# Riemann hypothesis and the arc length of the Riemann $Z(t)$-curve

Jan Moser  Department of Mathematical Analysis and Numerical Mathematics,
Comenius University, Mlynska Dolina M105, 842 48 Bratislava, SLOVAKIA
<EMAIL_ADDRESS>
###### Abstract.

On Riemann hypothesis it is proved in this paper that the arc length of the
Riemann $Z$-curve is asymptotically equal to the double sum of local maxima of
the function $Z(t)$ on corresponding segment. This paper is English remake of
our paper [9], with short appendix concerning new integral generated by
Jacob’s ladders added.

###### Key words and phrases:

Riemann zeta-function

##  1\. Introduction and result

### 1.1.

Main object of this paper is the study of the integral

(1.1) $\int_{T}^{T+H}\sqrt{1+\\{Z^{\prime}(t)\\}^{2}}{\rm d}t,$

i.e. the study of the arc length of the Riemann curve

$y=Z(t),\ t\in[T,T+H],\quad T\to\infty,$

where (see [13], pp. 79, 329)

(1.2)
$\begin{split}&Z(t)=e^{i\vartheta(t)}\zeta\left(\frac{1}{2}+it\right),\\\
&\vartheta(t)=-\frac{t}{2}\ln\pi+\text{Im}\ln\Gamma\left(\frac{1}{4}+i\frac{t}{2}\right)=\\\
&=\frac{t}{2\pi}\ln\frac{t}{2\pi}-\frac{t}{2}-\frac{\pi}{8}+\mathcal{O}\left(\frac{1}{t}\right).\end{split}$

######  Remark 1.

Let us remind that the formula

(1.3) $\begin{split}&\\{Z(t)=\\}\
e^{i\vartheta(t)}\zeta\left(\frac{1}{2}+it\right)=\\\
&=2\sum_{n\leq\sqrt{\bar{t}}}\frac{1}{\sqrt{n}}\cos\\{\vartheta(t)-t\ln
n\\}+\mathcal{O}(t^{-1/4}),\ \bar{t}=\sqrt{\frac{t}{2\pi}}\end{split}$

was known to Riemann (see [11], p. 60, comp. [12], p. 98).

Next, we will denote the roots of the equations

$Z(t)=0,\ Z^{\prime}(t)=0,\ t_{0}\not=\gamma$

by the symbols

$\\{\gamma\\},\ \\{t_{0}\\},$

correspondingly.

######  Remark 2.

On the Riemann hypothesis, the points of the sequences $\\{\gamma\\}$ and
$\\{t_{0}\\}$ are separated each from other (see [3], Corollary 3), i.e. in
this case we have

$\gamma^{\prime}<t_{0}<\gamma^{\prime\prime},$

where $\gamma^{\prime},\gamma^{\prime\prime}$ are neighboring points of the
sequence $\\{\gamma\\}$. Of course, $Z(t_{0})$ is local extremum of the
function $Z(t)$ located at $t=t_{0}$.

### 1.2.

In this paper we use the Riemann hypothesis together with some synthesis of
properties of the sequences

$\\{t_{0}\\},\ \\{h_{\nu}(\tau)\\},$

where the numbers $h_{\nu}(\tau)$ are defined by the equation (comp. (1.2))

(1.4) $\begin{split}&\vartheta_{1}[h_{\nu}(\tau)]=\pi\nu+\tau+\frac{\pi}{2},\
\nu=1,2,\dots,\ \tau\in[-\pi,\pi],\\\
&\vartheta_{1}(t)=\frac{t}{2}\ln\frac{t}{2\pi}-\frac{t}{2}-\frac{\pi}{8},\\\
&\vartheta(t)=\vartheta_{1}(t)+\mathcal{O}\left(\frac{1}{t}\right),\end{split}$

in order to obtain the following theorem.

######  Theorem.

On the Riemann hypothesis we have the asymptotic formula

(1.5) $\begin{split}&\int_{T}^{T+H}\sqrt{1+\\{Z^{\prime}(t)\\}^{2}}{\rm
d}t=2\sum_{T\leq t_{0}\leq T+H}|Z(t_{0})|+\\\ &+\Theta
H+\mathcal{O}\left(T^{\frac{\Delta}{\ln\ln T}}\right),\\\
&\Theta=\Theta(T,H)\in(0,1),\ H=T^{\epsilon},\ T\to\infty\end{split}$

for every fixed $\epsilon>0$.

######  Remark 3.

Geometric meaning of our asymptotic formula (1.5) is as follows: the arc
length of the Riemann curve

$y=Z(t),\ t\in[T,T+H]$

is asymptotically equal to the double of the sum of local maxima of the
function

$|Z(t)|,\ t\in[T,T+H].$

##  2\. Discrete formulae – Lemma 1

### 2.1.

In this part of the paper we use the following formula

(2.1)
$\begin{split}&Z^{\prime}(t)=-2\sum_{n<P}\frac{1}{\sqrt{n}}\ln\frac{P}{n}\sin\\{\vartheta-t\ln
n\\}+\\\ &+\mathcal{O}(T^{-1/4}\ln T),\ P=\sqrt{\frac{T}{2\pi}},\end{split}$

that we have obtained in our work [6], (see (2.1)). Next, we obtain from (2.1)
in the case

$\vartheta\to\vartheta_{1}$

(see (1.4)) that

(2.2)
$\begin{split}&Z^{\prime}(t)=-2\sum_{n<P}\frac{1}{\sqrt{n}}\ln\frac{P}{n}\sin\\{\vartheta_{1}-t\ln
n\\}+\\\ &+\mathcal{O}(T^{-1/4}\ln T),\ H\in(0,\sqrt[4]{T}].\end{split}$

Let $S(a,b)$ denotes elementary trigonometric sum

$S(a,b)=\sum_{a\leq n\leq b}n^{it},\quad 1\leq a<b\leq 2a,\
b\leq\sqrt{\frac{t}{2\pi}}.$

Then we obtain from (2.2) in the case of the sequence $h_{\nu}(\tau)$ (see
(1.4)) the following

######  Lemma 1.

If

(2.3) $|S(a,b)|\leq A(\Delta)\sqrt{a},\ \Delta\in(0,1/6]$

then ($h_{\nu}=h_{\nu}(0)$)

(2.4) $\begin{split}&\sum_{T\leq h_{2\nu}\leq
T+H}Z^{\prime}[h_{2\nu}(\tau)]=-\frac{1}{\pi}H\ln^{2}P\cos\tau+\mathcal{O}(T^{\Delta}\ln^{2}T),\\\
&\sum_{T\leq h_{2\nu+1}\leq
T+H}Z^{\prime}[h_{2\nu+1}(\tau)]=\frac{1}{\pi}H\ln^{2}P\cos\tau+\mathcal{O}(T^{\Delta}\ln^{2}T),\end{split}$

where $\mathcal{O}$-estimates are uniform for $\tau\in[-\pi,\pi]$.

###### Proof.

We obtain from (2.2) by (1.4)

(2.5) $\begin{split}&Z^{\prime}[h_{\nu}(\tau)]=2(-1)^{\nu+1}\ln P\cos\tau-\\\
&-2\sum_{2\leq n\leq P}\frac{1}{\sqrt{n}}\ln\frac{P}{n}\cos\\{\pi\nu-
h_{\nu}(\tau)\ln n+\tau\\}+\\\ &+\mathcal{O}(T^{-1/4}\ln T),\
h_{\nu}(\tau)\in[T,T+H].\end{split}$

∎

### 2.2.

Since (see [5], (23))

(2.6) $\sum_{T\leq h_{\nu}\leq
T+H}1=\frac{1}{2\pi}H\ln\frac{T}{2\pi}+\mathcal{O}(1)=\frac{1}{\pi}H\ln
P+\mathcal{O}(1),$

then we obtain from (2.5) (comp. [4], (59)-(61), [6], (51)-(53)) that

(2.7) $\sum_{T\leq h_{\nu}\leq
T+H}Z^{\prime}[h_{\nu}(\tau)]=-2\bar{w}(T,H;\tau)+\mathcal{O}(\ln^{2}T),$

where

$\begin{split}&\bar{w}=\frac{1}{2}(-1)^{\bar{\nu}}\sum_{n}\frac{1}{\sqrt{n}}\ln\frac{P}{n}\cos\varphi+\\\
&+\frac{1}{2}(-1)^{N+\bar{\nu}}\sum_{n}\frac{1}{\sqrt{n}}\ln\frac{P}{n}\cos(\omega
N+\varphi)+\\\
&+\frac{1}{2}(-1)^{\bar{\nu}}\sum_{n}\frac{1}{\sqrt{n}}\ln\frac{P}{n}\tan\frac{\omega}{2}\sin\varphi+\\\
&+\frac{1}{2}(-1)^{N+\bar{\nu}+1}\sum_{n}\frac{1}{\sqrt{n}}\ln\frac{P}{n}\tan\frac{\omega}{2}\sin(\omega
N+\varphi),\end{split}$

where

$\omega=\pi\frac{\ln n}{\ln P},\ \varphi=h_{\bar{\nu}}(\tau)\ln n-\tau,\
n\in[2,P),$

and

$\bar{\nu}=\min\\{\nu:\ h_{\nu}\in[T,T+H]\\},\ \bar{\nu}+N=\max\\{\nu:\
h_{\nu}\in[T,T+H]\\}.$

Of course, we have

$\sum_{T\leq h_{\nu}(\tau)\leq T+H}1=\sum_{T\leq h_{\nu}\leq
T+H}1+\mathcal{O}(1)$

for any fixed $\tau\in[-\pi,\pi]$. Now, it is clear that the method [6],
(54)-(64) implies by (2.3) that

$\bar{w}=\mathcal{O}(T^{\Delta}\ln^{2}T)$

uniformly for $\tau\in[-\pi,\pi]$, and consequently we obtain (see (2.7)) the
estimate

(2.8) $\sum_{T\leq h_{\nu}\leq
T+H}Z^{\prime}[h_{\nu}(\tau)]=\mathcal{O}(T^{\Delta}\ln^{2}T)$

uniformly for $\tau\in[-\pi,\pi]$.

### 2.3.

Next, we have (see (2.5), (2.6))

$\begin{split}&\sum_{T\leq h_{\nu}\leq
T+H}(-1)^{\nu}Z^{\prime}[h_{\nu}(\tau)]=-\frac{2}{\pi}H\ln^{2}P\cos\tau-2R+\mathcal{O}(\ln^{2}P),\\\
&R=\sum_{2\leq n<P}\frac{1}{\sqrt{n}}\ln\frac{P}{n}\sum_{T\leq h_{\nu}\leq
T+H}\cos\\{h_{\nu}(\tau)\ln n-\tau\\}.\end{split}$

Since by (2.3) and [6], (65)-(79) we have the estimate

$R=\mathcal{O}(T^{\Delta}\ln^{2}T)$

then we obtain the formula

(2.9) $\begin{split}&\sum_{T\leq h_{\nu}\leq
T+H}(-1)^{\nu}Z^{\prime}[h_{\nu}(\tau)]=\\\
&=-\frac{2}{\pi}H\ln^{2}P\cos\tau+\mathcal{O}(T^{\Delta}\ln^{2}T)\end{split}$

uniformly for $\tau\in[-\pi,\pi]$.

