In mathematics, physics, and art, a moiré pattern (UK: MWAH-ray, US: mwah-RAY, French: [mwaʁe] ) or moiré fringe is a large-scale wave interference pattern that can be produced when a partially opaque ruled pattern with transparent gaps is overlaid on another similar pattern. For the moiré interference pattern to appear, the two patterns must not be completely identical, but rather displaced, rotated, or have slightly different pitch. Moiré patterns appear in many situations. In printing, the printed pattern of dots can interfere with the image. In television and digital photography, a pattern on an object being photographed can interfere with the shape of the light sensors to generate unwanted artifacts. They are also sometimes created deliberately; in micrometers, they are used to amplify the effects of very small movements. In physics, its manifestation is wave interference like that seen in the double-slit experiment and the beat phenomenon in acoustics.
Etymology The term originates from moire (moiré in its French adjectival form), a type of textile, historically made of silk but now also made of cotton or synthetic fiber, with a rippled or "watered" appearance. Moire, or "watered textile", is made by pressing two layers of the textile when wet. The similar but imperfect spacing of the threads creates a characteristic pattern which remains after the fabric dries. In French, the noun moire is in use from the 17th century, for "watered silk". It was a loan of the English mohair (attested 1610). In French usage, the noun gave rise to the verb moirer, "to produce a watered textile by weaving or pressing", by the 18th century. The adjective moiré formed from this verb is in use from at least 1823.
Pattern formation
Moiré patterns are often an artifact of images produced by various digital imaging and computer graphics techniques, for example when scanning a halftone picture or ray tracing a checkered plane (the latter being a special case of aliasing, due to undersampling a fine regular pattern). This can be overcome in texture mapping through the use of mipmapping and anisotropic filtering. The drawing seen here shows a moiré pattern. The lines could represent fibers in moiré silk, or lines drawn on paper or on a computer screen. The nonlinear interaction of the optical patterns of lines creates a real and visible pattern of roughly parallel dark and light bands, the moiré pattern, superimposed on the lines. The moiré effect also occurs between overlapping transparent objects. For example, an invisible phase mask is made of a transparent polymer with a wavy thickness profile. As light shines through two overlaid masks of similar phase patterns, a broad moiré pattern occurs on a screen some distance away. This phase moiré effect and the classical moiré effect from opaque lines are two ends of a continuous spectrum in optics, which is called the universal moiré effect. The phase moiré effect is the basis for a type of broadband interferometer in x-ray and particle wave applications. It also provides a way to reveal hidden patterns in invisible layers.
Line moiré
Line moiré is one type of moiré pattern; a pattern that appears when superposing two transparent layers containing correlated opaque patterns. Line moiré is the case when the superposed patterns comprise straight or curved lines. When moving the layer patterns, the moiré patterns transform or move at a faster speed. This effect is called optical moiré speedup. More complex line moiré patterns are created if the lines are curved or not exactly parallel.
Shape moiré
Shape moiré is one type of moiré pattern demonstrating the phenomenon of moiré magnification. 1D shape moiré is the particular simplified case of 2D shape moiré. One-dimensional patterns may appear when superimposing an opaque layer containing tiny horizontal transparent lines on top of a layer containing a complex shape which is periodically repeating along the vertical axis. Moiré patterns revealing complex shapes, or sequences of symbols embedded in one of the layers (in form of periodically repeated compressed shapes) are created with shape moiré, otherwise called band moiré patterns. One of the most important properties of shape moiré is its ability to magnify tiny shapes along either one or both axes, that is, stretching. A common 2D example of moiré magnification occurs when viewing a chain-link fence through a second chain-link fence of identical design. The fine structure of the design is visible even at great distances.
Calculations
Moiré of parallel patterns
Geometrical approach
Consider two patterns made of parallel and equidistant lines, e.g., vertical lines. The step of the first pattern is p, the step of the second is p + δp, with 0 < δp < p. If the lines of the patterns are superimposed at the left of the figure, the shift between the lines increases when going to the right. After a given number of lines, the patterns are opposed: the lines of the second pattern are between the lines of the first pattern. If we look from a far distance, we have the feeling of pale zones when the lines are superimposed (there is white between the lines), and of dark zones when the lines are "opposed". The middle of the first dark zone is when the shift is equal to p/2. The nth line of the second pattern is shifted by n δp compared to the nth line of the first network. The middle of the first dark zone thus corresponds to
n ⋅ δ p = p 2 {\displaystyle n\cdot \delta p={\frac {p}{2}}}
that is
n = p 2 δ p . {\displaystyle n={\frac {p}{2\delta p}}.}
The distance d between the middle of a pale zone and a dark zone is
d = n ⋅ ( p + δ p ) = p 2 2 δ p + p 2 {\displaystyle d=n\cdot (p+\delta p)={\frac {p^{2}}{2\delta p}}+{\frac {p}{2}}}
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