Fig. 3.10. Spot-diagrams for the Schwarzschild telescope 1905 [3.1] for an aperture of 1 m with f/3.0
[3.23(a)], two such telescopes of significant size with 24-inch and 12-inch apertures were made between the world wars in the United States. It seems unlikely this system will be manufactured again today, except perhaps in the Couder modification discussed below. But the precise form of the original Schwarzschild proposal is completely unimportant. His aims were later satisfied by Schmidt telescopes, and either by the primary foci of large telescopes with field correctors, or by Cassegrain foci with focal reducers. The fundamental importance of his work was the theoretical formulation in third order theory which opened the design path to all modern telescope solutions.
Before we leave the subject of the original layout of aplanatic telescopes, it is instructive to follow the elegant graphical demonstration of the application of the sine condition to their design given by Danjon and Couder [3.24(a)], from which Fig. 3.11 has been reproduced in our nomenclature. The illustration refers to the RC modification of a classical Cassegrain, but the construction would apply to any other aplanatic modification. In Fig. 3.11, Mi and M2 represent the poles of the mirrors of a classical Cassegrain telescope forming an image at I2. The principal plane in the Gaussian sense is P0 Pq , giving f' = P0I'. A ray at finite height Yi from infinity strikes the RC primary at Arc and is reflected from the RC secondary at Brc to I'. From the sine condition, obeyed by the aplanatic RC telescope, the projection of I' Brc back to the incident ray must cut that ray at PRc , a point on the sphere P0PRc centered on I', according to the relation sin^ = f' (3.165)
from Eq. (3.85). Now, as stated in § 184.108.40.206 above, the elegant and simple form of the Abbe sine condition reveals that it takes account of all the aberration orders of coma in the aperture, although only the third order in the field. It is therefore a fundamentally different mathematical formulation of aperture aberrations from that of the Characteristic Function and third order Seidel theory. This is why the conversion from the sine condition to the corresponding third order coma, leading to the Staeble-Lihotzky condition, is by no means trivial. We require a similar transformation here to demonstrate the geometrical consequences of Fig. 3.11.
The treatment is much simplified if we apply an equivalent normalization to that of Table 3.2, but adapted to this special case, for which the second principal plane of the system P0 PG must be replaced by the reference sphere P0 PRc for aplanatism. For the aperture aberrations spherical aberration and coma, the only limitation to the aperture is when the reference sphere becomes a complete hemisphere. The maximum possible aperture angle U' is then 90°. We shall therefore start with the normalization f' = Yi = sin U' = 1, where Yi and sin U' are real ray parameters normalized to 1 at the maximum aperture. The enormous reduction of Y1 and sin U' in a real system will be applied later to link the conclusions with normal formulae for third order coma.
For the classical Cassegrain with its slightly steeper (parabolic) primary, the incident ray meets M1 earlier at the point Ac . The classical secondary is also less eccentric than the RC form and therefore also lies to the left of the RC secondary. The result is, as shown, that the classical Cassegrain gives a slightly smaller UC angle than URc, the difference being ¿U'. This leads to the error ¿f' in the sine condition of
Treating sin URC = 1 as a constant and differentiating Eq. (3.165) with respect to f' shows at once that
This is the excess of the length of the chord of the projection of the reference sphere P0PC to the vertical through 12 beyond the radius f' = 1 of the reference sphere P0PRC. We now require the angular error ¿U' corresponding to this offence against the sine condition ¿f'. Differentiating Eq. (3.165) with respect to U' with constant f' leads to an indeterminate result (infinity) for ¿U' because at U' = 90° the reference sphere is orthogonal to the Gaussian Y-axis P0PG! We must therefore replace the vertical axis by the reference sphere Pq PRc and measure Y1 along this circle up to URc = 90°. Setting then ¿Y1 as the increment at URc = 90°, we have at once
It follows that, for this extreme normalized case with U' = 90° and U' negative (see Table 2.2 and Fig. 3.11: with U' negative, #U' is positive, also sin U' is positive as shown in Eq. (3.165)),
Now the sine condition requires that this condition be met by all zonal apertures smaller than this maximum. However, as we have seen, the reference to a Gaussian principal plane represented by the Y1-axis breaks down completely at the limit angle U' = 90°. In fact, the definitions relating to are only strictly valid for the paraxial region. Beyond that, as Fig. 3.11 shows for a large angle U', the reference to the Y1-axis no longer gives the same simple results for f' and #U' in terms of ¿Y1, as revealed at once by Eqs. (3.165) and (3.166). This is because of higher order aberrations above the third order, which are automatically taken account of with the sine condition and the reference sphere Pq. For zonal rays, the terms emerging from Eqs. (3.165) and (3.166) involve trigonometrical functions which are inconvenient for conversion to normal third order aberration formulae. However, Eq. (3.167) provides a very convenient link. Since third order formulae are entirely based on paraxial parameters, the approximations involved are completely acceptable. We must also bear in mind for the construction of Fig. 3.11 that normal modern RC telescopes rarely have a final image f/no N < 10. Ritchey's first major RC telescope (see § 5.2) was exceptionally fast (f/6.8) for photography at that time. With N = 10, sin U^C = 0.05, U^C = 3.18° and cos U^c = 0.9987, a negligible difference from 1 and proving the high accuracy of third order theory for such telescopes when marginal ray data is used to establish the paraxial parameters used in third order aberration formulae.
We consider now Fig. 3.11 from the point of view of such real apertures. For the classical telescope, the projection of the emergent ray meets the incident ray at PC instead of P^c and the circle Pq PC has a longer radius than that required for aplanatism Pq 12 = 1. Since cos U' ~ 1, the extension f' is given by where cc is the curvature for the classical Cassegrain and 1 the curvature required for aplanatism.
If we now consider an oblique beam at a small field angle producing field coma in the classical Cassegrain, then the angular aberration of the equivalent oblique finite ray to the ray P^cArc which defines the sagittal coma is simply #U', the equivalent angular aberration to f' in Eq. (3.168).
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