Fig. 3.112. The MTF for a non-obstructed circular aperture with incoherent illumination in the presence of pure defocus aberration (after Hopkins [3.142])
Hopkins [3.145] proved the important principle that the introduction of aberration into a system will reduce the contrast for all frequencies compared with the non-aberrated diffraction MTF.
Further examples of the degradation of MTF by various combinations of aberrations are given by Wetherell [3.137].
It can be shown (see Wetherell [3.137]) from Eqs. (3.466), (3.491) and (3.494) that the Strehl Intensity Ratio is simply the area under the MTF function, normalized such that the area under the diffraction MTF in the absence of aberrations is defined as unity. O'Neill [3.146] first pointed out this important relation as a consequence of the fact that I(Q) and L(s,t) are Fourier transform pairs. This gives an excellent understanding of the physical link between the two approaches. For small aberrations, the MTF function is independent of the nature of the aberrations, only depending on their variance. But for large aberrations, both the form of the MTF and the diffraction image depend strongly on the nature of the aberrations. The integral under the function, i.e. the Strehl Intensity Ratio, cannot take account of these differences and is therefore an inadequate measure of the effect of strong aberrations.
The OTF may also be calculated as an approximation of geometrical optics: the larger the aberrations, the better the approximation will be. The formulation is given by Welford [3.6]. As was mentioned in § 3.3, the formulae of Nijboer [3.27] express the Fourier transform process of the rays from the pupil to the image. The intensity function Ig of the geometrical optical PSF is given by
Modern, sophisticated optical design programs can determine the geometrical PSF as a spot-diagram. The intensity function can then be used directly to give the geometrical optical approximation of the OTF; or combined with the diffraction integral to give the combined effect.
We shall return to these considerations in RTO II, Chap. 4, in connection with the optical specification of telescopes.
3.10.8 Diffraction effects at obstructions in the pupil other than axial central obstruction
The commonest obstruction is the "spider", or supporting cross, holding the secondary in place in a Cassegrain. Such effects are best handled by applying the Babinet principle of complementary screens - see Born-Wolf [3.120(g)] or Ditchburn [3.147]. The two screens are complementary in the sense that the opaque and transmitting parts are inverted. If the complex amplitude induced at a point is U if no screen is present and U1 and U2 when the screens are present, then the Babinet principle states that
If U = 0, then U2 = — U1. The principle was first enunciated by Babinet in 1837 [3.148]. The effect of a supporting (obstructing) rectangular spider is therefore the complement of two slits at right-angles placed over the pupil. The case of a rectangular aperture was treated in § 3.10.1.
Dimitroff and Baker [3.23(e)] give results from Scheiner and Hirayama [3.149] for diffraction effects at a large number of obstructed apertures, including a single-arm spider and the normal double-arm form shown in Fig. 3.113. Each arm produces a diffraction "spike" at right-angles to its own direction, the combination then producing the well-known "diffraction cross" present in most astronomical photographs with bright stars in the field. Figure 3.114 shows this in a typical plate taken with the ESO (MPIA) 2.2 m telescope.
The avoidance of such diffraction spikes is the second advantage of the Schiefspiegler. Alternatively, the spikes can be modified by other forms of
\ J I Fig. 3.113. Normal form of supporting spider for sec-
N. s y ondary mirrors shown here without central obstruction of
^----1---^ the secondary masking or support. An excellent resume of the possibilities and diffraction images produced is given by Cox [3.150]. He shows the evolution of the diffraction image with a normal spider with reduction of the spider projected area. The results of masking with 4, 5 and 16 round holes over the pupil are also shown. A 4-hole mask removes the spikes but affects the intensity distribution in a more symmetrical way. Any obstruction will cause loss of energy in the central maximum and corresponding loss of contrast. Only when an obstruction becomes small compared with the wavelength of light will its
presence become undetectable. Hence, if non-linear detectors such as photographic plates are operating with bright stars in a strongly saturated regime, even fine wires across the pupil can produce diffraction effects comparable with normal spider supports. This is also illustrated by the complementary character of the normal supporting spider and two slits at right angles referred to above from the Babinet principle. A spider arm with significant thickness is then complementary to a rectangular aperture, the case treated in § 3.10.1 above. Fig. 3.100 shows the minima of the diffraction pattern for one dimension at the values n, 2n, 3n,... of the function Su'. This function is defined for the y-direction by Eq. (3.436). For the first minimum at n, it follows that
where 2ym is the aperture (slit) width and Sr¡' is the linear distance in the image plane of the first minimum from the centre of the diffraction pattern at Su' = 0. Now if ym is doubled, then Sr¡' must be halved for the first minimum if Eq. (3.510) is maintained. In other words, a doubling of the slit width halves the length of the diffraction effect. For the complementary spider arm, this means that the length of the diffraction spike is inversely proportional to the thickness of the spider arm responsible. This inverse linear law is the same as that governing the case of diffraction at a circular aperture (see § 3.10.3) leading to the well-known Airy disk formula of Eq. (3.447) and the classical definition of the resolving power of a telescope given in Eq. (3.449).14
14 Footnote for corrected reprint 2007: Historical note concerning the Mersenne afo-cal telescope
The Mersenne afocal telescope, both aplanatic and anastigmatic, consisting of 2 confocal paraboloids, is the most fundamental form that exists. Paul (1935) invented its 3-mirror focal extension but, although the properties of the afocal form were implicit in his own equations, he failed to recognize its anastigmatic nature. With 3 colleagues, I have researched the question as to who first clearly recognized the anastigmatic property. The properties are treated in recent literature most thoroughly, e.g. Korsch, General Refs. . The key to the first clear statement of the anastigmatic property was given in a classic paper by Baker in 1969 [3.75]. He refers to the McCarthy slitless spectrograph designed for the McDonald Observatory 82-inch telescope in 1937. My great friend and collaborator in many historical matters, Don Osterbrock of Lick Observatory, has kindly sent me a copy of the original publication by E.L. McCarthy (1940, Contributions from the McDonald Observatory, 1, 97-102). He used the primary telescope paraboloid with a concave confocal paraboloid to produce the afocal collimator and proved clearly that the combination is anastigmatic. To make use of the excellent wide field of the collimator, he combined it with an anastigmatic Schmidt System camera in his slitless spectrograph. McCarthy was the true discoverer in a practical realisation of the anastigmatic property - but he had not heard of Mersenne!