Finally, from (2.8), (2.9) formulae (2.4) follow.

##  3\. Integrals over disconnected sets – Lemma 2

Let (comp. [7], (3))

(3.1) $\begin{split}&\mathbb{G}_{2\nu}(x)=\\{t:\ h_{2\nu}(-x)<t<h_{2\nu}(x),\
t\in[T,T+H]\\},\ x\in(0,\pi/2],\\\ &\mathbb{G}_{2\nu+1}(y)=\\{t:\
h_{2\nu+1}(-y)<t<h_{2\nu+1}(y),\ t\in[T,T+H]\\},\ y\in(0,\pi/2],\\\
&\mathbb{G}_{1}(x)=\bigcup_{T\leq h_{2\nu}\leq T+H}\mathbb{G}_{2\nu}(x),\\\
&\mathbb{G}_{2}(y)=\bigcup_{T\leq h_{2\nu+1}\leq
T+H}\mathbb{G}_{2\nu+1}(y).\end{split}$

The following lemma holds true.

######  Lemma 2.

(2.3) implies

(3.2) $\begin{split}&\int_{\mathbb{G}_{1}(x)}Z^{\prime}(t){\rm
d}t=-\frac{2}{\pi}H\ln P\sin x+\mathcal{O}(xT^{\Delta}\ln T),\\\
&\int_{\mathbb{G}_{2}(y)}Z^{\prime}(t){\rm d}t=\frac{2}{\pi}H\ln P\sin
y+\mathcal{O}(yT^{\Delta}\ln T).\end{split}$

###### Proof.

First of all we have (see (1.4), comp. [7], (51))

$\left(\frac{{\rm d}h_{2\nu}(\tau)}{{\rm
d}\tau}\right)^{-1}=\vartheta_{1}^{\prime}[h_{2\nu}(\tau)]=\ln
P+\mathcal{O}\left(\frac{H}{T}\right).$

Next, from (2.2) by (2.3) we obtain the estimate

$Z^{\prime}(t)=\mathcal{O}(T^{\Delta}\ln^{2}T),\ t\in[T,T+H]$

(Abel transformation). Then we have (comp. [7], (52)) that

(3.3) $\begin{split}&\int_{-x}^{x}Z^{\prime}[h_{2\nu}(\tau)]{\rm
d}\tau=\int_{-x}^{x}Z^{\prime}[h_{2\nu}(\tau)]\left(\frac{{\rm
d}h_{2\nu}(\tau)}{{\rm d}\tau}\right)^{-1}\frac{{\rm d}h_{2\nu}(\tau)}{{\rm
d}\tau}{\rm d}\tau=\\\ &=\ln
P\int_{h_{2\nu}(-x)}^{h_{2\nu}(x)}Z^{\prime}(t){\rm
d}t+\mathcal{O}\left(x\frac{H}{T}T^{\Delta}\ln^{2}T\frac{1}{\ln T}\right)=\\\
&=\ln P\int_{\mathbb{G}_{2\nu}(x)}Z^{\prime}(t){\rm
d}t+\mathcal{O}(xHT^{-5/6}\ln T).\end{split}$

Consequently, we obtain from the first formula in (2.4) by (2.6), (3.1), (3.3)
the following asymptotic equality

$\begin{split}&\int_{\mathbb{G}_{1}(x)}Z^{\prime}(t){\rm
d}t=-\frac{2}{\pi}H\ln P\sin x+\\\ &+\mathcal{O}(xT^{\Delta}\ln
T)+\mathcal{O}(xH^{2}T^{-5/6}\ln^{2}T),\end{split}$

i.e. the first integral in (3.2). The second integral can be derived by a
similar way. ∎

##  4\. An estimate from below – Lemma 3

The following lemma holds true.

######  Lemma 3.

From (2.3) the estimate

(4.1) $\int_{T}^{T+H}|Z^{\prime}(t)|{\rm d}t>\frac{4}{\pi}(1-\epsilon)H\ln P,\
P=\sqrt{\frac{T}{2\pi}},\ H\in[T^{\Delta+\epsilon},\sqrt[4]{T}]$

follows, where $\epsilon>0$ is an arbitrarily small number.

###### Proof.

Let (comp. [8], (10))

$\begin{split}&\mathbb{G}_{1}^{+}(x)=\\{t:\ Z^{\prime}(t)>0,\
t\in\mathbb{G}_{1}(x)\\},\\\ &\mathbb{G}_{1}^{-}(x)=\\{t:\ Z^{\prime}(t)<0,\
t\in\mathbb{G}_{1}(x)\\},\\\ &\mathbb{G}_{1}^{0}(x)=\\{t:\ Z^{\prime}(t)=0,\
t\in\mathbb{G}_{1}(x)\\},\end{split}$

and the symbols

$\mathbb{G}_{2}^{+}(y),\mathbb{G}_{2}^{-}(y),\mathbb{G}_{2}^{0}(y)$

have similar meaning. Of course

$m\\{\mathbb{G}_{1}^{0}(x)\\}=m\\{\mathbb{G}_{2}^{0}(y)\\}=0.$

Since the expressions (3.2) in the case

$H\in[T^{\Delta+\epsilon},\sqrt[4]{T}],\quad x,y\in(0,\pi/2]$

are asymptotic formulae then from them we obtain the following inequalities

(4.2) $\begin{split}&\frac{2}{\pi}(1-\epsilon)H\ln
P<-\int_{\mathbb{G}_{1}(\pi/2)}Z^{\prime}(t){\rm d}t\leq\\\
&\leq-\int_{\mathbb{G}_{1}^{-}(\pi/2)}Z^{\prime}(t){\rm
d}t=\int_{\mathbb{G}_{1}^{-}(\pi/2)}|Z^{\prime}(t)|{\rm d}t,\\\
&\frac{2}{\pi}(1-\epsilon)H\ln P<\int_{\mathbb{G}_{2}(\pi/2)}Z^{\prime}(t){\rm
d}t\leq\int_{\mathbb{G}_{2}^{+}(\pi/2)}|Z^{\prime}(t)|{\rm d}t.\end{split}$

Since

$\mathbb{G}_{1}^{-}(\pi/2)\cup\mathbb{G}_{2}^{+}(\pi/2)\subset[T,T+H],\
\mathbb{G}_{1}^{-}(\pi/2)\cap\mathbb{G}_{2}^{+}(\pi/2)=\emptyset$

then by (4.2) needful estimate

$\begin{split}&\int_{T}^{T+H}|Z^{\prime}(t)|{\rm
d}t\geq\int_{\mathbb{G}_{1}^{-}(\pi/2)}|Z^{\prime}(t)|{\rm
d}t+\int_{\mathbb{G}_{2}^{+}(\pi/2)}|Z^{\prime}(t)|{\rm d}t>\\\
&>\frac{4}{\pi}(1-\epsilon)H\ln P.\end{split}$

follows. ∎

##  5\. Quadrature formula – Lemma 4

The following lemma holds true.

######  Lemma 4.

On Riemann hypothesis we have the following asymptotic formula

(5.1) $\begin{split}&\int_{T}^{T+H}|Z^{\prime}(t)|{\rm d}t=2\sum_{T\leq
t_{0}\leq T+H}|Z(t_{0})|+\\\ &+\mathcal{O}\left(T^{\frac{A}{\ln\ln
T}}\right),\quad H\in[T^{\mu},\sqrt[4]{T}],\end{split}$

where $0<\mu$ is an arbitrary small number.

###### Proof.

First of all, we have on Riemann hypothesis the following two Littlewood’s
estimates

(5.2) $\gamma^{\prime\prime}-\gamma^{\prime}<\frac{A}{\ln\ln\gamma^{\prime}},\
\gamma^{\prime}\to\infty$

(see [2], p. 237), and

(5.3) $Z(t)=\mathcal{O}\left(t^{\frac{A}{\ln\ln t}}\right),\ t\to\infty$

(see [13], p. 300). Next, on Riemann hypothesis we have the following basic
configuration (see Remark 2)

(5.4) $\gamma^{\prime}<t_{0}<\gamma^{\prime\prime};\ t_{0}\in[T,T+H].$

Now, there are following possibilities (see (5.4)): either

(5.5) $\begin{split}&Z(t)>0,\ t\in(\gamma^{\prime},\gamma^{\prime\prime})\
\Rightarrow\\\ &Z^{\prime}(t)>0,\ t\in(\gamma^{\prime},t_{0}),\
Z^{\prime}(t)<0,\ t\in(t_{0},\gamma^{\prime\prime}),\end{split}$

or

(5.6) $\begin{split}&Z(t)<0,\ t\in(\gamma^{\prime},\gamma^{\prime\prime})\
\Rightarrow\\\ &Z^{\prime}(t)<0,\ t\in(\gamma^{\prime},t_{0}),\
Z^{\prime}(t)>0,\ t\in(t_{0},\gamma^{\prime\prime}).\end{split}$

Consequently, (5.5) and (5.6) imply that

(5.7) $\int_{\gamma^{\prime}}^{\gamma^{\prime\prime}}|Z^{\prime}(t)|{\rm
d}t=2|Z(t_{0})|,\ \forall t_{0}\in[T,T+H].$

Similarly, we obtain (see (5.2), (5.3)) the estimates

(5.8)
$\int_{\bar{\gamma}^{\prime}}^{\bar{\gamma}^{\prime\prime}}|Z^{\prime}(t)|{\rm
d}t,\
\int_{\bar{\bar{\gamma}}^{\prime}}^{\bar{\bar{\gamma}}^{\prime\prime}}|Z^{\prime}(t)|{\rm
d}t=\mathcal{O}\left(\frac{T^{\frac{A}{\ln\ln T}}}{\ln\ln T}\right)$

in the following cases

$\bar{\gamma}^{\prime}<T\leq t_{0}<\bar{\gamma}^{\prime\prime},\
\bar{\bar{\gamma}}^{\prime}<t_{0}\leq T+H<\bar{\bar{\gamma}}^{\prime\prime}.$

Now, our formula (5.1) follows from (5.7), (5.8). ∎

##  6\. Proof of Theorem

We use the following formula

(6.1) $\begin{split}&\int_{T}^{T+H}\sqrt{1+\\{Z^{\prime}(t)\\}^{2}}{\rm
d}t=\\\ &=\int_{T}^{T+H}|Z^{\prime}(t)|{\rm
d}t+\int_{T}^{T+H}\frac{1}{\sqrt{1+\\{Z^{\prime}(t)\\}^{2}}+|Z^{\prime}(t)|}{\rm
d}t.\end{split}$

Since

$0<\frac{1}{\sqrt{1+\\{Z^{\prime}(t)\\}^{2}}+|Z^{\prime}(t)|}\leq 1$

and

(6.2)
$\left.\frac{1}{\sqrt{1+\\{Z^{\prime}(t)\\}^{2}}+|Z^{\prime}(t)|}\right|_{t=t_{0}}=1,\
t_{0}\in[T,T+H],$

i.e. the inequality (6.2) holds true for the finite set of values, then the
mean-value theorem gives

(6.3)
$\int_{T}^{T+H}\frac{1}{\sqrt{1+\\{Z^{\prime}(t)\\}^{2}}+|Z^{\prime}(t)|}{\rm
d}t=\Theta H,\ \Theta=\Theta(T,H)\in(0,1).$

Next, we obtain by (4.1), (5.1), ($\mu\leq\epsilon$), the inequality

(6.4) $\begin{split}&\frac{4}{\pi}(1-\epsilon)H\ln
P<\int_{T}^{T+H}|Z^{\prime}(t)|{\rm d}t=\\\ &=2\sum_{T\leq t_{0}\leq
T+H}|Z^{\prime}(t_{0})|+\mathcal{O}\left(T^{\frac{A}{\ln\ln
T}}\right).\end{split}$

Hence, by (6.1)-(6.4) the formula (1.5) follows for

(6.5) $H\in[T^{\Delta+\epsilon},\sqrt[4]{T}].$

Since the Riemann hypothesis implies Lindelöf hypothesis a it implies that
$\Delta=\epsilon$ (comp. [1], p. 89), then we obtain from (6.5) that

$H=T^{2\epsilon};\ 2\epsilon\rightarrow\epsilon,$

(see (1.5)).

