According to Rayleigh's definition, this angle is the radius of the first ring of null intensity and known as the Rayleigh two-point resolution criterion. The angular resolution may as well be defined as the diameter of the Airy function at half-maximum which then is q = 1.04 X/D and sometimes referred to simply as q = X/D. If this separation is reduced to q = 0.947X/D, the dip between the two peaks just disappears; this is known as the Sparrow two-point resolution criterion (Wetherell [168]).

For comparisons of a wave obtained from optical testing, it is useful to consider as an absolute reference the encircled energy of a perfectly stigmatic wave (Fig. 1.31).

Fig. 1.31 Function y = 1 — Jq (x) — J2 (x) representing the fractional encircled energy into a radius r = x/ka of Airy's pattern (after Born and Wolf [17])

1.11.3 Diffraction from an Annular Aperture

An important case with telescopes is to consider the diffracted image of a wavefront obstructed at its center, such as caused by a secondary mirror (Born and Wolf [17]). Let us define the boundaries of this aperture area as the radii a and ea of two concentric circles, where e is a positive number less than infinity. In this Fraunhofer case, the light distribution at the focal plane is represented by

2J1(kar) 2 2J1(ekar)

Compared to a full aperture, this result shows that the resolving power is slightly increased but the maximum intensity of the rings is also increased. With e = 1/2, the first root of (1.89) is for x = kar = 1.00 n instead of 1.22 n for e =0, and the intensity of the second maximum is 0.092 instead of 0.018 for e=0.

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