguishing them.

The inside of the gravastar has the dark energy matter which has the equation
of state $\rho=-p$, while the outside of the star is considered as a vacuum
described by Reissner-Nordström geometry. There is also an intermediate
region, which consists of a thin shell containing a stiff matter satisfying
the equation of state $\rho=p$. The thin shell prevents the formation of the
event horizon and the phase transition surface appears at or near the region.
The surface separates the two phases of the space-time vacuum by quantum phase
transition Chapline2002 ; Chapline2003  and then the information paradox at
black holes is resolved by removing the event horizon. The dynamical stability
of the gravastar configuration was discussed by Visser and Witshire Visser2004
by assuming the infinitely small thin shell as a mathematical simplification.
Then the stability was further investigated for different exterior geometries
Carter2005  and dark energy star models Lobo2006 .

Moreover, even a very small electric charge relative to its mass, which may
form just inside the surface of the star, prevents the nuclear matter from
falling into the core and make the star more stable Cheng1998 ; Stettner ;
Krasinski ; Sharma ; mak-harko-2004 . The charged gravastar models for
Einstein-Maxwell theory were studied in Horvat2009 ; Usmani2011 ; Bhar2014 .
Then they were obtained also in the alternative gravitation models such as
Born-Infeld phantom Bilic2006  and nonlinear electrodynamics Lobo2007 . The
infinitely small thin shell was replaced by a continuous pressure profile with
an anisotropic fluid in Cattoen2005 ; DeBenedic . Meanwhile, exact interior
and exterior solutions describing mixed relativistic stars which contain dark
energy and ordinary matter were found in Yazadjiev2011  for the model with
phantom scalar field and they matched continuously at the surface of the star.

Despite all its achievements, Einstein’s theory of gravity has very important
astrophysical and cosmological problems such as dark matter and dark energy.
Thus, alternative theories such as $f(R)$ Chiba2003 ; Capozziello2006 ;
Carroll2004 ; Sotiriou2010 ; Lobo2009  gravity, $f(R,T)$ gravity Harko2011
etc. were proposed to solve the challenges, where $R$ is the Ricci curvature
scalar and $T$ is the trace of the stress-energy tensor. Spherically symmetric
gravastar solutions for these theories were studied in Das2017 ; Bhar2021  for
$f(R,T)$ gravity and in Shamir2018  for $f(G,T)$ gravity, where $G$ denotes
Gauss-Bonnet invariant. Furthermore, the effects of the electric charge on the
isotropic spherical gravastar models were investigated in Ghosh2017  for
higher dimensional Einstein-Maxwell theory and in Yousaf2019  Maxwell-$f(R,T)$
gravity by using the Mazur-Mottola conjecture. Also, a charged gravastar model
and its stability were studied in Ovgun2017  by using the noncommutative
geometry. Since gravastar is an ultra compact object like black hole, it has
extremely dense gravitational and Maxwell fields. Thus, it is possible to
consider the non-minimal $Y(R)F^{2}$ couplings Dereli201130111 ;
Dereli201130112 ; Sert12Plus ; Sert13MPLA ; bamba1 ; bamba2 ; Dereli901 ;
Baykal ; Sert16regular ; Dereli20113 ; AADS  to describe the objects. Some
spherically symmetric isotropic and anisotropic compact star solutions of the
theory are investigated in Sert2017 ; Sert2018 ; Sert20182 . In this study we
give two exact non-singular gravastar solutions to the non-minimal $Y(R)F^{2}$
gravity by considering the dark energy equation of state, $p=-\rho$. This
model turns out to be the Einstein-Maxwell model at the exterior of the
gravastar and their solutions to be the Reissner-Nordström solution.

##  II  A Non-minimal Model for The Gravastars

In order to describe gravastars in the non-minimal model, we consider the
following Lagrangian in differential forms Sert2017 ; Sert2018

$L=\frac{1}{2\kappa^{2}}R*1-Y(R)F\wedge*F+2A\wedge J+L_{mat}+\lambda_{a}\wedge
T^{a}\;$ (1)

where $\kappa$ is a coupling constant, $R$ is the Ricci scalar, $Y(R)$ is a
function of $R$ and corresponds to a non-minimal coupling between
electromagnetism with gravity, $A$ is the electromagnetic potential 1-form,
$F=dA$ is the Maxwell 2-form, $L_{mat}$ denotes the matter Lagrangian and $J$
denotes the electromagnetic source 3-form and $\lambda_{a}$ denotes the
Lagrange multiplier setting torsion to zero. $d$ denotes the exterior
derivative, $*$ is the Hodge star operator and $\wedge$ denotes the exterior
product in the exterior algebra. One can uses the Hodge map for fixing the
orientation of a manifold in terms of orthonormal (Lorentzian) coframe $e^{a}$
by writing down the volume form $*1=e^{0123}\equiv e^{0}\wedge e^{1}\wedge
e^{2}\wedge e^{3}$.

