© 1972 by Oxford University Press
ON MAXWELL'S EQUATIONS IN THREE-DIMENSIONAL ANISOTROPIC PERIODIC MEDIA: TENSOR FORMULATION OF THE PROBLEM AND THE N-BEAM APPROXIMATION
( Department of Mathematics, The University Dundee, Scotland )
The equations for diffraction of electromagnetic waves in an anisotropic periodic structure as presented in this paper have been developed for detailed investigation of problems connected with X-ray diffraction in perfect single crystals, particularly dispersion surfaces and polarization directions.
As in some earlier literature on the dynamical theory of X-ray diffraction, use is made of Fourier expansions for certain crystal properties and of plane wave approximations for waves travelling through the medium. However, a special feature of the present approach is that a tensor formulation is used, as opposed to the well-known vector notation of the dynamical theory of X-ray diffraction. Tensor formulation recommends itself by its simplicity and becomes feasible after a suitable generalization of the concept of direct and reciprocal lattice is introduced, but is nevertheless new in this context. As in previous papers some properties of dispersion surfaces are investigated. However, the formalism presented here, together with a few theorems of linear algebra, allows a generally valid discussion without restrictive geometrical approximations.
Examples are given for special N-beam cases in order to illustrate the advantages of this formulation and to show how this very general method can be applied usefully to actual problems of X-ray diffraction.