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Physical Review A

Gap 2π pulse with an inhomogeneously broadened line and an oscillating solitary wave

Boris I. Mantsyzov

Published Jun 1, 1995• 63 citations
Publisher

Abstract

The gap 2\ensuremath{\pi} pulse or gap solitary wave of self-induced transparency at an inhomogeneously broadened line of a resonant Bragg periodical structure has been found theoretically as an exact solution of the coupled-mode Maxwell-Bloch equations. In the case of a slight violation of the Bragg condition at exact two-level resonance, the analytical solution for the gap 2\ensuremath{\pi} pulse with phase modulation has been obtained. Furthermore, the existence of an oscillating pulse that periodically changes in amplitude and in the sign of its velocity has been numerically shown.

DOI
10.1103/physreva.51.4939

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Self-induced transparency solitary waves in a doped nonlinear photonic band gap material

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Journal of Experimental and Theoretical Physics Letters•Nov 1, 2001•9 citations
The gap 2\ensuremath{\pi} pulse or gap solitary wave of self-induced transparency at an inhomogeneously broadened line of a resonant Bragg periodical structure has been found theoretically as an exact solution of the coupled-mode Maxwell-Bloch equations. In the case of a slight violation of the Bragg condition at exact two-level resonance, the analytical solution for the gap 2\ensuremath{\pi} pulse with phase modulation has been obtained. Furthermore, the existence of an oscillating pulse that periodically changes in amplitude and in the sign of its velocity has been numerically shown.
Summary form only given. In recent years, there has been considerable interest in nonlinear and quantum optics of photonic bandgap structures. Some of this attention has centered on the properties of gap solitary waves, which are light pulses tuned at a Bragg frequency within a forbidden bandgap of the periodic structure. These solitary waves may occur in a periodically modulated medium with a Kerr-like nonlinearity, and in the resonance grating as a two-wave soliton of self-induced transparency. In the later case, the coupled-mode Maxwell-Bloch equations which describe the on-resonance coherent interaction of the intense field with a periodic structure of thin layers of two-level atoms in the ground state, lead to an integrable sine-Gordon equation. In our study, we concentrate on the nonintegrable case, that is, amplification in a resonant Bragg structure by means of a comprehensive analysis that includes and numerical modelling of Dicke superradiance and coherent amplification.
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.
We derive the properties of self-induced transparency (SIT) solitary waves in a one-dimensional periodic structure doped uniformly with resonance two-level atoms. In our model, the electromagnetic field is treated classically and the dopant atoms are described quantum mechanically. The resulting solitary waves take the form of ultrashort (picosecond) laser pulses which propagate near the band edge of the nonlinear photonic band gap (PBG) material doped with rare-earth atoms such as erbium. Solitary wave formation involves the combined effects of group velocity dispersion (GVD), nonresonant Kerr nonlinearity, and resonant interaction with dopant atoms. We derive the general Maxwell-Bloch equations for a nonlinear PBG system and then demonstrate the existence of elementary solitary wave solutions for frequencies far outside the gap where GVD effects are negligible and for frequencies near the photonic band edge where GVD effects are crucial. We find two distinct new types of propagating SIT solitary wave pulses. Far from Bragg resonance, we recapture the usual McCall-Hahn soliton with hyperbolic secant profile when the nonlinear Kerr coefficient ${\ensuremath{\chi}}^{(3)}=0.$ However, when the host nonresonant Kerr coefficient is nonzero, we obtain the first new type of soliton. In this case, the optical soliton envelope function deviates from the hyperbolic secant profile and pulse propagation requires nontrivial phase modulation (chirping). We derive the dependence of the solitary wave structure on the Kerr coefficient ${\ensuremath{\chi}}^{(3)},$ the resonance impurity atom density, and the detuning of the average laser frequency from the atomic transition. When the laser frequency and the atomic transition frequencies are near the photonic band edge we obtain the second type of soliton. To illustrate the second type of soliton we consider two special cases. In the first case, GVD facilitates the propagation of an unchirped SIT-gap soliton moving at a velocity fixed by the material's parameters. The soliton structure changes dramatically as the laser frequency is tuned through the atomic resonance. In the second illustrative case we set the Kerr coefficient ${\ensuremath{\chi}}^{(3)}=0.$ In this case, the solution is a chirped pulse which arises from the balance between GVD and the resonance interaction with the dopant atoms. Finally, we show that under certain circumstances, these solitary wave solutions may persist even in the presence of (subpicosecond) dipolar dephasing of the dopant atoms and absorption losses of the host PBG material, provided that the system is incoherently pumped. These results may be relevant to the application of PBG materials as optical devices in fiber-optic networks.
We consider an optical pulse propagation in a one-dimensional nonlinear periodic structure doped uniformly with inhomogeneously broadening two-level atoms wherein the pulse propagation is governed by the nonlinear coupled mode-Maxwell Bloch equations. We investigate the pulse-train like periodic waves as well as bright and dark solitons near the photonic bandgap structure under the influence of self-induced transparency (SIT) effect. Further, we demonstrate that the resulting bright and dark SIT Bragg solitons may be realized both in anomalous and normal dispersion regimes, and this novel concept is ultimately the crux of this paper.