Since the search for spherically symmetric exact solutions to the
gravitational field equations in the interior region of an isotropic fluid has
been a longstanding issue of theoretical physics, we assume a perfect fluid
which has the pressure $p$ and energy density $\rho$ inside the stars, that
is,

$\frac{\partial L_{mat}}{\partial e^{a}}:=(\rho+p)u_{a}*u+p*e_{a}\
,\qquad\frac{\partial L_{mat}}{\partial\omega^{a}{}_{b}}:=0\ ,$ (2)

where $\omega^{a}{}_{b}$ denotes the connection 1-form and $u$ the time-like
velocity 1-form, $u=u_{a}e^{a}$. Thus one can compute $\lambda_{a}$ from
$\omega^{a}{}_{b}$-varied equation and then substitute the findings into
$e^{a}$-varied equation of the action (1). In conclusion we obtain the
gravitational field equation

$\displaystyle-\frac{1}{2\kappa^{2}}R^{bc}\wedge*e_{abc}=$ $\displaystyle
Y(F_{a}\wedge*F-F\wedge\iota_{a}*F)+Y_{R}F_{mn}F^{mn}*R_{a}$
$\displaystyle+D[\iota^{b}\;d(Y_{R}F_{mn}F^{mn})]\wedge*e_{ab}+\
(\rho+p)u_{a}*u+p*e_{a}$ (3)

where $R^{a}{}_{b}:=d\omega^{a}{}_{b}+\omega^{a}{}_{c}\wedge\omega^{c}{}_{b}$
is the Riemann curvature 2-form, $R_{b}:=\iota_{a}R^{a}{}_{b}$ denotes Ricci
curvature 1-form, $\iota_{a}$ denotes the interior product of the exterior
algebra, $D$ is the covariant exterior derivative and $Y_{R}:={dY}/{dR}$,
$F_{a}:=\iota_{a}F$, $F_{ab}:=\iota_{ab}F$. As we have shown in our previous
papers Sert2017 ; Sert2018 , our gravitational field equation (II) can be
simplified considerably by the constraint

$\displaystyle Y_{R}F_{mn}F^{mn}=-\frac{k}{\kappa^{2}}\ .$ (4)

Consequently the above gravitational field equation turns into the following

$\displaystyle-\frac{1}{2\kappa^{2}}R^{bc}\wedge*e_{abc}=Y(F_{a}\wedge*F-F\wedge\iota_{a}*F)-\frac{k}{\kappa^{2}}*R_{a}+(\rho+p)u_{a}*u+p*e_{a}\
.$ (5)

We notice that as the equation (II) contains higher order derivatives than
two, now the new equation (5) consists of the second order differential
equations. Here $k$ is an arbitrary coupling constant between the
electromagnetic and gravitational fields. It is worth to calculate the trace
of this equation

$\displaystyle\frac{1-k}{\kappa^{2}}R=(\rho-3p)\;.$ (6)

In this non-minimal model, while the case with $k=1$ corresponds to the
radiation fluid stars with $\rho=3p$ Sert2017 , the other case $k\neq 1$,
corresponding to $\rho\neq 3p$, has been studied in Ref. Sert2018  for certain
equations of state between $\rho$ and $p$. In this study we deal with a new
model for the dark energy stars defined by the equation of state

$p=-\rho\ .$ (7)

Accordingly, the Ricci scalar must be proportional to energy density via (6)
for our model

$R={4\kappa^{2}\rho}/{(1-k)}\ .$ (8)

Before ending this section, in order to close the system of equations, we
write down the modified Maxwell equation by varying the action (1) for the
electromagnetic potential 1-form, $A$

$\displaystyle d(*YF)=J\;.$ (9)

##  III SPHERICALLY SYMMETRIC EXACT SOLUTIONS

We study solutions with the following spherically symmetric, static line
element

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-f^{2}(r)dt^{2}+g^{2}(r)dr^{2}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta
d\phi^{2}$ (10)

and the Maxwell 2-form representing a spherically symmetric static electric
field in the radial direction

$\displaystyle F$ $\displaystyle=$ $\displaystyle E(r)e^{1}\wedge e^{0}.$ (11)

Under these ansatz our subtle condition (4) takes the form

$\displaystyle\frac{dY}{dR}=\frac{k}{2\kappa^{2}E^{2}}\;.$ (12)

Thus the components of the gravitational field equation (II) together with the
assumption of the equation state (7) read explicitly

