Stationary scattering from a nonlinear network

Transmission through a complex network of nonlinear one-dimensional leads is discussed by extending the stationary scattering theory on quantum graphs to the nonlinear regime. We show that the existence of cycles inside the graph leads to a large number of sharp resonances that dominate scattering. The latter resonances are then shown to be extremely sensitive to the nonlinearity and display multistability and hysteresis. This work provides a framework for the study of light propagation in complex optical networks.

Figure 1

A graph with $V=4$ vertices, $B=6$ bonds and $N=2$ leads. The incoming, reflected, and transmitted waves are shown on the respective leads.

Figure 2

Upper panel: reflection probability $|r(k,0)|{}^2$ for the graph depicted in Fig. 1 in the linear limit. The arrow marks the narrow resonance which is discussed in the text. Lower panel: intensity amplification factor $\alpha (k)$ in logarithmic scale in the linear limit $a_{\mathrm{L}}^{\mathrm{in}}\to 0$ on the bond connecting vertices 1 and 2 in Fig. 1.

Figure 3

Double logarithmic plot of the numerically obtained distribution of the amplification factor on the nonlinear bond for the graph depicted in Fig. 1.

Figure 4

Reflection probability near the narrow resonance marked in Fig. 2. The central (blue) curve is the resonance in the linear limit. The green (red) curves left (right) from the central one correspond to 8 increasing values of the incoming intensity $\left|a_{\mathrm{L}}^{\mathrm{in}}\right|{}^2$ (in equal steps from $0.0006$ to $0.0048$) in the attractive ($g=1$) and repulsive ($g=-1$) case. The inset shows the corresponding amplification factor.