## B7 Numerical Limitations on Programs

Although APERTURE is also a numerical model, it is ASYMM which has the greatest opportunity to fail. Experience showed that the 129 points on the aperture's diameter had difficulty simulating defocusing aberration exceeding 10 wavelengths. Along each diameter under such conditions, a minimum of about 5 samples occur within a Fresnel zone. The errors have a chance to average to zero only because many such diameters are added together statistically.

ASYMM simulates an aperture with an approximate circular pattern 64 points in radius. These points are distributed in a rectangular pattern. The edge of such a sampled aperture is necessarily ragged. We can perhaps

Fig. B-5. Reproduction of coma and astigmatism image patterns. Left images are not at the same scale as the contour plots to the right. (Coma contours appear with permission of R. Kingslake and the astigmatism contour appears courtesy of Elsevier Science Publishing.)

Fig. B-5. Reproduction of coma and astigmatism image patterns. Left images are not at the same scale as the contour plots to the right. (Coma contours appear with permission of R. Kingslake and the astigmatism contour appears courtesy of Elsevier Science Publishing.)

estimate how much error is induced by inspecting Fig. B-6, which shows the differences between APERTURE and ASYMM on a defocused unaberrated aperture expanded until the errors are obvious. Because APERTURE can be set to sum over a finer one-dimensional grid (here it is 500 points), it suf-

Fig. B-6. Limited sample-size errors in ASYMM.

fers less corruption. The model has smooth edges in APERTURE because the angular integral is done analytically.

The maximum difference is about 0.000075, or 1 part in 13,300. Thus, we are seeing less than 1 erroneous sample of the 12,853 points on the aperture contributing to the intensity.

Think about the way those ragged edges are working for a moment. If the point is farther from the center than a radius of unity, the model ignores it. Each point represents a square area around its feet. Thus, if this little tile is more than halfway outside the radius, its contribution is neglected. If it extends less than halfway outside the boundary, its full contribution is counted, even for the area that stretches outside. A V13,000 error is a very small mistake and is perhaps better than the program deserves. One suspects that errors as large as 4 parts in 13,000 appear occasionally. Such an error would be 0.0003 or —35 dB. Since this intensity is about that of an image defocused 9 wavelengths, we should anticipate that the accuracy beyond defocus values of 8 to 10 wavelengths is lessened.

In practical use, ASYMM did not fail from rough pupil edges. Figure B-7 shows a comparison between two perfect images, each defocused by 8 wavelengths. One was produced with ASYMM and the other with APERTURE. They were both printed at extremely low contrast to show the feathery appearance induced by the coarse edges of the ASYMM pupil. This spurious detail was much dimmer than —35 dB. In all real cases, contrast was high enough so that such wrinkling was nearly invisible.

B. 7. Numerical Limitations on Programs 8

APERTURE output

ASYMM output

B. 7. Numerical Limitations on Programs 8

ASYMM output

APERTURE output

Fig. B-7. Extremely low-contrast image patterns showing tiny errors in ASYMM calculation.

The final problem with the sparsely sampled pupil of ASYMM is grating interference. If the phase shift between adjacent contributing elements is one wavelength, they once again act as if they are in phase. For our 128-point diameter pupil, this condition occurs at D0A = 128.

This effect is an artifact of pretending that the wonderfully smooth surface could be represented by a 129 x 129 square grid of points. In FFT terminology, we can call such a phenomenon "aliasing." It is an effect of changing the continuous Fourier transform to a discretely sampled one.

If we haven't defocused far enough to divert significant light to distant portions of the image space, no trouble results. In other words, the image and the twin image (way over at a reduced angle of 128) aren't interfering because they don't throw much light that far from their centers. Nevertheless, a limit exists as to how far the image can be defocused until interference becomes severe. We must know that limit so that it can be intelligently avoided.

Figure B-8 shows the pattern out to D0A = 128 for three cases: de-focusing aberrations of 8, 12, and 16 wavelengths. Clearly, 16 wavelengths is too far, and 12 is questionable because of dark gray tendrils between the images. Only the frame depicting 8 wavelengths seems to show the diffraction disks as truly isolated.

We should only expect the images to be free of interference if we can fit a whole defocused disk into the darkness between the images. The geometric radius of an image in reduced angle is 4 times the number of wavelengths of defocus. This result may be derived by considering Eq. 5.1,

Fig. B-8. Interference between side-order images in ASYMM. Note that the reduced angle is very large at 128