Deformable Mirrors: Design Fundamentals for Force Actuation of Continuous Facesheets

Deformable Mirrors: Design Fundamentals for Force Actuation of Continuous Facesheets S.K. Ravensbergen a, R.F.H.M. Hamelinck b, P.C.J.N. Rosielle a and M. Steinbuch a a Technische Universiteit Eindhoven, Den Dolech, 56 MB, The Netherlands b TNO Science and Industry, Stieltjesweg 1, 68 CK Delft, The Netherlands ABSTRACT Adaptive Optics is established as essential technology in current and future ground based (extremely) large telescopes to compensate for atmospheric turbulence. Deformable mirrors for astronomic purposes have a high number of actuators (> 1k), a relatively large stroke (> 1μm) on a small spacing (< 1mm) and a high control bandwidth (> 1Hz). The availability of piezoelectric ceramics as an actuator principle has driven the development of many adaptive deformable mirrors towards inappropriately stiff displacement actuation. This, while the use of force actuation supersedes piezos in performance and longevity while being less costly per channel by a factor of 1-. This paper presents a model which is independent of the actuator type used for actuation of continuous facesheet deformable mirrors, to study the design parameters such as: actuator spacing & coupling, influence function, peak-valley stroke, dynamical behavior: global & local, etc. The model is validated using finite element simulations and its parameters are used to derive design fundamentals for optimization. Keywords: deformable mirror, continuous facesheet, plate model, influence function 1. INTRODUCTION Large telescopes are affected by time varying wave front distortions due to air turbulence and inhomogeneity. To compensate for distortions, adaptive mirrors are used in such ground-based telescopes. Adaptive (and thus deformable) mirrors are a construction of many actuators on a support structure, actuating a reflective facesheet. Distortions are corrected by shifting the phase of the wavefront via a local change in optical path length. Actuators, which produce the required displacement, require a fast response time, controlled accuracy, stability and low power dissipation. The availability of piezoelectric ceramics has driven the development of such mirrors towards displacement actuation. However, recent research has focused on force actuation. 1 The continuous facesheet is manipulated via actuators with a low axial stiffness ( 1 3 N/m) compared to piezoelectric actuation ( 1 6-1 7 N/m). To describe the facesheet deformation, a semi-analytical model is presented (Sec. ). This model is relatively easy to implement compared to a finite element approach and leads to basic design equations (Sec. 3) which are used for parameter optimization (Sec. 5).. FORCE ACTUATED MIRROR MODEL Assume a thin facesheet (with thickness h) placed on discrete force actuators with a given pitch d act and stiffness C act, see Fig 1a. Each actuator covers a given plate area A, as shown for a hexagonal configuration in Fig 1b. The thin plate theory assumes isotropic material, a constant thickness, small out of plane displacements (<.5h) based on flexural deformation. The deflection w(r, φ) of a plate on an elastic foundation with load P, is described by the biharmonic plate equation: 4 w(r, φ) = Further author information: (Send correspondence to.) S. Ravensbergen: E-mail: s.k.ravensbergen@tue.nl, Telephone: +31 4 47458 P (r, φ) kw(r, φ). (1) D Advanced Wavefront Control: Methods, Devices, and Applications VII, edited by Richard A. Carreras, Troy A. Rhoadarmer, David C. Dayton, Proc. of SPIE Vol. 7466, 7466G 9 SPIE CCC code: 77-786X/9/$18 doi: 1.1117/1.8583 Proc. of SPIE Vol. 7466 7466G-1

