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Physical Review A

Rotating-wave approximation in high-gain lasers

Lee W. Casperson

Published Jul 1, 1992• 12 citations
Publisher

Abstract

Semiclassical models for lasers typically incorporate a fundamental simplification called the rotating-wave approximation. In this study the rotating-wave approximation is reexamined, and its implications for the simplest problem of a steady-state laser oscillator are considered in detail. It is found that for practical laser operating conditions the errors resulting from this approximation may not always be negligible.

DOI
10.1103/physreva.46.401

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Semiclassical models for lasers typically incorporate a fundamental simplification called the rotating-wave approximation. In this study the rotating-wave approximation is reexamined, and its implications for the simplest problem of a steady-state laser oscillator are considered in detail. It is found that for practical laser operating conditions the errors resulting from this approximation may not always be negligible.
In a related study the equations governing a high-gain laser amplifier have been solved numerically without making the usual slowly-varying-amplitude derivative approximation in Maxwell's equations, and thus the field amplitudes are not restricted to vary negligibly in a distance of one wavelength. The results reveal an instability that involves reflections and oscillatory growth of perturbations away from steady state. In the present study steady-state solutions of the field equations are obtained analytically, and transformations are described for converting the equations to alternative intensity-equation sets. These intensity equations are linearized and solved for the growth rate and oscillation period of the perturbations.
The rate-equation approximation is one of the most fundamental and universally employed simplifications in laser analyses. The accuracy and regions of applicability of this approximation are explored in comparisons with more rigorous semiclassical models. Higher-order rate-equation approximations are also developed, and these improved models can yield much better accuracy than conventional rate equations with little added complexity. The modified adiabatic elimination methods reported here would also be useful in reducing the mathematical models governing other physical systems.
Abstract This present paper is an attempt to describe the operation of a laser by a method patterned closely after the classical theory of dispersion. There, the electromagnetic field is treated as a classical system interacting with a collection of classical harmonic oscillators which are at rest. In the present case the radiation field is still treated classically, but, in accordance with the correspondence principle, the oscillators are now virtual oscillators associated with the up-and-down transitions in effective two-state atoms which are described quantum mechanically. The atoms carrying these virtual oscillators are not at rest but move with different velocities. We show how the Lorentz averaging can be performed for such systems, and derive a closed set of equations linking the average electric field E, with the average polarization density P„ and the average population inversion density Mp associated with atoms moving with the velocity v. Next, we investigate the conditions for the existence of a steady state if pumping is present, and if the electric field is represented by a standing wave, or a travelling wave of a single frequency (single-mode case). It turns out that, notwithstanding the nonlinear nature of the equations, the steady-state conditions give a simple complex dispersion relation; moreover, the real and imaginary part of this dispersion relation are equivalent to the usual heuristic expressions which specify the operating frequency in terms of the index of refraction and cavity length, and which balance the gain against the losses. The single-mode case is analysed in detail without any smallness assumptions for the resultant intensity. The small-intensity case gives the usual results exhibiting the tuning dip. For high intensities the relative depth of the dip tends to zero. At the end of the paper we discuss proposed extensions and additional applications.
In this note, I introduce a mysterious approximation called the rotating wave approximation (RWA) to undergraduates or non-experts who are interested in both Mathematics and Quantum Optics.In Quantum Optics, it plays a very important role in order to obtain an analytic approximate solution of some Schrödinger equation, while it is curious from the mathematical point of view.I explain it carefully with two coherent oscillations for them and expect that they will overcome the problem in the near future.