This paper deals with the scattering of a plane wave from a two-dimensional random thin film. For a Gaussian random disorder, a first order solution is derived explicitly by a probabilistic method. It is then found that ripples appear in angular distributions of the incoherent scattering. Furthermore, the incoherent scattering is enhanced in the directions of backscattering and specular reflection. Physical processes that yield such an enhanced scattering are discussed. Numerical examples of the coherent and incoherent scattering are illustrated in figures.

This paper deals with a probabilistic formulation of the diffraction and scattering of a plane wave from a periodic surface randomly deformed by a binary sequence. The scattered wave is shown to have a stochastic Floquet's form, that is a product of a periodic stationary random function and an exponential phase factor. Such a periodic stationary random function is then represented in terms of a harmonic series representation similar to Fourier series, where `Fourier coefficients' are mutually correlated stationary processes rather than constants. The mutually correlated stationary processes are written by binary orthogonal functionals with unknown binary kernels. When the surface deformations are small compared with wavelength, an approximate solution is obtained for low-order binary kernels, from which the scattering cross section, coherently diffracted power and the optical theorem are numerically calculated and are illustrated in figures.

This paper deals with an orthogonal functional expansion of a non-linear stochastic functional of a stationary binary sequence taking 1 with unequal probability. Several mathematical formulas, such as multivariate orthogonal polynomials, recurrence formula and generating function, are given in explicit form. A formula of an orthogonal functional expansion for a stochastic functional is presented; the completeness of expansion is discussed in Appendix.

This paper deals with an orthogonal functional expansion of a non-linear stochastic functional of a stationary binary sequence taking 1 with equal probability. Several mathematical formulas, such as multi-variate orthogonal polynomials, recurrence formula and generating function, are given in explicit form. A simple example of orthogonal functional expansion and stationary random seqence generated by the stationary binary sequence are discussed.