Abstract
Traditionally, interactions between laser beams or filaments were considered to be deterministic. We show, however, that in most physical settings, these interactions ultimately become stochastic. Specifically, we show that in the nonlinear propagation of laser beams, the shot-to-shot variation of the nonlinear phase shift increases with distance, and ultimately becomes uniformly distributed in [0, 2π]. Therefore, if two beams travel a sufficiently long distance before interacting, it is not possible to predict whether they would intersect in- or out-of-phase. Hence, if the underlying propagation model is non-integrable, deterministic predictions and control of the outcome of the interaction become impossible. Because the relative phase between the two beams becomes uniformly distributed in [0, 2π], however, the statistics of these stochastic interactions are universal and fully predictable. These statistics can be efficiently computed using a novel universal model for stochastic interactions, even when the noise distribution is unknown.
Highlights
We show that all laser beams gradually lose their initial phase information in nonlinear propagation
Nonlinear interactions between two or more laser beams, pulses, and filaments [1] are related to applications ranging from modulation methods in optical communication [2], to coherent combination of beams [3,4,5,6,7], interactions between filaments in atmospheric propagation [8] and ignition of nuclear fusion using up to 192 beams [9]
Nonlinear interactions between solitary waves were studied in other physical systems [18, 19], such as fiber optics [20, 21], waveguide arrays [22], water waves [23, 24], plasma waves [25] and Bose-Einstein condensates [26]
Summary
In a physical setting the noise distribution is typically unknown. the on-axis phase of each beam core is given by (8), where κ(α) is a random variable. We showed that when laser beams or pulses interact after a sufficiently long propagation distance, their relative phase at the crossing point cannot be predicted or controlled. In such cases, the notion of a ”typical experiment” or a ”typical solution” may be misleading, and one should adopt a stochastic approach. The loss of phase can explain some of the difficulties in phasedependent methods in optical communications such as Quadrature Amplitude Modulation (QAM) [2], and in coherent combining of hundreds of laser beams in a small space for ignition of nuclear fusion [14], and for creating a more powerful laser beam [6] In these applications, controlling the phases of the input beams or pulses might not provide a good control over their interaction or combination, due to the loss of phase. Loss of phase is relevant to the loss of polarization for elliptically-polarized beams [35]
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