2.2.1.

Multimode deformable mirror design

The chosen mirror geometry has been developed by Lemaître.²⁷ As presented in Fig. 1, it is a monolithic piece of Zerodur, made of a circular pupil with an external thicker ring and 12 arms. This design is based on the similarity between the Zernike polynomials used in optical aberration theory²⁸ and the Clebsch polynomials used in elasticity theory.²⁹ The application of 24 discrete forces on both extremities of each arm allows the generation of Zernikes defined by $n=m$ and $n=m+2$; $n$ and $m$ being the radial and azimuthal polynomials’ orders. With 12 arms, $m$ is included between 0 and 6. In addition a central clamp holds the system, notably for launch, and allows the generation of spherical aberration ($m=0$ and $n=4$).

Fig. 1

(a) Principle of a multimode deformable mirror (MDM). (b) Finite element model of the MADRAS mirror, with actuators represented by springs in red (63,708 hexaedral elements, 77,979 nodes –5881 nodes on the surface).

In this design, the forces that deform the mirror are applied far from the optical surface. This way, it avoids actuator print-through.³⁰ Moreover, it decouples the number of actuators from the mirror diameter: the number of required actuators is only driven by the maximal spatial frequency to be corrected. Finally, this design is not limited to a single actuator technology: any actuator applying discrete forces can be used with this type of mirror.

2.2.2.

Optimization with finite element analysis

The mirror geometry is optimized with FEA for the generation of each specified mode. The parameters are the thickness of the central meniscus and the radii and thicknesses of the outer ring, the arms and the central clamping. For a given geometry, a finite element model is created and its 24 influence functions (IF) are recovered by applying a unit command to each actuator, while the others are fixed. Then, the correction of a given WFE ${\varphi }_{\text{in}}$ is characterized by three main criteria:

The design optimization consists of minimizing these three criteria for each required mode. A classical least squares algorithm is used to converge to the optimal geometry.³¹

The finite element model, presented in Fig. 1, has 63,708 hexaedral elements and 77,979 nodes. The optical surface contains 100 nodes on a diameter and 120 angular sectors, providing sufficient sampling to characterize its deformation: up to 50 cycles per pupil. The actuators are modeled by springs with a given stiffness and length. The top extremity of the actuator is a node of the model, representing the force location (either under an arm or under the ring), and the bottom extremity is represented by a node clamped in the $(x,y)$ plane. A displacement along the $z$ axis is applied on the bottom spring node to simulate the actuation. The other boundary condition on the model is the central clamping: the bottom nodes of this part are fixed in the three directions.

2.3.1.

Eigen modes

The system eigen modes are deduced from the set of influence functions by performing a singular value decomposition.³² The eigen modes, shown in Fig. 2, constitute an orthogonal basis, representing all of the deformations that the system can achieve. As expected, the modes are similar to the Zernike polynomials.

Fig. 2

Finite element analysis (FEA) results: (a) system influence functions (unit: μm). (b) System eigen modes.

2.3.2.

Correction performance

Specified Zernike mode correction

Each specified Zernike, deduced from the expected deformation maps of primary mirrors in space (Table 1), is decomposed over the set of influence functions’ basis, in order to characterize mirror correction performance (see Sec. 2.2.2). The correction of tip, tilt and focus will be addressed by a five degrees of freedom mechanism on the secondary mirror: it is simulated by adding three virtual influence functions corresponding to these modes.

Figure 3 presents the study of the astigmatism3 correction and the performance of the correction of each mode is summarized in Fig. 4 and Table 1. All the modes are corrected with a precision better than 5 nm rms, except for spherical3 and pentafoil9 which are slightly worse. Fore these two modes, the residuals are due to the central clamping and to a symmetry mismatch between the system and the mode. In conclusion, the modal correction efficiency has been demonstrated with FEA.

Fig. 3

Characterization of the astigmatism3 correction: (a) Required mode (unit: μm). (b) Actuators’ displacements, deduced from the mode projection on the influence functions. (c) Precision of correction: generated mode and residues (unit: μm). (d) Resulting mirror deformation, from FEA (unit: mm). (e) Resulting stress, from FEA (unit: MPa).

Fig. 4

Correction of each specified mode: amplitude of the wavefront error before (in blue) and after (in green) correction.

2.3.3.

