Revisiting the nonlinear Gaussian noise model for hybrid fiber spans

doi:10.23919/ICN.2020.0018

2021, Vol. 2

Issue (1): 30-49 doi: 10.23919/ICN.2020.0018

Optical communication

Revisiting the nonlinear Gaussian noise model for hybrid fiber spans

Ioannis Roudas^*(

),Jaroslaw Kwapisz(

),Xin Jiang(

)

Department of Electrical and Computer Engineering, Montana State University, Bozeman, MT 59717, USA
Department of Mathematical Sciences, Montana State University, Bozeman, MT 59717, USA
Department of Engineering and Environmental Science, College of Staten Island, City University of New York, Staten Island, NY 10314, USA

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Abstract

We rederive from first principles and generalize the theoretical framework of the nonlinear Gaussian noise model to the case of coherent optical systems with multiple fiber types per span and ideal Nyquist spectra. We focus on the accurate numerical evaluation of the integral for the nonlinear noise variance for hybrid fiber spans. This task consists in addressing four computational aspects: (1) Adopting a novel transformation of variables (other than using hyperbolic coordinates) that changes the integrand to a more appropriate form for numerical quadrature; (2) Evaluating analytically the integral at its lower limit, where the integrand presents a singularity; (3) Dividing the interval of integration into subintervals of size $π$ and approximating the integral over each subinterval by using various algorithms; and (4) Deriving an upper bound for the relative error when the interval of integration is truncated in order to accelerate computation. We apply the proposed analytical model to the performance evaluation of coherent optical communications systems with hybrid fiber spans composed of quasi-single-mode and single-mode fiber segments. More specifically, the model is used to optimize the lengths of the optical fiber segments that compose each span in order to maximize the system performance. We check the validity of the optimal fiber segment lengths per span provided by the analytical model by using Monte Carlo simulation, where the Manakov equation is solved numerically using the split-step Fourier method. We show that the analytical model predicts the lengths of the optical fiber segments per span with satisfactory accuracy so that the system performance, in terms of the Q-factor, is within 0.1 dB from the maximum given by Monte Carlo simulation.

Key words： nonlinear Gaussian Noise (GN) model perturbation theory hybrid fiber spans

Received: 23 October 2020 Online: 19 August 2021

Corresponding Authors: Ioannis Roudas E-mail: ioannis.roudas@montana.edu;jarek@math.montana.edu;jessica.jiang@csi.cuny.edu

About author: Ioannis Roudas received the BS degree in physics and the MS degree in electronics and radio-engineering from the University of Athens, Greece in 1988 and 1990, respectively, and the MS and PhD degrees in coherent optical communication systems from the Ecole Nationale Supérieure des Télécommunications (currently Télécom ParisTech), Paris, France in 1991 and 1995, respectively. During 1995-1998, he worked in the Optical Networking Research Department, Bell Communications Research (Bellcore), Red Bank, NJ. At the same time, he taught for two semesters, as an adjunct professor, at Columbia University. He was subsequently with the Photonic Modeling and Process Engineering Department, Corning Inc., Somerset, NJ, from 1999 to 2002. He spent an eight-year period in Greece, during 2003-2011, working at the Department of Electrical and Computer Engineering, University of Patras as an associate professor of optical communications. In addition, he taught, as an adjunct professor, at the City University of New York and the Hellenic Open University. During 2011-2016, he was a research associate with the Science and Technology Division of Corning Inc., Corning, NY. Since July 2016, he has been with the Department of Electrical and Computer Engineering, Montana State University as the Gilhousen Telecommunications chair professor. He is the author or co-author of more than 100 papers in scientific journals and international conferences and holds five patents. He served as an associate editor for the IEEE Photonics Journal during 2013-2019. His current research focuses on optical communications systems with multimode and multicore fibers and on quantum networking.|Jaroslaw Kwapisz is a Polish-American mathematician with background in theoretical dynamical systems. He received the MS (1991) degree from University of Warsaw and the PhD degree (1995) from State University of New York at Stony Brook. Since July 2008, he has been a professor of mathematics at the Department of Mathematical Sciences, Montana State University. He has worked on problems in several subject areas, including integral and differential equations, iterated maps modeling coupled non-linear oscillators, pattern formation in fourth-order Hamiltonian systems, ergodic theory and entropy in smooth and symbolic dynamics, cohomological Conley index and cocyclic subshifts, almost-periodic tiling spaces and quasi-crystals, abelian-Nielsen classes and geometry of translation surfaces, and conformal dimension of fractal sets. He is currently interested in Anosov maps on infra-nil manifolds, non-Meyer substitution Delone sets, and problems in classical and quantum multi-mode fiber-optic communication.|Xin Jiang received the BS, MS, and PhD degrees in electronic engineering from Tsinghua University, Beijing, China, the former one in 1990, and the later two in 1995. She is currently an associate professor at the Department of Engineering and Environmental Science, College of Staten Island (CSI), City University of New York, NY, USA. Prior joining CSI, she has worked in R&D and engineering departments of several high-tech and telecommunication companies. She has authored and co-authored over 60 publications in peer-reviewed journals and international conferences. Her current areas of research include advanced optical fiber transmission technology and photonic systems and networks.


