R Rs Rt

So, by solving this equation system,

where the radius of curvature Rs belongs to the sagittal plane (Rx = Rs).

For a concave mirror where both principal curvatures have the same sign, this result leads to a less well-known property.

^ At given incidence angle i, Astm 3 corrected mirrors have an identical shape whatever the magnification M.2

The mirror shape is the co-addition of a sphere and a saddle, that is represented by the expanded series z = _L r2 - 1 -cos2i — r2cos20 + -L r4 - ^ r4cos20 + ..., (3.68) 2R 1 + cos2 i 2R 8R3 16R3

where the 0 origin is the x-axis.

• Optical design of an aplanatic image transport mirror (M = -1): Given an incidence angle i, the mirror shape of the image-transport is represented by (3.68). Neglecting the fifth-order astigmatism - Astm 5 mode - which is of small saddle sag, the Astm 3 mode to be generated by flexure is

Compared to Fig. 3.23, the optical design of the mounting must be modified as follows. We set s = t = s' = t' which provides a magnification M = -1. From the expressions of Rs and Rt in (3.67), we obtain the mounting geometry s = t = s = t' = + -8- + R - R. (3.70)

2 This property may be also demonstrated by considering an oblique pencil reflected at the sur face of a conicoid which principal ray passes through one of the geometrical focii: there is no astigmatism for conjugates at any position along the principal ray (cf. for instance in Chap. 1 [16], p. 161).

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