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Journal of Physics B Atomic and Molecular Physics

A simple model for multiphoton ionisation

S. Swain

Published Oct 14, 1979• 23 citations
Publisher

Abstract

The author examines some simple models for multiphoton ionisation which can be constructed when the bound-free transition rates are much smaller than the bound-bound transition rates and considers how to proceed when this condition is relaxed. The author identifies a regime where the transition rate is time independent. If the laser has a sufficiently broad linewidth one can use a rate equation approach; this assertion is justified and applied to two-, three- and N-photon ionisation. If the laser is narrow-band, a density matrix method has to be used and the effects of different laser lineshapes are discussed. In three-photon ionisation the role of optical pumping effects is emphasised. Explicit expressions for the multiphoton ionisation probabilities are presented and finally some extensions of the method are briefly discussed.

DOI
10.1088/0022-3700/12/19/007

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The author examines some simple models for multiphoton ionisation which can be constructed when the bound-free transition rates are much smaller than the bound-bound transition rates and considers how to proceed when this condition is relaxed. The author identifies a regime where the transition rate is time independent. If the laser has a sufficiently broad linewidth one can use a rate equation approach; this assertion is justified and applied to two-, three- and N-photon ionisation. If the laser is narrow-band, a density matrix method has to be used and the effects of different laser lineshapes are discussed. In three-photon ionisation the role of optical pumping effects is emphasised. Explicit expressions for the multiphoton ionisation probabilities are presented and finally some extensions of the method are briefly discussed.
The authors treat a model for photoionisation in which the atom has many bound atomic levels and the laser may be either monochromatic or possess a finite bandwidth. The treatment is restricted to the case where the laser is not sufficiently intense to cause Stark splitting of the atomic energy levels. They obtain an expression for the two-photon ionisation probability as a function of time which covers both resonant and non-resonant excitation. They identify several time regimes where the probability takes on a particularly simple form. In particular they discuss the short time behaviour and the conditions under which a time-independent transition rate exists. Numerical plots of the ionisation probability against laser frequency for a wide range of pulse durations are given for the two-level model.
The theory of resonant and non-resonant three-photon ionisation of a multi-level atomic system illuminated by a not too intense source (in a sense which is defined precisely in the text) with a Lorentzian spectrum is presented. The following new results are obtained: (i) the authors find different expressions for the non-resonant ionisation probability under monochromatic and broad band excitation; (ii) in addition to the usual one- and two-photon resonances connecting the ground state to excited states, the resonance which occurs when the photon energy is equal to the difference in energy between two excited states is discussed; and (iii) the triple resonance (that is the resonance occurring when the conditions for a one-photon and two-photon resonance are simultaneously satisfied) is examined.
According to laser theory the lineshape of a laser broadened by Ornstein-Uhlenbeck phase fluctuations is non-Lorentzian. The equations describing the interaction of such a laser with atomic systems are cast into a form which has the same structure as the density-matrix equations in the absence of fluctuations but with modified detunings, damping parameters and interaction matrix elements. By these means the authors derive substitution rules for taking into account the effect of these non-Lorentzian lineshapes which generalize those already known for Lorentzian lineshapes arising from Wiener-Levy fluctuations. The theory is applied to two-photon ionisation and analytic expressions are obtained for the rate of this process. For the extended two-level model the authors' results agree with those of Yeh and Eberly (1981) in the appropriate limit, but they also consider a more general model which leads to enhanced single-photon ionisation rates.
A theoretical procedure is presented for the evaluation of resonant multiphoton ionisation rates for atoms interacting with intense monochromatic lasers. The essence of the procedure is to treat first the strong coupling by solution of the truncated Schrodinger equation to obtain a time-dependent superposition state. This is then used as the lowest order approximation to iterate the full time-dependent Schrodinger equation to obtain the transition amplitude to the continuum. A model-atom study of the time evolution of the ionisation probability is carried out and the approach to rate behaviour in the long-time limit is demonstrated. Ionisation rates are evaluated for a variety of resonant processes in H, Na and Cs involving one or two lasers, as a function of intensities and detunings. Comparison with experiment is still in a fragmentary stage, and some large discrepancies are found to exist.
The authors investigate two-photon ionisation using a generalised rate equation technique for a model in which the atom is assumed to have only two bound levels (the ground state and an excited state) together with a simple continuum of ionised states. Spontaneous emission is allowed from the excited to the ground state, and the laser is assumed to have a finite bandwidth due to phase diffusion. The photoelectron energy distribution is calculated; it is found that if the Rabi frequency exceeds the ionisation rate the energy spectrum is a sensitive function of the linewidth and spontaneous emission rate, whereas if the ionisation rate exceeds the Rabi frequency then the sensitivity of the energy spectrum to these parameters depends upon the initial state of the system. A quite small value of laser linewidth is found to considerably reduce the heights of peaks and increase the widths. Analytic and numerical results are presented.
A time-independent single-rate theory for multiphoton ionization processes is developed to cover cases with a partially coherent (finite-bandwidth) driving laser. We use the Ornstein-Uhlenbeck fluctuating-phase model to describe the driving field, which has therefore a non-Lorentzian spectrum, and the effective two-level model to represent the atomic system. We present analytic and numerical methods that permit discussion of the effects of non-Lorentzian band-shape laser light, and we give analytically the bandwidth dependence of the rate in the wings of the ionization curve. Our model contains a simple scaling relationship that permits the rate for NNN-photon-resonant (N+1N+1N+1)-photon ionization to be obtained directly from the one-photon-resonant two-photon ionization rate. We treat explicitly the cases N=1N=1N=1 and N=2N=2N=2.