Controlling Soliton Refraction in Optical Lattices

We show in the framework of the 1D nonlinear Schrödinger equation that the value of the refraction angle of a fundamental soliton beam passing through an optical lattice can be controlled by adjusting either the shape of an individual waveguide or the relative positions of the waveguides. In the case of the shallow refractive index modulation, we develop a general approach for the calculation of the refraction angle change. The shape of a single waveguide crucially affects the refraction direction due to the appearance of a structural form factor in the expression for the density of emitted waves. For a lattice of scatterers, wave-soliton interference inside the lattice leads to the appearance of an additional geometric form factor. As a result, the soliton refraction is more pronounced for the disordered lattices than for the periodic ones.

Figure 1

The intensity of a soliton beam (log scale) scattered over a periodic comb of identical Gaussian-shaped waveguides for $a=1$, $M=0.2$. The inset shows the profile of a soliton beam scattering on a single defect in the comoving reference frame after a large propagation distance: For the soliton with velocity $2b=4$ ($\alpha =a/b=0.5$), the refraction angle decreases (soliton is decelerated, magenta), but for a faster soliton, $2b=5$ ($\alpha =0.4$), the opposite effect of acceleration takes place (cyan curve). The initial pulse profile is shown as a dashed line for reference.

Figure 2

(a) Velocity change of a soliton with $a=1$, as a function of $\alpha$ for the Gaussian- (red solid line) and sech- (blue solid line) potential shapes (both with $M=0.2$) and the $\delta$ potential (dashed black line). The two dots correspond directly to soliton acceleration and deceleration scenarios shown in the inset in Fig. 1. (b),(c) The phase diagram of the sign of the velocity change for a soliton with $a=1$ for the Gaussian- (b) and sech- (c) scatterer shapes. The red regions correspond to the soliton deceleration and the blue ones to the acceleration.

Figure 3

The dependence of the velocity change on the period $T$ for the periodic system containing 10 waveguides. Red lines correspond to Gaussian individual shapes; blue lines, to sech shapes, both with width $M=1$ and $a=1$; dashed lines, to $\delta$ scatterers. The parameter $\alpha$ defines the profile of $n_s(k)$: for (a) $\alpha =1$ and (b) $\alpha =0.1$.

Figure 4

The dependence of the average velocity change on $T$ for the random 10-waveguide lattices of $\delta$ (dashed line), Gaussian (blue line), and sech (red line) shapes, $M=0.1$, $\alpha =1$, $a=1$. The inset shows the dependence of RD $n(k)$ for $T=1$ for the periodic (green line) and random (magenta line) system of 10 Gaussian (solid line) and $\delta$ (dashed line) scatterers.