Classical Noise. VI. Noise in Self-Sustained Oscillators near Threshold

Because of the relative narrowness of the threshold region, a general model for spectrally pure self-sustained oscillators (both classical and quantum, including gas lasers) can be reduced, in the threshold region, to a rotating-wave Van der Pol (RWVP) oscillator. By a scaling transformation, we reduce to the normalized RWVP oscillator which contains only one dimensionless parameter, a net pump rate $p$, which determines the operating point. The power spectra of phase and amplitude fluctuations and of amplitude (intensity) fluctuations in the normalized RWVP oscillator near threshold are calculated "exactly" by numerical Fokker-Planck methods. Using the appropriate scaling transformation, our results yield these power spectra for any oscillator of this general type. In particular, for gas lasers our results yield the one-sided Fourier transform of $〈b^{†}\mathrm{}\left(t\right)b\left(0\right)\mathrm{}〉$ (the spectrum) and of $〈b^{†}\mathrm{}\left(0\right)\mathrm{}{b}^{†}\mathrm{}\left(t\right)b\left(t\right)b\left(0\right)\mathrm{}〉-{〈b^{†}b〉}^2$ (the intensity spectrum), where $b^{†}$ and $b$ are the creation and destruction operators for the radiation field. Except for intensity fluctuations just above threshold, the power spectra were found to be nearly Lorentzian, with half-widths at half power approximately equal to the lowest nonzero temporal eigenvalue of the Fokker-Planck equation. For intensity fluctuations above threshold, the second-lowest nonzero eigenvalue was found to yield a significant contribution to the power spectrum as well as the lowest nonzero eigenvalue. These two eigenvalues become nearly degenerate for operation well above threshold. Thus the intensity fluctuation spectrum is Lorentzian below and well above threshold, but more complex in the threshold region.