Fig. 1.24 On-axis beam satisfying the Abbe sine condition. The Abbe sphere of center Ca is shown by a dotted line for a convergent beam (Left), and for an object at infinity (Right)
point is free from spherical aberration. It stood for the word stigmatic, which did not exist in the language of opticians before ~1900. Hence, in his memoir of 1873 and the following ones, Abbe specified that the sine condition allows one to obtain, in addition to an axial "aplanatic" point, any other "aplanatic" points in the field of view provided their distance from the axis is infinitesimally small.
After introducing the term stigmatic, the modern significance of the term aplanatic was generalized into the literature during the period 1920-40.
The sine condition was brought to the attention of astronomers in stipulations decided by the Paris Congress for the "Carte du Ciel" (1889), mainly on the instigation of H.A. Steinheil acting as German delegate, for the standardization of astrographs. This recommendation was to avoid the dissymmetric coma images of solely stigmatic objectives which caused errors in the position of stars and required use of an inaccurate magnitude equation for those corrections.
These notes are based on the historical account by Chretien , Sect. 379.
The Abbe sine condition entails that both all-order spherical aberration terms and linear coma terms are zeroed. Thus from (1.48) and Table 1.2, the latter condition implies that the terms in linear dependence on field height fj - whose sum is the so-called linear coma - are also null, ai,n,ir¡pncose = 0, Vn = 3, 5, 7, ••• . (1.66)
As demonstrated by Welford , the above condition is derived from the Staeble-Lihotzky isoplanatism condition, simultaneously published in 1919 by F. Staeble and E. Lihotzky, which is also a condition for zero linear coma in the presence of spherical aberration.
Since Abbe's sine condition does not involve non-linear terms on the height as ñ3p3cos e coma or f\2p4 spherical aberration, the stationarity of this condition is valid for field image points which have an off-axis distance infinitesimally close to the system axis.
• Numerical aperture: The numerical aperture (N.A.) allows characterizing the light-gathering power of an optical system in the object space (whilst the f-ratio is preferably used in the image space). This number is defined from the maximum value of the finite aperture angle U in medium n by
The terminology N.A. is a universal use for characterizing the input beam of microscope objectives. Such objectives generally satisfy Abbe's sine condition; their front part may be immersed in oil.
• Isoplanatic singlet lens: Any single thick lens with one or two aspheric surfaces can be designed aplanatic for one wavelength. In convergent beams, a single lens with both spherical surfaces also can be. Such lenses are used as microscope objective components (cf. Sect. 9.1.3). If a singlet lens is corrected for coma only, then
Such a design is called an isoplanatic system.
For a lens, the condition SII = 0 requires a separate location of the pupil. This remote pupil and lens configuration - isoplanatic lens mounting - is sometimes used for spectrograph collimators.
• Isoplanatic single mirror: Analytic relations expressing the Seidel aberrations of one- or two-mirror systems are known (cf. for instance Wilson ). Considering a single concave mirror and an object at infinity, the two first ones are
SI = "4 (4(1 + k) f, SI = "4 ( Ji)V [2f - (1 + K) s], (1.68)
where q is the field angle defined by the principal ray, and s the axial separation of the pupil from the mirror vertex. If a spherical mirror (k = 0) is used with a pupil located at s = 2f', i.e. at its center of curvature, then Sn = 0 and SI = 0. This iso-planatic mirror mounting is the basic first property of Schmidt telescopes.
• Aplanatic Ritchey-Chretien telescopes: H. Chretien  published, in a second paper of 1922, the theory of two-mirror telescopes satisfying the sine condition, then obtaining the so-called exact mirror parametric equations.10 Within the mirror class represented by conicoids, the Cassegrain aplanatic form have since been built in very large size. In the third-order theory, this requires satisfying the stigmatism condition £ SI = 0, expressed by (1.59), and also the coma free condition which then reduces, whatever the pupil position (cf. Wilson ), to
This determines k2, and substituting it in (1.59) we obtain
which are the conic constants for the Cassegrain and Gregory aplanatic forms.
Referring to the signs of M, f', d and I as shown in the previous section, M-3£d-1 is positive for Cassegrain and negative for Gregory. Therefore the primary mirror of aplanatic telescopes is a hyperboloid for the Cassegrain form, and an ellipsoid for the Gregory form. Both slightly departs from the paraboloid (cf. Table 1.4).
• Parametric representation of two-mirror telescopes satisfying the sine condition: The exact shapes of the mirror surfaces for a two-mirror telescope satisfying the sine condition are not conicoid aspherics. In a certain manner it is fortunate that, even for all usual cases of very large Ritchey-Chretien telescopes, a truly accurate representation of those surfaces can be obtain by use of conicoids.
For instance the classical representation of optical surfaces by even polynomials - such as for conicoids - cannot be use for the raytrace of grazing incidence telescopes strictly satisfying the sine condition because high-order coma terms cannot be neglected (cf. Sect. 10.1.3).
