Abstract

The thermally deformable mirror is a device aiming at correcting beam-wavefront distortions for applications where classical mechanical methods are precluded by noise considerations, as in advanced gravitational wave interferometric detectors. This moderately low-cost technology can be easily implemented and controlled thanks to the good reproducibility of the actuation. By using a flexible printed circuit board technology, we demonstrate experimentally that a device of 61 actuators in thermal contact with the back surface of a high-reflective mirror is able to correct the low-order aberrations of a laser beam at 1064 nm and could be used to optimize the mode matching into Fabry–Perot cavities.

© 2013 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2013

R. Day, on behalf of the Virgo Collaboration, “Central heating radius of curvature correction for use in large scale gravitational wave interferometers,” Classical Quantum Gravity 30, 055017 (2013).
[CrossRef]

2012

B. Canuel, R. Day, E. Genin, P. La Penna, and J. Marque, “Wavefront aberration compensation with a thermally deformable mirror,” Classical Quantum Gravity 29, 085012 (2012).
[CrossRef]

2011

LIGO Scientific Collaboration, “A gravitational wave observatory operating beyond the quantum shot-noise limit,” Nat. Phys. 7, 962–965 (2011).
[CrossRef]

2010

2009

J.-Y. Vinet, “On special optical modes and thermal issues in advanced gravitational wave interferometric detectors,” Living Rev. Relativity 12, 2009-5 (2009).
[CrossRef]

2007

2006

2005

2004

R. Lawrence, D. Ottaway, M. Zucker, and P. Fritschel, “Active correction of thermal lensing through external radiative thermal actuation,” Opt. Lett. 29, 2635–2637 (2004).
[CrossRef]

J. Degallaix, C. Zhao, L. Ju, and D. Blair, “Thermal lensing compensation for AIGO high optical power test facility,” Classical Quantum Gravity 21, S903–S908 (2004).
[CrossRef]

H. Luck, A. Freise, S. Gobler, S. Hild, K. Kawabe, and K. Danzmann, “Thermal correction of the radii of curvature of mirrors for GEO 600,” Classical Quantum Gravity 21, S985–S989 (2004).
[CrossRef]

T. Corbitt, N. Mavalvala, and S. Whitcomb, “Optical cavities as amplitude filters for squeezed fields,” Phys. Rev. D 70, 022002 (2004).
[CrossRef]

2003

E. Calloni, J. T. Baker, F. Barone, R. DeRosa, L. Di Fiore, L. Milano, and S. R. Restaino, “Adaptive optics approach for prefiltering of geometrical fluctuations of the input laser beam of an interferometric gravitational waves detector,” Rev. Sci. Instrum. 74, 2570–2574 (2003).
[CrossRef]

2002

R. Lawrence, M. Zucker, P. Fritschel, P. Marfuta, and D. Shoemaker, “Adaptive thermal compensation of test masses in advanced LIGO,” Classical Quantum Gravity 19, 1803–1812 (2002).

G. Mueller, R. S. Amin, D. Guagliardo, D. McFeron, R. Lundock, D. H. Reitze, and D. B. Tanner, “Method for compensation of thermally induced modal distortions in the input optical components of gravitational wave interferometers,” Classical Quantum Gravity 19, 1793–1801 (2002).
[CrossRef]

2001

P. Hello, “Compensation for thermal effects in mirrors of gravitational wave interferometers,” Eur. Phys. J. D 15, 373–383 (2001).
[CrossRef]

2000

H. Luck, K. O. Muller, P. Aufmuth, and K. Danzmann, “Correction of wavefront distortions by means of thermally adaptive optics,” Opt. Commun. 175, 275–287 (2000).
[CrossRef]

G. Mueller, Q. Shu, R. Adhikari, D. B. Tanner, D. Reitze, D. Sigg, N. Mavalvala, and J. Camp, “Determination and optimization of mode matching into optical cavities by heterodyne detection,” Opt. Lett. 25, 266–268 (2000).
[CrossRef]

1996

T. F. Coleman and Y. Li, “A reflective Newton method for minimizing a quadratic function subject to bounds on some of the variables,” SIAM J. Optim. 6, 1040–1058 (1996).
[CrossRef]

