While not superficially looking much simpler than Eq. B.l, the Fraunhofer approximation has done away with those very messy distances r and s (or at least they are now neatly tucked away). The angle (x is the angle to the image point in the x-direction, while (f>y is the angle toward the image point in the y-direction. The second exponential in the integrand is just the phase difference induced by the tilt angle to the image location being calculated. This term does the same thing as picking up the corner of a table and asking how much higher the table is all along its surface. Over the leg still touching the floor, the extra height is zero. At the lifted corner, it has its full value. Everywhere over the aperture, this "tilt" aberration may be easily calculated. For all cases of interest here, sin( is very close to (, which makes it even simpler.

In the Fraunhofer approximation, the inclination factor is ignored because of the very tiny deflections of the wavefront from its spherically converging path. N is the new normalization constant.

The Fraunhofer formula is not supposed to apply to non-zero values of defocusing aberration. However, the next term of the expansion of Eq. B.l used in deriving the Fraunhofer approximation is just this defocusing. Including this term changes the integral to the Fresnel approximation. However, an external defocusing cannot be imposed on the Fresnel approximationâ€”that step would doubly count the defocusing. It is tidier to encapsulate all such terms into the pupil function, which is then applied to the skeletal Fraunhofer formula.

The light touch of small amounts of defocusing can perhaps be gauged by the focus shift in Table 5-1 divided by the focal length of the instrument (which is the same as the fractional change in the sagitta of the wavefront). For example, at 12 wavelengths defocusing aberration on an 8-inch (200-mm) f/6 telescope, hf/f = 0.0016, a very tiny fraction.

The star test of a focused aperture does not precisely reproduce at equal distances inside and outside of focus, but the difference is small. For example, a 200-mm f/6 telescope defocused by 1.9 mm is listed in Table 5-1 as having a defocusing aberration of 12 wavelengths. A careful calculation shows that 1.9mm inside focus has a more precise aberration of 12.02 wavelengths. On the outside, an equal eyepiece motion becomes 11.98 wavelengths defocusing aberration. If we were to adjust the focuser until the aberration were precisely 12 wavelengths, then the image inside focus would be ever-so-slightly smaller and brighter, and the image on the outside of focus would be incrementally bigger and dimmer. Thus, the effect is manifested as a magnification difference. Perhaps if we were to defocus very precisely with a measurement screw, we could barely detect such changes, but most star testers will never notice the difference. We must simply ensure that the fraction hf/f is small (Bachynski and Bekefi 1957; Li 1982; Erkkila and Rogers 1981).

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