Application of the uniform approximation to integrals occurring in ionization by a strong elliptically polarized laser field

Above-threshold detachment of electrons from negative ions by a strong low-frequency elliptically polarized laser field is considered using the strong-field approximation. The detachment probability amplitude is expressed via integral over times of highly oscillatory functions. Particular attention is devoted to application of the asymptotic methods to solve these integrals. For the direct detachment only the integral over the detachment time appears, while for the high-order above-threshold detachment the double integral over the detachment and rescattering times should be solved. Depending on the ellipticity of the laser field, a critical photoelectron energy exists for which the standard saddle-point method fails. The problem can be solved by properly deforming the integration contour in the complex time plane and, for energies higher than this critical energy, taking into account only one of the two saddle-point solutions. However, this procedure still leaves a spike in the photoelectron spectrum near this critical energy. This problem is cured applying the uniform approximation. A formula for the transition amplitude in the uniform approximation is derived, and it is shown how this formula should be modified for the energies higher than the critical one. For high-order above-threshold detachment many more saddle-point solutions contribute. They are classified into pairs. For the saddle-point method each pair produces a spike in the spectrum which spoils the total spectrum. When the contribution of each pair is treated using the uniform approximation with a careful choice of the phase factors after the anti-Stokes transition the agreement with the exact numerical results becomes excellent.