1993

1991

W. Winkler, K. Danzmann, A. Rudiger, and R. Schilling, “Heating by optical absorption and the performance of interferometric gravitational-wave detectors,” Phys. Rev. A 44, 7022 (1991).
[CrossRef]

1990

P. Hello and J. Y. Vinet, “Analytical models of thermal aberrations in massive mirrors heated by high power laser beams,” J. Phys. France 51, 1267–1282 (1990).
[CrossRef]

P. Hello and J. Y. Vinet, “Analytical models of transient thermoelastic deformations of mirrors heated by high power cw laser beams,” J. Phys. France 51, 2243–2261 (1990).
[CrossRef]

Adhikari, R.

Alda, J.

Amin, R.

Amin, R. S.

G. Mueller, R. S. Amin, D. Guagliardo, D. McFeron, R. Lundock, D. H. Reitze, and D. B. Tanner, “Method for compensation of thermally induced modal distortions in the input optical components of gravitational wave interferometers,” Classical Quantum Gravity 19, 1793–1801 (2002).
[CrossRef]

Arain, M.

Arain, M. A.

Aufmuth, P.

H. Luck, K. O. Muller, P. Aufmuth, and K. Danzmann, “Correction of wavefront distortions by means of thermally adaptive optics,” Opt. Commun. 175, 275–287 (2000).
[CrossRef]

Avino, S.

S. Avino, “Adaptive optics techniques for gravitational wave interferometers,” Ph.D. thesis (Universita Degli Studi di Napoli Federico II, 2006).

Baker, J. T.

E. Calloni, J. T. Baker, F. Barone, R. DeRosa, L. Di Fiore, L. Milano, and S. R. Restaino, “Adaptive optics approach for prefiltering of geometrical fluctuations of the input laser beam of an interferometric gravitational waves detector,” Rev. Sci. Instrum. 74, 2570–2574 (2003).
[CrossRef]

Barone, F.

E. Calloni, J. T. Baker, F. Barone, R. DeRosa, L. Di Fiore, L. Milano, and S. R. Restaino, “Adaptive optics approach for prefiltering of geometrical fluctuations of the input laser beam of an interferometric gravitational waves detector,” Rev. Sci. Instrum. 74, 2570–2574 (2003).
[CrossRef]

Blair, D.

J. Degallaix, C. Zhao, L. Ju, and D. Blair, “Thermal lensing compensation for AIGO high optical power test facility,” Classical Quantum Gravity 21, S903–S908 (2004).
[CrossRef]

Boreman, G. D.

Born, M.

M. Born and E. Wolf, Principle of Optics (Cambridge University, 1999).

Calloni, E.

E. Calloni, J. T. Baker, F. Barone, R. DeRosa, L. Di Fiore, L. Milano, and S. R. Restaino, “Adaptive optics approach for prefiltering of geometrical fluctuations of the input laser beam of an interferometric gravitational waves detector,” Rev. Sci. Instrum. 74, 2570–2574 (2003).
[CrossRef]

Camp, J.

Canuel, B.

B. Canuel, R. Day, E. Genin, P. La Penna, and J. Marque, “Wavefront aberration compensation with a thermally deformable mirror,” Classical Quantum Gravity 29, 085012 (2012).
[CrossRef]

Cohen, M.

Coleman, T. F.

T. F. Coleman and Y. Li, “A reflective Newton method for minimizing a quadratic function subject to bounds on some of the variables,” SIAM J. Optim. 6, 1040–1058 (1996).
[CrossRef]

Corbitt, T.

T. Corbitt, N. Mavalvala, and S. Whitcomb, “Optical cavities as amplitude filters for squeezed fields,” Phys. Rev. D 70, 022002 (2004).
[CrossRef]

Cruz, R. J.

Danzmann, K.

