Active Optics with Multimode Deformable Mirrors MDM Vase and Meniscus Forms

7.1 Introduction - Clebsch-Seidel Deformation Modes

This chapter treats the general case of obtaining aspherical mirrors such as off-axis mirrors locally defined from an axisymmetric shape, or on-axis mirrors for aberration correction of a non-centered system.

This requires the simultaneous correction of several wavefront modes such as 1st-order curvature, 3rd-order coma, astigmatism and spherical aberration, 5th-order triangle, etc. Although a mirror belonging to the variable thickness distribution (VTD) class may be bent within a diffraction limited tolerance for generating some superposed modes - such as, for instance, the Cv 1 and Astm 3 modes with the cycloid-type VTD T20 = T22 = (1 - p2)1/3 (see Sects. 2.1.2 and 3.5.2) -, in general, it is clear that a mirror belonging to the constant thickness distribution (CTD) class provides a higher superposition capability than a mirror of the VTD class.

From the analysis of an optical system showing aberrations, the optical wavefront function or the normalized Zernike form allows determining the various aberration modes to be corrected by a mirror. On the other hand, the elastic deformation modes of a CTD plate are represented by Clebsch's modes. These latter modes are perfectly similar to Seidel's optics modes, thus in the Fourier form XAnmrn cos mQ.

For some high-order Clebsch's modes, generating the elastic deformation implies that all the mirror clear aperture must be submitted to a non-uniform loading as a prismatic load, quadratic load, etc. Such loads are extremely difficult to generate for practical applications. However, restraining to the two cases where no load and uniform load are applied to the mirror clear aperture, we will see hereafter that a large variety of Clebsch's modes are easily practicable. The elastic modes generated from loads q = 0 or q = constant belong to a subclass of the optics triangle matrix modes; we propose to call them Clebsch-Seidel modes.

Multimode deformable mirrors (hereafter MDMs) have been developed for the purpose of generating and superposing Clebsch-Seidel modes.

G.R. Lemaitre, Astronomical Optics and Elasticity Theory, Astronomy and Astrophysics Library, DOI 10.1007/978-3-540-68905-8.7, @ Springer-Verlag Berlin Heidelberg 2009

7.2 Elasticity and Vase-Form MDMs

Given a continuous deformation mode, the elasticity theory of thin plates applied to a constant thickness disk - flat or with a small curvature - allows deriving the forces and moments acting on it. Some of the optical aberration modes in the form z = Anmrn cos mQ can be easily obtained by a convenient distribution of axial forces and bending moments acting along the disk perimeter. Generating a radial bending moment distribution requires use of two axial force sets. When discrete forces are applied close to the mirror edge - and equally distributed in azimuth angles -, the shear components of the elastic deformation due to the point forces provide slope discontinuities at these regions. In order to generate a smooth and continuous surface, it is much preferable to avoid point forces applied near the clear aperture edge of the mirror. Hence, the best continuity is obtained with point forces applied at maximal distances from the optical surface.

Following Saint Venant's principle [23, 57] (see Sect. 1.13.12), leads to a vaseform design, i.e. a mirror having two concentric zones of constant rigidity as proposed by Lemaitre [38,41]. Similar drum-shaped mirrors were suggested in the past by Couder [12] for obtaining low-weight axisymmetric mirrors. In a vase form, the outer zone is thicker than that of clear aperture which is the inner zone. The two zones are clamped together in a single - or holosteric - piece. Whatever the symmetry of the modes to generate, the radial bending moments to distribute along the outer ring may reach important values. Therefore, the vase form is completed with external radial arms built-in to the ring on which axial forces are applied to the outer ends. Apart from minimizing the shear flexure effect to negligible value, another advantage of the ring is to provide a smooth angular modulation of the perimeter flexure from the discrete forces. The number of radial arms is strongly dependent on the highest order of the flexural mode z(r, Q) to be generated and must also be optimized for all the required modes. A design with a vase form and radial arms is called multimode deformable mirror (Fig. 7.1).

Let us consider a plane MDM with a clear aperture zone defined by 0 < r < a, a built-in ring zone defined by a < r < b, where ti, t2 and Di, D2 are the thicknesses and associated rigidities of the inner and outer zones, respectively. The axial forces applied to the ring inner radius, r = a, are denoted Fa,k; those applied to the arm outer-end, at r = c, are Fc,k. With a total number of km arms, each arm is numbered by k e [1, 2,..., km] and k = 1 Q = 0. In addition, positive or negative uniform loads q can be superposed into the vase inner zone by mean of air pressure or de-pressure.

