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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Geometric phase in multimode lasers

Vladislav Yu. Toronov, Vladimir Leonardovich Derbov

Published Apr 29, 1996• 0 citations
Publisher

Abstract

In recent years a remarkable similarity between semiclassical laser equations and Schrodinger equation was noticed, which results, in particular, in `geometric' properties of the phase of laser field. The goals of the present paper are to realize the origin of this analogy and to reveal some of its new physical outcomes for multimode lasers.

DOI
10.1117/12.239164

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In recent years a remarkable similarity between semiclassical laser equations and Schrodinger equation was noticed, which results, in particular, in `geometric' properties of the phase of laser field. The goals of the present paper are to realize the origin of this analogy and to reveal some of its new physical outcomes for multimode lasers.
The semiclassical interaction of a two-level system with a multimode electromagnetic field is discussed in detail. The equations of motion for the macroscopic density matrix are transformed into a single integral equation for the population density, with a complex kernel which can take into account both running- and standing-wave configurations. The iterative solution to this integral equation, which corresponds to an expansion in terms of powers of the electromagnetic field amplitudes, turns out to be divergent when the field amplitudes are greater than some critical value. In the case in which the kernel is a periodic or quasiperiodic function of time, a general solution to the integral equation is found, and this solution turns out to be unique and convergent for every value of the electromagnetic (e.m.) field amplitude. This solution is expressed through a continued-fraction expansion, whose terms are matrices, which can be readily obtained from the kernel. This solution is a generalization of the continued-fraction expansion usually met in the strong-signal semiclassical theory of lasers. Some particular cases are then treated in detail; among them, the solution to the multimode operation in laser devices is given in which the excited level lifetime is long compared with the period of the beating term between two adjacent modes. Although the theory presented is directed towards the steady-state regime in multimode lasers, it may also treat the transient regime of laser operations, or it may be extended to include phenomena where a strong e.m. field, made up of several modes equally spaced in frequency, interacts with a two-level system.
During the last few years, considerable interest has been focused on the phase that waves accumulate when the equations governing the waves vary slowly. The recent flurry of activity was set off by a paper by Michael Berry, where it was found that the adiabatic evolution of energy eigenfunctions in quantum mechanics contains a phase of geometric origin (now known as 'Berry's phase') in addition to the usual dynamical phase derived from Schrödinger's equation. This observation, though basically elementary, seems to be quite profound. Phases with similar mathematical origins have been identified
According to De Broglie's idea of analogy, the relation between quantum mechanics and classical mechanics is similar to that between wave optics and geometric optics, we have given the quantum equation of the gravitational field intensity Eg(r⃗,t)E_g(\vec{r},t)Eg​(r,t) for matter particles, since the gravitational field intensity Eg(r⃗,t)E_g(\vec{r},t)Eg​(r,t) relatives to the particle position distribution function ψ(r⃗,t)ψ(\vec{r},t)ψ(r,t), the quantum equation convert into the Schrodinger equation. In addition, we have further studied the essence and origin of quantum theory, and obtained some new results.
We study the general problem of matter-laser interaction within the framework of nonrelativistic quantum electrodynamics, with particular emphasis on strong laser field effects. Consequently, we formulate a well-defined approximation leading in a straightforward manner toward the conventional semiclassical mode of description. Namely, we arrive naturally to two coupled equations of motion: (i) the Schr\"odinger equation which governs the quantum dynamics of an atomic system driven by classically described radiation field (composed of an incoming laser pulse plus radiation emitted from the atom), and (ii) the classical Maxwell wave equation which describes the emission of radiation from the mentioned atomic source. Employing the formalism of adiabatic Floquet theory, we derive a simple criterion of validity of the just described semiclassical approach. It shows that the semiclassical treatment is justified in most situations. On the other hand, it turns out that the semiclassical approximation breaks down completely in certain special but realistic cases, regardless of the fact that the incoming laser pulse contains a huge number of photons. Under such special circumstances, we anticipate new effects arising due to the quantized nature of the radiation field, to be observable, for example, in harmonic generation spectra. Our considerations are illustrated more explicitly using a simple model of a two-level atom strongly driven by a laser. The quantum dynamics of this model problem is resolved within the framework of quantum electrodynamics while adopting well-defined and physically justifiable approximations. As an outcome, analytic formulas are found serving as a quantitative criterion of (non)applicability of the semiclassical approach and demonstrating the breakdown of semiclassical theory under well-defined conditions. An illustrative numerical calculation is provided.