where Sq is the total aberration for one of the five monochromatic Seidel aberrations and (Sq)v the contribution of surface v.

Because the physical interpretation is the most direct, and because wave-front aberration (in general, not just for third order) can be added up algebraically through an optical system, we have given the formulation of the Characteristic Function above in terms of wavefront aberration. In telescope optics, both wavefront and lateral (angular) aberration (see Fig. 3.1) are essential: we shall constantly be making use of both systems and switching from one to the other. It is useful first to consider the wavefront formulation of the Seidel contributions. The derivation may be found in the works of Hopkins [3.3] and Welford [3.6]. The coefficients Sj, Sjj, ... correspond to the third order constants 0k40, ik3i, ... of the Characteristic Function of Table 3.1. They are expressed in terms of the following paraxial parameters deduced from paraxial ray traces at each surface v through the system:

the paraxial ray height (often, but not necessarily, set equal to the actual marginal ray height of the beam)

nviv = nviv, the Snell Invariant for the paraxial ray v prv

, the Snell Invariant for the paraxial principal ray the so-called "aplanatic" parameter = uv, the Lagrange Invariant cv ( \ โ€” 1) = cvA(1) , the Petzval sum of surface v, where cv v \n' n J v v \n) ' v is the curvature 1/rv of the surface

C3(nv โ€” )bsvy4, the influence of the aspheric form defined by the Schwarzschild constant &ยปโ€ž

(HE) v : y'yj v, defining the effect of the pupil position relative to surface v. Then the wavefront aberration

can be expressed in terms of the five Seidel coefficients defined by:

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