Abstract

This paper develops the theory of the aberrations of symmetrical reversible optical systems (sometimes called “symmetric” or “holosymmetric”). It is shown that if such a system is not working at unit magnification (m2 ≠ 1), then not all of its aberrations can be removed simultaneously. In particular it is shown that if s is the magnification associated with the pupil planes, f the focal length, σ1, σ2, ⋯,σ5 the primary (Seidel) coefficients, then

(1s2)(1sm)σ1+[3(1sm)2(sm)2]σ2+(1m2)(1sm)(3σ3+σ4)+(1m2)2σ5=12(1m2)/f2.

The variables entering into the aberrations here are the ideal image height and polar coordinates in the plane of the exit pupil. Analogous relations hold between coefficients of higher order, and some of them are derived explicitly. The number of relations in any order is counted. The usual results concerning systems working at unit magnification are derived, and the special case m = 0 (object at infinity) is discussed separately. The third- and fifth-order relations are transcribed (in the case m = 0) into the language of the “algebraic” coefficients which have been investigated in previous papers of this series and elsewhere. Finally, the problem of the imaging of plane objects is examined in some detail both when m ≠ 0 and when m = 0; and it is shown that when m2 ≠ 1 a sharp image can be formed only on an ellipsoid or hyperboloid of revolution unless m = 0. In the latter case one is of course left with a large amount of distortion when the image is plane.

© 1961 Optical Society of America

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