where A^ is the jumpsize in image angle and eTR is the total energy fraction that enters the system. If the obstruction is 50%, em is 0.75, and the encircled energy increases enough so that it reaches unity far beyond the Airy disk.
Actually, if the encircled energy is accumulated in the simple-minded manner of Eq. B.6, results are poor. The energy in focused images sharply increases just as the circle radius starts to open. Errors made here can be large. The program used Simpson's rule integration with a sophisticated starter.
The modulation transfer function (MTF) was not calculated for circularly symmetric apertures in its more straightforward form as an autocorrelation (see below). Such a calculation is too slow. Instead, once I (<p) had been carried out far beyond the Airy disk, it could be used to infer the transfer function. The image is merely the convolution of a perfect sinusoidal target with the point spread function. For spatial frequency v, the integral is (as adapted from
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