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The minimum transverse aberration will result for that plate profile in Fig. 3.28 giving the same numerical value of the slope at the edge of the pupil with ppi = ± 1 as for the minimum given by (3.246). This is the case for kpi = 1.5, as is easily confirmed as follows.

Inserting kpi = 1.5 in (3.246) gives (ppi)min = ± 0.5. The slope function at the point (ppi)min = +0.5 is then

At the point ppi = +1, the slope is +1. This shows that the form given in Fig. 3.28 with kpi = 1.5 is the most favourable for the focus balance of sphe-rochromatism if this is measured as the transverse (or angular) aberration. The effect of this optimum focus term can be understood in the following way. The central wavelength A0 is corrected for a certain axial focus position. For any other wavelength A1, the focus shift is such that the disk of least confusion of its spherical aberration arising from the spherochromatism is at the same focus position. This brings the same factor-of-4 advantage compared with a pure y4-plate with kpi = 0 that we had in Eqs. (3.186) and (3.190).

The diameter 5r/BF of the transverse aberration at this optimum focus can be derived from

VA dppi the factor 2 in this equation arising from the fact that the maximum slope of the wavefront slope function is present at ppi = ± 0.5 and ± 1.0 with opposite signs. From (3.245) this gives

Since, with the optimum focus kpi = 1.5, the maximum slope is also at the edge of the pupil (plate), we can put ppi = 1 in (3.248), giving

for the diameter of the transverse aberration.

This important result can be confirmed from the general formula (3.189) for the transverse diameter of the disk of least confusion

5rBf = f1 (SW'i*)gf , where GF refers to the Gaussian focus and BF to the best focus. At the Gaussian focus kpi = 0, giving with y1 = D/2 from (3.243) and (3.244)

since ppi = 1 for the edge of the pupil, thus confirming (3.249).

In the usual measure of angular diameter, the diameter of the spherochro-matic aberration is

In using this formula, it should be remembered that va is only the classical Abbe number if the same dispersion range is used. The general form is given from (3.244) as

in which n0 refers to the correction wavelength A0 in laying down the plate profile and n1 to any given subsidiary wavelength A1 for which the spherochro-matism is to be calculated. Bahner [3.5] gives the example of a Schmidt plate made of the Schott glass UBK7, for which the value of va in (3.251) is 100 between A0 = 430 nm and A1 = 386 nm in the blue or 490 nm in the red. With a typical value of N = 3, Eq. (3.251) gives an aberration diameter (100% geometrical energy) of 0.60 arcsec. Bearing in mind that the spectral bandwidth of 104 nm is not very large, it is clear that spherochromatism is by far the most significant residual error in the normal Schmidt telescope with a singlet corrector plate.

The largest Schmidt telescopes have apertures of 1 m or more. The two largest are at Palomar Observatory with dimensions for plate diameter (aper-ture)/mirror diameter/focal length of 1.22/1.83/3.07 m and a maximum field of 6.5° x 6.5° [3.33], and the Universal Telescope of the Karl Schwarzschild Observatory at Tautenburg in Germany with 1.34/2.00/4.00 m and a maximum field of 3.4° x 3.4° [3.34]. This latter instrument can also be used as a quasi-Cassegrain.

It is instructive to see the extent of the aspheric deformation required for a Schmidt plate of large size. If the values kpi = 1.5, N = 3, n' = 1.5 and D = 1000 mm are inserted in Eq. (3.242) defining the plate profile, the maximum departure from the plane-parallel plate occurs at the "neutral zone" where the plate thickness is a minimum. This radius is given from Eq. (3.245) for the slope function by

4pfi - 2kpiPpi = 0 , with kpi = 1.5. This gives the neutral zone at ppi = 0.866. Inserting this value into (3.242), with the other parameters defined as above, gives the maximum asphericity of the plate as only (dpi) - (dpi)0 = 0.081 mm.

3.6.2.3 Aberrations and compensation possibilities of a plane-parallel plate (normally filter) in the convergent beam. Plane-parallel glass plates in the form of colour filters are an essential adjunct of astronomical observations. Such a plate in the incident (parallel) beam has no effect, as mentioned above in connection with the finite thickness of Schmidt corrector plates. However, a plane-parallel plate placed in a converging (or diverging) beam has important aberration effects, all of which depend linearly on its thickness d. These are illustrated here with reference to the Schmidt telescope but are quite general for any telescope system. Since the commonest use is for filters, we will characterize the aberrations of such a plate by the suffix Fi.

A plate of thickness d and refractive index n' produces a focus shift

It also produces the following third order aberrations [3.3] [3.6]:

Apart from the specific form of (Si)Fi, the influence on (Sii)Fi ... (Sv)f% is determined by the same factor (A/A) which we encountered in (3.20) and in the general third order theory of telescopes: for a plate in a centered system, i pr upr and i'

&', where u' is the semi-angle of the exit beam. In most practical cases with small angular fields, the factor (upr/u') ^ 1 so that the effect on the spherical aberration is dominant. The sign of (Si)fi implies overcorrection i.e. the marginal ray focuses too long compared with the paraxial ray. The wavefront aberration is, from (3.21): 