# The Gaussian Noise Model in the Presence of Inter-channel Stimulated Raman
Scattering

Daniel Semrau, and Polina Bayvel This work was supported by a UK EPSRC
programme grant UNLOC (EP/J017582/1) and a Doctoral Training Partnership (DTP)
studentship for Daniel Semrau.D. Semrau and P. Bayvel are with the Optical
Networks Group, University College London, London WC1E 7JE, U.K. (e-mail:
{uceedfs<EMAIL_ADDRESS>
###### Abstract

A Gaussian noise (GN) model is presented that properly accounts for an
arbitrary frequency dependent signal power profile along the link. This
enables the evaluation of the impact of inter-channel stimulated Raman
scattering (ISRS) on the optical Kerr nonlinearity. Additionally, the
frequency dependent fiber attenuation can be taken into account and
transmission systems that use hybrid amplification schemes can be modeled,
where distributed Raman amplification is partly applied over the optical
spectrum. To include the latter two cases, a set of coupled ordinary
differential equations must be numerically solved in order to obtain the
signal power profile yielding a semi-analytical model. However for lumped
amplification and negligible variation in fiber attenuation, a less complex
and fully analytical model is presented which is referred to as the ISRS GN
model. The derived model is exact to first-order for Gaussian modulated
signals and extensively validated by numerical split-step simulations. A
maximum deviation of $0.1$ dB in nonlinear interference power between
simulations and the ISRS GN model is found. The model is applied to a
transmission system that occupies an optical bandwidth of $10$ THz,
representing the entire C+L band. At optimum launch power, changes of up to
$2$ dB in nonlinear interference power due to ISRS are reported. Furthermore,
comparable models published in the literature are benchmarked against the ISRS
GN model.

###### Index Terms:

Optical fiber communications, Gaussian noise model, Nonlinear interference,
nonlinear distortion, Stimulated Raman Scattering, First-order perturbation,
C+L band transmission

##  I Introduction

Analytical models that predict the performance degradation in optical fiber
communications due to Kerr nonlinearity have enjoyed significant popularity in
recent years. Most approaches analytically solve the nonlinear Schrödinger
equation using a first-order perturbation approach with respect to the
nonlinearity coefficient. The resulting expressions offer unique insight into
the underlaying parameter dependencies and are key enablers for efficient
system design [1], rapid achievable rate estimations of point-to-point links
[2, 3, 4] and physical layer aware network optimization. The latter is
essential for optical network abstraction and virtualization leading to
optimal and intelligent techniques to maximize optical network capacity [5].
Analytical models also offer a significant reduction in computational
complexity with minor inaccuracies compared to split-step simulations and
experiments [6, 7, 8, 9, 10, 11].

The literature offers a wide range of analytical models varying in accuracy
and complexity [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23]. The first
approaches in the context of modern coherent receivers and dispersion
uncompensated links date back to 1993 and 2002 [12, 13], enabling the
computation of the perturbation caused by Kerr nonlinearity. Similar results
were independently derived by other groups and the model became widely known
as the Gaussian noise (GN) model [14, 24, 15, 16, 22]. A key assumption of the
works is the signal Gaussianity assumption, which is that the signal can be
written as a Gaussian process at the fiber input. As a result of this
assumption, the GN model relies on large accumulated dispersion [20, Sec. 6]
and signals with high cardinality [19, Sec. 4]. Two conditions that are
satisfied in most cases of modern coherent fiber communication.

The popularity of the GN model undoubtedly origins in its moderate complexity.
However, as a result it fails to predict certain properties of nonlinear
interference such as modulation format dependence [17, 19, 18], symbol rate
dependence [25, 26, 27, 28, 29], nonlinear phase noise [19, 30], long temporal
correlations [18, 19] and the dependence on the memory length of the fiber-
optic channel [31]. In order to account for those properties, significantly
more complex models have been proposed [17, 18, 19, 20, 21, 31]. Comprehensive
overviews can be found in [32, 33].

The impact of the mentioned properties are usually small for lumped,
dispersion unmanaged, multi-span systems that use high-order modulation
formats and the GN model can be considered sufficiently accurate. Recently,
the conventional GN model was experimentally validated for the central channel
and optical bandwidths up to $7.3$ THz with a deviation of only $0.4$ dB in
nonlinear interference (NLI) power [34, 6].

An assumption of all above-mentioned works is that every frequency component
experiences the same power evolution along the link. They are therefore
inaccurate in the prediction of ultra-wideband transmission systems where the
variation of the fiber attenuation is not negligible and for bandwidths where
inter-channel stimulated Raman scattering (ISRS) is significant.

Inter-channel stimulated Raman scattering (ISRS) is a non-parametric nonlinear
process that amplifies low frequency components at the expensive of high
frequency components within the same optical signal. In modern optical
communications that use coherent technology in combination with high
dispersive links, ISRS effectively introduces a different power profile for
each frequency component [35, 36, 37]. For C band transmission (approximately
$5$ THz), as defined from the availability of the erbium doped fiber amplifier
(EDFA), ISRS is not significant and its impact is negligible in most cases.
However, for systems that occupy the entire C+L band (approximately $10$ THz)
or beyond, ISRS becomes significant and it must be taken into account. Beyond
C+L band transmission could be enabled by lumped Raman amplification or the
use of other dopants such as bismuth [38] or quantum dots in semiconductor
amplifiers [39] [40].

The first approach to include ISRS in the GN model was published in [41]. The
authors approximated the signal power profile with an exponential decay using
an effective attenuation coefficient that matches the (frequency dependent)
effective length in the presence of ISRS. Although first conclusions on the
impact of ISRS could be drawn, this method exhibits two shortcomings. First,
when ISRS is significant, the resulting signal power profile does not resemble
an exponential decay, particularly in the beginning of the fiber span, where
Kerr nonlinearity prevails. Second, the work assumes that, during the four-
wave mixing (FWM) process, every participating frequency component attenuates
in the same manner as the channel of interest. In other words, for the induced
nonlinear perturbation at frequency $f$, every frequency component in the
triplet $\left(f,f_{1},f_{2}\right)$ attenuates as $f$ during the nonlinear
mixing. This does not accurately represent the FWM process, as each frequency
component attenuates in a different manner.

