Abstract
Beginning with a critical examination of the Lorentz-Abraham (LA) classical equation of motion for an extended charge and the closely related Lorentz-Abraham-Dirac (LAD) equation of motion for a mass-renormalized point-charge, the Landau-Lifshitz (LL) approximate solution to the LAD equation of motion is determined for an electron subject to a counterpropagating linearly or circularly polarized plane-wave pulse with an arbitrarily shaped envelope. A convenient three-vector formulation of the LL equation is used to derive closed-form expressions for the velocities and associated powers of the electron directly in terms of the time in the laboratory frame. The three-vector formulation also reveals definitive criteria for the LL solution to be an accurate approximation to the LAD equation of motion and for the LL solution to reduce to the solution of the Lorentz force equation of motion that ignores radiation reaction. Semiclassical analyses are used to obtain simple conditions for determining the regimes where the quantum effects of either Compton electron scattering by the incident photons or electron recoil produced by the emitted photons is significant. It is proven that the LL approximation becomes an inaccurate solution to the LAD equation of motion only for large enough electron velocities and plane-wave intensities that quantum recoil effects on the electron can greatly alter the classical solution. Comparisons are made with previously published analytical and numerical solutions to the LL equation of motion for the velocity of an electron in a counterpropagating plane wave.10 MoreReceived 18 August 2021Accepted 1 November 2021DOI:https://doi.org/10.1103/PhysRevAccelBeams.24.114002Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasElectromagnetismElectron beams & opticsLaser driven electron accelerationQuantum theorySingle-particle dynamicsAccelerators & BeamsGeneral PhysicsParticles & FieldsNonlinear DynamicsCondensed Matter, Materials & Applied PhysicsAtomic, Molecular & Optical
Highlights
The deceleration of relativistic electrons by intense counterpropagating optical laser beams has produced X-rays and, more recently, γ-rays in the laboratory [1,2,3,4,5,6,7,8,9]
Because the classical Lorentz-Abraham-Dirac (LAD) equation of motion does not have a closed-form solution to the problem of an electron in a plane wave, nor is it amenable to a numerical solution, especially when many charges are involved [12], the more readily solvable Landau-Lifshitz (LL) approximation to the LAD equation of motion has become the classical equation of choice for this problem within much of the physics community [11,12,13,14]
Di Piazza [15,16] and later Hadad et al [17] have derived a closed-form solution to the LL approximate equation of motion for the problem of a plane-wave pulse scattered by a moving electron. These authors use a four-vector formulation of the LL equation of motion to determine the solution for the velocity components of the electron in terms of the retarded time parameter ξ 1⁄4 ωðt þ z=cÞ, where ω is the angular frequency of the plane wave, t is the time, c is the speed of light, and z is the time-dependent longitudinal coordinate of the electron
Summary
The deceleration of relativistic electrons by intense counterpropagating optical laser beams has produced X-rays and, more recently, γ-rays in the laboratory [1,2,3,4,5,6,7,8,9]. The three-vector formulation reveals an explicit closed-form expression for the time t in terms of the retarded time parameter ξ without having to deal with the proper time τ It facilitates the derivation of a simple useful formula for the error in the radiated power introduced by the LL approximation. The derivation manifestly separates the LL approximate radiation momentum-energy from the LL approximate Schott acceleration momentum-energy, the latter of which is no longer perfectly reversible in the LL approximate solution
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