H. Luck, A. Freise, S. Gobler, S. Hild, K. Kawabe, and K. Danzmann, “Thermal correction of the radii of curvature of mirrors for GEO 600,” Classical Quantum Gravity 21, S985–S989 (2004).
[CrossRef]

H. Luck, K. O. Muller, P. Aufmuth, and K. Danzmann, “Correction of wavefront distortions by means of thermally adaptive optics,” Opt. Commun. 175, 275–287 (2000).
[CrossRef]

W. Winkler, K. Danzmann, A. Rudiger, and R. Schilling, “Heating by optical absorption and the performance of interferometric gravitational-wave detectors,” Phys. Rev. A 44, 7022 (1991).
[CrossRef]

Day, R.

R. Day, on behalf of the Virgo Collaboration, “Central heating radius of curvature correction for use in large scale gravitational wave interferometers,” Classical Quantum Gravity 30, 055017 (2013).
[CrossRef]

B. Canuel, R. Day, E. Genin, P. La Penna, and J. Marque, “Wavefront aberration compensation with a thermally deformable mirror,” Classical Quantum Gravity 29, 085012 (2012).
[CrossRef]

Degallaix, J.

J. Degallaix, C. Zhao, L. Ju, and D. Blair, “Thermal lensing compensation for AIGO high optical power test facility,” Classical Quantum Gravity 21, S903–S908 (2004).
[CrossRef]

DeRosa, R.

E. Calloni, J. T. Baker, F. Barone, R. DeRosa, L. Di Fiore, L. Milano, and S. R. Restaino, “Adaptive optics approach for prefiltering of geometrical fluctuations of the input laser beam of an interferometric gravitational waves detector,” Rev. Sci. Instrum. 74, 2570–2574 (2003).
[CrossRef]

Di Fiore, L.

E. Calloni, J. T. Baker, F. Barone, R. DeRosa, L. Di Fiore, L. Milano, and S. R. Restaino, “Adaptive optics approach for prefiltering of geometrical fluctuations of the input laser beam of an interferometric gravitational waves detector,” Rev. Sci. Instrum. 74, 2570–2574 (2003).
[CrossRef]

Franzen, K. Yoshiki

Freise, A.

H. Luck, A. Freise, S. Gobler, S. Hild, K. Kawabe, and K. Danzmann, “Thermal correction of the radii of curvature of mirrors for GEO 600,” Classical Quantum Gravity 21, S985–S989 (2004).
[CrossRef]

Fritschel, P.

R. Lawrence, D. Ottaway, M. Zucker, and P. Fritschel, “Active correction of thermal lensing through external radiative thermal actuation,” Opt. Lett. 29, 2635–2637 (2004).
[CrossRef]

R. Lawrence, M. Zucker, P. Fritschel, P. Marfuta, and D. Shoemaker, “Adaptive thermal compensation of test masses in advanced LIGO,” Classical Quantum Gravity 19, 1803–1812 (2002).

Genin, E.

B. Canuel, R. Day, E. Genin, P. La Penna, and J. Marque, “Wavefront aberration compensation with a thermally deformable mirror,” Classical Quantum Gravity 29, 085012 (2012).
[CrossRef]

Gleason, J.

Gobler, S.

H. Luck, A. Freise, S. Gobler, S. Hild, K. Kawabe, and K. Danzmann, “Thermal correction of the radii of curvature of mirrors for GEO 600,” Classical Quantum Gravity 21, S985–S989 (2004).
[CrossRef]

Guagliardo, D.

G. Mueller, R. S. Amin, D. Guagliardo, D. McFeron, R. Lundock, D. H. Reitze, and D. B. Tanner, “Method for compensation of thermally induced modal distortions in the input optical components of gravitational wave interferometers,” Classical Quantum Gravity 19, 1793–1801 (2002).
[CrossRef]

Guerineau, N.

Hello, P.

P. Hello, “Compensation for thermal effects in mirrors of gravitational wave interferometers,” Eur. Phys. J. D 15, 373–383 (2001).
[CrossRef]

P. Hello and J. Y. Vinet, “Analytical models of transient thermoelastic deformations of mirrors heated by high power cw laser beams,” J. Phys. France 51, 2243–2261 (1990).
[CrossRef]

P. Hello and J. Y. Vinet, “Analytical models of thermal aberrations in massive mirrors heated by high power laser beams,” J. Phys. France 51, 1267–1282 (1990).
[CrossRef]

Hild, S.

