page_title: Airy disk
section_title: Mathematical formulation
hierarchical_section_title: Airy disk / Mathematical formulation
context_page_description: In optics, the Airy disk and Airy pattern are descriptions of the best-focused spot of light that a perfect lens with a circular aperture can make, limited by the diffraction of light. The Airy disk is of importance in physics, optics, and astronomy.
  The diffraction pattern resulting from a uniformly illuminated, circular aperture has a bright central region, known as the Airy disk, which together with the series of concentric rings around is called the Airy pattern. Both are named after George Biddell Airy. The disk and rings phenomenon had been known prior to Airy; John Herschel described the appearance of a bright star seen through a telescope under high magnification for an 1828 article on light for the Encyclopedia Metropolitana:
  ...the star is then seen as a perfectly round, well-defined planetary disc, surrounded by two, three, or more alternately dark and bright rings, which, if examined attentively, are seen to be slightly coloured at their borders. They succeed each other nearly at equal intervals round the central disc....
  Airy wrote the first full theoretical treatment explaining the phenomenon.
context_section_description: The intensity of the Airy pattern follows the Fraunhofer diffraction pattern of a circular aperture, given by the squared modulus of the Fourier transform of the circular aperture:
  where  is the maximum intensity of the pattern at the Airy disc center,  is the Bessel function of the first kind of order one,  is the wavenumber,  is the radius of the aperture, and  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. , where q is the radial distance from the observation point to the optical axis and R is its distance to the aperture. Note that the Airy disk as given by the above expression is only valid for large R, where Fraunhofer diffraction applies; calculation of the shadow in the near-field must rather be handled using Fresnel diffraction. 
  However the exact Airy pattern does appear at a finite distance if a lens is placed at the aperture. Then the Airy pattern will be perfectly focussed at the distance given by the lens's focal length (assuming collimated light incident on the aperture) given by the above equations.
  The zeros of  are at . From this, it follows that the first dark ring in the diffraction pattern occurs where , or
  .
  If a lens is used to focus the Airy pattern at a finite distance, then the radius  of the first dark ring on the focal plane is solely given by the numerical aperture A (closely related to the  f-number) by
  where the numerical aperture A is equal to the aperture's radius d/2 divided by R', the distance from the center of the Airy pattern to the edge of the aperture. Viewing the aperture of radius d/2 and lens as a camera (see diagram above) projecting an image onto a focal plane at distance f, the  numerical aperture A is related to the commonly-cited f-number N= f/d (ratio of the focal length to the lens diameter) according to ; for N>>1 it is simply approximated as . This shows that the best possible image resolution of a camera is limited by the numerical aperture (and thus f-number) of its lens due to diffraction. 
  The half maximum of the central Airy disk (where ) occurs at ; the 1/e² point (where ) occurs at , and the maximum of the first ring occurs at .
  The intensity  at the center of the diffraction pattern is related to the total power  incident on the aperture by
  where  is the source strength per unit area at the aperture, A is the area of the aperture () and R is the distance from the aperture. At the focal plane of a lens, . The intensity at the maximum of the first ring is about 1.75% of the intensity at the center of the Airy disk.
  The expression for  above can be integrated to give the total power contained in the diffraction pattern within a circle of given size:
  where  and  are Bessel functions. Hence the fractions of the total power contained within the first, second, and third dark rings (where ) are 83.8%, 91.0%, and 93.8% respectively.
caption: The Airy Pattern on the interval kasinθ = [−10, 10]