P d act d act w(r) h A r Figure 1. A model of a simply supported thin facesheet mirror with discrete force actuators. Area covered by each actuator assuming a hexagonal configuration. Using the biharmonic operator ( 4 =, the Laplacian squared) in polar coordinates and using axial symmetry (w(r, φ) =w(r) [m]) with a central load P (r = ) [N], the above equation becomes ( d dr + 1 )( d d w r dr dr + 1 ) dw P (r) kw(r) =. () r dr D The flexural rigidity of the plate is given by Eh 3 D = 1(1 ν [Nm], (3) ) with the Youngs modulus E and poisson ratio ν. The foundation modulus is approximated by k C act A = C [ ] act N 1 3d act m 3. (4) By introducing the notation k D = [ 1 1 ] l 4 m 4 the solution of (), using the Kelvin function 3 (kei) is C act, x = r l [ ], (5) w(r) = Pl kei(x). (6) πd Within Matlab the Kelvin function can be calculated as: kei(x) = π I { bessely (,xe π 4 i)} π R { besselj (,xe π 4 i)}. (7) The maximal deflection under load P and the combined stiffness reads: w max = Pl 8D, C plate+act = P w max = 8D l =8 kd. (8) The actuator coupling η is defined as the ratio of the faceplate deflection at an immediately adjacent actuator, to that of the peak deflection due to the energized actuator: 4 η =.1 Model Validation Using Finite Element Modeling w(r) w(r ± d act ). (9) To validate the model described above, a finite element model (FEM) is made with 61 uniaxial springs (C act )in a 3mm spaced hexagonal configuration. The plate boundary is simply supported and the facesheet is modeled using D triangular elements with a specified constant thickness. In Fig. a the response of a 1μm thick Pyrex and a 5μm thick beryllium plate is given for a 1mN loading on the central spring. Material properties used, are given in Table 1. The 5μm beryllium plate model and FEM deviate less than 3% (at the center). By summing two opposite responses, the inter-actuator response is approximated. A result for a 5μm beryllium plate is shown in Fig. b, the difference between model and FEM is below 3%. Proc. of SPIE Vol. 7466 7466G-

Table 1. Material properties used Youngs modulus [GPa] Poisson ratio [-] Density Beryllium 33.1 1844 Pyrex 63. 3 [ ] kg m 3 Deflection [μm] 1.5 1.5.5 Pyrex 1μm Beryllium 5μm 4 6 8 1 1 Radial distance [mm] Deflection [μm] 1.5 1.5.5 1 Beryllium 5μm 1.5 15 1 5 5 1 15 Radial distance [mm] Figure. Thin plate deflection under 1mN central load as function of the radial distance for Pyrex: h = 1μm (solid line) and beryllium: h = 5μm (dotted line), actuator locations indicated with triangles. Inter-actuator peak-valley response FEM (points) and analytical (solid line): beryllium h =5μmunder +1mN at r = mm, and 1mN at r = 3mm.. Lower Bound on Thickness To determine the lower bound on the thickness two factors are considered: the facesheet stress at an energized actuator and the deflection between the actuators under self-load...1 Facesheet stress The maximal tensile stress in the plate at an actuator location is given by: σ r,max =.75(1 + ν) P h log 1 Eh 3 kb 4, (1) 6 5 g 4.1mm load area diameter Beryllium Yield stress quilting 3 1.mm load area diameter 5 1 15 5 3 Facesheet thickness [μm] Figure 3. Mirror cross section of the deflection under self-load with a maximum quilting of.85nm using h =16μm beryllium. The facesheet stress as function of the thickness for beryllium with load area diameters of.1 and.mm under a constant load of 1mN and an actuator pitch of 3mm. Facesheet stress [MPa] Proc. of SPIE Vol. 7466 7466G-3

where b = 1.6c + h.675h when c<1.74h = c when c>1.74h with the radius of the load area c, see Fig. 3b. (11).. Deflection under self-weight The deflection under self-weight can be approximated with a simply supported circular plate with the actuator pitch as diameter: ξ max = (5+ν)ρgh ( dact ) 4 64(1+ν)D, ξ rms = 1 ξmax, (1) where ξ rms is called the facesheet gravitational quilting. Note that the deflection under self-weight is proportional to d 4 act/h,sinced h 3. In Fig. 3a, the mirror is simply supported at the actuator locations and subjected to a gravitational load. The maximal quilting is.85nm, where Eq. (1) gives 1nm for the same conditions. So the approximation is a usable measure for a thickness lower bound. 3. DYNAMIC MODEL The first order dynamical response is a measure for the maximal bandwidth of the deformable mirror. Structural analyses, using the FEM model (as also used in Sec..1) are given in Fig. 4. The actuator stiffness is varied for a beryllium faceplate with 3 μm thickness and for a 1 μm Pyrex plate. Both cases clearly show a break point in first Eigenfrequency. This behavior is caused by the transition from an overall vibration of the faceplate (with the boundary as only nodal diameter, see Fig. 5) towards local modes between the actuators (Fig. 6). So for a low actuator stiffness, the first Eigenfrequency is determined by the combination of actuator stiffness and the faceplate. Whereas for high actuator stiffnesses, it is only determined by the plate and the actuators appear as fixed points. A key distinction is made between force and displacement actuation. Force actuation is defined for situations in which the actuators are less stiff than the faceplate. Displacement or position actuators are equal or stiffer than the plate. 5 Using force actuation, the first Eigenfrequency can be estimated by calculating the frequency of a local spring-mass system. The spring is given by C act and the mass by m faceplate ρha, wherea is the plate area per actuator: f = 1 C act. (13) π m faceplate 3.1 Break Point Estimation As shown in Figs. 4 and 6 displacement actuation is characterized by local faceplate vibrations between the actuators. The break point is estimated using a model similar to the one presented in Sec.. Assume a circularly clamped plate with a radius a 1 3dact the deflection is given by: w(r) = P ( a r ). (14) 64D Using a uniform gravity loading, the maximal deflection and the faceplate stiffness reads: w max = Pa4 64D = ρgha4 64D, C faceplate = P w max = ρghπa w max = 64πD a. (15) Displacement actuation is the regime in which the stiffness of the actuator is beyond this local faceplate stiffness. The frequency of free vibration of a clamped circular plate is given by 6, 7 f = 1 ζ π D a ρh where ζ 1.. (16) Proc. of SPIE Vol. 7466 7466G-4

