Moiré fringes of higher-order harmonics versus higher-order moiré patterns.

This work presents a very simple and comprehensive approach for classification of the combinational spatial frequencies of the superimposed periodic or quasi-periodic structures. The reciprocal vectors of the structures are used to express their respective spectral components, and a unique reciprocal vectors equation is introduced for presenting the corresponding combinational frequencies. By the aid of the reciprocal vectors equation we classify moiré patterns of combinational frequencies into four classes: the conventional moiré pattern, moiré fringes of higher-order harmonics, higher-order moiré patterns, and pseudo-moiré patterns. The difference between the moiré fringes of higher-order harmonics and higher-order moiré patterns is expressed in the formulas. By some typical examples, conditions for simultaneous formation of moiré patterns of different harmonics of the superimposed gratings are investigated. We show that in the superimposition of two gratings, where at least one has a varying period and another has a non-sinusoidal profile, different moiré patterns are formed over different parts of the superimposed area, where a distinct pair of spatial frequencies of the superimposed structures contributes to the formation of each of the patterns. We use the same procedure in the analysis of simultaneously produced defected moiré patterns in the superimposition of a linear grating and a zone plate, where one or both consist of some topological defects at specific locations and at least one of the gratings has a non-sinusoidal profile. The topological defects of resulting moiré fringes are similar to those appearing in the interference patterns of optical vortices. It is shown that the defect number of resulting moiré fringes depends on the defect numbers and order of frequency harmonics of the gratings. The dependency of the defect number of the moiré fringes and its sign to the defect numbers of the gratings and their contributed frequency harmonics is derived for both additive and subtractive terms of moiré fringes, and the results are verified with several examples based on computational simulations.