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Journal of Modern Optics

Optical soliton propagation in erbium-doped fibre with variable dispersion and nonlinear effects

A. Uthayakumar, Arumugam Mahalingam, Young‐Geun Han, Sang Bae Lee

Published Apr 28, 2006• 16 citations
Publisher

Abstract

We propose a coupled system of generalized nonlinear Schrödinger and Maxwell–Bloch equations with variable coefficients which describes propagation of an optical soliton in an inhomogeneous nonlinear waveguide doped with two-level resonant atoms. The Lax pair of this model is derived using the Ablowitz–Kaup–Newell–Segur (AKNS) formalism. The one-soliton solution is constructed through a Darboux–Bäcklund transformation (BT). The coupled system is found to allow soliton propagation. We explain the soliton control by the effect of distributed amplification with varying group velocity dispersion, nonlinearity and gain or loss parameters.

DOI
10.1080/09500340600581546

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Physical Review A•Nov 1, 1991•60 citations
We propose a coupled system of generalized nonlinear Schrödinger and Maxwell–Bloch equations with variable coefficients which describes propagation of an optical soliton in an inhomogeneous nonlinear waveguide doped with two-level resonant atoms. The Lax pair of this model is derived using the Ablowitz–Kaup–Newell–Segur (AKNS) formalism. The one-soliton solution is constructed through a Darboux–Bäcklund transformation (BT). The coupled system is found to allow soliton propagation. We explain the soliton control by the effect of distributed amplification with varying group velocity dispersion, nonlinearity and gain or loss parameters.
We propose the system of generalized inhomogeneous coupled Hirota–Maxwell–Bloch equations which describes propagation of an optical soliton in an inhomogeneous erbium-doped fiber with two-level resonant atoms. For this system, higher-order dispersion, self-steepening and self-Raman scattering are assumed to be inhomogeneous, like the group velocity dispersion. The exact analytical multisoliton solutions are obtained by employing the simple, straightforward Darboux transformation based on the obtained 3 × 3 Lax pair. As an example, the one- and two-soliton solutions in explicit forms are generated. Then, we give the relative numerical evolution and discuss the propagation characteristics of the pulse.
The simultaneous propagation of two optical pulses through a doped nonlinear dispersive medium modelled by a resonant three-level system was investigated numerically, within the framework of a pair of coupled extended nonlinear Schrödinger equations. These included the contribution of the dopant resonances whose dynamics is governed by Bloch equations. In this work, we review the interesting possibilities on the manipulation of fields such as cloning, breakup and soliton interactions, that the combination of coherent population trapping with nonlinear dispersive media offers.
We theoretically show that a self-induced transparency (SIT) soliton and a Bragg soliton can coexist in a nonlinear photonic band gap (PBG) medium doped uniformly with inhomogeneous-broadening two-level atoms. The Maxwell-Bloch equations for the pulse propagating through such a uniformly doped PBG structure are derived first and further reduced to an effective nonlinear Schrödinger equation. This model describes an equivalent physical mechanism for a Bragg-soliton propagation resulting from the effective quadratic dispersion balancing with the effective third-order nonlinearity. Because the resonant atoms are taken into account, the original band gap can be shifted both by the dopants and the instantaneous nonlinearity response originating from an intense optical pulse. As a result, even if a SIT soliton with its central frequency deep inside the original forbidden band, it still can propagate through the resonant PBG medium as long as this SIT soliton satisfies the effective Bragg-soliton propagation. An approximate soliton solution describing such coexistence is found. We also show that the pulse width and group velocity of this soliton solution can be uniquely determined for given material parameters, atomic transition frequency, and input central frequency of the soliton. The numerical examples of the SIT soliton in a one-dimensional As2S3-based PBG structure doped uniformly with Lorentzian line-shape resonant atoms are shown. It is found that a SIT soliton with approximately 100-ps width in such a resonant PBG structure can travel with the velocity being two orders of magnitude slower than the light speed in an unprocessed host medium.
Fiber guided optical signal propagating in a Erbium doped nonlinear resonant medium is known to produce cleaner solitonic pulse, described by the self induced transparency (SIT) coupled to nonlinear Schrödinger equation. We discover two new possibilities hidden in its integrable structure, for amplification and control of the optical pulse. Using the variable soliton width permitted by the integrability of this model, the broadening pulse can be regulated by adjusting the initial population inversion of the dopant atoms. The effect can be enhanced by another innovative application of its constrained integrable hierarchy, proposing a system of multiple SIT media. These theoretical predictions are workable analytically in details, correcting a well known result.
Fiber guided optical signal propagating in a Erbium doped nonlinear resonant medium is known to produce cleaner solitonic pulse, described by the self induced transparency (SIT) coupled to nonlinear Schroedinger equation. We discover two new possibilities hidden in its integrable structure, for amplification and control of the optical pulse. Using the variable soliton width permitted by the integrability of this model, the broadening pulse can be regulated by adjusting the initial population inversion of the opant atoms. The effect can be enhanced by another innovative application of its constrained integrable hierarchy, proposing a system of multiple SIT media. These theoretical predictions are workable analytically in details, correcting the limitation of a well known result.
The possibility of observing a self-induced transparency (SIT) in an erbium-doped optical fiber is investigated. A general equation that includes the SIT, the group-velocity dispersion (GVD) of the fiber, and the nonlinear refractive index is discussed. It is shown that for one particular value of waveguide parameters the nonlinear index can balance the GVD, that is, the nonlinear Schr\"odinger equation (NLS), while maintaining a SIT soliton. This is called a SIT-NLS soliton. The phase change of the SIT-NLS soliton is governed solely by the NLS component, and the pulse delay due to resonance is determined solely by the SIT component when the detuning is zero. Practical parameters for silica-based fibers do not allow the coexistence of a mixed soliton state. A simple derivation of a condition for a SIT-NLS soliton is also presented. The SIT-NLS soliton is computer-run, and it is shown that a stable 2\ensuremath{\pi}/N=1 SIT-NLS soliton exists. However, high-order SIT-NLS solitons always split into multiple 2\ensuremath{\pi}/N=1 solitons. The NLS property can be preserved when Zsp{\mathit{Z}}_{\mathrm{sp}}Zsp​\ensuremath{\ll}Labs.{\mathit{L}}_{\mathrm{abs}.}Labs.​ Finally, the NLS soliton that interacts coherently with erbium ions is studied, and the coherent pulsation that produces multiple narrow pulses is described.