A more rigorous approach to include ISRS in the GN model was published in [42,
43]. In both works, the channel under test (i.e. frequency component $f$)
attenuates precisely according to the signal power profile $g\left(f\right)$
resulting from ISRS (or any arbitrary profile), lifting the exponential decay
assumption in [41]. However, for an attenuation profile that is linear in
frequency (like the one resulting from ISRS), the frequencies in the triplet
$\left(f,f_{1},f_{2}\right)$ also attenuate according to $f$ during the FWM
process. Therefore, this approach overestimates the impact of ISRS on the Kerr
nonlinearity and the frequency dependent signal power profile is not
accurately taken into account.

In this paper, a Gaussian noise model is presented that properly accounts for
any arbitrary frequency dependent signal power profile. This enables the
modeling of nonlinear interference in ultra-wideband regimes where ISRS is
significant. The model is referred to as the ISRS GN model. Additionally, the
variation in fiber attenuation and hybrid amplification schemes can be
included. In general, the signal power profile is obtained by numerically
solving a set of coupled ordinary differential equations (ODE). This yields a
semi-analytical ISRS GN model which relies on a numerical ODE solver.

However, for lumped amplification and negligible variation in fiber
attenuation, a fully analytically model is derived based on a linear
approximation on the ISRS gain function. This reduces model as well as
computational complexity. The analytical ISRS GN model holds for bandwidths up
to approximately $15$ THz after which the Raman gain function cannot be
considered linear anymore.

This paper is organized as follows. In Sec. II, the ISRS GN model is presented
and its key derivation steps are briefly outlined. The detailed derivation can
be found in the Appendix A. The model is extensively validated by split-step
simulations in Sec. III and applied to a C+L band transmission system based on
standard single mode fiber (SMF) spans in Sec. IV. In Sec. III and IV, the
results in [41, 42, 43] are benchmarked against the ISRS GN model.

##  II The ISRS GN model

In order to maximize the information throughput of an optical communication
system, it is vital to evaluate and maximize the performance of each
individual channel that is transmitted. After coherent detection and
electronic dispersion compensation, the channel dependent signal-to-noise
ratio (SNR) can be calculated as

$\begin{split}\textnormal{SNR}_{i}\approx\frac{P_{i}}{nP_{\textnormal{ASE}}+\eta_{n}P_{i}^{3}},\end{split}$
(1)

where $P_{i}$ is the launch power of channel $i$, $P_{\textnormal{ASE}}$ is
the amplified spontaneous emission (ASE) noise power over the channel
bandwidth and $\eta_{n}$ is the nonlinear interference coefficient after $n$
spans. The SNR is a function of the spectral location of the channel within
the optical signal as $P_{\textnormal{ASE}}$ and $\eta_{n}$ are frequency
dependent quantities. When the channel bandwidth $B_{\text{ch}}$ is small
compared to the total optical bandwidth $B$, the power spectral density (PSD)
of the NLI can be considered locally flat and $\eta_{n}$ can be approximated
as

$\begin{split}\eta_{n}\left(f_{i}\right)=\int_{-\frac{B_{\text{ch}}}{2}}^{\frac{B_{\text{ch}}}{2}}\frac{G\left(\nu+f_{i}\right)}{P_{i}^{3}}d\nu\approx\frac{B_{\text{ch}}}{P_{i}^{3}}G\left(f_{i}\right),\end{split}$
(2)

where $f_{i}$ is the center frequency of channel $i$ and $G\left(f\right)$ is
the PSD of the nonlinear interference. For later use, we further define the
total optical launch power as

$\begin{split}P_{\text{tot}}=\int G_{\text{Tx}}(\nu)d\nu=\sum_{\forall
i}P_{i},\end{split}$ (3)

where $G_{\text{Tx}}$ is the input PSD of the entire optical signal. The aim
of the next sections is to find an analytical expression for the nonlinear
interference PSD $G\left(f\right)$, in order to compute the channel dependent
SNR or similar performance metrics.

###  II-A The nonlinear interference power

Various models have been proposed in the past in order to calculated the NLI
power, where it is generally assumed that all frequency components attenuate
in the same manner along a fiber span [12, 13, 14, 15, 16, 17, 18, 19, 20, 21,
22, 23, 24]. However, this assumption is no longer satisfied when transmission
systems operate at large optical bandwidths (C+L band and beyond). This is
because each frequency component undergoes a different power evolution during
propagation as a result of ISRS and a frequency dependent attenuation
coefficient. In addition, frequency dependent signal power profiles are
present in hybrid amplification schemes, where part of the spectrum is
amplified using distributed Raman amplifiers in order to reduce the ASE noise
power for longer wavelengths or extend the amplification window beyond
conventional EDFA’s [44, 45, 46].

