NationMaster - Encyclopedia: Airy disc

A computer-generated image of an Airy disc. The gray scale intensities have been adjusted to enhance the brightness of the outer rings of the Airy pattern.

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). Image File history File links Image of an airy diffraction pattern File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Image File history File links Image of an airy diffraction pattern File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... For the book by Sir Isaac Newton, see Opticks. ... This article does not cite any references or sources. ... a big (1) and a small (2) aperture For other uses, see Aperture (disambiguation). ... The intensity pattern formed on a screen by diffraction from a square aperture Diffraction refers to various phenomena associated with wave propagation, such as the bending, spreading and interference of waves passing by an object or aperture that disrupts the wave. ... Interference of two circular waves - Wavelength (decreasing bottom to top) and Wave centers distance (increasing to the right). ...

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. George Biddell Airy Sir George Biddell Airy (or Airey) FRS (July 27, 1801â€“January 2, 1892) was British Astronomer Royal from 1835 to 1881. ...

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 ability to produce optical images with angular separations as small as the instruments theoretical limit. ...

The Airy disc is of importance in physics, optics and astronomy This is a discussion of a present category of science. ... For the book by Sir Isaac Newton, see Opticks. ... For other uses, see Astronomy (disambiguation). ...

1 Size of the Airy disc
2 Examples
- 2.1 Cameras
- 2.2 The human eye
3 Mathematical details
4 Obscured Airy pattern
5 See also
6 Notes and references
7 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

$sin theta = 1.22 frac{lambda}{d}$

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. Angular resolution describes the resolving power of any optical device such as a telescope, a microscope, a camera, or an eye. ...

Examples

Cameras

The smallest angular separation two objects can have before they significantly blur together is given as stated above by

$sin theta = 1.22 frac{lambda}{d}$ .

Since θ is small we can approximate this by

$frac{x}{f} = 1.22 frac{lambda}{d}$ ,

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 This article is about focal length related to lenses and systems of lenses. ...

$x = 1.22 frac{lambda f}{d}$ ,

but $frac{f}{d}$ is the f-number of a lens. A typical setting for use on a sunny day would be f/16.^[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. A 35mm lens set to f/11, as indicated by the white dot above the f-stop scale on the aperture ring In photography the f-number (focal ratio) expresses the diameter of the diaphragm aperture in terms of the effective focal length of the lens. ... A 35mm lens set to f/11, as indicated by the white dot above the f-stop scale on the aperture ring In photography the f-number (focal ratio) expresses the diameter of the diaphragm aperture in terms of the effective focal length of the lens. ... Hello--80. ... Image resolution describes the detail an image holds. ...

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. This article is about cellular photoreceptors. ...

Mathematical details

Light from a uniformly illuminated circular aperture (or from a uniform, flattop beam) which is focused by a lens will exhibit an Airy diffraction pattern at the focus of the lens due to Fraunhofer diffraction. Fraunhofer diffraction is diffraction of light through an aperture for small values of the Fresnel number, F<<1. ...

The intensity of the Fraunhofer diffraction pattern of a circular aperture is given by: In physics, intensity is a measure of the time-averaged energy flux. ... Fraunhofer diffraction is diffraction of light through an aperture for small values of the Fresnel number, F<<1. ...

$I(theta) = I_0 left ( frac{2 J_1(ka sin theta)}{ka sin theta} right )^2 = I_0 left ( frac{2 J_1(x)}{x} right )^2$

where $I 0$ is the intensity in the center of the diffraction pattern, $J 1$ is the Bessel function of the first kind of order one, $k = 2π / λ$ is the wavenumber, $a$ is the radius of the aperture, and $x = k a sinθ$ . Here $θ$ 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 $theta rightarrow 0$ is $I (0) = I 0$ . In mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are canonical solutions y(x) of Bessels differential equation: x2 for an arbitrary real or complex number Î±. The most common and important special case is where Î± is an integer n, then Î± is referred...

