Further simplification of the Fraunhofer approximation results if the pupil function is circularly symmetric. If the integral in Eq. B.2 is rewritten in circular coordinates, the angle integral can be performed to yield
N p=o where p is the normalized radial coordinate of the aperture, r' is the radial coordinate on the focal plane, Jo is the zeroth-order Bessel function, and N" is another normalization constant (Luneburg 1964, p. 345; Schroeder 1987, pp. 181-182).
In this book, all circularly-symmetric images were calculated using this simplification. The particular algorithm used divided the radius into Np equally-spaced points (typically 300-500) and calculated the intensity sum
f pn knp J
Here the value y is the reduced image angle D0/X, where 0 is the true angle, D is the diameter of the aperture, and 1 is the wavelength. (This angle conveniently reaches the edge of the Airy disk for a uniform circular aperture at 1.22.) Wd is the defocusing aberration A2(j/Np)2 (defined in Chapter 10) in wavelengths, and Wj contains the rest of the aberrations. Tj is just the transmission coefficient at a sampled radial point j. J0 is calculated using a subroutine adapted from Numerical Recipes (Press et al. 1986).
It was straightforward also to keep track of the encircled energy of the image as y was increased. Because Eq. B.5 is a radial sum, the intensity of the outer portions of the image had to be weighted for the increased perimeter. The normalized encircled energy increment is approximately
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