H. Luck, A. Freise, S. Gobler, S. Hild, K. Kawabe, and K. Danzmann, “Thermal correction of the radii of curvature of mirrors for GEO 600,” Classical Quantum Gravity 21, S985–S989 (2004).
[CrossRef]

Ju, L.

J. Degallaix, C. Zhao, L. Ju, and D. Blair, “Thermal lensing compensation for AIGO high optical power test facility,” Classical Quantum Gravity 21, S903–S908 (2004).
[CrossRef]

Kawabe, K.

H. Luck, A. Freise, S. Gobler, S. Hild, K. Kawabe, and K. Danzmann, “Thermal correction of the radii of curvature of mirrors for GEO 600,” Classical Quantum Gravity 21, S985–S989 (2004).
[CrossRef]

Korth, W.

La Penna, P.

B. Canuel, R. Day, E. Genin, P. La Penna, and J. Marque, “Wavefront aberration compensation with a thermally deformable mirror,” Classical Quantum Gravity 29, 085012 (2012).
[CrossRef]

Lawrence, R.

R. Lawrence, D. Ottaway, M. Zucker, and P. Fritschel, “Active correction of thermal lensing through external radiative thermal actuation,” Opt. Lett. 29, 2635–2637 (2004).
[CrossRef]

R. Lawrence, M. Zucker, P. Fritschel, P. Marfuta, and D. Shoemaker, “Adaptive thermal compensation of test masses in advanced LIGO,” Classical Quantum Gravity 19, 1803–1812 (2002).

Lee, J.

Leidel, C.

Li, Y.

T. F. Coleman and Y. Li, “A reflective Newton method for minimizing a quadratic function subject to bounds on some of the variables,” SIAM J. Optim. 6, 1040–1058 (1996).
[CrossRef]

Luck, H.

H. Luck, A. Freise, S. Gobler, S. Hild, K. Kawabe, and K. Danzmann, “Thermal correction of the radii of curvature of mirrors for GEO 600,” Classical Quantum Gravity 21, S985–S989 (2004).
[CrossRef]

H. Luck, K. O. Muller, P. Aufmuth, and K. Danzmann, “Correction of wavefront distortions by means of thermally adaptive optics,” Opt. Commun. 175, 275–287 (2000).
[CrossRef]

Lundock, R.

G. Mueller, R. S. Amin, D. Guagliardo, D. McFeron, R. Lundock, D. H. Reitze, and D. B. Tanner, “Method for compensation of thermally induced modal distortions in the input optical components of gravitational wave interferometers,” Classical Quantum Gravity 19, 1793–1801 (2002).
[CrossRef]

Marfuta, P.

R. Lawrence, M. Zucker, P. Fritschel, P. Marfuta, and D. Shoemaker, “Adaptive thermal compensation of test masses in advanced LIGO,” Classical Quantum Gravity 19, 1803–1812 (2002).

Marque, J.

B. Canuel, R. Day, E. Genin, P. La Penna, and J. Marque, “Wavefront aberration compensation with a thermally deformable mirror,” Classical Quantum Gravity 29, 085012 (2012).
[CrossRef]

Martin, R.

Mavalvala, N.

McFeron, D.

G. Mueller, R. S. Amin, D. Guagliardo, D. McFeron, R. Lundock, D. H. Reitze, and D. B. Tanner, “Method for compensation of thermally induced modal distortions in the input optical components of gravitational wave interferometers,” Classical Quantum Gravity 19, 1793–1801 (2002).
[CrossRef]

Milano, L.

E. Calloni, J. T. Baker, F. Barone, R. DeRosa, L. Di Fiore, L. Milano, and S. R. Restaino, “Adaptive optics approach for prefiltering of geometrical fluctuations of the input laser beam of an interferometric gravitational waves detector,” Rev. Sci. Instrum. 74, 2570–2574 (2003).
[CrossRef]

Mueller, G.

Muller, K. O.