##  Appendix A Influence of Jacob’s ladders

If

$\varphi_{1}\\{[\mathring{T},\widering{T+H}]\\}=[T,T+H],$

then from (1.5) we obtain (see [10], (9.7)) the formula

(A.1)
$\begin{split}&\int_{\mathring{T}}^{\widering{T+H}}\sqrt{1+\\{Z^{\prime}_{\varphi_{1}}[\varphi_{1}(t)]\\}^{2}}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm
d}t\sim\\\ &\sim\left\\{2\sum_{T\leq t_{0}\leq T+H}|Z(t_{0})|+\Theta
H+\mathcal{O}\left(T^{\frac{A}{\ln\ln T}}\right)\right\\}\ln T,\
T\to\infty.\end{split}$

From (A.1) we obtain by mean-value theorem that

(A.2)
$\begin{split}&\int_{\mathring{T}}^{\widering{T+H}}\sqrt{1+\\{Z^{\prime}_{\varphi_{1}}[\varphi_{1}(t)]\\}^{2}}{\rm
d}t\sim\\\ &\sim\frac{\ln
T}{\left|\zeta\left(\frac{1}{2}+i\alpha\right)\right|^{2}}\left\\{2\sum_{T\leq
t_{0}\leq T+H}\left|\zeta\left(\frac{1}{2}+it_{0}\right)\right|+\Theta
H+\mathcal{O}\left(T^{\frac{A}{\ln\ln T}}\right)\right\\},\\\
&\alpha\in(\mathring{T},\widering{T+H}).\end{split}$

######  Remark 4.

Since we have (see [10], (8.5))

$\rho\\{[T,T+H];[\mathring{T},\widering{T+H}]\\}\sim(1-c)\pi(T)>(1-\epsilon)(1-c)\frac{T}{\ln
T},\ T\to\infty,$

where $\rho$ denotes the distance of corresponding segments and $\pi(T)$ is
the prime-counting function and $c$ is the Euler constant, then the formula
(A.2) gives strongly non-local expression for the integral on the left-hand
side of (A.2).

I would like to thank Michal Demetrian for helping me with the electronic
version of this work.

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# A Closed-Form Approximation of the Gaussian Noise Model in the Presence of
Inter-Channel Stimulated Raman Scattering

Daniel Semrau, Robert I. Killey, and Polina Bayvel This work was supported by
a UK EPSRC programme grant TRANSNET (EP/R035342/1) and a Doctoral Training
Partnership (DTP) studentship for Daniel Semrau.D. Semrau, Robert I. Killey
and P. Bayvel are with the Optical Networks Group, University College London,
London WC1E 7JE, U.K. (e-mail: {uceedfs; r.killey<EMAIL_ADDRESS>
###### Abstract

An accurate, closed-form expression evaluating the nonlinear interference
(NLI) power in coherent optical transmission systems in the presence of inter-
channel stimulated Raman scattering (ISRS) is derived. The analytical result
enables a rapid estimate of the signal-to-noise ratio (SNR) and avoids the
need for integral evaluations and split-step simulations. The formula also
provides new insight into the underlying parameter dependence of ISRS on the
NLI. The proposed result is applicable for dispersion unmanaged, ultra-
wideband transmission systems that use optical bandwidths of up to 15 THz.

The accuracy of the closed-form expression is compared to numerical
integration of the ISRS Gaussian Noise model and split-step simulations in a
point-to-point transmission, as well as in a mesh optical network scenario.

###### Index Terms:

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

##  I Introduction

Analytical models to estimate nonlinear interference (NLI) are key for rapid
and efficient system design [1], 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].

Most approaches analytically solve the nonlinear Schrödinger equation using
first-order perturbation theory with respect to the Kerr nonlinearity. The
resulting integral expressions offer a significant reduction in computational
complexity with minor inaccuracies compared to split-step simulations and
experiments [6, 7, 8, 9, 10, 11].

Particularly, the Gaussian Noise (GN) model offers reasonable accuracy while
exhibiting a moderate computational complexity [12, 13]. Numerical integration
of the GN model to obtain the nonlinear interference has a typical computation
time of a few minutes per WDM channel [14, 15]. For ultra-wideband signals
that consist of 200 WDM channels or more, the computation time quickly amounts
to a few hours to obtain the NLI distribution across the entire optical
signal.

Figure 1: A section from the British Telecommunications (BT) 20+2 topology of
the United Kingdom (UK) core network [16]. In section III-B, the nonlinear
interference of the example light path (A-B) is modeled.

For some applications, such time frames are not acceptable and closed-form
approximations, that yield performance estimations in picoseconds, are
preferable. Such applications include e.g. physical layer aware network
optimization and network performance estimation. Additionally, closed-form
approximations offer a unique insight into the underlying parameter
dependencies and provide useful design and scaling rules.

An optical mesh network consists of a vast amount of point-to-point
transmissions; two between two adjacent reconfigurable optical add-drop
multiplexer (ROADM). As an example, the British Telecommunications (BT) 20+2
topology of the United Kingdom core network [16] exhibits 34 bidirectional
network edges. A section of the network is shown in Fig. 1. For ultra-wideband
transmission, assuming a WDM grid spacing of 50 GHz (ITU grid) over the entire
C+L band (10 THz), a full performance estimation of a single network state
requires up to $\frac{10\text{THz}}{50\text{GHz}}\cdot 2\cdot 34=13600$ NLI
function evaluations. Furthermore, network operators often seek for optimum
channel allocations and optimum launch powers for new incoming demands,
requiring performance computations of numerous potential network states. It is
evident that the problem is intractable using numerical integrations. However,
using closed-form approximations, the performance estimation of a given
network state can be reduced to only a few microseconds.

Closed-form approximations, that predict the nonlinear interference power in
coherent transmission systems, were derived for lossless fibers [17, 3] and
lossy fibers using lumped amplification [18, 19, 13, 20, 21, 22] as well as
for distributed Raman amplified links [23].

However, all aforementioned formulas are not applicable for optical bandwidths
beyond C-band (5 THz) where inter-channel stimulated Raman scattering (ISRS)
becomes significant. ISRS is a non-parametric nonlinear effect that
effectively amplifies low frequency components at the expense of high
frequency components within the same optical signal. This significantly alters
the NLI distribution across the received spectrum [24].

Recently, we introduced the ISRS GN model [25, 14], which is a GN model that
rigorously accounts for ISRS and is, therefore, suitable for optical
bandwidths beyond C-band. Comparable models were published in [24, 26, 27,
15]. The ISRS GN model relies on numerically solving an integral of at least
three dimensions and an approximation in closed-form has yet to be reported.

In this paper, a closed-form approximation of the ISRS GN model is presented
which accurately accounts for the impact of ISRS on the nonlinear interference
power. The proposed formula is applicable to dispersion unmanaged ultra-
wideband transmission systems with optical bandwidths of up to 15 THz.
Additionally, the proposed formula accounts for the dispersion slope and
variably loaded fiber spans, enabling performance estimations in optical mesh
networks. The analytical result is extensively validated by split-step
simulations.

The paper is organized as follows: The ISRS GN model is briefly revised in
Section II-A. Section II-B addresses the key steps in the derivation of the
closed-form approximation and the result is presented in Section II-C. In
Section III, the proposed formula is validated via split-step simulations and
via the ISRS GN model in integral form. The validation is carried out in a
point-to-point scenario in Section III-A and in a mesh optical network
scenario in Section III-B.

##  II The ISRS GN model in closed-form

In this section, the proposed closed-form approximation of the ISRS GN model
is presented. The main derivation steps are outlined and its key assumptions
are addressed and discussed.

###  II-A The ISRS GN model

In the following the ISRS GN model, a Gaussian model that accounts for inter-
channel stimulated Raman scattering, is briefly revised. After coherent
detection, electronic dispersion compensation and neglecting the impact of
transceiver noise, the signal-to-noise ratio (SNR) of the channel of interest
(COI) $i$ can be calculated as

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

where $P_{i}$ is the launch power of channel $i$ and $P_{\textnormal{ASE}}$ is
the accumulated amplified spontaneous emission (ASE) noise originating from
optical amplifiers. The nonlinear interference coefficient
$\eta_{n}\left(f_{i}\right)$ after $n$ spans is dependent on the center
frequency $f_{i}$ of the COI. When the impact of ISRS is negligible, it is
convenient to write the total nonlinear interference power as
$P_{\text{NLI}}=\eta_{n}P_{i}^{3}$, as the NLI coefficient is independent of
the absolute launch power and the cubic law in (1) holds within the first-
order perturbation regime.

However, for bandwidths beyond C-band, where ISRS cannot be neglected, the NLI
coefficient itself becomes a function of the total launch power
$P_{\text{tot}}$ and its absolute launch power distribution
$G_{\text{Tx}}\left(f\right)$. Similar to [25, Eq. (9)], the NLI coefficient
in the presence of ISRS is given by

$\begin{split}&\eta_{1}\left(f_{i}\right)=\frac{B_{i}}{P_{i}^{3}}\frac{16}{27}\gamma^{2}\int
df_{1}\int df_{2}\ G_{\text{Tx}}(f_{1})G_{\text{Tx}}(f_{2})\\\ &\cdot
G_{\text{Tx}}(f_{1}+f_{2}-f_{i})\\\ &\cdot\left|\int_{0}^{L}d\zeta\
\frac{P_{\text{tot}}e^{-\alpha\zeta-
P_{\text{tot}}C_{\text{r}}L_{\text{eff}}(f_{1}+f_{2}-f_{i})}}{\int
G_{\text{Tx}}(\nu)e^{-P_{\text{tot}}C_{\text{r}}L_{\text{eff}}\nu}d\nu}e^{j\phi\left(f_{1},f_{2},f_{i},\zeta\right)}\right|^{2},\end{split}$
(2)

where $B_{i}$ is the bandwidth of the COI $i$, $\gamma$ is the nonlinearity
coefficient, $\alpha$ is the attenuation coefficient,
$L_{\text{eff}}=\frac{1-e^{-\alpha\zeta}}{\alpha}$ (the $\zeta$ dependence is
suppressed throughout this paper for notational brevity), $C_{r}$ is the slope
of the linear regression of the normalized Raman gain spectrum and
$\phi=-4\pi^{2}(f_{1}-f)(f_{2}-f)\left[\beta_{2}+\pi\beta_{3}(f_{1}+f_{2})\right]\zeta$
is a phase mismatch term, with the group velocity dispersion (GVD) parameter
$\beta_{2}$ and its linear slope $\beta_{3}$. Eq. (2) is valid for optical
bandwidths of up to 15 THz as it assumed a linear Raman gain spectrum. The
ISRS GN model in integral form for multi-span systems ca be found in [14, Eq.
(2)]. Eq. (2) is obtained from the result derived in [25, Eq. (9)] with the
difference that (2) assumes a uniform NLI PSD over the channel bandwidth (in
the literature sometimes referred to as the local white noise assumption).
This assumption is required in order to avoid the integration of the NLI PSD
over the receiver filter (i.e. the matched filter), which is analytically
difficult for arbitrary filter shapes.