$\displaystyle\frac{1}{\kappa^{2}g^{2}}\left(\frac{2g^{\prime}}{gr}+\frac{g^{2}-1}{r^{2}}\right)+\frac{k}{\kappa^{2}g^{2}}\left(\frac{f^{\prime\prime}}{f}-\frac{f^{\prime}g^{\prime}}{fg}+\frac{2f^{\prime}}{fr}\right)=$
$\displaystyle YE^{2}+\rho\;,$ (13)
$\displaystyle\frac{1}{\kappa^{2}g^{2}}\left(-\frac{2f^{\prime}}{rf}+\frac{g^{2}-1}{r^{2}}\right)+\frac{k}{\kappa^{2}g^{2}}\left(\frac{f^{\prime\prime}}{f}-\frac{f^{\prime}g^{\prime}}{fg}-\frac{2g^{\prime}}{rg}\right)=$
$\displaystyle YE^{2}+\rho\;,$ (14)
$\displaystyle\frac{1}{\kappa^{2}g^{2}}\left(\frac{f^{\prime\prime}}{f}-\frac{f^{\prime}g^{\prime}}{fg}+\frac{f^{\prime}}{rf}-\frac{g^{\prime}}{rg}\right)+\frac{k}{\kappa^{2}g^{2}}\left(\frac{g^{\prime}}{rg}-\frac{f^{\prime}}{rf}+\frac{g^{2}-1}{r^{2}}\right)=$
$\displaystyle YE^{2}-\rho\;,$ (15)

where prime denotes the derivative with respect to $r$. Additionally we have
checked that the exterior covariant derivative of the gravitational field
equation (II) generates the conservation of the total energy-momentum tensor.
When the equations (13) and (14) are compared, it is seen

$\displaystyle g(r)={1}/{f(r)}\ .$ (16)

Then the following two equations are left at our disposal

$\displaystyle\frac{1}{\kappa^{2}}\left(-\frac{2ff^{\prime}}{r}+\frac{1-f^{2}}{r^{2}}\right)+\frac{k}{\kappa^{2}}\left(ff^{\prime\prime}+f^{\prime
2}+\frac{2ff^{\prime}}{r}\right)=$ $\displaystyle YE^{2}+\rho\;,$ (17)
$\displaystyle\frac{1}{\kappa^{2}}\left(ff^{\prime\prime}+f^{\prime
2}+\frac{2ff^{\prime}}{r})+\frac{k}{\kappa^{2}}(-2\frac{ff^{\prime}}{r}+\frac{1-f^{2}}{r^{2}}\right)=$
$\displaystyle YE^{2}-\rho\;.$ (18)

Here it is important to notice that the constraint equation (12) is not an
independent equation, because it can be derived by adding equation (17) and
(18), then by taking differential. Also, one can show that the trace equation
(6) is arrived by subtracting equation (17) from (18).

At the end there are only three independent equations, (8), (17) and (18), in
our hand to be solved, but four unknown functions $f(r),Y(R),\rho(r)$ and
$E(r)$. In these cases literally two strategies are possible. One strategy is
firstly to adopt a non-minimal coupling function such as
$Y(R)=c_{1}R+c_{2}R^{2}$ etc, then to determine the other three unknowns. In
the reverse order, one firstly assumes a metric function such as
$f(r)=c_{1}+c_{2}r^{2}$ etc, and then calculates the others. In this work we
follow the second approach.

###  III.1 Solution-1

A simple regular interior solution of these differential equations could be
found by considering the metric function

$\displaystyle f(r)=1-ar^{2}\;$ (19)

where $a$ is a real constant. Then the Ricci scalar is calculated as

$\displaystyle R=6a(4-5ar^{2})$ (20)

Correspondingly the other three unknown functions are computed as

$\displaystyle E^{2}(r)$ $\displaystyle=$
$\displaystyle\frac{5(1+k)}{2\kappa^{2}C_{1}}a^{2}r^{(14k+2)/{(k+1)}}\;,$ (21)
$\displaystyle\rho(r)$ $\displaystyle=$
$\displaystyle-p(r)=\frac{3a(1-k)}{2\kappa^{2}}(4-5ar^{2})\;,$ (22)
$\displaystyle Y(r)$ $\displaystyle=$ $\displaystyle
C_{1}r^{-{12k}/{(k+1)}}\;,$ (23)

where $C_{1}$ is an integration constant which will be determined by the
boundary conditions. In order for writing down the non-minimal coupling
function explicitly in terms of the Ricci scalar we compute firstly the
inverse function of (20) as $r^{2}={4}/{5a}-{R}/{30a^{2}}$. Thus we could
rewrite (23) as

$\displaystyle
Y(R)=C_{1}\left(\frac{4}{5a}-\frac{R}{30a^{2}}\right)^{-{6k}/{(1+k)}}\;.$ (24)

On the other hand, since the outside of the star is empty, it is described by
the minimal Einstein-Maxwell theory and the Reissner-Nordström metric which
has the property $R=0$. Since we require that $Y(R)$ must be continuous, the
equation (24) becomes $Y(R)=C_{1}({4}/{5a})^{-{6k}/{(k+1)}}$ for $R=0$ on the
boundary. For the minimal coupling, i.e. $Y(R)=1$, on the boundary this result
produces