For a frequency dependent power evolution, the PSD of the NLI after one span
is derived in Appendix A and it is found to be

$\begin{split}&G(f)=\frac{16}{27}\gamma^{2}\int df_{1}\int df_{2}\
G_{\text{Tx}}(f_{1})G_{\text{Tx}}(f_{2})G_{\text{Tx}}(f_{1}+f_{2}-f)\\\
&\cdot\left|\int_{0}^{L}d\zeta\
\sqrt{\frac{\rho(\zeta,f_{1})\rho(\zeta,f_{2})\rho(\zeta,f_{1}+f_{2}-f)}{\rho(\zeta,f)}}e^{j\phi\left(f_{1},f_{2},f,\zeta\right)}\right|^{2},\\\
\end{split}$ (4)

where
$\phi=-4\pi^{2}\beta_{2}\left[(f_{1}-f)(f_{2}-f)+\pi\beta_{3}(f_{1}+f_{2})\right]\zeta$,
$\gamma$ is the nonlinearity coefficient and $\rho(z,f)$ is the normalized
signal power profile. For example, the normalized signal power profile of a
passive fiber with constant attenuation coefficient $\alpha$ is
$\rho(z,f)=e^{-\alpha z}$. For multi-span systems, where each span has
identical fiber parameters and signal power profiles, the phased-array term

$\begin{split}&\left|\frac{\sin\left[\frac{1}{2}n\phi\left(f_{1},f_{2},f,L\right)\right]}{\sin\left[\frac{1}{2}\phi\left(f_{1},f_{2},f,L\right)\right]}\right|^{2}\\\
\end{split}$ (5)

must be inserted into the integral in LABEL:eq:general_G. For multi-span
systems with non-identical signal power profiles for each span (e.g. when non-
ideal gain flattening filters are considered), the phased-array term cannot be
used. Instead Eq. (LABEL:eq:general_G) must be considered, where $L$ must be
reinterpreted as the link length and $\rho(z,f)$ as the signal power profile
for the entire link. However, identical spans are considered for the remainder
of this work.

We note that (LABEL:eq:general_G) is different than the result derived in [42,
Eq. (16)] and [43, Eq. (18)], where $\rho(\zeta,f_{1}+f_{2}-f)$ and
$\rho(\zeta,f)$ are swapped. 111As a consequence of the different result, the
$f_{1}$ and $f_{2}$ dependence vanishes, for power profiles of the form
$\rho(z,f)=e^{a(z)\cdot f+b(z)}$ (as the one resulting from ISRS). This means
that in the nonlinear process all three frequencies in the triplet
($f$,$f_{1}$,$f_{2}$) attenuate according to frequency $f$ which overestimates
the impact of ISRS. In contrast, in Eq. (LABEL:eq:general_G) each frequency
($f_{1}$,$f_{2}$,$f$) correctly attenuates with its respective power profile.
However, in section III, it is shown by split-step simulations that
(LABEL:eq:general_G) is the correct formula.

Eq. (LABEL:eq:general_G) can be used to model any arbitrary frequency
dependent signal power profile and it is therefore suitable to evaluate the
impact of ISRS on the optical Kerr nonlinearity.

###  II-B Inter-channel stimulated Raman scattering

In the following, the PSD of the NLI in the presence of ISRS is presented,
which is hereafter referred to as the ISRS GN model. In modern dispersion
uncompensated links, ISRS effectively amplifies high wavelength components at
the expense of low wavelength components [35, 36, 37]. The resulting frequency
dependent signal power profile can be obtained by solving a set of coupled
ordinary differential equations [47, Eq. (3)]

$\displaystyle\begin{split}&\frac{\partial P_{i}}{\partial z}=\\\
&-\underbrace{\displaystyle\sum_{j=i+1}^{M}\frac{1}{2}\frac{f_{j}}{f_{i}}g_{\text{r}}(\Delta
f)P_{j}P_{i}}_{\textnormal{ISRS
loss}}+\underbrace{\displaystyle\sum_{j=1}^{i-1}\frac{1}{2}g_{\text{r}}(\Delta
f)P_{j}P_{i}}_{\textnormal{ISRS
gain}}-\alpha\left(f_{i}\right)P_{i},\end{split}$ (6)

where $M$ is the total number of WDM channels, $g_{r}(\Delta f)$ is the
normalized (by the effective core area $A_{\text{eff}}$) Raman gain spectrum
for a frequency separation $\Delta f=\left|f_{i}-f_{j}\right|$ and
$\alpha\left(f\right)$ is the frequency dependent attenuation coefficient. The
index of the channel with the highest center frequency is $i=1$. Eq. (6) can
be extended to include distributed Raman amplification using [48, Eq. 1]. Eq.
(6) has no general analytical solution and must be solved numerically. The
obtained power profile can then (after normalization) be used in
(LABEL:eq:general_G) to yield a semi-analytical model that accurately accounts
for ISRS, a frequency dependent attenuation coefficient and distributed Raman
amplification.

The variation of the attenuation coefficient does typically not exceed $0.01$
dB/km across the C+L band ranging from 1530 nm to 1625 nm and might be
negligible depending on accuracy and computational complexity requirements
[49]. The impact of a frequency dependent attenuation on the NLI coefficient
can be loosely upper bounded by assuming that every frequency component
attenuates according to the minimum in one case and according to the maximum
attenuation coefficient in another case. The resulting maximum deviation in
NLI coefficient is then

$\begin{split}\Delta\eta_{1}\left[\text{dB}\right]<\left(\frac{\alpha_{\text{min}}}{\alpha_{\text{max}}}\right)\left[\text{dB}\right],\end{split}$
(7)

where $\left(\cdot\right)\left[\text{dB}\right]$ means conversion to decibel
and $\alpha_{\text{min}}$ and $\alpha_{\text{max}}$ is the minimum and maximum
attenuation coefficient, respectively. In (7) it was assumed that $e^{-\alpha
L}\ll 1$,
$\text{ln}\left(\pi^{2}B^{2}|\beta_{2}|/\alpha_{\text{min}}\right)\approx\text{ln}\left[\pi^{2}B^{2}|\beta_{2}|/\alpha_{\text{max}}\right]$
and [50, Eq. 5] was used. For an attenuation deviation of $0.01$ dB/km over 95
nm, (7) yields a maximum deviation of
$\Delta\eta_{1}\left[\text{dB}\right]<0.2\ \text{dB}$.

