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Physical Review A

Calculation of resonant cross sections for multiphoton ionization using very-narrow-bandwidth sources

Burke Ritchie

Published Feb 1, 1978• 7 citations

Abstract

We consider the infinite-order set of coupled, integral equations for the states of an atom in a very-narrow-bandwidth classical radiation field. In the vicinity of a one-photon resonance for two-photon ionization, this set is reduced to a pair of coupled equations describing the response of the atom to the radiation field. This set is derived by dropping terms of the original set whose lowest-order contributions to the radiation-induced shift and width of the resonance are quadratic in the intensity. In the absence of simultaneous absorption and emission, impossible except for two-photon processes from excited states, this pair can be decoupled. The remaining uncoupled integral equation describes the ac Stark effect based on contributions to the shift and width linear in the intensity. These contributions derive from radiative corrections to the intermediate atomic state and depend on the virtual processes of ionization from and recombination into, as well as emission from and reabsorption into, this state. A method for obtaining the exact numerical solution is discussed. An approximate solution is obtained by use of the single-eigenfunction approximation to the Green's function belonging to the response function. This Green's function has a pole at the static (unshifted) position of the resonance; thus, this approximation is quite accurate very near the pole. The order of nonlinearity is calculated as a function of the static detuning through the resonance and is shown to undergo rapid fluctuations about the nonresonance value of 2. The error inherent in the single-eigenfunction approximation to the Green's function is estimated and found to range from about 1% to 27% corresponding to static detunings ranging from about 1 to 25 \AA{}, respectively.

DOI: 10.1103/physreva.17.659

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•May 20, 1991•0 citations

We consider the infinite-order set of coupled, integral equations for the states of an atom in a very-narrow-bandwidth classical radiation field. In the vicinity of a one-photon resonance for two-photon ionization, this set is reduced to a pair of coupled equations describing the response of the atom to the radiation field. This set is derived by dropping terms of the original set whose lowest-order contributions to the radiation-induced shift and width of the resonance are quadratic in the intensity. In the absence of simultaneous absorption and emission, impossible except for two-photon processes from excited states, this pair can be decoupled. The remaining uncoupled integral equation describes the ac Stark effect based on contributions to the shift and width linear in the intensity. These contributions derive from radiative corrections to the intermediate atomic state and depend on the virtual processes of ionization from and recombination into, as well as emission from and reabsorption into, this state. A method for obtaining the exact numerical solution is discussed. An approximate solution is obtained by use of the single-eigenfunction approximation to the Green's function belonging to the response function. This Green's function has a pole at the static (unshifted) position of the resonance; thus, this approximation is quite accurate very near the pole. The order of nonlinearity is calculated as a function of the static detuning through the resonance and is shown to undergo rapid fluctuations about the nonresonance value of 2. The error inherent in the single-eigenfunction approximation to the Green's function is estimated and found to range from about 1% to 27% corresponding to static detunings ranging from about 1 to 25 \AA{}, respectively.

The resonant multiphoton resonances shifts and widths and ionization cross-sections for multielectron atoms in a intense laser radiation field are studied. It is carried out a new consistent approach to atom in a strong realistic laser field, based on the relativistic energy formalism (S-matrix adiabatic formalism), relativistic Dirac equation Green function method and relativistic many-body perturbation theory with the Dirac-Kohn-Sham zeroth approximation. In relativistic theory, the Gell-Mann and Low adiabatic formula for energy shift is connected with electrodynamical scattering matrice, which includes interaction with as a laser field as a photon vacuum field (radiative decay). The optimized basis of the relativistic orbitals is generated with using a minimization procedure for the gauge-non-invariant contribution (the fourth-order of the QED perturbation theory) to the radiation width of atomic state. An approach to treating the multiphoton atomic processes is outlined on example of H, Cs, Kr,Mg etc. Analysis shows that the shift and width of the multi-photon resonance line for the interaction “atom- multimode laser pulse” is greater than the corresponding shift and width for a case of the “atom- single-mode pulse” (the Lorenz pulse model) interaction. From the physical point of view it is naturally provided by action of the photon-correlation effects and influence of the laser pulse multi-modity.

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.

Resonant multiphoton ionization by finite bandwidth multimode radiation is investigated without the restriction of either a "long" or a "short" correlation time of the field. By approximating the multimode radiation by a stochastic model of a chaotic field, the stochastic atomic-density-matrix equation is reduced to a tractable infinite set of differential equations. For a given bandwidth this set can be suitably truncated and numerically integrated. For large-bandwidth fields it turns out that these truncated equations constitute a systematic improvement of the method usually employed of decorrelating the atom-field variables. For zero-bandwidth fields the well-known result is recovered in that the statistical averaging of the ionization probability reduces to an average with respect to the Glauber

P

-distribution function of the chaotic field. Detailed results of numerical calculations for two-photon ionization are presented which reveal a number of new interesting features.

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.

The non linear interaction between an intense laser radiation and atoms leads to ionization through the absorption of N photons from the laser radiation via laser-induced virtual states. The multiphoton ionization rate varies as a function of the laser intensity I as IN. We discuss the two most important effects which govern multiphoton ionization processes : resonance effects and laser-coherence effects. In a moderate laser intensity range (107 - 109 W cm-2) corresponding to the two, three or four-photon ionization of atoms, resonance effects play a dominant role while the photon statistics of the laser radiation are relatively unimportant. On the contrary, in the very high intensity range (1012 - 1013 W cm-2) required to observe large Nth - order ionization processes, coherence effects play a dominant role through N! enhancement in the N-photon ionization rate, while resonance effects are completely damped due to the high laser intensity. We conclude by reviewing the possible applications of multiphoton absorption processes in other fields.

A general program has been developed to calculate multiphoton processes in atomic hydrogen to arbitrarily high order in perturbation theory. This program can be used to calculate quantities in intense field multiphoton processes, such as the generalized cross section of multiphoton absorption, the a.c. Stark effect (shift and the broadening of the level due to the field interaction), and the parameters in nonlinear optics such as the nonlinear index of refraction and the nonlinear susceptibility. Effective parametrization of high-order susceptibilities has been achieved over a wide range of frequencies, utilizing concepts from quantum defect theory to express the results of lengthy calculations in a compact form that also permits extrapolation across thresholds and resonance poles. Significant progress has been made in direct numerical solution of the time-dependent Schroedinger equation. In addition, contributions have been made in: understanding the role of spatial dimensionality in the solution of model systems (e.G., the one-dimensional delta-function potential); the development of a general R-matrix code for perturbative computations of multiphoton processes in many-electron atoms; and multiphoton ionization with two commensurate laser frequencies.