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lC 8D 3 + 2cos2p0 + 3cos4p0

Since ZSub is opposite to ZOpt, the submaster optical sag from center to contour C is easily expressed as a function of the optics parameters by substitution of (5.92) in (5.76b). This gives

where, due to the constant thickness of the built-in inner plate, only the two coefficients A2, A4 are possible which, then, provides a spherical aberration correction limited to the Sphe 3 term. For a meridian flexure with k = 3/2 in accordance to (5.75), the approximations A2 ~ M = 3/64Q2 = r^/2R2 and A4 = -1/4 leads to

1C cos p0+cos a 4R3

Clear aperture of final grating replica

Fig. 5.23 Theoretical isolevel lines of the flexure for Sphe 3 correction only. The constant thickness elliptical plate is built-in to an infinitely large outer ring. Full line: built-in contour ellipse of semi-axes (ro/cosft, ro) of the submaster. Dashed line: clear aperture ellipse of semi-axes (rm/ cos a, rm) of the final grating replica

### Clear aperture of final grating replica

Fig. 5.23 Theoretical isolevel lines of the flexure for Sphe 3 correction only. The constant thickness elliptical plate is built-in to an infinitely large outer ring. Full line: built-in contour ellipse of semi-axes (ro/cosft, ro) of the submaster. Dashed line: clear aperture ellipse of semi-axes (rm/ cos a, rm) of the final grating replica

Equalizing the elastic and optical sags of the substrate, we obtain q = cos2qm(3 + 2cos2ft + 3cos4ft) t3 E = 6 (1 - v2)(cos ft + cos a) R3''

which provides the following thickness of the built-in plate

6 (1-v2)(cosft + cosa) q cos2 çm (3 + 2cos2ft + 3cos4ft)) E ro = \J3/2 rm and R = 4 Q rm.

This determines the execution conditions for a grating with Sphe3 correction mode only and illuminated by a circular collimated beam of semi-aperture rm. The two ellipses representing the built-in edge of the submaster and the clear aperture contour of the final grating replica are displayed by Fig. 5.23.

• Variable thickness submasters for fast focal ratio cameras: Diffraction gratings for fast camera mirrors require higher-order corrections. Similarly as axisymmetric submasters for ft = 0 (cf. Sect. 5.4.3), this can be achieved by a built-in plate whose thickness is decreasing from center to edge. The thickness function must follow an homothetic ellipse distribution.

Let us denote variables v, 6, and Z respectively defined by v2 = cos2Po X2 + y2, 6 = v and Z = — ■

The dimensionless flexure Z of the submaster plate must be, from (5.76b), z = -

where An(M) coefficients in Table 4.1 can be considered up to a large n-value. The boundary ellipse of semi axes ro/cos ft, ro, which determines the clamped edge location, is derived from the positive root of

With the exact representation of the A„(M) coefficients in Table 4.1, this root gives the accurate ellipse of the null powered zone. This leads to a k-ratio, k = r^/r^, which is slightly smaller than the k = 3/2-value of the third-order condition. Since $ in (5.98) would correspond to p in (5.81) for the axisymmmetric case, the exact k-ratios as a function of Q are identical to those of an axisymmetric submaster (see Table 5.5, 2nd-column).

Comparing the required q/E ratios in (5.78b) and (5.95a), which corresponds to constant thickness built-in plates of circular and elliptical contour respectively, one notes that the expression of the load transforms as

Then, from (5.82), let us define dimensionless geometry parameter g and rigidity D as cos2$m _ 3 + 2cos2ft + 3 cos4 ft) ,c.nm g =--r—— and D =-——-— gD. (5.100)

In the transformation from circular to elliptic symmetry, p2 ^ $2, hence from (5.85), the rigidity is a solution of d D

" £ n(n - 1 + v)A« $n-2 + D £ (n - 2)n2An $n-4 = 1. (5.101)

Representing it, up to the ellipse of built-in radius parameter = vo/R, by

a dimensionless solution D( Q, $) shows the same variation of the rigidity as D( Q, p) in (5.86) for a circular plate. The derived thickness t(Q,x,y) and the built-in boundary completely determine the execution conditions of a deformable submaster generating bisymmetric gratings for the aberration correction of a fast focal-ratio camera mirror and field flattener lens. A typical design of an elliptical vase from the submaster uses a similar vase enclosure either welded at the ring rear surface or linked via discrete points on an homothetic elliptic line. The ring outer edge of both submaster and enclosure can be made circular (Fig. 5.24).

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