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Canadian Journal of Physics

Multiphoton ionization of the hydrogen atom

S. N. Haque, M. G. Mustafa

Published Sep 1, 1987• 1 citation
Publisher

Abstract

In this paper, we have presented the analytical and numerical calculations of the ionization rate (σ/I in units of cm4∙W−1) for the hydrogen atom in the ground state. The summation over infinite intermediate states, discrete plus continuous, has been carried out using our modified technique based on Bebb and Gold's concept of average frequency. Our method improves the results of Bebb and Gold at all frequencies, and the results obtained are in good agreement with the results of Gontier and Trahin, Chan and Tang, and Klarsfeld.

DOI
10.1139/p87-192

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In this paper, we have presented the analytical and numerical calculations of the ionization rate (σ/I in units of cm4∙W−1) for the hydrogen atom in the ground state. The summation over infinite intermediate states, discrete plus continuous, has been carried out using our modified technique based on Bebb and Gold's concept of average frequency. Our method improves the results of Bebb and Gold at all frequencies, and the results obtained are in good agreement with the results of Gontier and Trahin, Chan and Tang, and Klarsfeld.
The general theory of multiphoton ionization of hydrogen atoms by simultaneous absorption of several photons is discussed. In the nonrelativistic dipole approximation, the calculations lead to a recursion relationship concerning angular functions and involve radial matrix elements containing many summations over intermediate states. A general method for exactly evaluating several sums is described in detail and applied to the photo-ionization of atomic hydrogen in the ground state. Numerical results are presented as dispersion curves of the transition rate $\frac{{\ensuremath{\sigma}}_{n}}{{I}^{N\ensuremath{-}1}}$ for a range of wavelengths from 1000 up to 7200 \AA{}.
In this paper we show that for an intensity parameter much less than unity, the Reiss approximation for the multiphoton ionization of atoms is equivalent to the neglect of the mean-frequency term in the treatment of Bebb and Gold. We have also improved upon the result of Reiss by including a first-order term. As an example of the above result, numerical values for many-photon $1s\ensuremath{\rightarrow}2s$ transitions in hydrogen atoms are presented and are compared with the exact values of Gontier and Trahin.
The infinite summations over the complete set of unperturbed atomic states appearing in the NNNth-order perturbation theory for multiphoton ionization are performed using the Dalgarno-Lewis method. The relevant transition matrix elements are written in a closed integral form exhibiting all the analytic properties of the amplitude as a function of incident photon energy. The cross sections for two- and three-photon ionization for atomic hydrogen are calculated for both linearly and circularly polarized light with a wide range of photon energy spectrum including the near resonance, and numerical comparison is made with the values obtained by different methods.
This paper illustrates the study on asymmetry of the angular distribution of the ejected electrons in a multiphoton process from the ground state of a hydrogen atom. General expressions for the multiphoton ionization rates have been derived based on a variation of the Dalgarno–Lewis method for the evaluation of matrix elements in higher order perturbation theory. In addition, these analytical expressions for the various multiphoton transition rates and their variation as a function of polarization, phase, frequency and intensity of the incident radiation have been studied extensively. The general expression obtained can be used to analyse any number of incident beams with arbitrary polarization, phase and intensity.
Analyses of the resonant multiphoton ionization of atoms require knowledge of ac Stark energy shifts and of multiphoton, bound-to-bound state, transition amplitudes. In this paper, we consider the three-photon photoionization of hydrogen atoms at frequencies that are at and surrounding the two-photon 1s to 2s resonance. AC energy shift sums of both the 1s and 2s states are calculated as a function of the laser frequency along with two-photon 1s ? 2s resonant transition amplitude sums. These quantities are calculated using an extended version of a method, which has often been employed in a variety of ways, of calculating these sums by expressing them in terms of solutions to a variety of differential equations that are derived from the different sums being evaluated. We demonstrate how exact solutions are obtained to these differential equations, which lead to exact evaluations of the corresponding sums. A variety of different cases are analysed, some involving analytic continuation, some involving real number analysis and some involving complex number analysis. A dc Stark sum calculation of the 2s state is carried out to illustrate the case where analytic continuation, pole isolation and pole subtraction are required and where the calculation can be carried out analytically; the 2s state, ac Stark shift sum calculations involve a case where no analytic continuation is required, but where the solution to the differential equation produces complex numbers owing to the finite photoionization lifetime of the 2s state. Results from these calculations are then used to calculate three-photon ionization probabilities of relevance to an analysis of the multiphoton ionization data published by Kyrala and Nichols (1991 Phys. Rev. A 44, R1450).