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American Journal of Physics

Additional formulas for stimulated atomic transitions

Anthony E.. Siegman

Published Dec 1, 1983• 1 citation
Publisher

Abstract

A number of additional formulas for stimulated atomic transitions are presented to supplement the collection of formulas recently assembled by Hilborn [Am. J. Phys. 50, 982–986 (1982)]. Additional topics covered include resonant atoms imbedded in a dielectric medium, the vector and tensor susceptibility properties of atomic transitions, and the often-discussed ‘‘factor of 3.’’

DOI
10.1119/1.13346

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A number of additional formulas for stimulated atomic transitions are presented to supplement the collection of formulas recently assembled by Hilborn [Am. J. Phys. 50, 982–986 (1982)]. Additional topics covered include resonant atoms imbedded in a dielectric medium, the vector and tensor susceptibility properties of atomic transitions, and the often-discussed ‘‘factor of 3.’’
All optical oscillators and amplifiers can be analyzed as an electromagnetic structure, such as a cavity or a transmission line, which contains an amplifying medium. The operating characteristics of lasers are governed by both the electromagnetic properties of the structures and the properties of the amplifying medium. Stimulated emission and absorption of radiation from energy states are the physical basis of amplification in all laser materials. In order to study lasers, it is necessary to understand the energy levels of the amplifying medium and the stimulated transitions involving them. Quantum mechanical analysis of the energy states and the stimulated emission and absorption is presented in most books on lasers. However, quantum mechanical analyses cover only the analysis of individual atoms. We need to relate the effects of the quantum mechanical interactions to the macroscopic properties of the materials so that properties of the lasers may be analyzed. Therefore, the macroscopic susceptibility of materials related to stimulated emission and absorption is the focus of discussion in this chapter. It is not the purpose of this book to teach quantum mechanics. There are already many excellent books on this topic. The readers are assumed to have a fundamental knowledge of quantum mechanics. However, it is necessary first to review some of the major steps in order to understand precisely the notation used and the meaning of the results. This is presented in Section 5.1.
The problem of electromagnetic-field quantization in time-dependent nonuniform linear nondispersive media is investigated. The explicit formulas for the number of photons generated from the initial vacuum state due to the change in time of dielectric permeability of the medium are obtained in the case when the spatial and temporal dependences are factorized. The concrete time dependences include adiabatic and sudden changes of permeability, the parametric resonance at twice the eigenfrequency of the mode, Epstein's symmetric and transition profiles, ``temporal Fabry-P\'erot resonator,'' and some others. The upper and lower bounds for the squeezing and correlation coefficients of the field in the final state are given in terms of the reflection coefficient from an equivilent potential barrier or the number of created quanta. The problem of impulse propagation in a spatially uniform but time-dependent dielectric medium is discussed.