An investigation of the small slope approximation for scattering from rough surfaces. Part I. Theory

The small slope approximation (SSA) of Voronovich [Soy. Phys. JETP 62, 65-70 (1985)] is a promising method for application to scattering from many natural surfaces. The theory gives a systematic expansion that can be interpreted as a series in generalized surface slope. The SSA series for the T matrix satisfies the appropriate reciprocity condition at each order and reduces to the standard perturbation series for small surface roughness. In this paper we examine in detail the derivation of the SSA for surfaces ubject to the Dirichlet (zero field) boundary condition. A number of points are discussed, including the requirements for determining the series terms. In addition, questions have been raised recently about he SSA: It has been argued that (1) an assumption i the derivation contradicts the exact formulation of the problem and (2) there is an arbitrariness in determining the series terms. These two points are refuted and the assumptions eeded to determine the series terms unambiguously are clarified. The meaning of slope orders in the SSA series expansion are examined and the concept of generalized slope is discussed. A future companion paper (Part II. Numerical studies) will present an investigation of the accuracy of the SSA through comparison with exact results.