Airy disc

From Exampleproblems

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, 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 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.

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, $a$ is the radius of the disc, and $k = \frac{2 \pi}{\lambda}$ . The first zero of $J 1 (x)$ is where $x = 3.83$ , so the first zero of the diffraction pattern is where