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Fig. 7.3 (Up) Distribution of Clebsch-Seidel modes into the optics triangle matrix (piston z00 not shown). ^ q = 0 provides m = n modes, and m = n - 2 modes (D\ and D2 diagonals). ^ q = constant provides m = 0, n = 4 mode (z40 coupled with z20). (Down) Deformation interferograms obtained with the 12-arm vase MDM described and displayed by Fig. 7.2. From left to right, the interferograms are as follows: (Up diagonal boxes) Tilt 1, Astm3, Tri5, and Squa7 modes, (Middle line boxes) Cv 1 and Coma 3 modes, (Bottom line boxes) Mirror at rest with respect to a plane, co-addition of Cv 1 and Sphe 3 modes, co-addition of Cv 1, Sphe 3, and Astm 3 modes, and co-addition of Astm 3 and Squa 7 modes [Loom]

Fig. 7.3 (Up) Distribution of Clebsch-Seidel modes into the optics triangle matrix (piston z00 not shown). ^ q = 0 provides m = n modes, and m = n - 2 modes (D\ and D2 diagonals). ^ q = constant provides m = 0, n = 4 mode (z40 coupled with z20). (Down) Deformation interferograms obtained with the 12-arm vase MDM described and displayed by Fig. 7.2. From left to right, the interferograms are as follows: (Up diagonal boxes) Tilt 1, Astm3, Tri5, and Squa7 modes, (Middle line boxes) Cv 1 and Coma 3 modes, (Bottom line boxes) Mirror at rest with respect to a plane, co-addition of Cv 1 and Sphe 3 modes, co-addition of Cv 1, Sphe 3, and Astm 3 modes, and co-addition of Astm 3 and Squa 7 modes [Loom]

7.3 Elasticity and Meniscus-Form MDMs

In order to reduce the flexure singularities due to the shear component of the flexure, the number of discrete forces and moments applied along the mirror perimeter has to be optimized. This arm number depends on the order of the considered modes and on the stress level in the material. Low curvature meniscuses and flat plates can generate Clebsch-Seidel modes (Fig. 7.4).

Fig. 7.4 Meniscus-form multimode deformable mirror deflected by clamped radial arms. The constant rigidity allows one to achieve the superposition of Clebsch-Seidel modes by action of discrete axial forces Fak and Fck that are equivalent to the bending moment Mr and net shearing force Vr per unit length on r = a. For the single curvature mode z = A20?"2, if A 20 > 0, then the sign convention gives Mr > 0

Fig. 7.4 Meniscus-form multimode deformable mirror deflected by clamped radial arms. The constant rigidity allows one to achieve the superposition of Clebsch-Seidel modes by action of discrete axial forces Fak and Fck that are equivalent to the bending moment Mr and net shearing force Vr per unit length on r = a. For the single curvature mode z = A20?"2, if A 20 > 0, then the sign convention gives Mr > 0

Denoting D the mirror rigidity and representing the elastic flexure to generate by

Z = £ Anmrn cos mQ , from (7.8) and (7.9), the bending moments Mr, shearing forces Qr and net shearing forces Vr are

Mr = D £[n(n - 1) + v(n - m2)] Anmrn-2 cos mQ (7.15)

Vr = -D £ [(n - 2)(n2 - m2) + ( 1 - v)(n - 1 )m2] Anm rn-3 cos mQ (7.17)

For each of first modes these moments and forces at the mirror perimeter r = and at Q = 0 are the following.

Mode

n

m

Mr (a, 0)

Qr(a,0)

Vr (a,0)

0 0

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