Space Active Optics: toward optimized correcting mirrors forfuture large spaceborne observatories

Marie Laslandesa, Emmanuel Hugota, Marc Ferraria, Gerard Lemaitrea and Arnaud Liotardb

aCNRS and Universite de Provence - Laboratoire d’Astrophysique de Marseille38 rue F. Joliot Curie, 13388 Marseille cedex 13, France;

bThales Alenia Space - 100 bd du Midi, 06156 Cannes la Bocca, France.

ABSTRACT

Wave-front correction in optical instruments is often needed, either to compensate Optical Path Differences,off-axis aberrations or mirrors deformations. Active optics techniques are developed to allow efficient correctionswith deformable mirrors. In this paper, we will present the conception of particular deformation systems whichcould be used in space telescopes and instruments in order to improve their performances while allowing relaxingspecifications on the global system stability.A first section will be dedicated to the design and performance analysis of an active mirror specifically designedto compensate for aberrations that might appear in future 3m-class space telescopes, due to lightweight primarymirrors, thermal variations or weightless conditions. A second section will be dedicated to a brand new design ofactive mirror, able to compensate for given combinations of aberrations with a single actuator. If the aberrationsto be corrected in an instrument and their evolutions are known in advance, an optimal system geometry can bedetermined thanks to the elasticity theory and Finite Element Analysis.

Keywords: Active Optics, Deformable Mirrors, Space Telescopes, Aberrations Correction, Elasticity

1. INTRODUCTION

Active optics consists in controlling mirrors’ deformations. It is used to correct optical aberrations appearingin an instrument with a deformable mirror, but also to produce aspherical mirror with an excellent opticalquality thanks to Stress Mirror Polishing. These two applications are based on deformations systems, designedwith the elasticity theory which describes the behavior of mechanical structures1. In this paper we present twotypes of deformable mirrors able to generate surface corresponding to the first optical aberrations such as focus,astigmatism, coma, etc.In astronomy, the first application of active optics was to maintain the shape of large telescopes’ primary mirrors;indeed, ground-based telescopes are deformed under their own weight and a set of actuators, pushing or pullingthe optical surface, allows keeping the mirrors’ optimal shape2. A similar problematic is appearing in the fieldof space-based telescopes: primary mirror size is increasing in order to obtain high angular resolution and largecollecting power, while their weight must be low enough to allow a reasonable cost. These large lightweight mirrorswill be sensitive to environment variations. Preliminary studies performed using Finite Element Analysis (FEA)have shown that two main sources will introduce non-negligible form errors. The first major source will be theprediction error of the mirror shape difference between integration on earth, at 1g and operations in space, at 0g.The second source will be the thermal variations, inducing periodic form variations during operations. Primarymirrors’ deformations will induce the apparition of optical aberrations of low spatial frequencies and thereforewill become a limiting factor for the instruments’ performances3. For that reason, inserting an active deformablemirror in telescopes’ optical train will be mandatory to correct in-situ these static and dynamic deformations.In the next section, we present the design and analysis of a 24-actuators deformable mirror based on Vase formMultimode Deformable Mirror developpements pursued at LAM.Some instruments have also some specific active optics needs. For instance Variable Curvature Mirrors have beendeveloped for the Very Large Telescope Interferometric mode to adjust the delay lines tertiary mirror radius ofa curvature. The application of a pressure on the mirror backface allows changing the radius of curvature from

Further author information: marie.laslandes@oamp.fr

0 to 84 mm, leading to a pupil positionning better than 15 cm over 350 m4. Starting from an instrument design,it is possible to specify a needed correction and conceive a deformation system as simple as possible dedicated tothis use. We present in Section 3 a system called Correcting Optimized Mirror with a Single Actuator (COMSA)able to generate given combinations of focus plus other aberrations with only one degree of freedom. The methodconsists in generating a mechanical mode of deformation corresponding to a linear combination of optical modes.It requires a precise knowledge of the instrument but can significantly improve performances and simplify designs.Moreover, it is really easy to set up and monitor, which is always important for a space application.

