Random dither modulation of a ring-laser gyroscope

In this paper we discuss the possibilities to circumvent the lock-in nonlinearities in a dithered ring-laser gyroscope by random modulation. We review the Cayley matrix formulation of the theory of a dithered ring laser and show that it is particularly suitable to give both an intuitive understanding and a quantitative calculational tool for the effects of randomness on the gyroscope performance. The relationship of our problem to the related physical problems of an overdamped Josephson junction and a random atomic chain is pointed out. The Saxon–Hutner result is utilized to suggest a method to circumvent all locking and trapping phenomena in the laser gyroscope. The effects of amplitude or frequency noise are discussed, but it is found that the introduction of a randomly varying phase is most efficient in eliminating phase trapping. This can be used to provide an even distribution of the Cayley-mapping fixed points around the unit circle that provides the most even phase motion possible. The response of the gyroscope is then found to be optimally linear. All numerical calculations reported are based on the analytically solvable square-wave dither case, but, using the intuitive picture of the process provided by the Cayley transformation method, we argue that the results are qualitatively valid also in the case of a phase-modulated sinusoidal dither signal. The work is related to earlier work on ring lasers and Josephson junctions.