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.
We will review computational and physical aspects of the ionization of an atom by N photons with N>P, where P is the minimum number of photons required to ionize the atom. In perturbation theory the matrix element, M(N), for N-photon absorption can be decomposed as M+(N) + M−(N), which results from the decomposition of the stationary wave of the emergent photoelectron into its outgoing ( + ) and ingoing (−) parts. Each of the M±(N)are N-dimensional integrals over N radial coordinates ri, i=1,…,N. Although these integrals are not formally convergent for N>P, the integrand of M+(N) consists of functions that either decay exponentially or are outgoing waves for ri~∞.
We present an atomic model to study above-threshold ionization (ATI). We show that when the laser field is sufficiently intense the usual perturbation theory is not valid. We solve the model dynamics exactly, and several interesting quantities, including the initial-state decay rate and photoelectron spectrum, are calculated. These quantities are related to recent ATI experiments, and qualitatively they agree with these experiments. We specify the strong-field condition for continuum–continuum transitions and explore its physical meaning. Finally some comments on perturbation theory are made.
The strong-field multiphoton ionization of atoms is considered and a theoretical approach dealing nonperturbatively with the radiation field formulated. The general computational scheme is the conventional perturbation theory, but the intermediate states are dressed by the field. We present in detail a method to dress the continuum states and to study the dipole transitions within the continuum. In the high-frequency domain, the proposed procedure rapidly converges over a wide range of field intensity and offers an interesting framework for calculating ionization rates for arbitrary numbers of absorbed (above-threshold) photons and field polarization.
In the final state of a multiphoton-ionization process the ejected electron may interact simultaneously with the external laser field and with the residual atomic system. A procedure for calculating transition probabilities for such processes is described which incorporates the effects of resonances in the final continuum states of the atom. By using standard methods of scattering theory, it is shown how these resonant states may be separated out leaving a "background" amplitude which is a smooth function of the energy, and which therefore lends itself more readily to numerical evaluation. Such an evaluation may be based on the generalized Lippmann-Schwinger integral equation presented here. The constraint imposed by unitarity on the background amplitude is derived, and it is pointed out how approximations may be set up which satisfy unitarity automatically. A simplified model is described in order to illustrate this and other features of the theory and to make contact with some earlier work on this subject. Finally, a low-frequency approximation is developed in which final-state interaction effects are accounted for in a relatively simple analytical form.