## N An I R

Defining dimensionless quantities

the substitutions and simplifications lead to

I pdp I dp2 + p dp J + D pdp\ dp2 + p dp J = g, (583)

I 2,4,..., p from which the dimensionless rigidity is a solution of d D

^ I n(n - 1 + v)Anpn-2 + D I (n - 2)n2Anpn-4 = 1. (5.85)

Let us represent the rigidity D(p) up to the null power zone po = ro/R which will be the dimensionless clamping radius of the active plate at the outer thick ring. From the even expansion

the substitution in (5.85) and identifications of the constant-, p2-, p4-terms, etc, lead to a linear system,

18A6 Xo + (7 + v)A4 X2 + (1 + v)A2 X4 = 0, were the next null sums successively include the unknown X6, X8, etc.

If A6 = 0 and the higher-order coefficients are null, then Xn = 0 for n > 2 and D = X0 = 1/(32A4) which, from (5.82) and (5.78a), provides the well-known constant thickness plate solution correcting Sphe3. If A6 = 0 and the higher-order coefficients are null, then the third identification equation for p4 terms, in (5.85), is needed to determine the rigidity that generates the additional Sphe 5 mode correction. In this latter three-equation set, X6 = 0; therefore the dimensionless rigidity is of the form D = X0 + X2p2 + X4p4. A general result is that, for higher-order spherical aberration corrections than Sphe 3, a polynomial flexure represented by n/2 = p/2 terms is generated by a rigidity represented by p/2 - 1 even terms.

For the design of active optics submasters generating corrected gratings from f/3 to f/1.22 in the nebular direction, computational resolutions of the equation set (5.85) were carried out up to the order including n = 8 in (5.86). These rigidities are for normal- or quasi-normal-diffraction reflective grating spectrographs that have been built (Fig. 5.17 and Table 5.5). For instance since pm = rm/R, for the clear aperture edge Q = 1/4pm = 1.22 - i.e. an f-ratio of f/1.22 in the nebular direction, we obtain D(po)/D(0) = 0.785; since the thickness and rigidity are related by t ^ D1/3, this corresponds to a thinner plate edge than at the center in a ratio t (po)/t (0) = 0.922.

The final design of the submaster must take into account the stress level at the clamping boundary by appropriate set up of the plate thickness. The maximum stresses caused by the inner uniform pressure load q (>0) come from the radial bending moment and stand at the built-in radius ro. At the outer surfaces of the plate these stresses are

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