Paraxial is not Perfect

Summary

A perfect lens forms a sharp, undistorted image of a planar object. Newcomers to optics sometimes think that a perfect lens should obey the rules for paraxial ray tracing. This is not true. Although paraxial rays can be used to predict the location and magnification of images, paraxial ray trajectories do not obey Fermat's principle, except in a region near the optical axis. A perfect lens can't form a perfect image at more than one magnification, and the trajectories of rays through it cannot be found using paraxial ray tracing.

Fortunately, it is possible to use a perfect lens model that is more accurate than the paraxial model, without undue computational complexity. The aplanatic model used in OSLO considers that for a perfect lens, the ratio of the sines of axial ray angles in object and image space is constant. This leads to curious pictures showing that if rays in object and image space are projected on a single surface, the trajectories appear to be discontinuous. The fact that this obviously can't occur means that there is no such thing as a perfect thin lens, not that the aplanatic model is incorrect.

Discussion

The paraxial approximation applies strictly to rays that are infinitesimally displaced from the optical axis of a system. Such rays may be called true paraxial rays. In everyday optical design, however, paraxial concepts and terminology are used to describe rays that are far removed from the optical axis. For example, it is common to describe a lens by its paraxial f-number, defined to be the ratio of the focal length to the diameter, even when this ratio is quite small, e.g. f/2. The rays through the edge of the lens obviously don't satisfy the paraxial requirement. However, if surfaces are replaced by their tangent planes, the equations for true paraxial rays can be applied to rays having large slopes in a self consistent geometry that predicts perfect imagery, as shown below. These rays can be called formal paraxial rays.

parxreal.gif (1173 bytes)

The problem is that in the real world, light doesn't travel along those paths. Sometimes the discrepancy between the real ray trajectories and the paraxial trajectories is minimal; sometimes (as in the drawing in the note) it is quite large. In systems containing field lenses, the tradeoffs between pupil aberrations and image aberrations often make it impossible to correct the system to perfection, and the use of paraxial lenses can be quite misleading. In fact, it can be shown that formal paraxial ray trajectories do not obey Fermat's principle [1].

It has been known for about 170 years that a perfect lens can only be perfect at a single magnification. This is a consequence of the Herschel condition, named for its discoverer William Herschel, a brilliant astronomer (who discovered Uranus) and musician (oboist and composer). Curiously, the Herschel condition predates the better-known Abbe sine condition by more than 50 years. For a derivation and discussion of the Herschel condition, see chapter 6 of Walther [2] or section 4.5 of Born & Wolf [3].

OSLO defines perfect lenses correctly, requiring the specification of both the focal length and magnification (this is not the case in some other programs). An example of a perfect lens is provided in the OSLO User's Guide.

1. A. Walther, "Teaching the theory of real lenses", Am. J. Phys. 64 (9), 1161-1165 (1996).

2. A. Walther, "The Ray and Wave Theory of Lenses", Cambridge University Press 1995, ISBN 0-521-45.

3. Max Born and Emil Wolf, "Principles of Optics", Pergamon Press, ISBN 0-08-018018 3.

Copyright 1998 Sinclair Optics, Inc. All rights reserved.