# Rayleigh criterion

Superposition of two diffraction images that can barely be resolved according to Rayleigh

The Rayleigh criterion is a heuristic condition for the distance between two light sources in order to be able to recognize them as separate. According to John William Strutt, 3rd Baron Rayleigh , this minimum distance is equal to the distance of the first minimum from the center of the diffraction pattern. With this reference, the criterion can only be used if the resolving power is limited by diffraction and the diffraction pattern has a minimum at all. There are more generally applicable criteria.

## Diffraction at the slit

If the diffraction-limited resolution in only one direction is of interest, as is the case with optical incremental encoders , the diffraction at the slit must be considered. For the single-color illuminated single slit, for example, this results in the still separable angle (in radians ):

${\ displaystyle \ alpha = \ arcsin \ left ({\ frac {\ lambda} {d}} \ right) \ approx {\ frac {\ lambda} {d}}}$

With

• the wavelength ${\ displaystyle \ lambda}$
• the gap width ${\ displaystyle d.}$

At a distance from the gap, this results in the observable half- width${\ displaystyle l}$ ${\ displaystyle b:}$

${\ displaystyle b = l \ cdot \ tan (\ alpha) \ approx l \ cdot {\ frac {\ lambda} {d}}}$

Both approximations for the angle (in radians) apply if the wavelength of the light used is much smaller than the slit width:

${\ displaystyle \ lambda \ ll d}$

## Diffraction at an aperture

For imaging optics important is the case of diffraction at a circular aperture of diameter  d , z. B. the opening of a telescope , see diffraction disks . Then the following applies to the angular distance of the first minimum:

${\ displaystyle \ alpha = \ arcsin {\ Bigl (} 1 {,} 22 \ cdot {\ frac {\ lambda} {d}} {\ Bigr)} \ approx 1 {,} 22 \ cdot {\ frac {\ lambda} {d}}}$

This formal result is close to the empirically found Dawes criterion for visual observations of binary stars .

## Optical microscopy

A microscope is called the Abbe resolution limit , which is determined by the numerical aperture and the wavelength. Here, the resolution is usually described using the smallest distance between two (point) objects (not using angles as above). As described above, according to Rayleigh, two (point) objects can just be resolved with the distance when the diffraction disk of the first object falls on the first minimum of the diffraction disk of the second object. Mathematically this leads to: ${\ displaystyle {\ text {NA}}}$${\ displaystyle a}$

${\ displaystyle a = {\ frac {1 {,} 22} {2}} \ cdot {\ frac {\ lambda} {\ text {NA}}} = 0 {,} 61 \ cdot {\ frac {\ lambda } {n \ cdot \ sin \ theta}}}$

Here is the refractive index of the medium between the lens and the image. The factor 2 comes from the fact that here or refer to half the diameter of the lens , in contrast to the equations above. ${\ displaystyle n}$${\ displaystyle {\ text {NA}}}$${\ displaystyle \ theta}$${\ displaystyle d}$