Numerical simulations of the LBT adaptive secondary mirror Ciro Del Vecchioa and Daniele Gallienib a Osservatorio Astro sico di Arcetri, Largo Enrico Fermi 5, I-50125 Firenze, Italy b ADS International S.r.L., C.so Promessi Sposi 23/d, I-23900 Lecco, Italy
ABSTRACT
In this paper we describe the design of the deformable mirror of the Large Binocular Telescope adaptive secondary unit. Starting from the optical design, a numerical model of the ultra-thin, aspherical glass shell, accommodating the 918 magnets on the selected actuator geometry, has been run. As a results, we can evaluate the response of this crucial component of the telescope optics with great accuracy. The DM is analyzed from the mechanical standpoint | gravity deformations, wavefront residue, residue of low-order Zernike aberrations, corrections of magnetic interactions | in order to compute the optical performances in the most demanding operational circumstances. Keywords: adaptive optics, deformable mirror, adaptive secondary mirror, electromagnetic actuator, FEA
1. INTRODUCTION
The purpose of this paper is the discussion of the results of the mechanical studies of the deformable mirror (DM) of the Large Binocular Telescope (LBT) adaptive secondary unit, in order to determine the e ectiveness of the actuator geometry and the mechanical performances of the design (see Ref. 1 for a complete description of the LBT; an overview of the adaptive secondary unit is given in Ref. 2). The DM thickness is parametrically varied, in order to select the nal design. Such a choice is performed by comparing the response of the DM to four sets of loads, as described in Ref. 3. These loads | gravity, some low-order Zernike deformations, an atmospheric set of turbulent wavefronts, and the magnetic interaction between the magnets, discussed in Sections 4, 5, 6, and 7, respectively | are applied to three DM's of di erent \nominal" thicknesses | 1, 1.5, and 2 mm. In case of a at, circular plate, the deformation under a pressure load, like its own weight, depends on the second power of the thickness, while the reaction forces due to an arbitrary set of imposed deformations depend on the third power of it. As the DM is not at, and because the permanent magnets can be considered as \quasi-concentrated" loads, no scaling law can be applied, so that each case is computed separately. The gravitational loads are evaluated by comparing the DM distortion and actuator forces. In the other three load cases, some sets of displacements are applied to the models, so that the DM performances are evaluated by comparison of the residues of the surface displacements and the actuator (correction) forces for the three sample thicknesses. The computation of the residue at high spatial frequencies is used to evaluate the optical performances, while the actuator forces re relevant for power and thermal evaluations. The DM geometry, described in Sec. 2, is translated into a numerical model according to the method depicted in Sec. 3. The power dissipation is computed in Sec. 8, in order to evaluate the thermal load a ecting the system.
2. THE THICKNESS PROFILE
The z coordinate of the front and back surfaces of the DM is de ned by
z=? p
2 p 2r 2 2 ; R+ R ?r ?r K
(1)
where r = x2 + y2 is the distance from the optical axis, R is the surface radius of curvature (on the front surface its value is Rf = ?1974:242 mm), and K is the conic constant (equal to 0 for the back, spherical surface and to ?:7328021 for the front, aspherical, concave surface of the LBT gregorian secondary mirror). r is de ned between Further author information: (Send correspondence to Ciro Del Vecchio) Ciro Del Vecchio: E-mail:
[email protected]; Telephone: +39 055 2752261; Fax: +39 055 2752292 Daniele Gallieni: E-mail:
[email protected]; Telephone: +39 0341 259231; Fax: +39 0341 259235
back centers of curvature [mm] @ vertex mean front back radii of curvature [mm] @vertex mean front max min thickness [m] for ptv Ri r R o mean rms std glass mass [kg] magnets total
t0 = 1 mm t0 = 1:5 mm t0 = 2 mm -7.347
7.347 1989.935 1982.057 1974.242 1078.3 888.5 189.7 1033.0 1033.8 41.2 1.714 4.009
-6.847 0.000 6.847 1989.435 1981.809 1974.242 1568.3 1362.1 206.1 1523.6 1524.3 44.6 2.524 2.295 4.819
-6.347
6.347 1988.935 1981.561 1974.242 2059.0 1835.7 223.3 2014.3 2014.9 49.0 3.334 5.629
Main optical and physical parameters of the DM: centers and radii of curvature at vertex, maximum, minimum, peak-to-valley (ptv), mean, rms, and standard deviation (std) of thickness, and mass.
