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Australian Journal of Physics

Renormalised Perturbation Theory of Resonant Multiphoton Ionisation

K. Unnikrishnan

Published Oct 1, 1995• 0 citations

Abstract

Perturbation theory of multiphoton ionisation due to a classical electromagnetic field is modified to allow for intermediate resonances with bound states. Complex energies, generally associated with resonances, do not enter into this formalism. For a monochromatic field of frequency u, a constant ionisation rate can then be defined unambiguously and only such continuum states are excited which correspond to the absorption of energy in integral multiples of hu. As an application, differential and total cross sections for the two-photon ionisation of hydrogen, for frequencies below the n = 3 resonance region, are obtained in closed form. Existing data for generalised cross sections, calculated numerically using the complex coordinate method, are in good agreement with the present results. Finally, the mean fractional ionisation resulting from a pulse of finite duration is estimated on the basis of the associated power spectrum. For short pulses, the time dependence of ionisation exhibits a departure from that expected of a time-independent rate.

DOI: 10.1071/ph950835

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Perturbation theory of multiphoton ionisation due to a classical electromagnetic field is modified to allow for intermediate resonances with bound states. Complex energies, generally associated with resonances, do not enter into this formalism. For a monochromatic field of frequency u, a constant ionisation rate can then be defined unambiguously and only such continuum states are excited which correspond to the absorption of energy in integral multiples of hu. As an application, differential and total cross sections for the two-photon ionisation of hydrogen, for frequencies below the n = 3 resonance region, are obtained in closed form. Existing data for generalised cross sections, calculated numerically using the complex coordinate method, are in good agreement with the present results. Finally, the mean fractional ionisation resulting from a pulse of finite duration is estimated on the basis of the associated power spectrum. For short pulses, the time dependence of ionisation exhibits a departure from that expected of a time-independent rate.

Non-resonant k-photon ionisation cross sections may be calculated by kth order perturbative treatment. This method is generally considered to be good when the detuning from the closest resonance is much larger than the expected shift and width of this resonance. The aim of this paper is to discuss the validity of this criterion for several realistic cases. By diagonalising an effective Hamiltonian built on a few quasi-resonant states, we have calculated multiphoton ionisation cross sections for ground state caesium by linearly polarised light in frequency ranges containing the tuning range of the ruby laser, the Nd glass laser and its first harmonic. The study is carried out up to an intensity of 10 11 W cm -2 .

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.

N-photon ionisation cross sections obtained by perturbation theory treatments are expressed in terms of Fourier transforms of appropriate autocorrelation functions. As expected, the N=1 case is related to the well known electric dipole-dipole autocorrelation function, while correlation of increasingly complicated operators is involved as N=1, 2, 3, . . .. A simple application of the formalism to the evaluation of one- and two-photon cross sections for a schematic model of the negative ion H - is presented.

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.

The two-photon ionisation cross section of potassium is calculated from a quasi-degenerate perturbation theory expansion in the atom-radiation field interaction. The authors assume a narrow bandwidth radiation field of moderate photon flux (1025-1029 photons/cm2 s). Resonances are treated by first diagonalising the Hamiltonian among the strongly coupled resonant states and then applying perturbation theory to include nonresonant background effects. All bound and continuum atomic states are calculated in the non-relativistic Hartree-Fock approximation and the various summations over intermediate states are handled by the inhomogeneous differential equation technique. Results for potassium show that resonance structures, including fine structure, are well resolved for a photon flux of 1027 photons/cm2 or less. As the flux increases to 1029, the resonance enhancement is substantially quenched.