From the Rose-DuBois Ansatz of Hot Spot Fields to the Instanton Solution: a Pedestrian Presentation

TL;DR

The paper addresses how laser hot-spot fields description in backscattering with a smoothed beam fails in the far tail of the amplification distribution $p(U)$. By formulating a functional-integral (MSR) approach and performing a saddle-point (instanton) analysis, it derives the tail behavior $p(U) \, ot o\sim$ typical hot-spot fields and identifies filamentary instanton structures as the dominant realizations for large $ abla ext{log}U$, yielding a critical coupling $g_c(L)=\frac{1}{2\mu_{ m max}(L)}$ and a tail $p(U)\sim f(U)U^{-\zeta}$ with $\zeta=1+\frac{1}{2g\mu_{ m max}(L)}$. Numerical testing with biased sampling confirms the algebraic tail and the instanton-like realizations in the far tail, while near the upper tail exhibits coexistence between hot spots and instanton–hot spot complexes. The work suggests updating the Rose–DuBois framework to account for instanton–hot spot complexes, and outlines future directions toward a statistical theory of $S(x,z)$ in the near tail and a nonlinear, time-dependent extension of the linear background. Overall, it provides a rigorous mechanism for understanding extreme amplification events in laser–plasma interaction and clarifies when hot-spot-only descriptions remain valid versus when filamentary instanton structures dominate.

Abstract

This paper gives a pedestrian presentation of some technical results recently published in mathematical physics with non-trivial implications for laser-plasma interaction. The aim is to get across the main results without going into the details of the calculations, nor offering a specialist's user guide, but by focusing conceptually on how these results modify the commonly-held description -- in terms of laser hot spot fields -- of backscattering instabilities with a spatially smoothed laser beam. The intended readers are plasma physicists as well as graduate students interested in laser-plasma interaction. No prior knowledge of scattering instabilities is required. Step by step, we explain how the laser hot spots are gradually replaced with other structures, called instantons, as the amplification of the scattered light increases. In the amplification range of interest for laser-plasma interaction, instanton--hot spot complexes tend to appear in the laser field (in addition to the expected hot spots), with a non-negligible probability. For even larger amplifications and systems longer than a hot spot length, the hot spot field description is clearly invalidated by the instanton takeover.

Figures (17)

• Figure 1: Schematic representation of a backscattering instability of a spatially smoothed laser beam in the strongly damped regime considered by Rose and DuBois in Ref. RD1994.
• Figure 2: Regions in the $(g,\, L)$ plane where $\langle U\rangle$ is well defined (below the curve), and where $\langle U\rangle$ diverges (above the curve). Adapted from http://dx.doi.org/10.1103/PhysRevLett.72.2883 with permission.
• Figure 3: Illustration of the Rose– DuBois ansatz: (left) typical realizations of $\vert S(x,z)\vert^2$ in the bulk of $p(U)$ are hot spot fields; (right) realizations in the tail of $p(U)$ are assumed to be hot spot fields with a few high-intensity hot spots.
• Figure 4: Illustration of the aim of Refs. Mounaix2023Mounaix2024. The Rose– DuBois ansatz of hot spot fields in the tail of $p(U)$ (see Fig. \ref{['figure3']}) is replaced with the problem to be solved: determine the realizations of $S(x,z)$ in the tail of $p(U)$ with no a priori assumptions.
• Figure 5: Artist's rendering of $\vert S_{\rm inst}^{x_{\rm inst}(\cdot)}(x,z)\vert^2$ for two non-degenerate single-filament instantons in the symmetric case $C(-x,z)=C(x,z)$ with two different paths $x^{\pm}_{\rm inst}(\cdot)$ maximizing $\mu_1\lbrack x(\cdot)\rbrack$ (dashed lines). Note that $C(-x,z)=C(x,z)$ implies $\mu_1\lbrack -x(\cdot)\rbrack =\mu_1\lbrack x(\cdot)\rbrack$ and $x^{\pm}_{\rm inst}(\cdot)$ are symmetric with respect to $x=0$. (See defs. 1 to 3 and points 1 to 3 in the text.)