Abstract

In recent years, there has been vast interest in nonlinear pulse propagation in photonic band gap structures, or photonic crystals [1]. It has been shown [2,3], that due to nonlinear light-matter interaction, intensive laser pulse can propagate at frequency within the linear forbidden Bragg gap band through the structure with different types of nonlinearity, so called gap soliton. The steady gap soliton moves in periodical structure like optical soliton in homogeneous medium keeping its shape and constant velocity. However, the existence of photonic band gap gives rise to specific features of the gap soliton dynamics, for instance, the pulse can stand with zero velocity [2,3] or oscillate periodically changing its amplitude and sign of velocity [4,5]. These oscillations have been investigated in the framework of the generalized massive Thirring model for gap solitons in periodic cubic materials [5]. However, the oscillations of gap 2π pulse of self-induced transparency in a resonant periodic structure were only numerically demonstrated in the case of complicated set of equations in complex functions [4], and the physical nature of the oscillations has not been described. Physically, it is clear that a reason of the oscillations is in photonic band gap. If an optical soliton is formed by arbitrary pulse in homogeneous medium, then the part of energy, which has not been trapped by the soliton, leaves fast the space region of the slow soliton as free linear radiation. In the case of gap soliton, this untrapped energy is fixed in excited atoms and in weak field that can not propagate through the structure because of linear photonic band gap. As a result, if the initial soliton velocity is slow enough, the gap soliton, interacting with the perturbation, cannot leave the region of interaction because its kinetic energy is smaller than the potential energy of the interaction. This gives rise to gap soliton oscillations. In the present paper, we study the instability of gap 2π-pulse of self-induced transparency in a resonantly absorbing Bragg grating. It is shown that initial problem for simple two-wave Maxwell-Bloch equations in real functions are reduced to modified sine-Gordon equation. This allows one to obtain an equation of motion to describe the evolution of stable oscillating gap 2π-pulse and unstable excited gap 2π-pulse, which decays to a steady soliton and perturbation. The oscillating pulse is physically stable because it does not decay, and is unsteady because this solution is within a region of oscillatory instability on phase-plane of equation of motion. Solving a boundary problem, we explain the physical nature of delayed reflection and delayed transmission of gap 2π-pulse, when an incident pulse forms the gap 2π-pulse at low velocity near the boundary.

© 2002 Optical Society of America