Abstract

An optical system is called symmetric if it possesses an axis of symmetry <i>A and</i> a plane of symmetry containing <i>A</i>. A system <i>K</i> will be called semi-symmetric if it is merely axially symmetric, i.e., if it possesses a screw sense pointing along <i>A</i>. Previous work concerning the consequences of reversibility of symmetric systems is extended to the semi-symmetric case, a “reversal” of <i>K</i> being understood to be its rotation through 180° about a line through <i>A</i> and normal to it, together with a reversal of its screw-sense. It is shown that among the <i>n</i>(<i>n</i>+2) aberration coefficients of order 2<i>n</i>-1 there exist altogether ½ (<i>n</i>-1) (<i>n</i>+2) relations. These divide themselves into a set of relations, previously obtained in the symmetric case, between the “proper coefficients” alone, and a new set of homogeneous relations between the “skew coefficients” alone. The third- and fifth-order relations are exhibited explicitly, and some special points relating to all orders are considered. As a contribution towards a proper appreciation of the meaning of the results obtained, a fairly detailed discussion is included of the geometrical significance of the various types of aberrations possessed by semi-symmetric systems. This part of the work has been shorn of all irrelevancies and it is essentially an extension of Steward’s elegant presentation.

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  1. H. A. Buchdahl, J. Opt. Soc. Am. 51, 608 (1961). This paper will be referred to as IX, and a knowledge of its notation and terminology is presupposed.
  2. In the optical field, examples of systems with this property are likely to be rather artificial. However, since the formalism of this paper does not distinguish between optical and particle focusing devices, and so constitutes a unified theory, an electron microscope with magnetic focusing is a realistic example of an image-forming system which is merely axially symmetric.
  3. This term is borrowed from the theory of linearly connected spaces; e.g., J. A. Schouten, Ricci Calculus (Springer-Verlag, Berlin, 1954), 2nd ed., Ch. 3, p. 126.
  4. G. C. Steward, Trans. Camb. Phil. Soc. 23, 235 (1926), also his tract The Symmetrical Optical System, Cambridge Tracts No. 25 (Cambridge University Press, London, 1928), Ch. 3, pp. 30–49.
  5. Provided only that the part of K which lies between B and B' is not telescopic.
  6. Bold-face type is used in accordance with the remark in IX following Eq. IX(8.2). To use this notation to advantage when considering semi-symmetric systems, it is convenient to adjoin to any quantity q= (qy,qz) the quantity q = (-qz,qy). If we regard q and q as euclidean two-vectors, then q has the same length as q but is orthogonal to it.
  7. For explanation of this terminology, see H. A. Buchdahl, J. Opt. Soc. Am. 48, 747 (1958), in particular Sec. 7.
  8. It might be more appropriate to introduce separate "meridional planes" in object and image space, namely the planes y = 0 and y′= 0, respectively; the latter, of course, relate to adapted coordinates.

Buchdahl, H. A.

H. A. Buchdahl, J. Opt. Soc. Am. 51, 608 (1961). This paper will be referred to as IX, and a knowledge of its notation and terminology is presupposed.

For explanation of this terminology, see H. A. Buchdahl, J. Opt. Soc. Am. 48, 747 (1958), in particular Sec. 7.

Schouten, J. A.

This term is borrowed from the theory of linearly connected spaces; e.g., J. A. Schouten, Ricci Calculus (Springer-Verlag, Berlin, 1954), 2nd ed., Ch. 3, p. 126.

Steward, G. C.

G. C. Steward, Trans. Camb. Phil. Soc. 23, 235 (1926), also his tract The Symmetrical Optical System, Cambridge Tracts No. 25 (Cambridge University Press, London, 1928), Ch. 3, pp. 30–49.

Other (8)

H. A. Buchdahl, J. Opt. Soc. Am. 51, 608 (1961). This paper will be referred to as IX, and a knowledge of its notation and terminology is presupposed.

In the optical field, examples of systems with this property are likely to be rather artificial. However, since the formalism of this paper does not distinguish between optical and particle focusing devices, and so constitutes a unified theory, an electron microscope with magnetic focusing is a realistic example of an image-forming system which is merely axially symmetric.

This term is borrowed from the theory of linearly connected spaces; e.g., J. A. Schouten, Ricci Calculus (Springer-Verlag, Berlin, 1954), 2nd ed., Ch. 3, p. 126.

G. C. Steward, Trans. Camb. Phil. Soc. 23, 235 (1926), also his tract The Symmetrical Optical System, Cambridge Tracts No. 25 (Cambridge University Press, London, 1928), Ch. 3, pp. 30–49.

Provided only that the part of K which lies between B and B' is not telescopic.

Bold-face type is used in accordance with the remark in IX following Eq. IX(8.2). To use this notation to advantage when considering semi-symmetric systems, it is convenient to adjoin to any quantity q= (qy,qz) the quantity q = (-qz,qy). If we regard q and q as euclidean two-vectors, then q has the same length as q but is orthogonal to it.

For explanation of this terminology, see H. A. Buchdahl, J. Opt. Soc. Am. 48, 747 (1958), in particular Sec. 7.

It might be more appropriate to introduce separate "meridional planes" in object and image space, namely the planes y = 0 and y′= 0, respectively; the latter, of course, relate to adapted coordinates.

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