W. Brouwer, Matrix Methods in Optical Instrument Design (Benjamin, New York, 1964).

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 15.

O. N. Stavroudis, The Optics of Rays, Wavefronts and Caustics (Academic, New York, 1972), Chaps. IX and X.

O. N. Stavroudis, “Caustic as an expression of the image errors of a lens,” in International Optical Design, G. W. Forbes, ed., Vol. 22 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), pp. 87–91.

J. K. Davis, “Geometric optics in ophthalmic lens design,” in Applications of Geometrical Optics II, W. J. Smith, ed., Proc. SPIE39, 65–100 (1973).

G. M. Fuerter, “Spline surfaces as means for optical design,” in 1985 International Lens Design Conference, D. T. Moore, W. H. Taylor, eds., Proc. SPIE554, 118–127 (1985).

As is customary, the direction of the normal to the surface is taken in the direction of the incident ray.

W. T. Welford, Aberrations of Optical Systems (Adam Hilger, Bristol, UK, 1986), Chap. 4.

Formally, in a one-dimensional expansion o(hn) means terms such that limh→ 0o(hn)/hn= 0.

H. Goldstein, Classical Mechanics, 2nd ed. (Addison-Wesley, Reading, Mass., 1980), Chap. 4.

At normal incidence we cannot define a plane of incidence, and the interpretation of the Euler angles that we have chosen must be reconsidered. When β= I= 0, one of the two other rotations is evidently redundant. Since we have shown that α1= α, for consistency we must set α′= α= 0 when β′= β= 0. This means that in this case P1= T1(γ) (an orthogonal matrix), where γ is now the angle from the O¯usaxis to the O¯uaxis. Clearly P1′=T1(γ′), and in general γ′≠ γ. But none of these considerations affects the subsequent arguments of our proof; they are included here only for the purpose of giving a clear interpretation to the results of Section 3 in the special case of normal incidence.

See Ref. 20, Chap. 9.

N. V. Yefimov, Quadratic Forms and Matrices (Academic, New York, 1964), Chap. III.

The other diagonal element of matrix U, 1−a31′2=a32′2+a33′2, cannot be zero unless a33′=0.

D. J. Struik, Lectures on Classical Differential Geometry, 2nd ed. (Addison-Wesley, Reading, Mass., 1961). Chap. 2.

We can verify this condition in Eq. (54) with the equation tr Z= z11+ z22= λ1+ λ2. We have already sketched a proof in Section 4 to show that z12= z21.