Quantum noise in ring-laser gyros. I. Theoretical formulation of problem

This is the first of three papers dealing with the effects of quantum noise on the mean beat frequency and the spectrum of the beat signal produced by a ring-laser gyro. The behavior of a noise-free ring laser is analyzed from a novel point of view, and the characteristic response curve, demonstrating the existence of the well-known locked region, is derived. The noise-free spectrum is calculated in both the locked and unlocked regions, and the existence of higher harmonics in the unlocked region is demonstrated. The general eqution for the phase of the beat signal is then derived from a quantum-noise perspective and the general approach to the problem of calculating the spectrum in the presence of noise is presented from a Fokker-Planck point of view. Approximate analytic expressions for the spectrum are derived in the locked region and well into the unlocked region. In the latter case, the usual laser spectrum is obtained, while in the former case, the spectrum is shown to separate into two parts: a "coherent" part consisting of a function centered at zero frequency which represents the continued effect of locking even in the presence of noise, and a weak background "incoherent" part representing the effects of noise.