Coupled-mode theory for spatial gap solitons in optically induced lattices

We derive two systems of coupled-mode equations for spatial gap solitons in one-dimensional (1D) and quasi-one-dimensional (Q1D) photonic lattices induced by two interfering optical beams in a nonlinear photorefractive crystal. The models differ from the ordinary coupled-mode system (e.g., for the fiber Bragg grating) by saturable nonlinearity and, if expanded to cubic terms, by the presence of four-wave-mixing terms. In the 1D system, solutions for stationary gap solitons are obtained in an implicit analytical form. For the Q1D model and for tilted (“moving”) solitons in both models, solutions are found in a numerical form. The existence of stable tilted solitons in the full underlying model of the photonic lattice in the photorefractive medium is also shown. The stability of gap solitons is systematically investigated in direct simulations, revealing a nontrivial border of instability against oscillatory perturbations. In the Q1D model, two disjointed stability regions are found. The stability border of tilted solitons does not depend on the tilt. Interactions between stable tilted solitons are investigated too. The collisions are, chiefly, elastic, but they may be inelastic close to the instability border.

Figure 1

The integral power of the soliton vs the propagation constant, in the first finite band gap of Eq. (2) (the shaded areas are the Bloch bands which confine the band gap). The parameters are $I_0=25.5$ and $K=0.5$. The solid and dashed lines show, respectively, a direct numerical solution of the stationary version of Eq. (2) and the analytical one, as obtained from Eqs. (18, 17, 3). Examples of numerical (a) and analytical (b) profiles of the soliton, taken close to the edge of the band gap (at $q=0.39$, point $A$), are shown in the bottom part of the figure.

Figure 2

A gap soliton found in the full CMT system (4), at the same values of the parameters, $I_0=25.5$ and $K=0.5$, which were used in Fig. 1. Here, $q=-0.2$ [cf. the value $Q\approx 0.4$ at the edge of the corresponding band gap (9)]. In fact, this soliton is quite accurately approximated by the analytical expressions (17) (with ${\varphi }_0=-3\pi ∕4$) and (20). The lower panel illustrates the dynamical stability of the soliton.

Figure 3

The dependences $N\left(q\right)$ for the stationary solitons, found in a numerical form from the full system of the the 1D coupled-mode equations (4) and for the analytical solutions of the simplified system, which are given, in an implicit form, by Eqs. (18, 17).

Figure 4

The dashed and dotted lines are, respectively, stability borders of the gap solitons in the full coupled-mode system (4) and in its simplified version which amounts to Eq. (8). The upper solid border is the band gap’s edge, $q=2∕\sqrt{I_0}$; see Eq. (9).

Figure 5

The lower panel shows an example of the spontaneous onset of oscillatory instability in the $u$ component of the gap soliton in the case of $I_0=25.5$, $K=0.5$, and $q=-0.05$ (the initial profile of the soliton is shown in the upper panel). The instability grows from a very small numerical noise. In this figure and below, the dynamics is shown only in the $u$ component if the picture in the $v$ component turns out to be very similar.

Figure 6

A result of sudden multiplication of the $u$ and $v$ components of the stable soliton shown in Fig. 2 by phase-shifting factors $0.8\pm 0.6i$. The parameters are $I_0=25.5$, $K=0.5$, and $q=-0.2$.

Figure 7

Typical examples of stationary solitons found in the quasi-1D model based on Eqs. (10, 24) with $K=0.5$ and $I_0=1$: (a) $q=-1.125$ and (b) $q=-1.4$.

Figure 8

Instability of the soliton in the quasi-1D model, whose stationary form is shown in Fig. 7b.

Figure 9

A typical example of a stable tilted soliton in the 1D model (4), found for $K=0.5$ and $I_0=25.5$. The soliton corresponds to $q=-0.3$ and $c=0.3$ in Eq. (26).

Figure 10

The stability region for titlted solitons in the proper-1D model with $K=0.5$ and $I_0=25.5$. Up to the accu-racy of the numerical results, the border does not depend on the tilt $c$.

Figure 11

The application of the initial shove, in the form of the multiplication by $\exp \left(ikx\right)$, to a stationary soliton in the underlying equation (2) with $K=0.5$ and $I_0=25.5$. The results are presented by dint of contour plots of the local power, ${\mid u(z,x)\mid }^2+{\mid v(z,x)\mid }^2$. The initial stable soliton, taken with $q=-0.388$, is shown in (a) (with its profile in the inset). The shove with $k=0.05$ (b) and $k=0.2$ (c) generates stable titled solitons, after some loss; the shove with $k=0.4$ (d) destroys the soliton.

Figure 12

Typical examples of elastic and inelastic collisions between titled solitons in the 1D model (4). In this case, $I_0=25.5$, $K=0.5$, both solitons have $q=-0.2$, and their velocities are $c_1=0.3$, $c_2=-0.1$. The difference is that, in the case of the elastic collision (upper panel), the phase difference between the solitons is $\Delta \varphi =\pi$, while in the case of the inelastic collision, $\Delta \varphi =0$.

Figure 13

Regions of elastic and inelastic head-on collisions between two solitons, with tilts $c_1>0$ and $c_2<0$, and $q_1=q_2=-0.2$ (close to the instability border, which is at $q\approx -0.19$, in this case), in the 1D model (4) with $I_0=25.5$ and $K=0.5$.