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Physical Review

Green's Function Theory of Multimode Cavities

Gábor Kemény

Published Jan 6, 1964• 12 citations
Publisher

Abstract

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.

DOI
10.1103/physrev.133.a69

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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.
The spectrum of power radiated by a solid-state optical maser in steady-state operation is obtained by considering each atomic system to be a source of randomly fluctuating dipole moment which drives every mode of the cavity. The nonlinear behavior of the collection of atomic systems is treated in such a way that a detailed examination of the distribution of power in the various modes is possible. A number of examples are considered which show how critically dependent on the relative loss rates of the various modes are the characteristics of the output from a multimode oscillator. An interesting result of this calculation is the appearance in some cases of an abrupt transition as the pumping power is increased: above this point a large fraction of energy goes into the mode which is resonant with the atomic transition and which has the lowest loss rate. Another case is studied which does not have this feature, but which could be useful in analyzing the behavior of solid-state optical masers which have substantial scattering due to crystal imperfections.
We extend the investigation of the long-time behavior of a model consisting of N two-level atoms interacting with a single electromagnetic cavity mode to include interaction with many cavity modes. We show that, as a consequence of the coupling between radiation modes produced by spatial density variations of the population inversion, there is no strictly stationary state possible for multimode behavior. However we obtain a stationary state by neglecting the rapidly oscillating terms. The steady-state population inversion is then a solution of an eigenvalue problem. The eigenvalue determinate is a function of the number of modes and the coupling between modes in addition to the usual dependence on frequencies and relaxation times. We explicitly solve for the unique eigenvalue in special cases. The corresponding eigenvector gives the steady-state mode intensity ratios. The absolute values of the steady-state intensities are determined by the energy conservation equation generalized to include pumping and dissipation. We also calculate the steady-state frequency shifts for each mode. The mode frequency shifts are practically independent of each other and have the same functional form as the single-mode frequency shifts.
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.
Simple interference experiments identify correlation functions which describe the response of maser amplifiers and the output of maser oscillators. General operator techniques are used to evaluate these functions in cavity maser systems for which the coupling of the maser electromagnetic field to external energy sources or sinks is mediated by systems whose short-term response depends primarily upon average or macroscopic system properties. The dielectric theory which results includes only those short-term non-linearities associated with the external pumping fields. In the long term both the external and the cavity fields can modify the susceptibility functions. The techniques utilized permit a detailed analysis of approximations and suggest generalizations for increased precision. As one important result of this type the theory indicates that macroscopic rate equations are valid if modulation frequencies are much less than the width of the narrowest coupling system spectral line.
The theory of the previous articles of this series is extended to allow analysis of the spectral distribution of induced and spontaneous emission in a cavity and to study the steady-state condition when a molecular beam traverses the cavity.It is shown that the induced emission energy, when averaged over a reasonable time, depends only on the energy spectrum of the driving field. Expressions are obtained for the spectral distribution of induced, spontaneous, and thermally induced emission. These results are then applied to a molecular beam analysis and conditions for a steady state in the cavity are obtained.The amplification by the molecular beam of the driving field (signal) and of the thermal and spontaneous emission energies (noise) is studied. Signal and noise energies are compared and the dependence of the results on both the bandwidth of the signal and the bandwidth of detection is examined.