As an illustration of simple scaling, we will take Case 1 of Table 3.3, a single-mirror telescope with a spherical primary, giving for the normalized case
Suppose a practical system works at f/10 with a semi-field angle upr1 of 1/100 rad and has f = 1000 mm. Then
These results illustrate how the very large coefficients for the normalized systems of Table 3.3, corresponding to the enormous aperture and semi-field of f/0.5 and upr = 1 radian respectively, reduce to values comparable with the wavelength of visible light when scaled to the dimensions of real systems, even though the real focal lengths may be much greater in the units chosen. The actual wavefront aberrations are further reduced by the numerical factors in Eq. (3.181).
In the next section, the conversion formulae between wavefront aberration and the other common forms of aberration will be given.
3.3 Nature of third order aberrations and conversion formulae from wavefront aberration to other forms
The first four third order (Seidel) monochromatic aberrations are of fundamental importance in the optical layout and design of modern telescopes. We shall see in Reflecting Telescope Optics II, Chap. 3.5, that they are equally important in performance and maintenance aspects. It is therefore essential to have a clear idea of the physical meaning of these aberrations. Eq. (3.21) gave the general relation between the wavefront aberration and the third order aberration coefficients. Here we shall give the essential conversion formulae: a more complete treatment is given by Welford [3.6].
3.3.1 Spherical aberration (Sj)
The first term of Eq. (3.21) gives for the spherical aberration term at the Gaussian focus:
8 V VmJ
Fig. 3.14. Third order spherical aberration as wavefront aberration
W' varies, from Eq. (3.21), with y4, where y is the height of the ray in the aperture. ym will be the value of y taken for paraxial calculations and is the maximum value defining the aperture of the system. Figure 3.14 shows this wavefront aberration relative to the nominal reference sphere (Fig. 3.1). The choice of focus is free: it does not have to be the Gaussian focus represented by the y-axis as the reference sphere to that focus. A change of focus is represented by the parabola shown by the dashed curve, i.e. a function of y2. If the parabola is such that it cuts the W' curve at ym, the wavefront aberration is clearly reduced. It is easily shown that
where (W')BF is the optimum (best focus) value corresponding to the case shown and (W')GF is the basic value referred to the Gaussian focus. The height in the pupil corresponding to the maximum of the residual zonal aberration (W')bf is ym/V2 = 0.7071ym and the reference sphere shown cutting the wavefront at height ym in Fig. 3.14 corresponds to a focus exactly halfway between the paraxial and marginal ray foci. It is also the focus of the zonal ray. This result is in fundamental disagreement with the best focus according to geometrical optics based on focusing rays, given below, and leading to a best focus (disc of least confusion) three quarters of the distance from the paraxial to the marginal focus. However, both the wavefront aberration and ray aberration treatments give a reduction of a factor of four of the paraxial focus aberration at their respective "best foci". An admirably clear demonstration of the wavefront relationships is given by Conrady [3.167]. The wavefront aberrations of Eqs. (3.184) and (3.185) have historically been applied to tolerancing on the basis of the Rayleigh A/4 criterion. This matter is treated in detail in § 3.10.5. The Rayleigh criterion is only a rough criterion according to physical optics, the physically correct treatment being the Strehl Intensity Ratio based on the variance of the wavefront - see Eq. (3.464). However, Table3.26 shows that, in the cases of defocus, third order spherical aberration and their combination, there is close agreement between the Rayleigh approximation and the more rigorous Strehl approach. The fac-
tor 4 gain due to the optimum focus at the halfway focus is also given almost exactly by the Strehl treatment.
The fact that geometrical optics loses its validity as one approaches the diffraction limit should always be borne in mind in the geometrical treatment of aberrations using ray paths near the focus.
In elementary texts, spherical aberration is usually introduced as longitudinal aberration, a definition of value since there is no ambiguity of interpretation regarding focus compensation. Figure 3.15 shows the longitudinal and lateral forms of third order spherical aberration. The process of image formation from the pupil to the image is a Fourier transformation [3.26] following differential equations given by Nijboer [3.27] [3.3]. The paraxial rays focus at the Gaussian image position I', the marginal rays at A. The longitudinal aberration is
If all rays are considered, the smallest diameter of the "tube" containing the rays is at B where
The diameter of the image (containing 100% of the geometrical energy) at the paraxial focus is 5r/GF and the smallest diameter at B is 5r/BF. Then obviously
a result which corresponds to (3.185). 5^bf is called the disk of least confusion. From the Nijboer equations, it is easily shown that the longitudinal aberration is
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