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Physical Review A

Theory of excess noise in unstable resonator lasers

Claus Lamprecht, Helmut Ritsch

Published Nov 18, 2002• 4 citations
Publisher

Abstract

We theoretically investigate the quantum dynamics of an unstable resonator laser. Compared to a stable cavity laser of equal gain and loss it exhibits a K-fold enhanced linewidth. This excess noise factor K is a measure of the nonorthogonality of the resonator eigenmodes and amounts to an enlargement of the quantum vacuum fluctuations. Using a quantum treatment starting from first principles based on the nonorthogonal eigenmodes, we put previous theoretical predictions onto a more firm ground. While we find a position-dependent enhancement of the spontaneous emission rate into an empty mode of only $\sqrt{K},$ the constructive quantum interference of the spontaneous emission with a single oscillating mode lets the Petermann excess noise factor K reappear in the phase diffusion (laser linewidth). Hence locally enhanced spontaneous emission as well as noise enhanced by interference (amplified spontaneous emission) play an equal role in the origin of excess noise.

DOI
10.1103/physreva.66.053808

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We theoretically investigate the quantum dynamics of an unstable resonator laser. Compared to a stable cavity laser of equal gain and loss it exhibits a K-fold enhanced linewidth. This excess noise factor K is a measure of the nonorthogonality of the resonator eigenmodes and amounts to an enlargement of the quantum vacuum fluctuations. Using a quantum treatment starting from first principles based on the nonorthogonal eigenmodes, we put previous theoretical predictions onto a more firm ground. While we find a position-dependent enhancement of the spontaneous emission rate into an empty mode of only K,\sqrt{K},K​, the constructive quantum interference of the spontaneous emission with a single oscillating mode lets the Petermann excess noise factor K reappear in the phase diffusion (laser linewidth). Hence locally enhanced spontaneous emission as well as noise enhanced by interference (amplified spontaneous emission) play an equal role in the origin of excess noise.
A single inverted two-level atom is used as a theoretical model for a quantum noise detector to investigate fundamental properties of excess noise in an unstable optical resonator. For a symmetric unstable spherical mirror cavity, we develop an analytic quantum description of the field in terms of a complete set of normalizable biorthogonal quasimodes and adjoint modes. Including the interaction with a single two-level atom leads to a description analogous to the Jaynes-Cummings model with modified coupling constants. One finds a strong position and geometry-dependent atomic decay probability proportional to the square root K\sqrt{K}K​ of the excess noise factor K at the cavity center. Introducing an additional homogeneous gain one recovers the K-fold emission enhancement that has been predicted before for the linewidth of an unstable cavity laser. We find that excess noise may be viewed as a spatial redistribution of the field quantum noise inside the resonator. Taking a position average of the atomic decay rate over the cavity volume leads to a cancellation of the excess noise enhancement.
The results of a quantum-electrodynamical theory of excess spontaneous emission noise in lossy resonators will be presented. The " Petermann K factor" does not enter into the spontaneous emission rate of a single atom in the cavity. The QED theory allows different interpretations of the K factor and we use this fact to justify semiclassical analyses and to provide in one example a simple derivation of K in terms of the amplification of the quantum vacuum field entering the resonator through its mirrors.
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.
A second-quantized theory of the radiation field is used to study the origin of the excess noise observed in gain-guided amplifiers. We find that the reduction of the signal-to-noise ratio is a function of the length of the amplifier, and thus the enhancement of the noise is a propagation effect arising from longitudinally inhomogeneous gain of the noise rather than from an excess of local spontaneous emission. We confirm this conclusion by showing that the microscopic rate of spontaneous emission into a given non-power-orthogonal cavity mode is not enhanced by the Petermann factor. In addition, we illustrate the difficulties associated with photon statistics for this and other open systems by showing that no acceptable family of photon-number operators corresponds to a set of non-power-orthogonal cavity modes.