$C_{1}=({4}/{5a})^{{6k}/{(k+1)}}\ .$ (25)

In summary in the interior region the non-minimal Lagrangian and the metric of
the theory become, respectively,

$\displaystyle L_{in}=$
$\displaystyle\frac{1}{2\kappa^{2}}R*1-\left(1-\frac{R}{24a}\right)^{-{6k}/{(1+k)}}F\wedge*F+2A\wedge
J+L_{mat}+\lambda_{a}\wedge T^{a}\ ,$ (26) $\displaystyle ds^{2}_{in}=$
$\displaystyle-\left(1-ar^{2}\right)^{2}dt^{2}+\left(1-ar^{2}\right)^{-2}dr^{2}+r^{2}d\Omega^{2}\
,$ (27)

together with the electric field (21) to which (25) is inserted and the
pressure (22) where $d\Omega$ is the solid angle element. In the outer region
the same quantities are

$\displaystyle L_{out}=$
$\displaystyle\frac{1}{2\kappa^{2}}R*1-F\wedge*F+\lambda_{a}\wedge T^{a}\;,$
(28) $\displaystyle ds^{2}_{out}=$
$\displaystyle-\left(1-\frac{2M}{r}+\frac{\kappa^{2}Q^{2}}{r^{2}}\right)dt^{2}+\left(1-\frac{2M}{r}+\frac{\kappa^{2}Q^{2}}{r^{2}}\right)^{-1}dr^{2}+r^{2}d\Omega^{2}\;,$
(29)

and we have the electric field $E(r)=Q/r^{2}$ and the pressure $p(r)=0$, where
$M$ and $Q$ are the total mass and charge of the dark star, respectively. In
the interior of the star, there is a specific fluid which has electromagnetic
and gravitational fields with very high density. The charge of a dark star in
the volume with the radius $r$ is derived from the modified Maxwell equation
(9) by taking the volume integral of the current density 3-form $J$

$\displaystyle q(r)=\frac{1}{4\pi}\int_{V}J=YEr^{2}\;.$ (30)

Here the integral is evaluated with help of the Stokes theorem. So, we
calculate the total charge of the gravastar within the volume with the radius
$r$ by combining the equations (21) and (23) with the equation (30)

$\displaystyle
q^{2}(r)=\frac{5C_{1}a^{2}(1+k)r^{{6(1-k)}/{(1+k)}}}{2\kappa^{2}}\;.$ (31)

We see that $k$ must be in the range $-1<k\leq 1$ in order for the charge to
be regular at the origin. Additionally when we look at the solutions (21) and
(22), we discover that the parameter $k$ can take values in the range
$-1<k<1$.

As a result of the continuity condition of the metric, by matching the
exterior and the interior metrics at the surface, $r=r_{b}$, we arrive at

$\displaystyle
a=\frac{1-\sqrt{1-{2M}/{r_{b}}+{\kappa^{2}Q^{2}}/{r_{b}^{2}}}}{r_{b}^{2}}\;.$
(32)

In addition, by applying the condition that the pressure at the surface is
zero, the equation (22) gives one more equation among the parameters

$\displaystyle a=\frac{4}{5r_{b}^{2}}\;.$ (33)

Similarly, the continuity of the electric field on the boundary via the
equation (30) with $Y=1$ yields

$\displaystyle Q=E(r_{b})r_{b}^{2}$ (34)

where $Q$ denotes the total charge of the star with the radius $r_{b}$. If we
substitute (21) and (25) into (34), the the total charge could be re-expressed
as

$\displaystyle Q^{2}=\frac{8(1+k)r_{b}^{2}}{5\kappa^{2}}\;.$ (35)

We observe that the total charge increases linearly with the subtle constant
$k$. We are able to calculate the total gravitational mass of the star by
using (32), (33) and (35)

$\displaystyle M=\frac{4(5k+8)}{25}r_{b}\ .$ (36)

We also calculate the matter (or baryonic) mass of the star defined by the
integral of the energy density

$\displaystyle
M_{mat}=\frac{\kappa^{2}}{2}\int_{0}^{r_{b}}\rho(r)r^{2}dr=\frac{8(1-k)r_{b}}{25}\
.$ (37)

Finally we search the gravitational redshift on the surface of the star

$\displaystyle z=(1-ar_{b}^{2})^{-1}-1=4\;.$ (38)

We see that all the possible $k$ values with different mass and charge
configurations give only one redshift $z=4$. Experimentally high redshifts are
observed in dense celestial objects such as quasars. Despite the high
redshifts in general may be due to huge distance, extreme velocity or metric
expansion of space, a common alternative explanation was that they were caused
by extreme mass (gravitational redshifting) and electric charge. Thus, the
high redshift may be due to quasars formed from ultra compact dark energy
condensation at the center of galaxies, or they may be observational evidence
for gravastars.

