From Table 3.5, the field curvature is given by
giving for the normalized system S/y) = -1.6
From the last column of Table 3.3, the effective field curvature is given by
(EFC)schw. =+1.5 - 1.6 = -0.1 , a value accepted by Schwarzschild as negligible. This small undercorrection of E S/y by 2 E S/// accounts for the small imbalance of the angular sagittal and tangential astigmatism values quoted in Table 3.7. The very large residuals (by today's standards) of —7 arcsec and +9 arcsec respectively reflect the coarse emulsions of the time. They also demonstrate how desirable was reduction of astigmatism in addition to aplanatism.
If m2 is maintained unchanged (that is, the final f/no is unchanged) while L and Ra are reduced to improve the obstruction, then |dx| increases from (2.75) and E S/// reduces from (3.162). But the negative value of E S/y increases rapidly, giving a rapid increase in the negative EFC residue from (3.164). Thus the effective flat-field condition forced Schwarzschild to accept the relatively high obstruction ratio. We shall see below that the Couder modification, which abandons the flat-field condition, opens up further possibilities.
The asphericities represented by bs1 and bs2 are relatively extreme forms. However, the hyperbolic form of the primary is not excessive in its deformation because the curvature is weak (f/7.5), as given by Eq. (3.11). The oblate spheroid of the secondary is, in practice, much more difficult because of test problems of this form.4
Table 3.8 gives a comparison of some fundamental parameters showing the evolution of the aplanatic telescope from Schwarzschild to modern telescopes, taken from ref. [3.13]. The modern RC telescope has a high value of m2 and a primary form scarcely departing from the parabola. The hyperbola on the secondary is also modest compared with the original RC of Chretien. The trend is determined by the increasing value of |m2|, resulting from steeper primaries and a final f/no which would have been too slow for the emulsions available to Schwarzschild. With large |m2|, the aplanatic solution is converging rapidly towards the common aplanatic/classical form of the afocal telescope.
Having derived his aplanatic, effective flat-field solution from third order theory, Schwarzschild applied the Abbe sine condition and his "Winkelei-konal" to the calculation of the total optical path of the rays forming the image in the field - see § 188.8.131.52. He thus calculated the higher order aberration effects, a beautiful and complete analysis of the imaging properties of the system in the pre-computer age. The constant optical path requirement was expressed as a differential equation in terms of polar coordinates of the secondary mirror. This led to an explicit analytical form for the polar equation of the secondary and then for the primary. Conversion to Cartesian coordinates and application of the sine condition of Eq. (3.84) led to explicit expressions for the forms of the two mirrors as a function of the aperture angle U'. These were then converted into infinite series giving the requirements for the correction of successive aperture orders of spherical aberration and coma, the terms up to y4 agreeing with those deduced from third order theory. An excellent treatment of the general approach of such methods, all a consequence of the Fermat principle discussed briefly in § 2.2.2, is given by Schroeder [3.22].
Figure 3.10 shows the spot-diagrams for the original Schwarzschild telescope for an aperture of 1 m with f/3.0 and a field of ±1°, plotted on a flat field as he intended. The astigmatic limitation, growing with the square of the field according to Eq. (3.21), is very evident.
Schwarzschild's proposed form was the logical solution for his time, a fast system with a relatively slow primary. According to Dimitroff and Baker
4 The reason why an oblate spheroid form is normally more difficult to test lies in the technology of null-testing. This fundamentally important test technology is treated in §§ 1.3.4 and 1.3.5 of RTO II. A negative Schwarzschild constant bs (the normal case for telescope mirrors) requires a positive lens to compensate its aberration. A positive lens also forms a real image, which is essential in the test procedure. However, a positive bs, corresponding to an oblate spheroid, requires a negative lens for compensation, producing a virtual image. A real image can then, in most cases, only be produced by a further (spherical) concave mirror, since positive lenses neutralize the desired aberration of the negative lens. The natural compensation for a negative bs is illustrated by Eq. (1.77) in RTO II.
File : C:\ZEMAX-EE\SCHWARZ.RAY Title: SCHWARZSCHILD Date : Fri Mar 03 1995
GENERAL LENS DATA:
System Aperture Ray aining a F/#
sing F/# Obj . Space N.A. Stop Radius Parax.Ima. Hgt. Parax. Mag. Entr. Pup. Dia. Entr. Pup. Pos. Exit Pupil Dia. Exit Pupil Pos. Maximum Field Primary Wave Lens Units Angular Mag.
Field Type: Angle # X-Value
1 0.000000 2 0.000000 3 0.000000
3 Pupil Diameter
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