Percolation in Laser Filamentation
Author: Wahb Ettoumi
Affiliation: GAP-Biophotonics, University of Geneva, Switzerland.
Other coauthors of the PRL paper: Jérôme Kasparian (left) and Jean-Pierre Wolf.
The discovery of laser filamentation can be attributed to M. Hercher , who observed damage tracks along the laser path in crystals. Later, the filamentation phenomenon was shown for a laser propagating in air (For a review, see Ref.). For the first time, the optical power at hand could allow one to witness a new type of light propagation based on the Kerr effect, a non-linear phenomenon which acts as a focusing lens and overcomes the beam natural diffraction. As a consequence, the propagation medium is ionized, and produces a plasma filament of tens of microns wide, which can be sustained over meters in air.
The beam collapse is eventually stopped by this newly created plasma, which acts as a defocusing lens, and counter-balances the Kerr effect. This subtile equilibrium is broken when the energy losses along the propagation cause the Kerr effect to be negligible again, and the beam finally diffracts.
For powers largely exceeding the critical power needed for the observation of a single filament, the initial beam inhomogeneities seed the emergence of many single filaments, as if many small beamlets were each undergoing filamentation. In 2010, an experimental campaign in Dresden  was aimed at characterizing the number of filaments with respect to the initial power (Image 1). However, we only noticed until recently the similarity between the laser burns obtained there on photographic paper and the numerical simulations of systems relevant to the statistical physics community. More particularly, we decided to probe the resemblance of the experimental recordings with percolation patterns.
Initially, the laser beam exhibits a noisy profile, but with rather small fluctuations around an average fluence. As the laser propagates, the Kerr effect drives the light to concentrate more and more around the peaks of the highest amplitude, leading to the clustering of light into islands of different sizes, each one potentially holding one or multiple filaments.
At larger distances, typically of several meters in usual experimental setups, the energy flux towards the inner cores of the multiple filaments causes the fluence islands to shrink in size, destroying the previously well held light clusters into smaller, disconnected parts (Image 2). At higher distances, the losses due to the medium's absorption eventually wipe out the smallest clusters.
Because of the lack of experimental data, we turned to the numerical simulation of the non-linear Schrödinger equation, well-known for its remarkable agreement with real filamentation experiments. We showed  that the precise way light clusters depending to each other is a phase transition: we measured a set of seven critical exponents governing the pattern dynamics at the vicinity of the transition between a fully connected state and a non-connected one. The similarity with the percolation universality class is striking, but the clusters' size distribution in the laser case exhibits a finite cut-off physically associated to fluence islands withholding a single filament (their area is approx. 2 mm2).
An interesting issue subsists, however. The finite-size scaling techniques we used are intrinsically equilibrium methods, so that we implicitely assumed that each slice during the laser propagation could be treated as a statistical equilibrium of a given system. But the laser obviously evolves in time, and is not trapped into a quasi-stationary state, nor a fluctuating equilibrium. A hand waving argument can be drawn by saying that the evolution is quasi-static, but a correct theoretical argument remains to be found.
 M. Hercher, "Laser-induced damage in transparent media". Journal of Optical Society of America, 54, 563 (1964).
 A. Couairon, A. Mysyrowicz, "Femtosecond filamentation in transparent media". Physics Report, 441, 47-189 (2007). Abstract.
 S. Henin, Y. Petit, J. Kasparian, J.-P. Wolf, A. Jochmann, S. D. Kraft, S. Bock, U. Schramm, R. Sauerbrey, W. M. Nakaema, K. Stelmaszczyk, P. Rohwetter, L. Wöste, C.-L. Soulez, S. Mauger, L. Bergé, S. Skupin, "Saturation of the filament density of ultrashort intense laser pulses in air". Applied Physics B, 100, 77 (2010). Abstract.
 W. Ettoumi, J. Kasparian, J.-P. Wolf, "Laser Filamentation as a New Phase Transition Universality Class". Physical Review Letters, 114, 063903 (2015). Abstract.
Labels: Photonics 8