Dramatic Acceleration of Wave Condensation Mediated by Disorder in Multimode Fibers

Classical nonlinear waves exhibit a phenomenon of condensation that results from the natural irreversible process of thermalization, in analogy with the quantum Bose-Einstein condensation. Wave condensation originates in the divergence of the thermodynamic equilibrium Rayleigh-Jeans distribution, which is responsible for the macroscopic population of the fundamental mode of the system. However, achieving complete thermalization and condensation of incoherent waves through nonlinear optical propagation is known to require prohibitive large interaction lengths. Here, we derive a discrete kinetic equation describing the nonequilibrium evolution of the random wave in the presence of a structural disorder of the medium. Our theory reveals that a weak disorder accelerates the rate of thermalization and condensation by several order of magnitudes. Such a counterintuitive dramatic acceleration of condensation can provide a natural explanation for the recently discovered phenomenon of optical beam self-cleaning. Our experiments in multimode optical fibers report the observation of the transition from an incoherent thermal distribution to wave condensation, with a condensate fraction of up to 60% in the fundamental mode of the waveguide trapping potential.

Figure 1

Disorder-induced condensation. (a) Numerical simulations of the modal NLS equation (1) in the absence of structural disorder (a) and in the presence of structural disorder (b) [NLS equation (2)], starting from the same initial coherent condition. Evolutions of the modal populations $n_p/N$: fundamental mode $p=0$ (red), $p=1$ (dark blue), $p=2$ (blue), $p=3$ (light blue), $p=4$ (cyan), $p=5$ (light green), $p=6$ (green), $p=7$ (yellow), $p=8$ (orange). A structural disorder breaks the coherent modal interaction (a) and induces condensation in the fundamental mode $n_0$ (b), which relaxes to the theoretical value $n_0^{\text{eq}}/N=0.67$ at thermal equilibrium (dashed black). The dotted red line in (b) reports $n_0(z)$ from the simulation of the kinetic Eq. (3). (Insets) Corresponding intensity patterns $|\psi {|}^2(\mathit{r},z)$ where the circles depict the core of the multimode fiber ($N_{*}=120$ modes without including polarization degeneracy).

Figure 2

Condensation to equilibrium. (a) Simulation of the modal NLS equation (2) (accounting for structural disorder) showing the evolution of $n_p(z)$ for the fundamental mode $p=0$ (red), and $p=3$ (blue), and corresponding simulation of the kinetic equation (3) (dotted red and blue lines) starting from the same initial incoherent condition: $n_0(z)$ reaches the theoretical equilibrium value $n_0^{\text{eq}}/N\simeq 0.57$ (dashed black). (b) Corresponding energy per mode ${\mathcal{E}}_p(z)=({\beta }_p-{\beta }_0)n_p(z)$ at $z=0$ (blue dots), and at $z=20\mathrm{m}$ (red dots) showing that the beam reaches thermal equilibrium with “energy equipartition” ${\mathcal{E}}_p^{\text{eq}}=T$. (c) Condensate fraction at equilibrium $n_0^{\text{eq}}/N$ vs energy $E$ (keeping constant the power $N$): For $E\le E_{\text{crit}}$ the system undergoes a phase transition to condensation. The NLS simulations of Eq. (2) (red circles) are in quantitative agreement with the theory (blue line) without adjustable parameters.

Figure 3

Experiments. Intensity patterns recorded at $Z_0=20\mathrm{cm}$ (i)–(l) and at the output of the whole fiber length $L=13\mathrm{m}$ (e)–(h): by reducing the number of modes initially excited (i.e., by decreasing $E$), a transition occurs from $n_0=0$ to a condensate fraction of $n_0/N\simeq 0.63$. (a)–(d) The fundamental Gaussian mode $u_0(\mathit{r})$ of the MMF gets macroscopically populated as evidenced by the intensity lineouts $I(x,y=0)$ recorded experimentally at $z=L$ (blue line), the corresponding fits of the condensate contribution $I_{\text{cond}}(\mathit{r})$ (orange-filled region), and the incoherent contribution $I_{\text{inc}}(\mathit{r})$ (gray-filled region). The red dashed line denotes $I_{\text{cond}}(\mathit{r})+I_{\text{inc}}(\mathit{r})$. Above the transition to condensation $n_0=0$ (d), we computed the angle averaged distance $|\mathit{r}|$ of the intensity recorded experimentally (blue), which is in agreement with the Rayleigh-Jeans distribution $I_{\text{inc}}(\mathit{r})$ (red dashed). $R=26\mu \mathrm{m}$ is the fiber core radius [corresponding to the circles (e)–(l)], $N_{*}=120$ modes, $N=19\mathrm{kW}$.