YJJ P x y2 dxdy

where the integral is taken over the aperture plane. Here P is a shorthand notation for the pupil function defined in the text following Eq. B.l. The aperture coordinates are defined so that the perimeter is at x2 + y2 = 1, and the spatial frequency fractions are similarly normalized. One can envision the integral above as the overlap between the pupil function and the pupil function moved sideways by the spatial frequency fractions as in Fig. B-2. For non-aberrated pupils, the OTF is the dark area divided by the uncovered area of the whole aperture.

3 The MTF as defined in Eq. B.10 is always positive, but the sign component of the phase transfer function is sometimes appended to the MTF when the OTF is mostly real. The resulting "MTF" is smoother and tells more about the optical performance.

Fig. B-2. The MTF as the overlap with a shifted pupil.

The actual algorithm took into account the sampling of the aperture on a rectangular grid. Since the offsets have to land on sampled positions, the OTF was calculated only for integer offsets along the x and y directions and a 45° offset with equal shifts. The algorithm for a shift in only the x direction was

Z64 T 2

t,u=-64 tu where indices s (for "shift"), u, s, and t run from —64 to 64 and vx = (s + 64)/128. The maximum spatial frequency corresponds to vx = 1. Here the pupil array is measured from its center.

Encircled energy can be defined for an asymmetric aperture, but no such calculations appear in this book.

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