  We study the statistical properties of the scattering matrix S(q|k) for the
problem of the scattering of light from a randomly rough one-dimensional
surface, defined by the equation $x_3 = \zx$, where the surface profile
function $\zx$ constitutes a zero-mean, stationary, Gaussian random process,
through the effects of $S(q|k)$ on the angular intensity correlation function
C(q,k|q',k'). The existence of both the C^{(1)} and C^{(10)} correlation
functions is consistent with the amplitude of the scattered field obeying
complex Gaussian statistics in the limit of a long surface. We show that the
deviation of the statistics of the scattering matrix from circular Gaussian
statistics and the C^{(10)} correlation function are determined by exactly the
same statistical moment. As the random surface becomes rougher, the amplitude
of the scattered field no longer obeys complex Gaussian statistics but obeys
complex circular Gaussian statistics instead. In this case, the C^{(10)}
correlation function should vanish. This result is confirmed by numerical
simulation calculations.