The strength of ISRS for a given system can be assessed by calculating the net
power transfer $\Delta\rho\left(z\right)$ between the outer channels of the
WDM signal. This is the ISRS net gain of the lowest frequency channel added to
the ISRS net loss of the highest frequency channel, which is given by [28, Eq.
(8)]

$\begin{split}\Delta\rho\left(z\right)\left[\text{dB}\right]\triangleq
10\log_{10}\left[\Delta\rho\left(z\right)\right]=4.3\cdot
P_{\text{tot}}C_{\text{r}}L_{\text{eff}}B_{\text{tot}},\end{split}$ (3)

where $B_{\text{tot}}$ is the total optical bandwidth.

###  II-B The XPM assumption

In the following, the key steps in deriving the proposed closed-form
approximation of (2) are addressed and discussed.

We first evaluate the nonlinear perturbation of the COI $i$, caused by a
single interferer (INF) $k$, which is denoted as
$\eta_{\text{XPM}}^{\left(k\right)}(f_{i})$. The special case where the NLI is
caused by the COI itself (i.e. $k=i$), is denoted as SPM (sometimes referred
to as SCI). The NLI contribution of all other interferer is denoted as XPM
(sometimes referred to as XCI). An illustration of the SPM and XPM
contribution of a COI and single INF is schematically shown in Fig. 2. In more
detail, the set of all XPM interferers, with respect to the COI $i$, is given
as

$\begin{split}A_{i}=\left\\{k\in\mathbb{N}\ |\ 1\leq k\leq N_{\text{ch}}\
\text{and}\ k\neq i\right\\}.\end{split}$ (4)

In the literature, this assumption is often referred to as XPM assumption [29,
30, 31, 21, 32] and it neglects NLI contributions that are jointly generated
by two interfering channels, which is denoted as FWM (sometimes referred to as
MCI). However, this contribution is typically very small in high dispersive
links, where high baud rates or channel spacings are used [33, 34].

Figure 2: Illustration of a transmitted spectrum $G_{\text{Tx}}\left(f\right)$
subject to the XPM assumption. Shown are the channel under test (COI) and a
single interferer (INF) with arbitrary power levels, bandwidths and center
frequencies. The total NLI is then obtained by summing over all interferers
$k\in A_{i}$ as in (4).

Using the XPM assumption, the NLI coefficient can be written as

$\begin{split}&\eta_{n}\left(f_{i}\right)\approx\eta_{\text{SPM}}\left(f_{i}\right)n^{1+\epsilon}+\eta_{\text{XPM}}\left(f_{i}\right)n,\end{split}$
(5)

where $\eta_{\text{SPM}}\left(f_{i}\right)$ is the SPM contribution and
$\eta_{\text{XPM}}\left(f_{i}\right)$ is the total XPM contribution, which is
the summation over all XPM interferer, as

$\begin{split}&\eta_{\text{XPM}}\left(f_{i}\right)=\sum_{\forall k\in
A_{i}}\eta_{\text{XPM}}^{\left(k\right)}(f_{i}),\end{split}$ (6)

where $\eta_{\text{XPM}}^{\left(k\right)}(f_{i})$ is the NLI (XPM)
contribution of a single channel $k$ on channel $i$.

The coherent accumulation along multiple fiber spans is included using the
coherence factor $\epsilon$. The coherence factor can be obtained in closed-
form [13, Eq. (22)]. The coherence factor is typically defined for the entire
optical signal [13, 23]. However in this work, only the SPM contribution is
assumed to accumulate coherently and the coherence factor is redefined over
the channel bandwidth $B_{i}$ [35]. The XPM contribution is assumed to
accumulate incoherently. The advantage of this approach is that the coherent
accumulation is independent of the transmitted spectrum and only a function of
the channel bandwidth. This significantly simplifies the modeling of NLI in
optical mesh networks as fiber spans are variably loaded. The approach is
consistent with the observations in [33, 22].

Additionally, it is assumed that the coherence factor itself is not altered by
ISRS. In [25], it was shown that for standard single mode fiber (SMF) based
spans and an ISRS power transfer of
$\Delta\rho\left(z\right)\left[\text{dB}\right]=6.5$ dB over $10$ THz the
coherence factor changes by up to $\Delta\epsilon=0.012$. Neglecting this
effect, results in an approximation error of $\Delta\epsilon\cdot
10\log\left(10\right)=0.1$ dB in NLI after 10 spans and $\Delta\epsilon\cdot
10\log\left(50\right)=0.2$ dB after 50 spans. In this work, this effect is
neglected and the coherent accumulation is assumed to be unaffected by ISRS.

In the following, the NLI caused by an interferer $k$ on the COI $i$, is
analytically evaluated. It is assumed that the channel of interest $i$ has
normalized pulse shape
$g_{i}(f-f_{i})=\frac{1}{B_{i}}\Pi\left(\frac{f-f_{i}}{B_{i}}\right)$, launch
power $P_{i}$, channel bandwidth $B_{i}$ and is centered around frequency
$f_{i}$. The function $\Pi\left(x\right)$ denotes the rectangular function.
The interfering channel has normalized pulse shape
$g_{k}(f-f_{k})=\frac{1}{B_{k}}\Pi\left(\frac{f-f_{k}}{B_{k}}\right)$, launch
power $P_{k}$, bandwidth $B_{k}$ and is centered around frequency $f_{k}$. The
transmitted spectrum is then given by

$\begin{split}G_{\text{Tx}}(f)=P_{i}g_{i}(f-f_{i})+P_{k}g_{k}(f-f_{i}-\Delta
f),\end{split}$ (7)

where $\Delta f=f_{k}-f_{i}$ is the frequency separation between COI and INF.
An illustration of (7) with the resulting nonlinear interactions on the COI is
shown in Fig. 2.

Substituting the transmitted spectrum (7) in the ISRS GN model (2) yields six
non-identical cross terms where only two are non-zero and contribute to the
NLI of the COI. These two terms are the SPM and the XPM contribution. The XPM
contribution is

$\begin{split}&\eta_{\text{XPM}}^{\left(k\right)}(f_{i})=\frac{32}{27}\frac{\gamma^{2}}{B^{2}_{k}}\left(\frac{P_{k}}{P_{i}}\right)^{2}\int_{-\frac{B_{i}}{2}}^{\frac{B_{i}}{2}}df_{1}\int_{-\frac{B_{k}}{2}}^{\frac{B_{k}}{2}}df_{2}\
\Pi\left(\frac{f_{1}+f_{2}}{B_{k}}\right)\\\ &\cdot\left|\int_{0}^{L}d\zeta\
\frac{P_{\text{tot}}C_{\text{r}}L_{\text{eff}}B_{\text{tot}}}{2}\frac{e^{-\alpha\zeta-
P_{\text{tot}}C_{\text{r}}L_{\text{eff}}(f_{1}+f_{2}+f_{i}+\Delta
f)}}{\text{sinh}\left(\frac{P_{\text{tot}}C_{\text{r}}L_{\text{eff}}B_{\text{tot}}}{2}\right)}\right.\\\
&\left.\cdot e^{j\phi\left(f_{1}+f_{i},f_{2}+f_{i}+\Delta
f,f_{i},\zeta\right)}\right|^{2},\end{split}$ (8)

and the SPM contribution is

$\begin{split}&\eta_{\text{SPM}}(f_{i})=\frac{1}{2}\eta_{\text{XPM}}^{\left(i\right)}(f_{i}).\end{split}$
(9)

It should be noted that the quantities $P_{\text{tot}}$ and $B_{\text{tot}}$
refer to the launch power and bandwidth of the entire transmitted WDM signal
and not to the transmitted spectrum of a single COI and INF pair.

ISRS alters the effective attenuation for a given WDM channel during
propagation and hence introduces a change in the signal power profile within a
fiber span. This change is a function of the total transmitted launch power,
mathematically expressed as
$P_{\text{tot}}e^{-P_{\text{tot}}C_{\text{r}}L_{\text{eff}}(f_{1}+f_{2}-f_{i})}$
in (2), and its spectral distribution, mathematically expressed as $\int
G_{\text{Tx}}(\nu)e^{-P_{\text{tot}}C_{\text{r}}L_{\text{eff}}\nu}d\nu$ in
(2).

In deriving (8) and (9), it was assumed that the total launch power of the
entire WDM signal is uniformly distributed over the entire optical bandwidth.
This effectively assumes that the effective channel power attenuation (the
signal power profile) is independent of the launch power distribution and only
dependent on the total launch power. The impact of this assumption is
discussed in more detail in the next section and quantified for mesh optical
networks in Section III-B and sloped launch power distributions in Section IV.

###  II-C The ISRS GN model in closed-form

The SPM and the XPM contribution are solved separately yielding two formulas,
one for each contribution. The total NLI is then obtained using (5). The
reader is referred to Appendix A and Appendix B for the detailed derivations.

The closed-form approximation for the SPM contribution is

$\begin{split}&\eta_{\text{SPM}}\left(f_{i}\right)\approx\frac{16}{27}\frac{\gamma^{2}}{B^{2}_{i}}\left[\frac{\pi\left(T_{i}^{2}-\frac{4}{9}\right)}{\alpha\phi_{i}}\text{asinh}\left(\frac{B_{i}^{2}\phi_{i}}{16\alpha}\right)+\frac{B_{i}^{2}}{9\alpha^{2}}\right],\end{split}$
(10)

with $\phi_{i}=12\pi^{2}\left(\beta_{2}+2\pi\beta_{3}f_{i}\right)$ and
$T_{i}=2-\frac{f_{i}P_{\text{tot}}C_{\text{r}}}{\alpha}$.

The closed-form approximation for the total XPM contribution is

$\begin{split}&\eta_{\text{XPM}}\left(f_{i}\right)\approx\frac{32}{27}\frac{\gamma^{2}}{\alpha}\sum_{\forall
k\in A_{i}}\left(\frac{P_{k}}{P_{i}}\right)^{2}\frac{1}{B_{k}\phi_{i,k}}\\\
&\left[\frac{T_{k}^{2}-1}{3}\text{atan}\left(\frac{B_{i}\phi_{i,k}}{\alpha}\right)+\frac{4-T_{k}^{2}}{6}\text{atan}\left(\frac{B_{i}\phi_{i,k}}{2\alpha}\right)\right]\\\
\end{split}$ (11)

with
$\phi_{i,k}=2\pi^{2}\left(f_{k}-f_{i}\right)\left[\beta_{2}+\pi\beta_{3}(f_{i}+f_{k}\right]$.
The sum in (LABEL:eq:XPM_CF) represents the summation over the XPM
contribution of each individual interferer as in (4). Both formulas were
derived from (8) with three key assumptions:

1) For the XPM contribution, the frequency separation between the channel of
interest and the interfering channel is much greater than half of the channel
bandwidth, i.e. $\left|f_{k}-f_{i}\right|=\left|\Delta
f\right|\gg\frac{B_{k}}{2}$.

