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^ Sagittal ray fans provide tangential focii that are on the tangential focal surface, ^ Tangential ray fans provide sagittal focii that are on the sagittal focal surface.

• Astigmatism of an axi-symmetric concave mirror around its center: A concave mirror or a singlet lens illuminated off-axis provide a third-order astigmatism if having a rotational symmetry. For instance, let us consider the case of a concave spherical mirror of curvature 1/R with its vertex V corresponding to a mechanical center, and C its center of curvature (Fig. 3.23).

After reflection on the mirror, an object point-source O taken sideways at distance h from C, and into a P plane normal in C to VC, gives a conjugated image showing Astm 3. The reflected beam leans against two distant perpendicular segments centred on the principal ray in T' and S' points. These segments are the tangential and sagittal focii, respectively. Denoting i = h/R the incident angle of the principal ray at the mirror vertex, the astigmatism length - axial distance of these segments - is a function of i. Since the intersection of the input beam with the output beam is a sphere - the mirror -, the design satisfies the Abbe sine condition of aplanatism [1] i.e. Sphe 3 = 0 and Coma 3 = 0.

Saggital focal surface Image least confusion surface Tangential focal surface

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