Table 1.
Ri = 28 mm, the inner hole, and Ro = 455:486 mm, the outer edge. If zBF , de ned by Eq. 2, is the equation of the \best t circumference" in the 1st quadrant,
q
2 ? r2 ; (2) zBF = CBF ? RBF we can determine the center CBF , laying on the z axis, and the radius RBF of zBF by minimizing the sum of (zi ?zBF )2 | both Equations 1 and 2 are sampled in a nite number of points of the domain ri , where Ri ri Ro |
R code. We de ne as the DM with the unconstrained nonlinear optimization algorithm provided by the Matlab \nominal" thickness t0 the value of zBF at r = 0, the vertex of the DM | i.e., from Eq. 2, CBF ? RBF . Such a condition can be satis ed by varying CBF by c and/or RBF by r, that is: t0 = CBF + c ? RBF ? r : (3) Because it must be c = 0, in order to allow that zBF and zf are concentric at r = 0 and that the thickness is constant if Kf = 0, Eq. 3 gives: r = CBF ? RBF ? t0 : (4) Eq. 2 gives the back surface equation zb , provided that RBF is increased by r , as de ned by Eq. 4: p zb = CBF ? (CBF ? t0 )2 ? r2 : (5) As the center of zb in Eq. 5 is CBF and the center of zf in Eq. 1 for r = 0 is Rf , the radius of the mean surface zm = (zb + zf )=2 for r=0 is Rm0 = c ? t0 =2, where c = (jRf j + CBF )=2. The equation of the mean surface for small values of r around 0 is given by q (6) zm = Cm0 Rm2 0 ? r2 ; where the sign + is related to a concave surface and the sign ? to a convex surface. As zm must be t0 =2 for r = 0 in Eq. 6, the center of the mean surface is given by Cm0 = c for a convex surface, and Cm0 = ?c + t0 for a concave surface. If zm is o set by Cm0 , its center of curvature is in the origin of the Cartesian coordinate system for r = 0. Such a de nition of zm | properly sampled in a certain number of points | has been used to generate the Finite Element Analysis (FEA) model of the DM, choosing three values of t0 | 1, 1.5, and 2 mm | as nominal thicknesses at the mirror vertex. The main optical and physical parameters of the DM model are summarized in Tab. 1. 918 actuators are arranged into 17 rings, concentric with the optical axis. The actuator axes that lay on each ring are normal to the back surface and intersect the optical axis z at the points referred as \centers of curvature @ vertex" of the back surface in Tab. 1. The angles between the actuator axes and z range from 1.16 to 13, with a constant equal to .74, according to the geometry described in Ref. 2. The actuator magnets are the same described in Ref. 3. i
nodes (⋅) and actuator (×) location
0.4
0.3
0.2
y axis
0.1
0
−0.1
−0.2
−0.3
−0.4
−0.4
−0.3
Figure 1.
−0.2
−0.1
0 x axis
0.1
0.2
0.3
0.4
FEA model of the 1 mm thick DM: glass nodes and actuator geometry.
3. THE NUMERICAL MODEL
R code, used also for all the Three FEA models | one for each thickness| were set up by means of the Ansys computations described in this paper. Pending on t0 , the three models consists of 37272, 37248, and 38784 shell elements that model the glass, with the element thickness varying according to the optical design discussed in Sec. 2, and using the material properties of the Zerodur. The total numbers of nodes de ning the glass are 25308, 25296, and 25158, respectively. The DM glass nodes and actuator locations of the 1 mm thick model are plotted in Fig. 1. In a separate run, the model of an octagonal magnet has been built with 24 brick elements and 51 nodes. Such a model, properly displaced and rotated 918 times in the actuator locations, allows the \mount" of the magnets on the
gravity direction wrt opt axis
?