This contribution might be deemed negligible and the prevailing effect that
causes a frequency dependent signal power profile is inter-channel stimulated
Raman scattering. Eq. (6) can then be solved analytically when the Raman gain
spectrum is assumed to be linear up to approximately $14$ THz (i.e. up to the
Stokes shift). The normalized signal power profile for a spectral component
$f$ is then given by [51, Eq. (7)]

$\begin{split}\rho(z,f)=\frac{P_{\text{tot}}e^{-\alpha
z-P_{\text{tot}}C_{\text{r}}L_{\text{eff}}f}}{\int
G_{\text{Tx}}(\nu)e^{P_{\text{tot}}C_{\text{r}}L_{\text{eff}}\nu}d\nu},\end{split}$
(8)

where $C_{\text{r}}$ is the slope of a linear regression of the normalized
Raman gain spectrum $g_{r}(\Delta f)$ and
$L_{\text{eff}}=\frac{1-\exp\left(-\alpha z\right)}{\alpha}$. The $z$
dependence in $L_{\text{eff}}$ is suppressed for notational convenience.

The ISRS gain of a $10$ THz signal after $100$ km propagation, obtained from
numerically solving (6) using the Raman gain spectrum as in [52] and its
analytical approximation (8) are shown in Fig. 1. The precise functions that
have been used can be found in [41, Fig. 1]. For a relatively high optical
launch power of $28$ dBm, the average deviation between the numerical solution
and its approximation is $0.18$ dB which can be considered negligible.
Therefore, the analytical solution (8) is sufficiently accurate for modeling
the nonlinear interference power.

Substituting (8) in (LABEL:eq:general_G) yields the reference formula of the
analytical ISRS GN model (9) as

$\begin{split}G(f)&=\frac{16}{27}\gamma^{2}\int df_{1}\int df_{2}\
G_{\text{Tx}}(f_{1})G_{\text{Tx}}(f_{2})G_{\text{Tx}}(f_{1}+f_{2}-f)\\\
&\cdot\left|\int_{0}^{L}d\zeta\ \frac{P_{\text{tot}}e^{-\alpha\zeta-
P_{0}C_{\text{r}}L_{\text{eff}}(f_{1}+f_{2}-f)}}{\int
G_{\text{Tx}}(\nu)e^{P_{\text{tot}}C_{\text{r}}L_{\text{eff}}\nu}d\nu}e^{j\phi\left(f_{1},f_{2},f,\zeta\right)}\right|^{2}.\end{split}$
(9)

Eq. (9) is a key result of this work which is extensively validated in Sec.
III and further applied to a C+L band case study in Sec. IV.

Figure 1: The net gain due to ISRS as a function of channel frequency obtained
by solving the set of coupled differential equations (6) shown in dotted lines
and its analytical approximation (8) shown in solid lines for a variety of
total optical launch powers $P_{\text{tot}}$.

It is useful to analyze Eq. (8) in more detail. After trivial algebraic
manipulations, we obtain that the power transfer between the spectral edges
can be computed as

$\begin{split}\Delta\rho\left(z\right)\left[\text{dB}\right]=4.3\cdot
P_{\text{tot}}C_{\text{r}}L_{\text{eff}}B,\end{split}$ (10)

which is independent of the spectral distribution of the input power. For
modern fiber parameters $L_{\text{eff}}=26$ km, $C_{\text{r}}=0.008$
$1$/W/km/THz (approximately corresponding to a Corning® Vascade® EX2000 fiber
with $A_{\text{eff}}=111$ $\mu$m) and an optimum launch power of $40$ fW/Hz
(corresponding to $2$ dBm over $40$ GHz) as in [6], an ISRS power transfer of
$\Delta\rho=1$ dB is present at a bandwidth of $5.3$ THz and $\Delta\rho>4$ dB
for bandwidths larger than $10.6$ THz.

It should be further noted that the ISRS gain is independent of the Raman gain
slope and the optical launch power as long as their product is kept constant.
For a given optical bandwidth, this also holds for the nonlinear interference
coefficient. This will be useful for relating the results in Sections III and
IV to fibers with different Raman gain slopes.

##  III Numerical Validation

In this section, the analytical ISRS GN model (9) is validated by split-step
simulations for an optical fiber communication system with parameters listed
in Table I (a). Numerically solving the Manakov equation for the entire C+L
band (approximately 10 THz) is extremely challenging, due to high memory
requirements and the excessive use of very large fast Fourier transforms.

Therefore, the validation is carried out over a bandwidth of $B=1$ THz with an
artificially tenfold increased Raman gain slope $C_{\text{r}}$. Using (10),
the resulting ISRS gains for $1$ THz bandwidth coincide with the ones shown in
Fig. 1 as the product $B\cdot C_{\text{r}}$ is kept constant. The reader can
therefore conveniently refer to Fig. 1 for the ISRS gains that are present at
a particular total launch power (where the abscissa must be scaled down by
$10$).

TABLE I: System Parameters

Parameters | (a) | (b)
---|---|---
for section | III | IV
Loss ($\alpha$) [dB/km] | 0.2
Dispersion ($D$) [ps/nm/km] | 17
Dispersion slope ($S$) [ps/$\text{nm}^{2}$/km] | 0 | 0.092
NL coefficient ($\gamma$) [1/W/km] | 1.2
Raman gain slope ($C_{\text{r}}$) [1/W/km/THz] | 0.28 | 0.028
Raman gain ($C_{\text{r}}\cdot 14$ THz) [1/W/km] | 4 | 0.4
Fiber length ($L$) [km] | 100
Noise figure [dB] | 5
Symbol rate [GBd] | 10 | 50
Channel spacing ($B_{\text{ch}}$) [GHz] | 10.001 | 50.001
Number of channels | 101 | 201
Optical bandwidth ($B$) [THz] | $1.01$ | $10.05$
Roll-off factor [%] | 0.01
Number of symbols [$2^{x}$] | 15 |
Step size [m] | 5 |

Figure 2: The nonlinear interference coefficient after 1 span as a function of
channel frequency for different total launch powers is shown in a) and the NLI
change due to ISRS as a function of total launch power is shown in b). Solid
lines represent the analytical ISRS GN model (9) and markers represent results
obtained by split-step simulations. For comparison, the model in [41] is shown
with a dotted line and the model in [42,43] is shown with a dashed line for
$25$ dBm launch power in a).  Figure 3: The accumulation of NLI as a function
of span number for the most outer frequencies of the signal with and without
ISRS. Lines represent the analytical ISRS GN model (9) and markers represent
split-step simulations. The inset a) shows the coherence factor after 10 spans
and the inset b) shows a magnified area of the figure.

