Collapse of the wave field in a one-dimensional system of weakly coupled light guides

The analytical and numerical study of the radiation self-action in a system of coupled light guides is fulfilled on the basis of the discrete nonlinear Schrödinger equation (DNSE). We develop a variational method for qualitative study of DNSE and classify self-action modes. We show that the diffraction of narrow (in grating scale) wave beams weakens in discrete media and, consequently, the “collapse” of the one-dimensional wave field with power exceeding the critical value occurs. This results in the ability to self-channel radiation in the central fiber. Qualitative analytical results were confirmed by numerical simulation of DNSE, which also shows the stability of the collapse mode.

Figure 1

Dynamics of the beam amplitude $\left|\mathrm{\Psi }\right(z,n\left)\right|$ for different values of the parameters $\delta$ and $\mathcal{N}$: (a) $\delta =0.1,\mathcal{N}=2$ ($\mathcal{P}=1.6<{\mathcal{P}}_{cr}$); (b) $\delta =0.05,\mathcal{N}=3$ ($\mathcal{P}=1.8<{\mathcal{P}}_{cr}$); (c) $\delta =0.1,\mathcal{N}=3$ ($\mathcal{P}=3.6>{\mathcal{P}}_{cr}$); and (d) $\delta =0.1,\mathcal{N}=5$ ($\mathcal{P}=10>{\mathcal{P}}_{cr}$).

Figure 2

Dependence of the transverse size $a$ of the wave packet on the evolution variable $z$ for two different initial beam size values: (a) $a_0=10$, (b) $a_0=2$, and for different power values $\mathcal{P}$.

Figure 3

Dynamics of the beam amplitude $\left|\mathrm{\Psi }\right(z,n\left)\right|$ in an array of coupled light guides for the case when the initial distribution (26) is specified at the medium input at different noise levels: (a) $\mu =0.001$, (b) $\mu =0.01$, and (c) $\mu =0.1$. (d) The amplitude distribution at $z=396$ for different values of $\mu$. Here, $\delta =0.1$ and $\mathcal{Q}$ is the noise signal.

Figure 4

Dynamics of the beam amplitude ${\left|\mathrm{\Psi }(z,n)\right|}^2$ in an array of coupled light guides for the case when the initial distribution $\mathrm{\Psi }(z=0,n)={\mathcal{A}}_0{\delta }_{n,n_0}$ is specified at the medium input at different values of ${\mathcal{A}}_0$: (a) ${\mathcal{A}}_0=1.7$, (b) ${\mathcal{A}}_0=2$, (c) ${\mathcal{A}}_0=3$, and (d) ${\mathcal{A}}_0=4$.