2. MULTIMODE DEFORMABLE MIRRORS FOR AN ACTIVE CORRECTION INSPACE TELESCOPES

2.1 Description and interests in space

The concept of Vase Form Multimode Deformable Mirror (VMDM) allows generating optical aberrations usingthe similarity between Clebsch polynomials in elasticity theory and Zernike polynomials in optical aberrationstheory5. Such a mirror is composed of a circular meniscus, a thicker outter ring at the meniscus’ edges and kmclamped arms regularly distributed along the ring. The central pupil is deformed through a set of 2km actuators,located on the ring and at the end of each arms, applying discrete forces and moments. In addition, a uniformload can be applied under the optical surface. This pressure application can be replaced by a central clamping,providing a simpler system and a convenient way to hold the structure.Figure 1 presents the Finite Element model of a 100 mm diameter correcting mirror designed to compensateWave-Front Errors (WFE) induced by large primary mirror deformations in space. On the Figure right areshown mirror’s eigen modes which are well similar to Zernike polynomials and correspond to the aberrations tobe corrected6. Thus, inserting such a system in a space telescope optical train would allow the correction of anylinear combinations of these modes. Moreover, this design suits spatial specifications: it is quite simple, light andcompact; with only 24 actuators, it does not need too much power and the central clamping allows the systemto hold launch vibrations.The main interest of such a deformation system consists in the actuators’ location: actions to deform the opticalsurface are applied around the pupil instead of under it (like in classical adaptive optics mirrors). Thus, thenumber of actuators does not depend on the mirror size but only on the needed deformation’s spatial frequency.Moreover, due to the action far from the optical surface, no high frequency defects are generated.

Figure 1. Left: Back face view of a Vase form Multimode Deformable Mirror model designed for a space application(63708 hexaedrical elements - 77979 nodes) - Right: System eigen modes (from the less to the more energetic)

2.2 Design optimization

Finite Element Analysis (FEA) of the correcting mirror has been performed to optimize its geometry. This opti-mization is based on the system Influence Function (IF) and is driven by precision of correction and mechanical

strength.An actuator Influence Function, φIFi , is the phase map generated on the mirror by a unit command on theactuator. For a VMDM, there are two different types of IF; corresponding to an action on the ring or on onearm (Figure 2, left). The 2km system’s Influence Functions constitute a characteristic base7; projecting a phasemap to be corrected on this base provides two information (see Figure 2, right):

- Commands to be given to the actuators directly correspond to the projection coefficients- Expected precision of correction can be deduced by comparing initial and reconstructed phase maps.

Then, injecting actuators’ commands on the FEA model allows characterizing system mechanical behavior whenit generates the required deformation. If the level of stress is too important in the material, it is possible tominimize the energy by filtering some eigen modes.This analysis method provides a complete system characterization for a given wave-front compensation: precisionof correction, level of stress in the material, amplitude of deformation and actuators strokes, forces and precision.These data can be combined for all the needed phase map to determine the overall system performances.

Figure 2. Left: Types of Influence Functions for a VMDM (actuator on the ring (up) and on the arm (bottom)) - Right:Method to determine a VMDM performances with the IF base

2.3 Performance analysis

2.3.1 Wave-front correction

Correction capacity for each Zernike polynomials is computed with the method described in Figure 2 and theresults are presented in the left of Figure 3. Precision of correction is defined as the ratio between the RMSamplitudes of the residual phase and the phase to be corrected. With a precision of correction better than 2%,coma3, astigmatism3, trefoil5 and hexafoil11&13 are precisely generated. With residues around 5%, generationof focus, spherical3, astigmatism5, trefoil7 and tetrafoil7 is also quite performant. On the other hand, pentafoilare not corrected precisely due to the symmetry difference between the system and these modes. Residual phaseare mainly harmonics of the considered aberrations. We can also see arms footprint on some modes. Moreover, apart of residual phases comes from the central clamping, inducing parasite mechanical moments. This clampingslightly damages the system performance but adds a significant gain in simplicity: it allows correcting sphericalaberration without using a pressure system and also provides an efficient clamping, designed to resist launchvibrations.From this quantitative study we can define a set of WFE corrigible by the system based on the amount ofresidues, but also on actuators needed stroke and stress level in material. For instance, pentafoils and hexafoilsare energetic modes that must be filtered in order to limit breakage probability. Then the maximal amplitudeto be corrected is chosen for each mode by fixing an acceptable quantity of residues. Conveniently it is specifiedin fraction of λ. Table 1 presents the capacity of the studied system with a precision of correction better thanλ/30 for each mode.In a real operation mode, the mirror will correct wave-fronts composed of these Zernike polynomials, plus other

