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.
Superposition of layers with periodically repeating parallel lines
Simple moiré patterns can be observed when superposing two transparent layers comprising periodically repeating opaque parallel lines as shown in Figure 1. The lines of one layer are parallel to the lines of the second layer. The superposition image does not change if transparent layers with their opaque patterns are inverted. When considering printed samples, one of the layers is denoted as the base layer and the other one as the revealing layer. It is assumed that the revealing layer is printed on a transparency and is superimposed on top of the base layer, which can be printed either on a transparency or on an opaque paper. The periods of the two layer patterns are close. We denote the period of the base layer as pb and the period of the revealing layer as pr. The superposition image of Figure 1 outlines periodically repeating dark parallel bands, called moiré lines. Spacing between the moiré lines is much larger than the periods of lines in the two layers.
Light bands of the superposition image correspond to the zones where the lines of both layers overlap. The dark bands of the superposition image forming the moiré lines correspond to the zones where the lines of the two layers interleave, hiding the white background. The labels of Figure 2 show the passages from light zones with overlapping layer lines to dark zones with interleaving layer lines. The light and dark zones are periodically interchanging.
Figure 3 shows a detailed diagram of the superposition image between two adjacent zones with overlapping lines of the revealing and base layers (i.e., between two light bands). The period pm of moiré lines is the distance from one point where the lines of both layers overlap (at the bottom of the figure) to the next such point (at the top). Let us count the layer lines, starting from the bottom point. At the count 0 the lines of both layers overlap. Since in our case pr<pb, for the same number of counted lines, the base layer lines with a long period advance faster than the revealing layer lines with a short period. At the halfway of the distance pm, the base layer lines are ahead the revealing layer lines by a half a period (pr/2) of the revealing layer lines, due to which the lines are interleaving, forming a dark moiré band. At the full distance pm, the base layer lines are ahead of the revealing layer lines by a full period pr, so the lines of the layers again overlap. The base layer lines gain the distance pm with as many lines (pm/pb) as the number of the revealing layer lines (pm/pr) for the same distance minus one: pm/pr = pm/pb + 1. From here we obtain the well known formula for the period pm of the superposition image:
p m = p b ⋅ p r p b − p r . {\displaystyle p_{m}={\frac {p_{b}\cdot p_{r}}{p_{b}-p_{r}}}.}
For the case when the revealing layer period is longer than the base layer period, the distance between moiré bands is the absolute value computed by the formula. The superposition of two layers comprising parallel lines forms an optical image comprising parallel moiré lines with a magnified period. According to the formula for computing pm, the closer the periods of the two layers, the stronger the magnification factor is. The thicknesses of layer lines affect the overall darkness of the superposition image and the thickness of the moiré bands, but the period pm does not depend on the layer lines’ thickness.
Speedup of movements with moiré The moiré bands of Figure 1 will move if we displace the revealing layer. When the revealing layer moves perpendicularly to layer lines, the moiré bands move along the same axis, but several times faster than the movement of the revealing layer.
The GIF animation shown in Figure 4 corresponds to a slow movement of the revealing layer. The GIF file repeatedly animates an upward movement of the revealing layer (perpendicular to layer lines) across a distance equal to pr. The animation demonstrates that the moiré lines of the superposition image move up at a speed, much faster than the movement speed of the revealing layer. When the revealing layer is shifted up perpendicularly to the layer lines by one full period (pr) of its pattern, the superposition optical image must be the same as the initial one. It means that the moiré lines traverse a distance equal to the period of the superposition image pm while the revealing layer traverses the distance equal to its period pr. Assuming that the base layer is immobile (vb=0), the following equation represents the ratio of the optical speed to the revealing layer’s speed:
v m v r = p m p r . {\displaystyle {\frac {v_{m}}{v_{r}}}={\frac {p_{m}}{p_{r}}}.}
By replacing pm with its formula, we have
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