Moiré speedup

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.

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 p_b and the period of the revealing layer as p_r.

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 p_m 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 p_r<p_b, 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 p_m, the base layer lines are ahead the revealing layer lines by a half a period (p_r/2) of the revealing layer lines, due to which the lines are interleaving, forming a dark moiré band. At the full distance p_m, the base layer lines are ahead of the revealing layer lines by a full period p_r, so the lines of the layers again overlap. The base layer lines gain the distance p_m with as many lines (p_m/p_b) as the number of the revealing layer lines (p_m/p_r) for the same distance minus one: p_m/p_r = p_m/p_b + 1. From here we obtain the well known formula for the period p_m of the superposition image:

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