RF spectrum of a carrier with a random phase modulation of arbitrary slope

In this paper we review the theory concerning the radiofrequency (RF) spectrum of a carrier phase modulated by a stochastic process with spectral density of the type vertical bar omega vertical bar(-alpha) and 0 <= alpha < 3. This problem is of particular interest in the fields of atomic frequency standards and, in general, in high resolution spectroscopy. We also extend the theory to non-integer slopes (alpha real) and examine the role of the low-pass filter shape in the RF lineshape, pointing out the peculiar behaviour of the Lorentzian filter. The singular features of the RF spectrum in the case of flicker phase noise are examined as well as the lineshape in the high- and low-noise limits. Analytical expressions for the RF spectrum are also given in terms of special functions for the white phase, flicker phase and white frequency noises.