minor terms. To have a representative idea of the global system capacity, we study the generation of randomphase maps, described as linear combinations of modes listed in Table 1. Statistics on 1000 random draws givethe expected precision of correction: the system presented in Figure 1 is able to compensate for WFE composedof the first optical aberrations at amplitudes up to 1λ rms with a precision of λ/25.

Table 1. System optimal performances (λ = 632.8 nm)

- Focus Sphe3 Coma3 Astm3 Astm5 Tref5 Tref7 Tetraf7 Tetraf9

Amplitude (rms) λ λ/2 λ λ λ/2 λ λ/2 λ/2 λ/5

Residues (rms) λ/31 λ/40 λ/308 λ/59 λ/32 λ/124 λ/38 λ/42 λ/33

Figure 3. Left: Modal precision of correction of the studied MDM: residues rms are given in percentage of the correctedphase - Right: Example of random phase map correction (in λ unit)

2.3.2 Actuators specifications

We have seen in Section 2.2 that a phase map projection on the correcting system Influence Functions directlygives commands to actuators. This information allows defining actuators specifications in terms of force, strokeand precision for a given application. Indeed, projection coefficients values give the necessary stroke. Actuators’precision influence can also be performed by adding a random term to each command and analysing the effectson the phase map reconstruction.It is also interesting to study the impact of a dead actuator on system performance. A dead actuator is simulatedby the loss of an Influence Function. So the phase map projection is done on a degraded base. Actuatorsdisplacements are modulated according to the mode to be generated so that the influence of a dead actuatordepends on that actuator location and on the considered mode. A statistical study has been performed byrandomly choosing a dead actuator among the 24 and computing the corresponding residues for each mode(using the method detailed in Figure 2). In Figure 4, left, we can see the mean amount of residues for oursystem with one dead actuator compared to the performances of a fully functionnal system. Performances aresignificantly damaged and we can notice that odd modes are the most affected. Then, on the same principle wecan study the mean correction capacity evolution with the number of dead actuators by randomly choosing a setof defective actuators. The results are presented in the right of Figure 4. The precision of correction logicallydecreases with the number of dead actuators and effects are more important on the most energetic modes. Thisstudy confirm the need for 24 actuators to correct the modes expected to appear in large space telescopes andalso highlight the necessity to have redundancy in such systems to always insure the most performant correction.

Figure 4. Left: performances comparison between a system fully functionnal and a system with one dead actuator - Right:Evolution of modal precision of correction with the number of dead actuators

3. CORRECTING OPTIMIZED MIRROR WITH A SINGLE ACTUATOR (COMSA)

The COMSA concept8 allows generating given linear combinations of optical modes with a single actuator andan adapted warping harness attached to a mirror. In this section we present the one actuator, one mode principleand its adaptation to the generation of complex mechanical modes for dedicated functions.

3.1 Principle: One actuator = One mode

As we have seen in Section 2, a 24-actuators Multimode Deformable Mirror is able to correct any linear com-binations of its 24 Eigen Modes. It is usefull for an active compensation of large primary mirror deformationsbut for some applications there is no need for so many modes. Indeed, in some instruments, the wave-front tobe corrected and its evolution can be predicted and the number of actuators can be minimized by matchingInfluence Function and optical mode. That means that the mechanical mode could correspond to a combinationof classical optical aberrations.For instance, in an interferometer, there are different aberrations between the instrument’s two arms. These noncommon aberrations could damage the beams recombination. Being directly linked to Optical Path Difference,this WFE can be predicted and compensated with a Correcting Optimized Mirror with a Single Actuator. Sucha technique is based on bending moment transmission at the system edges an consists in adapting contour, thick-ness and actuator location to generate the required optical modes combination with the application of a singleforce. Minimizing the number of actuators allows the conception of simple efficient systems, which is essentialfor space instrumentation.