Figure 4. First Eigen frequency as function of elastic support stiffness (e.g. actuator stiffness). Figure 5. Low-frequency phenomena: plate deflection between the discrete actuator locations (less stiff actuators). Top view, actuator locations are indicated. Proc. of SPIE Vol. 7466 7466G-5

7 \.,- / / ' T --' \ \ t S \ ) I I \ \ \\\.>'//// \\\'J/l \\\\, jj/ / / \ \\\\\-////// ;:-Y/l// <'\\\'\ \ \\\\\'--' / /////7/ \\\\\\\ \ I ////i'/ /I/// "\\l( I \ (I/I/-\\\ii ( 111/ / / 1 / \ (/ /7.--- \ I I 7,'-Z-N \ r / /ZN.. \ ----.---- ----- \_------ J \N --..-..-.. ------ Figure 6. High-frequency phenomena: plate deflection between the discrete actuator locations (stiffer actuators). Top view, actuator locations are indicated with the effective area. 4. POWER OPTIMIZATION To calculate the dissipated power, an assumption on the mean wave front variance is made, based on Kolmogorov s turbulence theory: 4, 8 ( ) 5 σmodesremoved =1.3 Dprimary 3 [ r rad ] ( ) 5 and σmodesremoved( axis tilt) =.134 Dprimary 3 [ r rad ]. (17) Where D primary istheaperturediameterandr the coherence length of the atmosphere (Fried parameter 9 )and is specified at a given wavelength λ. So 87% of the phase variance is compensated by a separate tip-tilt mirror. Assuming that most of the turbulence (99.4%) will be within ±.5σ and due to mirror reflection (one unit of motion of the mirror results in two units of wavefront correction) the total required actuator range is: PV stroke = 1 5.134 ( Dprimary r ) 5 6 λ π [m]. (18) The maximal actuator force using Eq. (8) is: F max = 1 PV stroke C plate+act.66 ( ) 5 Dprimary 6 DC plate+act λ r d 1 act. (19) Note that the actuator rms force is a factor.5 lower. The material properties are of great influence on the actuator forces: F max Eh 3. And thus also on the energy dissipated: P diss F, for a reluctance actuator with permanent magnet and for a voice-coil actuator (P = I R, I F ). 1 5. DESIGN FUNDAMENTALS For the design of a deformable mirror, different parameters must be determined: facesheet thickness, actuator spacing & coupling, influence function and peak-valley stroke. Figure 7a shows the rms facesheet quilting (Eq. (1)) and actuator coupling (Eq. (9)) for beryllium and Pyrex as function of the thickness. A constant actuator pitch of d act = 3mm and a stiffness of C act = 1N/m is used. By preferring the facesheet quilting to be below nm and the actuator coupling below.1 (so 1%), the design window of the thickness is determined. In Fig. 7b the actuator coupling as function of actuator stiffness is given for a 3μm beryllium and 1μm Pyrex plate. Again the transition from force to displacement actuation is visible at an actuator stiffness of 1 4 N/m for beryllium and 1 1 5 N/m for Pyrex. When the facesheet thickness is determined using Fig. 7a, the inter-actuator stroke and actuator stiffness follow from the preferred actuator coupling (eg. 5 or 1%). These are shown in Fig. 8a for beryllium, as function Proc. of SPIE Vol. 7466 7466G-6

