Dynamical theory of moire fringe patterns

The dynamical theory of moiré patterns is developed on the basis of the two-beam approximation. The method used is the column approximation introduced by Howie and Whelan. First, the geometry of the moiré patterns is discussed. Then the intensity profile of the fringes is calculated. Assuming that the anomalous absorption is negligible, the effect of a step on the surface and of tilting the crystal is examined. During tilting the fringes will be displaced and the contrast between dark and bright fringes will die away, possibly in an observable fluctuating way. The kinematical approximation is also discussed. It is suggested that under favourable conditions the crystal thicknesses may in principle be determined from the fluctuations of the contrast during the tilting of the crystal. The anomalous absorption effects have then been considered; they do not influence the geometry of the pattern, but introduce corrections for the fringe profile if the considered crystal is of intermediate thickness. For thick crystals the anomalous absorption effects become predominant. It is shown that the results of the column approximation can be obtained and generalized in a simple way. The transmitted and diffracted waves are described as arising by the interference of two waves with slightly different wave vectors. The column approximation reveals itself as a good approximation for sandwiches of crystals of the same material.