# The Rayleigh Criterion

## The Rayleigh criterion determines the separation angle between two light sources which are distinguishable from each other.

#### Key Points

• Diffraction plays a large part in the resolution at which we are able to see things. There is a point where two light sources can be so close to each other that we cannot distinguish them apart.

• When two light sources are close to each other, they can be: unresolved (i.e., not able to distinguish one from the other), just resolved (i.e., only able to distinguish them apart from each other), and a little well resolved (i.e., easy to tell apart from one another).

• In order for two light sources to be just resolved, the center of one diffraction pattern must directly overlap with the first minimum of the other diffraction pattern.

#### Terms

• The bending of a wave around the edges of an opening or an obstacle.

• The degree of fineness with which an image can be recorded or produced, often expressed as the number of pixels per unit of length (typically an inch).

#### Examples

• Calculate the minimum angular spreading of a flashlight beam that is originally 5.00 cm in diameter with an average wavelength of 600 nm. We are given: D - 5 cmλ - 600 nm. We are looking for: θ $\theta = 1.22 \frac{\lambda}D\\\theta = 1.22 \frac{600 nm}{5 cm}\\\theta = 1.46*10^{-5} rad$

#### Figures

1. ##### Resolution Limits

(a) Monochromatic light passed through a small circular aperture produces this diffraction pattern. (b) Two point light sources that are close to one another produce overlapping images because of diffraction. (c) If they are closer together, they cannot be resolved or distinguished.

2. ##### Rayleigh Criterion

(a) This is a graph of intensity of the diffraction pattern for a circular aperture. Note that, similar to a single slit, the central maximum is wider and brighter than those to the sides. (b) Two point objects produce overlapping diffraction patterns. Shown here is the Rayleigh criterion for being just resolvable. The central maximum of one pattern lies on the first minimum of the other.

### Resolution Limits

Along with the diffraction effects that we have discussed in previous atoms, diffraction also limits the detail that we can obtain in images. Figure 1 shows three different circumstances of resolution limits due to diffraction:

• (a) shows a light passing through a small circular aperture. You do not see a sharp circular outline, but a spot with fuzzy edges. This is due to diffraction similar to that through a single slit.
• (b) shows two point sources close together, producing overlapping images. Due to the diffraction, you can just barely distinguish between the two point sources.
• (c) shows two point sources which are so close together that you can no longer distinguish between them.

This effect can be seen with light passing through small apertures or larger apertures. This same effect happens when light passes through our pupils, and this is why the human eye has limited acuity.

### Rayleigh Criterion

In the 19th century, Lord Rayleigh invented a criteria for determining when two light sources were distinguishable from each other, or resolved. According to the criteria, two point sources are considered just resolved (just distinguishable enough from each other to recognize two sources) if the center of the diffraction pattern of one is directly overlapped by the first minimum of the diffraction pattern of the other. If the distance is greater between these points, the sources are well resolved (i.e., they are easy to distingiush from each other). If the distance is smaller, they are not resolved (i.e., they cannot be distinguished from each other). The equation to determine this is:

$\theta = 1.22 \frac{\lambda}D$

θ - angle the objects are separated by, in radian λ - wavelength of light D - aperture diamete. Figure 2 shows this concept visually. This equation also gives the angular spreading of a source of light having a diameter D.

#### Key Term Glossary

angular
Relating to an angle or angles; having an angle or angles; forming an angle or corner; sharp-cornered; pointed; as in, an angular figure.
##### Appears in these related concepts:
aperture
The diameter of the aperture that restricts the width of the light path through the whole system. For a telescope, this is the diameter of the objective lens (e.g., a telescope may have a 100 cm aperture).
##### Appears in these related concepts:
atom
The smallest possible amount of matter which still retains its identity as a chemical element, now known to consist of a nucleus surrounded by electrons.
##### Appears in these related concepts:
average
The arithmetic mean.
##### Appears in these related concepts:
diffraction
The bending of a wave around the edges of an opening or an obstacle.
##### Appears in these related concepts:
equation
An assertion that two expressions are equal, expressed by writing the two expressions separated by an equal sign; from which one is to determine a particular quantity.
##### Appears in these related concepts:
light
The natural medium emanating from the sun and other very hot sources (now recognised as electromagnetic radiation with a wavelength of 400-750 nm), within which vision is possible.
##### Appears in these related concepts:
pupil
The hole in the middle of the iris of the eye, through which light passes to be focused on the retina.
##### Appears in these related concepts:
The angle subtended at the centre of a circle by an arc of the circle of the same length as the circle's radius.
##### Appears in these related concepts:
Rayleigh criterion
The angular resolution of an optical system can be estimated from the diameter of the aperture and the wavelength of the light, which was first proposed by Lord Rayleigh.
##### Appears in these related concepts:
resolution
The degree of fineness with which an image can be recorded or produced, often expressed as the number of pixels per unit of length (typically an inch).
##### Appears in these related concepts:
wavelength
The length of a single cycle of a wave, as measured by the distance between one peak or trough of a wave and the next; it is often designated in physics as λ, and corresponds to the velocity of the wave divided by its frequency.