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.
Classical nonlinear random waves can exhibit a process of condensation. It originates in the singularity of the Rayleigh-Jeans equilibrium distribution and it is characterized by the macroscopic population of the fundamental mode of the system. Several recent experiments revealed a phenomenon of spatial beam cleaning of an optical field that propagates through a graded-index multimode optical fiber (MMF). Our aim in this article is to provide physical insight into the mechanism underlying optical beam self-cleaning through the analysis of wave condensation in the presence of structural disorde…
We show that Kerr beam self-cleaning results from parametric mode mixing instabilities, that generate a number of nonlinearly interacting modes with randomized phases -- optical wave turbulence, followed by a direct and inverse cascade towards high mode numbers and condensation into the fundamental mode, respectively. This optical self-organization effect is analogue to wave condensation that is well-known in hydrodynamic 2D turbulence.
We show from ab initio laser theory that by choosing an appropriate spatial pump profile, many different spatial modes of a typical microlaser can be turned on at the same pump energy, substantially increasing the number, N, of simultaneous lasing modes. The optimal pump profile can be obtained simply from knowledge of the space-dependent saturated gain profile when the system is uniformly pumped up to the Nth modal threshold. We test this general result by applying it to a two-dimensional diffusive random laser and a microdisk laser. Achieving highly multimode lasing at reasonable pump powers…
The wave turbulence theory predicts that a conservative system of nonlinear waves can exhibit a process of condensation, which originates in the singularity of the Rayleigh-Jeans equilibrium distribution of classical waves. Considering light propagation in a multimode fiber, we show that light condensation is driven by an energy flow toward the higher-order modes, and a bi-directional redistribution of the wave-action (or power) to the fundamental mode and to higher-order modes. The analysis of the near-field intensity distribution provides experimental evidence of this mechanism. The kinetic …
We address the problem of thermalization in the presence of a time-dependent disorder in the framework of the nonlinear Schrödinger (or Gross-Pitaevskii) equation with a random potential. The thermalization to the Rayleigh-Jeans distribution is driven by the nonlinearity. On the other hand, the structural disorder is responsible for a relaxation toward the homogeneous equilibrium distribution (particle equipartition), which thus inhibits thermalization (energy equipartition). On the basis of the wave turbulence theory, we derive a kinetic equation that accounts for the presence of strong disor…
Since its first demonstration in graded-index multimode fibers, spatial beam self-cleaning has attracted a growing research interest. It allows for the propagation of beams with a bell-shaped spatial profile, thus enabling the use of multimode fibers for several applications, from biomedical imaging to high-power beam delivery. So far, beam self-cleaning has been experimentally studied under several different experimental conditions. Whereas it has been theoretically described as the irreversible energy transfer from high-order modes towards the fundamental mode, in analogy with a beam condens…
We show that coupling among multiple resonances can be conveniently introduced and controlled by boundary wave scattering. We demonstrate this principle in optical microcavities of quasi-circular shape, where the couplings of multiple modes are determined by the scattering from different harmonic boundary deformations. We analyze these couplings using a perturbation theory, which gives an intuitive understanding of the first-order and higher-order scattering processes. Different scattering paths between two boundary waves can either enhance or reduce their coupling strength. The effect of cont…
On-chip optical interconnect has been widely accepted as a promising technology to realize future large-scale multiprocessors. Mode-division multiplexing (MDM) provides a new degree of freedom for optical interconnects to dramatically increase the link capacity. Present on-chip multimode devices are based on traditional wave-optics. Although large amount of computation and optimization are adopted to support more modes, mode-independent manipulation is still hard to be achieved due to severe mode dispersion. Here, we propose a universal solution to standardize the design of fundamental multimo…
We introduce the concept of spatial and spectral control of nonlinear parametric sidebands in multimode optical fibers by tailoring their linear refractive index profile. In all cases, the pump experiences Kerr self-cleaning, leading to a bell-shaped profile close to the fundamental mode. Geometric parametric instability, owing to quasi-phase-matching from the grating generated via the Kerr effect by pump self-imaging, leads to frequency multicasting of beam self-cleaning across a wideband array of sidebands. Our experiments show that introducing a gaussian dip in the refractive index profile …
Theoretical studies on wave turbulence predict that a purely classical system of random waves can exhibit a process of condensation, in analogy with the quantum Bose-Einstein condensation. We report the experimental observation of the transition to condensation of classical optical waves propagating in a multimode fiber, i.e., in a conservative Hamiltonian system without thermal heat bath. In contrast to conventional self-organization processes featured by the non-equilibrium formation of nonlinear coherent structures (solitons, vortices...), here the self-organization originates in the equili…
Bosonic condensates offer exciting prospects for studies of non-equilibrium quantum dynamics. Understanding the dynamics is particularly challenging in the sub-picosecond timescales typical for room temperature luminous driven-dissipative condensates. Here we combine a lattice of plasmonic nanoparticles with dye molecule solution at the strong coupling regime, and pump the molecules optically. The emitted light reveals three distinct regimes: one-dimensional lasing, incomplete stimulated thermalization, and two-dimensional multimode condensation. The condensate is achieved by matching the ther…
Media featuring both optical gain and disorder, such as random lasers, represent formidable challenges as subjects of research due to the high complexity of the light propagation within them; however, dramatic advances in this nascent field have been furnished by the paradigm of applying wavefront shaping techniques to the beam pumping the system. We present here a theoretical and experimental study employing this approach in a gain medium where the disorder arises not from random scattering but rather from the multimodality of a waveguide: an amplifying multimode fiber. The shaping of the cor…
FLASH: Ultrafast beam quality characterization via spatial-to-temporal mapping
Wuhan National Laboratory for Optoelectronics (WNLO), School of Optical and Electronic Information, Huazhong University of Science and Technology（武汉光电国家实验室，华中科技大学光学与电子信息学院）
Spatio-temporal equilibrium thermodynamics of guided optical waves at positive and negative temperatures
Université Côte d’Azur（蔚蓝海岸大学）；Ecole polytechnique, Institut Polytechnique de Paris（巴黎综合理工学院，巴黎理工学院）；Institut Universitaire de France (IUF)（法国高等研究院）；University of Calabria（卡拉布里亚大学）；Sapienza University of Rome（罗马第一大学）；Università di Trento（特伦托大学）；Université Bourgogne Europe（勃艮第欧洲大学）
