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Optica Acta International Journal of Optics

Area Theorem for a Model of Classical Anharmonic Oscillators

Richard D’Souza, S. V. Lawande

Published Jan 1, 1983• 0 citations
Publisher

Abstract

The propagation effects of linearly polarized near-resonant laser radiation through a thermal ensemble of Doppler-broadened anharmonic oscillators are considered. The dynamical equations of motion are analytically solved by a perturbation approach to obtain the nonlinear polarization. Subsequent self-consistent analysis of the Maxwell wave equation then leads to a classical area theorem, which is in qualitative agreement up to third order with the corresponding quantum mechanical result. There is, however, a discrepancy between the respective coefficients of the third-order terms.

DOI
10.1080/713821050

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The propagation effects of linearly polarized near-resonant laser radiation through a thermal ensemble of Doppler-broadened anharmonic oscillators are considered. The dynamical equations of motion are analytically solved by a perturbation approach to obtain the nonlinear polarization. Subsequent self-consistent analysis of the Maxwell wave equation then leads to a classical area theorem, which is in qualitative agreement up to third order with the corresponding quantum mechanical result. There is, however, a discrepancy between the respective coefficients of the third-order terms.
The Doppler-broadened steady state and transient response of a thermal ensemble of anharmonic oscillators to incident laser radiation is investigated. Non-linear polarization of this classical system is obtained by analytically solving the dynamical equations up to third order in the electric field. Except for the sign of the third-order transient polarization, the results are in striking contrast to those obtained in semi-classical theory. In particular, the classical response at resonance is unsaturated up to this order. The Doppler-integrated third-order free induction decay (FID) signal vanishes in the usual approximation of the homogeneous width being smaller than the Doppler width. The physical conclusion of the paper is negative: resonant non-linear and coherent effects such as saturation and nutation typical of the quantum mechanical two-level atoms do not arise for classical anharmonic oscillators.
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Conventional weak-coupling perturbation theory suffers from problems that arise from the resonant coupling of successive orders in the perturbation series. Multiple-scale perturbation theory avoids such problems by implicitly performing an infinite reordering and resummation of the conventional perturbation series. Multiple-scale analysis provides a good description of the classical anharmonic oscillator. Here, it is extended to study the Heisenberg operator equations of motion for the quantum anharmonic oscillator. The analysis yields a system of nonlinear operator differential equations, which is solved exactly. The solution provides an operator mass renormalization of the theory.
Using semiclassical radiation theory, a formalism similar to that used by Lamb in his Theory of an Optical Maser is developed for studying the amplification of vector traveling waves in a laser-type medium. The effect of the medium on the waves is given in terms of space (or time) dependent field amplitudes and phases and a nonlinear index of refraction. With particular emphasis on typical gaseous media, the effects of Doppler broadening are treated in detail for arbitrary ratios of natural to Doppler linewidths. Polarization and propagation vectors in various directions are considered, and the nonlinear effects are found to make an isotropic medium effectively anisotropic. Lowest order nonlinear effects (due to a polarization cubic in the field amplitudes) are studied extensively, and the frequency dependence of several of these processes is presented in graphical form. In particular, the introduction of fields at new frequencies and polarization effects are considered. The characteristics of these nonlinear processes peculiar to Doppler broadened lines are discussed, and the processes are interpreted in terms of saturation and coherent modulation of the population inversion density. Strong nonlinear effects are considered in a more approximate way and are found to consist of saturation of the various linear and nonlinear processes previously considered. These strong nonlinear effects should occur at low enough intensities to be easily observed in practice on a CW basis. With the present formalism, the analytical results of Gordon, White and Rigden regarding gain saturation in laser amplifiers are obtained, and the extension is made to include frequencies away from line center and the effects of multiple spectral components. Again, the introduction of fields at new frequencies is considered in detail. These results are also discussed in terms of saturation and coherent modulation of the populations and hole burning.
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