An approximate non-perturbative theory of multiphoton ionization of a hydrogen atom by two strong radiation fields is outlined. The special case is then considered for when one of the fields is of low intensity and high frequency and the other field is of high intensity and low frequency. A set of representative calculations has been performed for this case; that is, (a) angular distributions for given numbers of absorbed and emitted low-frequency photons and different geometries; (b) total cross-sections versus the strong-field intensity; (c) angular distributions summed over all the low-frequency photon exchanges, showing a fourfold symmetry in a geometry in which each ionization channel exhibits, instead, only a twofold symmetry; (d) the average energy gained by the ejected electrons. Calculations are performed for a strong low-frequency field of arbitrary polarization. In many respects, the present calculations are expected to complement those of single-field multiphoton ionization. It is hoped that the results of the calculations reported here may prompt systematic measurements also in this new class of processes, as the predictions are well within the state of the art of present multiphoton-ionization experiments.
One-photon free-free absorption cross sections are evaluated by means of an integral relationship involving electron-atom off-shell scattering amplitudes. Off-shell effects are explicitly calculated and compared to various on-shell approximations. This calculation shows that off-shell effects become small with increasing energy as expected. One of the on-shell approximations agrees with the exact result over the entire energy range under consideration to within a few percent. The only integrals which appear in the calculations are absolutely convergent in contrast to methods used in previous calculations.
Two-photon above-threshold ionization processes in the x-ray regime are studied using atomic hydrogen as a model system. Within the minimal-coupling formalism of nonrelativistic quantum electrodynamics, two distinct interactions---$\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathbf{A}}\ensuremath{\cdot}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathbf{p}}$ in second order and ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathbf{A}}}^{2}$ in first order---contribute to the two-photon absorption amplitude. The relative importance of these two interactions is assessed. It is found that above a photon energy of 6.8 keV, the contribution from ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathbf{A}}}^{2}$ to the total two-photon absorption cross section dominates. In this high-energy regime, above-threshold ionization is a nonsequential purely nondipole process. Rate equations are employed to calculate the probabilities of ionization by Compton scattering, one-photon absorption, and two-photon absorption.
A perturbation theory of the ionization of atoms by simultaneous absorption of several photons, each of whose energy is less than the ionization potential, is developed from the evolution-operator formalism. A precise computation is made for the hydrogen atom, giving transition rates as a function of photon energy for two- through twelve-photon photoionization. The eighth-order ionization rate (in cgs units) at the 1.78-eV ruby-laser line is found to be \ensuremath{\sim}${10}^{\ensuremath{-}244}$\ifmmode\times\else\texttimes\fi{}
(photonflux)8 and should be observable using available techniques. Good agreement is obtained with Zernik's exact calculation of the two-photon ionization rate of metastable
2S hydrogen. Approximate calculations are made for the rare gases. Assuming "typical" experimental conditions of a gas density of \ensuremath{\sim}
1020 atoms ${\mathrm{cm}}^{\ensuremath{-}3}$ and a ruby laser focused into a volume of \ensuremath{\sim}${10}^{\ensuremath{-}8}$
cm3, we find that the flux required to liberate one electron during a 10-nsec pulse is \ensuremath{\sim}
1029 ${\mathrm{cm}}^{\ensuremath{-}2}$ ${\mathrm{sec}}^{\ensuremath{-}1}$ for Xe, Kr, and Ar and \ensuremath{\sim}5\ifmmode\times\else\texttimes\fi{}
1030 photons ${\mathrm{cm}}^{\ensuremath{-}2}$ ${\mathrm{sec}}^{\ensuremath{-}1}$ for Ne and He. These gases ionize with the simultaneous absorption of 7, 8, 9, 13, and 14 photons, respectively. The predicted rate for Xe is found to be in excellent agreement with the recent direct measurements of Voronov and Delone. We conclude that multiphoton ionization provides the initial electrons required for the optical breakdown of gases, though it does not account for the over-all growth of the discharge except possibly at very low pressures. Impurity atoms (particularly heavy rare gases) may be the source of "initiating" electrons in Ne and He.