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Physical Review A

Quantum theory of a one-dimensional optical cavity with output coupling. Field quantization

Kikuo Ujihara

Published Jul 1, 1975• 55 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.12.148

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IEEE Journal of Quantum Electronics•Jul 1, 1988•9 citations
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 \ensuremathδ\ensuremath{\delta}\ensuremathδ 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.
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 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.
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.
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.
A macroscopic, canonical quantization of the EM field and radiating atom system in quantum optics and cavity QED involving classical, linear optical devices, based on expanding the vector potential in terms of quasi mode functions is presented. The quasi mode functions approximate the true mode functions for the device, and are obtained by solving the Helmholtz equation for an idealized spatially dependent electric permittivity function describing the device. The Hamiltonian for the EM field and radiating atom system is obtained in multipolar form and the quantum EM field is found to be equivalent to a set of quantum harmonic oscillators, one oscillator per quasi mode. However, unlike true mode theory where the quantum harmonic oscillators are uncoupled, in the quasi mode theory they are coupled and photon exchange processes can occur. Explicit expressions for the coupling constants are obtained. The interaction energy between the radiative atoms and the quantum EM field depends on the amplitudes of the quasi mode functions at the positions of the radiating atoms, similar to that for the true mode approach. The simpler forms for the quasi mode functions enable the atom-field interaction energy to be written in a form in which the atoms are only coupled to certain types of modes—for example cavity quasi modes, which are large inside the optical cavity. In such cases the escape of energy from excited atoms in the cavity can be pictured in quasi mode theory as a two step process—the atom de-excites and creates a photon in a cavity quasi mode, the photon in the cavity quasi mode is then lost and appears as a photon in an external quasi mode. In this process the first step occurs via the atom-cavity quasi mode interaction, the second through coupling between cavity and external quasi modes. This may be contrasted with the true mode approach, where the excited atom loses its energy and the photon is created in one of the true modes. As all true modes have non-zero amplitudes outside as well as inside the cavity, the escape of energy from excited atoms in the cavity is seen as a one step process. An application of the quasi mode theory to the quantum theory of the beam splitter is outlined. The unitary operator used to describe this device is a scattering operator, relating initial and long time values of annihilation, creation operators for pairs of incident and reflected modes, interpreted here as quasi modes.
For pt.IV see J. Phys. Soc. Japan, vol.47, p.217-224 (1982). Spontaneous emission in a lossy one-dimensional cavity is analyzed under the perturbation approximation and the Wigner-Weisskopf approximation. The cavity loss due to output coupling is treated quantum mechanically by the use of a set of continuously distributed radiation modes. The enhancement and reduction factors in the spontaneous emission probability due to the presence of the cavity structure are derived. Interesting transients in spontaneous emission during a time of the order of the cavity decay time are predicted, which leads to an interpretation of the enhancement or the reduction in spontaneous emission in terms of feedback and self-interference effects. The detection of the emitted photon by a detector atom outside the cavity is also discussed.>