On stationarity and ergodicity of the field scattered by Gaussian random slightly rough surfaces

We study the field scattered by a dielectric slightly rough surface illuminated by a monochromatic plane wave. The surface is represented by a random function with a Gaussian probability density with zero mean value. The surface is assumed to be a stochastic stationary and ergodic process. In the mainframe of the small perturbation method, we derive the analytical expressions of the statistical and spatial variances of the field and demonstrate that the field is neither stationary nor ergodic at the second order.