Theoretical Interpretation of Moiré Patterns

TLDR

The paper presents a general mathematical technique for solving moiré patterns arising from overlapping figures. Moiré patterns can reveal functional variations, refractive index gradients, and lens distortions, and their mathematical solutions often mirror those in physical problems. The method is applied to various figure combinations—parallel, radial, concentric—and analyzes three positional arrangements relative to the sample. The technique reveals spacing variation functions, measures refractive index gradients, offers a compact arrangement yielding only the gradient, and is illustrated with examples from optics, hydrodynamics, and electrostatics.

Abstract

A general mathematical technique for the solution of moiré patterns produced by the overlapping of two figures is presented. The technique is applied to combinations of figures involving parallel lines, radial lines, and concentric circles including those in which the spacing is variable. When an equispaced parallel line figure is overlapped on a parallel line figure whose spacing is variable, the resultant moiré pattern reveals the functional form of the variation.The theory of the measurement of refractive index gradients by the moiré technique is presented. Three arrangements of the positions of the figures with respect to the sample are analyzed. One compact arrangement gives exclusively the refractive index gradient. The interpretation of moiré patterns distorted by lenses is considered in terms of the properties of the lenses.The mathematical solutions of moiré patterns are, in many cases, identical with those arising in physical problems. Examples are given for a number of phenomena arising in physical optics, hydrodynamics, and electrostatics.