For a description of the Scheimpflug condition see, for example, R. Kingslake, Optical System Design (Academic, Orlando, Fla., 1983), Chap. 5, Sec. VI.

The expression for M˜11 follows from Eq. (3.4b) of Ref. 2, while the expressions for L˜111,L˜221, and L˜122, follow from Eqs. (3.4e) of Ref. 2.

To apply the expressions that appear in Ref. 2, the following correspondence must be made here: In Eqs. (A1), θscorresponds to θ1, whereas θs′ corresponds to (π− θ1), tcorresponds to d0, the matrix S is diagonal, and the diagonal elements correspond to S11and S22associated with mirror 1 (which are denoted in what follows by S11I and S22I). The sum of t′ that appears in Eqs. (A1) and tthat appears in Eqs. (A5) corresponds to d1. In Eqs. (A5), θscorresponds to θ2, θs′ corresponds to (π− θ2), t′ corresponds to d2, the matrix S is diagonal, and the elements are denoted in what follows by S11II and S22II. Finally, all refractive indices that appear in Eqs. (A5) and (A1) are taken to be unity here.

The refractive indices in object and image space are also generally required for determination of a system’s first-order imaging properties, but these indices have been taken to be unity here.

For example, in Eq. (2.10), if F is not symmetric, it can be replaced by a symmetric tensor with components ⅙(Fijk+Fikj+Fjik+Fjki+Fkij+Fkji) without altering the value of the characteristic function.

This convention is more appropriate for this analysis than the more general parameterization that is used in Refs. 1, 2, and 6. In those works, the systems of interest were not necessarily plane symmetric, and only angles of incidence between 0 and π/2 were considered. However, the situation corresponding to a negative angle of incidence here corresponds to a change in the value of a separate variable that specifies the angle between the planes of incidence associated with the base ray at two consecutive surfaces. Since all these planes of incidence are coincident in a plane-symmetric system, these variables are abandoned here, and the information they contained is now carried by the signs of the angles of incidence.

For a description of Hamilton’s characteristic functions see, for example, H. A. Buchdahl, Introduction to Hamiltonian Optics (Dover, New York, 1993), Chap. 2.

For an axially symmetric system, the base ray typically coincides with the axis of symmetry. In such a case, the terms of degree three in the Taylor expansion of any of Hamilton’s characteristic functions vanish on account of the symmetry. For an asymmetric system, the base ray is chosen to be near the center of the bundle of rays passed by the system, and, in general, the terms of degree three in the Taylor expansion of any of Hamilton’s characteristic functions are nonzero.