Gaussian modulation was implemented in order to emulate the signal Gaussianity
assumption of the GN model. Additionally, a small channel bandwidth was chosen
such that $B\gg B_{\text{ch}}$. A matched root-raised-cosine (RRC) filter was
used to obtain the output symbols and the SNR was ideally estimated as the
ratio between the variance of the transmitted symbols $E[|X|^{2}]$ and the
variance of the noise $\sigma^{2}$, where $\sigma^{2}=E[|X-Y|^{2}]$ and $Y$
represents the received symbols after digital signal processing. The nonlinear
interference coefficient was then estimated via (1) and was compared with the
predictions of the ISRS GN model via (2) and (9). A spectrally uniform launch
power was assumed yielding

$\begin{split}G_{\text{Tx}}(f)=\dfrac{P_{\text{tot}}}{B}\Pi\left(\frac{f}{B}\right),\end{split}$
(11)

for the model calculations, where $\Pi\left(x\right)$ denotes the rectangular
function. In the simulation environment, a frequency dependent power profile
was implemented to emulate the power transfer between channels due to ISRS
based on (8).

The results for the nonlinear interference coefficient after one span as a
function of the channel frequency $f_{i}$ is shown in Fig. 2a) and as a
function of total launch power in Fig. 2b). The accuracy of the ISRS GN model
is remarkable with a maximum deviation of $<0.1$ dB. The deviation is slightly
higher at the exact spectral edges. At the exact spectral edges the NLI PSD
varies over the channel bandwidth and the NLI PSD cannot be considered locally
flat as in (2). This is not an inherit approximation of the ISRS GN model and
it can be lifted by properly integrating over the NLI PSD. As expected, the
ISRS GN model converges to the conventional GN model for low launch powers.
For increasing launch powers, the nonlinear interference PSD begins to tilt.
The NLI is decreased for channels that experience net ISRS loss and increased
for channels that experience net ISRS gain. Moreover, as shown in Fig. 2b),
the NLI interference coefficient depends exponentially on the launch power.
Fig. 2b) indicates further that the launch power dependence on the NLI
coefficient is stronger for an increasing number of spans.

For Gaussian modulation, the NLI accumulation in decibel as a function of
fiber spans can be written as [14, Sec. IX]

$\begin{split}\left(\eta_{n}\right)\left[\text{dB}\right]-\left(\eta_{1}\right)\left[\text{dB}\right]=\left(1+\epsilon\right)\cdot\left(n\right)\left[\text{dB}\right],\end{split}$
(12)

where $\left(\cdot\right)\left[\text{dB}\right]$ denotes conversion to decibel
scale and $\epsilon$ is the coherence factor that is a measure for coherent
accumulation of the NLI. As the coherence factor depends on the signal power
profile (cf. [14, Fig. 10] and [11, Fig. 3]), it is affected by ISRS. The
accumulation of NLI together with the resulting coherence factor obtained from
the ISRS GN model and simulation results are shown in Fig. 3. Indeed, ISRS
introduces a power dependent tilt on the coherent accumulation. This
corresponds to the increasing power dependence of the NLI coefficient with
increasing span number.

Figure 4: The nonlinear interference coefficient after 1 span as a function of
channel frequency for different total launch powers is shown in a) and the NLI
as a function of total launch power is shown in b) obtained by the analytical
ISRS GN model (9). The uniform optimum launch power for the system under test
is $24$ dBm. Figure 5: The coherence factor as a function of channel frequency
for a variety of total launch powers obtained by the analytical ISRS GN model
(9).

To benchmark our results against previous works, the results in [42, Eq. (16)]
and [43, Eq. (18)] are shown in dashed in Fig. 2a) using the same frequency
dependent power profile (8). The difference originates for the reason
described in Sec. II-A, effectively assuming that all frequencies in the
nonlinear process attenuate as the one that is evaluated (i.e. $f_{i}$). The
deviation therefore increases with increasing ISRS gain towards the spectral
edges.

The model in [41] is shown in dotted lines. The model implements the
conventional GN model, where an effective attenuation coefficient is used,
that matches the effective length of the evaluated channel $f_{i}$.
Consequently, the frequency dependent attenuation within the nonlinear process
is not properly accounted for, mainly resulting in an underestimation of the
ISRS impact.

Bases on the numerical validation carried out in this section, it is concluded
that the ISRS GN model (9) accurately predicts the nonlinear interference
resulting from inter-channel stimulated Raman scattering.

##  IV C+L band transmission

In this section, the ISRS GN model is used to evaluate the impact of ISRS on a
C+L band transmission system covering $10$ THz of optical bandwidth with
parameters listed in Table I (b). The NLI coefficient as a function of channel
frequency is shown in Fig. 4a). For a particular launch power, the
corresponding ISRS gains can be found in Fig. 1.

Moreover, the results can be converted to a different Raman slopes coefficient
by subtracting the deviation in decibel from the total launch power (cf. Eq.
(10)). For example, for halving the Raman gain slope and a total launch power
of $25$ dBm, one can find the resulting NLI coefficients and ISRS gains that
indicate $22$ dBm.

