Now representing its argument by x = kar, a perfectly spherical wavefront converges at the Gauss focus by radially distributing the intensity of the light according to the so-called Airy function [2/1(x)/x]2 (Fig. 1.29) which provides a pattern with successive dark rings (Fig. 1.30). This function has a maximum value equal to unity for x = 0, i.e. on the axis. The null intensities are given by the zeros of J1 (x), whilst the secondary maxima are given by the zeros of [2J1(x)/x]2 derivative (Table 1.5). The first dark ring occurs at x = 1.220n. Since the wave number is k = 2k/—, the corresponding angular radius is r = 0.610 - . (1.88a)

For any instrument with an aperture diameter D = 2a, the maximal angular resolution due to the diffraction limitation, called the resolving power, is defined from the Airy patterns of two object points - or stars - as the angle p = 1.220 — (1.88b)

separating their central peaks.

Fig. 1.29 Fraunhofer diffraction of a circular aperture with normalized intensity y = [2J (x) /x]2 (after Born and Wolf [17])
Fig. 1.30 Fraunhofer diffraction - Airy pattern - of a circular aperture 6 mm in diameter, magnification 50 x, X = 579 nm. The central intensity has been overexposed to show the weak subsidiary maxima (after Institute of Optics, Orsay)
Table 1.5 First minima and maxima of the function [2J (x)/x]2 (after Born and Wolf [17])
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