Airy Disk - Approximation Using A Gaussian Profile

Approximation Using A Gaussian Profile

The Airy pattern falls rather slowly to zero with increasing distance from the center, with the outer rings containing a significant portion of the integrated intensity of the pattern. As a result, the root mean square (RMS) spotsize is undefined (i.e. infinite). An alternative measure of the spot size is to ignore the relatively small outer rings of the Airy pattern and to approximate the central lobe with a Gaussian profile, such that

where is the irradiance at the center of the pattern, represents the radial distance from the center of the pattern, and is the Gaussian width. If we equate the peak amplitude of the Airy pattern and Gaussian profile, that is, and find the value of giving the optimal approximation to the pattern, we obtain

where N is the f-number. If, on the other hand, we wish to enforce that the Gaussian profile has the same volume as does the Airy pattern, then this becomes

In optical aberration theory, it is common to describe an imaging system as diffraction-limited if the Airy disk radius is larger than the RMS spotsize determined from geometric ray tracing (see Optical lens design). Since the RMS spotsize is equivalent to the standard deviation of a function, the Gaussian profile approximation provides a convenient means of comparison: here the RMS spotsize is just the Gaussian width parameter, . And, using the approximation above shows that the RMS spotsize of the Gaussian approximation to the Airy disk is about one-third that of the Airy disk radius, i.e. as opposed to .