The tilt in NLI coefficient in the absence of ISRS is due to the dispersion
slope $S$ (or $\beta_{3}$), where lower frequencies experience a higher amount
of dispersion and therefore experience reduced NLI. As the launch power is
increased, the effect of ISRS starts to balance the effect of the dispersion
slope in terms of NLI. At a launch power of $22$ dBm, the NLI PSD is almost
flat showing that ISRS and the dispersion slope are somewhat complementing
each other in flatting the NLI spectrum. The system under test exhibits an
optimum launch power of $24$ dBm, which was calculated for a flat input PSD
and only considering the center channel. The resulting ISRS power transfer
ranges from $-3.7$ dB to $2.9$ dB while the resulting NLI deviation due to
ISRS ranges from $-1.7$ dB to $2$ dB. The slope of the NLI spectrum is $-0.24$
$\frac{\text{dB}}{\text{THz}}$ over its linear-like part from $-4$ THz to $4$
THz.

The deviation of the NLI coefficient as a function of total launch power is
shown in Fig. 4b). The NLI PSD depends exponentially on the total launch power
like the ISRS gain itself and already discussed in Sec. III. The deviation at
the spectral edges of the signal is $0.5$ dB at $18$ dBm launch power. In Sec.
III it was shown that the coherence factor is changed as a result of ISRS. The
same effect is seen in Fig. 4b), where the deviation of the NLI is stronger
for an increased number of spans. At $24$ dBm the additional deviation of NLI
is $0.1$ dB at the spectral edges after $10$ spans.

The coherence factor as a function of channel frequency is shown in Fig. 5.
The coherence factor is relatively small ($\epsilon<0.07$) due to the large
bandwidth. In the absence of ISRS the average coherence factor is $0.027$.
Using (12) the average coherent NLI accumulation
$\epsilon\cdot\left(n\right)\left[\text{dB}\right]$ is $0.3$ dB and $0.5$ dB
after 10 and 50 spans, respectively. For $24$ dBm (optimum) launch power, the
maximum deviation in coherence factor is found to be $0.013$ at the spectral
edges. This corresponds to a deviation in coherent NLI accumulation of $0.1$
dB and $0.2$ dB after 10 and 50 spans, respectively. The change in coherence
factor due to ISRS might be deemed negligible depending on the accuracy
requirements of the application.

To relate our work to previously published results, the NLI coefficient after
one span obtained by the ISRS GN model is compared to the works [41] and [42,
43]. The signal power profile as in (8) was used for all comparisons. The
deviation of the NLI coefficient between the ISRS GN model and [41] is shown
in Fig. 6. For optimum launch power ($24$ dBm), the deviation stays below
$0.19$ dB. Even for high ISRS gains at $28$ dBm launch power, the maximum
deviation is $0.8$ dB.

The deviation of the NLI coefficient between the ISRS GN model and [42, Eq.
(13) and (16)] [43, Eq. (18)] is shown in Fig. 7. The maximum deviation is
$0.56$ dB and $2.1$ dB for $24$ dBm and for $28$ dBm launch power,
respectively. The reader is referred to sections II-A and III for the origin
of the discrepancy.

Figure 6: Deviation of the NLI coefficient after one span between the
analytical ISRS GN model (9) and [41]. The validity of the ISRS GN model is
shown in Sec. III. Figure 7: Deviation of the NLI coefficient after one span
between the analytical ISRS GN model (9) and [42, Eq. (13) and (16)] [43, Eq.
(18)]. The validity of the ISRS GN model is shown in Sec. III.

Based on the case study in this section, it is concluded that the impact of
ISRS on the Kerr nonlinearity is significant in C+L band systems. This is
strongly depending on launch power and the Raman gain slope due its
exponential relationship to the nonlinear interference power. It should be
noted that idealized gain flattening filters (GFF) were considered to
compensate the ISRS power transfer at the end of each span. When realistic
GFF’s are considered, the ISRS gain accumulates over distance and the impact
on the nonlinear interference power is more significant than as shown in this
section.

##  V Conclusion

The ISRS GN model was introduced and presented which analytically models the
impact of inter-channel stimulated Raman scattering on the nonlinear
perturbation caused by Kerr nonlinearity. Its accuracy was compared to split-
step simulations and a maximum deviation of $0.1$ dB in nonlinear interference
power was found. The model can further account for the frequency dependent
fiber attenuation, optical bandwidths beyond the Stokes shift (approximately
$14$ THz) and hybrid-amplified transmission systems at the expense of greater
computational complexity using a semi-analytical approach.

It was shown that ISRS changes the nonlinear interference power by up to $2$
dB at optimum launch power for the studied C+L band transmission system. For
such optical bandwidths and beyond, ISRS must be addressed in order to
maximize system performance. From a physical perspective, possible solutions
include the use of gain flattening filters, optimized launch power
distributions or tailored fiber designs. All of which can be modeled and
analyzed using the results in this paper.

The derived ISRS GN model is therefore a powerful tool for efficient design,
optimization, capacity calculations and physical-layer abstractions of ultra-
wideband transmission systems that operate over the entire C+L band and
beyond.

##  Appendix A Derivation of the ISRS GN model

In this section, Eq. (LABEL:eq:general_G) is derived for one fiber span based
on the nonlinear Schrödinger equation (NLSE) and a first-order regular
perturbation approach. The result for one span can then be extended to
multiple spans using the phased-array term (LABEL:eq:phasedarry) or
reinterpreting a span as the entire link length with an according signal power
profile. Instead of a constant attenuation coefficient $\alpha$, a generic
frequency and distance dependent gain coefficient $g\left(z,f\right)$ is used
to model the effect of inter-channel stimulated Raman scattering. For the sake
of brevity, only the key derivation steps are outlined.

