This formulation will enable us to transform Eq. (3.19) into an explicit form (Eq. (3.21) giving the corresponding coefficients of the Characteristic Function by tracing a paraxial ray and a paraxial principal ray through the system). Only in specific, simple cases is it feasible to express the Seidel coefficients in terms of the basic constructional parameters of the system, rather than the ray-trace derived parameters used in Eq. (3.20). Reflecting telescopes consisting of one or two mirrors are a case where explicit formulation is feasible.
The formulation of the Seidel coefficients of Eqs. (3.20) and the definition of the parameters involved given above, are identical with those of Welford [3.6]. Conventionally, field curvature S/y is defined as positive for a thin positive (biconvex) lens giving a real image. However, the curved image surface is concave to the incident light, i.e. negative in the Cartesian sense. The same property will hold for telescope systems with a positive focal length. Because of the close interplay between field curvature and astigmatism, we shall retain the Welford sign convention of S/y to maintain consistency in the total formulation of the aberrations. The sign convention of Eqs. (3.20) is based on the consistent principle that the wavefront aberration be positive if the coefficients are positive, i.e. a positive phase shift in the direction of the light as shown in Fig. 3.1. It is not possible to have a consistent Cartesian regime of aberrations in all three systems of aberration definition: wavefront, lateral, longitudinal. Of these, priority is given to the wavefront definition since it is physically the most meaningful.
3.2.3 Seidel coefficients of some basic reflecting telescope systems
The paraxial aperture ray and paraxial principal ray are traced according to Eqs. (2.36) to (2.38). Conventionally, the starting values for such ray traces are the same values as for the traces of the real rays according to the exact formulae [3.6] [3.7], as discussed in § 2.2.3, where it was shown that the linear nature of the paraxial equations makes it immaterial for the result what starting values are taken. Following Bahner [3.5] and others, we shall therefore derive the Seidel coefficients for some normalized telescope systems, in air, for which f = ±1
rrj = ±1 (since n = f 'upr1 in air) H = n'u'ri' = — 1
In these examples, the entrance pupil is taken to be at the conventional position, the primary mirror. This normalization is practical since it allows direct comparison of aberration values for different systems.
The normalized starting values for the paraxial aperture ray (y1) and the paraxial principal ray (upr1) given above correspond to a semi-aperture of f/0.5 and a semi-field of 1 radian, values far exceeding those of most real, practical systems. The resulting coefficients can easily be converted into the real coefficients of given systems by using the aperture and field dependencies of the third order aberrations as given by the Characteristic Function of Eq. (3.15) and Table 3.1. The conversion of the coefficients of Eq. (3.20) to wavefront aberration for third order aberrations W3 is then given from the normalized parameters ym1, rfm to the real parameters y1, rf by (see [3.3]
Tables 3.2 and 3.3 give, respectively, the paraxial data and third order aberration coefficients for the nine cases shown. The significance of the results will be discussed in the next sections in connection with the analytical formulations. The values in the tables can be used as a reference for checking the correct use of these formulations. Distortion (^ Sy) is not given as it is normally of little significance, but it can easily be derived from the values given and from Eqs. (3.20).
Equations (3.20) are the fundamental formulae of the aberration theory of telescopes. We saw in the definitions of the quantities involved that the parameter (HE)V defines the effect of the pupil position relative to the surface v, and is determined from the paraxial principal ray trace, giving (ypr)v, and the paraxial aperture ray trace, giving yv. This concept leads to an important set of equations known as the stop-shift formulae (see [3.3] [3.6]). These define the effect of a stop shift, or pupil shift, in any centered optical system. In these formulae, ^ Sj, ^ S11, etc. are the values for the original stop position
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