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Cite this article:

Ioannis Roudas,Jaroslaw Kwapisz,Xin Jiang. Revisiting the nonlinear Gaussian noise model for hybrid fiber spans. , 2021, 2: 30-49.

URL:

http://icn.tsinghuajournals.com/10.23919/ICN.2020.0018 OR http://icn.tsinghuajournals.com/Y2021/V2/I1/30

Fig. 1 Long-haul coherent optical communications system with hybrid fiber spans.

Main result (double integral form)
Nonlinear noise coefficient
$γ ~ = 6427 ? N s ? Δ ? ν res R s 3 ? ∫ 0 B 0 2 ∫ 0 B 0 2 ξ ? (f 1, f 2) ? d ? f 1 ? d ? f 2$			(45)
Integrand
$ξ ? (f 1, f 2) ? : = ? η ? (f 1, f 2) ? ϕ ? (f 1, f 2)$			(40)
Auxiliary function
FWM efficiency per span		Phased-array factor
$η ? (f 1, f 2) ? : = ? \| ∑ k = 1 N f γ^k ? (f 1, f 2) ? L^eff k ? (f 1, f 2) \| 2$	(42)	$ϕ ? (f 1, f 2) ? : = ? 1 N s 2 ? sin 2 [N s Δ β (f 1, f 2) ℓ s / 2)] sin 2 ? [Δ ? β ? (f 1, f 2) ? ℓ s / 2]$	(41)
Complex nonlinear fiber coefficient		Average phase mismatch
$γ^k ? (f 1, f 2) ? : = ? γ k ? e - ∑ m = 1 k - 1 α m ? (f 1, f 2) ? ℓ s m$	(28)	$Δ ? β ? (f 1, f 2) ? : = ? ℓ s - 1 ? ∑ k = 1 N f Δ ? β k ? (f 1, f 2) ? ℓ s k$	(32)
Complex effective length
$L^eff k ? (f 1, f 2) ? : = ? 1 - e - α k ? (f 1, f 2) ? ℓ s k α k ? (f 1, f 2)$	(29)

Table 1 Compendium of mathematical formulas summarizing the nonlinear Gaussian noise model for coherent optical communications systems with hybrid fiber spans. The numbers refer to the corresponding equations in the main text.

𝐥𝐧(?𝜻𝟎/𝜻?) (in green),

ϕ ? (? 𝜻 ?)

(in blue), the normalized

𝜼 ? (? 𝜻 ?) ← 𝜼 ? (? 𝜻 ?) / 𝜼 ? (0)

(in brown), and their product

g (? 𝜻 ?) = 𝐥𝐧 (? 𝜻 𝟎 / 𝜻 ?) ? ϕ ? (? 𝜻 ?) ? 𝜼 ? (? 𝜻 ?) / 𝜼 ? (0)

(in red). Conditions (for illustration purposes only):

N s = 𝟒

𝝂 = 1,

𝜻 𝟎 = 10 π,

one fiber type per span.
">

Fig. 2 Sketches of $𝐥𝐧 (? 𝜻 𝟎 / 𝜻 ?)$ (in green), $ϕ ? (? 𝜻 ?)$ (in blue), the normalized $𝜼 ? (? 𝜻 ?) ← 𝜼 ? (? 𝜻 ?) / 𝜼 ? (0)$ (in brown), and their product $g (? 𝜻 ?) = 𝐥𝐧 (? 𝜻 𝟎 / 𝜻 ?) ? ϕ ? (? 𝜻 ?) ? 𝜼 ? (? 𝜻 ?) / 𝜼 ? (0)$ (in red). Conditions (for illustration purposes only): $N s = 𝟒$ , $𝝂 = 1,$ $𝜻 𝟎 = 10 π,$ one fiber type per span.

𝟐; SMF effective mode area: 112 μm

𝟐

; No MPI compensation; Lines: Fitting using Eq. (1)).
">

Fig. 3 Q-factor as a function of the total launch power per channel for different QSMF lengths per span (Conditions: System length: 6000 km, 100 km spans; QSMF effective mode area: 250 μm $𝟐$ ; SMF effective mode area: 112 μm $𝟐$ ; No MPI compensation; Lines: Fitting using Eq. (1)).

1)).
">

Fig. 4 Q-factor as a function of the total launch power per channel. (a) No MPI compensation and (b) 100% MPI compensation (Points: Monte Carlo simulations; Lines: Fitting using Eq. (1)).

