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The Review of Laser Engineering

Quantum Theory of Laser with a Fabry-Perot Resonator I

Masahiko HIROOKA, Sigenobu Sunakawa

Published Jan 1, 1986• 0 citations
Publisher

Abstract

The semi-classical theory of laser has been reformulated on the basis of the Hartree average-field method. It is assumed that the light field is enclosed in a Fabry-Perot resonator and forms a standing wave in the cavity. The frequency pulling effect also has been taken into consideration.
DOI
10.2184/lsj.14.613

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We give a general equation which allows the description of the evolution of the laser properties when the gain is varied from below to above threshold. The method is general in the context of the semi-classical theory of lasers. It is numerically illustrated in the simplest case of a single mode laser with a source term characterized by a very narrow spectrum. The transformation of the Airy line shape of the passive Fabry-Perot cavity into the laser line shape is described. A classical linewidth which complements the usual quantum limit can thus be introduced, together with its control parameters.
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The velocity-dependent spontaneous emission of a two-level atom in a Fabry-P\'erot cavity in the strong-coupling regime and the deflection of a beam of two-level atoms in a classical standing wave inside a cavity are discussed using a renormalization group approach. In this way we are able to renormalize the leading-order solutions for both problems through calculations of the corrections at first order. In fact, the first-order terms are not bounded for large times and no sense can be attached to this higher-order correction unless small times are considered. These are like the divergences of quantum field theory. To make them harmless, the condition for the Raman-Nath regime is recovered. The renormalization group methods permit one to eliminate those divergences generating a renormalized leading-order wave function without any condition of applicability. For the spontaneous emission of a two-level atom in a Fabry-P\'erot cavity in the strong regime, using a Hamiltonian without losses, it is shown that the unperturbed levels are shifted by a term proportional to the zeroth-order Bessel function with an argument yielded by the ratio of the Rabi frequency and the Doppler-shifted frequency of the mode of the cavity. When the detuning is zero, the correction to the leading-order wave function is not present and known results are recovered. For the beam of two-level atoms in a classical standing wave, when the detuning is much larger than the Rabi frequency, it is shown that the renormalization group equation, which gives the correction for the renormalized leading-order wave function, is a time-dependent Schr\"odinger equation for a free particle that induces a spreading of the initial Gaussian wave packet.
We establish that the energy transfer between transverse modes in nonlinear Fabry-Perot resonators with curved mirrors, having degenerate or nearly degenerate modes, is governed by selection rules. Although these rules are derived in a perturbative limit, assuming a thin medium centered in the resonator, we have found numerically and experimentally that they still account for the otherwise unexpected enhancement or suppression of certain modes in more realistic cases. The existence of selection rules constitutes a fundamental difference between standing wave and traveling wave cavities and therefore prevents, in principle, the possibility of describing experiments on transverse effects performed in Fabry-Perot cavities with (simpler) ring resonator models.