## Other Aberrations The Pupil Function

The tilt and defocusing aberrations are correctable errors. True, you can't see much if the image is defocused, but you can regain a crisp image with only a turn of a knob. Tilt error merely requires you to redirect your gaze to the center of the stellar image. The observer can't get rid of the smearing of the image caused by diffraction, but if the magnification is small enough, that spread is acceptably small.

Up to now, the discussion has been mostly about perfect apertures. What happens when some imperfection causes the wavefront to buckle as it passes through the aperture? Our little secondary sources, or radiators, are no longer on the surface of a sphere. They are confined to some sort of surface shape for which no unique focus position can be found. Asymmetries form in the node patterns (as in Fig. 4-14).

The best focus position no longer sees these radiators at a single phase. The wave sum can never equal the whole area of the aperture. Of course, for most of the realistic aberrations considered in this book, the errors at best focus will never get so large as to demand more than one coarse Fresnel zone to show it. Not until aberration amounts to 1/2 wavelength peak-to-valley will more than one Fresnel zone appear on the aperture at best focus. A more accurate way of doing the wave sum will show that the peak intensity is always less than unity for aberrated apertures.

These Fresnel zones are a good educational device, but they don't allow sufficient gradation to permit detailed examination of what is happening with radiators at the aperture. For that, we need to introduce the pupil function. For now, we can define the pupil function as a surface containing information about both the phase and the transmission of the aberration. It is defined more precisely in Appendix B.

The aberration distortion of the wavefront for simple defocusing appears in Fig. 4-15. The aperture is shown below the wavefront. Fresnel zones are contours of such a surface. The transmission portion of the pupil function

4.5. Other Aberrationsâ€”The Pupil Function

looks like a flat tabletop because this aperture has uniform transmission to the edge. Most of the pupil functions appearing later will have similar on-off transmission behavior. The aberration function is shown most often because it is generally the most interesting component of the complete pupil function.

Fig. 4-15. The aberration function of defocusing, with the aperture indicated in black below the wavefront surface. The direction of focus is up.

Figure 4-15 shows the defocusing aberration function outside focus. The aberration function inside focus is a low mound. For a perfect aperture, the aberration function would be a flat plate. All of the optical difficulties appearing in this book will have an associated pupil function. Far from being just a conceptual device, the pupil function contains the essence of image formation in the telescope. Mathematical operations on the pupil function lead to the diffraction pattern and the modulation transfer function.