Crossed-laser-beam solutions for the Klein-Gordon equation

We present analytic solutions of the Klein-Gordon equation for an electron of momentum $p$ in $N$ crossed laser beams ${\mathbf{A}}_{j}\left({\ensuremath{\varphi}}_{j}\right)$ with phases ${\ensuremath{\varphi}}_{j}={\ensuremath{\omega}}_{j}t\ensuremath{-}{\mathbf{k}}_{j}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{r}+{\ensuremath{\delta}}_{j}.$ The solutions are of the form ${\mathrm{\ensuremath{\Psi}}}_{p}={e}^{ipx}\phantom{\rule{0.16em}{0ex}}{\mathit{\ensuremath{\digamma}}}_{p}\left(\mathbf{r},t\right).$ The determination of the distortion factor ${\mathit{\ensuremath{\digamma}}}_{p}$ is pursued within a method of ``auxiliary variables,'' which uses the ${\ensuremath{\varphi}}_{j}$ as variables, ${\mathit{\ensuremath{\digamma}}}_{p}\left(\mathbf{r},t\right)\ensuremath{\equiv}{\mathit{\ensuremath{\digamma}}}_{p}\left({\ensuremath{\varphi}}_{1},...,{\ensuremath{\varphi}}_{N}\right)$. The equation for ${\mathit{\ensuremath{\digamma}}}_{p}\left({\ensuremath{\varphi}}_{1},...,{\ensuremath{\varphi}}_{N}\right)$ is a linear second-order partial differential equation and it does not appear to be soluble analytically exactly. However, the second-order derivative terms are multiplied by coeffients that are small with respect to those of the first-order derivatives, if (A) the angles between all beams are small or (B) for all beams we have ${\ensuremath{\omega}}_{j}/m{c}^{2}\ensuremath{\ll}1.$ In the second case, their neglect would be quite justified for frequencies up into the x-ray range. With this approximation the equation would reduce to a first-order derivatives ``reduced equation.'' Mathematically, however, this cannot be done without further analysis. This is because we are in a situation typical of singular perturbations theory, in which the exact, perturbed solution might not be connected continuously to the unperturbed one. Nevertheless, on the basis of exactly soluble models (see the Appendix), we argue that the approximation is justified in our case and proceed to solve it. It turns out that the reduced equation can be solved exactly for certain crossed-beam geometries of interest. We consider first the case of laser pulses of arbitrary shape. The most general geometry we solve is that in which all beams have coplanar propagation directions and the fields are linearly polarized perpendicular to the propagation plane, except possibly for two that can be oblique and elliptically polarized. For two beams $(N=2)$ this covers the most general case possible. As an application, we calculate the closed-form solution for Gaussian-pulse crossed beams. Next, we treat the case of monochromatic beams as a limit of the laser-pulse case and derive closed-form solutions for some geometries, including standing waves. We then discuss the effect of the passage of crossed beams over an electronic wave packet and show that its momentum distribution is not modified and no pair production is possible (in the reduced equation approximation). We also show that the final wave packet displays the classical ponderomotive Lorentz shift, if low relativistic momenta are involved. The Appendix deals with soluble models of the exact equation for ${\mathit{\ensuremath{\digamma}}}_{p}$ which have the salient features of the original. For these models, the reduced equation solution is indeed a valid approximation to the exact one, if conditions similar to (A) or (B) above are met.