The zeros of $J 1 (x)$ are at $x = ka sin theta approx 0, 3.8317, 7.0156, 10.1735, 13.3268, 16.470...$ . From this follows that the first actual dark ring in the diffraction pattern occurs where

$sin theta = frac{3.83}{ka} = frac{3.83 lambda}{2 pi a} = 1.22 frac{lambda}{2a} = 1.22 frac{lambda}{d}$ .

The radius $q 1$ of the first dark ring on a screen is related to $θ$ by $q 1 = R sinθ$ , 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

$I_0 = frac{P_0 A}{lambda^2 R^2}$

where A is the area of the aperture ( $A = π a 2$ ) and R is the distance from the aperture. The expression for $I (θ)$ above can be integrated to give the total power contained in the diffraction pattern within a circle of given size:

$P(theta) = P_0 [ 1 - J_0^2(ka sin theta) - J_1^2(ka sin theta) ]$

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 $J 1 (k a sinθ) = 0$ ) are 83.8%, 91.0%, and 93.8% respectively ^[2]. In mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are canonical solutions y(x) of Bessels differential equation: x2 for an arbitrary real or complex number Î±. The most common and important special case is where Î± is an integer n, then Î± is referred...

Obscured Airy pattern

Similar equations can also be derived for the obscured Airy diffraction pattern ^[3] ^[4], which is the diffraction pattern of a uniform circular aperture (beam) obscured by a circular block at the center.

$I(theta) = frac{I_0}{ (1 - epsilon ^2)^2} left ( frac{2 J_1(x)}{x} - frac{2 epsilon J_1(x)}{x}right )^2$

where $ε$ is the annular aperture obscuration ratio, or the fraction of the diameter of the obscuring disk and the diameter of the aperture (beam). $left( 0 le epsilon < 1 right)$ , and x is defined as above: $x=ka sin(theta) approx frac {pi R}{lambda N}$ where R is the radial distance in the focal plane from the optical axis, $λ$ is the wavelength and N is the f-number of the system. The encircled energy (the fraction of the total energy contained within a circle of radius R centered at the optical axis in the focal plane) is then given by: A 35mm lens set to f/11, as indicated by the white dot above the f-stop scale on the aperture ring In photography the f-number (focal ratio) expresses the diameter of the diaphragm aperture in terms of the effective focal length of the lens. ...

$E(R) = frac{I_0}{ (1 - epsilon ^2)^2 } left( 1 - J_0^2(x) - J_1^2(x) + epsilon ^2 left[ 1 - J_0^2 (epsilon x) - J_1^2(epsilon x) right] - 4 epsilon int_0^x frac {J_1(t) J_1(epsilon t)}{t},dt right)$

For $epsilon rightarrow 0$ the formulas reduce to the unobscured versions above.

Notes and references

^ See Sunny 16 rule.
^ M.Born and E.Wolf, Principles of Optics (Pergamon Press, New York, 1965)
^ Rivolta, Applied Optics, 25, 2404 (1986)
^ Mahajan, J.Opt.Soc.Am.A, 3, 470 (1986)

In photography, the sunny f/16 rule is a method to obtain correct exposure without using a light meter. ...

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.

Categories: Physical optics | Diffraction

Results from FactBites:

Spider diffraction and apodization (1528 words)
	While the spikes caused by spider vanes can be visually distracting, the amount of energy lost from the disc is usually negligible for general observing (3-vane spider spikes are usually shorter, due to the vanes being generally thicker, as it is needed for mechanical stability in that spider configuration).
	Relative intensity of the central peak to that of the Airy disc is proportional to the ratio of the vane area to the area of the clear aperture.
	The encircled energy (within the Airy disc) increase due to the apodizing mask is 20% for unobstructed apertures, with the gain steadily decreasing as the central obstruction grows larger: it is ~10% at 0.2D obstruction, and ~7% at 0.3D obstruction.

Airy disc - Wikipedia, the free encyclopedia (273 words)
	The diameter of this disc is related to the wavelength of the illuminating light and the size (f-number) of the circular aperture.
	The Rayleigh criterion for barely resolving two objects is that the centre of the Airy disc for the first object occurs at the first minima of the Airy disc of the second.
	The Airy disc is used in astronomy as one of several methods used to determine the quality and alignment of the optical components of a telescope.

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