3.2 Geometry optimization

3.2.1 Bending moment generation

From elasticity theory, we know that a circulare plate can be deformed in a curvature mode by applying uniformbending moments at its edges9. These moments can be generated on an intermediate plate with a central forceand transmitted to the mirror via a flexible outer ring (see Figure 5). The COMSA concept is based on adeformation system able to transfer bending moments at a mirror edges to modify its curvature. If the mirroris circular, its curvature will be the same in all angular directions and the mirror’s warping will correspond toa pure focus mode. Finite Element Analysis allows characterizing such a system performance: focus is obtainedwith a precision better than 0.1%. We will see in next section that mirror curvature can vary with the mirrororientation, generating higher order optical aberrations.

Figure 5. Principle of bending moment generation and transmission on a circular system

3.2.2 Contour adaptation

Keeping the single actuator, other modes can be added to focus by modifying mirror contour. Indeed, theboundary condition allowing a uniform bending moment transmission is that the mirror edges displacement isequal to zero. So, starting from the required deformation on a circular pupil, z(ρ, θ) =

∑aiZi(ρ, θ), with Zi given

Zernike polynomials and ai their amplitudes, we can prolong it until it crossed the z = 0 plane. System contourwill be then defined as the intersection between this plane and the deformation surface. It can be expressed asa radius, ρc, in function of the angular coordinate, θ, and the modes’ amplitudes, ai:

z(ρc, θ) =∑

aiZi(ρc, θ) = 0 => ρc = f(θ, ai). (1)

Figure 6 presents a few example of contour to give to a COMSA system to generate a given combination ofaberrations.

Figure 6. Left: Contours for different optical modes combinations (circular pupil contour in blue) - Right: Contourevolution for different combinations of Focus + Astigmatism

3.2.3 Thickness distribution

System contour gives the bending moment modulation at mirror edges. This moment distribution will inducethe mirror deformation. In order to equalize generated and required bending moments, we add a thickness

distribution on the intermediate plate, calculated as explained below.Bending moments are generated with a central force application on the intermediate plate. These moments,Mrc , will then be modulated according to the system contour:

Mrc(θ) = Fρc(θ) = Frb(θ)

a, (2)

where F is the applied force, rc(θ) is the distance from centre to edge for a given orientation and a is the opticalpupil radius.Necessary moments to produce exactly the required deformation, z(r, θ) are given by the elasticity theory9:

Mrc(θ) = − Et3

12(1 − ν2)[∂2z(rc, θ)

∂rc(θ)2 + ν(

1

rc(θ)

∂z(rc, θ)

∂rc(θ)+

1

rc(θ)2∂2z(rc, θ)

∂θ2)], (3)

where t is the plate thickness, E its Young modulus and ν its Poisson ratio.Equalizing Equations 2 and 3, the thickness at intermediate plate edges can be determined:

tc(θ) = [12(1 − ν2)F

E

rc(θ)

a(∂2z(rc, θ)

∂rc(θ)2 + ν(

1

rc(θ)

∂z(rc, θ)

∂rc(θ)+

1

rc(θ)2∂2z(rc, θ)

∂θ2))−1]1/3. (4)

Thus, for a given deformation it is possible to compute a contour and a thickness distribution for the intermediateplate edges in order to generate bending moments inducing the right deformation on the mirror.

3.2.4 Actuator location

Instead of modifying the contour, Tilt and Coma can be generated by decentring the actuator. Indeed, transmitedbending moments depends on the distance between force location and edges, r′c(θ). Decentering the force of(xd, yd) induces a new bending moment generation and Equation 2 becomes:

Mrc(θ) = Fr′c(θ)

a=F

a

√rc(θ)2 + x2d + y2d − 2rc(θ)(xdcos(θ) + ydcos(θ)). (5)

Terms in cos(θ) and sin(θ) corresponds to generation of tilt and coma, their amplitudes depending on the shiftingdistance. In order not to damage other modes quality, the thickness distribution can be recalculated as describedbefore, equalizing Equations 3 and 5.