Denoting E and v the Young modulus and Poisson's ratio respectively, in a cylindrical coordinate system z, r, Q, the flexure surface Z is given by Poisson's equation (7.6)

where q and D are the load per unit surface area and the flexural rigidity,

Fig. 7.1 Elasticity design of a vase-form MDM based on two concentric rigidities and radial arms. The clear aperture zone is built-in at r = a into a thicker ring. This holosteric design allows generating and co-adding the Clebsch-Seidel deformation modes, Cv 1, Sphe3, Coma3, Astm3, etc, by axial forces Fak and Fck applied at the ring inner radius r = a and outer end r = c of km arms

Fig. 7.1 Elasticity design of a vase-form MDM based on two concentric rigidities and radial arms. The clear aperture zone is built-in at r = a into a thicker ring. This holosteric design allows generating and co-adding the Clebsch-Seidel deformation modes, Cv 1, Sphe3, Coma3, Astm3, etc, by axial forces Fak and Fck applied at the ring inner radius r = a and outer end r = c of km arms and V2- = d2 • /dr2 + d • /rdr + d2 • /r2d62 is the Laplacian. Inner and outer constant rigidities are defined by iD = D1 if 0 < r < a ' (7.2b)

• Inner zone (0 < r < a): At the inner zone of the vase form, we assume hereafter that a flexural mode to generate is with a polynomial representation having a symmetry with respect to z,x plane (6 = 0), i.e. we only consider cosmd terms; sinmd terms would give a similar representation. Thus, in a cylindrical coordinate system, a representation of the flexure is

Z = ^znm = ^Anmrn cosm6, n + m even , m < n, (7.3)

where n and m are positive integers and Anm coefficients are identical to those belonging to the optics triangular matrix (see Sect. 1.8.2) expressing the shape of a wavefront or a mirror, i.e. using the same composition rules for n and m. Given a mode znm, the substitution in (7.1) leads to

Anm(n2 - m2)[(n - 2)2 - m2]rn~4cosm6 = q/D, with n > 2. (7.4a)

The only combinations of n and m for which the equation can be solved for practicable applications are:

^m = n-2, i.e. z20,z3i,z42,... terms, (7.4b) case q = constant ^ n = 4, m = 0, i.e.the z40 term.

These cases define a subclass proposed to be called Clebsch-Seidel modes. Except for the z40 mode, these modes belong to the two lower diagonals of the optics triangle matrix. The generation of z20 = Cv 1, z40 = Sphe3, z3i = Coma3, z22 = Astm3, z42 = Astm5, z33 = Tri5, z53 = Tri7, z44 = Squa7, ... modes is obtained, while it is found not possible to generate the two other 5th-order modes z51 = Coma 5 or z60 = Sphe 5 by only using q = 0 or a uniform loading q = constant. Generating z51 would require a prismatic loading while for z60 a parabolic loading. Because of the extreme difficulties to achieve them in practice, such non-uniform loading distributions will not be considered hereafter.

• Outer zone (a < r < b): On the outer zone of the vase mirror, a uniform load is never applied, so that the equation to be solved for a < r < b is (7.1) with q = 0. The solutions are

Z = £ znm = Rn0 + £ Rnm cos m0 + £ R'nm sin m0, (7.5)

in which Rn0, Rn1, ..., R'n1, ... are a function of the radial distance only. In our case, one considers the same azimuth of the deformation as for (7.3), so that the third functions R'nm vanish.

The functions Rnm are Clebsch's solutions [11] of

For m = 0, m = 1 and m > 1, the functions Rnm - or Clebsch's polynomials - have the following forms

Rn0 = Bn0 + Cn0 ln r + Dnor2 + Enor2 ln r, Rni = Bnir + Cnir-1 + Dnir3 + Enir ln r,

• Bending moments and shearing Forces: The boundaries between the two zones at r = a must provide a continuity of the flexure znm, slope dznm/dr, bending moment Mr and net shearing force Vr. Denoting z = znm for a single mode, the bending moments Mr, Mt, and twisting moment Mrt are respectively defined by

The radial and tangential shearing force Qr and Qt derived from the static equilibrium [cf. Sect. 3.2, eqns. (3.3) and (3.4)], are dMr 1 ( dMrt

and their equilibrium with an external load q [cf. Sect. 3.2, (3.5)] demonstrates Pois-son's biharmonic equation

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