Symmetry breaking and manipulation of nonlinear optical modes in an asymmetric double-channel waveguide

We study light-beam propagation in a nonlinear coupler with an asymmetric double-channel waveguide and derive various analytical forms of optical modes. The results show that the symmetry-preserving modes in a symmetric double-channel waveguide are deformed due to the asymmetry of the two-channel waveguide, yet such a coupler supports the symmetry-breaking modes. The dispersion relations reveal that the system with self-focusing nonlinear response supports the degenerate modes, while for self-defocusing medium the degenerate modes do not exist. Furthermore, nonlinear manipulation is investigated by launching optical modes supported in double-channel waveguide into a nonlinear uniform medium.

Figure 1

The profile of a two-channel waveguide with different refractive indices.

Figure 2

Various different optical modes existing in self-defocusing medium ($\eta =-1$), where the dash-dotted red line is the optical mode for $V_1=2.500$, the solid green line is the optical mode for $V_1=2.525$, and the dashed blue line is the optical mode for $V_1=2.550$. Here $\beta =2.00$ in (a) and (b) and $\beta =0.85$ in (c) and (d).

Figure 3

Various different optical modes existing in self-focusing medium ($\eta =1$), where the dash-dotted red line is the optical mode for $V_1=2.500$, the solid green line is the optical mode for $V_1=2.525$, and the dashed blue line is the optical mode for $V_1=2.550$. Here $\beta =2.30$ in (a) and (b) and $\beta =1.10$ in (c) and (d).

Figure 4

Several optical modes with different $\beta$ values in a nonlinear asymmetrical double-channel waveguide for the self-defocusing medium ($\eta =-1$). Here the parameters are $V_1=2.500$ and $V_2=2.525$.

Figure 5

Several optical modes with different $\beta$ values in a nonlinear asymmetrical double-channel waveguide for the self-focusing medium ($\eta =1$). Here the parameters are the same as in Fig. 4.

Figure 6

The dependence of the total energy $P_0$ on the propagation constant $\beta$ for the modes existing in self-defocusing medium ($\eta =-1$). Here the parameters are $V_1=2.500$ and $V_2=2.525$. The shaded areas are enlarged in insets (a) and (b). The labels 2a and 2b, and so forth correspond to modes shown in Figs. 2a and 2b, and so forth, respectively.

Figure 7

The dependence of the total energy $P_0$ on the propagation constant $\beta$ for the modes existing in self-focusing medium ($\eta =1$). Here the parameters are $V_1=2.500$ and $V_2=2.525$. The shaded areas are enlarged in (a)–(d). The labels 3a, 3b, and so forth correspond to modes shown in Figs. 3a and 3b, and so forth, respectively.

Figure 8

Several symmetry-breaking optical modes, where the dash-dotted red lines are optical modes for $V_1=2.500$, the solid green lines are optical modes for $V_1=2.525$, and the dashed blue lines are optical modes for $V_1=2.550$. Here $\eta =-1$ in (a) and (c), $\eta =1$ in (b) and (d), $\beta =1.78$ in (a), $\beta =2.39$ in (b), $\beta =0.66$ in (c), and $\beta =1.15$ in (d).

Figure 9

Several symmetry-breaking optical modes with different $\beta$ values in a nonlinear asymmetrical double-channel waveguide. Here the parameters are the same as in Fig. 8.

Figure 10

The evolution of optical modes shown in Fig. 2 into the self-focusing Kerr medium without channels, where ${\eta }^{\prime }=10$ in (a) and (b) and ${\eta }^{\prime }=20$ in (c) and (d). Here the parameters are $V_1=V_2=2.525$ and ${\zeta }_0=10$. The labels (a)–(d) correspond to modes shown in Figs. 2a, 2b, 2c, 2d, respectively.

Figure 11

The evolution of optical modes shown in Fig. 3 into the self-focusing Kerr medium without channels, where ${\eta }^{\prime }=5$ in (a) and (b) and ${\eta }^{\prime }=10$ in (c) and (d). Here the parameters are $V_1=V_2=2.525$ and ${\zeta }_0=10$. The labels (a)–(d) correspond to modes shown in Figs. 3a, 3b, 3c, 3d, respectively.

Figure 12

The evolution of optical modes shown in Fig. 8 into the self-focusing Kerr medium without channels, where ${\eta }^{\prime }=10$. Here the parameters are $V_1=V_2=2.525$ and ${\zeta }_0=10$. The labels (a)–(d) correspond to modes shown in Figs. 8a, 8b, 8c, 8d, respectively.

Figure 13

Energy sharing (row 1) and escape angles (row 2) of optical beams as a function of parameter $V_1$. In all cases, the solid-blue and the dashed-red curves correspond to the left and right beams, respectively. Here the parameters are $V_2=2.525$, ${\eta }^{\prime }=10$ in (a) and (c), which correspond to the modes shown in Fig. 2b [namely Fig. 11b], and ${\eta }^{\prime }=5$ in (b) and (d), which correspond to the modes shown in Fig. 3b [namely Fig. 12b].