Theory of laser noise in the phase locking region

Abstract

We start from quantum mechanical laser equations which were derived in a previous paper for an inhomogeneously broadened laser and which contain in particular the noise sources due to cavity losses, vacuum fluctuations, interaction with phonons and nonlasing photons and the pump. For the example of frequency locking caused by the nonlinear polarization we derive a quantum mechanical Langevin equation for the relative phase angle ~,= 2q/2-q/i--N3, where q/l, ~,//2, 1[I3 are the total phases of three axial modes which would be equally spaced in the unloaded cavity. In the resulting equation = 6 --t7 sin ~ +f (t) (1) the fluctuating forcef(t) is Markoffian and Gaussian, the second moment being given by d (t --t') 2 (F 1 + F 3 + 4/'2), where Fj is the linewidth of the individual unlocked mode, j.Eq. (1) is solved by Fokker-Planck techniques and numerical results are represented for the characteristic function of order 1. Furthermore the mean locking time is represented in a graphical plot. The results are also applicable to two modes which are coupled to each other, e.g. by loss modulation. In this case the second moment off(t) is given by 8(t

Sinusoidal-signal detection by active, noisy oscillators on the brink of self-oscillation

“…Complex Gaussian white noise, which possesses uncorrelated real and imaginary components, can be added to the sinusoidally-forced Hopf normal form [18]. If the amplitude of oscillation is approximately constant, the phase dynamics is equivalent to that of a Brownian particle in an inclined potential and the degree of entrainment can be found analytically [19]. More generally, the response function of isochronous, selfoscillating Hopf oscillators to sinusoidal forcing at or near the natural frequency is constant for weak forcing and declines compressively for stimuli of increasing amplitude, with an exponent for frequency-tuned forcing and weak noise that can be calculated analytically [20].…”

Driven Polymer Translocation Through a Narrow Pore

“…Our approach follows several authors (Peskin et al, 1993;Simon et al, 1992;Park and Sung, 1998a;Park and Sung, 1998b;Sung and Park, 1996) in viewing the transloca-tion process as essentially diffusion in one dimension; we differ, however, in emphasizing the role that interactions with the pore itself play in this diffusion process. On the more microscopic level, we include the effects of these interactions through a tilted washboard potential, similar to models of laser mode locking (Haken et al, 1967) or phase dynamics in Josephson junctions (Ambegokar and Halperin, 1969) (see figures 5 and 6). The periodic modulation of the potential reflects the periodicity of the polynucleotide's sugar-phosphate backbone.…”

Local Exponents of Nonlinear Compression in Periodically Driven Noisy Oscillators

“…(2). We can selfconsistently calculate both d and he ic i invoking the characteristic function for a Brownian particle in an inclined cosine potential [15] and obtain for the sensitivity…”