Analytical design of dispersion-managed fiber system with map strength 1.65

doi:10.1016/S0375-9601(03)00085-9

Physics Letters A

Volume 308, Issues 5–6, 10 March 2003, Pages 417-425

https://doi.org/10.1016/S0375-9601(03)00085-9 Get rights and content

Abstract

We present an easy analytical method for designing dispersion-managed fiber systems with map strength of 1.65, where the transmission lines have minimal pulse–pulse interactions.

Introduction

Dispersion-managed (DM) fiber system is one of the promising technique that can be utilized for high-speed communications with many advantages [1]. The first step in the design of any DM fiber transmission line is to derive the pulse parameters of the fixed point (stationary solution) corresponding to the desired dispersion map. The Nijhof et al. [2] averaging method is the commonly used technique for finding the fixed point for any desired dispersion map. As Gaussian pulses represent the best approximation for the profile of DM solitons, the numerical solution of the variational equations derived using a Gaussian ansatz is also used for finding the parameters of the fixed point. Recently, we have reported a fully analytical method for designing the dispersion map for any desired input pulse width and energy [3]. That procedure is based on the exact solution of the variational equations derived using a Gaussian ansatz. Generally, in DM fiber systems, average dispersion and map strength parameter are the important parameters which help to define the stability of DM soliton propagation. Map strength of a DM fiber line is defined by a S parameter given by $S= L_{+} β_{+} −L_{−} β_{−} τ_{0}^{2},$ where τ₀ is the FWHM pulse width at chirp-free point, β₊ (β₋) and L₊ (L₋), respectively, represent the normal (anomalous) fiber dispersion parameter and length. Nakazawa et al. [4] proved that DM solitons propagating in average normal dispersion regime are unsuitable for high-speed long-distance communication. It has been [5], [6] shown that DM solitons have stable propagation when S≲4.79 for anomalous average dispersion, S≈4.79 for zero average dispersion and 4.79≲S≲9.75 for normal average dispersion and no stable DM solitons for S≳9.75. Yu et al. [7], have numerically found that DM solitons propagating in DM fiber lines with map strength S≈1.65 will have weaker interaction. Hence, we like to investigate the effectiveness of our analytical method [3] for designing DM transmission lines in the practically useful S parameter range (0<S<4). Also in this Letter we analyze the relation between the maximum pulse width (which is one of the input parameters required for our analytical design) and the S parameter. From that analysis, we present an easy way to analytically design the DM fiber system with map strength S≈1.65 where the transmission lines have minimal pulse–pulse interactions [7].

Section snippets

Effectiveness of our analytical design procedure

Pulse dynamics in DM fibers is governed by the nonlinear Schrödinger equation (NLSE) $ψ_{z} + iβ(z) 2 ψ_{tt} −iγ(z)|ψ|^{2} ψ=0,$ where ψ is the slowly varying envelope of the axial electrical field, β(z) and γ(z) represent the group-velocity dispersion and self-phase modulation parameters, respectively.

Recently, we have reported a fully analytical method for immediately designing the DM fiber systems, without any iterative or numerical procedure [3]. That analytical design is fundamentally based on the fact

Design of dispersion map with $S≈1.65$

From Fig. 1(d), we find that the x_3max is minimum for S≈1.65. To get more insight into the minimum of x_3max, we introduce a breathing parameter R given by $R= x_{3max}^{2} τ_{0}^{2} .$

Fig. 2 shows the variation of the breathing parameter R for different DM lines considered in Fig. 1. In Fig. 2 the solid and the dashed curves respectively represents the numerical and analytical results. Substituting τ₀ expressed from Eq. (1) in Eq. (12), and then from the resulting equation, x_3max can be written as a function of

Numerical simulations

We can also illustrate the effectiveness of our analytical design of dispersion map with S≈1.65, S<1.65 and S>1.65, by using the simulation of a bit pattern 〈011110101100100〉 propagation in a single-channel transmission line operating at 40 Gb/s, with a periodic dispersion management using two types of fiber, with dispersions: $±1 ps nm^{−1} km^{−1}$ (for S≈1.65), $±0.595 ps nm^{−1} km^{−1}$ (for S<1.65) and $±2.5 ps nm^{−1} km^{−1}$ (for S>1.65), nonlinear coefficient: $0.002 m^{−1} W^{−1}$ , fiber loss: 0.22 dB/km, coupling loss:

Conclusion

In this Letter, we have found some of the important points explaining why the DM solitons propagating in dispersion map with strength S=1.65 have minimal pulse–pulse interaction. We have also presented an easy and efficient way for analytically designing DM fiber lines with map strength around S≈1.65. Hence our analytical method is particularly useful for designing DM fiber lines with S≈1.65 without the help of any numerical procedure.

Acknowledgements

This work has been done under the contract URP/4.00 between the university of Burgundy and the Alcatel research corporation. The Ministère de l'Education Nationale de la Recherche et de la Technologie (contract ACI Jeunes No. 2015) is gratefully acknowledged for partially supporting this work. K. Nakkeeran acknowledges the support of the Research Grant Council of the Hong Kong Special Administrative Region, China (Project No. PolyU5132/99E). The authors wish to thank S. Wabnitz for fruitful

References (8)

M. Wald et al.
Opt. Commun.
(1999)
V.E. Zakharov et al.
Optical Solitons: Theoretical Challenges and Industrial Perspectives
(1998)
J.H.B. Nijhof et al.
Electron. Lett.
(1997)
K. Nakkeeran et al.
Opt. Lett.
(2001)

There are more references available in the full text version of this article.

