Condensation and thermalization of classsical optical waves in a waveguide

P. Aschieri, J. Garnier, C. Michel, V. Doya, and A. Picozzi

Phys. Rev. A 83, 033838 – Published 31 March 2011

Abstract

We consider the long-term evolution of a random nonlinear wave that propagates in a multimode optical waveguide. The optical wave exhibits a thermalization process characterized by an irreversible evolution toward an equilibrium state. The tails of the equilibrium distribution satisfy the property of energy equipartition among the modes of the waveguide. As a consequence of this thermalization, the optical field undergoes a process of classical wave condensation, which is characterized by a macroscopic occupation of the fundamental mode of the waveguide. Considering the nonlinear Schrödinger equation with a confining potential, we formulate a wave turbulence description of the random wave into the basis of the eigenmodes of the waveguide. The condensate amplitude is calculated analytically as a function of the wave energy, and it is found in quantitative agreement with the numerical simulations. The analysis reveals that the waveguide configuration introduces an effective physical frequency cutoff, which regularizes the ultraviolet catastrophe inherent to the ensemble of classical nonlinear waves. The numerical simulations have been performed in the framework of a readily accessible nonlinear fiber optics experiment.

Received 17 December 2010

DOI:https://doi.org/10.1103/PhysRevA.83.033838

Authors & Affiliations

P. Aschieri¹, J. Garnier², C. Michel³, V. Doya¹, and A. Picozzi³

¹Laboratoire de Physique de la Matière Condensée, CNRS, University of Nice Sophia-Antipolis, F-06108 Nice, France
²Laboratoire de Probabilités et Modèles Aléatoires, University of Paris VII, F-75251 Paris, France
³Laboratoire Interdisciplinaire Carnot de Bourgogne, CNRS, University of Burgundy, F-21078 Dijon, France

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Vol. 83, Iss. 3 — March 2011

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Images

Figure 1
(a) Refractive index profile $n (r)$ of the optical waveguide and (b) corresponding confining potential $V (r)$ in the NLS Eq. (3).Reuse & Permissions
Figure 2
(a) Initial field intensity $| ψ |^{2} (r, z = 0)$ , and (b) corresponding intensity distribution at $z = 7$ m obtained by integrating numerically the NLS Eq. (3) (in $10 {Log}_{10}$ scale). (c) Evolution of the fraction of condensed power $n_{0} / N$ vs the propagation distance $z$ . The horizontal dashed red line denotes the value of the condensate amplitude $n_{0}^{eq} / N$ predicted by the theory [see Eqs. (23) and (24)]. (d) Corresponding evolution of the nonequilibrium entropy $S$ . Parameters of the simulation are given in the text.Reuse & Permissions
Figure 3
Numerical simulation of the NLS Eq. (3) showing the establishment of energy equipartition among the modes of the waveguide: Energy per mode $E_{m} = β_{m} n_{m}$ [see Eq. (13)] vs the mode $m = (m_{x}, m_{y})$ , in (a) the initial condition and (b) averaged over the propagation once the equilibrium state is reached, i.e., $\partial_{z} S ≃ 0$ . The amount of power $n_{m}$ in the mode $m = (m_{x}, m_{y})$ is calculated by projecting the field amplitude into the corresponding eigenmode [see Eq. (12)]. Energy almost reaches an equipartition among all modes except the fundamental condensed mode $m_{x} = m_{y} = 0$ [not shown in (a–b)]. Parameters are the same as in Fig. 2 (see the text in Sec. 3a). In particular, we considered a truncated parabolic potential (Fig. 1), so that $β_{m_{x}, m_{y}} ≃ β_{0} (m_{x} + m_{y} + 1)$ and only modes whose eigenvalue verifies $β_{m_{x}, m_{y}} ⩽ V_{0}$ are guided.Reuse & Permissions
Figure 4
Fraction of power condensed in the fundamental mode at equilibrium $n_{0} / N$ vs the energy of the field $H$ for a truncated parabolic potential (parameters are given in Sec. 3a). The red points refer to the results of the numerical simulations. They have been obtained by averaging $n_{0} / N$ over the propagation distance once the equilibrium state is reached, i.e., $\partial_{z} S ≃ 0$ . The “error-bars” denote the amount of fluctuations (standard deviation) of $n_{0} / N$ once equilibrium is reached. The continuous blue line refers to the theoretical condensation curve given in Eqs. (23, 24, 25), while the dashed green line refers to the corresponding thermodynamic limit [ $μ̃ \to 0$ in Eqs. (23, 24, 25)]. In these plots the eigenvalues $β_{m}$ and eigenmodes $u_{m} (r)$ in Eqs.(23, 24, 25) account for the truncation of the potential ( $V_{0} < \infty$ ). Note that the units of the Hamiltonian $H$ are determined from its definition [see Eqs. (6) and (7)].Reuse & Permissions
Figure 5
Representation of the eigenvalues (energies) of the modes of the system as a function of the mode number. (a) Modes of the parabolic core and the first few modes of the cladding. The inset depicts the potential $V (r)$ including the parabolic core and the cladding of respective radii $a = 21 μ$ m and $a^{cl} = 120 μ$ m. (b) Eigenvalues of the modes of the cladding for an energy close to $β^{cl} = 12 \times β_{0}$ . The two circled modes can resonate with those modes in the core that verify the condition (A7), i.e., $β_{m_{11}} + β_{m_{11}} = β_{m_{10}} + β^{cl}$ .Reuse & Permissions

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