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Cassegrain Optical Mathematics |
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by Albert G. Ingalls |
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In this department in September, 1942, Capt. Alan E. Gee described his solution of the old problem of testing a Cassegrainian secondary. He proposed to grind and fine-grind the convex mirror to a sphere; then set it aside and polish, or partly polish, the concave tool; next, figure this concave tool to the required hyperboloid, using the formula at the top . in Figure 1 with the Foucault test. Finally, figure and test the convex against this prepared concave by means of interference fringes. Commenting on this proposal, F. A. Lucy, 3427 W. Penn St., Philadelphia, Pa., writes: "Capt. Gee's method of figuring a Cassegrainian secondary, appears to be the best yet offered, and deserves all the emphasis that can be given it. "Hindle's test for an hyperboloid ('A.T.M.' Part X) has advantages over Ritchey's autocollimation method: it shows more of the mirror surface and is less troubled by diffraction effects. Further, there is only one auxiliary surface instead of two. Gee's test has the same advantages more strongly: it shows all the mirror surface, is still legs troubled by diffraction, and uses no auxiliary surfaces, except insofar as the test-plate is considered one. "It is true that the tests of Hindle and Ritchey employ a double reflection from the mirror, thus doubling the sensitivity, but a test-plate set up for Gee's test will be much closer to the knife-edge and, therefore, will have a larger apparent angular diameter. "Geometrically, then, the tests are about equal in sensitivity but, as a matter of psychology, it is easier to judge contrasts in a large field. "On the whole, finally, Gee's method appears to be capable of yielding the most accurate results of any. "In like manner, an ellipsoidal mirror (for example, a Gregorian secondary) will appear more than twice as large when set up for c. of c. testing as when set up for the direct focal test-again with the least possible trouble from diffraction. Because of its greater speed, the direct focal method might be preferable for preliminary figuring, reserving the zonal survey for final work. "With regard to the application of this test to convertible telescopes, a compound telescope is generally made with a paraboloidal primary of fairly large focal K ratio, so that it may be used at the primary focus when desired. For looking at lunar or planetary detail, or for resolving close stars, the compound system is, however, preferable. The necessary high magnification can be obtained by using a short-focus paraboloid and a sufficiently small eyepiece; but if the latter has a Ramsden disk less than 2mm in diameter, the resolving power of the eye is reduced, and the use of still smaller eyepieces will give less detail rather than more in the image. It is better to use an objective of long focal length. If compactness is to be maintained, the Cassegrainian arrangement is generally considered the best way of doing this. "While the classical Gregorian arrangement is less advantageous than the Cassegrainian, an off-axis Gregorian would be superior in at least one way to an off-axis Cassegrainian. In the Gregorian, the active part of the secondary will be below the axis when the active part of the primary is above. Thus the primary can lie very close to the axis with no shadowing at all by the secondary. An off-axis Gregorian, indeed, can be made even closer to the axis than can an analogous Newtonian; thus securing the advantages (A. A. Michelson, Physical Review, 20, 391 [1905]; N. J. Schell, Sci. Am. Telescoptics, April 1939, May, 1940) of this type (no shadowing, minimum diffraction) with the least extreme focal ratios and least difficulty in figuring. Incidentally, Gregory's original design has the eyepiece a considerable distance behind the primary, with a tube long enough and narrow enough so that direct sky light is kept from striking the eyepiece: the secondary shades it. This detail usually appears to be neglected in diagrams of the Gregorian construction. "It follows that the convertible telescope of best performance would be an off-axis Gregorian with paraboloidal primary; although it would be easier to figure an on-axis Cassegrainian. In either case, the equations of Selby and Gee offer the best guide to the figuring (see below)." When Capt. Gee first offered his data he included a mathematical proof, but this was not published-complicated formula matter is difficult to set correctly in type. Now comes Lucy with a condensed derivation, which follows. In it, the numbers in parenthesis refer to the numbered equations in Figure 1, written out by Lucy and reproduced by line-cut (one way to get around the above problem, mathematician-compositors apparently having all marched off to war). Lucy states: "From a text on analytic geometry, one sees that, measured from the center of an hyperboloid, the x intercept of the normal is (1), where (x2, y2) is the zone on the test-glass through which the normal passes. The fundamental equation of the hyperbola is (2). "Substituting (2) in (1) gives an equation for the intercept as a function of the zonal radius y2. In discussions of mirrors, y2 is usually called r, and will be below. As design constants, it is convenient to use, not the semi-axes a and b, but p and p', the latter two being respectively the distances from the secondary vertex to the primary focus and to the focus of the combined mirrors. "By construction, (3) follows: "Referring again to a text on analytic geometry, one sees that a2 + b2 is the square of the distance between the center and a focus. By construction, and with the use of (3), we have (4). "With these substitutions, Capt. Gee obtains an equation for the radius of curvature (in the sense of the Foucault test). Going a step further, one may subtract Ro = 2p'p/ (p' - p), the radius of curvature for the central zone. Reduction gives directly the knife-edge displacement for the zone of radius r (5).
