Fig. 3.23 Third-order astigmatism of a spherical mirror around its center of curvature. The mounting is aplanatic, i.e. Sphe 3 = 0 and Coma 3 = 0

Since the mirror is spherical, Rs = Rt = R, and since the input beam is stigmatic s = t = R / cos i.

The distances from the vertex to the conjugated focal images are s' = s ~ R (1 + i2/2) and t' - R (1 - 3 i2/2).

Denoting I = s' -1', 2a the clear aperture diameter of the mirror and Q = R/2a the beam f-ratio, the astigmatism length and the diameter of the transverse least confusion image are respectively

The radii of curvature of the tangential, least confusion, and sagittal focal surfaces are respectively

These results apply also if the mirror is replaced by a diffraction grating of spherical shape. In the symmetry plane of the mounting, spectroscopists move the input slit along the principal ray from s = R/cos i to s = Rcos i, i.e. onto the circle of radius R/2; then, the tangential image also moves to reach the same sphere. This Rowland circle [3(c)] is currently used in optical mounting of concave gratings because it is free from coma.

• Astigmatism of an axi-symmetric concave mirror with object at infinity (M =

0): In this case, the distances of the object focii are s = t = and for a pupil at the mirror, conjugation relations become

If the mirror is axisymmetric Rt = Rs = R, and whatever is its aspherical shape, the resulting astigmatism length I = s' -1' is

^ Astigmatism lengths are four-times larger at magnification M = -1 than at M=0.

• Shape of a mirror correcting Astm 3 at any magnification (M): With the sign convention of (3.61), let us define the magnification as M = -s/s' = -t/t'. Thus, the main curvatures are

The correction provides stigmatic images if s' = t'. Then, M vanishes and we obtain

whatever is the magnification. Let us generate an elastic deformation in r2 cos 20 (saddle) which is co-added to an axi-symmetric mirror of curvature 1/R. We set the latter as the mean of sagittal and tangential curvatures, hence

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