In the author’s previous work on aberration coefficients the geometrical behavior of optical systems has been analyzed in terms of the displacement ɛ′ of the intersection points with the ideal image plane of arbitrary rays, relative to ideal intersection points, ɛ′ being expressed as series in ascending powers of suitably chosen variables. The presence of (geometrical) aberrations is entirely equivalently summed up in the equation of the wave front W′ in the image space, i.e. of that surface whose normals constitute the congruence into which the system has transformed the pencil of rays issuing from any object point. The equation of W′, when written in a suitable form, differs from that of a certain spherical surface W0′ only through the presence of a term D, here called the deformation of the wave front. D may be approximated by the terms of a power series up to a certain order, the coefficients of which (deformation coefficients) are obviously closely related to the aberration coefficients which earlier defined ɛ′. The principal object of this paper is to establish the relations between these two sets of coefficients, those of the third, fifth, and seventh orders being dealt with explicitly. In the diffraction theory of aberrations on the other hand, one is interested primarily in the normal displacement between corresponding points on W0′ and W′. A function R which describes this displacement may be called the retardation of the wave front. This function is also considered, and simple relations are established between the deformation coefficients and the retardation coefficients, i.e. the coefficients of the power series for R.
In so far as D, or R, may in fact be sufficiently closely approximated by the first three orders, one thus has incidentally a convenient set of 28 (monochromatic) performance numbers governing all pencils of rays simultaneously. The numerical values of the deformation and retardation coefficients of the first three orders are given for a certain triplet.
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