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

Abstract

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.

Elliptic dichroism in strong-field ionization of atoms subjected to tailored laser fields

“…Momentum distributions of the direct electrons only are analyzed in ref. 44 and 45. With increasing ellipticity the shapes of the rescattering momentum distributions appear to become closer to those of the direct electrons.…”

“…This is as predicted by the simple-man model. The curve is also an approximate solution of the saddle-point equation for the direct electrons (in the saddle-point approximation the integral over the ionization time in eqn (3) can be approximated by a sum over the saddle-point solutions t 0 s of the equation [ p + A ( t 0 s )] 2 = −2 I p 36,45 ). The momentum distributions in Fig.…”

Conservation laws for electron vortices in strong-field ionisation

“…(32). The solutions can also be found very efficiently by transforming the equation to a polynomial of order 2N + 2 and finding the roots, as outlined in [68]. This means there will be N +1 valid solutions, see Fig.…”

“…Interference between different paths can be seen as faint circular fringes. The following references [68][69][70] can give some further insight into these solutions and the method. Such distributions bear some similarity with the attoclock [71][72][73][74][75][76][77][78], where an elliptical nearly circular field is used to relate the electron emission angle to the tunnelling time.…”

Conservation laws for Electron Vortices in Strong-Field Ionisation

“…( 32). The solutions can also be found very efficiently by transforming the equation to a polynomial of order 2N + 2 and finding the roots, as outlined in [58]. This means there will be N + 1 valid solutions, see Fig.…”