This paper deals with the scattering of a TE plane wave by an apodised sinusoidal surface. The analysis starts with the extended Floquet solution, which is a 'Fourier series' with 'Fourier coefficients' given by band-limited Fourier integrals of amplitude functions. An integral equation for the amplitude functions is derived and solved by the small perturbation method to get single and double scattering amplitudes. Then, it is found that the beam shape generated by the single scattering is proportional to the Fourier spectrum of the apodisation function, but that generated by the double scattering is proportional to the spectrum of the squared apodisation. As a result, the single scattering beam and the double scattering beam may have different sidelobe patterns. It is demonstrated that the sidelobes are much reduced if Hanning window or Hamming window is used as an apodisation function.

As a new idea for analyzing the wave scattering and diffraction from a finite periodic surface, this paper proposes the periodic Fourier transform. By the periodic Fourier transform, the scattered wave is transformed into a periodic function which is further expanded into Fourier series. In terms of the inverse transformation, the scattered wave is shown to have an extended Floquet form, which is a 'Fourier series' with 'Fourier coefficients' given by band-limited Fourier integrals of amplitude functions. In case of the TE plane wave incident, an integral equation for the amplitude functions is obtained from the the boundary condition on the finite periodic surface. When the surface corrugation is small, in amplitude, compared with the wavelength, the integral equation is approximately solved by iteration to obtain the scattering cross section. Several properties and examples of the periodic Fourier transform are summarized in Appendix.

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.