Abstract

Two improvements have been made on a recently described 8×8 transfer-matrix formulation of dynamic x-ray diffraction theory for flat crystals with uniform lateral periodicity and nonuniform periodicity normal to the surface.¹ The method is particularly useful for epitaxial structures. One improvement is to break each differential transfer-matrix into a part that depends only on the average complex refractive index, which can be integrated in closed form over any thickness, plus another small part that varies sinusoidally in the oblique and normal directions over some number of cycles. A fast perturbation method can be used to combine these parts. When attenuation of the beam is very great in a thick uniform periodic substrate beneath a different or varying overlayer, we previously truncated the transfer matrix at a level where the wave amplitude is very small but still appreciable and matched boundary conditions with two transmitted and two forward-diffracted plane-polarized waves propagating into a fictitious homogeneous medium. Numerical stability is often greatly enhanced by describing the transmitted and forward-diffracted beams as four orthogonal Bloch waves in the periodic substrate, which is treated as semi-infinite, and using only the transfer matrix down to that substrate to obtain reflection parameters.

© 1988 Optical Society of America