Fourier-based analysis and synthesis of moirés in the superposition of geometrically transformed periodic structures

The best method for investigating moiré phenomena in the superposition of periodic layers is based on the Fourier approach. However, superposition moiré effects are not limited to periodic layers, and they also occur between repetitive structures that are obtained by geometric transformations of periodic layers. We present in this paper the basic rules based on the Fourier approach that govern the moiré effects between such repetitive structures. We show how these rules can be used in the analysis of the obtained moirés as well as in the synthesis of moirés with any required intensity profile and geometric layout. In particular, we obtain the interesting result that the geometric layout and the periodic profile of the moiré are completely independent of each other; the geometric layout of the moiré is determined by the geometric layouts of the superposed layers, and the periodic profile of the moiré is determined by the periodic profiles of the superposed layers. The moiré in the superposition of two geometrically transformed periodic layers is a geometric transformation of the moiré formed between the original layers, the geometric transformation being a weighted sum of the geometric transformations of the individual layers. We illustrate our results with several examples, and in particular we show how one may obtain a fully periodic moiré even when the original layers are not necessarily periodic.