On born approximation for the scattering by rough surfaces

In many modern industrial processes, diffractive optical elements employ light with small wavelengths. The current technological progress is based on the steady decrease in wavelengths (e.g. EUV light). For such wavelengths, the effect of random surface roughness of gratings has an increasingly high impact on the behaviour of the scattered wave. Consequently, the roughness should be included into the modelling of the scattering and, maybe, into the numerical solution process of inverse problems. Since the solution for the full Maxwell system for rough surfaces is extremely difficult, many authors employ simplified models. One way to do so is proposed by Stearns [1] and is based on the Born approximation of the Maxwell equations. We will discuss mathematically strict assumptions which allow the use of the manipulations in Stearns´ approach. Under these assumptions, we give a rigorous definition of an approximate solution and derive the corresponding formulas for the reflected far field.