Mathematical Details
The intensity of the Fraunhofer diffraction pattern of a circular aperture (the Airy pattern) is 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 optics axis in the observation (or focal) plane and (d=2a is the aperture diameter, R is the observation distance) is the f-number of the system.
If a lens after the aperture is used, the Airy pattern forms at the focal plane of the lens, where R = f (f is the focal length of the lens). Note that the limit for (or for ) is .
The zeros of are at . From this follows that the first dark ring in the diffraction pattern occurs where, or
- .
The radius of the first dark ring on a screen is related to and to the f-number by
where R is the distance from the aperture, and the f-number N = R/d is the ratio of observation distance to aperture size. 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.
Read more about this topic: Airy Disk
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