Interaction of ultrashort relativistically intense laser pulses with matter: conservative models and instabilities of the light field

TLDR: In this paper, the Lagrangian formulation of the problem is given below and the Lipschitz constant of a uniform light field with respect to a single laser pulse is analyzed.

Abstract: Recently, in connection with the development of superintense lasers, a number of researchers have considered the interaction of ultrashort laser pulses with matter at relativistic intensities of the light field. The optics of ultrashort pulses of nonrelativistic intensities have been completely worked out. The basic results in this field are summarized, for example, in the monographs of Akhmanov, Visloukh, and ~hirkin' and ~ukhorukhov.~ A number of questions on the physics of the interaction of strong laser radiation with matter have been discussed in the book of Koroteev and ~ h u m a i . ~ When laser radiation of relativistic intensity interacts with matter, the leading edge the pulse produces rapid ionization, and therefore the radiation propagates in an induced plasma. The pioneering papers on the corresponding plasma nonlinearities (relativistic and striction) are those of Akhiezer and ~olovin? ~ s k a r ' ~ a n ? ~ i tvak? and others. Recent work on the interaction of laser radiation of relativistic intensities with matter can be summarized as follows. A mathematical model of the interaction of long laser pulses with a plasma was formulated in Ref. 7. A detailed mathematical study of this modelg showed that the equations given in Ref. 7 have a denumerable set of eigenmodes. In addition, it was established in Ref. 8 by numerical simulation that at large depths, laser pulses of supercritical intensity experience stabilization and their asymptotic transverse profiles are described by the lowest eigenmodes of the problem. This phenomenon is called relativistic-striction selfchanneling of laser radiation. The model of Refs. 7 and 8 was extended in Refs. 9 and 10 to take into account the effect of higher-order dispersion on the nature of the propagation, and it was shown that under conditions of sharp self-focusing of light in the plasma there is strong self-modulation of an ultrashort laser pulse. Physical effects associated with the finite duration of the laser pulse have also been considered in a number of other For example, a model of the propagation of infinitely wide laser pulses of finite length was worked out in Ref. 11, and the equations of this model were extended in Ref. 12 to three spatial dimensions. However, these equations do not transform into the equations obtained in Ref. 7 in the limit of an infinitely long pulse. We recall that the nonrelativistic three-dimensional interaction of laser pulses with a plasma was considered in Ref. 13, where a number of dispersion terms in the equations describing the propagation of the radiation were neglected. Equations closest to those derived in the present paper were given in Ref. 14. However, a number of terms associated with the dependence of the Langmuir waves generated by the laser radiation on the transverse coordinates were omitted in Ref. 14, with the result that the problem became nonconservative. Most theoretical studies of the problem are based on Maxwell's equations and the equations of cold, collisionless, relativistic hydrodynamics of charged particles in an electromagnetic field in the absence of collisions and thermal effects. These equations can be written in relativistically invariant notation or in the usual three-dimensional form and have an energy-momentum tensor of matter and field whose components are conserved quantities (see Ref. 9, for example). Because of the complexity of these equations, in most papers they are averaged over the period of the laser radiation, and other approximations are introduced, leading to a system of simplified equations, which are also required to be conservative. The present paper is devoted to the derivation of a conservative, time-dependent, three-dimensional model taking the following physical effects into account: diffraction and refraction of the radiation, relativistic and striction nonlinearities in its interaction with the plasma, and the generation of plasma waves by the propagating laser pulses. The Lagrangian formulation of the problem is given below and the general instability of a uniform light field is analyzed. As a starting point we use Maxwell's equations in the Coulomb gauge and the equations of cold, relativistic hydrodynamics for the electronic component of the plasma: