The instabilities of a nonlinear optical medium with induced absorption in a ring cavity are theoretically analyzed. Stable, bistable, and unstable stationary solutions are found for various values of incident intensity and detuning. The dynamic treatment yields, in the unstable regime, oscillations of the transmitted intensity in various modes for constant incident intensity. If the medium relaxation time is short compared with the round-trip time, oscillatory solutions are obtained, which show strong locking of the oscillation frequency to multiples of the inverse round-trip time. If the round-trip time is comparable with the medium relaxation time, numerical integrations yield periodic solutions with substructures because of the mismatch of the two characteristic times.
Abstract We study the existence and stability of dark-gap solitons in linear lattice and nonlinear lattices. The results indicate that the combination of linear and nonlinear lattices gives dark-gap solitons unique properties. The linear lattice can stabilize dark-gap solitons, while the nonlinear lattice reduces the stability of dark-gap solitons. On the basis of numerical analysis, we investigate the effects of lattice depth, chemical potential, nonlinear lattice amplitude, and nonlinear lattice period on the soliton in mixed lattices with the same and different periods. The stability of dark-gap soliton is studied carefully by means of real-time evolution and linear stability analysis. Dark-gap solitons can exist stably in the band gap, but the solitons formed by the mixed lattices are slightly different when the period is the same or different.
We investigate stability of gap solitons (GSs) in the first two band gaps in the framework of the one-dimensional Gross-Pitaevskii equation, combining the repulsive nonlinearity and a moderately strong optical lattice (OL), which is subjected to ``management,'' in the form of time-periodic modulation of its depth. The analysis is performed for parameters relevant to the experiment, characteristic values of the modulation frequency being $\ensuremath{\omega}\ensuremath{\sim}2\ensuremath{\pi}\ifmmode\times\else\texttimes\fi{}20\phantom{\rule{0.3em}{0ex}}\mathrm{Hz}$. First, we present several GS species in the two band gaps in the absence of the management. These include fundamental solitons and their bound states, as well as a subfundamental soliton in the second gap, featuring two peaks of opposite signs in a single well of the periodic potential. This soliton is always unstable, and quickly transforms into a fundamental GS, losing a considerable part of its norm. In the first band gap, (stable) bound states of two fundamental GSs are possible solely with opposite signs, if they are separated by an empty site. Under the periodic modulation of the OL depth, we identify stability regions for various GS species, in terms of $\ensuremath{\omega}$ and modulation amplitude, at fixed values of the soliton's norm,
N. In either band gap, the GS species with smallest
N has a largest stability area; in the first and second gaps, they are, respectively, the fundamental GS proper, or the one spontaneously generated from the subfundamental soliton. However, with the increase of
N, the stability region of every species expands in the first gap, and shrinks in the second one. The outcome of the instability development is also different in the two band gaps: it is destruction of the GS in the first gap, and generation of extra side lobes by unstable GSs in the second one.
Multiple-scale analysis is employed for the study of nonlinear wave propagation in periodic layered media. In a first step, wave propagation in each individual layer is modeled by a corresponding equivalent nonlinear transmission line. The multiple-scale analysis is then employed to establish a system of nonlinear equations for the amplitudes of the forward and backward waves in the transmission lines of the model mentioned. This system of nonlinear equations is solved with the aid of a continuation technique for derivation of transmission characteristics of the periodic structure. To study optical bistability, a parameter-switching algorithm is utilized for obtaining the solutions of the system including both stable and unstable ones. For the sake of verification, we have also utilized a nonlinear finite-difference time-domain (NFDTD) method to analyze the wave propagation in the aforementioned structure. While there is an acceptable agreement between the bistability results obtained using our method and the NFDTD, there is a considerable difference between our results and the ones derived using the conventional coupled-mode theory.
In this paper, the numerical studies on optical-bistability phenomena in third-, fifth-, and seventh-order nonlinear periodic media are presented. By cross-phase modulation with strong Gaussian pump, the continuous-wave probe transmission that operates in bistability regime can be permanently switched from low to high and vice versa using the property of the unstable state. While there is only one hysteresis in the third-order nonlinear periodic media, double hysteresis are observed in the nonlinear-transmission characteristics of pi phase-shifted periodic media with third-fifth and third-fifth-seventh-order nonlinearities. In the context of double hysteresis, unique-bistability regime is defined where the high state can be reached as input comes from low to high with fast enough rise time such that the output rises above the unstable state during its transient. Finally, all-optical transistor operation is shown as a prominent application of optical bistability based on the unstable-state principle