AIRY DISC

The 'Airy disc' is a phenomenon in optics. Owing to the wave nature of light, light passing through an aperture is diffracted and forms a pattern of light and dark regions on a screen some distance away from the aperture (see interference).
The diffraction pattern resulting from a uniformly illuminated circular aperture, has a bright region in the center, known as the 'Airy disc' which together with a series of concentric rings is called the 'Airy pattern' (both named after George Airy). The diameter of this disc is related to the wavelength of the illuminating light and the size of the circular aperture.
The most important application of this concept is in cameras and telescopes. Due to diffraction, the smallest point to which one can focus a beam of light using a lens is the size of the Airy disk. Even if one were able to make a perfect lens, there is still a limit to the resolution of an image created by this lens. An optical system in which the resolution is no longer limited by imperfections in the lenses but only by diffraction is said to be diffraction limited.
The Airy disc is of importance in physics, optics and astronomy

Contents

Size of the Airy disc

Examples

Cameras

The human eye

Mathematical details

See also

Notes and references

External links

Size of the Airy disc

Far away from the aperture, the angle at which the first minimum occurs, measured from the direction of incoming light, is given by
:
where λ is the wavelength of the light and ''d'' is the diameter of the aperture. The Rayleigh criterion for barely resolving two objects is that the center of the Airy disc for the first object occurs at the first minimum of the Airy disc of the second. This means that the angular resolution of a diffraction limited system is given by the same formula.

Examples

Cameras

The smallest angular separation two objects can have before they significantly blur together is given as stated above by
:.
Since θ is small we can approximate this by
:,
where $x$ is the separation of the images of the two objects on the film and $f$ is the distance from the lens to the film. If we take the distance from the lens to the film to be approximately equal to the focal length of the lens, we find
:,
but is the f-number of a lens. A typical setting for use on a sunny day would be .^[1] For visible light, the wavelength λ is about 450 nanometers. This gives a value for $x$ of about 0.01 mm. In a digital camera, making the pixels of the image sensor smaller than this would not actually increase image resolution.

The human eye

The smallest f-number for the human eye is about 2.1. The resulting resolution is about 1μm. This happens to be about the distance between optically sensitive cells, photoreceptors, in the human eye.

Mathematical details

The intensity of the Fraunhofer diffraction pattern of a circular aperture is given by:
:
ight )^2
where $J\_1$ is a Bessel function of the first kind of order one, $a$ is the radius of the aperture, $I\_0$ is the intensity in the center of the diffraction pattern, and $k\; =\; \{2\; pi\}/\{lambda\}$ is the wavenumber. Here $heta$ is the angle of observation, i.e. the angle between the axis of the circular aperture and the line between aperture center and observation point. Note that the limit for $heta$
ightarrow 0 is $I(0)\; =\; I\_0$.
The zeros of $J\_1(x)$ are at , the first actual dark ring in the diffraction pattern occurs where
:.
The radius $q\_1$ of the first dark ring on a screen is related to $heta$ by $q\_1\; =\; R\; sin\; heta$, where R is the distance from the aperture.
The intensity $I\_0$ at the center of the diffraction pattern is related to the total power $P\_0$ incident on the aperture by
:
where A is the area of the aperture ($A=pi\; a^2$) and R is the distance from the aperture.
The expression for $I(\; heta)$ above can be integrated to give the total power contained in the diffraction pattern within a circle of given size:
:$P(\; heta)\; =\; P\_0\; [\; 1\; -\; J\_0^2(ka\; sin\; heta)\; -\; J\_1^2(ka\; sin\; heta)\; ]$
where $J\_0$ and $J\_1$ are Bessel functions. Hence the fractions of the total power contained within the first, second, and third dark rings (where $J\_1(ka\; sin\; heta)=0$) are 83.8%, 91.0%, and 93.8% respectively.

See also

★ George Biddell Airy

★ Fraunhofer diffraction

★ Amateur astronomy

Notes and references

1. See Sunny 16 rule.

External links

★ Diffraction Limited Photography understanding how airy discs, lens aperture and pixel size limit the absolute resolution of any camera.

★ Diffraction from a circular aperture Mathematical details to derive the above formula.