1.9 3mu beryllium 1mu Pyrex Actuator coupling.8.7.6.5.4.3..1 1 1 1 1 3 1 4 1 5 1 6 Actuator stiffness [N/m] Figure 7. Mirror design trades: rms facesheet quilting and actuator coupling as function of the thickness for both Pyrex and beryllium. Actuator coupling as function of the stiffness for h =3μm beryllium and h = 1μm Pyrex. 3 5 Beryllium Pyrex Maximal force [mn] 15 1 5 1 3 4 5 Facesheet thickness [µm] Figure 8. Peak-valley inter-actuator stroke and actuator stiffness at 5 and 1% actuator coupling as function of mirror thickness. Maximal actuator force necessary to compensate 99.4% of the wavefront error as defined in Eq. (19). of the thickness. The inter-actuator stroke is calculated via the model described in Sec..1 for a 1mN actuator force (see also Fig. b). As can be seen in Fig. 8a, increasing the actuator coupling from 5 to 1%, result in a lower inter-actuator stroke and a higher actuator stiffness. Figure 8b shows the maximal force that is necessary to compensate the wavefront for 99.4%, as defined in Eq. (19). The force is given for both Pyrex and beryllium as function of the thickness, using the following parameters: r =13[cm] D primary =4[m] λ = 55 [nm] d act =3[mm] C act = 1 [N/m]. () So the optimal choice is selecting the minimal facesheet thickness for a given material and a given gravitational quilting. Together with a desired actuator coupling (or stiffness) the forces and thus the energy dissipated are minimized. Proc. of SPIE Vol. 7466 7466G-7

6. SUMMARY & CONCLUSION Themodeldescribedby is adapted to deformable mirrors, and appeared well-suited to describe thin facesheets on discrete actuators. Specifications like inter-actuator stroke and actuator coupling are also predicted accurately. A mirror classification between force and displacement actuation is made, based on 1 st Eigen frequency results. Design fundamentals for optimization are derived under the assumption of a 1% actuator coupling. The equations provide a starting point for designing a specific mirror. Typical system specifications are elaborated to develop working design windows as follows for: Mirror quilting and actuator coupling as function of thickness Mirror thickness as function of actuator stiffness Inter-actuator stroke for different materials as function of thickness Power optimization Future work includes designing, building and testing a new force actuation based mirror. The aimed specifications are 61 actuators with 3mm pitch, μm stroke, μm inter-actuator stroke and an inter-actuator coupling below 15%. REFERENCES [1] Hamelinck, R., Ellenbroek, R., Rosielle, N., Steinbuch, M., Verhaegen, M., and Doelman, N., Validation of a new adaptive deformable mirror concept, Proc. SPIE 715, 715Q 1 (July 8). [] Timoshenko, S. and Woinowsky-Krieger, S., [Theory of Plates and Shells], McGraw-Hill (1959). [3] Abramowitz, M. and Stegun, I. A., [Handbook of Mathematical Functions], Dover Publications (1965). [4] Hardy, J. W., [Adaptive Optics for Astronomical Telescopes], Oxford University Press, US (1998). [5] Ealey, M. A. and Wellman, J. A., Deformable mirrors: design fundamentals, key performance specifications, and parametric trades, in [Active and Adaptive Optical Components], 1543, 36 51, SPIE, San Diego, CA, USA (199). [6] Reddy, J. N., [Theory and Analysis of Elastic Plates and Shells], CRC Press (6). [7] Blevins, R. D., [Formulas for Natural Frequency and Mode Shape], Van Nostrand Reinhold (1979). [8] Roddier, F., [Adaptive Optics in Astronomy], Cambridge University Press (1999). [9] Fried, D. L., Statistics of a geometric representation of wavefront distortion, JournaloftheOpticalSociety of America 55, 147 (Nov. 1965). [1] Furlani, E. P., [Permanent Magnet and Electromechanical Devices: Materials, Analysis, and Applications], Academic press, San Diego (1). Proc. of SPIE Vol. 7466 7466G-8