2) The impact of ISRS on the effective channel power attenuation (i.e. the
signal power profile) is small, which means that it can be approximated by a
first-order Taylor series.

3) The effective channel power attenuation (i.e. the signal power profile) is
only a function of the total launch power and independent of its spectral
distribution. This assumption has no impact on a uniform launch power
distribution.

Figure 3: XPM contribution in NLI power for channel $i=25$ with $f_{i}=-4000$
as a function of frequency separation between channel of interest (COI) and
interferer (INF), obtained from numerically solving the ISRS GN model in
integral form (8) and its proposed approximation in closed-form (10) and
(LABEL:eq:XPM_CF) (for $k\in\left\\{26\text{,}\ 27\text{,}\ 28\text{,}\
29\right\\}$). A WDM signal with an optical bandwidth of 10 THz is assumed
with $0$ dBm launch power per channel.

Assumption 1) is valid when the interferer has a sufficient frequency
separation from the COI. For very densely spaced channels, e.g. Nyquist-
spacing, $\left|\Delta f\right|\gg\frac{B_{k}}{2}$ holds for all interferers
that are separated by more than a few channel bandwidths. A separation of five
channel bandwidths yields $5\gg 0.5$. For non-Nyquist spacing, e.g. a ITU grid
configuration of 32 GBd channels ($B_{i}\approx 32$ GHz) on a 50 GHz grid
fulfills the same condition after only 3 channel separations. In reality, the
impact of this assumption is even smaller (see Appendix A for details).

Figure 4: The NLI coefficient after one span as a function of channel
frequency is shown in a). The deviation of the NLI coefficient after one span
as a function of ISRS power transfer for different channels within the
transmitted WDM signal is shown in b). The results were obtained by numerical
simulations, numerically solving the ISRS GN model in integral form (2) and
its proposed approximation in closed-form (10) and (LABEL:eq:XPM_CF).

The SPM/XPM contribution as a function of channel separation, obtained by
numerically integrating (8) and its proposed approximation in closed-form are
shown in Fig. 3. The COI has a center frequency of $f_{i}=-4000$ GHz and is
transmitted with a single interferer over 100 km with parameters listed in
Table I. Despite assumption 1), Eq. (LABEL:eq:XPM_CF) yields remarkable
accuracy with negligible error ($<0.1$ dB) for all XPM contributions with
channel separations $\Delta f\geq 40$ GHz. The SPM contribution exhibits a
discrepancy of only $0.1$ dB. However, the accuracy of the total NLI is
dominated by XPM which yields an accuracy of the proposed closed-form of
$<0.1$ dB on the NLI (neglecting FWM) in the case of no ISRS. In the case of
ISRS, the discrepancy is slightly higher due to assumption 2).

Assumption 2) holds when the impact of ISRS on the the effective channel power
attenuation (i.e. the signal power profile) can be considered low. The SPM/XPM
contribution for a COI with $f_{i}=-4000$ GHz and a launch power of $0$ dBm
accounting for ISRS is shown in Fig. 3. For this launch power, assuming a
fully occupied 10 THz WDM signal, the power transfer between the outer
channels is $\Delta\rho\left(L\right)\left[\text{dB}\right]=6.3$ dB and the
net ISRS gain of the COI is $2.3$ dB. The only additional inaccuracy, compared
to the case of no ISRS, originates from the weak ISRS assumption. The
discrepancy for the SPM contribution is $0.3$ dB and $0.1$ dB for a frequency
separation of $\Delta f=40$ GHz. The accuracy of this assumption depends on
the strength of ISRS for a given transmission, where stronger ISRS reduces the
accuracy. In order to obtain an approximate validity range, we compare the
first-order and second-order terms of the Taylor expansion used to approximate
ISRS. The reader is referred to Appendix C for a detailed derivation. It is
found that the second-order term is negligible if and only if

$\begin{split}\Delta\rho\left(L\right)\left[\text{dB}\right]=4.3\cdot
P_{\text{tot}}C_{\text{r}}L_{\text{eff}}B_{\text{tot}}\ll 26.\end{split}$ (12)

For an ISRS power transfer of
$\Delta\rho\left(L\right)\left[\text{dB}\right]=6.3$ dB, Eq. (12) yields
$6.3\ll 26$. For this case, (12) is not fully satisfied and assumption 2) has
a small impact on the accuracy of the closed-form expression.

Assumption 3) introduces no approximation error when the launch power
distribution is uniform. When the launch power distribution deviates from a
uniform one, the signal power profile is slightly changed due to ISRS. In
practice, non-uniform launch power distributions may be present due to a
variety of reasons. In mesh optical networks, non-continuous launch power
distributions arise from unoccupied wavelengths as a result of traffic demands
and routing algorithms. Additionally, launch power distributions might be
optimized and sloped in order to achieve higher information throughput. The
former case is addressed in Section III-B while the latter case is addressed
in Section IV.

##  III Numerical Validation

In this section the proposed closed-form approximations (10) and
(LABEL:eq:XPM_CF) are validated in an optical transmission system with
parameters listed in Table I. The validation is performed for a point-to-point
transmission in III-A and for a mesh optical network scenario in III-B.

The validation was carried out by numerically solving the Manakov equation
using the well established split-step Fourier method (SSFM). Inter-channel
stimulated Raman scattering was included in the SSFM by applying a frequency
dependent loss at every linear step, so that the signal power profile altered
by ISRS, is obtained.

A logarithmic step size distribution was implemented, where $0.25\cdot 10^{6}$
simulation steps were found to be sufficient for launch powers as high as $0$
dBm per channel and $1\cdot 10^{6}$ for launch powers as high as $3$ dBm/ch.
Launch powers of up to $3$ dBm/ch. were considered in order to check the
validity of the weak ISRS assumption (see assumption 2 in Section II-C) for
power transfers of up to $\Delta\rho\left(L\right)\left[\text{dB}\right]=13$
dB. At the beginning of the fiber, the step size was as short as $21.5$ cm
($86$ cm), while the step size was $2.15$ m ($8.6$ m) for $3$ dBm/ch. ($0$
dBm/ch.) after 100 km of propagation.

Gaussian symbols, drawn from a circular-symmetric Gaussian distribution and
uniform 64-QAM symbols were used for transmission. The former was chosen in
order to verify the closed-form approximation while the latter was chosen to
compare the performance to a standard modulation format.

The receiver consisted of digital dispersion compensation, ideal root-raised-
cosine (RCC) matched filtering and constellation rotation. 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 Eq. (1). In order to improve the simulation accuracy, four
different data realizations were simulated and averaged for each transmission.

Ideal, noiseless amplifiers were considered to ease the NLI computation and
for a fair comparison between numerical simulation and ISRS GN model.

TABLE I: System Parameters Parameters |
---|---
Loss ($\alpha$) [dB/km] | 0.2
Dispersion ($D$) [ps/nm/km] | 17
Dispersion slope ($S$) [ps/$\text{nm}^{2}$/km] | 0.067
NL coefficient ($\gamma$) [1/W/km] | 1.2
Raman gain slope ($C_{\text{r}}$) [1/W/km/THz] | 0.028
Raman gain ($C_{\text{r}}\cdot 14$ THz) [1/W/km] | 0.4
Symbol rate [GBd] | 40
Channel Bandwidth ($B_{i}$) [GHz] | $40.004$
Channel Launch Power ($P_{i}$) [dBm] | 0
Total Launch Power ($P_{\text{tot}}$) [dBm] | 24
Channel spacing [GHz] | 40.005
Number of channels | 251
Optical bandwidth ($B_{\text{tot}}$) [THz] | $10.05$
Reference Wavelength [nm] | $1550$
Roll-off factor [%] | 0.01
Number of symbols [$2^{x}$] | 17
Simulation steps per span [$10^{6}$] | 0.25 to 1
Figure 5: The distribution of the NLI coefficient after 6 spans ($600$ km)
without a) and with b) ISRS. A launch power of $0$ dBm/ch. was considered
yielding an ISRS power transfer of
$\Delta\rho\left(L\right)\left[\text{dB}\right]=6.3$ dB. The results were
obtained by numerical simulations, the ISRS GN model in integral form (2) and
its proposed approximation in closed-form (10) and (LABEL:eq:XPM_CF), in
coherent and incoherent form.

The spectral distribution of the NLI coefficient after a single span obtained
by the SSFM, the ISRS GN model in integral form and its proposed approximation
in closed form are shown in Fig. 4a). Launch powers of $0$ dBm/ch. and $2$
dBm/ch. are shown which results in an ISRS power transfer of
$\Delta\rho\left(L\right)\left[\text{dB}\right]=6.3$ dB and
$\Delta\rho\left(L\right)\left[\text{dB}\right]=10.3$ dB, respectively. A
case, where no ISRS is considered is shown for comparison.

The tilt in NLI, in the case of no ISRS, is due to the dispersion slope $S$
(or $\beta_{3}$), where low frequency components exhibit a higher amount of
dispersion resulting in lower nonlinear penalties. With increasing launch
powers, low frequency components are increasingly amplified, at the expense of
high frequency components, leading to increased (reduced) NLI for low (high)
frequency components.

Not surprisingly, the ISRS GN model in integral form matches the simulation
results with negligible error except at the most outer channels due to the
local white noise assumption (which could be lifted by properly integrating
the NLI PSD over the channel bandwidth). The proposed closed-form
approximation is in good agreement with the ISRS GN model in integral form and
the simulation results. The average gap, in the case of no ISRS, is $0.1$ dB.
This discrepancy is due to the XPM assumption (see Section II-B), as the
individual SPM and XPM contributions are approximated with negligible error,
as shown in Fig. 3. The average discrepancy is $0.1$ dB for $0$ dBm/ch. and
$0.2$ dB for 2 dBm/ch. launch power. The increasing discrepancy with
increasing launch power is due to the weak ISRS assumption (see Section II-C).
This assumption has more impact on the outer channels as the net ISRS gain is
larger. Not surprisingly, the ISRS GN model in integral form matches the
simulation results with negligible error except at the most outer channels due
to the local white noise assumption (which could be lifted by properly
integrating the NLI PSD over the channel bandwidth). The proposed closed-form
approximation is in good agreement with the ISRS GN model in integral form and
the simulation results. The average gap, in the case of no ISRS, is $0.1$ dB.
This discrepancy is due to the XPM assumption (see Section II-B), as the
individual SPM and XPM contributions are approximated with negligible error,
as shown in Fig. 3. The average discrepancy is $0.1$ dB for $0$ dBm/ch. and
$0.2$ dB for 2 dBm/ch. launch power. The increasing discrepancy with
increasing launch power is due to the weak ISRS assumption (see Section II-C).
This assumption has more impact on the outer channels as the net ISRS gain is
larger for those channels.

