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Journal of Physics B Atomic and Molecular Physics

The theory of non-Lorentzian laser bandshapes in atomic processes

R I Jackson, S. Swain

Published Dec 14, 1982• 17 citations
Publisher

Abstract

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.

DOI
10.1088/0022-3700/15/23/020

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Journal of the Optical Society of America B•Oct 1, 1985•13 citations
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.
The effect of a cut off of the Lorentzian laser lineshape (as predicted by the phase-diffusion model of ideal laser theory) on the laser-atom interaction is investigated. The damping terms in the averaged density matrix equation of the atom, arising from the finite laser bandwidth are shown to be multiplied by a detuning-dependent cut-off. For detunings of the laser well within the Lorentzian part of the spectrum, the authors recover the previously derived results, while these damping terms are seen to drop out for detuning larger than the cut-off. Within their model, therefore, a laser with a lineshape falling faster than a Lorentzian, and for large detunings, appears monochromatic to the atom. In particular, the authors obtain qualitative agreement with recent experiments in which the reversed peak asymmetry in the doublet of double optical resonance persists only within a certain range of detuning from resonance.
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.
The object of this paper is to devise a relatively simple ways of describing the effect of laser field fluctuations on the dynamics of atomic systems when the fluctuations are due to phase fluctuations only. Previous methods have involved the solution of awkward partial differential equations or the evaluation of continued fractions whose terms are matrices. The authors use an exact rate equation approach, where the rates are expanded in a series whose terms are continued fractions (with scalar coefficients). It is found that the lowest-order approximation to these rates gives accurate results except near the Gaussian limit, but here an exact procedure (Greenland, Rzazewski) may be used. The method is used to describe the excitation of a two-level atom and to discuss the optical double resonance experiment. Its accuracy is assessed.
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 effect of amplitude and phase fluctuations on N-photon excitation from N independent lasers is investigated. Each laser is assumed to have a multi-lorentzian line shape. The Fokker-Planck equation is solved to yield atomic density matrix equations which incorporate the fluctuations. Most of the results discussed are for the lowest-order approximation. No enhancement in the Rabi frequency or transition probability appears, but the Stark shift is found to be enhanced by a factor which is independent of the degree of non-linearity of the process. A general expression for the transition probability in the rate equation regime is presented and its characteristic features discussed. The difference between N-photon excitation from N independent lasers and from a single laser is examined.
Cross correlation refers to the situation in which the fluctuations in the fields of two or more lasers are correlated in some way. Recently, Thomas et al. [ Opt. Lett.6, 298 ( 1981); Phys. Rev. Lett.48, 867 ( 1982)] proposed the generation of cross correlation between two laser fields interacting with a three-level system as a means of reducing noise and went on to exploit this property in the observation of narrow Ramsay fringes. Dalton and Knight [ J. Phy. B15, 3997 ( 1982)] discussed the theoretical aspects of cross correlation and showed that it produced interesting effects in population trapping for example. We have demonstrated that it has significant effects on the optical double-resonance (ODR) spectra [ J. Phys. B17, L389 ( 1984)]. On the other hand, the effects of non-Lorentzian laser line shapes have been shown by various workers to have significant effects on a wide range of phenomena, such as multiphoton ionization, resonance fluorescence, and ODR. In this paper we present a theory of the interactions of lasers whose fluctuations are governed by the phase-diffusion process with atomic systems that includes the effects of cross correlation and non-Lorentzian line shapes. We derive the basic equations (which in general have to be solved numerically) and obtain analytic results in the case in which the line shape is approximately Lorentzian near line center but falls off faster than a Lorentzian in the wings. We believe that this analytic model shows all the essential features of the interplay between cross correlation and non-Lorentzian line shapes. The theory is applied to the ODR experiment. In particular, we find that the asymmetry of the spectra is strongly affected by these two features, and so is the ODR spectral line shape when the saturating laser intensity is close to the threshold for producing two peaks.