###  III.2 Solution-2

In this section we consider the following regular metric function inside the
dark star

$\displaystyle f(r)=\sqrt{1-b_{1}r^{2}+b_{2}r^{3}}$ (39)

where $b_{1}$ and $b_{2}$ are arbitrary real constants. Again the equations
(8), (17) and (18) are used to determine the other unknown functions

$\displaystyle E^{2}(r)=$
$\displaystyle\frac{b_{2}(1+k)}{\kappa^{2}C_{2}}r^{{(11k+1)}/{(k+1})}\;,$ (40)
$\displaystyle\kappa^{2}\rho(r)=$
$\displaystyle-\kappa^{2}p(r)=(k-1)(5b_{2}r-3b_{1})\;,$ (41) $\displaystyle
Y(r)=$ $\displaystyle C_{2}r^{-{10k}/{(k+1)}}$ (42)

where $C_{2}$ is an integration constant that will be determined by the
boundary conditions. Now after calculating the Ricci scalar

$\displaystyle R=12b_{1}-20b_{2}r$ (43)

we compute the radial coordinate $r$ in terms of $R$ as
$r={3b_{1}}/{5b_{2}}-{R}/{20b_{2}}$ and then the non-minimal coupling function
(42)

$\displaystyle
Y(R)=C_{2}\left(\frac{3b_{1}}{5b_{2}}-\frac{R}{20b_{2}}\right)^{-{10k}/{(1+k)}}\;.$
(44)

Remembering that the exterior region of the star was described by the minimal
Einstein-Maxwell theory which has the Reissner-Nordström solution with the
property $R=0$, we apply the boundary condition for the non-minimal coupling
function as $Y(R)=1$ for $R=0$. It determines the integration constant

$C_{2}=\left({3b_{1}}/{5b_{2}}\right)^{{10k}/{(k+1)}}$ (45)

In conclusion, the Lagrangian and the metric take the forms, respectively, in
the interior of the star

$\displaystyle L_{in}=$
$\displaystyle\frac{1}{2\kappa^{2}}R*1-\left(1-\frac{R}{12b_{1}}\right)^{-{10k}/{(1+k)}}F\wedge*F+2A\wedge
J+L_{mat}+\lambda_{a}\wedge T^{a}\ ,$ (46) $\displaystyle ds^{2}_{in}=$
$\displaystyle-\left(1-b_{1}r^{2}+b_{2}r^{3}\right)dt^{2}+\left(1-b_{1}r^{2}+b_{2}r^{3}\right)^{-1}dr^{2}+r^{2}d\Omega^{2}\
,$ (47)

and we have the electric field (40) and the pressure (41). In the outer space,
they are

$\displaystyle L_{out}=$
$\displaystyle\frac{1}{2\kappa^{2}}R*1-F\wedge*F+\lambda_{a}\wedge T^{a}\ ,$
(48) $\displaystyle ds^{2}_{out}=$
$\displaystyle-\left(1-\frac{2M}{r}+\frac{\kappa^{2}Q^{2}}{r^{2}}\right)dt^{2}+\left(1-\frac{2M}{r}+\frac{\kappa^{2}Q^{2}}{r^{2}}\right)^{-1}dr^{2}+r^{2}d\Omega^{2}\
,$ (49)

with the electric field $E(r)=Q/r^{2}$ and the pressure $p(r)=0$ where $Q$ and
$M$ are the total charge and mass of the dark star, respectively. By using
(30) and (40), we calculate the electric charge of the star inside the sphere
with radius $r$

$\displaystyle
q^{2}(r)=\frac{C_{2}b_{2}(1+k)}{\kappa^{2}}r^{-{5(k-1)}/{(1+k)}}\;.$ (50)

Continuities of the metric functions and the pressure on the boundary surface
at $r=r_{b}$ generate

$\displaystyle 1-b_{1}r^{2}+b_{2}r^{3}=$ $\displaystyle
1-\frac{2M}{r_{b}}+\frac{\kappa^{2}Q^{2}}{r_{b}^{2}}\;,$ (51) $\displaystyle
b_{2}=$ $\displaystyle\frac{3b_{1}}{5r_{b}}\;.$ (52)

By inserting (45) and (52) into (50) on the surface we can calculate the total
charge of the star $Q$

$\displaystyle Q^{2}=\frac{3\beta(1+k)r_{b}^{2}}{5\kappa^{2}}$ (53)

where we defined a new dimensionless constant $\beta$ instead of the
physically dimensional constant $b_{1}$ as $\beta=b_{1}r_{b}^{2}$. We remark
that the increasing $k$ and $\beta$ values increase the total charge. Besides
we calculate the total gravitational mass which is the mass appeared in the
metric (49) and the matter mass defined by the volume integral of the energy
density (41) over the star as follows