4 Field correctors and focal reducers or extenders
The term "field corrector" in telescope systems implies some system placed inside the image plane, but relatively close to it, whose primary function is to correct the field aberrations of the mirror system. If the mirror system does not correct the axial image at the focus in question (e.g. the prime focus of an RC telescope), the field corrector will also be required to achieve this. Normally, field correctors are substantially afocal, i.e. they only have a minor effect on the f/no of the incident beam from the mirror system.
The term "focal reducer" refers to a transfer system (which may, in principle, be before or after the real telescope image) whose primary function is to reduce the f/no of the mirror system, e.g. convert an f/8 Cassegrain image to f/3. A focal reducer will normally have the task of maintaining the quality of imagery over a larger angular field. Inverse focal reducers are also of practical importance, i.e. focal extenders which reduce the angular field and increase the focal length. In most practical cases, the focal reducers are closely related to the optical systems of spectrographs.
Field correctors fall, in principle, into two groups: those designed to work with an existing (fixed) mirror system and those optimized with one or more free parameters of the mirror system. In the latter case, there is no real distinction between an optimized telescope with field corrector and the optimized telescopes of Chap. 3 in which additional plates, mirrors or lenses were added to the 1- or 2-mirror basic system. Since the theory is closely linked, the two field corrector groups will be considered together.
The first reference to field correctors was apparently made by Sampson [4.1] [4.2]. In two remarkable papers based on Schwarzschild's theory, he laid down some of the basic principles of field correctors with lenses, both for Cassegrain and Newton foci, the Newton case being optically the same as the prime focus (PF). This work was extended in 1922 by Violette [4.3] to the field correction of the newly invented RC telescope of Chretien. This was followed in 1935 by a classic paper by Paul [4.4], who systematically analysed the possibilities of lens correctors for both Cassegrain and prime foci and introduced the concept of aspheric plate correctors as the theoretical equivalent of a deformed plane mirror. He also treated the general case of 3 aspheric mirrors leading to the Paul telescope discussed in Chap. 3.
The period from 1945 to about 1975 saw intensive activity in the design of field correctors to give larger fields for photographic plates. An excellent review was given by Wynne [4.5]. Their significance declined with the replacement in normal telescopes of the photographic plate by modern electronic detectors, above all the CCD. These detectors had - and still have - relatively small fields compared with photographic plates. However, arrays of detectors are becoming possible with larger fields, and techniques such as multi-image spectroscopy at the Cassegrain focus using fibres also place higher demands on field correction.
Compared with 3- or 4-mirror solutions, refracting field correctors are usually more light-efficient but suffer from ghost reflections and chromatic aberrations. Aspheric plates are, in general, superior for ghost reflections because they are essentially similar to a flat, thin filter glass: the ghost image of a bright object is, for the small angular fields of telescopes, close to its primary image and normally swamped by the latter's over-exposure (Fig. 4.1). For a plane-parallel plate in air, the ghost image displacement in the image
plane 5gh is given for a small principal ray angle upr by
ngh where dgh is the plate thickness and n'gh its refractive index. In a Cassegrain telescope upr is magnified by the telephoto effect typically by a factor of the order of 4 compared with the object field angle. For a semi-field of 0.25°, this gives upr ~ 1°, say 0.02 rad. For dgh = 10 mm and ng}i = 1.5
$gh = 0.267 mm , which is only of the order of 2 arcsec or less with the plate scale of a typical 3.5 m telescope in the Cassegrain focus. By contrast, a lens can produce a ghost image widely separated from the primary image. It will also, in general, be strongly defocused whereas the ghost image of the plate is only defocused by -2dgh/ngh relative to the primary image.
4.2 Aspheric plate correctors
4.2.1 Prime focus (PF) correctors using aspheric plates
Paul [4.4] was the first to recognize, on the basis of Schwarzschild-Chretien theory, the correction possibilities of aspheric plates for both the PF and Cassegrain foci.
For the prime focus, he considered first the case of a parabolic primary, showing that a single plate could correct the field coma at the cost of introducing spherical aberration. With 2 separated plates the second condition of field astigmatism could be corrected, again at the cost of introducing spherical aberration which Paul proposed to correct by changing the form of the primary into a hyperbola. Implicitly, the complete theory of correctors using aspheric plates was laid out in this work by Paul. However, the practical significance was apparently not appreciated until the definitive work of Meinel [4.6] and Gascoigne [4.7] [4.8] [4.9], the two latter papers giving perhaps the best review of modern telescope optics available.
The theory of aspheric plates was given in § 3.4, the conditions for the correction of the first three Seidel aberrations, Sj, Sjj and Sjjj, being given by Eq. (3.220) for a 2-mirror telescope and aspheric plate near the pupil:
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