## Focuser Motion Related to Defocusing Aberration

The defocusing aberration was mentioned briefly at the end of Chapter 4, but no derivation of it was made. It is a simple expression that describes the differences in the sagittae of two different wavefront spheres.1 The difference between two eyepiece positions for a given number of wavelengths of defocusing aberration is derived in Appendix E. The result is where F is the focal ratio, X is the wavelength, and An is the change in defocusing aberration in wavelengths.

If n is allowed to go from + 1/4 to — 1/4, this quantity becomes Af = 4XF2. This is an expression for the depth of focus, or the maximum range for improperly setting the focuser. Since the diameter of the diffraction image is 2.44XF, the image spot is much longer than it is wide. In fact, the length-to-width ratio f-f = ¿f= 8F2Ank,

1 The sagitta is the amount by which the sphere protrudes through the aperture. Because the radii to the edges of this partial sphere look like stretched bowstrings, it was natural to name this quantity sagitta, or arrow. (See Figure in Appendix E.)

is nearly 25 for an f/15 instrument.

The length of this sausage-shaped region is very helpful. It permits small amounts of error in the setting of focus in accessory instruments (such as cameras). It makes less difference in adjusting visual focus because focus is usually fine-tuned by the eye itself. Only those observers who have had cataract surgery or who have little flexibility in their eyes may be forced to rely exclusively on the focusing action of the telescope.

Sidgwick gave another formula for depth of focus: Af = 4(1.22XF2). This factor is 1.22 larger than the one derived here (Sidgwick 1955, p. 425). The seeming discrepancy comes from the methods used to derive the expressions. Neither formula is meant to be a crisp limit, only the point where the image begins to degrade noticeably. Both expressions are proportional to the focal ratio squared. Thus, an f/5 telescope has only a quarter of the focusing tolerance of an f/10 telescope.

Tables 5-1a and 5-1b list eyepiece motions for varying focal ratios and defocusing aberrations. For example, if we defocus the image of an f/6 Newtonian by 8 wavelengths, we can see from the tables that we must change the focus by 0.050 inches (or 1.27 mm). In the convention used here, one must focus outward when the defocus aberration is given as a positive number and focus inward if defocus aberration is given as negative.

Much can be learned by carefully examining these tables. They show that defocus distances are vanishingly small for fast focal ratios. The first tabulated column is labeled by 0.5 wavelength of defocusing aberration, or about the depth of focus mentioned above. Yet to achieve focusing within wavelength at f/4 (or An = 0.5), one must hold focus to within 0.0014 inches, or 0.035 mm. Clearly, if our eyes weren't somewhat internally adjustable, we would struggle to focus fast instruments. Slow motion helical or motorized focusers would seem to be justified for these low focal ratio telescopes.

At the other end of the chart are extremely long focal ratios like f/22, which would describe two-mirror Kutter schiefspieglers. To induce 12 wavelengths of defocusing aberration in such instruments, we would have to move the eyepiece over an inch. It is apparent that on such slow instruments, we will scrutinize only small defocusing aberrations before running out of focuser travel. Long focal length telescopes, however, are usually lunar-planetary instruments. They are deliberately tested to higher standards, so small amounts of defocus are the most interesting. A focal ratio of 50 is included because you might mask down your instrument to see a supposedly perfect image.

On fast instruments, the star-test image will probably be evaluated at high values of defocus, beyond even 12 wavelengths. This is not too much of a problem because the test is still sensitive to the relatively severe

Table 5-1a Defocus distances for different focal ratios and defocusing aberrations (distances in inches) Wavelength is 2.165 x 10-5 in

Defocusing Aberration (wavelengths)

Table 5-1a Defocus distances for different focal ratios and defocusing aberrations (distances in inches) Wavelength is 2.165 x 10-5 in

Defocusing Aberration (wavelengths)

 0.5 1 4 8 12 Focal ratio 4 0.0014 0.0028 0.011 0.022 0.033 4.5 0.0018 0.0035 0.014 0.028 0.042 5 0.0022 0.0043 0.017 0.035 0.052 6 0.0031 0.0062 0.025 0.050 0.075 7 0.0042 0.0085 0.034 0.068 0.102 8 0.0055 0.011 0.044 0.089 0.133 9 0.0070 0.014 0.056 0.112 0.168 10 0.0087 0.017 0.069 0.139 0.208 11 0.010 0.021 0.084 0.168 0.252 12 0.012 0.025 0.100 0.200 0.299 15 0.019 0.039 0.156 0.312 0.468 22 0.042 0.084 0.335 0.671 1.006 50 0.217 0.433 1.732 3.465 5.197

Table 5-1b Defocus distances for different focal ratios and defocusing aberrations (distances in millimeters) Wavelength is 550 nm

 0.5 1 4 8 12 Focal ratio 4 0.035 0.070 0.282 0.563 0.845 4.5 0.045 0.089 0.356 0.713 1.069 5 0.055 0.110 0.440 0.880 1.320 6 0.079 0.158 0.634 1.267 1.901 7 0.108 0.216 0.862 1.725 2.587 8 0.141 0.282 1.126 2.253 3.379 9 0.178 0.356 1.426 2.851 4.277 10 0.220 0.440 1.760 3.520 5.280 11 0.266 0.532 2.130 4.259 6.389 12 0.317 0.634 2.534 5.069 7.603 15 0.495 0.990 3.960 7.920 11.880 22 1.065 2.130 8.518 17.037 25.555 50 5.500 11.000 44.000 88.000 132.000

aberrations that pester these instruments.

The best way of using Tables 5-1a and 5-1b is to look up the values corresponding to your focal ratio and write them down somewhere. It might even be convenient to calibrate your focuser knob. Rack it a full turn and see how much it advances focus. This procedure is easy on Newtonians and refractors. You just measure the change in the amount of protrusion in the focuser tube. For example, if one turn of the knob yields 3/4 inch (19.05 mm) of focuser travel, a 30° twist gives about 1/16 inch (1.6 mm).

This motion is equivalent to 10 wavelengths defocusing aberration for a telescope working at f/6.

On Cassegrain-type catadioptrics, it is less obvious how to tell which direction focus is tracking or how far it moves. These instruments usually achieve focus not by physically transporting the eyepiece, but by internally moving the primary mirror toward the secondary. First, focus the telescope with an eyepiece sitting firmly in its socket. Then, loosen the eyepiece and draw it 10 mm or so outside of the socket. Now, tighten the set screw. Find focus once again, being aware of the direction and angle that you have turned the focuser knob (it may help to stick a temporary pointer on the end of the knob). You have found the direction and amount of an effective 10 mm inward focus change. On my Schmidt-Cassegrain, this motion was counterclockwise.