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.
By using a perturbative treatment based on the central field approximation, the authors have calculated multiphoton ionisation probabilities in several cases of experimental interest (two-, three- and four-photon ionisation of the caesium and potassium ground states at the doubled Nd glass laser frequency and one-, two- and three-photon ionisation of some s, p, d and caesium Rydberg states). Some of these probabilities have been calculated beyond the minimum non-vanishing order. They have also performed some calculations for hydrogen. The comparison of their results with previous ones obtained by analytical methods provides a good test of the numerical techniques involved in the calculations.
By focusing an excimer pumped dye laser on an atomic beam, the authors have observed multiphoton multiple ionisation of the neutral strontium atom in the tuned wavelength range 532-41 nm. Sr + and Sr ++ yields were measured with a time of flight spectrometer. At least three photons are required to produce Sr + and eight photons to produce Sr ++ . As a function of the wavelength, the Sr + yield exhibits resonances which correspond to two-photon excitation of bound states (5s 2 -5p 2 1 D 2 and 5s 2 -5p 2 1 S 0 ) and three-photon excitation of autoionising states 4d5f J=1 and J=3 of the neutral atom. The Sr ++ yield exhibits four broad smooth resonances that they interpret, from known spectroscopic data, as two-photon (5p 1/2 -4f), three-photon (5s-6p 1/2, 3/2 ) and four-photon (5s-7d) resonances in the singly charged ion. The latter transitions start from a singly charged ion in the 5s or 5p 1/2 state reached after three- or four-photon ionisation of the neutral ground state. The variation of the Sr ++ pattern as a function of the dye laser beam intensity shows that resonances are shifted and broadened when the intensity is increased. These features have been interpreted by a perturbative treatment of Sr + ionisation based on the use of an effective Hamiltonian which takes into account higher-order terms involved in the calculation of resonance effects. Calculated multiphoton ionisation probabilities are in good agreement with experimental data.