Eyepiece Travel and Defocusing Aberration

This book uses the generic unit of defocusing aberration when referring to distance out of focus. However, the most convenient way of thinking about defocus is in terms of eyepiece travel. Here, we wish to determine the amount of difference between two spheres at the aperture, one centered on focus f and the other centered on a defocus position/. We will then relate that very tiny distance at the aperture (the defocusing aberration) to the relatively large amount we have moved the eyepiece between f and/. This situation is depicted in Fig. 4-9.

The derivation proceeds from the difference between the two wavefront sagittae, or how much the wavefront is "cupped" at the aperture. Another common sagitta involves surface shape (or the shallow depth of the mirror itself). It is half the wavefront sagitta. Don't confuse surface sagitta with wavefront sagitta.

Fig E-1. Geometry of the sagittal relation. Focal length f is the radius of curvature of the wavefront.

Fig E-1. Geometry of the sagittal relation. Focal length f is the radius of curvature of the wavefront.

338 Appendix E. Eyepiece Travel andDefocusing Aberration

If the distance to a focus position is f and if D is the diameter of the aperture, then by the Pythagorean theorem, the sagitta s is

If the focal length is much greater than the aperture diameter, this equation may be approximated by performing a Taylor expansion. Such an approximation results in D2

Taking the difference between two different wavefront sagittae in Eq. E.2,

Next we demand that the quantity s — s' be thought of as a certain number of wavelengths "n1" out of focus, where 1 is the common symbol for wavelength. Noticing that ff is to a very high degree of precision just" equal to the average focal length squared, then

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