Airy Disk - Size

Size

Far away from the aperture, the angle at which the first minimum occurs, measured from the direction of incoming light, is given by the approximate formula:

or, for small angles, simply

where θ is in radians and λ is the wavelength of the light and d is the diameter of the aperture. Airy wrote this as

where s was the angle of first minimum in seconds of arc, a was the radius of the aperture in inches, and the wavelength of light was assumed to be 0.000022 inches (the mean of visible wavelengths). The Rayleigh criterion for barely resolving two objects that are point sources of light, such as stars seen through a telescope, is that the center of the Airy disk for the first object occurs at the first minimum of the Airy disk of the second. This means that the angular resolution of a diffraction limited system is given by the same formulae.

However, while the angle at which the first minimum occurs (which is sometimes described as the radius of the Airy disk) depends only on wavelength and aperture size, the appearance of the diffraction pattern will vary with the intensity (brightness) of the light source. Because any detector (eye, film, digital) used to observe the diffraction pattern can have an intensity threshold for detection, the full diffraction pattern may not be apparent. In astronomy, the outer rings are frequently not apparent even in a highly magnified image of a star. It may be that none of the rings are apparent, in which case the star image appears as a disk (central maximum only) rather than as a full diffraction pattern. Furthermore, fainter stars will appear as smaller disks than brighter stars, because less of their central maximum reaches the threshold of detection. While in theory all stars or other "point sources" of a given wavelength and seen through a given aperture have the same Airy disk radius characterized by the above equation (and the same size diffraction pattern), differing only in intensity (the "height" of the surface plot at upper right), the appearance is that fainter sources appear as smaller disks, and brighter sources appear as larger disks. This was described by Airy in his original work:

The rapid decrease of light in the successive rings will sufficiently explain the visibility of two or three rings with a very bright star and the non-visibility of rings with a faint star. The difference of the diameters of the central spots (or spurious disks) of different stars ... is also fully explained. Thus the radius of the spurious disk of a faint star, where light of less than half the intensity of the central light makes no impression on the eye, is determined by, whereas the radius of the spurious disk of a bright star, where light of 1/10 the intensity of the central light is sensible, is determined by .).

Despite this feature of Airy's work, the radius of the Airy disk is often given as being simply the angle of first minimum, even in standard textbooks. In reality, the angle of first minimum is a limiting value for the size of the Airy disk, and not a definite radius.