For a general discussion of line foci see, for example, M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, England, 1989), Section 4.6.

For a description of Coddington’s equations, see, for example, R. Kingslake, Lens Design Fundamentals (Academic, New York, 1978), Section 10.1.

For a detailed discussion of second-order aberrations at a line focus see, for example, J. L. Synge, Geometrical Optics, An Introduction to Hamilton’s Method (Cambridge U. Press, Cambridge, England, 1937), Section 10.

In this paper, the subscript 0 is used to identify quantities associated with the base ray.

It is assumed here (and in Section 4) that the cone of rays does not include the axial ray, so that ω cannot equal zero (i.e., the pupil of the system is located completely off axis). In cases in which the axis is included in the cone, the end points of the line image are determined from the ray with angle ω0 + β and the paraxial image location of the object point.

The R number is defined as the reciprocal of the product of the curvature and the clear aperture (CA) of the mirror: R number = (c CA)-1.

If the cone of rays contains the axis (so that ω is zero for some ray in the system) then (dF/dω)|ω=0 and the ray density becomes infinite at the paraxial image location. In another example, if d0 = ∞ and the mirror is parabolic, all rays will merge to a single focal point on the axis, resulting in dω/dδ′ = 0, which also gives an infinite ray density.

J. M. Howard, B. D. Stone are preparing the following paper for publication: “Imaging a point with two spherical mirrors.”