In his second paper of 1922, introducing the sine condition, Chretien  was the first to derive from integrations a parametric representation of the optical surfaces for the general case of two-mirror telescopes, hence deriving systems that are
10 G. Ritchey encouraged Chrétien in investigating the case of the coma free two-mirror telescopes. Without knowledge of Schwarzschild's third-order theory of aplanatic two-mirror telescopes (1905), Chretien derived apparently before 1910 the all-order theory of two-mirror telescopes by integrations of differential equations that he laid down from the sine condition (see also historical note in Sect. 1.1).
completely free from all-order spherical aberrations and linear coma terms. In a cylindrical coordinate frame z, r, Chretien expressed the mirror surfaces by
Z1,2 = Fi2 [t (U'),l/f, d/f], ri,2 = Gi,2 [t (U'),l/f, d/f], (1.71)
where t (U') is the parametric variable of the finite aperture angle U'.
Also using a similar parametric representation, Lynden-Bell  carried out a general study of the two-mirror case for finite aperture angle U' that may reach extremely large values such as ±n/2 (or even ±n) for the purpose of determining and drawing the shape of the most typical telescopes of this family. In several cases, also including virtual image fields, the mirror surfaces were found to be cusp- or trumpet-shaped at their vertices.
This was followed up by Willstrop and Lynden-Bell  in a classification review of all two-mirror telescopes satisfying the sine condition for varied driver parameters i/f and d/f. The results proved that all telescopes of practical interest have already been discovered.
If an optical system is free from Sphe 3, Coma 3, and Astm 3 aberration terms, the three Seidel sums are and the design is said to be an anastigmatic system in the third order.
In this case, the tangential, mean curvature and sagittal surfaces merge into one, the Petzval surface.
• Anastigmatic singlet lens: Any single thick lens with one or two aspheric surfaces can be designed anastigmatic for one wavelength. A special case of anastigmatism is a monocentric lens of index n with front and back spherical radii R and R/n centered in C: if the incident beams converge behind C on a sphere of radius nR centered at C, then all the refracted beams converge on its back surface (cf. Sect. 9.1.3).
• Anastigmatic two-mirror telescopes: With the two first Seidel sums set to zero, the astigmatism of an aplanatic telescope is
and if this sum is also set to zero, then d + 2 f ' = 0,
which is the anastigmatism condition for two-mirror telescopes.
By substitution in (1.70b), the conic constants of an anastigmatic telescope are
In an attempt to find a perfect telescope, K. Schwarzschild in his celebrated paper of 1905  formulated the complete third-order theory of one- and two-mirror telescopes including all the above conditions. His eikonal method allowed him to determine telescopes satisfying X SI = X S¡I = X SI¡I = 0, and particularly the Schwarzschild telescope which in addition satisfies XSIV = 0 for a flat field. Unfortunately, his telescope was impracticable because the theory leads to a convex
Various designs of the anastigmatic telescope family are shown in Sect. 4.1.
• All reflective Schmidt telescopes: The Schmidt telescope is with no doubt the most important design in the anastigmat instrument class. It must be considered as an extrapolate design from the two-mirror anastigmat family. Although not knowing the Schwarzschild results, B. Schmidt  in 1930 indirectly found that the primary mirror could be substituted by a pseudo-flat refracting corrector plate which then is located at the center of curvature of the secondary. This satisfies the apla-natism and anastigmatism conditions, i.e. both (1.65c) and (1.74).
The case of purely reflective Schmidts has been investigated (i) for spectrographs with normal diffraction aspherized gratings which hold for the primary, (ii) for spectroscopy sky surveys with a large telescope by tilt of the primary (cf. Chaps. 4, 5 and Lamost).
Let us consider as an example of interest, a fully obstructed design which is the basic case of reflective Schmidts. From the anastigmatism condition, the focus is midway between the two mirrors (Fig. 1.25).
For all Schmidt telescopes the transverse magnification is M ~ 0 because generally the primary element (corrector plate, mirror, or grating) shows a very low optical power. Assuming that M = 0 entails that the primary Mi is a purely plane-aspheric surface; then from (1.75) the conic constants are K1 ^ ^ (since the primary curvature is null) and k2 = 0 which gives a spherical secondary in the third-order. Starting from the condition d + 2f' = 0, we will determine the shape of the mirrors with higher-order terms in two different ways:
(C1) by using a perfectly spherical secondary and stigmatism condition, (C2) by using the Abbe sine condition with Abbe's sphere construction.
In order to determine the shape of the mirrors for each condition, let us denote 1/R2 the curvature of the secondary, f' = R2/2 the efl, and Z = z/ f, p = r/f'
11 Schwarzschild was widely involved in many important fields of physics. Within a year of Einstein's publication of the theory of general relativity, he discovered in 1916 the first and one of the most important of rare solutions for the metric. This probably explains why Schwarzschild did not focus his attention on the Seidel conditions for aplanatic two-mirror telescopes, conditions which were implicitly included in his theory.
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