$\displaystyle M$ $\displaystyle=$
$\displaystyle\frac{(3k+5)\beta}{10}r_{b}\;,$ (54) $\displaystyle M_{mat}$
$\displaystyle=$ $\displaystyle\frac{(1-k)\beta}{8}r_{b}\;.$ (55)

Besides, the gravitational redshift for the surface of the star is computed as

$\displaystyle
z=(1-b_{1}r_{b}^{2}+b_{2}r_{b}^{3})^{-1}-1=\frac{1}{1-\frac{2}{5}\beta}-1\;.$
(56)

Compared with our previous model which has only one free parameter $k$, in
this model we have a new arbitrary parameter $\beta$ along with $k$. As usual
the observational extreme values will guide us to determine the limits of
$\beta$. According to the literature F_Wang2021 ; B_W_Lyke2020 ; E_Momjian2021
more than 750,000 quasars have been found as for today. All observed quasar
spectra have redshifts between 0.056 and 7.642. Accordingly we obtain the
interval for $\beta$ as $0<\beta<2.47$. In addition, a much larger redshift
can be obtained in the range $2.47\leq\beta<2.5$. Future observation results
in this direction may help determine the parameter $\beta$ for each dark
quasar.

##  IV Conclusion

We have constructed two exact, new gravastar solutions to the non-minimally
coupled $Y(R)F^{2}$ theory. The interior of the gravastar consists of the dark
energy condensate which has the equation of state $p=-\rho$ and the exterior
is vacuum describing by the Reissner-Nordström geometry. Then the thin shell
proposed by Mazur and Muttola Mazur20042  is replaced by an infinitely thin
surface, which continuously connects these two regions. Thus, in this study,
all the physical quantities are continuous and the exterior geometry
continuously matches with the interior geometry on the surface. We obtain the
total gravitational mass, matter mass, total electric charge and gravitational
surface redshift in terms of the free parameters of the model. In the first
solution we have only one free parameter $k$ and in the second solution two
parameters $k$ and $\beta$. We determined the ranges of these parameters from
regularity of the physical quantities. The second solution gives a wide
redshift range from zero to infinity depending on the $\beta$ parameter, while
the first solution gives a single redshift value. In the literature, although
there is an upper limit of the surface redshift for uncharged general
relativistic objects as $z<2$ Buchdahl , the surface redshift can reach much
higher values such as $z=5$ Bohmer-Harko2006  and $z=5.211$ Ivanov2002 ;
Guven1999  for anisotropic stars. Furthermore, this limit can be exceeded in
the charged case and it is proportional to the charge parameter $Q$ Mak1 .
Moreover, by considering alternative gravity models such as Kaluza-Klein
theory this limit can reach much higher values Zhang2006  which is very close
to our result $z=4$. The authors of a very recent paper inform the discovery
of a luminous quasar at $z=7.642$, J0313-806 and the most distant quasar
observed so far F_Wang2021 . Besides, the surface redshift can be indefinitely
large for quasiblack holes Arbanil2014 ; Lemos2004 ; Bronnikov2014 .
Therefore, these high redshift values in our model may be due to quasars,
quasi-black holes or gravastars formed from ultra compact dark energy
condensation at the center of galaxies.

