Perturbation of Gaussian optical solitons in dispersion-managed fibers

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Abstract

The dispersion-managed optical solitons is studied in this paper in presence of various perturbation terms. The Gaussian solitons are considered and the dynamics of such pulses are obtained in presence of such perturbation terms. The integrals of motion are evaluated. The soliton perturbation theory is exploited to formulate the parameter dynamics of the Gaussian solitons.

Introduction

The propagation of solitons through optical fibers has been a major area of research given its potential applicability in all optical communication systems [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20]. The field of telecommunications has undergone a substantial evolution in the last couple of decades due to the impressive progress in the development of optical fibers, optical amplifiers as well as transmitters and receivers. In a modern optical communication system, the transmission link is composed of optical fibers and amplifiers that replace the electrical regenerators. But the amplifiers introduce some noise and signal distortion that limit the system capacity. Presently the optical systems that show the best characteristics in terms of simplicity, cost and robustness against the degrading effects of a link are those based on intensity modulation with direct detection (IM-DD). Conventional IM-DD systems are based on non-return-to-zero (NRZ) format, but for transmission at higher data rate the return-to-zero (RZ) format is preferred. When the data rate is quite high, soliton transmission can be used. It allows the exploitation of the fiber capacity much more, but the NRZ signals offer very high potential especially in terms of simplicity.

There are limitations, however, on the performance of optical system due to several effects that are present in optical fibers and amplifiers. Signal propagation through optical fibers can be affected by group velocity dispersion (GVD), polarization mode dispersion (PMD) and the nonlinear effects. The chromatic dispersion that is essentially the GVD when waveguide dispersion is negligible, is a linear effect that introduces pulse broadening generates intersymbol interference. The PMD arises due the fact that optical fibers for telecommunications have two polarization modes, in spite of the fact that they are called monomode fibers. These modes have two different group velocities that induce pulse broadening depending on the input signal state of polarization. The transmission impairment due to PMD looks similar to that of the GVD. However, PMD is a random process as compared to the GVD that is a deterministic process. So PMD cannot be controlled at the receiver. Newly installed optical fibers have quite low values of PMD that is about 0.1 ps/km.

The main nonlinear effects that arises in monomode fibers are the Brillouin scattering, Raman scattering and the Kerr effect. Brillouin is a backward scattering that arises from acoustic waves and can generate forward noise at the receiver. Raman scattering is a forward scattering from silica molecules. The Raman gain response is characterized by low gain and wide bandwidth namely about 5 THz. The Raman threshold in conventional fibers is of the order of 500 mW for copolarized pump and Stokes’ wave (that is about 1 W for random polarization), thus making Raman effect negligible for a single channel signal. However, it becomes important for multichannel wavelength-division-multiplexed (WDM) signal due to an extremely wide band of wide gain curve.

The Kerr effect of nonlinearity is due to the dependence of the fiber refractive index on the field intensity. This effect mainly manifests as a new frequency when an optical signal propagates through a fiber. In a single channel the Kerr effect induces a spectral broadening and the phase of the signal is modulated according to its power profile. This effect is called self-phase modulation (SPM). The SPM-induced chirp combines with the linear chirp generated by the chromatic dispersion. If the fiber dispersion coefficient is positive namely in the normal dispersion regime, linear and nonlinear chirps have the same sign while in the anomalous dispersion regime they are of opposite signs. In the former case, pulse broadening is enhanced by SPM while in the later case it is reduced. In the anomalous dispersion case the Kerr nonlinearity induces a chirp that can compensate the degradation induced by GVD. Such a compensation is total if soliton signals are used.

If multichannel WDM signals are considered, the Kerr effect can be more degrading since it induces nonlinear cross-talk among the channels that is known as the cross-phase modulation (XPM). In addition WDM generates new frequencies called the Four-Wave mixing (FWM). The other issue in the WDM system is the collision-induced timing jitter that is introduced due to the collision of solitons in different channels. The XPM causes further nonlinear chirp that interacts with the fiber GVD as in the case of SPM. The FWM is a parametric interaction among waves satisfying a particular relationship called phase-matching that lead to power transfer among different channels.

To limit the FWM effect in a WDM it is preferable to operate with a local high GVD that is periodically compensated by devices having an opposite sign of GVD. One such device is a simple optical fiber with opportune GVD and the method is commonly known as the dispersion-management. With this approach the accumulated GVD can be very low and at the same time FWM effect is strongly limited. Through dispersion-management it is possible to achieve highest capacity for both RZ as well as NRZ signals. In that case the overall link dispersion has to be kept very close to zero, while a small amount of chromatic anomalous dispersion is useful for the efficient propagation of a soliton signal. It has been demonstrated that with soliton signals, the dispersion-management is very useful since it reduces collision-induced timing jitter [3] and also the pulse interactions. It thus permits the achievement of higher capacities as compared to the link having constant chromatic dispersion.

Section snippets

Governing equations

The relevant equation, for studying the propagation of solitons through polarization-preserving optical fibers, is the nonlinear Schrödinger’s equation (NLSE) with damping and periodic amplification [1], [3]. In the dimensionless form, the NLSE is given byiuz+D(z)2utt+|u|2u=-iΓu+i[eΓza-1]n=1Nδ(z-nza)u.Here, Γ is the normalized loss coefficient, za is the normalized characteristic amplifier spacing and z and t represent the normalized propagation distance and the normalized time, respectively,

Polarization preserving fibers

In a polarization preserved optical fiber, the propagation of solitons is governed by DM-NLSE given by (12). If D(z)=g(z)=1 in (12), the NLSE is recovered. It is possible to integrate NLSE by the method of Inverse Scattering Transform (IST) [3]. The IST is the nonlinear analog of Fourier transform that is used for solving linear partial differential equations. Moreover, the NLSE has an infinite number of conserved quantities. However, (12), as it appears, is no longer integrable and thus it

Soliton perturbation

In presence of perturbation terms, the perturbed DM-NLSE is given byiqz+D(z)2qtt+g(z)|q|2u=iϵR[q,q],where R represents the spatio-differential operator, although sometimes it could, very well, represent an integral operator. Also, the perturbation parameter ϵ is the relative width of the spectrum and 0<ϵ1 by virtue of quasi-monochromaticity [3]. In presence of these perturbation terms, the adiabatic variation of soliton parameters are given bydEdz=ϵ(qR+qR)dt.From (20), (21), (22), one

Conclusions

This paper talks about the adiabatic parameter dynamics of dispersion-managed Gaussian optical solitons in presence of perturbation terms that are both local as well as non-local. In the local ones, both Hamiltonian as well as non-Hamiltonian type perturbations are taken into account.

These results are going to be used for further study of dispersion-managed solitons. One immediate application of this is in the study of intra-channel collision of solitons by virtue of quasi-particle theory.

Acknowledgements

The research work of the first (RK) and second author (AB) were fully supported by NSF Grant No: HRD-0630388 and the support is genuinely and sincerely appreciated.

The research work of the fourth author (EZ) is financially supported by the Army Research Office (ARO) under the award number W911NF-05-1-0451 and this support is thankfully acknowledged.

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