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Physical Review A

Quantum-statistical properties of a laser with output coupling

I. Guedes, B. Baseia

Published Dec 1, 1990• 8 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.42.6858

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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.

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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.
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.
Subthreshold operation of a one-dimensional laser having output coupling is analyzed quantum mechanically under the assumption of constant population inversion. The conventional quantum Langevin theory of the laser is improved in that the spatial behavior of the field is treated exactly by use of a multimode formulation of the field of the universe. This formulation allows the thermal noise to be derived automatically without introducing any artificial damping mechanism for the field; this noise comes from the statistical nature of the initial state of the field. The analysis shows that the cavity quasimodes are excited, that the linewidth formula differs from that of a conventional theory by a factor determined by the field distribution excited in the cavity, and that, outside the cavity, the quantum-mechanical coherence function is propagated with the velocity of light.
In a previous paper we considered the linear problem of a one-dimensional laser having output coupling. In this paper we derive the nonlinear equation for the same model of the laser and solve it for steady state. The output coupling is exactly taken into account and the field is treated as having a continuous spectrum composed of the normal modes of the universe. Damping terms for the atomic polarization and for the population inversion are introduced phenomenologically. The fluctuating forces due to these damping terms and to the initial field are ignored. The resultant field inside the cavity consists of two oppositely travelling waves whose amplitudes are functions of the distance from the coupling surface. Outside the cavity we have a monochromatic, outgoing wave whose amplitude does not change with distance. The threshold inversion and the oscillation frequency are the same as those derived in the linear theory.
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.