+ -8C (1+ e2 + e4 + e6)
Third order Astigmatism
(1 + e2 + e4)
sinusoidal with (nz - nZ) complete wavelengths over annular pupil
2^2 = 2.83
that the annular radius contains a complete integral number (nz — n'z) of ripple (zonal) wavelengths, i.e. both nz and n'z are integers, where n'z is the number of obscured ripple waves.
As in Table 3.26, the result for astigmatism in Table 3.27 differs from that given by Schroeder because of the difference in the wavefront functions with cos 2^> and cos2 ^ respectively. Both formulations are valid and can readily be transformed into each other.
For the combined function of spherical aberration and defocus kdp2 + ksip4
Table 3.27 shows that the ratio ptv/rms is unchanged by the optimum focus combination with kd = — ks1, since both quantities are reduced by the factor 4 (ptv = zonal aberration in this case). Similarly for the coma-tilt combination
the ratio ptv/rms is also unchanged for the optimum tilt ky = — 3 kc. The maximum aberration here is not the zonal aberration but the aberration at the edge of the pupil (Wq)p=1 = + | kc, giving ptv = | kc. With rms = kc/V72, the ratio remains 2^8 = 5.66. These two cases of invariance to changes of the reference sphere are illustrations of the displacement theorem discussed above in the derivation of Eq. (3.459).
3.10.7 The diffraction PSF in the presence of larger aberrations: the Optical Transfer Function (OTF)
The effect of aberrations in optical systems may broadly be classified in three main groups: first, aberrations near the diffraction limit; second, aberrations above the diffraction limit but acceptable for the detector; third, very large aberrations relative to the diffraction limit. The first group has been dealt with in §§ 3.10.5 and 3.10.6 above. It assumes more and more importance with the improvement in quality of ground-based telescopes and sites, and the steadily increasing advance of space telescopes. The third group represents the domain where diffraction effects become negligible and geometrical optical image sizes give a good description of the image. Until recently, accepted values for external "seeing" (non-local air turbulence) were so far above the diffraction limit, that geometrical-optical interpretations of the image quality were reasonable, above all, if the detector resolution is much lower than the diffraction image size given by (3.447). This is no longer the case if accurate assessment is desired. The second group is the most complex situation: neither the series approximations used in the first group nor the asymptotic developments of the third are adequate. The best tool for this group is the Optical Transfer Function (OTF).
The theory of the OTF, essentially a systematic application of Fourier theory to optical imagery, was initiated by Duffieux in 1946 [3.136]. The application to television, with analogy to filter theory in communication systems, was a powerful impetus. A very extensive literature exists. General treatments of varying depth, all excellent within the aims of the work in question, are given by Marechal-Francon [3.26(e)], Welford [3.6], Schroeder [3.22(h)] (particularly well adapted to the case of telescopes) and Wetherell [3.137]. For complete treatments of the Fourier theory required, see Marechal-Francon [3.26(f)] or Goodman [3.138]. Here, we shall confine ourselves to the essential aspects applicable to normal use of telescopes, i.e. the case of incoherent light.
The OTF is based on the concept of spatial frequency of the intensity in the object plane, represented as the sum of a continuous infinity of sinusoidal components. Each component is transmitted by the optical system with reduced contrast, the contrast being defined by
I0 max + I0 min for the object and by Cj for the image, as shown in Fig. 3.107. If we consider an object consisting of only one sinusoidal frequency in the t-section, then the Modulation Transfer Function (MTF) for this frequency s is given simply by
Following Welford [3.6], we take coordinates in the object plane as £0 and r/0 and consider these as transmitted to the image plane with the magnification factor m, then for the image £ = m£0, r/ = rnr/0. For a sinusoidal object whose "lines" are parallel to the n0-axis, we can write
for the function of the object intensity (incoherently illuminated), in which s = 1/A is the spatial frequency in mm-1, if £0 is expressed in mm, and a0 is the normalized amplitude. According to (3.483), the contrast of the object C0 = 1. The transfer through the system then gives at the image
where the factor aj/a0 represents the constant intensity loss due to absorption etc.
As with the diffraction integral, it is most convenient in the general formulation to use the complex form from the Euler relation. This general formulation is given, for example, by Marechal-Francon [3.26]. For a simple frequency s, (3.486) then becomes
where R is the real part of the complex quantity L. It follows that Tc = R, the real part of the OTF L, is the MTF and the phase shift is arg L(s).
In the general case, we have a two-dimensional situation depending not only on s but on the position in the field and the aberrations affecting that image point. The OTF will depend on whether the intensity function is in the t- or s-section. For symmetrical aberrations, the phase shift p will be zero; for coma it is not zero, though it may be small in practice. An important condition for the generalisation is that the aberration function be invariant over the range of the lateral aberration at any point: this is the isoplanatism condition. It will only break down if aberrations become very large, in which case the optical system is useless in practice.
If the wavefront aberration in the pupil is W(x, y), then as for (3.459) the complex amplitude in the exit pupil induced by a point object is proportional to eikW(x'y) with k = 2n/A as before. We define
as the pupil function, having the form of (3.488) over the area of the pupil and zero outside it. The diffraction integral (3.439) can be written to give the complex amplitude at the image point £, n in terms of rectangular coordinates, if constants are ignored, as
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