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(HdE)SI

Si , Sii, Siii, Siv , Sv are the aberrations present with the original stop position. We have already encountered the factor (HdE) in Eqs. (3.20) and (3.22). dE is a normalized factor expressing the shift in stop position, H the Lagrange Invariant (Chap. 2) which is a measure of the throughput of the system. If Si ... Siv are all zero, a stop shift has no effect on any of these third order aberrations. If - as in the case of the error in HST which is on the primary mirror - the system has a finite Si but the other terms are zero, then a change in coma, astigmatism and distortion occurs through the right-hand terms containing Si in (3.213). The other terms will be zero in such a case.

The above "stop-shift" effect is closely linked to the effect of inserting an aspheric plate into the stop of a system and then shifting the plate away from the stop by a normalized distance dE. This is shown in Fig. 3.23. If the

Stop or pupil

-Profile z ay4

Fig. 3.23. Stop-shift effect for a single third order aspheric plate shifted from the pupil aspheric plate is inserted at the stop (or its image, the pupil), it acts, to third order accuracy, as a "pure" element in its monochromatic function: it only affects Sj (wavefront aberration is also a fourth power function of aperture), all field effects are zero, so that we have:

Sj

= SS/

SJI

= 0

Sjji

= 0

Sjv

= 0

Sv

If now the plate is shifted from the stop by dE, then substituting (3.214) in (3.213) gives:

This result shows the origin of the right-hand, aspheric, terms in Eqs. (3.20). The term "aspheric plate" has a much more general significance than the normal refracting plate shown in Fig. 3.23. It applies equally to any fourth power figuring modification to a refracting lens surface, plane mirror or mirror with optical power (curvature).

The influence of a given corrector plate, producing a change Sj in spherical aberration, on the field aberrations depends then only on Sj and the parameter (HdE). This has a very simple geometric optical significance. It can be shown ([3.3] [3.6]) that

where ypr and y are simply the heights of the paraxial principal ray and normal (aperture) paraxial ray at the plane of the system where the plate is inserted. Fig. 3.24 shows schematically the way the heights of these rays vary as they pass through a typical Cassegrain telescope. The aperture ray is shown by a continuous line, the principal ray by the dashed line, the stop being placed at the primary. Clearly, the ratio ypr/y is small in the object space above the primary for normal small semi-fields upr and grows linearly with the distance above the primary. After reflection at the primary, ypr at the secondary is the same as in the object space, but y suffers a dramatic reduction as the aperture ray converges towards the prime focus. The smaller the secondary (i.e. the higher the secondary magnification m2), the bigger the increase of ypr/y at the secondary reflection. In such circumstances, SSji in (3.215) can be quite large. As will be shown in RTO II, Chap. 3, this is the y

Image

Fig. 3.24. Heights of the paraxial aperture and paraxial principal rays as they pass through a Cassegrain telescope reason why the theoretical possibility of correcting the spherical aberration of the HST primary by modifying the secondary must be ruled out, even if it were possible in practice, because the field coma SSu in (3.215) is far too large at a field radius of 10 arcmin to be acceptable. However, ypr/y is in this case still ^ 1. This means the field astigmatism SSm is much smaller than SSn.

After reflection at the secondary, the principal ray diverges rapidly because of the telephoto effect of the Cassegrain form, whereas the aperture ray converges to the final image. At some point, depending on the field chosen, the rays intersect giving ypr/y = 1. Further down, ypr/y > 1 and the effect on the astigmatism increases rapidly.

From (3.214), if correction of spherical aberration can be achieved at the pupil, monochromatically and to a third order the correction is perfect. But if a single corrector plate is inserted elsewhere with HdE = ypr/y = 0, then field aberrations appear according to (3.215). However, if two plates with significant separation are introduced, two conditions can be fulfilled; with three separated plates, three conditions. In other words, with three separated plates in any geometry, monochromatic residues of si, Su and Sm can theoretically be corrected in any telescope system. Whether the solution is practical will depend on the aberration values and the separations available in practice. This general theorem is of great importance in understanding the possibilities of wide-field telescopes (§ 3.6).