Cited by (11)

A scheme for pre-shaping of dispersion-managed solitons
2007, Optics Communications
We demonstrate efficacy of shaping Gaussian pulses into dispersion-managed (DM) solitons by dint of a device which includes two fibers with the same group-velocity-dispersion (GVD) coefficients as in the DM system, a frequency-domain filter, and an amplifier. The shaping of a given input pulse is optimized by adjusting values of four free parameters of the cell, viz., lengths of the two fibers, filtering coefficient, and amplification gain, with the objective to achieve the best fit of the transformed pulse to the (numerically found) DM soliton, including systems with the third-order GVD, and with three-step DM maps. Launching reshaped pulses into the DM system, we demonstrate, in direct simulations, their complete stability over indefinitely long propagation distances. We examine sensitivity of the reshaping to deviations of the control parameters from their optimum values, a noteworthy results being very weak dependence on variations of the filtering strength. The dependence of the four control parameters on the width and chirp of the Gaussian input signal is also studied in detail.
Transmission of pulses in a dispersion-managed fiber link with extra nonlinear segments
2005, Optics Communications
Citation Excerpt :
Thus, there are strong reasons to look for RZ regimes in the DM links which use pulses that are different from solitons. This line of the theoretical and experimental studies has recently drawn renewed interest [7,8]. The objective of the present work is to put forward a promising scheme, in which non-soliton pulses are transmitted through a link composed of DM cells including an extra segment, with strong nonlinearity and negligible GVD.
We introduce an extended version of the dispersion-management (DM) model, which includes an extra nonlinear element, and consider transmission of return-to-zero pulses in this system (they are not solitons). The pulses feature self-compression, accompanied by generation of side peaks (in the temporal domain). An optimal transmission distance, z_opt, is identified, up to which the pulse continues to compress itself (the eventual width-compression factor is ≃2), while the amplitude of the side peaks remains small enough. The distance z_opt virtually does not depend on the strength S of the DM part of the system in the interval 1.5 < S < 11, but it is sensitive to the nonlinearity strength in the extra segment. The system provides essentially stronger suppression of the noise-induced jitter of the pulses than the ordinary DM model. The most important issue is interaction between adjacent pulses, which is a basic difficulty in the case of DM solitons. In a broad parameter region, the system provides effective isolation between pulses. The minimum initial temporal distance between them, necessary for the isolation, is quite small, slightly larger than 1.5 the pulse’s width. The transmission actually improves the quality of multi-pulse arrays, as it leads to deepening of hiatuses between originally overlapping pulses.
An iterative numerical method for dispersion-managed solitons
2005, Optics Communications
Citation Excerpt :
Interactions between DM-controlled RZ pulses inside one channel is a bigger problem, as large variations of the pulses’ width in the regime of strong DM contribute toward accumulation of interaction-induced perturbations. Nevertheless, the effects of the intra-channel interactions can be minimized in the regime of relatively weak or moderate DM [3,4]. A pulse transmitted through a DM fiber link, which is built as a periodic alternation of fiber segments with anomalous and normal group-velocity dispersion (GVD), undergoes compression in the former element and broadening in the latter one, as it was showed in detail both theoretically and experimentally (see, e.g. [5]).
We present a new numerical method for finding regimes of stable transmission of pulses in nonlinear dispersion-managed (DM) systems. Unlike the known averaging method, whose iterations require finding the pulses with the minimum and maximum width, our method merely starts with an initial guess for the pulse. Even if the initial guess is far from the correct shape, the method rapidly converges to a DM soliton after several iterations. We show that this method is especially relevant for the application to realistic models including filtering and amplification, where the soliton plays the role of an attractor. A new result obtained by means of the iterative method and reported in this work is a global stability region for the solitons in the DM system with lumped filters and amplifiers. We also demonstrate the utility of the method, applying it to a more sophisticated three-step DM system.
Integration of nonlinearity-management and dispersion-management for pulses in fiber-optic links
2004, Optics Communications
We introduce a model of a long-haul fiber-optic link that uses a combination of the nonlinearity- and dispersion-compensation (management) to stabilize nonsoliton pulses. The compensation of the accumulated fiber nonlinearity, and simultaneously pulse reshaping, which helps to suppress the inter-symbol interference (ISI, i.e., blurring of blank spaces between adjacent pulses), are performed by second-harmonic-generating modules, which are periodically inserted together with amplifiers. We demonstrate that the dispersion-management (DM), which was not included in an earlier considered model, drastically improves stability of the pulses. The stable-transmission length for an isolated pulse, which was less than 10 fiber spans with the use of the nonlinearity-management only, becomes indefinitely long. It is demonstrated too that the pulse is quite robust against fluctuations of its initial parameters, and the scheme operates efficiently in a very broad parameter range. The interaction between pulses can be safely suppressed for the transmission distance exceeding 16 spans (≃1000 km). The smallest temporal separation between adjacent pulses, which is necessary to prevent the ISI, attains a minimum in the case of moderate DM, similar to known results for the DM solitons. The mutually-induced distortion of co-propagating pulses being accounted for by the emission of radiation, a plausible way to further increase the stable-transmission limit is to introduce bandpass filters.
Analytical solutions for the variational equations derived for the nonlinear Schrödinger equation: Dispersion-managed fiber system
2008, Journal of Nonlinear Optical Physics and Materials
Energy diagram and stability range of families of soliton molecules in fibers
2019, Optical and Quantum Electronics

View all citing articles on Scopus

View full text

Analytical design of dispersion-managed fiber system with map strength 1.65

Abstract

Introduction

Section snippets

Effectiveness of our analytical design procedure

Design of dispersion map with S≈1.65

Numerical simulations

Conclusion

Acknowledgements

Opt. Commun.

Optical Solitons: Theoretical Challenges and Industrial Perspectives

Electron. Lett.

Opt. Lett.

Design of dispersion map with $S≈1.65$