B. M. Boshnyak, A. N. Korolev, “Synthesis of astigmatism-free condensers consisting of spherical mirrors,” Opt. Spectrosc. 43, 204–207 (1977).

B. M. Boshnyak, “Meridional aberrations of a spherical surface at large angles of incidence,” Opt. Spectrosc. 40, 518–521 (1976).

B. M. Boshnyak, “Extrameridional aberrations of a spherical surface at large angles of incidence,” Opt. Spectrosc. 41, 386–390 (1976).

B. M. Boshnyak, “Summation of meridional aberrations of a spherical surfaces with arbitrary decentering angles,” Opt. Spectrosc. 41, 632–634 (1976).

B. M. Boshnyak, “Addition of extrameridional aberrations of a spherical surfaces with large angles of decentering,” Opt. Spectrosc. 42, 106–110 (1977).

For a description of Hamilton’s characteristic functions see, for example, H. A. Buchdahl, Introduction to Hamiltonian Optics (Dover, New York, 1993), Chap. 2. In that reference the point-angle mixed characteristic is denoted by W2.

The geometrical interpretation associated with Hamilton’s characteristic functions can be found in Chapter 2 of the reference cited in note 14, or alternatively in J. L. Synge, Geometrical Optics, An Introduction to Hamilton’s Method (Cambridge U. Press, Cambridge, UK, 1937), Sec. 6.

A detailed example of a Taylor expansion of a characteristic function can be found in Section 2 of the reference cited in note 15.

See Chapter 2 of the reference cited in note 14.

For a description of Coddington’s equations, see, for example, Rudolf Kingslake, Lens Design Fundamentals (Academic, San Diego, Calif., 1978), Sec. 10.1.

A discussion of the Schwarzschild configuration for both infinite and finite conjugates can be found in W. J. Smith, Modern Optical Engineering: The Design of Optical Systems, 2nd ed. (McGraw-Hill, New York, 1990), Sec. 13.2.The original paper on the subject discusses the infinite-conjugate case: K. Schwarzschild, “Untersuchungen zur geometrischen Optik, II: Theorie der Spiegeltelescope,” Abh. Königl. Ges. Wiss. Göttingen, Math.-Phys. Klasse 9, Neue Folge, Bd. IV, No. 2, 2–28 (1905).

D. Shafer, “Optical design with only two surfaces,” in International Lens Design Conference, R. E. Fischer, ed., Proc. SPIE237, 256–261 (1980).

[CrossRef]

Although the systems of this paper are designed for imaging a single point, a magnification can be defined for these systems by considering the angle that a ray makes with the axis and taking the ratio of the object- and image-space angles of this ray. In the limit as the ray angles approach zero, this ratio approaches the transverse magnification. The axial NA is defined as the sine of the maximum angle (in object space) between the axis and the rays that are passed by the system.

See Figs. 6 and 8 of Ref. 6.