The deviation of the NLI coefficient as a function of the ISRS power transfer
$\Delta\rho\left(L\right)\left[\text{dB}\right]$ for different channels within
the WDM signal is shown in Fig. 4b). The discrepancy between the ISRS GN model
in integral form and the SSFM is negligible for the shown range of power
transfers. Due to the weak ISRS assumption, the accuracy of the closed-form
expression decreases with increasing ISRS. This is because higher order terms
of the Taylor expansion are becoming significant.

###  III-A A point-to-point transmission scenario

In this section, a multi-span transmission system, consisting of six identical
100 km SMF fiber spans, is studied with parameters listed in Table I. A
uniform launch power of $0$ dBm/ch. was used which is the optimum launch power
for the central channel in the presence of Erbium-doped fiber amplifiers
(EDFA) with a noise figure of 5 dB. The ISRS power transfer was equalized by a
gain flattening filter after every fiber span.

The NLI coefficient after six spans is shown in Fig. 5a) without accounting
for ISRS and in Fig. 5b) accounting for ISRS. Simulation results using
Gaussian modulation as well as uniform 64-QAM are shown together with the ISRS
GN model in integral form and its proposed approximation in closed-form. To
account for coherent accumulation and variably loaded fiber spans, the ISRS GN
model takes a slightly different form which was published in [14, Eq. (2)] and
used to obtain the results in Fig. 5.

The closed-form approximation is considered with an incoherent ($\epsilon=0$)
and a coherent ($\epsilon\neq 0$) accumulation of NLI along multiple fiber
spans (see (5)). The coherence factor of the given system configuration is
$\epsilon=0.15$. The average gap between the closed-form, including a coherent
accumulation, and the ISRS GN model in integral form is 0.1 dB and 0.2 dB
without and with ISRS, respectively. The accuracy is similar to the single
span case (see Fig. 4), indicating that Eq. (5) sufficiently approximates the
coherent accumulation of NLI.

A majority of the NLI originates from XPM, which is accumulating incoherently.
The formalism may therefore be simplified by assuming an incoherent
accumulation and setting $\epsilon=0$. The average accuracy loss, of assuming
incoherent accumulation, is $0.2$ dB for the studied system. Depending on
accuracy requirements, this error may be deemed negligible. It should be
noted, however, that this accuracy loss (with respect to Gaussian modulation)
increases with the number of spans.

A key assumption of the model is that each frequency component carries a
symbol drawn from a symmetric circular Gaussian distribution which leads to an
overestimation of the NLI power with respect to square QAM formats. To compare
the model predictions to a standard modulation format, the NLI coefficient
using 64-QAM obtained by the SSFM is shown in Fig. 5. The average gap between
SSFM using 64-QAM and the closed-form approximation in coherent form is $1.6$
dB in both cases, without and with ISRS. This gap decreases with increasing
accumulated dispersion, hence, with increasing transmission distance.
Additionally, modern transmission systems utilize probabilistic or geometric
shaping which further decreases this gap as shaped signals partially resemble
Gaussian modulated signals [36, 37].

The modulation format dependence can be partly accounted for with the
following heuristic: First, it is assumed that ISRS has negligible impact on
the relative modulation format dependence, which is supported by the shown
results. For lumped-amplified, multi-span transmission systems, where ISRS is
negligible, the relative impact of the modulation format on the NLI has been
derived in closed-form [22, Eq. (8)]. Using this result, the average deviation
can be reduced from $1.6$ dB to only $0.8$ dB. The modulation format
dependence is not fully accounted for, as the formula is derived
asymptotically for high span counts and it only corrects for XPM. Therefore,
this approach is still conservative but with a reduced margin.

In summary, the proposed closed-form approximation models the impact of ISRS
on the NLI with excellent accuracy in fully occupied point-to-point
transmission scenarios. In can, therefore, be used for system design,
optimization and real-time performance estimations of ultra-wideband
transmission point-to-point links.

###  III-B A mesh optical network scenario

In this section, the closed-form approximation (10) and (LABEL:eq:XPM_CF) is
applied and validated in a mesh optical network. The fundamental difference in
a mesh network, as opposed to a point-to-point transmission, is that not all
channels within a WDM signal are transmitted along the entire lightpath. At
each reconfigurable optical add-drop multiplexer (ROADM), channels are added
and dropped according to traffic demands and as the result of wavelength
routing and lightpath assignment algorithms (RWA).

We introduce the following two definitions. For a given lightpath, channels
that are transmitted along the entire lightpath are denoted as channels of
interest. On the other hand, channels that are added and/or dropped at any
point along the lightpath are denoted as interfering channels. Due to variably
loaded network edges, the NLI of the channels of interest is different with
respect to an equivalent point-to-point transmission as different interfering
channels are emphasizing different XPM contributions
$\eta_{\text{XPM}}^{\left(k\right)}(f_{i})$ at each network edges.
Additionally, most interfering channels have already propagated through part
of the network, resulting in different amounts of accumulation dispersion
compared to a point-to-point transmission.

The British Telecommunications 20+2 topology of the United Kingdom core
network [16] is considered for the analysis. For a given network topology and
a given traffic demand, a vast number of feasible lightpath combinations are
possible. For the sake of validation, only one lightpath is analyzed with two
different network utilizations. We define network utilization as the average
spectrum occupancy out of the entire available optical bandwidth.

The lightpath under test is the path between node A and B as indicated in Fig.
1. It is assumed that the first two edges, the first edge with $197$ km length
and the second edge with $203$ km length, are each split into two fiber spans.
The resulting lightpath of interest is illustrated in Fig. 6, where after each
ROADM a different spectrum is launch into the fiber due to the adding and
dropping of interfering channels. The channels of interest, that are
propagated along the entire lightpath, are shown in black where interfering
channels are shown in color.

Figure 6: Example transmission link, between nodes 15 and 13, from the BT 20+2
topology of the UK core network Fig. 6, showing interfering channels (in
color) added and dropped at each ROADM. The channels of interest, that are
transmitted along the entire lightpath, are shown in black. Figure 7: The NLI
coefficient of every fifth channel (i.e. a channel of interest) after six
spans where interfering channels are continuously added and dropped along the
transmission with a network utilization of 80% shown in a) and 90% shown in
b). The results were obtained by numerical simulations and using the proposed
closed-form approximation (10) and (LABEL:eq:XPM_CF).

For this work, a few assumptions on the traffic and the established lightpaths
are made. Every fifth channel (51 out of a total of 251 channels slots) was a
channel of interest and their NLI coefficients were obtained by the SSFM and
by the proposed closed-form approximation using (10) and (LABEL:eq:XPM_CF).
The remaining 200 channel slots were partially filled with interfering
channels which were continuously dropped and added at each ROADM. At each
ROADM, 80% of the interfering channels were randomly dropped and interfering
channels were added by randomly choosing an empty channel slot. The unoccupied
channel slots were randomly filled until a certain network utilization was
reached. The considered network utilizations were 80% and 90%.

The added channels exhibit a random power offset between $\pm 1$ dB with
respect to the COI to simulate potential non-ideal power equalization.
Additionally, this was done in order to test the impact of assumption 3) (see
Section II-C) in a network scenario. Interfering channels were using the same
modulation format as the channel under test and were randomly pre-dispersed
corresponding to a transmission distance between 0 and 1000 km, to emulate the
propagation from different lightpaths in the network. The wavelength dependent
gain due to ISRS was ideally compensated after each span to ease a comparison
to the point-to-point case in Section III-A.

The NLI coefficient for a network utilization of $80$% is shown in Fig. 7a)
and a network utilization of $90$% is shown in Fig. 7b). The SSFM results in
Fig. 7a) were first published in [14]. The ISRS power transfers were
$\Delta\rho\left(L\right)\left[\text{dB}\right]=5$ dB and
$\Delta\rho\left(L\right)\left[\text{dB}\right]=5.7$ dB, which is less than in
the point-to-point case as less average power was launched into a span.

The fluctuating behavior of the NLI coefficient is a direct consequence of the
variably loaded network edges. The fluctuations are weaker in the case of $90$
% network utilization as a larger average spectral occupation yields more
averaging. The change in NLI due to ISRS was $-1.6$ dB to $1.5$ dB for 80% of
network utilization and $-1.8$ dB to $1.6$ dB for 90% of network utilization.
The proposed closed-form approximation is in good agreement with the
simulation results with an average discrepancy of $0.1$ dB and $0.2$ dB for
80% and 90% network utilization, respectively. Assumption 3) in II-C seems to
have a negligible impact on the accuracy of the formula in variably loaded
mesh optical networks. The average gap between the closed-form approximation
and the SSFM using uniform 64-QAM is $1$ dB which is less than in the point-
to-point case (cf. Fig. 5) as interfering channels exhibit, in average, an
higher amount of accumulated dispersion.

Based on the validation carried out in this section, it is concluded that the
proposed closed-form approximation models the NLI in mesh optical network
scenarios with excellent accuracy. The results in this paper, therefore,
enable the performance evaluation of complicated light path configurations for
an entire network topology within only a few micro seconds. This is an
essential step in the modeling of optical network performance in the ultra-
wideband regime.

##  IV Accuracy impact of sloped launch power distributions

In this section, the impact of a sloped launch power distribution on the
accuracy of the proposed closed-form is addressed. In the derivation of the
proposed closed-form approximation, it is assumed that the effective channel
power attenuation is only a function of the total launch power and independent
of its spectral distribution (see assumption 3 in Section II-C). The accuracy
loss in the case of variable loaded fiber spans was found to be negligible in
Section III-B.

Sloped launch power distributions may be used as in order to increase the
achievable information throughput or to improve the SNR margin [38, 26, 39].

The deviation of the SPM contribution using (8) as a function of launch power
slope with respect to a uniform launch power distribution is shown in Fig. 8.
A positive launch power slope means that high frequency channels have a larger
launch power than low frequency channels. The plot shows maximum, minimum and
average deviation of all channels within the WDM signal. Fig. 8 can be
interpreted as in the following. For a launch power slope of e.g. $\pm 2$ dB,
assumption 3) in section II-C introduces an approximation error of about $\mp
0.2$ dB on the predicted NLI.

A positively sloped launch power distribution increases the amount of measured
NLI of all channels. This means that the gap between the NLI, predicted by the
closed-form approximation, and the NLI of a square QAM signal actually
decreases as a function of input slope. This is due to a cancellation of
approximation errors between assumption 3) in Section II-C and the Gaussian
modulation assumption. The opposite is true for negatively sloped launch power
distributions.

The extension of the proposed closed-form approximation to fully account for
arbitrary launch distributions is left for future research.

Figure 8: Deviation between the SPM contribution by considering a uniform and
a sloped launch power distribution as a function of input power slope for a 10
THz signal over a 100 km SMF span. The results were obtained using Eq. (8).

##  V Conclusion

A closed-form approximation of the Gaussian noise model in the presence of
inter-channel stimulated Raman scattering was presented.

