Light and Complexity

Haus, Gross and Pitaevskii, the legacy of random lasers. Our proposed equation for describing the linewidth of random lasers is based on the work of these three scientists. In arXiv:1005.2082 (Josa B 27, 1446 (2010)), the derivation of our model (originally introduced here, Phys. Rev. Lett. 101, 143901 (2008)) is detailed and it is shown how a single equation quantitatively predicts the observed random laser linewidth.

The model is not based on the diffusive approximation for light; conversely it starts from the general Haus model for passively mode-locked lasers and arrives to an equation that is formally identical to that describing one-dimensional zero-temperature Bose-Einstein condensation in an external potential (the Gross-Pitaevskii equation).

The basic idea follows that of a stochastic resonator, originally introduced by Lethokov in 1968, i.e., an open system sustaining a large number of electromagnetic resonances with overlapping linewidths, as sketched in the following figure

By extending the ideas of Haus for multi-mode lasers to such a kind of resonator we derive the following equation:

The Fourier transform of the solution gives the spectrum of the Random Laser.

The equation contains only one parameter: the "nonlinear eigenvalue" E, which depends on the gain bandwidth and the coefficients that measure the spread of the mode decay-time distribution. When the nonlinear eigenvalue E is greater than 1, a bell-shaped linewidth is attained.

This allows to define rigorously a threshold for random lasers, and is in quantitative agreement with our experiments.

This also shows how random lasing can be understood as a classical condensation process of electromagnetic resonances in open disordered resonators.

The investigation of the interaction between a nonlocal nonlinear response and disorder is still at its infancy. In the manuscript arXiv:1005.0578 [Phys.Rev.Lett.104,193901(2010)], we consider the interplay between nonlocality and disorder in solitary wave propagation. Such a topic has relevance in practically all the problems concerning nonlinear wave propagation in soft-matter, as the material density fluctuactions always introduce a certain degree of disorder and, at the same time, long range correlations in the material response function turn out in a nonlocal nonlinear response. In addition nonlocality is also relevant in Bose-condensed gases, plasma physics and many other fields.

The outcome of our analysis is that, for any kind of nonlocality, as the correlation length increases (i.e., the degree of nonlocality increases), the Brownian motion of the solitons is frustrated, and nonlocality filters out fluctuactions. This happens even if nonlocality involves large material regions, if compared with the local case, and hence, in principle, the solitons feel an enhanced disorder. In some sense, nonlocality acts as a cooling mechanism of soliton dynamics.

The figure below shows trajectories of the solitons in a nonlocal disorder medium (details in arXiv:1005.0578).