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Physical Review A

Nonresonant multiphoton ionization in atomic hydrogen

Rajesh Radhakrishnan, R. B. Thayyullathil

Published Mar 22, 2004• 12 citations
Publisher

Abstract

The infinite summations over the complete set of unperturbed atomic states appearing in the $N$th-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.

DOI
10.1103/physreva.69.033407

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Journal of the Optical Society of America B•Jun 1, 1986•37 citations
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.
The infinite summations over the complete set of unperturbed atomic states appearing in the Nth-order time-dependent perturbation theory are performed using the Green's function formalism. In order to test the method extended calculations of the cross section for two- and three-photon ionization for atomic hydrogen are made and numerical comparison is shown with the values obtained by different methods. The resonant character of the process is made evident since the poles of the radial part of the Green's function represent the actual energy spectrum of the atom. The relevant transition matrix elements are written in closed form and then computed exactly without averaging over the intermediate states. The cross sections vanish for circularly polarized light near some energies which explains the form of the cross sections for circularly and linearly polarized light.
In the present thesis we have formulated the Dalgarno-Lewis procedure\nfor two-and three-photon processes and an elegant alternate expressions\nare derived. Starting from a brief review on various multiphoton processes\nwe have discussed the difficulties coming in the perturbative treatment\nof multiphoton processes. A small discussion on various available methods\nfor studying multiphoton processes are presented in chapter 2. These\ntheoretical treatments mainly concentrate on the evaluation of the higher\norder matrix elements coming in the perturbation theory. In chapter 3 we\nhave described the use of Dalgarno-Lewis procedure and its implimentation on second order matrix elements. The analytical expressions for twophoton\ntransition amplitude, two-photon ionization cross section, dipole\ndynamic polarizability and Kramers-Heiseberg are obtained in a unified\nmanner. Fourth chapter is an extension of the implicit summation technique\npresented in chapter 3. We have clearly mentioned the advantage of\nour method, especially the analytical continuation of the relevant expressions\nsuited for various values of radiation frequency which is also used for\nefficient numerical analysis.\nA possible extension of the work is to study various multiphoton processcs\nfrom the stark shifted first excited states of hydrogen atom. We\ncan also extend this procedure for studying multiphoton processes in alkali\natoms as well as Rydberg atoms. Also, instead of going for analytical expressions,\none can try a complete numerical evaluation of the higher order\nmatrix elements using this procedure.
The cross section for multiphoton ionization of hydrogen is computed using the momentum translation method (MTA) in the cases of linearly and circularly polarized light, alternatively. Extended numerical comparison is made with well-established results obtained from the Nth-order time-dependent perturbation theory in the weak intensity limit. The MTA does not reproduce the values given by perturbation theory and considerable differences are observed. Higher order iterations than the lower one are introduced and an analysis of the leading terms of the development is made: at weak intensities the zeroth-order term that defines the MTA cancels with higher order terms and the final result obtained after making (N-1 iterations is none other than the well-known transition amplitude deduced from standard perturbation theory, defined by the remaining terms of the last iteration after mutual cancellation with lower-order ones. According to the results obtained in this work, it is not possible to affirm that the MTA is adequate to calculate atomic multiphoton ionization cross sections.
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.
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{}.
Within the framework of nonrelativistic quantum mechanics, the analysis of the two-photon absorption by atomic hydrogen is generalized to the case of two incident beams with arbitrary direction and polarization (including circular polarization). In the dipole approximation, the second-order matrix elements responsible for two-photon absorption are transformed into a finite sum consisting of the product of a radial part and an angular part. Exact calculation of the angular part predicts that, for the s–s transition only, transparency is obtained regardless of the frequencies of the two beams with perpendicular polarizations. In addition, circularly polarized light is found to be more efficient (up to a factor of 1.5) for the s–d two-photon transition. The radial parts of the matrix elements are accurately evaluated by using the Coulomb Green’s function technique and an implicit technique of Dalgarno and Lewis. Their calculation indicates that “zeros” exist in the two-photon absorption spectrum, thereby predicting that absorption is not possible at certain frequencies. It should be noted that in the calculated spectrum, near or at resonance points, the linewidths of the intermediate levels must be included in order to obtain accurate results.