We begin with the NLSE in the frequency domain which is given by [53, Ch. 2]

$\begin{split}\frac{\partial}{\partial z}E(z,f)&=\\\
&\widetilde{\Gamma}(z,f)E(z,f)+j\gamma E(z,f)*E^{*}(z,-f)*E(z,f),\end{split}$
(13)

with
$\widetilde{\Gamma}(z,f)=\frac{g\left(z,f\right)}{2}+j2\pi^{2}\beta_{2}f^{2}+j\frac{4}{3}\pi^{3}\beta_{3}f^{3}$
and $u(x)*v(x)$ denoting the convolution operation. The complex envelope of
the electric field $E(z,f)$ is expanded in a regular perturbation series with
respect to the nonlinearity coefficient $\gamma$. The series is then truncated
to first-order and we have

$\begin{split}E(z,f)=E^{\left(0\right)}(z,f)+\gamma
E^{\left(1\right)}(z,f).\end{split}$ (14)

Inserting (14) in (13), we obtain

$\begin{split}E^{\left(0\right)}(z,f)=E(0,f)\cdot
e^{\Gamma\left(z,f\right)},\end{split}$ (15)

with
$\Gamma\left(z,f\right)=\int_{0}^{z}\widetilde{\Gamma}\left(\zeta,f\right)d\zeta$
as the solution for the zeroth-order terms and a linear ordinary differential
equation for the first-order terms as

$\begin{split}\frac{\partial}{\partial
z}E^{\left(1\right)}(z,f)&=\widetilde{\Gamma}(z,f)E^{\left(1\right)}(z,f)+Q(z,f),\end{split}$
(16)

with
$Q(z,f)=jE^{\left(0\right)}(z,f)*E^{\left(0\right)*}(z,-f)*E^{\left(0\right)}(z,f)$.
The initial condition for the first-order solution is
$E^{\left(1\right)}(0,f)=0$ and we obtain

$\begin{split}E^{\left(1\right)}(z,f)=e^{\Gamma\left(z,f\right)}\int_{0}^{z}\frac{Q(\zeta,f)}{e^{\Gamma\left(\zeta,f\right)}}d\zeta,\end{split}$
(17)

as the solution of (16).

In order to compute $Q(z,f)$, we assume that the input signal can be modeled
as a periodic Gaussian process, a key assumption of the GN model, which is
[24, Eq. 13]

$\begin{split}E(0,f)=\sqrt[]{f_{0}G_{\text{Tx}}(f)}\sum_{n=-\infty}^{\infty}\xi_{n}\delta\left(f-nf_{0}\right),\end{split}$
(18)

where $G_{\text{Tx}}(f)$ is the power spectral density of the input signal,
$\xi_{n}$ is a complex circular Gaussian distributed random variable,
$T_{0}=f_{0}^{-1}$ is the period of the signal and $\delta(x)$ denotes the
Dirac delta function. For notational convenience, we write $nf_{0}$ as $f_{n}$
and $\sum_{n=\infty}^{\infty}$ as $\sum_{\forall n}$ for the remainder of this
derivation. Using (18), $Q(z,f)$ can be written as

$\begin{split}&Q(z,f)=jf^{\frac{3}{2}}_{0}\sum_{\forall m}\sum_{\forall
n}\sum_{\forall
k}\sqrt[]{G_{\text{Tx}}(f_{m})G_{\text{Tx}}(f_{n})G_{\text{Tx}}(f_{k})}\\\
&\xi_{m}\xi^{*}_{n}\xi_{k}\delta\left(f-f_{m}+f_{n}-f_{k}\right)e^{\Gamma\left(z,f_{m}\right)+\Gamma^{*}\left(z,f_{n}\right)+\Gamma\left(z,f_{k}\right)}.\end{split}$
(19)

To first order, it can be shown that only non-degenerate frequency triplets in
(19) contribute to the nonlinear interference power. Degenerate frequency
triplets merely introduce a constant phase shift of the first-order solution
$E^{\left(1\right)}(z,f)$, which cancels out when the PSD of
$E^{\left(1\right)}(z,f)$ is computed. For more details, the reader is
referred to [54, Ch. IV.B and IV.D]. Therefore, we neglect degenerate
frequency triplets in order to keep the derivation concise. Similar to [54],
we define the triplets of non-degenerate frequency components as

$\begin{split}A_{i}=\left\\{(m,n,k):[m-n+k]=i\>\text{and}\>[m\neq
n\>\text{or}\>k\neq n]\right\\},\end{split}$ (20)

and rewrite (19) as

$\begin{split}&Q(z,f)=jf^{\frac{3}{2}}_{0}\sum_{\forall
i}\delta\left(f-f_{i}\right)\sum_{\forall(m,n,k)\in
A_{i}}\xi_{m}\xi^{*}_{n}\xi_{k}\\\
&\sqrt[]{G_{\text{Tx}}(f_{m})G_{\text{Tx}}(f_{n})G_{\text{Tx}}(f_{k})}e^{\Gamma\left(z,f_{m}\right)+\Gamma^{*}\left(z,f_{n}\right)+\Gamma\left(z,f_{k}\right)}.\end{split}$
(21)

Inserting (21) in (17) yields the first-order solution as

$\begin{split}&E^{\left(1\right)}(z,f)=jf^{\frac{3}{2}}_{0}e^{\Gamma\left(z,f\right)}\sum_{\forall
i}\delta\left(f-f_{i}\right)\\\ &\sum_{\forall(m,n,k)\in
A_{i}}\xi_{m}\xi^{*}_{n}\xi_{k}\sqrt[]{G_{\text{Tx}}(f_{m})G_{\text{Tx}}(f_{n})G_{\text{Tx}}(f_{k})}\\\
&\int_{0}^{z}d\zeta\
e^{\Gamma\left(\zeta,f_{m}\right)+\Gamma^{*}\left(\zeta,f_{n}\right)+\Gamma\left(\zeta,f_{k}\right)-\Gamma\left(\zeta,f_{m}-f_{n}+f_{k}\right)}.\end{split}$
(22)

In order to obtain the nonlinear interference power, we compute the average
PSD of the first-order solution $\gamma E^{\left(1\right)}(z,f)$. Similar to
[54, Ch. IV.D], the average PSD of (22) multiplied by $\gamma$ is