3.3 Application: Off-Axis Parabola generation

COMSA system is interesting to generate Off-Axis Parabola (OAP). An OAP is defined as a sphere plus termscorresponding to the first optical aberrations, mainly focus, astigmatism and coma. Those terms can be deducedfrom the OAP characteristics10: pupil semi-diameter a, radius of curvature k, conic constant C and off-axisdistance R.Starting from one required OAP we can compute the aberrations to be generated, leading to a system contourand a thickness distribution. Then, with such a system, all the aberrations combinations with constant ratioscan be generated (in the range of actuator stroke and system mechanical strenght):

z(ρ, θ, t) = A(t)[a20Z20(ρ, θ) + a22Z22(ρ, θ) + a31Z31(ρ, θ)], (6)

where aij are the initial optical modes amplitudes, Zij the Zernike polynomials and A(t) a coefficient evolvingwith time and directly linked to the actuator applied force.So, the mirror deformation evolves and each case corresponds to a different OAP: the system consists on avariable OAP.As a simple case study, we can start from the generation of an Off-Axis Parabola with the following character-istics:

- 100 mm diameter: a = 50 mm- Focal ratio of 1: F#1 => k = 200 mm- 45◦ Off-Axis angle: R = 141.4 mm

- Conic constant: C = −1This OAP can be described as a shape constitute of:

- a sphere of radius L = k = 200 mm- a focus term: a20 = −1.73 µm rms- an astigmatism term: a22 = −2.03 µm rms- a coma term: a31 = −0.35 µm rms- other minors terms

From there, the COMSA system can be dimensionned with the method presented in Section 3.2. Design char-acteristics are shown in Figure 7. In this Figure, we also present the combination of aberrations achievable withsuch a system and the corresponding OAP. A final optimization is then done with Finite Element Analysis; asdescribed in Section 2.2, residual deformations, level of stress and needed force are minimized by adjusting othersystem dimensions (mirror thickness and ring lenght and thickness). Performances of the FEA model are reallypromising: with only one actuator, our system is able to generate the required OAP, whose shape is 2.7 µmrms, with a residual deformation of 12.7 nm rms (see Figure 8). Moreover, it can generate other combinationsof aberrations, corresponding to different OAPs, always with a 0.5% precision.

Figure 7. OAP generation with COMSA: System contour and thickness distribution defined by a required OAP - Aber-rations achievable with such a system - Corresponding OAP (shape deduced from generated aberrations)

Figure 8. Previous system performances: Required deformation (deduced from the OAP characteristics) - Optical surfacedeformation (due to the actuator) - Residual deformation on pupil (Unit = µm)

4. CONCLUSION

Active optics provides performant and relatively simple means to generate, and hence correct, optical aberrations.Such techniques find numerous applications such as active correction of instrument intrinsic defects, generationof high optical quality aspherical surfaces with Stress Mirror Polishing, or conception of original optical designusing the capacity of active mirrors.With the increase of telescope aperture size, active wave-front correction will soon be mandatory in space inorder to compensate for primary mirrors’ deformation. We have developped a concept of active MultimodeDeformable Mirror to be included in satellites’ design able to correct efficiently these expected deformations.With only 24 actuators, this compact and light system can compensate combinations of the low order opticalaberrations at amplitudes up to 1λ with a residual Wave-Front Error of λ/25. Such a system has been entirelydesigned, optimized and characterized with Finite Element Analysis and the constraints due to a space use havebeen taken into account. Correcting mirror testing and characterization in a representative in-flight operationmode are on-going.Multimode Deformable Mirrors are performant to correct random sets of aberrations but when correction needsare known in advance, it is interesting to minimize the system’s degrees of freedom. In this paper we havepresented an original concept with one actuator transfering bending moments to an optical surface to deformit: the Correcting Optimized Mirror with a Single Actuator. It has been developped in order to simplify bothactive systems and instruments design. It is mainly based on a contour adaptation to equalize mechanical modeand optical mode on the pupil: thus, the correcting system is designed to suit a specific needed correction. Wehave seen that such a system can be used to generate variable Off-Axis Parabolas with 0.5 % precision.Active deformable mirrors are really interesting for spaceborne instrument thanks to the significant gain in termof wave-front quality provided by simple systems, easy to set up and monitor.

ACKNOWLEDGMENTS

This study is performed with the support of a Ph.D grant from CNES (Centre National d’Etudes Spatiales) andThales Alenia Space.

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