k
max min ptv rms std max min ptv rms std
t0 = 1 mm t0 = 1:5 mm t0 = 2 mm DM actuator DM actuator DM actuator displ forces displ forces displ forces [nm] [N ] [nm] [N ] [nm] [N ] 548.1 0.171 209.0 0.213 106.0 0.253 -548.1 -0.171 -209.0 -0.213 -106.0 -0.253 1096.2 0.342 417.9 0.425 212.0 0.505 18.8 0.014 7.2 0.017 3.6 0.020 18.8 0.014 7.2 0.017 3.6 0.020 0.0 0.055 0.0 0.069 0.0 0.083 -41.8 0.037 -18.5 0.044 -10.3 0.050 41.8 0.018 18.5 0.026 10.3 0.033 6.6 0.044 2.9 0.052 1.6 0.061 3.9 0.003 1.7 0.004 0.9 0.005
Table 2. DM displacements and actuator reaction forces due to the gravity for horizon (gravity perpendicular to the optical axis) and zenith (gravity parallel to the optical axis) pointing: the maximum, minimum, peak-to-valley (ptv), rms, and standard deviation (std) values are listed.
glass as \superelements", dramatically improving the computation time. Each glass-magnet interface is provided by 9 spring elements, whose axial sti ness is 3:556 106 N m?1 . The interface sti ness in the directions normal to the magnet axis is set as in nite. Such an interface, whose diameter is 6 mm, is a good approximation of the glue static and dynamic properties, as experimentally evaluated. Due to the negligible glue volume, no mass is associated to the 8262 spring elements. As in Ref. 3, the nodes belonging to the inner hole of the DM model are restrained in the radial and tangential directions, approximating the central, non-linear membrane described in Ref. 4 as in nitely rigid in its plane and neglecting its out-of-plane sti ness. All the input and output displacements have been considered as radial in the spherical coordinate system centered at the point (0; 0; 0).
4. GRAVITY
Two static analyses have been performed in order to evaluate the DM response under the gravity when it is applied along the optical axis and perpendicularly to it | the zenith and the horizon pointing, respectively. The results, summarized in Tab. 2, show that the rms values of the actuator forces | some hundredths of newtons at zenith and roughly 4 times less at the horizon | are low enough in terms of power dissipation, as it will be discussed in Sec. 8. The resulting displacement elds | always below 20 nm | are within the optical speci cations for thicknesses from 1.5 to 2 mm. We note that the peak values of the displacements, that occur when the gravity is perpendicular to the optical axis, fall in the central obscuration of the mirror. In the optically used portion of the DM, the deformation induced by the gravity is therefore signi cantly less than the values reported in Tab. 2. This e ect is larger for the 1 mm thickness. The gravity \quilting" can be used to derive the wind \quilting". Assuming an air density equal to 1 kg m?3 , the wind velocities that give the same de ections as gravity are 7.2, 8.7, and 10.0 m sec?1, as the thickness increases from 1 to 2 mm.
5. LOW-ORDER ZERNIKE DEFORMATIONS
The Zernike aberrations of orders 4 to 9 | focus, astigmatism, coma, and spherical, calculated in such a way that all of them have a ptv of .5 m | have been imposed to the DM at the 918 actuator positions, in order to compute the residues. The results, summarized in Tab. 3, show that removing the rst low-order Zernike aberration implies residues as low as few nanometers and correction forces of few thousandths of newtons. We note that in the spherical case | the most demanding aberration one | the ratios between residues and displacements ranges from 65 to 97, as the thickness increases from 1 to 2 mm | a relevant improvement with respect to the MMT value of 32.