## References

  * (1) Pawel O. Mazur and Emil Mottola, “Gravitational condensate stars: An alternative to black holes,” (2001), arXiv:gr-qc/0109035.
  * (2) P. O. Mazur and E. Mottola, “Dark energy and condensate stars: Casimir energy in the large,” arXiv:gr-qc/0405111.
  * (3) P. O. Mazur and E. Mottola, “Gravitational vacuum condensate stars,” Proc. Nat. Acad. Sci. 111, 9545 (2004) [arXiv:gr-qc/0407075].
  * (4) Tiberiu Harko, Zoltan Kovacs, Francisco S. N. Lobo, Can accretion disk properties distinguish gravastars from black holes? Class. Quant. Grav. 26:215006, (2009).
  * (5) G. Chapline, E. Hohlfeld, R. B. Laughlin and D. I. Santiago, “Quantum phase transitions and the breakdown of classical general relativity,” Int. J. Mod. Phys. A 18 (2003) 3587 [arXiv:gr-qc/0012094].
  * (6) G. Chapline, R. Laughlin, amd D. Santiago, “Emergent relativity and the physics of black hole interiors”, published in Artificial black holes, edited by Mario Novello, Matt Visser, and Grigori Volovik. (World Scientific, Singapore, 2002).
  * (7) M. Visser and D. L. Wiltshire, “Stable gravastars — an alternative to black holes?,” Class. Quant. Grav. 21, 1135 (2004) [arXiv:gr-qc/0310107].
  * (8) B. M. N. Carter, Stable gravastars with generalised exteriors Class. Quant. Grav. 22 4551-62 (2005)
  * (9) F.S.N. Lobo, Stable dark energy stars Class. Quant. Grav. 23 1525-41, (2006)
  * (10) R. Stettner, On the stability of homogeneous, spherically symmetric, charged fluids in relativity, Ann. Phys. 80, 212 (1973).
  * (11) A. Krasinski, Inhomogeneous cosmological models, Cambridge University Press (1997).
  * (12) R. Sharma, S. Mukherjee and S.D. Maharaj, General Solution for a Class of Static Charged Spheres, Gen. Relativ. Gravit. 33, 999 (2001).
  * (13) K. S. Cheng, Z. G. Dai and T. Lu, Strange stars and related astrophysical phenomena, Int. J. Mod. Phys. D 7, 139, (1998).
  * (14) M. K. Mak, and T. Harko, Quark stars admitting a one-parameter group of conformal motions, Int. J. Mod. Phys. D, 13, 149 (2004).
  * (15) D. Horvat, S. Ilijic and A. Marunovic, ”Electrically charged gravastar configurations”, Class. Quantum Grav. 26 (2009) 025003, [arXiv:gr-qc/0807.2051].
  * (16) A. A. Usmani, F. Rahaman, S. Ray, K. K. Nandi, P. K. F. Kuhfittig, Sk. A. Rakib and Z. Hasan, Charged gravastars admitting conformal motion, Phys. Lett. B 701 (2011) 388.
  * (17) . P. Bhar, Higher dimensional charged gravastar admitting conformal motion, Astrophys. Space Sci. 354 (2014) 457
  * (18) N. Bilic, G. B. Tupper, and R. D. Viollier, Born-Infeld phantom gravastars JCAP 2-13 (2006)
  * (19) F.S.N. Lobo and A.V.B. Arellano, Gravastars supported by nonlinear electrodynamics, Class. Quant. Grav. 24 1069-88, (2007).
  * (20) C. Cattoen, T. Faber and M. Visser, Gravastars must have anisotropic pressures, Class. Quant. Grav. 22, 4189 (2005) doi:10.1088/0264-9381/22/20/002, gr-qc/0505137.
  * (21) A. DeBenedictis, D. Horvat, S. Ilijic, S. Kloster and K. S. Viswanathan, Gravastar Solutions with Continuous Pressures and Equation of State, Class. Quant. Grav. 23, 2303 (2006).
  * (22) Stoytcho S. Yazadjiev, Exact dark energy star solutions, Phys. Rev. D 83, 127501, (2011).
  * (23) T. Chiba, $1/R$ gravity and Scalar-Tensor Gravity, Phys. Lett. B 575, 1 (2003)
  * (24) S. Capozziello, S. Nojiri, S. D. Odintsov, and A. Troisi, Cosmological viability of f(R)-gravity as an ideal fluid and its compatibility with a matter dominated phase, Phys. Lett. B 639, 135 (2006);
  * (25) S. M. Carroll, V. Duvvuri, M. Trodden, and M. S. Turner, Is cosmic speed-up due to new gravitational physics?, Phys. Rev. D 70, 043528 (2004).
  * (26) T. P. Sotiriou and V. Faraoni, $f(R)$ theories of gravity Rev. Mod. Phys. 82, 451 (2010);
  * (27) F. S. N. Lobo, The dark side of gravity: Modified theories of gravity, Dark Energy-Current Advances and Ideas, 173-204 (2009), arXiv:0807.1640 [gr-qc]
  * (28) T. Harko, F.S.N. Lobo, S. Nojiri, S.D. Odintsov, $f(R,T)$ gravity, Phys. Rev. D 84, 024020 (2011). arXiv:1104.2669
  * (29) A. Das, S. Ghosh, B. Guha, S. Das, F. Rahaman, S. Ray, Gravastars in $f(R,T)$ gravity, Phys. Rev. D 95 (2017) 124011.
  * (30) P. Bhar and P. Rej, Stable and self-consistent charged gravastar model within the framework of f(R,T) gravity, Eur. Phys. J. C 81 763, (2021)
