Complex light: Dynamic phase transitions of a light beam in a nonlinear nonlocal disordered medium

The dynamics of several light filaments (spatial optical solitons) propagating in an optically nonlinear and nonlocal random medium is investigated using the paradigms of the physics of complexity. Cluster formation is interpreted as a dynamic phase transition. A connection with the random matrices approach for explaining the vibrational spectra of an ensemble of solitons is pointed out. General arguments based on a Brownian dynamics model are validated by the numerical simulation of a stochastic partial differential equation system. The results are also relevant for Bose condensed gases and plasma physics.

Figure 1

Filaments trajectories vs the normalized propagation coordinate $t$ for a given noise realization and various values of the interaction range $u∕w$ (here $N=10$).

Figure 2

Filaments trajectories vs the normalized propagation coordinate $t$ for a given noise realization and various values of the interaction range $u∕w$ (here $N=30$).

Figure 3

(Color online) Crosses, filament positions at $t_{\mathit{max}}=1000$ for ten noise realizations; thick black line, average position (see text); and white line, inherent structures (here $N=10$).

Figure 4

(Color online) Crosses, filament positions at $t_{\mathit{max}}=6000$ for ten noise realizations; thick black line, average position (see text); and white line, inherent structures (here $N=30$).

Figure 5

Average potential energy $\Phi$ of the inherent structure $e_{\mathit{IS}}$ in units of $V_0$ vs $u∕w$; 1000 noise realizations have been considered $(N=10)$.

Figure 6

Average potential energy $\Phi$ of the inherent structure $e_{\mathit{IS}}$ in units of $V_0$ vs $u∕w$; 100 noise realizations have been considered $(N=30)$.

Figure 7

Average saddle-order for the case $N=10$, other parameters as in Fig. 5.

Figure 8

Average saddle-order for the case $N=30$, other parameters as in Fig. 6.

Figure 9

Standard deviation of the energy of the inherent structures for 100 noise realizations vs $u∕w$ ($N=10$, $t_{\mathit{max}}=1000$).

Figure 10

Standard deviation of the energy of the inherent structures for 100 noise realizations vs $u∕w$ ($N=30$, $t_{\mathit{max}}=6000$).

Figure 11

Maximum relative deviation from the average position, averaged over all the $N=10$ filaments vs $u∕w$. Parameters as in Fig. 9.

Figure 12

Maximum relative deviation from the average position, averaged over all the $N=30$ filaments vs $u∕w$. Parameters as in Fig. 10.

Figure 13

Pseudo-color plot of the squared modulus of the Fourier transform of the course grained density (overall intensity profile) when $N=10$, $u∕w=1$, and $t_{\mathit{max}}=200$, averaged over 100 noise realizations.

Figure 14

Pseudo-color plot of the squared modulus of the Fourier transform of the course grained density (overall intensity profile) when $N=10$, $u∕w=8$, and $t_{\mathit{max}}=200$, averaged over 100 noise realizations.

Figure 15

(Color online) Three realizations of the numerical solution of the stochastic PDE (15) for ${\sigma }^2=0.12$.

Figure 16

(Color online) Three realizations of the numerical solution of the stochastic PDE (15) for ${\sigma }^2=0.45$.

Figure 17

(Color online) Three realizations of the numerical solution of the stochastic PDE (15) for ${\sigma }^2=1$.

Figure 18

Average intensity distribution at $\zeta =4$ over 100 noise realizations as a function of $\xi$ and ${\sigma }^2$.

Figure 19

Maximum relative deviation from the average profile vs the nonlocality parameter calculated at $\zeta =4$.

Figure 20

Two-dimensional Fourier transform of the intensity profile, average over 100 realizations, when ${\sigma }^2=1$.