$\begin{split}&G_{\text{NLI}}(z,f)=2\gamma^{2}f^{3}_{0}e^{2\text{Re}\left[\Gamma\left(z,f\right)\right]}\sum_{\forall
i}\delta\left(f-f_{i}\right)\\\ &\sum_{\forall(m,n,k)\in
A_{i}}G_{\text{Tx}}(f_{m})G_{\text{Tx}}(f_{n})G_{\text{Tx}}(f_{k})\\\
&\left|\int_{0}^{z}d\zeta\
e^{\Gamma\left(\zeta,f_{m}\right)+\Gamma^{*}\left(\zeta,f_{n}\right)+\Gamma\left(\zeta,f_{k}\right)-\Gamma\left(\zeta,f_{m}-f_{n}+f_{k}\right)}\right|^{2}.\end{split}$
(23)

In the following, we transform the inner summation appearing in (23) into a
summation over two independent variables. For the non-degenerate set $A_{i}$,
we have that $f_{m}-f_{n}+f_{k}=f_{i}$ and for a given frequency triplet
$\left(f_{i},f_{m},f_{k}\right)$ it follows that $f_{n}=f_{m}+f_{k}-f_{i}$.
Therefore, (23) can be written as

$\begin{split}&G(z,f)=2\gamma^{2}f^{3}_{0}e^{2\text{Re}\left[\Gamma\left(z,f\right)\right]}\sum_{\forall
i}\delta\left(f-f_{i}\right)\\\ &\sum_{\forall m}\sum_{\forall
k}G_{\text{Tx}}(f_{m})G_{\text{Tx}}(f_{k})G_{\text{Tx}}(f_{m}+f_{k}-f)\\\
&\left|\int_{0}^{z}d\zeta\
e^{\Gamma\left(\zeta,f_{m}\right)+\Gamma^{*}\left(\zeta,f_{m}+f_{k}-f\right)+\Gamma\left(\zeta,f_{k}\right)-\Gamma\left(\zeta,f\right)}\right|^{2}.\end{split}$
(24)

Finally, we define the normalized signal power profile of a frequency
component as $\rho(z,f)=e^{\int_{0}^{z}g\left(\zeta,f\right)d\zeta}$ and
rewrite (24) as an integral expression by letting $f_{0}\to 0$

$\begin{split}&G(z,f)=2\gamma^{2}\rho(z,f)\int df_{1}\int df_{2}\ \\\
&G_{\text{Tx}}(f_{1})G_{\text{Tx}}(f_{2})G_{\text{Tx}}(f_{1}+f_{2}-f)\\\
&\left|\int_{0}^{z}d\zeta\
\sqrt[]{\frac{\rho(\zeta,f_{1})\rho(\zeta,f_{2})\rho(\zeta,f_{1}+f_{2}-f)}{\rho(\zeta,f)}}e^{j\phi(f_{1},f_{2},f,\zeta)}\right|^{2}.\end{split}$
(25)

As (25) was derived for single polarization, $2\gamma^{2}$ must be replaced by
$\frac{16}{27}\gamma^{2}$ to obtain the nonlinear interference power for dual
polarization. Furthermore, the term $\rho(z,f)$ outside of the integral can be
removed when the span loss is compensated. The result is Eq.
(LABEL:eq:general_G).

## Acknowledgment

The author would like to express sincere gratitude to Dr. T. Fehenberger from
Technical University of Munich for fruitful discussions on the GN model
derivation. Moreover, the author thanks G. Saavedra from University College
London for valuable feedback on previous drafts of the paper and Dr. D. Lavery
from University College London for help with the simulation environment.
Financial support from UK EPSRC programme grant UNLOC (EP/J017582/1) and a
Doctoral Training Partnership (DTP) studentship to Daniel Semrau is gratefully
acknowledged.

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Daniel Semrau received the B. Sc degree in electrical engineering from the
Technical University of Berlin (TUB), Berlin, Germany, in 2013, and the M. Sc.
degree in Photonic Networks Engineering from Scuola Superiore Sant’Anna
(SSSUP), Pisa, Italy, and Aston University, Birmingham, U.K., in 2015. In 2015
he joined the Optical Networks Group at University College London (UCL) where
he is currently working towards a Ph.D. degree. His research interests are
mainly focused on channel modeling, nonlinear compensation techniques and
ultra-wideband transmission for long-haul coherent optical communications.
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Polina Bayvel received the B.Sc. (Eng.) and Ph.D. degrees in electronic and
electrical engineering from University College London (UCL), London, U.K., in
1986 and 1990, respectively. Her Ph.D. research focused on nonlinear fiber
optics and their applications. In 1990, she was with the Fiber Optics
Laboratory, General Physics Institute, Moscow (Russian Academy of Sciences),
under the Royal Society Postdoctoral Exchange Fellowship. She was a Principal
Systems Engineer with STC Submarine Systems, Ltd., London, U.K., and Nortel
Networks (Harlow, U.K., and Ottawa, ON, Canada), where she was involved in the
design and planning of optical fiber transmission networks. During 1994–2004,
she held a Royal Society University Research Fellowship at UCL, and, in 2002,
she became a Chair in Optical Communications and Networks. She is currently
the Head of the Optical Networks Group, UCL. She has authored or co-authored
more than 290 refereed journal and conference papers. Her research interests
include optical networks, high-speed optical transmission, and the study and
mitigation of fiber nonlinearities. Prof. Bayvel is a Fellow of the Royal
Academy of Engineering (F.R.Eng.), the Optical Society of America, the U.K.
Institute of Physics, and the Institute of Engineering and Technology. She is
a member of the Technical Program Committee (TPC) of a number of conferences,
including Proc. ECOC and Co-Chair of the TPC for ECOC 2005. She was the 2002
recipient of the Institute of Physics Paterson Prize and Medal for
contributions to research on the fundamental aspects of nonlinear optics and
their applications in optical communications systems. In 2007, she was the
recipient of the Royal Society Wolfson Research Merit Award.
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