Airy disc

From Wikipedia, the free encyclopedia

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.

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

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

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 minimum 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.

[edit] Mathematical Details

The intensity of the Fraunhofer diffraction pattern of a circular aperture is given by:

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

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 $I (0) = I 0$ .

The zeros of $J 1 (x)$ are at $x = ka \sin \theta \approx 0, 3.832, 7.016, 10.173, 13.324, ...$ , 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 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.

[edit] See also

[edit] 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.

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Category: Physical optics

Airy disc

From Wikipedia, the free encyclopedia

[edit] Mathematical Details

[edit] See also

[edit] External links

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