Statistical analysis for beam smoothing and some applications

Abstract

We develop statistical tools for beam smoothing analysis. As applications we study the respective performances of two-dimensional smoothing by spectral dispersion and of smoothing by optical fiber. The calculations are valid in the asymptotic framework of a large number of elements of the random-phase plate and of excited optical modes of the fiber. Theoretical results and closed-form expressions for the contrast and the spatial spectrum of the integrated intensity of the speckle pattern are derived so as to produce evidence of performance differences between these methods, which are essentially based on the longer time delay induced by the multimode fiber with respect to that induced by the gratings and on the nature of the spectral broadening.

On the Validity of the Independent Hot-Spot Model

“…For the sake of completeness, it should be noticed that beside the laser-plasma interaction context, models in which the physics is determined by the occurrence of rare and intense hot spots are also relevant to the interaction of a smoothed laser beam with other nonlinear media like, e.g., liquids and crystals. As examples, one can mention the problems of optic damaging by a partially incoherent laser, stimulated Brillouin scattering in lens, and stimulated Raman scattering in crystals [6]. The IHS model is characterized by the following three steps.…”

Statistics of the hottest spot of speckle patterns generated by smoothing techniques

“…In particular, the mean value of the random part of I max is ¡hln Zi ' 1:92, and its variance is h ln Z ¡ hln Zi … † 2 i ' 1:99. The tails of the random variable ¡ ln Z can be estimated from (8). The left tail, corresponding to the probabilities of exceptional low values, decays very fast:…”

“…In particular, if f has Gaussian shape, then D Sˆ2 . The statistical distribution of Z is described by equation (8).…”

“…This aspect of incoherent pulses was not addressed in our previous works [8,9] which dealt with a statistical approach to the ampli®cation of incoherent pulses. Indeed the energy contained in the hottest spot of a speckle pattern is very negligible compared with the total energy of the pulse, so that the role of the hottest spot was then found negligible in terms of the ampli®cation e ciency.…”

“…In particular the instantaneous spatial intensity distribution of such a pulse is a speckle pattern [12]. These statistical characteristics are actually very usual, and correspond to the pulses generated by the smoothing techniques that we quote here [8,14].…”