## Oiy

Fig. 3.24 Location of stigmatic images with the same Astm 3 corrected mirror. M = — 1: object and image on the sphere R. M = 0: image on the sphere R/2

Therefore, after elastic deformation, the stigmatic object and image now belong to the sphere of radius R centered on the mirror vertex (Fig. 3.24 - dotted lines). Compared to the initial position of point source O in Fig. 3.23, its axial displacement is Az = —Ri2/2 towards the mirror assuming small i (Rs ~ (1 + i2/4)R). In addition, to obtain a best imaging, the conjugate fields of view must be conveniently tilted about the y-axis.

The above Gaussian approximation in the quasi-normal incidence case is equivalent to assume that the shape of the mirror results from the co-addition of two quadric surfaces - a paraboloid and a hyperbolic paraboloid - which gives a local region of a toroidal surface [see also (1.38d)].

• Object at infinity (M = 0): The mirror is of the same shape as for M = — 1 [see (3.68)], and the distance of the stigmatic image from the mirror vertex is

Thus, in first approximation, the locus of the stigmatic images belongs to a sphere of radius R/2, centered at the mirror vertex (Fig. 3.24).

### 3.5.6 Aspherization of Concave Mirrors - Examples

• Off-axis astigmatism correction by VTD concave mirrors: Image transport systems designed with a single optical surface - a mirror used off-axis - are useful in ultraviolet, visible, and infrared. For instance, such a system has been proposed by the author as a star tracker at telescope focus by viewing the reflected field at the spectrograph slit input [18]. For multi-object spectroscopy, reflective slits cannot be tilted for the off-axis guiding; but now using YAG lasers for the 2-D mask cuts of

rectilinear or curvilinear micro-slits [5], it should be possible to replace the slit tilted plane by normal incidence mask-gratings with multi-aperture cuts, thus providing an output beam clearance. The low line density of such field gratings allows the transport mirror to reimage all order beams, thus recovering the field of view without light loss due to the reflective slit-mask-grating.

The design of a concave-saddle mirror requires the adaptation of the f-ratio and of the scaling of its mean curvature R in order to maintain the transported imaging quality of the field. The sagittal and tangential curvature of the mirror define the elastic deformation to generate by stress polishing or in situ. From (3.69) we obtain the A22 coefficient, and substituting in (3.57b) we obtain the elasto-optics coupling that gives the perimeter force Vr = V0cos29 . For a moderate i angle, these are

Defining the f-ratio of the image-transport beams by Q

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