NationMaster - Encyclopedia: Airy pattern

A computer generated image of an Airy Disc. The greyscale intensities have been adjusted to enhance the brightness of the outer rings of the pattern.

Due to the wave nature of light, light passing through apertures is diffracted, and the diffraction increases with decreasing aperture size. 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. ... A wave is a disturbance that propagates through space, often transferring energy. ... Prism splitting light Light is electromagnetic radiation with a wavelength that is visible to the eye (visible light) or, in a technical or scientific context, electromagnetic radiation of any wavelength. ... Definitions of Aperture in the 1707 Glossographia Anglicana Nova For other uses, see Aperture (disambiguation). ... To meet Wikipedias quality standards, this article or section may require cleanup. ...

The resulting diffraction pattern of a uniformly illuminated circular aperture has a bright region in the centre, known as the Airy disc or Airy pattern (after its discoverer George Airy) which is surrounded by concentric rings. The diameter of this disc is related to the wavelength of the illuminating light and the size (f-number) of the circular aperture. The angle from the center at which the first minimum occurs is George Biddell Airy Sir George Biddell Airy (July 27, 1801 – January 2, 1892) was British Astronomer Royal from 1835 to 1881. ... 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. ...

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

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 centre of the Airy disc for the first object occurs at the first minima of the Airy disc of the second. Angular resolution describes the resolving power of a telescope. ...

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. Radio telescopes are among many different tools used by astronomers Astronomy (Greek: Î±ÏƒÏ„ÏÎ¿Î½Î¿Î¼Î¯Î± = Î¬ÏƒÏ„ÏÎ¿Î½ + Î½ÏŒÎ¼Î¿Ï‚, astronomia = astron + nomos, literally, law of the stars) is the science of celestial objects (such as stars, planets, comets, and galaxies) and phenomena that originate outside the Earths atmosphere (such as auroras and cosmic background radiation). ... 50 cm refracting telescope at Nice Observatory. ...

Mathematical Details

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. ...

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

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 $k = 2π / λ$ is the wavenumber. 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 named after Friedrich Bessel, are canonical solutions y(x) of Bessels differential equation: for an arbitrary real number Î± (the order). ...

The zeros of $J 1 (x)$ are at , so the first 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 diffration 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 total power $P$ 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. In mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli and named after Friedrich Bessel, are canonical solutions y(x) of Bessels differential equation: for an arbitrary real number Î± (the order). ...

External links

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

Category: Physical optics

Results from FactBites:

Angular resolution - Wikipedia, the free encyclopedia (772 words)
	The lens' aperture is a "hole" that is analogous to a two-dimensional version of the single-slit experiment; light passing through it interferes with itself, creating a ring-shaped diffraction pattern, known as the Airy pattern, that blurs the image.
	The factor 1.22 is derived from a calculation of the position of the first dark ring surrounding the central Airy disc of the diffraction pattern.
	Thus it is worth bearing in mind that the Rayleigh criterion is an empirical estimate of resolution based on the assumption of a human observer, and may slightly underestimate the resolving power of a particular optical train.

Olympus FluoView Resource Center: Airy Pattern Basics - Interactive Java Tutorial (692 words)
	The three-dimensional diffraction pattern formed by a circular aperture near the focal point in a well-corrected microscope is symmetrically periodic along the axis of the microscope as well as radially around the axis.
	Appearing beneath the Airy patterns is a simulation of the illumination cone projected into the microscope objective the by substage condenser, and a set of sliders that are used to control the tutorial.
	The Airy pattern radius is governed by the wavelength of illumination and the combined numerical apertures of both the objective and condenser.

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