# Moiré effect

The moiré effect (from the French moiré [ mwaˈʀe ], "moiriert, marbled") is an optical effect in which an apparently coarse grid is created by superimposing regular, fine grids . The resulting pattern, which looks similar to the patterns from interference , is a special case of aliasing due to undersampling .

## Explanations and occurrences

Moiré effect when two dot patterns of the same pitch are superimposed, twisted against each other

Possible causes of the moiré effect are:

1. a rotation of the superimposed grids with the same pitch against each other (mutual shifting only causes local brightness or color changes ( color printing )),
2. a (minimal) unequal division of the superimposed grids, or
3. an additional rotation of superimposed grids of unequal pitch against each other.

When multi-color screen printing moiré effects are a known bug. The grids then do not have the same pitch and / or the individual prints do not exactly meet one another.

During printing, television, scanning and other image-generating raster processes, moiré effects occur when the object itself is finely screened (clothing fabrics, but also if the object is already a raster or pixel image).

## Moiré effect in line screens

Flat grids are usually grid-shaped, i.e. two-dimensional . The line grid is the reduction of the general grid to one -dimensionality .

1. In the case of two non  -   parallel (twisting   ) line grids with the same pitch   , one observes a brightness modulation in the form of more or less diffuse appearing parallel lines (moiré lines, Fig. 1) with the distance  .${\ displaystyle \ alpha}$${\ displaystyle a_ {1}}$
${\ displaystyle a = {\ frac {a_ {1}} {2 \ cdot \ sin (\ alpha / 2)}}}$
2. If two line grids with the divisions and parallel (   ) are placed on top of each other, the moiré lines (Fig. 2) have the distance  . Line grids ( ≈ ) that differ little lead to moiré lines that are far apart. If you define the reciprocal of the line spacing as the line density    , you get    . This corresponds entirely to the expression   for the low-frequency beats  that in superposition of waves with similar frequencies and arise.${\ displaystyle a_ {1}}$${\ displaystyle a_ {2}}$${\ displaystyle \ alpha = 0}$
${\ displaystyle a = {\ frac {a_ {1} \ cdot a_ {2}} {| a_ {2} -a_ {1} |}}}$
${\ displaystyle a_ {1}}$${\ displaystyle a_ {2}}$
${\ displaystyle d_ {i} = {\ frac {1} {a_ {i}}}}$${\ displaystyle d = d_ {2} -d_ {1}}$${\ displaystyle f = f_ {2} -f_ {1}}$ ${\ displaystyle f}$${\ displaystyle f_ {1}}$${\ displaystyle f_ {2}}$
3. If two line grids are laid one on top of the other with the divisions and rotated by an angle , moiré lines appear with the distance  . That is the equation for the general case. With the conditions of the special cases 1 and 2, their shorter equations arise from it.${\ displaystyle a_ {1}}$${\ displaystyle a_ {2}}$${\ displaystyle \ alpha}$
${\ displaystyle a = {\ frac {a_ {1} \ cdot a_ {2}} {\ sqrt {a_ {1} ^ {2} + a_ {2} ^ {2} -2 \ cdot a_ {1} \ cdot a_ {2} \ cdot \ cos (\ alpha)}}}}$

## Examples

• Image 1: Moiré structures arise when two line grids with the same pitch are twisted on top of one another (pitch 4 pixels, twist 2 °, division of the apparent grid about 115 pixels).
• Figure 2: Two superimposed line grids show long-period brightness modulations if the divisions differ slightly from each other (division of the left raster 4 pixels, the right 0.95 × 4 pixels, the apparent middle 76 pixels).
• Image 3: Examples of moiré effects that can arise when images are rasterized (here: portrait of Sarah Bernhardt ).
The original image is inserted as a reduced halftone image. The large image at the top left is the first raster (halftone → raster). If the raster image is reduced to a new raster image, moiré lines are created that overlay the image (raster → raster). The top right image has been reduced by 1%, the image below by 20%. In comparison, the raster image at the bottom left, which was created from a halftone image reduced by 20%, shows no interference whatsoever (halftone → raster).
• Fig. 4: Electron microscope image of graphite . The resolution is too low to recognize the basal planes running vertically in the image (in the object superimposed line raster with a division of about 0.3 nm). But you can see dark horizontally running bands that result from a moiré overlay of slightly tilted planes.
• Image 5: Digital photography of Lötzen Castle . Here the periodic structures of the image converter overlap with those of the brick pattern, a problem that occurs relatively frequently.

## Applications

The effect is used as a design element, for example with fabrics and papers (see moiré ).

By superimposing suitable structures (for example line patterns of different periods) on transparent substrates, the Moiré effect can be used to calculate the relative displacement of the substrates while maintaining the same resolution. This process is used in photolithography to align the mask and wafer.

The vernier for exact length determination with a caliper also works according to this principle. Two line patterns of different periods collide. The line pair that is best aligned is evaluated during the measurement.

Moiré clock for minute and second simulation, animation with approx. 30-fold time lapse

The moiré clock shown on the left is a playful application .
Minute display: a perforated black hour disc rotates over a black dial with white lines. The rotating moiré pattern simulates the missing minute hand.
Seconds display: a minute disc rotates above the hour disc of the clock. The analog moiré pattern simulates the missing seconds hand.

Changing characters can also be generated through a higher-level structure in at least one of the two line patterns. One example is the so-called moiré beacon, in which the initially uniform image changes into an arrow-like display when the viewing angle is changed. The arrow indicates that the ship has deviated from a direct course in the direction of the fire and which side to correct to.

Another example of a superordinate structure in one of the two line patterns is a digital sundial (figure on the right). The sunlight changes direction during the day. The light shining through two line patterns generates a digital time display that changes every five minutes.