Reliability analysis: dead actuator impact

The characterization of the system’s reliability can be performed by studying the impact of one or several dead actuators on the system performance. A dead actuator is not supplied any more, but keeps its stiffness. The occurrence of a dead actuator can be modeled in two different ways, depending if there is a system recalibration or not. Without recalibration, the mode to be corrected is still projected on the 24 influence functions but the command of the dead actuator is forced to 0. With a recalibration, the mode projection is carried out on the 23 remaining influence functions so the dead actuator will be compensated by its neighbors.

The impact of a dead actuator depends on the actuator location and on the mode to be corrected. The performance of correction is computed for each mode and for each actuator. The results advocate that a recalibration is essential to maintain reasonable performance. Without recalibration, the residual WFE is increased by a factor 7.7, on average; this factor is reduced to 1.3 with recalibration.

For a chosen azimuthal order $m$, modes with the greatest radial order $n$, are less influenced by the loss of one actuator. The corrections of the modes defined by $n=m$ are more damaged by the loss of an internal actuator, while the modes defined by $n=m+2$ are more dependent on external actuators. This fact will allow balancing the impact of a dead actuator with respect to a global WFE correction. On the left of Fig. 5, the mean resulting residuals and the worst and best cases are compared to the nominal performance: with a recalibration, the loss of one actuator degrades the performance in a reasonable way, the correction stays within the specification.

Fig. 5

(a) System performance with one dead actuator (FEA results, considering a recalibration): comparison of the mean precision of correction of a system fully functional (in blue) and a system with one dead actuator (in green). (b) Evolution of the mean residual wavefront with the number of dead actuators, with a recalibration (statistics on 5000 random WFEs and sets of dead actuators).

The evolution of the mean correction performance with the number of dead actuators can also be studied: for a given number of dead actuators, 50 random sets of dead actuators are drawn and the correction of 100 random WFEs is performed for each deteriorated set of influence functions basis (Fig. 5, right). Logically, the residues increase with the number of dead actuator, but we can see that with two dead actuators the system is still functioning within the specification: the mean precision is 9.7 nm rms, with a standard deviation of 2.1 nm rms.

This is possible due to the intrinsic redundancy of the system: 24 actuators are used for the generation of only 17 modes (the specified Zernike polynomials in any orientation). This performance is a major advantage for space applications, ensuring the robustness and reliability of the MADRAS concept.

2.3.4.

Assembled system

Definition

The correcting system, shown in Fig. 6, weighs 4 kg and is 80 mm in height, with a diameter of 150 mm. It is composed of four main pieces:

• 1. The Zerodur mirror, with a central meniscus, an outer ring, 12 arms and a fixed central cylinder. It is polished in the 100 mm diameter aperture.

• 2. An Invar supporting structure, composed of a cone and a reference plate. The reference plate is stiffened with thin ribs so that any deformation of the plate will be negligible. The mirror is glued to the top part of the cone, on the bottom perimeter of the central clamp.

• 3. Twenty-four piezoelectric actuators. They are linked to the mirror and to the reference plate. Their required stroke is deduced from the worst case correction: the actuators must provide a $\pm 10\mu \mathrm{m}$ displacement in order to compensate for all the specified WFEs. The chosen actuators are the Cedrat PPA40M which have a 40 μm stroke, at 80 V.

• 4. The three bipods holding the system. These fixation devices, attached at three points on the reference plate, have been designed to provide an isostatic condition.

Fig. 6

(a) Model of the assembled mirror (mirror in green, reference plate in purple, cone in gray, actuators in blue and fixation devices in light blue). (b) Integrated system.

3.2.1.

Mirror control

For the demonstration, the wavefront sensing and control is performed through a Shack–Hartmann WFS. The sensor, placed in a plane conjugated to the active mirror, measures local wavefront slopes within subapertures defined by a microlenses array.³⁵ A Real-Time Computer processes the measurements and computes the mirror commands in order to match the measured wave-front with a reference one. Once the control law, loop gain and reference wavefront are defined, the active correcting loop operates autonomously, at a 1 Hz frequency.

This wavefront sensing and control strategy is typically used in adaptive optics systems.³⁶ But a different wavefront sensing approach could be adopted, optimized according to the mission.

3.2.2.

Test bed

Description

The test bench, shown in Fig. 10, is composed of the following elements:

• 1. A point source and collimating lens, simulating the observation of a distant object.

• 2. A telescope simulator generating WFE expected in a 3-m-class telescope. It is an adaptive optics loop, composed of a 20-mm-diameter, 88-actuator magnetic DM (DM88³⁷), a 3-mm-diameter Shack–Hartmann WFS1 with 784 subapertures and a RTC1. The DM pitch is 2.5 mm and the actuators are positioned on a 10 by 10 grid. This sampling allows the generation of all the required spatial frequencies and oversampling the WFS1 (28 by 28 subapertures) allows for an accurate measurement of the residual wavefront, up to 14 cycles per pupil.

