Interactions of spatial solitons with fused couplers

We study dynamical and stationary states of solitons in dual-core waveguides which are locally coupled (fused) at one or several short segments. The model applies to planar optical waveguides, and to Bose-Einstein condensate in dual traps. Collisions of an incident soliton with single and double locally fused couplers are investigated by means of systematic simulations and several analytical methods (quasilinear, fast-soliton, and adiabatic approximations). Excitation dynamics of a soliton trapped by a local coupler is studied by means of the variational approximation, and verified by simulations. Shuttle motion of a soliton trapped in a cavity between two local couplers, and in a finite array of couplers, is studied too.

Figure 1

A schematic structure of the fused planar coupler.

Figure 2

Collision of an incident soliton with the fused coupler, for $\eta =0.8$ and $q=1$ [see Eq. (10)], in the straight (a) and cross (b) cores.

Figure 3

Fractions of the power of the incident soliton reflected (a) and transmitted (b) in the straight core, scattered (transmitted = reflected) in the cross core (c), and trapped by the fused coupler (d), vs the velocity and amplitude of the incident soliton, $q$ and $\eta$. The powers obey the balance condition (11).

Figure 4

Comparison of the power ratios, given by the analytical solution of the linear scattering problem, as per Eqs. (16) and (17) (curves), to their counterparts produced by direct simulations (chains of symbols) for the small-amplitude soliton with $\eta =0.1$.

Figure 5

Comparison of the analytical results, given by Eqs. (20) and (21), to their numerical counterparts for the absolute values of the trapped (main plot) and radiation (inset) components of the field in the cross core at $x=0$, produced by the passage of a fast heavy soliton (18), with $Q=8,\eta =10$, through the local coupler. The comparison is shown for $z>0$, as at $z<0$ (before the soliton passes the fused coupler) it is difficult to separate the numerically generated field into the radiation and trapped components.

Figure 6

The comparison of the amplitude of the $v$ component of the soliton, predicted by Eq. (23) in the adiabatic approximation, to its counterpart produced by direct simulations, for amplitude $\eta =4$ and initial velocity $q=0.456$. The inset shows the comparison of full profiles of the $v$ component at $z=0$, when the soliton is passing the local coupler.

Figure 7

Exact stationary solutions for the symmetric and asymmetric trapped modes, as given by Eqs. (26) and (27), are represented by continuous curves. Numerically found counterparts of these solutions are shown by chains of symbols [the respective values of $\xi$ and $\zeta$ are identified via $U(x=0)$ and $V(x=0)$ as per Eq. (26)]. The dashed line shows the unstable portion of the symmetric-solution family.

Figure 8

The comparison of the VA prediction for eigenfrequencies of small positional oscillations of the components of the symmetric ($\eta =\theta$, upper lines) and asymmetric ($\eta \ne \theta$, lower lines) trapped modes, as given by Eq. (32), to results extracted from direct simulations.

Figure 9

Splitting of the power of the incident soliton, with amplitude $\eta =1$ and velocity $q$, hitting the pair of local couplers separated by distance $D$. (a), (b) Shares of the power reflected and transmitted, respectively, in the straight core. (c) The same for the cross core. In (c), the plots for the reflected and transmitted shares are butt joined to corroborate the equality of these shares for the single coupler ($D=0$). (d) The trapped-power share.

Figure 10

Splitting of the power of the incident soliton, with amplitude $\eta =2$ and velocity $q$, which hits the pair of local couplers separated by distance $D$. (a), (b) Shares of the power reflected and transmitted in the straight core. (c) The same for the cross core. In (c), the plots for the reflected and transmitted shares are butt joined to corroborate the equality of these shares for the single coupler ($D=0$). (d) The trapped-power share.

Figure 11

The wave field in the symmetric state, trapped by the dual coupler of a small width, $D=0.5$, as a result of the passage of a soliton with amplitude $\eta =1.0$ and initial velocity $q=0.8$.

Figure 12

An example of persistent shuttle dynamics of a soliton in a broad cavity ($D=40$) bounded by two local couplers. The soliton was launched in the $u$ component, with initial amplitude $\eta =0.6$ and velocity $q=0.4$. Panels (a) and (b) display absolute values of fields $u$ and $v$.

Figure 13

An example of the persistent shuttle motion of a symmetric soliton, created with amplitude $\eta =1$ and velocity $q=0.8$, in the array defined as per Eq. (7), with $C=1/4$ and $D=0.354$. The number of local couplers in the array is $2N+1=65$. Note that these parameters satisfy the average trapping condition (39).