B4 Image Calculations for Nonsymmetric Apertures

Although the field calculations of Eq. B.4 are long, they are a great deal shorter than the double integral required for a nonsymmetric pupil function. Others have noted that generating these patterns is tedious (Allred and Mills 1989); before computer time became inexpensive, they were not often attempted at all. Each aperture is modeled by a square grid of 129 x 129 points, with points farther than 64 from the center set to zero. Within the aperture are 12,853 points. The image plane is a grid of 129 x 129 points as well. The integral in Eq. B.3 must be done for each of these 16,641 image locations. For each nonsymmetric image frame in this book, the integrand of Eq. B.3 needed to be evaluated over 210 million times.

The image plane may be mapped out in one grand sweep using another technique. If a two-dimensional discrete Fourier transform were taken of the complex pupil function, the result would be (after rearrangement and further processing) a complete image. The fast Fourier transform (FFT) is useful to reduce the computational load (Brigham 1988).

This procedure involves considerable computation itself. In this book the angle y was sampled at D0A values as small as 0.05 or 0.1. To achieve a grid with such fine spacing, the pupil function array would have to be blank-padded (filled with zeroes) out to 1024 x 1024 or 2048 x 2048. The storage of just this single-precision array takes 8 to 32 megabytes. Much of the speed advantage of the FFT would be gobbled up by the relatively slow disk-access times to virtual memory (large RAM sizes were not common when this program was designed). One software company offered a two-dimensional FFT routine that could turn around a complex 1024 x 1024 array in a few minutes. However, this routine did not use the disk as virtual memory during the benchmark test; the array was held entirely in fast electronic memory.

Once such an FFT is finished calculating, the array has to be reorganized and the image intensity extracted. All 8 megabytes need not be kept, but the additional processing to sample and compress the image would take more computer time. In the end, it was decided that the extra effort required to process the image by simple integration of Eq. B.3 was not a severe burden, and was more than compensated by the complete transparency of the code.

The algorithm that was used at the core of the program was a discrete version of Eqs. B.2 and B.3. It had the form

where Wd is the defocusing aberration, Wd = A2(t2 + u2)/642 and values Wtu contain the rest of the aberrations. N here is the number of pupil points, i.e., 12,853. This sum is done for each image point in a grid that runs indices m and n from —64 to 64.

Asymmetric pupils have complex transfer functions. The full description of the performance of asymmetric apertures is called the optical transfer function (OTF) and has the following form:

The imaginary exponent ¥(v) is the phase transfer function. For small aberrations the real part of the OTF is much larger than the imaginary portion. Thus, phase is commonly neglected. The modulus MTF(v) is the quantity usually equated with the optical quality of the system.

The effect of a small imaginary part of the OTF on the bar pattern image is to shift it sideways a tiny amount but not enough to reverse it completely. Reversal is adequately handled by a negative MTF.3

Clearly, we cannot use the circularly symmetric formulation of Eq. B.7 to calculate the OTF of an asymmetric pupil. Instead, we may use a tidy formulation described well in a number of places (Born and Wolf 1980, p. 485; Parrent and Thompson 1969, p. 22; Luneburg 1964, p. 356). The OTF is calculated as the autocorrelation of the pupil function:

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