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Physical Review A

Quantum theory of a one-dimensional laser with output coupling. II. Phase noise above threshold

Kikuo Ujihara

Published Jun 1, 1984• 13 citations
Publisher

Abstract

Phase noise of a one-dimensional laser with output coupling that is oscillating well above threshold is analyzed on the basis of a multimode formulation of the field. The phase diffusion, both in time and space, is calculated for the output beam taking into account the lowest-order moments of the thermal and the quantum noise forces. Thermal noise is introduced automatically without assuming any absorbing atoms or heatbaths, and is interpreted as the amplified, ambient thermal radiation that penetrates the laser cavity. Quantum noise is treated by the quantum Langevin method. The resulting linewidth formula differs from that of a conventional theory by factors that are determined by the degree of output coupling and by the nonuniform distribution of the saturated gain in the cavity.

DOI
10.1103/physreva.29.3253

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Phase noise of a one-dimensional laser with output coupling that is oscillating well above threshold is analyzed on the basis of a multimode formulation of the field. The phase diffusion, both in time and space, is calculated for the output beam taking into account the lowest-order moments of the thermal and the quantum noise forces. Thermal noise is introduced automatically without assuming any absorbing atoms or heatbaths, and is interpreted as the amplified, ambient thermal radiation that penetrates the laser cavity. Quantum noise is treated by the quantum Langevin method. The resulting linewidth formula differs from that of a conventional theory by factors that are determined by the degree of output coupling and by the nonuniform distribution of the saturated gain in the cavity.
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.
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.
The problem of the quantum-limited or intrinsic linewidth of a good-cavity laser is revisited. Starting from the Scully-Lamb master equation, we present a fully analytical treatment to determine the correlation function and the spectrum of the cavity field at steady state. For this purpose, we develop an analytical approximation method that implicitly incorporates the microscopic fluctuations of both the phase and intensity of the field, and, in addition, takes full account of the saturation of the nonlinear gain. Our main result is a simple formula for the quantum-limited linewidth that is valid from near to far above threshold and also includes the presence of thermal photons. Close to the threshold, the linewidth is twice as large as predicted by the standard phase-diffusion treatment neglecting intensity fluctuations, and even 50% above threshold the increase is still considerable. In general, quantum fluctuations of the intensity are present and continue to influence the linewidth as long as the photon-number distribution is not strictly Poissonian. This inherent relationship is displayed by a formula relating the linewidth and the Mandel Q parameter. More than 100% above treshold the linewidth is found to be smaller than predicted by the standard treatment, since the simple phase-diffusion model increasingly overestimates the rate of phase fluctuations by neglecting gain saturation. In the limit of a very large mean photon number the expected perfectly coherent classical field is obtained.
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.
1 A one-dimensional optical cavity with output coupling-classical analysis- 2 A one-dimensional optical cavity with output coupling-quantum analysis- 3 A one-dimensional quasimode laser: General formulation 4 A one-dimensional quasimode laser-Semiclassical and quantum analysis- 5 A one-dimensional laser with output coupling -Derivation of the laser equation of motion- 6 A one-dimensional laser with output coupling-Contour integral method- 7 A one-dimensional laser with output coupling-Semiclassical linear gain analysis 8 A one-dimensional laser with output coupling-Semiclassical nonlinear gain analysis 9 A one-dimensional laser with output coupling-Quantum linear gain analysis- 10 A one-dimensional laser with output coupling -Quantum nonlinear gain analysis- 11 Analysis of a one-dimensional laser with two-side output coupling -The propagation method- 12 A one-dimensional laser with output coupling -Summary and interpretation of the results- 13 Spontaneous emission in a one-dimensional optical cavity with output coupling 14 Theory of excess noise 15 Quantum theory of the output coupling of an optical cavity Appendices
The theory of the laser linewidth is formulated to account for arbitrarily large output couplings and spatial hole burning. We show explicitly that the linewidth can be interpreted in terms of either spontaneous-emission noise or the amplification of vacuum field modes leaking into the cavity, depending on the ordering of operators in the correlation function determining the laser spectrum. This allows us to derive the Petermann K factor associated with ``excess spontaneous-emission noise'' in a physically transparent and mathematically simple way, without the need to introduce adjoint modes of the resonator. It also allows us to straightforwardly include spatial-hole-burning effects, which are found to increase the K factor and the linewidth in high-gain systems appreciably.