6. TURBULENT WAVEFRONT
Adopting the method described by Ref. 5, and the procedure adopted in Ref. 3, we apply at the DM models 100 wavefront deformations assuming the Kolmogorov turbulence spectrum of the atmosphere, with a Fried's parameter
aberration max min focus ptv rms std max min astigmatism ptv rms std max min coma ptv rms std max min spherical ptv rms std
DM displ [m] 0.244 -0.248 0.492 0.148 0.148 0.248 -0.248 0.495 0.102 0.102 0.239 -0.239 0.479 0.089 0.089 0.318 -0.165 0.483 0.155 0.154
t0 = 1 mm
t0 = 1:5 mm actuator DM DM actuator DM forces residue displ forces residue [N ] [nm] [m] [N ] [nm] 0.00304 28.02 0.244 0.00437 12.86 -0.01118 -2.55 -0.247 -0.01702 -1.13 0.01422 30.58 0.491 0.02138 13.99 0.00250 1.50 0.148 0.00371 0.70 0.00250 1.49 0.148 0.00371 0.69 0.00003 0.18 0.248 0.00007 0.19 -0.00003 -0.18 -0.248 -0.00007 -0.19 0.00005 0.36 0.495 0.00015 0.37 0.00001 0.02 0.102 0.00002 0.02 0.00001 0.02 0.102 0.00002 0.02 0.00280 3.29 0.239 0.00459 2.74 -0.00280 -3.29 -0.239 -0.00459 -2.74 0.00559 6.58 0.478 0.00918 5.48 0.00146 0.45 0.089 0.00217 0.31 0.00146 0.45 0.089 0.00217 0.31 0.01501 3.32 0.318 0.02246 2.16 -0.00293 -36.51 -0.164 -0.00428 -15.64 0.01794 39.84 0.482 0.02674 17.80 0.00274 2.38 0.155 0.00431 1.71 0.00274 2.35 0.154 0.00431 1.69
DM displ [m] 0.244 -0.247 0.491 0.147 0.147 0.248 -0.248 0.495 0.102 0.102 0.239 -0.239 0.478 0.090 0.090 0.320 -0.164 0.484 0.154 0.154
t0 = 2 mm
actuator DM forces residue [N ] [nm] 0.00605 7.03 -0.02262 -0.89 0.02867 7.92 0.00492 0.39 0.00492 0.39 0.00017 0.19 -0.00017 -0.19 0.00033 0.38 0.00005 0.02 0.00005 0.02 0.00685 2.58 -0.00685 -2.58 0.01370 5.15 0.00291 0.28 0.00292 0.28 0.02929 2.51 -0.00560 -10.00 0.03489 12.50 0.00644 1.59 0.00645 1.58
Correction of some low-order Zernike aberrations. The input displacements, the forces needed to perform the correction, and the DM residues are shown for each sample thickness in terms of maximum, minimum, peak-tovalley (ptv), rms, and standard deviation (std).
Table 3.
of 100 cm at = 2m. The results for the three selected DM thicknesses are shown in Fig. 2(a), that reports the rms of the uncorrected and tip-tilt-removed wavefronts and their relative residues, and in Fig. 2(b) in terms of actuator forces | rms, mean, and maximum absolute values. The averages of 100 wavefronts for each term of Figs. 2(a) and 2(b), reported in Tab. 4, show that the average rms residue of the DM surface, equal to 39m for the three thicknesses, is 2% of the average value of the rms uncorrected wavefront and 6% of average rms of the tip-tilt-removed wavefront, while the actuator forces strongly depend on the thickness: their rms values are .036, .107, and .242 N when the thickness is 1, 1.5, and 2 mm, respectively, and their absolute values are always < 1 N. We note that the rms residue of the 1.6 mm tick DM of the MMT corresponds to 4% of the rms uncorrected wavefront and to 10% of the rms tip-tilt-removed wavefront, and the rms correction force is .11 N rms. 1
t0 [mm]
1.5 2 DM uncorrected wf 1723.0 rms displacements tt-removed wf 670.0 [nm] residue 38.8 38.7 38.6 actuator rms 0.036 0.107 0.242 forces max of absolute values 0.130 0.389 0.875 [N ] mean 1:695 10?6 2:488 10?6 3:270 10?6 Table 4.
Averages values of rms displacements, from Fig. 2(a), and actuator forces, from Fig. 2(b).
