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 moire patterns of combinational frequencies into four classes: the conventional moire pattern, moire fringes of higher-order harmonics, higher-order moire patterns, and pseudo-moire patterns. The difference between the moire fringes of higher-order harmonics and higher-order moire patterns is expressed in the formulas. By some typical examples, conditions for simultaneous formation of moire 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 moire 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 moire 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 moire fringes are similar to those appearing in the interference patterns of optical vortices. It is shown that the defect number of resulting moire fringes depends on the defect numbers and order of frequency harmonics of the gratings. The dependency of the defect number of the moire 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 moire fringes, and the results are verified with several examples based on computational simulations.