Application of the saddle-point method to strong-laser-field ionization

The quantum-mechanical transition amplitude of an ionization process induced by a strong laser field is typically expressed in the form of an integral over the ionization time of a highly oscillatory function. Within the saddle-point (SP) approximation this integral can be represented by a sum over the contributions of the solutions of the SP equation for complex ionization time. It is shown that, for the general case of an elliptically polarized polychromatic laser field, these solutions can be obtained as zeros of a trigonometric polynomial of the order n and that there are exactly n relevant solutions, which are to be included in the sum. The results obtained are illustrated by examples of various tailored laser fields that are presently used in strong-field physics and attoscience. For some critical values of the parameters two SP solutions can coalesce and the topology of the 'steepest descent' integration contour changes so that some SPs are bypassed. Around the critical parameters a uniform approximation should be used instead of the SP method. (paper)