rms of displacements [nm] due to 100 wavefronts (λ=2µm, r =100cm) 0
tt−removed wf
residue
95
95
95
90
90
90
85
85
85
80
80
80
75
75
75
70
70
70
65
65
65
60
60
60
55
55
55
50
50
45
45
45
40
40
40
35
35
35
30
30
30
25
25
25
20
20
20
15
15
15
10
10
10
5 1
5 1
5 1
40.8
40.3
2.0
39.8
39.3
38.8
38.3
36.8
1.5 −3 thk [m×10 ]
37.8
1.0
1248
1144
2.0
1040
936
832
728
1.5 −3 thk [m×10 ]
624
416
520
1.0
3619
3243
2.0
2866
2490
2113
1.5 −3 thk [m×10 ]
1360
984
607
1.0
37.3
50
wf #
100
wf #
100
1737
wf #
uncorrected wf 100
(a) displacements actuator forces [N] due to 100 wavefronts (λ=2µm, r =100cm) 0
mean
rms
95
95
95
90
90
90
85
85
85
80
80
80
75
75
75
70
70
70
65
65
65
60
60
60
55
55
55
50
50
45
45
45
40
40
40
35
35
35
30
30
30
25
25
25
20
20
20
15
15
15
10
10
10
5 1
5 1
5 1
0.26
0.23
2.0
0.20
0.18
0.15
0.12
1.5 −3 thk [m×10 ]
0.09
0.03
1.0
2.1×10−4
1.6×10−4
2.0
1.1×10−4
6.3×10−5
1.3×10−5
−3.6×10−5
1.5 −3 thk [m×10 ]
−8.6×10−5
−1.4×10−4
−1.8×10−4
1.0
1.16
1.02
2.0
0.89
0.76
0.63
1.5 −3 thk [m×10 ]
0.37
0.23
0.10
1.0
0.06
50
wf #
100
wf #
100
0.50
wf #
max of absolute values 100
(b) forces Figure 2. Correction of 100 wavefronts. Rms values of uncorrected and tip-tilt-removed wavefronts and residues and rms, maximum absolute values (a), and mean of actuator forces (b) for the three DM thicknesses.
t0 = 1 mm t0 = 1:5 mm t0 = 2 mm DM actuator DM actuator DM actuator residue forces residue forces residue forces [nm] [N ] [nm] [N ] [nm] [N ] max 34.3 0.01086 10.4 0.01081 4.5 0.01075 min -15.7 -0.00321 -4.6 -0.00312 -2.0 -0.00314 ptv 50.0 0.01407 15.0 0.01393 6.5 0.01389 rms 5.1 0.00370 1.6 0.00368 0.7 0.00367 std 5.1 0.00370 1.5 0.00368 0.7 0.00367 DM residue and actuator correction forces needed to correct the distortion due to the magnetic coupling between permanent magnets glued to the DM: maximum, minimum, peak-to-valley (ptv), rms, and standard deviation (std) values are listed. Table 5.
7. MAGNETIC INTERACTION
Forces and moments act between each pair of magnets. Such forces and moments depend on the relative distance between the magnet centroids (measured perpendicularly to the optical axis), to their z o set (measured in a direction parallel to the optical axis) and to their angular separation, as discussed in Ref. 3. The DM distortion caused by this loads can be compensated by applying to each actuator a stroke equal to the displacement due the magnetic load multiplied by ?1. The resulting residues are summarized in Tab. 5, along with the actuator forces needed to compensate the DM distortion. As the rms values of such correction forces are in all cases 3:7 10?3 N (with a ptv value as low as .014 N), and the rms residue of the DM ranges from :7 nm (for a thickness of 2 mm) to 5:1 nm (for a 1 mm thick DM), all the three selected thicknesses allow a successful correction of the magnetic forces.
8. POWER BUDGET
The actuator force needed to balance the gravity load is equal to .061 N rms in the worst case. Correcting the turbulent wavefronts causes a maximum rms actuator force of .875 N. The correction of the magnetic coupling can be actuated with rms forces (at least) two order of magnitude below that. Furthermore, the static correction of the Zernike aberrations, at the moment unknown, are likely to be negligible. As a consequence, only the gravity and wavefront correction forces are considered in the power budget. We de ne the \static" forces due to the gravity as fj (j = 1; 2; : : :; N , where N = 918 is the total number of actuators; fj = const = WN , where Wtot is the sum of the total magnet weight and of the DM one), and the \wf" forces needed to correct the turbulent wavefront as 'j . Thus, for a given , as de ned in Ref. 6, the total dissipated power is equal to actuator eciency 2 PN 2 2 W2 2 (1=) j=1 (fj + 'j ) = (1=) N + N h'j i , because h'j i = 0. The total dissipated power is reported in Tab. 6 as a function of the thickness. Such a table provides the power budget for two values of the eciency , de ned in Ref. 5 | .5 and .75 NW? 12 . The rst value is the eciency measured in Ref. 2, the second one was computed in Ref. 3. Although the relative small reduction of the eciency increases the dissipated powers by a factor of 2.25, from Tab. 6 we can conclude that a DM whose thickness ranges from 1.5 to 2 mm requires a total power of around 100 W for the adaptive operations, and even less in static conditions. As the thermal conductance of the actuator is equal to 6.7 K W?1 , the worst case of power dissipation | 228 W | implies a T of 1.66 K between the actuator tip and the cold plate temperature | a value within the acceptable range. This T decreases to .38 K when the thickness is 1.5 mm. tot
tot
[N W ? 12 ] 0.50 0.75
t0 [mm]
1.5 2 total dissipated power [W ] static wf total static wf total static wf total 6.7 4.8 11.5 9.7 42.0 51.8 13.3 215.0 228.4 3.0 2.1 5.1 4.3 18.7 23.0 5.9 95.6 101.5
Table 6.