It was verified using split-step simulations and numerical integrations of the
ISRS GN model in integral form, reporting an average deviation of 0.2 dB in
nonlinear interference power for SMF based spans operating over the entire C+L
band. This discrepancy, which could be addressed in the future, is primarily
because the NLI contribution, that is jointly generated by two interfering
channels (FWM or MCI), is neglected and due to the first-order description of
ISRS. However, this has little impact for systems using dispersion unmanaged
links, operating around optimum launch power.

The results in this paper allow for rapid evaluation of performance (e.g. SNR,
maximum reach, optimum launch power) in ultra-wideband transmission systems,
an essential step towards dynamic optical network capacity optimization and
intelligent information infrastructure design.

##  Appendix A Derivation of the XPM contribution

In this section, the closed-form approximation of the XPM contribution
(LABEL:eq:XPM_CF) is derived. The derivation consists of finding an analytical
approximation of the integral form (8) which models the nonlinear interference
caused on channel $i$ by a single interfering channel $k$. The total XPM
contribution is then obtained by summing over all XPM contributions as

$\begin{split}\eta_{\text{XPM}}\left(f_{i}\right)=\sum_{\forall
k}\eta_{\text{XPM}}^{\left(k\right)}\left(f_{i}\right).\end{split}$ (13)

For notational brevity, we define
$x\left(\zeta\right)=P_{\text{tot}}C_{\text{r}}L_{\text{eff}}\left(\zeta\right)$
and a pre-factor of
$\frac{32}{27}\frac{\gamma^{2}}{B^{2}_{k}}\left(\frac{P_{k}}{P_{i}}\right)^{2}$
is suppressed throughout the derivation. Eq. (8) is then written as

$\begin{split}&\eta_{\text{XPM}}^{\left(k\right)}(f_{i})=\int_{-\frac{B_{i}}{2}}^{\frac{B_{i}}{2}}df_{1}\int_{-\frac{B_{k}}{2}}^{\frac{B_{k}}{2}}df_{2}\
\Pi\left(\frac{f_{1}+f_{2}}{B_{k}}\right)\\\ &\cdot\left|\int_{0}^{L}d\zeta\
\frac{xB_{\text{tot}}e^{-\alpha\zeta-x(f_{1}+f_{2}+f_{i}+\Delta
f)}}{2\text{sinh}\left(\frac{xB_{\text{tot}}}{2}\right)}e^{j\phi\left(f_{1}+f_{i},f_{2}+f_{i}+\Delta
f,f_{i},\zeta\right)}\right|^{2}\\\
&\approx\int_{-\frac{B_{i}}{2}}^{\frac{B_{i}}{2}}df_{1}\int_{-\frac{B_{k}}{2}}^{\frac{B_{k}}{2}}df_{2}\
\left|\int_{0}^{L}d\zeta\
\frac{xB_{\text{tot}}e^{-\alpha\zeta-x(f_{1}+f_{i}+\Delta
f)}}{2\text{sinh}\left(\frac{xB_{\text{tot}}}{2}\right)}\right.\\\
&\cdot\left.e^{j\phi\left(f_{1}+f_{i},f_{i}+\Delta
f,f_{i},\zeta\right)}\right|^{2}\\\ &=2B_{k}\int_{0}^{\frac{B_{i}}{2}}df_{1}\
\left|\int_{0}^{L}d\zeta\
\frac{xB_{\text{tot}}e^{-\alpha\zeta-x(f_{1}+f_{i}+\Delta
f)}}{2\text{sinh}\left(\frac{xB_{\text{tot}}}{2}\right)}\right.\\\
&\left.\cdot e^{j\phi\left(f_{1}+f_{i},f_{i}+\Delta
f,f_{i},\zeta\right)}\right|^{2},\end{split}$ (14)

where $\Delta f=f_{k}-f_{i}$ is the (center) frequency separation between
channel $k$ and $i$. In (14), it is assumed that the frequency separation is
much larger than half of the bandwidth of channel $k$ (i.e. $\left|\Delta
f\right|\gg\frac{B_{k}}{2}$). This assumption allows to approximate
$f_{2}+\Delta f\approx\Delta f$. This approximation has only a small impact on
the phase mismatch term $\phi$ for channels that are close to the COI. It has
negligible impact on the ISRS term (the signal power profile) as it is
essentially constant over one channel bandwidth $B_{k}$. Additionally, the
approximation partly benefits from error cancellation, as the phase mismatch
term is quasi affine with respect to $f_{2}$ over its integration domain
$f_{2}\in\left[-\frac{B_{k}}{2},\frac{B_{k}}{2}\right]$. In ultra-wideband
transmission a large number of channels contribute to the total NLI, where the
vast majority of interfering channels are spaced such that $\left|\Delta
f\right|\gg\frac{B_{k}}{2}$ holds. Therefore the error of this assumption on
the total NLI is expected to be small (see Fig. 3). Additionally, the term
$g_{k}(f_{1}+f_{2})$ in (14) is neglected.

The phase mismatch factor $\phi$, we obtain

$\begin{split}&\phi\left(f_{1}+f_{i},f_{i}+\Delta f,f_{i},\zeta\right)\\\
&=-4\pi^{2}f_{1}\Delta f\left[\beta_{2}+\pi\beta_{3}(f_{1}+2f_{i}+\Delta
f)\right]\zeta\\\ &\approx-4\pi^{2}f_{1}\Delta
f\left[\beta_{2}+\pi\beta_{3}(2f_{i}+\Delta f)\right]\zeta\\\
&=-4\pi^{2}f_{1}\left(f_{k}-f_{i}\right)\left[\beta_{2}+\pi\beta_{3}(f_{i}+f_{k}\right]\zeta\\\
&=\phi_{i,k}f_{1}\zeta,\end{split}$ (15)

with
$\phi_{i,k}=-4\pi^{2}\left(f_{k}-f_{i}\right)\left[\beta_{2}+\pi\beta_{3}(f_{i}+f_{k}\right]$
and it is assumed that the impact of the dispersion slope is constant over one
channel bandwidth $B_{i}$.

In order to simplify (14), the ISRS term is expanded into a Taylor series and
truncated to first-order, assuming weak ISRS. The validity range of this
approximation is analyzed in Appendix C in more detail and numerically
validated in section III. Additionally, it is assumed that the signal power
profile is constant over one channel bandwidth $B_{i}$, mathematically
$e^{-x(f_{1}+f_{i}+\Delta f)}\approx e^{-x(f_{i}+\Delta f)}$. The Taylor
expansion of the ISRS term is then given by

$\begin{split}\frac{B_{\text{tot}}xe^{-x(f_{i}+\Delta
f)}}{2\text{sinh}\left(\frac{B_{\text{tot}}}{2}x\right)}=1-(f_{i}+\Delta
f)x+\mathcal{O}(x^{2}),\end{split}$ (16)

and the signal power profile (to first-order) as

$\begin{split}&\frac{B_{\text{tot}}xe^{-\alpha\zeta-x(f_{i}+\Delta
f)}}{2\text{sinh}\left(\frac{B_{\text{tot}}}{2}x\right)}\approx\left(1+T_{k}\right)e^{-\alpha\zeta}-T_{k}e^{-2\alpha\zeta},\end{split}$
(17)

with

$\begin{split}T_{k}=-\frac{f_{i}+\Delta
f}{\alpha}P_{\text{tot}}C_{\text{r}}=-\frac{P_{\text{tot}}C_{\text{r}}}{\alpha}f_{k}.\end{split}$
(18)

Enabled by the first-order assumption of ISRS, the following simplification is
obtained

$\begin{split}&\left|\ \int_{0}^{L}d\zeta\
\frac{B_{\text{tot}}xe^{-\alpha\zeta-x(f_{i}+\Delta
f)}}{2\text{sinh}\left(\frac{B_{\text{tot}}}{2}x\right)}e^{j\phi_{i,k}f_{1}\zeta}\right|^{2}\\\
&\approx\left|\ \int_{0}^{L}d\zeta\
\left(1+T_{k}\right)e^{-\alpha\zeta+j\phi_{i,k}f_{1}\zeta}-T_{k}e^{-2\alpha\zeta+j\phi_{i,k}f_{1}\zeta}\right|^{2}\\\
&\approx\left|-\frac{\left(1+T_{k}\right)}{-\alpha+j\phi_{i,k}f_{1}}+\frac{T_{k}}{-2\alpha+j\phi_{i,k}f_{1}}\right|^{2}\\\
&=\frac{(2+T_{k})^{2}\alpha^{2}}{4\alpha^{4}+5\alpha^{2}\phi_{i,k}^{2}f_{1}^{2}+\phi_{i,k}^{4}f_{1}^{4}}+\frac{\phi_{i,k}^{2}f_{1}^{2}}{4\alpha^{4}+5\alpha^{2}\phi_{i,k}^{2}f_{1}^{2}+\phi_{i,k}^{4}f_{1}^{4}},\end{split}$
(19)

where it is assumed that $e^{-\alpha L}\ll 1$. Substituting the simplification
LABEL:eq:1 in (14) and using the exact integral identities
(LABEL:eq:integral1) and (LABEL:eq:integral2) yields

$\begin{split}&2B_{k}\int_{0}^{\frac{B_{i}}{2}}df_{1}\
\left|\int_{0}^{L}d\zeta\ \frac{xB_{\text{tot}}e^{-\alpha\zeta-x(f_{i}+\Delta
f)}}{2\text{sinh}\left(\frac{xB_{\text{tot}}}{2}\right)}e^{j\phi_{i,k}f_{1}\zeta}\right|^{2}\\\
&\approx 2B_{k}\\\
&\left[\frac{(2+T_{k})^{2}-1}{3\alpha\phi_{i,k}}\text{atan}\left(\frac{\phi_{i,k}B_{i}}{2\alpha}\right)+\frac{4-(2+T_{k})^{2}}{6\alpha\phi_{i,k}}\text{atan}\left(\frac{\phi_{i,k}B_{i}}{4\alpha}\right)\right].\end{split}$
(20)

In order to obtain the XPM contribution of channel $k$ on channel $i$, the
suppressed pre-factor
$\frac{32}{27}\frac{\gamma^{2}}{B^{2}_{k}}\left(\frac{P_{k}}{P_{i}}\right)^{2}$
must be included. Finally, $T_{k}$ and $\phi_{i,k}$ are redefined and the
individual XPM contributions $\eta_{\text{XPM}}^{\left(k\right)}(f_{i})$ are
summed up in order to obtain the total XPM contribution
$\eta_{\text{XPM}}(f_{i})$ as in (LABEL:eq:XPM_CF).

##  Appendix B Derivation of the SPM contribution

In this section, the closed-form SPM contribution of the NLI (10) is derived.
The derivation consists of finding an analytical approximation of the integral
expression (8) which models the nonlinear interference caused by channel $i$
on itself. The reader is reminded that, for the SPM contribution, a factor of
$\frac{1}{2}$ must be multiplied to $\eqref{eq:XPM_integral}$. For notational
brevity, we define
$x\left(\zeta\right)=P_{\text{tot}}C_{\text{r}}L_{\text{eff}}\left(\zeta\right)$
and a pre-factor of $\frac{16}{27}\frac{\gamma^{2}}{B^{2}_{i}}$ is suppressed
throughout the derivation.

