Following Schroeder [3.22(e)], it is possible to transform the above formulae by replacing the parameter Ra by b = b/fi. From (2.65) and (2.72) one can deduce
a useful relation, since b is often a defining parameter in the layout of a telescope. In general, Eq. (3.415) shows that the despace coma will increase, for a fixed image position, if |m2| increases, since |Ra| will be reduced. This is simply another example demonstrating the increased sensitivity of small secondaries with high magnification |m2|.
Equation (3.420) is similar but not identical with the corresponding RC equation given by Schroeder [3.22(e)]. If this is transformed into the parameters of (3.420), the difference is in the first term of the square bracket i.e. (3 — 4Ra)/2(1 — Ra). Schroeder's form is (3 — RA)/2. With RA = 0.25, these two forms give 1.333 and 1.375 respectively. Since, in the RC case, this term involving £ is small compared with the residual terms corresponding to the classical Cassegrain if |m2| ^ 1, the difference in [d(#up)comat]Rc is very small in practical cases. For a case given by Schroeder with N = —2.5, N =10 (giving m2 = —4), b = —0.25 (giving Ra = 0.25), upr 1 = 18 arcmin and dd1 = — 0.001f, [d(<5up)Comat] RC is —0.25245 arcsec from Eq. (3.420) and —0.25312 arcsec from Schroeder's formula. A ray-tracing check performed with the highly versatile ACCOS V optical design program and using the above parameters, to give an exact calculation of the third order field coma arising from despace, gives agreement with Eq. (3.420) to better than 0.00005 arcsec.
Table 3.21 gives the despace angular field coma for the different telescope forms under the same conditions as for the despace spherical aberration with dd ]_//' = —4 ■ 10-3. The semi-field angle in Eq. (3.415) is upr 1 = 15 arcmin, a typical field for RC telescopes. As is expected from the large field coma, the SP and SPG forms are by far the worst, the SPG value being very large. This arises not only because the SPG supplement term in (3.415) is larger, but also because it is additive to the classical telescope terms, whereas in the SP case it is subtractive. The Cassegrain SP form is, for all aberration effects, superior.
One reason for this is that the Gregory secondary, for the normalized geometry of Table 3.2, is markedly stronger than the Cassegrain secondary. However, this normalization is somewhat unfair to the Gregory. Since the Cassegrain and Gregory telescopes of Table 3.2 have the same values of |Ra| and |m2|, the final image position is not the same, because L = |Ra| in the normalized system and d 1 is not the same in the two cases. From (3.421) we have b = —0.125 for the Cassegrain systems with m2 = —4 and +0.325 for the Gregory systems. For an aplanatic Gregory (AG) with b = —0.125, the value of Ra is —0.375. Since Ra enters into the denominator of (3.420) linearly, this larger axial obstruction gives a more favourable value of [d(5u'p)Comat] AG for the same image position as the Cassegrain, namely +0.621 arcsec compared with the value in Table 3.21 of +0.898 arcsec with Ra = —0.225. This is a further illustration of the price paid in sensitivity with small obstruction, high magnification secondaries.
In practice, the despace sensitivity to field coma is of little significance in non-aplanatic 2-mirror telescopes, since the field coma is not corrected. However, this is no longer true if field correctors are added which are also intended to correct the field coma (see Chap. 4).
The important effect of introducing decentering coma, uniform over the field, was treated in detail in § 3.7.2. This adds vectorially to the field coma in non-aplanatic telescopes, giving a transverse shift of the image point in the field which is free from coma. For aplanatic telescopes, the decentering coma is not compensated at any field point and the coma-free property of the system is lost. Thus decenter is particularly serious for aplanatic telescopes, even though the formal sensitivity to decentering coma may not be significantly more than with classical forms.
Lateral decenter of the secondary (see Fig. 3.97) also produces a Gaussian lateral shift of the image. The change in the direction of the beam is twice the slope of the secondary corresponding to the height of the decenter S. Now, from (3.2)
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