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\( du + + (1 + v)S + (3  v)p2 A\dp + Vp) + (S  p2)3
which allows the general determination of ar. The tangential maximum stresses at would be derived similarly.
For instance, if p = 0, (9.37) entails (p1 /p = 1, hence du i T + Vdp p p=
Using the dimensionless thickness T = t/a, the maximum radial stresses at the center of the lens is
and since p is related to the pseudopower m of the lens and to the load q by (9.29), using also ^ = 2 (1 + v) from (9.32), this finally writes
where a and q are the only dimensioned quantities.
For lenses in fused silica, some values of the ratio ar (0)/q with respect to T0 and m are displayed in the last column of Table 9.1.
• Degenerated case of null powered lenses: The stress formula for ar(0) also includes the case of a nullpower lens, m = K = 0, which is a plate or a meniscus of thickness t = constant = to. In this case 8 = T0/W ^ ^ so the above formula is indeterminate. For 8 > 1, the constant A can be expanded. Since u(8  1) + 6 = + 2(u + 3) + (8  1)3 82 + 83 + '
from (9.40), we obtain the expansion of the terms in parentheses in eqs. (9.44) as A+82 = JI±^+oi(M)h + . (9.52a)
Since W8 remains finite, the maximum radial stress at the center is
3 + U 1 3 a2 Or (0) = ± q = ± 8 (3 + v) 2 q, (9.53)
4 J 0 8 to a wellknown formula (cf. for instance [9]) which is verified in Table 9.1 when W = 0. For a low power lens, one can show that or(0) > ot (0) and that or(0) is maximal over 0 < r < a.
Expansions (9.52) lead to a flexure where the coefficient of the p6 term is zeroed as also are the higherorder coefficients. Substituting these expansions into (9.44), the flexure reduces to
'Em3
3 + ß 2 + _L 4 2ß83p + 483p and since m8 = T0 = constant when the pseudopower m = 0, this may be written, with the rigidity Do = Et0/[12(1  v2)] = constant,
By identification we obtain for the coefficients in (9.45) representing z/aq, 3(1  v)(3 + v) A 3(1  v2) A 0
These coefficients are listed in Table 9.1 for the particular cases where W = 0. Equation (9.54) represents the wellknown bending of a constant thickness plate  or of a meniscus of moderate cambrure  when the uniform load is simply supported at the edge.
9.2.5 Lenses with Particular Thickness Distributions
In the sequences listed in Table 9.1, the coefficient A2 is, of course, always negative for a positive load since it represents the curvature mode which is the first mode of the flexure. Considering next the A4 and A6 coefficients, we obtain the following results.
First, one notices that the A4 coefficient vanishes for particular lenses,
A4 = 0 if S ~ 3, which are all positive lenses such that the thickness is of the form t(r) = 2(ci  C2)(3a2  r2), (9.56a)
hence with an edge thickness of twothirds the central thickness. This result is in complete accordance with the thickness distribution t ^ (1  p2)1/3 = 1  p2 H
obtained for a variable curvature mirror designed with the same loading configuration (cf. Sect. 2.1.2).
Second, in addition to the case of a null power lens or meniscus where the A6 coefficient vanishes because S ^ ^ for which t(r) = constant [cf. (9.55)], one finds from Table 9.1 that this coefficient also vanishes for particular lenses,
A6 = 0 if S ~ 2, which are all positive lenses such that the thickness is of the form t(r) = (c1  C2)(a2  1 r2), (9.56b)
hence with a central thickness twice the edge thickness.
9.2.6 Conclusions for Active Optics Aspherization
From the previous results and Table 9.1 for fused silica, the following conclusions for the aspherization of single lenses by active optics methods  stress figuring with spherical tools  are inferred from parameter S = 70/m.
• Negative power lens: For a negative power lens, it appears that if, say, S < 4, the contribution of the p6term in the expansion of the flexure becomes nonnegligible compared to that of the p4term. For instance, for a fusedsilica lens with S = 2, the A4 coefficient reaches ~ A2/2. This also entails that the next order terms also have significant relative values. Hence, unless such significant terms of higher order than Sphe 3 are required by a more complex system in order for the lens to compensate such aberrations, a conclusion is as follows:
^ The correction of Sphe 3 for a virtual stigmatic conjugate can be achieved for an aspherized singlet lens if < 5 < 5, and there may be an optimum value of 5 for which Sphe 5 is also corrected.
• Positive power lens: For a positive power lens, the most promising geometries for correcting Sphe 3 of the lens also correspond to the cases where A6 is at least several times smaller than A4. From Table 9.1, we find that A6 = 0 if 5 ~ 2, i.e. lenses where the edge thickness is twice smaller than the central thickness. From condition 5 > 1 stated in (9.43), the flexure of sharpedge lenses cannot be expanded  since 5 = 1  and thus must be excluded for an active optics aspherization and also for practicable reasons. On the other hand, positive values of 5 such as 5 > 3 lead to A4 coefficients with an unacceptable sign. Hence for obtaining A6 and higherorder coefficients of moderate values relative to usable values of A4, a conclusion is as follows:
^ The correction of Sphe 3 for a real stigmatic conjugate can be achieved for an aspherized singlet lens if 1.666 < 5 < 2.250, and there exists an optimum value of 5 for which Sphe 5 is also corrected.
• Remark: Lenses with a higher refractive index show a substantially smaller Sphe 3 aberration (cf. Fig. 9.2). If the power K is the same and if n = 1.75, from (9.18a) the asphericity is 3.18times smaller than for a fused silica lens (nd = 1.458). Hence, for the case of positive lenses, the somewhat restrictive latter 5range for fused silica lenses may be relaxed for lenses with a high refractive index. However, except for exotic materials such as fused sapphire, it seems difficult to find a material which shows both high refractive index and high ultimate strength.
9.3 Spectrograph with Single Lens and Corrector Plate
Astronomical spectrographs are usually designed with a parabolic mirror as a collimator. For a collimated beam of slow fratio and a design with a reflective grating, the collimator mirror is sometimes substituted by a convexoplane lens plus an aspherical corrector plate which are airseparated. The plate is located at the position of the remote pupil of the isoplanatic mounting in Fig. 9.6Left. If the lens is assumed thin, the convex spherical surface of the lens of curvature c1 is located at the distance d = 1/nci from the plate. The first Seidel sums are Sj = Sjj = Sjjj = 0, the system is anastigmatic and we know that in this case the location of the input pupil is free. The only remaining aberrations are axial chromatism, field curvature, and the chromatic aberration variations. For moderate spectral ranges, i.e. high spectral resolutions, the aberration variations are small and the others can be compensated by a lens flattener and a tilt of the focal plane at the output of the spectrograph.
A Littrow mounting  a = [see grating law (4.23)]  of the grating is always preferable to achieve the highest possible dispersion. Hence the dispersed light is retroreflected through the system which then is directly used as camera optics if no addition of a focal reducer is necessary (Fig. 9.9).

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