The NLI coefficient of the SPM contribution is then written as

$\begin{split}&\eta_{\text{SPM}}(f_{i})=\frac{1}{2}\eta^{\left(i\right)}_{\text{XPM}}(f_{i})\approx\int_{-\frac{B_{i}}{2}}^{\frac{B_{i}}{2}}df_{1}\int_{-\frac{B_{i}}{2}}^{\frac{B_{i}}{2}}df_{2}\
\\\ &\cdot\left|\ \int_{0}^{L}d\zeta\ \frac{xB_{\text{tot}}e^{-\alpha\zeta-
xf_{i}}}{2\text{sinh}\left(\frac{xB_{\text{tot}}}{2}\right)}e^{j\phi\left(f_{1}+f_{i},f_{2}+f_{i},f_{i},\zeta\right)}\right|^{2}\\\
&\approx 4\int_{0}^{\frac{B_{i}}{2}}df_{1}\int_{0}^{\frac{B_{i}}{2}}df_{2}\ \
\left|\ \int_{0}^{L}d\zeta\ \frac{xB_{\text{tot}}e^{-\alpha\zeta-
xf_{i}}}{2\text{sinh}\left(\frac{xB_{\text{tot}}}{2}\right)}\right.\\\
&\left.\cdot
e^{j\phi\left(f_{1}+f_{i},f_{2}+f_{i},f_{i},\zeta\right)}\right|^{2},\end{split}$
(21)

where, again, it was assumed that the power transfer due to ISRS is constant
over one channel bandwidth $B_{i}$.

For the phase mismatch factor $\phi$, we obtain

$\begin{split}&\phi\left(f_{1}+f_{i},f_{2}+f_{i},f_{i},\zeta\right)\\\
&=-4\pi^{2}f_{1}f_{2}\left[\beta_{2}+\pi\beta_{3}(f_{1}+f_{2}+2f_{i})\right]\zeta\\\
&\approx-4\pi^{2}f_{1}f_{2}\left(\beta_{2}+2\pi\beta_{3}f_{i}\right)\zeta\\\
&=\phi_{i}f_{1}f_{2}\zeta,\end{split}$ (22)

with $\phi_{i}=-4\pi^{2}\left(\beta_{2}+2\pi\beta_{3}f_{i}\right)$, where it
is assumed that the impact of the dispersion slope is negligible over one
channel bandwidth $B_{i}$.

Similar to the derivation of the XPM contribution, the ISRS term is expanded
into a first-order Taylor series as in (17). In order to derive an analytical
approximation of the SPM contribution, it is necessary to further assume that
$2\alpha^{2}\gg\phi_{i}^{2}f_{1}^{2}f_{2}^{2}$. Eq. (21) can then be
approximated as

$\begin{split}&4\int_{0}^{\frac{B_{i}}{2}}df_{1}\int_{0}^{\frac{B_{i}}{2}}df_{2}\
\ \left|\ \int_{0}^{L}d\zeta\ \frac{xB_{\text{tot}}e^{-\alpha\zeta-
P_{\text{tot}}C_{\text{r}}L_{\text{eff}}f_{i}}}{2\text{sinh}\left(\frac{xB_{\text{tot}}}{2}\right)}\right.\\\
&\left.\cdot e^{j\phi_{i}f_{1}f_{2}\zeta}\right|^{2}\\\ &\approx
4\int_{0}^{\frac{B_{\text{ch}}}{2}}df_{1}\int_{0}^{\frac{B_{\text{ch}}}{2}}df_{2}\
\
\frac{(2+T_{i})^{2}\alpha^{2}+\left(\phi_{i}f_{1}f_{2}\right)^{2}}{\left[2\alpha^{2}-\left(\phi_{i}f_{1}f_{2}\right)^{2}\right]^{2}+9\alpha^{2}\left(\phi_{i}f_{1}f_{2}\right)^{2}}\\\
&\approx
4\int_{0}^{\frac{B_{\text{ch}}}{2}}df_{1}\int_{0}^{\frac{B_{\text{ch}}}{2}}df_{2}\
\
\frac{(2+T_{i})^{2}\alpha^{2}+\left(\phi_{i}f_{1}f_{2}\right)^{2}}{4\alpha^{4}+9\alpha^{2}\left(\phi_{i}f_{1}f_{2}\right)^{2}}\\\
&=4\int_{0}^{\frac{B_{i}}{2}}df_{1}\left[\frac{\left(\frac{1}{6}(2+T_{i})^{2}-\frac{2}{27}\right)}{\phi_{i}\alpha}\frac{\text{atan}\left(\frac{3\phi_{i}}{4\alpha}B_{i}f_{1}\right)}{f_{1}}+\frac{B_{i}}{18\alpha^{2}}\right]\\\
&\approx
4\left[\frac{\left(\frac{1}{6}(2+T_{i})^{2}-\frac{2}{27}\right)}{\phi_{i}\alpha}\frac{\pi}{2}\text{asinh}\left(\frac{3\phi_{i}B_{i}^{2}}{16\alpha}\right)+\frac{B_{i}^{2}}{36\alpha^{2}}\right].\end{split}$
(23)

In the last derivation step, the exact integral identities (30) and the
approximate integral solution (LABEL:eq:integral4) were used. Finally, the
pre-factor $\frac{16}{27}\frac{\gamma^{2}}{B^{2}_{i}}$ must be included and
$\phi_{i}$ is redefined in order to obtain the SPM contribution in closed-form
as in (10).

##  Appendix C Derivation of the validity range

In order to derive a validity range of the weak ISRS assumption, the ISRS term
to first-order is compared to the ISRS term to second-order at a frequency
component $f_{k}$. The first-order approximation is then valid when the
second-order term is negligible. The second coefficient of the Taylor series,
as in (16), is given by

$\begin{split}T^{(2)}_{k}&=\frac{f_{k}^{2}}{2}-\frac{B^{2}}{24}.\end{split}$
(24)

Requiring that the second-order term is negligible to the first-order
approximation yields

$\begin{split}\left|f_{k}x\right|&\gg\left|T^{(2)}_{k}x^{2}\right|=\left|\frac{f_{k}^{2}}{2}-\frac{B_{\text{tot}}^{2}}{24}\right|x^{2}.\end{split}$
(25)

The channel that is most impacted by ISRS is the channel with center frequency
$f_{k}=\frac{B}{2}$ for which we will evaluate (25) and obtain

$\begin{split}6\gg
B_{\text{tot}}P_{\text{tot}}L_{\text{eff}}C_{r}.\end{split}$ (26)

Eq. 26 can be related to the power transfer due to ISRS at the end of a fiber
span $\Delta\rho\left(L\right)\left[\text{dB}\right]$ in decibels using (3) as

$\begin{split}25.8\gg\Delta\rho\left(L\right)\left[\text{dB}\right].\end{split}$
(27)

It should be stressed that although
$\Delta\rho\left(L\right)\left[\text{dB}\right]$ is expressed in decibels, the
relation in 27 compares two numerical numbers in a linear manner. This means
that for an ISRS power transfer of
$\Delta\rho\left(L\right)\left[\text{dB}\right]=6$ dB, we have $25.8\gg 6$ not
$25.8\text{ dB}\gg 6\text{ dB}$.

##  Appendix D Integrals

This section contains a lists of integral identities that were used in order
to derive the proposed closed-form expression.

$\begin{split}&\int_{0}^{X}dx\ \frac{1}{A+Bx^{2}+x^{4}}\\\
=&\frac{\sqrt[]{2}}{C\sqrt[]{B-C}}\text{atan}\left(\frac{\sqrt[]{2}X}{\sqrt[]{B-C}}\right)-\frac{\sqrt[]{2}}{C\sqrt[]{B+C}}\text{atan}\left(\frac{\sqrt[]{2}X}{\sqrt[]{B+C}}\right),\\\
\end{split}$ (28) $\begin{split}&\int_{0}^{X}x\
\frac{x^{2}}{A+Bx^{2}+x^{4}}\\\ &=\frac{\sqrt[]{B+C}}{\sqrt[]{2}C}\
\text{atan}\left(\frac{\sqrt[]{2}X}{\sqrt[]{B+C}}\right)-\frac{\sqrt[]{B-C}}{\sqrt[]{2}C}\
\text{atan}\left(\frac{\sqrt[]{2}X}{\sqrt[]{B-C}}\right),\\\ \end{split}$ (29)

with $C=\sqrt[]{B^{2}-4A}$.

$\begin{split}\int_{0}^{X}df_{2}\frac{1+A^{2}f_{2}^{2}}{1+B^{2}f_{2}^{2}}=\frac{\left(B^{2}-A^{2}\right)\text{atan}\left(Bx\right)+A^{2}BX}{B^{3}},\end{split}$
(30) $\begin{split}&\int_{0}^{x}df_{1}\
\frac{\text{atan}\left(Df_{1}\right)}{f_{1}}=\frac{1}{2}j\left[\text{Li}_{2}\left(-jDx\right)-\text{Li}_{2}\left(jDx\right)\right]\\\
&\approx\frac{\pi}{2}\text{asinh}\left(\frac{Dx}{2}\right),\end{split}$ (31)

where the approximation in (LABEL:eq:integral4) was originally proposed in
[13]. A comparable asymptotic expansion of (LABEL:eq:integral4) based on the
natural logarithm was proposed in [21, Appendix A].

## Acknowledgment

Financial support from UK EPSRC through programme grant TRANSNET
(EP/R035342/1) and a Doctoral Training Partnership (DTP) studentship to Daniel
Semrau is gratefully acknowledged.

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Daniel Semrau (S’16) received the B.Sc. degree in electrical engineering from
the Technical University of Berlin, Berlin, Germany, in 2013, the M.Sc. degree
in photonic networks engineering from Scuola Superiore Sant’Anna, Pisa, Italy,
and Aston University, Birmingham, U.K., in 2015. In 2015, he joined the
Optical Networks Group, University College London, U.K., where he is currently
working toward the Ph.D. degree. In 2018, Daniel was presented with the
Graduate Student Fellowship award of the IEEE Photonics Society. His research
interests are mainly focused on channel modeling, nonlinear compensation
techniques, and ultra-wideband transmission for long-haul coherent optical
communications.
---
Robert I. Killey (SM’17) received the B.Eng. degree in electronic and
communications engineering from the University of Bristol, Bristol, U.K., in
1992, the M.Sc. degree from University College London (UCL), London, U.K., in
1994, and the D.Phil. degree from the University of Oxford, Oxford, U.K., in
1998. He is currently a Reader in optical communications with the Optical
Networks Group with UCL. His research interests include nonlinear fiber
effects in WDM transmission, advanced modulation formats, and digital signal
processing for optical communications. He has participated in many European
projects, including ePhoton/ONe, Nobel, BONE and ASTRON, and national
projects. He is currently a Principal Investigator in the EPSRC funded UNLOC
project. He was with the technical program committees of many international
conferences including European Conference on Optical Communication, Optical
Fiber Communication Conference ACP, and OECC. He was an Associate Editor of
the IEEE/OSA Journal of Optical Communications and Networking and is currently
an Associate Editor of the Journal of Lightwave Technology.
---
Polina Bayvel (F’10) 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 with 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
coauthored 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. Payvel 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. He 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 Proceedings of European Conference on Optical
Communication (ECOC) and Co-Chair of the TPC for ECOC 2005.
---