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外部結合のある光共振器およびレーザーの理論

Kikuo Ujihara

Published Dec 25, 2006• 0 citations
OpenAlex

Abstract

A quantum-mechanical analysis of an optical cavity having output coupling is presented withapplications to the laser theory. The rigorous treatment of the output coupling allows unifiedanalysis of the optical field inside and outside of the cavity. This treatment had lead to a newexpression for the laser line-width that contained the influences of non-uniform oscillating fielddistribution of the real cavity mode as well as the deviation from the cavity field mode due to non-uniformgain saturation.

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A quantum-mechanical analysis of an optical cavity having output coupling is presented withapplications to the laser theory. The rigorous treatment of the output coupling allows unifiedanalysis of the optical field inside and outside of the cavity. This treatment had lead to a newexpression for the laser line-width that contained the influences of non-uniform oscillating fielddistribution of the real cavity mode as well as the deviation from the cavity field mode due to non-uniformgain saturation.
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.
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 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.
This paper is a natural extension of a previous paper [Guedes, Penaforte, and Baseia, Phys. Rev. A 40, 2463 (1989)], where a quantum theory of laser transmission loss was developed to give the solutions for the laser field, inside and outside an optical cavity. Here we calculate in the linear and nonlinear analysis the coherence functions and spectral distributions for the internal and external fields, for a laser operating below and above threshold. The comparison of our results with those found in the current literature shows that with the use of a more realistic model of optical cavity, together with the assumption that the field is initially in a random-phase state, we are able to avoid the use of the ``fictitious'' heat bath of the conventional treatments.
A semiclassical treatment of the radiation loss in a laser, previously investigated by Baseia and Nussenzveig [Opt. Acta 31, 39 (1984)], is extended to the quantum-theory level. The model employs an optical cavity having an output coupling that includes the radiation loss due to the beam extraction through appropriate boundary conditions, instead of the unrealistic loss reservoir of the conventional treatment. This allows one to work entirely within the continuous spectrum and obtain the equation for the radiation field both inside and outside the cavity. Inside the cavity the results agree, in the coherent representation and absence of quantum noise, with those of the previous semiclassical theory. Also, the results of both treatments for the field inside the cavity coincide with those of the conventional theory, since we reinterpret its ``mode amplitude'' as a collective variable. Outside the cavity (an inaccessible region for a standard approach) the explicit solution for the laser field exhibits a natural relationship between the intensities inside and outside the cavity. Further considerations, such as those concerning the role played by the noise term in the field buildup, are also discussed.
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.