Fig. 3.19. Third order astigmatism: wavefront aberration reversal in the t- and .s-sections due to the cos term
Fig. 3.20 shows marginal rays, but the effect is also present for rays paraxial to the principal ray since defocus is a Gaussian error (first order), as was shown in Table 3.1. The longitudinal astigmatism It Is is therefore independent of aperture. The lateral astigmatism, by contrast, depends on the aperture: it reveals itself as the astigmatic lines.
At the tangential focus It, the rays in the s-section are out of focus and form a line perpendicular to the plane of the paper; at Is the rays in the t-section are out of focus and form a line in the plane of the paper along the s-surface. Figure 3.21 shows the effect in a symmetrical field and explains the origin of the terms "tangential" and "sagittal". If we start by focusing well inside the tangential focus, the image will be a round defocus patch made slightly elliptical by the astigmatism with the long axis tangential. At the t-focus, it becomes a pure tangential line. Beyond this it becomes elliptical
t-section ii, t-section ii, s-section
Fig. 3.21. Tangential and radial astigmatic lines at the t-focus and s-focus respectively t-focus s-section s-focus
Fig. 3.21. Tangential and radial astigmatic lines at the t-focus and s-focus respectively again with growing minor axis: at Im the geometrical image is exactly round (if diffraction is considered, this is not true for small values of astigmatism - see § 3.10). After Im the image becomes elliptical again with long axis radial, passing through the sagittal astigmatic line at Is. The behaviour of the astigmatic patch is most important in the practical analysis of telescopes since the changes through focus permit a good qualitative assessment of the astigmatism in the image.
The distances ImIt and I Im are given by
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