• 3. An active correction loop, composed of the MADRAS mirror, a second 3 mm diameter $28\times 28$ subaperture Shack–Hartmann WFS2 and a second RTC2.

• 4. Four beam expanders, relaying the pupil between the DM88, which is the entrance pupil, and the WFS1, the MADRAS mirror and the WFS2.

• 5. Two imaging cameras located in focal planes before and after the correction.

• 6. A Fizeau interferometer, directly looking at the mirror in order to monitor its deformation in real time.

Fig. 10

MADRAS testbed: optical design and picture.

3.2.3.

Mode correction

The correction performance in closed loop is measured for each specified mode. They are all corrected with a precision better than 8 nm rms (see Fig. 9 and Table 1). The expected precision of correction, deduced from simulations and measurements with the Fizeau interferometer is recovered with a difference lower than 1.5 nm for seven of the modes.

The correction of astigmatism3&5, trefoil5&7 and tetrafoil7&9 is highly efficient, the residual WFE is below 5 nm rms, and the pentafoil9 correction precision is around 7 nm rms. The amplitude of the residuals measured for all these modes (except for tetrafoil7) is slightly higher than the expected one. This difference has several causes such as the numerical errors from the phase reconstruction using the measured slopes, and the noncommon path aberrations between the interferometric path and the MADRAS loop.

Coma3 and spherical3 are currently corrected with a precision of 6 and 8 nm rms, but this result could be improved up to the values expected from the interferometric measurements by working on the control matrix, or by adding a real system to actuate the tip, tilt and focus modes. Both coma and tilt modes have a component in $\cos (\theta )$ in their mathematical expressions, and both focus and spherical aberrations have a component in ${\rho }^2$. Thus, the generations of these modes are linked and the method used for the tip, tilt and focus handling impacts the correction. In a representative configuration, a five degrees of freedom mechanism will address the three modes. Tip, tilt and focus influence functions will then be real, automatically solving this problem. So, the mismatch between simulations and measurements for these two modes is not due to the active mirror itself but to the current control method. As seen in Sec. 3.1, the interferometric measurements have validated the efficient generation of these modes and, as for the other modes, the performance will be recovered with the new experimental set-up.

In Fig. 11 we present the wavefront shapes before and after the correction of the specified astigmatism3. The residual shape and amplitude are close to the ones expected from simulation (see Fig. 3). For each specified mode, the PSF before and after correction have been recorded, illustrating well the correction. The optical shape of the mirror has been measured with the Fizeau interferometer, constituting another validation of the correction: the measured deformation corresponds well to the injected WFE.

Fig. 11

Astigmatism3 correction: WFE before (147.4 nm rms) and after (3.3 nm rms) correction—Zernike decomposition of the WFEs—point spread function (PSF) before and after correction—Optical shape of the mirror (76.5 nm rms).

In conclusion, the simulated precision of correction is efficiently recovered, which validates the functioning of the correcting mirror within the active loop.

3.2.4.

Expected WFE correction

After having demonstrated the system’s ability to correct each specified mode separately, the correction performance is studied regarding global WFE expected in space telescopes, as defined in Sec. 2.1. In order to perform a statistical study as in Sec. 2.3.2, the linearity of the system must be verified. This is validated by studying influence functions: each influence function amplitude evolves linearly with applied voltage, and the wavefront resulting from the command on several actuators corresponds to the sum of the wavefront resulting from individual commands. So, the experimental mean precision of the MADRAS system is deduced from a statistical study on the correction of 1000 random WFE, representative of a 3-m-class primary mirror deformation in space: 8.2 nm rms, with a standard deviation of 1.8 nm rms. This correction performance is well within the 10 nm rms specification.

By working on the tip, tilt and focus handling, this performance could be improved to 6.2 nm rms, deduced from the interferometric measurements. To reach this ultimate performance, a real tip, tilt and focus correction will be implemented on the test bench, with a motorized platform.

Finally, an example of random WFE correction is presented in Fig. 12: the injected wavefront is 129 nm rms and the corrected wavefront is 8.6 $\pm 0.5\mathrm{nm}\mathrm{rms}$. Once again, the gain for the PSF measurement is obvious.

Fig. 12

Correction of a random WFE (the given wavefront is an average of 100 measurements, tip, tilt and focus substracted).