1
DM power budget. See the text for a discussion.
9. CONCLUSIONS
The optical, magnetic, and mechanical design of the DM for the LBT is an evolution of the DM studies for the MMT. The LBT DM is not only physically larger (its diameter is .911 m, whereas the MMT diameter is .642 m), but also it has a smaller actuator separation ( 25 mm instead of 30 mm); furthermore, the actuator pattern is signi cantly di erent near the outer edge. The results of the current calculations can be summarized as follows.
The correction of the turbulent wavefront is more e ective: the rms wavefront is reduced by a factor of 17,
whereas the correspondent value for the MMT is 10. The power per actuator of the wavefront correction remains basically unchanged with respect to the one of the MMT DM if we consider similar thicknesses. This behavior can be explained considering that a smaller separation samples high spatial order terms of the turbulence spectrum, but these terms contain little power; lower terms, whose power is greater, are sampled by a larger number of actuators, therefore reducing the force per actuator. The removal of low-order Zernike aberrations is more ecient; as an example, the correction of the spherical aberration is obtained with a factor of 100 with respect to the residue (the MMT corresponding reduction factor is 30). Although the average actuator separation of the LBT DM is smaller than the MMT one, the e ect of the magnetic coupling remains very small and easy to correct.
We are therefore oriented to adopt the actuator distribution suggested here and to select a thickness in the range 1.5 to 1.8 mm. The current MMT choice is 1.8 mm. The nal selection will depend on mirror manufacturing issues.
REFERENCES
1. J. M. Hill and P. Salinari, \The Large Binocular Telescope project," in Adaptive Optics Systems Technology, P. L. Wizinowich, ed., Proc. SPIE 4007, (Munich, Germany), 2000. Proceedings of this conference. 2. D. Gallieni and C. Del Vecchio, \LBT adaptive secondary preliminary design," in Adaptive Optics Systems Technology, P. L. Wizinowich, ed., Proc. SPIE 4007, (Munich, Germany), 2000. Proceedings of this conference. 3. C. Del Vecchio, G. Brusa, D. Gallieni, M. Lloyd-Hart, and W. B. Davidson, \Static and dynamic responses of an ultra-thin, adaptive secondary mirror," in Adaptive Optics Systems and Technology, D. Bonaccini and R. K. Tyson, eds., Proc. SPIE 3762, pp. 330{340, (Denver, Colorado), 1999. 4. C. Del Vecchio, \Supporting a magnetically levitated, very thin meniscus for an adaptive secondary mirror: summary of nite element analyses," in Adaptive Optics and Applications, R. K. Tyson and R. Q. Fugate, eds., vol. 3126, pp. 397{404, SPIE, (San Diego, California), 7 1997. 5. G. Brusa and C. Del Vecchio, \Designing an adaptive secondary mirror: a global approach," Applied Optics 37(21), pp. 4656{4662, 1998. 6. C. Del Vecchio, \Aluminum reference plate, heath sink, and actuator design for an adaptive secondary mirror," in Astronomical Telescopes and Instrumentation, D. Bonaccini and R. K. Tyson, eds., Proc. SPIE 3353, pp. 839{849, (Kona, Hawaii), 3 1998.
