Stochastic Dynamics of Incoherent Branched Flow

Abstract

Waves propagating through weakly disordered smooth linear media undergo a universal phenomenon called branched flow. Branched flow has been observed and studied experimentally in various systems by considering coherent waves. Recent experiments have reported the observation of optical branched flow by using an incoherent light source, thus revealing the key role of coherent phase-sensitive effects in the development of incoherent branched flow. By considering the paraxial wave equation as a generic representative model, we elaborate a stochastic theory of both coherent and incoherent branched flow. We derive closed-form equations that determine the evolution of the intensity correlation function, as well as the value and the propagation distance of the maximum of the scintillation index, which characterize the dynamical formation of incoherent branched flow. We report accurate numerical simulations that are found in quantitative agreement with the theory without free parameters. Our theory highlights the important impact of coherence and interference on branched flow, thereby providing a framework for exploring branched flow in nonlinear media, in relation with the formation of freak waves in oceans.

Figures (6)

• Figure 1: Coherent initial plane wave with a medium with Gaussian correlation: (a) Numerical simulation of Eq.(\ref{['eq:parax0']}) showing the evolution of $|\psi_z(x)|^2$ starting from $\psi_o(x)=1$. Parameters: $\ell_c/\lambda=100$, $\sigma^2 \lambda^2=10^{-4}$ ($X_c\approx 12.4$). (b) Scintillation index $S_z$ versus $z/z_c$ for different values of $X_c$: the black dashed lines report the theory, Eq.(\ref{['eq:expressS']}); the dotted line is the small $z$ prediction $S_z \simeq 2 \sqrt{\pi} (z/z_c)^3$; the colored lines are the results of the numerical simulations, averaged over $1000$ independent realizations of the disordered potential. Parameters: from the bottom, $\ell_c/\lambda=10, 25, 50, 75$, with $\sigma^2 \lambda^2=10^{-4}$ for all curves, except for the top yellow curve ($X_c \approx 12.4$) where $\ell_c/\lambda=50$, $\sigma^2 \lambda^2=8 \times 10^{-4}$.
• Figure 2: Coherent initial plane wave with a medium with Gaussian correlation: (a) Theoretical intensity correlation function $C^{{\cal I}}_z(x)$ from Eq.(\ref{['eq:C_I_coh']}). (b) Comparison of $C^{{\cal I}}_z(x)$ from Eq.(\ref{['eq:C_I_coh']}) (black dashed lines), with the numerical simulations of Eq.(\ref{['eq:parax0']}) (colored lines), for different propagation lengths $z/z_c$. An average over $1000$ simulations with different realizations of the random potential $V(z,x)$ has been carried out. Parameters: $\ell_c/\lambda=100$, $\sigma^2 \lambda^2=10^{-4}$ ($X_c \approx 12.4$).
• Figure 3: Incoherent initial wave with a medium with Gaussian correlation: (a) Numerical simulation of Eq.(\ref{['eq:parax0']}) showing the evolution of $|\psi_z(x)|^2$ starting from a coherent speckle field [situation (c)], with $\rho_o/\lambda=10$, $\ell_c/\lambda=100$, $\sigma^2 \lambda^2=10^{-4}$. (b-d) Evolution of $S_z^{(c)}$ versus $z/z_c$, by varying different parameters: the black dashed lines report the theory, Eq.(\ref{['eq:Szii1']}); the colored lines are the results of the numerical simulations, averaged over $1000$ independent realizations of the disordered potential and of the initial random field. Parameters: (b) $\rho_o/\lambda=10$, $\sigma^2 \lambda^2=10^{-4}$; (c) $\ell_c/\lambda=100$, $\sigma^2 \lambda^2=10^{-4}$; (d) $\rho_o/\lambda=10$, $\ell_c/\lambda=100$.
• Figure 4: Incoherent initial wave with a medium with Gaussian correlation: Theoretical intensity correlation function $C^{{\cal I}}_z(x)$: from Eq.(\ref{['eq:C_I_c']}) for a coherent speckle field (a) [situation (c)], and from Eq.(\ref{['eq:C_I_pc']}) for a partially coherent speckle field (b) [situation (pc)]. Corresponding comparison of the theoretical correlation function $C^{{\cal I}}_z(x)$ (black dashed lines), with the numerical simulations of Eq.(\ref{['eq:parax0']}) (colored lines), for different propagation lengths $z/z_c$. In situation (c), an average over $1000$ realizations of $V(z,x)$ and of $\psi_o(x)$, has been considered. In situation (pc) an average over $300$ realizations of $V(z,x)$, each with $400$ realizations of $\psi_o(x)$. Parameters: $\ell_c/\lambda=100$, $\rho_o/\lambda=10$, $\sigma^2 \lambda^2=10^{-4}$.
• Figure 5: Coherent initial plane wave. Example of a congruence of rays for a single realization of the random potential. Panel (a) shows a realization of the random potential $V(z,x)$ together with the rays associated with an initial coherent plane wave at $z=0$. Panel (b) shows the BF obtained by solving the paraxial Eq.(1) (main text) with the potential shown in (a), superimposed with the congruence of rays. We can see the formation of caustics which are associated with increases in wave intensity. Parameters are the same as in Fig. 1(a) (main text).