Fig. 5 Q-factor as a function of the total launch power per channel for the case of 45/55 QSMF/SMF mix (Condition: No MPI compensation).

Q𝟎 vs. QSMF length

ℓ s 𝟏

per span (Condition: No MPI compensation).
">

Fig. 6 Peak Q-factor $Q 𝟎$ vs. QSMF length $ℓ s 𝟏$ per span (Condition: No MPI compensation).

ℓs𝟏/ℓs as a function of the percentage of MPI compensation at the coherent optical receiver.
">

Fig. 7 Variation of the optimal normalized QSMF length per span $ℓ s 𝟏 / ℓ s$ as a function of the percentage of MPI compensation at the coherent optical receiver.

$β 2$	GVD parameter
$β ~ ? P$	Multipath crosstalk variance
$γ$	Nonlinear coefficient
$γ ˉ$	Averaged nonlinear coefficient $γ ˉ = 8 ? γ / 9$
$γ^$	Effective nonlinear coefficient
$γ ~ ? P 3$	Nonlinear noise variance
$Γ$	Worst-case (real) effective nonlinear coefficient
$δ$	Small number in the vicinity of zero
$Δ$	Step size of Simpson’s quadrature
$Δ ? β i ? j ? k ? (z)$	Phase mismatch
$Δ ? β$	Average propagation constant mismatch
$Δ ? ν$	Frequency spacing of WDM channels
$Δ ? ν res$	Resolution bandwidth
$ε$	Perturbation parameter
$ϵ r$	Relative error
$η ? (f 1, f 2)$ , $η ? (ζ)$	Four-wave mixing efficiency
$∂ x$	Partial derivative $∂ / ∂ ? x$
$λ$	Carrier wavelength of central WDM channel
$λ k$	WDM channel wavelength
$λ ˉ k$	Auxiliary multiplicative coefficient
$ℓ s$	Span length
$ν k$	Normalized electric field attenuation coefficient
$σ k$	Normalized, chromatic dispersion-adjusted, and real attenuation coefficient for the $k$ -th fiber segment
$ϕ ? (f 1, f 2)$ , $ϕ ? (ζ)$	Normalized phased-array term
$Ψ m$	Set of index triplets for the ODE for the $m$ -th order perturbation
$ξ ? (f 1, f 2)$	Nonlinear noise coefficient integrand
$ζ k ? (ζ)$	Normalized electric field phase shift
$Ω n$	Set of index triplets for FWM combinations
$ω n$	Angular frequencies, $ω n = 2 ? π ? f n$

$a$	Attenuation coefficient
$A eff$	Mode effective area
$a ˉ i ? j ? k ? (z)$	Complex attenuation coefficient
$a ˉ n$	Complex attenuation coefficient
$a ~$	ASE noise variance
$B 0$	Optical bandwidth of the WDM signal
$𝒄 n ? 0$	Complex envelope of the unperturbed Fourier coefficient of the $n$ -th spectral component at the fiber input
$𝒄 n ? 1$	Complex amplitude of the nonlinear noise
$D$	Chromatic dispersion parameter
$D x$	Regular derivative $d/d ? x$
$F A$	Amplifier noise figure
$f ϕ$	Average phased-array bandwidth
$f ϕ k$	Phased-array bandwidth for the $k$ -th fiber segment
$f 0$	Pseudorandom signal fundamental frequency
$G$	Amplifier gain
$G NLI ? (f)$	Nonlinear noise psd
$I$	Various definite integral
$J ? (μ, ζ 0)$	Auxiliary integral, $J ? (μ, ζ 0) ? : =$ $∫ μ ζ 0 N s ϕ (ζ) ⋅$ $ln ? (ζ 0 ζ) ? η ? (ζ) ? d ? ζ$
$K N s ? (δ)$	Auxiliary integral, $K N s ? (δ) ? : =$ $∫ 0 δ ln (δ / ζ) ⋅$ $N s ? ϕ ? (ζ) ? d ? ζ$
$L$	Link length
$L^eff$	Normalized (i.e., dimensionless) complex effective length
$N int$	Number of periods of $ϕ ? (ζ)$ in the interval $[0, ζ 0]$
$n 2$	Nonlinear index coefficient
$N ch$	Number of wavelength channels
$N n$	Number of integration nodes in a $π$ subinterval
$N s$	Number of spans
$N f$	Number of fiber segments per span
$R$	Region of integration in the hyperbolic $u ? v$ -plane
$T 0$	Pseudorandom signal period
$X i ? j ? k, X i ? j$	Complex FWM efficiency
$x k ? (ζ)$	Normalized power complex attenuation coefficients
$𝒖 n ? (z)$	Fourier coefficients
$𝒖 n ? k ? (z) ? ε k$	$k$ -th order correction to the unperturbed solution $𝒖 n ? 0 ? (z)$
$𝒚 ? (z, t)$	Complex envelope WDM PDM signal


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