Let us consider again the practical example of the spherical aberration of the HST. One possible means of correction is a corrector in the baffle ("stovepipe") above the image. A single aspheric plate generates unacceptable field aberrations because of the large ypr/y values. But three separated plates can fulfil the three third order correction conditions

E Sii = ki(Si )i + k2(Si )2 + k3(Si ) 3 = 0 ESiii = k?(Si )i + k22(Si )2 + k32(Si )3 = 0

in which ki, k2, k3 represent HdE = ypr/y for the positions of the three plates. Inversion of the matrix (3.217) gives the required aspheric powers of the plates. As a plausible example, k1, k2 and k3 were given values 1, 1.5 and 2 respectively. Solution of (3.217) gives

(Si )i = +6(SSi ) (Si )2 = -8(SSi ) (Si )3 = +3(SSi ),

giving E Si = +SSi as required. However, the price paid for the field correction with plates in these positions is an increase of individual aspheric powers up to eight times the required correction. This is simply the equivalent phenomenon to the fact that the powers of the individual lenses of an achromatic doublet or triplet camera objective are much higher than the resulting total power of the system. Of course, in practical optical design, compensations with quadratic and even sixth power terms must also be taken into account rather than the simple plate shown in Fig. 3.23 with a pure fourth power term. Aspheric plates have the merit of correcting aberrations without affecting the Gaussian optical terms associated with the optical power of the system: specifically they do not change the f/no of the transmitted or reflected beam.

Following Bahner [3.5], we can use the general formulation of § 3.2.4 for the third order aberrations of 1-mirror and 2-mirror telescopes to derive relations for such telescopes when an aspheric plate is added. Tables 3.4 and 3.5 gave the general formulae, whereby the terms in the square brackets correspond to the normalization of Tables 3.2 and 3.3 with

The formulation of (3.215) and (3.216) above assumes that the pupil is initially at the primary (i.e. spri = 0), the plate shift from it being expressed by HdE = ypr/y. (We show below that, finally, the pupil position has no effect on the first three third order aberrations, provided that the first two are corrected to zero). With the normalization, the quantity ypr/'V = spi/f', where spl is the plate shift relative to the primary. If the plate introduces third order spherical aberration SSf, then the equations of Table 3.4 for a 1-mirror telescope with an aspheric plate become from (3.215), with spri = 0:

ESiii =

upr 1

pr 1

We shall see in § 3.6 that these equations lead at once to the Schmidt telescope.

Similarly, if the same addition of an aspheric plate is made to a 2-mirror telescope, the equations of Table 3.5 give, by the same reasoning:

upr 1

If these three equations are set to zero, then such a system of 2 aspheric mirrors and an aspheric plate can always give a solution with SI = SII = SIII = 0 for any geometry (generalized Schwarzschild theorem [3.13]). Now we can apply a very important consequence of the stop-shift equations (3.22). If the first three Seidel aberrations have been made zero in Eqs. (3.220), then all three are independent of a stop shift in the total system. This means that the entrance pupil can be shifted to the aspheric 'plate (or elsewhere) without changing the third order correction of E Si, E Sn, ES///. This is a very important general principle in the theory of wide-field telescopes. For the 1-mirror telescope with an aspheric plate of Eqs. (3.219), in the general case, only two conditions can be fulfilled with two aspherics, giving SI = SII = 0. But a stop shift will still not affect SI, E S/i and E S/ii, even if E S/ii = 0. Of course, higher order aberrations are not independent of stop shifts and may have a determinant influence.

If the aspheric plate is in the object space, as in the Schmidt telescope, then sp; can be applied directly as a negative quantity, i.e. with the same sign as f' in (3.219). For a plate inside the system, i.e. to the right of the primary for the positive light direction from left to right, a virtual positive value of sp; can be calculated from the Gaussian image of the plate backwards into the object space [3.28] [3.29]. The necessary expressions are given in Chap. 4 (see Fig. 4.2) in connection with field correctors consisting of aspheric plates, applied to 1-mirror or 2-mirror telescope foci.

Since third order spherical aberration varies, as wavefront aberration, with y4, the plate profile must obey the same law. From Eq. (3.21), its coefficient is given for a plate of refractive index n' placed in a medium of index n by

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