H. Luck, K. O. Muller, P. Aufmuth, and K. Danzmann, “Correction of wavefront distortions by means of thermally adaptive optics,” Opt. Commun. 175, 275–287 (2000).
[CrossRef]

Ottaway, D.

Primot, J.

Quetschke, V.

Rakhmanov, M.

Reitze, D.

Reitze, D. H.

Restaino, S. R.

E. Calloni, J. T. Baker, F. Barone, R. DeRosa, L. Di Fiore, L. Milano, and S. R. Restaino, “Adaptive optics approach for prefiltering of geometrical fluctuations of the input laser beam of an interferometric gravitational waves detector,” Rev. Sci. Instrum. 74, 2570–2574 (2003).
[CrossRef]

Rudiger, A.

W. Winkler, K. Danzmann, A. Rudiger, and R. Schilling, “Heating by optical absorption and the performance of interferometric gravitational-wave detectors,” Phys. Rev. A 44, 7022 (1991).
[CrossRef]

Schilling, R.

W. Winkler, K. Danzmann, A. Rudiger, and R. Schilling, “Heating by optical absorption and the performance of interferometric gravitational-wave detectors,” Phys. Rev. A 44, 7022 (1991).
[CrossRef]

Shoemaker, D.

R. Lawrence, M. Zucker, P. Fritschel, P. Marfuta, and D. Shoemaker, “Adaptive thermal compensation of test masses in advanced LIGO,” Classical Quantum Gravity 19, 1803–1812 (2002).

Shu, Q.

Sigg, D.

Tanner, D.

Tanner, D. B.

Velghe, S.

Verpoort, S.

Vinet, J. Y.

P. Hello and J. Y. Vinet, “Analytical models of thermal aberrations in massive mirrors heated by high power laser beams,” J. Phys. France 51, 1267–1282 (1990).
[CrossRef]

P. Hello and J. Y. Vinet, “Analytical models of transient thermoelastic deformations of mirrors heated by high power cw laser beams,” J. Phys. France 51, 2243–2261 (1990).
[CrossRef]

Vinet, J.-Y.

J.-Y. Vinet, “On special optical modes and thermal issues in advanced gravitational wave interferometric detectors,” Living Rev. Relativity 12, 2009-5 (2009).
[CrossRef]

Wattellier, B.

Whitcomb, S.

T. Corbitt, N. Mavalvala, and S. Whitcomb, “Optical cavities as amplitude filters for squeezed fields,” Phys. Rev. D 70, 022002 (2004).
[CrossRef]

Williams, L.

Williams, L. F.

Winkler, W.

W. Winkler, K. Danzmann, A. Rudiger, and R. Schilling, “Heating by optical absorption and the performance of interferometric gravitational-wave detectors,” Phys. Rev. A 44, 7022 (1991).
[CrossRef]

Wittrock, U.

Wolf, E.

M. Born and E. Wolf, Principle of Optics (Cambridge University, 1999).

Zhang, L.

Zhao, C.

J. Degallaix, C. Zhao, L. Ju, and D. Blair, “Thermal lensing compensation for AIGO high optical power test facility,” Classical Quantum Gravity 21, S903–S908 (2004).
[CrossRef]

Zucker, M.

R. Lawrence, D. Ottaway, M. Zucker, and P. Fritschel, “Active correction of thermal lensing through external radiative thermal actuation,” Opt. Lett. 29, 2635–2637 (2004).
[CrossRef]

R. Lawrence, M. Zucker, P. Fritschel, P. Marfuta, and D. Shoemaker, “Adaptive thermal compensation of test masses in advanced LIGO,” Classical Quantum Gravity 19, 1803–1812 (2002).

Appl. Opt.

Classical Quantum Gravity

R. Lawrence, M. Zucker, P. Fritschel, P. Marfuta, and D. Shoemaker, “Adaptive thermal compensation of test masses in advanced LIGO,” Classical Quantum Gravity 19, 1803–1812 (2002).

