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Physical Review

Poles of Green's Functions for an Atomic System in a Multimode Cavity

John H. Williamson

Published Sep 9, 1966• 1 citation
Publisher

Abstract

Schwinger's action principle has been applied by Kemeny to a set of harmonic oscillators representing the cavity modes of a laser. It is shown here that the radiation emitted into each mode has two terms, a narrow line near the unloaded mode frequency and a modified spontaneous emission line.
DOI
10.1103/physrev.149.342

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Schwinger has applied his generalized quantum action principle and the method of thermodynamic Green's functions to an harmonic oscillator. He has shown how this technique describes the buildup of cavity oscillations in a simple model of the maser. This paper applies Schwinger's technique to multimode cavities. The energy, autocorrelation (or coherence), and spectral distribution of individual cavity modes in the steady state, their rate of buildup and their response to external signals have been calculated. Conditions for the buildup and for the steady state of laser action are stated. It has been shown that the steady-state laser radiation in each cavity mode can be described in terms of spontaneous and induced emissions, the latter one containing a coherent and an incoherent part.
Previous theories of laser radiation have described the electromagnetic field in terms of a discrete set of quasimodes (Fox and Li "modes") of the laser cavity. Each quasimode was assumed to have a finite quality factor Q. In steady state (when gain due to lasing atoms cancels the losses) and in the absence of noise sources, the semiclassical theory predicts a $\ensuremath{\delta}$-function line shape. In the present semiclassical analysis (fluctuations are ignored), this theory is generalized for a maser with a cavity having a semitransparent wall as one of the mirrors so that there are now many modes of the universe corresponding to each Fox-Li-type quasimode. It is demonstrated that the normal modes of the universe associated with a single Fox-Li "mode" may, under proper conditions, lock together and the $\ensuremath{\delta}$-function laser line shape may be regained. Specifically, we investigate the conditions under which the coupled multimode equations associated with laser oscillation can be reduced to an equation of the form found in the usual single-quasimode theory.
The spectrum of spontaneously emitted photons from an atom interacting with a single cavity mode depends on the relative sizes of the mode coupling constant and the cavity loss rate. If the coupling constant is not too large, perturbation theory can he used to show that the atomic linewidth is either increased (enhanced spontaneous emission) or decreased (inhibited spontaneous emission) by the presence of the cavity. When the coupling constant is so large that perturbation theory cannot be used, the spontaneous- emission spectrum forms a doublet, with peaks located at the normal-mode frequencies for the coupled-dipole and cavity-mode oscillators (the so-called "vacuum" Rabi splitting).
The value of the Einstein BBB coefficient for stimulated emission for a collection of atoms interacting with a radiation field is shown to depend upon the velocity distribution of the excited atoms and also upon the variation of field intensity with frequency. An emission line narrowing phenomenon for radiating matter enclosed in a nonresonant cavity with reflecting walls is predicted. It is shown that a loss mechanism must be introduced if significant line narrowing is to result from a single pass of radiation through an amplifying medium. An additional contributing effect to the mode pushing and mode pulling phenomena in a gas laser is demonstrated. A method of setting a gas laser to a predetermined frequency is described.