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V Q! \^12 45 S1 112 S2 6 20 dS 6 ) m78x + / k2 + kT kc + ki \ + / kA \ (. ]

Applying (3.465) to the various groupings leads to the results of Ta-ble3.26. It should be noted that all these cases refer to incoherent illumination for point sources, the normal case for astronomical telescopes. Other cases are dealt with by Marechal and Francon [3.26(d)]. If the third order astigmatism is defined as kAp2 cos2 $ (Welford [3.6] and Schroeder [3.22(g)]), the transformation cos2 $ = 2 (1 + cos 2$) produces a shift of the mean focus giving different values for (AWq)2 and kA.

It is noteworthy that the definition of I(Q)/I0 = 0.8 leads to coefficients for defocus and third order spherical aberration, considered individually, which are about A/4. Rayleigh [3.131] deduced this tolerance for defocus on an experimental basis and suggested it as a general rule. Table 3.26 confirms that it gives a reasonable approximation for the individual aberrations, but the Strehl Intensity Ratio puts the matter on a more rigorous physical basis.

The combination of defocus with third order spherical aberration shows that, as in the geometrical-optical case treated in ยง 3.3.1, a multiplication of the tolerance by a factor of 4 takes place. However, in the present diffraction case, the optimum focus is not at the disk-of-least-confusion of the geometrical-optical case: Table 3.26 shows it is half way from the paraxial to the marginal focus, not three quarters of the way. It is easily shown that the reference sphere cuts the wavefront at the edge of the pupil.

The case of the balance of defocus with spherical aberration of both third and fifth orders is instructive in showing the large individual coefficients which can appear when non-orthogonal terms are balanced.

The final case shown in Table 3.26 is a departure from the normal aberration polynomial but represents, in idealized form, a very important practical error in the figure of large optics: ripple. Ripple is a succession of concentric zones and arises from resonance effects in the motion of the polishing tool. The steeper the aspheric function, the greater the danger of ripple, if conventional figuring methods are used. Figure 3.106 shows an idealized sinusoidal ripple with the wavefront function

in which Az is the normalized ripple wavelength and kz is the amplitude. Then

Az Az where nz is the number of complete zones over the radius. Our treatment assumes nz is any whole positive integer.

Table 3.26. Coefficients of various aberrations giving a Strehl Intensity Ratio of 0.8 (incoherent light, point source, circular unvignetted pupil)

Aberration

Aberration polynomial

I (Q)/1o = 0.80

Defocus

kdp2

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