# Terminology/Rayleigh criterion

## Rayleigh Criterion/Angular resolution

Angular resolution describes the ability of any image-forming device such as an optical or radio telescope, a microscope, a camera, or an eye, to distinguish small details of an object, thereby making it a major determinant of image resolution. It is used in optics applied to light waves, in antenna theory applied to radio waves, and in acoustics applied to sound waves.

The closer you are to two objects, the greater is the angular separation between them. Up close, two objects are easily resolved. As your distance from the objects increases, their images become less well resolved and eventually merge into one image.

Rayleigh criterion / Angular resolution is the minimum angular distance between two distant objects that an instrument can discern resolvable detail. As an example, if a person holds two pens 4 inches (10cm) apart and stands 6.5 feet (2m) from you, you can discern there are two pencils. As the other person moves away, the pencils appear to move closer together or the angular separation decreases. The calculation of this angle is very important in optics. This angle represents the resolving power and precision of optical instruments such as your eye, a camera, and even a microscope.

The Rayleigh criterion shows that the minimum angular spread that can be resolved by an image forming system is limited by diffraction to the ratio of the wavelength of the waves to the aperture width. For this reason, high resolution imaging systems such as astronomical telescopes, long distance telephoto camera lenses and radio telescopes have large apertures.

Resolving power is the ability of an imaging device to separate (i.e., to see as distinct) points of an object that are located at a small angular distance or it is the power of an optical instrument to separate far away objects, that are close together, into individual images. The term resolution or minimum resolvable distance is the minimum distance between distinguishable objects in an image, although the term is loosely used by many users of microscopes and telescopes to describe resolving power.

Telescopes, for example, enhance our ability to see distant objects in a number of ways. First, they can gather more light than our eyes. Second, with the help of an eyepiece, they can magnify an image. Lastly, they can help distinguish objects that are close together. This last enhancement is called a telescope's resolving power. In general, the resolving power of a telescope increases as the diameter of the telescope increases.

### Formula

This formula is known as the angular resolution formula and is the mathematical representation of the Rayleigh criterion. The Rayleigh criterion basically says that two different points are resolved when the diffraction maximum of one image coincides with the first minimum diffraction of a second image. If the distance is greater, the two points are resolved and if it is smaller they are not resolved.

Calculate the wavelength of the light waves used to focus the image. This number is represented by W in the angular resolution formula.

The wavelength for yellow light is about 577nm. This number can be looked up. To get a more precise answer you will need to know the frequency of the light you are using and the speed of light. The wavelength equation is:

Find the value of the entrance pupil diameter (D) or the diameter of the lens aperture (D) of the imaging system you are using. For telescopes and most other optical instruments, the diameter of the aperture can be found in the user manual or you can contact the manufacturer who will be able to tell you the correct value.

Ensure your wavelength and diameter have been converted into the same units of measure. For example, if your wavelength is in meters then your diameter needs to be converted to meters or vice versa.

Manipulate the formula to solve for A by dividing both sides of the equation by sin. The manipulated formula should appear as the following:

Use your calculator to do the math to find out what the angular resolution (A) is equal to. The units of the wavelength and the diameter cancel so the answer is expressed in radians. For astronomy purposes, you can convert radians to seconds of arc.