  * (31) M. Shamir, M. Ahmad, Gravastars in $f(G,T)$ gravity, Phys. Rev. D 97 (2018) 104031.
  * (32) S. Ghosh, F. Rahaman, B. Guha, S. Ray, Charged gravastars in higher dimensions, Phys. Lett. B 767 (2017) 380.
  * (33) Z. Yousaf, K. Bamba, M. Bhatti, U. Ghafoor, Charged gravastars in modified gravity, Phys. Rev. D 100 (2019) 024062.
  * (34) A. Övgün, A. Banerjee, K. Jusufi, Charged thin-shell gravastars in noncommutative geometry. Eur. Phys. J. C 77, 566 (2017).
  * (35) T. Dereli, Ö. Sert, Non-minimal $ln(R)F^{2}$ couplings of electromagnetic fields to gravity: static, spherically symmetric solutions, Eur. Phys. J. C 71, 3, 1589 (2011).
  * (36) Ö. Sert, Gravity and electromagnetism with $Y(R)F^{2}$-type coupling and magnetic monopole solutions, Eur. Phys. J. Plus 127: 152 (2012).
  * (37) Ö. Sert, Electromagnetic duality and new solutions of the non-minimally coupled Y(R)-Maxwell gravity, Mod. Phys. Lett. A 28, 12, 1350049 (2013).
  * (38) M. Adak, Ö. Akarsu, T. Dereli, Ö. Sert, Anisotropic inflation with a non-minimally coupled electromagnetic field to gravity, JCAP 11 026 (2017).
  * (39) K. Bamba and S. D. Odintsov, Inflation and late-time cosmic acceleration in non-minimal Maxwell-F(R) gravity and the generation of large-scale magnetic fields, JCAP (04), 024 (2008).
  * (40) K. Bamba, S. Nojiri and S. D. Odintsov, Future of the universe in modified gravitational theories: approaching to the finite-time future singularity, JCAP, (10), 045, (2008).
  * (41) T. Dereli, Ö. Sert, Non-minimal $R^{\beta}F^{2}$-coupled electromagnetic fields to gravity and static. Spherically Symmetric Solutions, Mod. Phys. Lett. A 26, 20, 1487-1494 (2011).
  * (42) T. Dereli, G. Üçoluk, Kaluza-Klein reduction of generalised theories of gravity and non-minimal gauge couplings, Class. Q. Grav. 7, 1109 (1990).
  * (43) A. Baykal, T. Dereli, Nonminimally coupled Einstein-Maxwell model in a non-Riemann spacetime with torsion, Phys.Rev. D92 6, 065018 (2015).
  * (44) Ö. Sert, Regular black hole solutions of the non-minimally coupled $Y(R)F^{2}$ gravity, Journal of Mathematical Physics 57, 032501 (2016).
  * (45) T. Dereli, Ö. Sert, Nonminimally coupled gravitational and electromagnetic fields: pp-wave solutions, Phys. Rev. D 83, 065005 (2011).
  * (46) Ö. Sert, Radiation fluid stars in the non-minimally coupled $Y(R)F^{2}$ gravity, Eur. Phys. J. C 77:97 (2017).
  * (47) Ö. Sert, Compact stars in the non-minimally coupled electromagnetic fields to gravity, Eur. Phys. J. C 78: 241, (2018).
  * (48) Özcan Sert, Fatma Çeliktaş, Muzaffer Adak, Anisotropic Stars in the Non-minimal $Y(R)F^{2}$ Gravity Eur. Phys. J. C, 78 10 (2018) 824
  * (49) B.W. Lyke et al., ApJS 250 (2020) 8, DOI: 10.3847/1538-4365/aba623, arXiv:2007.09001
  * (50) E. Momjian et al., AJ 161 (2021) 207, DOI: 10.3847/1538-3881/abe6ae, arXiv:2103.03481.
  * (51) F. Wang et al., ApJL 907 (2021) L1, DOI: https://doi.org/10.3847/2041-8213/abd8c6, arXiv:2101.03179
  * (52) H.A. Buchdahl, General Relativistic Fluid Spheres, Phys. Rev., 116 1027 (1959).
  * (53) C. G. Boehmer, T. Harko, Bounds on the basic physical parameters for anisotropic compact general relativistic objects, Class.Quant.Grav. 23 (2006) 6479-6491, DOI: 10.1088/0264-9381/23/22/023, arXiv:gr-qc/0609061
  * (54) B. V. Ivanov, Maximum bounds on the surface redshift of anisotropic stars, Phys. Rev. D 65, 104011 (2002), DOI:https://doi.org/10.1103/PhysRevD.65.104011
  * (55) J. Guven and N. O. Murchadha, Bounds on $2m/R$ for static spherical objects, Phys. Rev. D 60, 084020, (1999).
  * (56) M. K. Mak, Peter N. Dobson Jr., T. Harko, Maximum Mass-Radius Ratios for Charged Compact General Relativistic Objects, Europhys.Lett. 55 310-316 (2001).
  * (57) T. X. Zhang, Electric Redshift and Quasars, ApJ 636 L61, (2006).
  * (58) J.D. Arbanil, J. P. S. Lemos, and V. T. Zanchin, Incompressible relativistic spheres: Electrically charged stars, compactness bounds, and quasiblack hole configurations, Phys. Rev. D, 89 (2014) 104054.
  * (59) J. P. S. Lemos and E. Weinberg, “Quasiblack holes from extremal charged dust”, Phys. Rev. D 69, 104004 (2004); arXiv:gr-qc/0311051.
  * (60) K. A. Bronnikov, J. C. Fabris, R. Silveira, and O. B. Zaslavskii, “Dilaton gravity, (quasi)black holes, and scalar charge” (2014), arXiv:1312.4891 [gr-qc].