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Physical Review A

Quantum theory of laser transmission loss

I. Guedes, J.C. Penaforte, B. Baseia

Published Sep 1, 1989• 11 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.40.2463

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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.
Conventional laser theory deals with discrete ‘cavity modes’ and introduces artificial mechanisms to simulate radiation field losses caused by beam extraction. A more realistic model of a laser with a transmitting window, previously investigated by Lang, Scully and Lamb, is reanalysed semiclassically. In the previous work, the continuous radiation field spectrum was treated as the limit of a discrete but very dense spectrum of modes, and the narrowness of the laser linewidth was attributed to the locking together of these modes. Working entirely within the continuous spectrum, we find that the treatment is considerably simplified, leading to an explicit solution for single-quasimode operation within the linear approximation which exhibits the role of the excitation conditions in the build-up of the laser field. As was pointed out by Lang and Scully, the transmission loss plays the role of an effective noise source, satisfying the fluctuation-dissipation theorem. However, the continuum treatment does not support the mode-locking mechanism, but rather the usual explanation of the laser linewidth in terms of gain narrowing. The nonlinear analysis agrees with the main conclusion of Lang, Scully and Lamb that the ‘mode amplitude’ of the conventional theory should be interpreted as a collective variable that satisfies the well-known rotating-wave Van der Pol oscillator equation. The domain of validity of this result and of the laser quasimode concept depends on an approximation, analogous to the Weisskopf-Wigner approximation, which is well satisfied for the usual case of very long-lived quasimodes.
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 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 recent result of a previous paper describing the field loss in an optical cavity in a natural way is applied to laser theory. The field loss, due to the transmitted beam light, and the field gain, now attributed to the presence of active atoms but previously attributed to two noninteracting mechanisms, are correctly added to provide the equation of motion for the density operator of the radiation field. The replacement of individual operators, which appear in the conventional treatment, with collective operators is a conceptual difference that emerges from the present approach.
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 inclusion of damping in the quantized radiation field in pure and mixed states is accomplished in a natural way, without using traditional loss reservoirs, through the application of appropriate boundary conditions in a simple model of an optical cavity coupled to the outside world. By using collective operators (which are suitable weighted superpositions of the usual creation and annihilation operators for modes of the continuous spectrum, extended over each Fox–Li band) and convenient initial conditions, we are able to describe the radiation loss from an optical cavity to the outside world. The physical situations corresponding to initial conditions leading to the exponential damping are also investigated.