Emergence of long-range phase coherence in nonlocal fluids of light

A. Fusaro, J. Garnier, G. Xu, C. Conti, D. Faccio, S. Trillo, and A. Picozzi

Phys. Rev. A 95, 063818 – Published 14 June 2017

Abstract

The emergence of long-range phase coherence among random nonlinear waves is a fascinating effect that characterizes fundamental phenomena such as the condensation of nonlinear waves and the related manifestation of superfluidity of certain turbulent flows. Here we report a previously unrecognized phenomenon of spontaneous emergence of long-range phase coherence among incoherent waves interacting through a nonlocal nonlinearity. The theory reveals that the establishment of long-range phase coherence constitutes a generic property of a conservative (Hamiltonian) system of highly nonlocal random waves that evolve in the strongly nonlinear regime. Aside from phase coherence, the field exhibits intensity fluctuations whose coherence length is shown to increase in a dramatic way during the evolution of the system. The analysis can be transposed to the temporal domain, revealing the emergence of temporal phase coherence among incoherent waves propagating in nonlinear materials featured by a noninstantaneous nonlinear response.

Received 20 March 2017
Revised 9 May 2017

DOI:https://doi.org/10.1103/PhysRevA.95.063818

Physics Subject Headings (PhySH)

Nonlinear optics Optical coherence Spatiotemporal chaos

Nonlinear Dynamics

Authors & Affiliations

A. Fusaro¹, J. Garnier², G. Xu^1,3, C. Conti^4,5, D. Faccio⁶, S. Trillo⁷, and A. Picozzi¹

¹Laboratoire Interdisciplinaire Carnot de Bourgogne (ICB), UMR 6303 CNRS–Université Bourgogne Franche-Comté, F-21078 Dijon, France
²Centre de Mathématiques Appliquées, Ecole Polytechnique, 91128 Palaiseau Cedex, France
³Univ. Lille, CNRS, UMR 8523, PhLAM–Physique des Lasers Atomes et Molécules, F-59000 Lille, France
⁴Dipartimento di Fisica, Università di Roma La Sapienza, 00185 Rome, Italy
⁵Institute for Complex Systems, National Research Council (ISC-CNR), Via dei Taurini 19, 00185 Rome, Italy
⁶School of Engineering and Physical Sciences, SUPA, Heriot-Watt University, Edinburgh EH14 4AS, United Kingdom
⁷Department of Engineering, University of Ferrara, Via Saragat 1, 44122 Ferrara, Italy

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Vol. 95, Iss. 6 — June 2017

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Images

Figure 1
Numerical simulation of the NLSE starting from an initial random wave with linear and nonlinear effects of the same order ( $U_{0} ≃ E_{0}$ , or $λ_{c} ≃ Λ$ ). Spatial distribution of the intensity ${| ψ |}^{2} (r)$ (a), and corresponding (modulo $2 π$ ) phase $ϕ (r)$ (b), at $z = 30 L_{0}$ : No long-range phase coherence emerges in the system. Parameters are $U_{0} / E_{0} ≃ 0.85$ , $L = 1200 Λ$ , $σ = 100 Λ$ , width of the initial incoherent beam $Δ ≃ 120 Λ$ .
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Figure 2
Numerical simulation of the NLSE starting from an initial random wave in the strongly nonlinear regime, $U_{0} / E_{0} ≫ 1$ (or $λ_{c} ≫ Λ$ ). Spatial distribution of the intensity ${| ψ |}^{2} (r)$ (first column) and the (modulo $2 π$ ) phase $ϕ (r)$ (second column) at $z = 0$ (a–b), $z = 4 L_{0}$ (c–d), $z = 30 L_{0}$ (e–f). At variance with Fig. 1, in the strongly nonlinear regime long-range phase coherence emerges in the system. Parameters are $U_{0} / E_{0} ≃ 480$ (or $λ_{c} ≃ 66 Λ$ ), $L = 1200 Λ$ , $σ = 100 Λ$ , width of the initial incoherent beam $Δ ≃ 120 Λ$ .
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Figure 3
Numerical simulation of the NLSE showing the effective potential $V (r)$ (first line), and unwrapped spatial phase $ϕ (r)$ of the random field $ψ (r, z)$ (second line), at $z = 30 L_{0}$ for (a) and (c) and $z = 55 L_{0}$ for (b) and (d). These plots correspond to the simulation reported in Fig. 2 in the strongly nonlinear regime ( $U_{0} ≫ E_{0}$ ): The emergence of phase coherence is mediated by the long-range effective potential, $V (r)$ .
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Figure 4
Intensity fluctuations: (a–b) Evolution of the intensity pattern obtained by simulation of the NLSE (1) in the strongly nonlinear regime ( $U_{0} ≫ E_{0}$ or $λ_{c} ≫ Λ$ ): $z = 0$ (a), $z = 50 L_{0}$ (b). (c–d) Corresponding evolutions of the sizes $[ln (δ) vs z]$ of two particular speckles (brown- and red-encircled), for six different values of their intensity levels: The dramatic increase of the coherence length $\sim exp (ν α V_{0} z^{2})$ is in agreement with the theory [Eq. (19)]; see the parabolic dot-dashed red lines for $z \leq 50 L_{0}$ . The dashed dark line shows the corresponding evolution of the width of $V (r)$ – note that $V (r) ≃$ const for $z ≲ 40 L_{0}$ , so that the coherence enhancement of intensity fluctuations is not due to the beam expansion. Parameters are width of the initial incoherent beam $Δ ≃ 80 Λ$ , $U_{0} / E_{0} ≃ 330$ , $L = 1500 Λ$ , $σ = 150 Λ$ .
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Figure 5
Properties of phase coherence: Numerical simulations of the NLSE (1) in the strongly nonlinear regime $U_{0} ≫ E_{0}$ (or $λ_{c} ≫ Λ$ ). (a) Unwrapped phase $ϕ (x, y = 0)$ at $z = 15 L_{0}$ (blue line), $z = 45 L_{0}$ (red line); the spatial phase exhibits a parabolic-shaped profile (dashed blue and red lines) for large $z$ , in agreement with the theoretical prediction of Eq. (20). (b) Evolution of the phase with respect to the propagation length $z$ , for $r = 10, 20, 30, 50, 100$ in units of $Λ$ (from the bottom to the top). The dashed dark line shows the linear behavior predicted by Eq. (20); note that $V (r) ≃$ const for $z ≲ 25 L_{0}$ . Parameters are $U_{0} / E_{0} ≃ 220$ , $σ = 200 Λ, Δ ≃ 80 Λ, L = 1500 Λ$ .
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Figure 6
Emergence of temporal phase coherence: Simulation of the one-dimensional temporal version of the NLSE in the strongly nonlinear regime ( $t_{c} ≫ τ_{0}$ ), showing the intensity profile ${| ψ |}^{2} (t, z)$ at $z = 0$ (dot-dashed blue line), $z = 80 L_{0}$ (red line) (a), and corresponding unwrapped phases $ϕ (t, z)$ at $z = 0$ (dot-dashed blue line), $z = 25 L_{0}$ (green line), $z = 80 L_{0}$ (red line) (b). The dashed-dark line shows a parabolic fit. The initial phase has been multiplied by a factor $\times 5$ for its visibility. Parameters are $τ_{R} = 200 τ_{0}$ , size of the temporal numerical window $2048 τ_{0}$ , $Δ ≃ 120$ , $β_{2} γ > 0$ , $R (t) = H (t) τ_{R}^{- 1} exp (- t / τ_{R})$ , $H (t)$ being the Heaviside function.
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