This is for use with a testing set-up in which light source and knife-edge move together. If the knife-edge moves while the light source is stationary, the displacements are doubled, and the 2 in the denominator of (5) should be cancelled. "H. H. Selby ('A.T.M.A.,' p. 134) has already given the corresponding equation for an ellipsoidal mirror. His equation may be derived by the process just described, using the ellipse analogues of (1) (2), and (3). The foregoing equations are all exact." Subsequent to receipt of the above note by Lucy, there came from Captain (now Major) Gee, Corps of Engineers, APO 702, care of Postmaster, Seattle, Wash. the following communication. "In the June, 1938, number of Scientific American, Kirkham furnished equations and dope concerning the construction Cassegrainians with spherical secondaries and modified primaries of predetermined amount of spherical aberration to match MacIntosh, Conner, and I applied this method to the 20 1/2" telescope we made in Portland; also to a 12 1/2" telescope now being completed there. On both of these telescopes I am interested in making high magnification secondaries to use alternately with the 3X secondaries for which both instruments were designed. With this in mind, I have worked out the necessary math to determine difference in radii of curvature of the zones of such a secondary, as measured by the King test or on the polished tool (for test of the convex by interference fringes). My equations are general and simple, so the dope may be of use to other ATMs who desire alternate secondaries for their modified Cassegrainians. "Let a1 be the spherical aberration of the primary (this would have been determined by Kirkham's equation when designing the original telescope); r2 the radius of the new secondary to the margin of the area reached by light incident parallel to the axis, p' the distance from new secondary to secondary focus; and the distance from new secondary to prime focus. Then "R = (2p'p)/(p' - p) = radius of curvature of new secondary (actually R of central zone) and a2 = [(R + p,)2 /(R + 2p')2] (r2)2 R. (The formulas are recast somewhat to permit them to be set on a linotype machine.-Ed. ) "Desired knife-edge movement between edge zone and central zone of new
secondary then is d' = "The above equations are all derived for direct substitution of absolute values. In other words a1, r, p', p, R, and a2 are all positive values. "If a1 is greater than a2, edge zone will have shorter R than center zone. If a1 is less than a2, edge zone will have longer R than center zone. "Derivation assumes a stationary pinhole and moving knife-edge or grating. "For intermediate zones, measured correction is proportional to square of the radius of the zone; that is, for a zone half way between center and edge, one fourth of the correction, d'/4, would be applied. "The curve on these secondaries would approximate an oblate spheroid on secondaries smaller than the one for which the primary was designed, and a prolate spheroid for secondaries larger than the original. "The above method could be applied to make a secondary for a Cass with spherical primary. In that case a1 = r2/4R, where r is the marginal radius of primary and R the radius of curvature of primary. Secondary would approximate an oblate spheroid. "It is actually possible to make a properly corrected telescope with both primary and secondary spherical but unfortunately, the proportions are highly impractical. "Here is one more suggestion. Some ATMs who have conventional f/8 or f/10 Newtonians could convert them into RFTs by making a concave secondary for them to be placed inside focus. For a paraboloidal primary, the secondary hyperboloidal and the various knife-edge readings would be given by the equation I furnished in the September, 1942, number (first equation, Figure 1). For this special case, p' would be the distance from secondary to prime focus, and p the distance from secondary to secondary focus (just the reverse of the usual Ritchey notations). R = (2p'p) /(p' - p) would still give the radius of curvature. "If it were desired deliberately to introduce spherical aberration to
correct for RFT eyepiece aberration, as suggested by Kirkman, my equation
d' = Maj. Gee's comments were next shown to Lucy, who then added: "His equation for the hyperboloid to be used with a given paraboloid is exact; whereas the corrections in the modified cases are approximate. For that matter, Kirkham's equation is also approximate, as Kirkham himself made clear. The adequacy of the approximation depends on the size of the telescope, the focal ratio of the elements used, and on the severity of the selected tolerances. A modified telescope, made precisely to these equations, might require further correction by the aid of optical tests on the assembled instrument. "I should be interested in hearing how these modified Cassegrainians actually perform when first assembled, especially when they are made to the extreme specifications of the Walkden RFT ("one-gallon" Cassegrainian)." By accident, your scribe has just discovered that Lucy is the author of a two-part, eight-page article entitled "Exact and Approximate Computation of Schmidt Cameras," Journal of the Optical Society of America, June 1940 and May 1941. In this two-part article, the first half discusses the Schmidt design in a mathematical manner, while the second part discusses several modifications: a reversed plate Schmidt, solid Schmidt, thick-mirror Schmidt. Your scribe is collecting names of ATMs who go in for the math stuff. Such a group might become a sort of "Design Club," not formally organized but at least kept in closer mutual touch than at present, simply by being made known to each other. ATMs have asked this department about the content of some sets of salvaged lenses advertised elsewhere (page 275) in the June number. The Edmund Salvage Co. kindly gave us the requested data and a 70-lens set for inspection. The first set contains two lenses, 31mm diameter and 92mm f.l., two 33 x 221 ditto; two 37 x 393 ditto, two 42 x 152 ditto; two 18 x 50 biconvex; two 17 x 58 plano-concave, two 14 x 35 ditto; and one 14 x 33 biconvex.
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THIS MONTH, let's turn over the meeting to the math sharks. For the rest
of us, what follows may have to be taken in low gear.