See, for example, H. A. Buchdahl, An Introduction to Hamiltonian Optics (Cambridge U. Press, Cambridge, 1970), Sec. 10.

R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Los Angeles, Calif., 1964), Sec. 36.

The exclusion of cases where ℳis singular precludes only those systems for which the derivative of the base ray configuration in image space with respect to the base ray configuration in object space is ill defined. Such cases occur, for example, when the base ray grazes an interface.

The form of the characteristic function written in Eqs. (3.1) is a simple consequence of the axial symmetry about the base ray. The appearance of N in Eq. (3.1b) accounts for the form taken by the characteristic function (for an axially symmetric system) when the Xand the X′ axes are antiparallel, as opposed to parallel. With the conventions described here, the Xand X′ axes are antiparallel only when the base ray undergoes an odd number of reflections. For more details see, for example, Ref. 1, Secs. 10 and 14.

As defined in Eq. (3.6), the magnification is independent of the number of reflecting surfaces present in the system. If the magnification is defined as the ratio between zd′ and z, the sign on the magnification depends on whether an even or an odd number of mirrors are present in the system. Similarly, the back focal length could be defined as the ratio between ydand βy′. However, if this is done, the definition of magnification would have to be changed to the alternative definition just mentioned; otherwise it would not be possible to determine the value of (−1)r. Also note that when the focal length is chosen to be negative 1 times the ratio between zdand βz′, the focal length of what is commonly referred to as a single, positive, refracting element is itself positive.

The constant ais simply the tangent of the angle between the normal to the tilted plane and the base ray. The tilted plane intersects the plane x= 0 in a line, and ωrepresents the angle between the Yaxis and this line of intersection. The constants a′ and ω′ have similar interpretations. Note that a′ and ω′ are related to a′ of Eq. (3.9):a′2=a′2,tan(ω′)=(az′/ay′).

Including the term sign(m) in this definition ensures that sign(m†) equals sign(m).

For a description of the Scheimpflug condition see, for example, R. Kingslake, Optical System Design (Academic, Orlando, Fla., 1983), Chap. 5, Sec. VI. Briefly, the Scheimpflug condition states that an axially symmetric system forms the sharpest image with tilted object and image planes when the following two conditions hold:ω′-(-1)rω=π[1-sign(M)]/2,a′=∣M∣a.

Keystone distortion is described by Luneburg in Sec. 34 of Ref. 4.

See, for example, R. Kingslake, Lens Design Fundamentals, (Academic, Orlando, Fla., 1978), p. 185. Also see the references cited in Ref. 9.

See, for example, Ref. 9. As another example, Buchdahl (Ref. 1, p. 29) states that “the angles between the focal lines and ℜ [the base ray] cannot be calculated on the basis of the quadratic terms of V[a characteristic function] alone.” This statement suggests that there are unique image lines but that to find the angle between these lines and the base ray requires higher-order terms. A similar discussion of focal lines in the context of first-order optics is presented in M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), Sec. 4.6.1.

The focal length, l1, associated with the plane x= Δ1can be defined analogously: l1=sign[Det(M˜)](μ/F¯˜11)(M˜112+M˜122)1/2.However, if this quantity is used in place of l2as the eleventh geometric entity, the procedure discussed in Subsection 4.B for measuring sign(−M˜22) {which is equal to sign[Det(M˜)]} less straightforward.