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Physical Review A

Quantum theory of a one-dimensional optical cavity with output coupling. III. Cavity quasimodes as the field eigenvalue

Kikuo Ujihara

Published Sep 1, 1979• 9 citations
Publisher

Abstract

The quantum theory is developed for an arbitrarily stratified one-dimensional optical cavity which transmits light at one of its end The field is expanded in terms of the modes of the universe and each universal mode is quantized separately by means of the usual method. A class of coherent states which yields oscillation wave shapes of the cavity quasimodes as the field eigenvalue is studied. A state of this class gives an eigenvalue for the annihilation operator of a universal mode which is an analytic function of the mode frequency except for a mode-dependent phase factor. The mathematical meaning of the phase factor is studied. A new scheme of field quantization is developed which absorbs the above phase factor and yields simpler forms of the eigenvalue equations for the above class of states. For a state of the class, the local intensity, its fluctuation, and the various correlations of the field decay with time. For the local field inside or outside of an optical structure which is coupled to the free space, the almost definite phase relationship among the universal modes plays a decisive role.

DOI
10.1103/physreva.20.1096

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The quantum theory is developed for an arbitrarily stratified one-dimensional optical cavity which transmits light at one of its end The field is expanded in terms of the modes of the universe and each universal mode is quantized separately by means of the usual method. A class of coherent states which yields oscillation wave shapes of the cavity quasimodes as the field eigenvalue is studied. A state of this class gives an eigenvalue for the annihilation operator of a universal mode which is an analytic function of the mode frequency except for a mode-dependent phase factor. The mathematical meaning of the phase factor is studied. A new scheme of field quantization is developed which absorbs the above phase factor and yields simpler forms of the eigenvalue equations for the above class of states. For a state of the class, the local intensity, its fluctuation, and the various correlations of the field decay with time. For the local field inside or outside of an optical structure which is coupled to the free space, the almost definite phase relationship among the universal modes plays a decisive role.
In this chapter, a one-dimensional optical cavity with output coupling is con-sidered. The optical cavity has transmission loss at one or both of the end surfaces. The classical, natural cavity mode is defined, and decaying or growing mode functions are derived using the cavity boundary conditions. A series of resonant modes appears. But these modes are not orthogonal to each other and are not suitable for quantum-mechanical analysis of the optical field inside or outside of the cavity. Hypothetical boundaries are added at infinity in order to obtain orthogonal wave mode functions that satisfy the cavity and infinity boundary conditions. These new mode functions are suitable for field quantization, where each mode is quantized separately and the electric field of an optical wave is made up of contributions from each mode. Some results of quantization are described in the next chapter. Chapter 3 deals with the usual quasimode model: a perfect cavity with distributed internal loss or with a fictitious loss reservoir.
The quantum theory of a one-dimensional optical cavity is developed with emphasis on the presence of output coupling. First the resonant mode is defined and calculated classically. Then the field is decomposed into a sequence of modes by introducing an imaginary boundary at a large distance. The mode functions are proved to be orthogonal with respect to an integration with the dielectric constant as a weighting factor. The Hamiltonian of the field is shown to be equivalent to that of a collection of independent harmonic oscillators, the mass of which is a function of the frequency of oscillation. This equivalent of mass appears on normalizing the mode functions and proves to carry information on the structure of the cavity. Quantization of the field is carried out, and the commutation relation for the electric fields inside and outside the cavity is derived. The commutator is composed of an infinite set of derivatives of $\ensuremath{\delta}$ functions, which discloses the effect of the presence of output coupling and that of the small size of the cavity on the radiation field. Also, the commutator is shown to have some properties common with the classical resonant mode, i.e., the property of exponential decay and its rate.
The spatial behavior of the field of a one-dimensional laser having a partial transmission in its cavity wall is analysed. The partial transmission allows coupling between the field inside the laser cavity and that of outer space (output coupling). A formulation is developed in which the field is described in terms of many normal modes of the "universe", i.e. , the whole space incluing the cavity. The transient building-up of the field is considered under the assumption that the atoms have a constant population inversion. Noise is ignored. The equations for the modes of the universe are converted into a single equation for the electric field which displays very clearly the space-time structure of the laser action. Cavity resonant modes (lossy quasi-modes) are shown to be excited and to produce outgoing waves outside the cavity. The threshold population inversion and the oscillation frequency at the threshold are derived.
Chaotic fields in a general class of one-dimensional optical cavity having output coupling is considered. The fluctuation-dissipation theorem is derived assuming that the cavity materials are lossless. The cavity admittance probed by a prescribed current source for an arbitrarily stratified optical cavity is calculated and using the explicit expression for the admittance the “dissipation” is proved to be equal to the radiation loss. The form of the fluctuation-dissipation theorem for a restricted frequency region around a cavity mode is discussed with particular attention to the cavity mode degeneracy.
A simple model of a one-dimensional optical cavity having output coupling is used to develop a fully quantized linear laser theory. The present model and procedure, which work directly within the continuous spectrum, allow us to find a simple treatment that leads to explicit solutions for the field inside and outside the laser cavity. The threshold condition for laser light onset is obtained by assuming the field operating in a single collective mode and the usual adiabatic hypothesis. A development that removes this later restriction is also discussed and outlined.
The purpose of this paper is to compare two unitary, deterministic approaches to the study of the interaction of quantized fields with atomic systems in optical cavities. In particular it is shown that the ``modes of the universe'' approach, in which each mode has both an intracavity and an outside part, is formally equivalent, in an appropriate limit, to the better known ``input-output'' formalism in its unitary (i.e., Hermitian-Hamiltonian based) form; the latter may be called a ``quasimode'' theory, since it treats the field inside the cavity as a separate degree of freedom. Differences that arise between the two approaches for numerical calculations involving a finite set of discrete modes are also pointed out. The formalism is illustrated with an in-depth discussion of the explicit solution for a single-photon pulse, of arbitrary shape and detuning, incident on a cavity containing a two-level atom.