Recursive algorithm for arrays of generalized Bessel functions: Numerical access to Dirac-Volkov solutions

TL;DR

This paper introduces a stable recursive algorithm for efficiently computing large arrays of generalized Bessel functions, crucial for quantum electrodynamics calculations in laser-matter interactions.

Contribution

A novel recursive method for numerically stable evaluation of generalized Bessel functions without initial value computation, applicable to Dirac-Volkov solutions.

Findings

Efficient evaluation of large generalized Bessel function arrays.

Numerical stability achieved through recurrence relation and normalization.

Application demonstrated in quantum-classical correspondence studies.

Abstract

In the relativistic and the nonrelativistic theoretical treatment of moderate and high-power laser-matter interaction, the generalized Bessel function occurs naturally when a Schr\"odinger-Volkov and Dirac-Volkov solution is expanded into plane waves. For the evaluation of cross sections of quantum electrodynamic processes in a linearly polarized laser field, it is often necessary to evaluate large arrays of generalized Bessel functions, of arbitrary index but with fixed arguments. We show that the generalized Bessel function can be evaluated, in a numerically stable way, by utilizing a recurrence relation and a normalization condition only, without having to compute any initial value. We demonstrate the utility of the method by illustrating the quantum-classical correspondence of the Dirac-Volkov solutions via numerical calculations.