1.9 Stigmatism, Aplanatism, and Anastigmatism
The third-order spherical aberration, coma, and astigmatism - Sphe 3, Coma 3, Astm 3 - are sometimes called pupil aberrations, they should preferably be corrected on or near the aperture stop.
If an optical system is free from Sphe 3 aberration, then the first Seidel sum is null,
and the design is said to be a stigmatic system in the third order.
• Stigmatic single mirror: The first known stigmatic surface is a paraboloid mirror for an object at infinity. The discovery of this geometrical property belongs to the Greek heritage. Considering a meridian plane, Diocles (cf. Sect. 1.1.1) demonstrated that a parabola provides the convergence of lines that are parallel to its axis at a single point. Ellipsoid and hyperboloid mirrors provide the stigmatic property for an object at finite distances.
Adding the plane and the sphere to these three shapes defines the conicoid family (cf. Sect. 1.7.1).
• Stigmatic singlet lens: In Descartes' attempt to develop refractive telescopes with a singlet lens, he invented the ovoid surfaces - known as Descartes' ovoids  (cf. Sect. 9.1.2) - where the other surface of the lens is a sphere of the same curvature as the incident or refracted wavefront. This provides stigmatic singlet lenses. The difficulty encountered with the axil chromatism of a singlet lens stopped these developments although in fact Descartes invented analytical geometry for the purpose of solving the stigmatism problem.
Descartes' ovoids include the subfamily of conicoids.
• Any centered system: From the above stigmatic single mirror or lens, we have the following important theorem.
^ in any dioptric, catoptric, or catadioptric centered system, Descartes' ovoids -which include the conicoid family - provide stigmatism at all orders. Spherical aberration at all orders, Sphe 3, Sphe 5, etc, can be completely corrected for finite aperture angles U, U' reaching large values.
The Descartes theory of stigmatism is the first theory of abberrations including either lenses, mirrors or both of them. For more than two centuries this theory, where the meridian section of an optical surface can be a fourth degree curve, remained the only one.
• Stigmatic two-mirror telescopes: Let us consider two-mirror telescopes in all generality, i.e. the Cassegrain and Gregory forms (Fig. 1.23).
Denote d the algebraic distance M1M2 from the vertex of the primary to that of the secondary (d < 0 for both forms in Fig. 1.23 since opposite to the z-direction), and R1 = 2f, R2 = 2f2 their radii of curvature (R1 < 0 and f < 0 in both forms). Setting n = -1 in (1.28a) for the reflection case, the power of the system is
f' V R1 R2 R1R2J \< 0 for Gregory, since the sign of R2 = 2f2 is opposite in the two forms (R2 < 0 for Cassegrain and R2 > 0 for Gregory). The secondary generates a transverse magnification
fi R1 \ > 0 for Gregory, where the sign means that the image is erect for Gregory and inverted for Cassegrain. Denoting 7 the distance M2F' from the secondary to the resulting focus (7 > 0 in both forms). A representation for d/f' with these parameters is d 1 ( e\
Following Wilson  in the theory of two-mirror focal systems, a representation of the first Seidel sum is
where x1 is the marginal ray height at the primary. The stigmatism condition, £ = 0, entails
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