J. Degallaix, C. Zhao, L. Ju, and D. Blair, “Thermal lensing compensation for AIGO high optical power test facility,” Classical Quantum Gravity 21, S903–S908 (2004).
[CrossRef]

H. Luck, A. Freise, S. Gobler, S. Hild, K. Kawabe, and K. Danzmann, “Thermal correction of the radii of curvature of mirrors for GEO 600,” Classical Quantum Gravity 21, S985–S989 (2004).
[CrossRef]

R. Day, on behalf of the Virgo Collaboration, “Central heating radius of curvature correction for use in large scale gravitational wave interferometers,” Classical Quantum Gravity 30, 055017 (2013).
[CrossRef]

G. Mueller, R. S. Amin, D. Guagliardo, D. McFeron, R. Lundock, D. H. Reitze, and D. B. Tanner, “Method for compensation of thermally induced modal distortions in the input optical components of gravitational wave interferometers,” Classical Quantum Gravity 19, 1793–1801 (2002).
[CrossRef]

B. Canuel, R. Day, E. Genin, P. La Penna, and J. Marque, “Wavefront aberration compensation with a thermally deformable mirror,” Classical Quantum Gravity 29, 085012 (2012).
[CrossRef]

Eur. Phys. J. D

P. Hello, “Compensation for thermal effects in mirrors of gravitational wave interferometers,” Eur. Phys. J. D 15, 373–383 (2001).
[CrossRef]

J. Phys. France

P. Hello and J. Y. Vinet, “Analytical models of thermal aberrations in massive mirrors heated by high power laser beams,” J. Phys. France 51, 1267–1282 (1990).
[CrossRef]

P. Hello and J. Y. Vinet, “Analytical models of transient thermoelastic deformations of mirrors heated by high power cw laser beams,” J. Phys. France 51, 2243–2261 (1990).
[CrossRef]

Living Rev. Relativity

J.-Y. Vinet, “On special optical modes and thermal issues in advanced gravitational wave interferometric detectors,” Living Rev. Relativity 12, 2009-5 (2009).
[CrossRef]

Nat. Phys.

LIGO Scientific Collaboration, “A gravitational wave observatory operating beyond the quantum shot-noise limit,” Nat. Phys. 7, 962–965 (2011).
[CrossRef]

Opt. Commun.

H. Luck, K. O. Muller, P. Aufmuth, and K. Danzmann, “Correction of wavefront distortions by means of thermally adaptive optics,” Opt. Commun. 175, 275–287 (2000).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. A

W. Winkler, K. Danzmann, A. Rudiger, and R. Schilling, “Heating by optical absorption and the performance of interferometric gravitational-wave detectors,” Phys. Rev. A 44, 7022 (1991).
[CrossRef]

Phys. Rev. D

T. Corbitt, N. Mavalvala, and S. Whitcomb, “Optical cavities as amplitude filters for squeezed fields,” Phys. Rev. D 70, 022002 (2004).
[CrossRef]

Rev. Sci. Instrum.

E. Calloni, J. T. Baker, F. Barone, R. DeRosa, L. Di Fiore, L. Milano, and S. R. Restaino, “Adaptive optics approach for prefiltering of geometrical fluctuations of the input laser beam of an interferometric gravitational waves detector,” Rev. Sci. Instrum. 74, 2570–2574 (2003).
[CrossRef]

SIAM J. Optim.

T. F. Coleman and Y. Li, “A reflective Newton method for minimizing a quadratic function subject to bounds on some of the variables,” SIAM J. Optim. 6, 1040–1058 (1996).
[CrossRef]

Other

M. Born and E. Wolf, Principle of Optics (Cambridge University, 1999).

S. Avino, “Adaptive optics techniques for gravitational wave interferometers,” Ph.D. thesis (Universita Degli Studi di Napoli Federico II, 2006).

ET Science Team, “Einstein gravitational wave telescope conceptual design study,” ET-0106C-10, 2011, https://tds.ego-gw.it/ql/?c=7954 .

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Figures (10)

Fig. 1.
Fig. 1.

Schematic representation of the TDM principle: the wavefront of a laser beam is modified by reflection on the back side of a substrate for which its temperature is tuned by an array of resistors.

Fig. 2.
Fig. 2.

(a) Scheme of the prototype array with the 61 resistors in blue squares and the pupil of observation defined by the area enclosed by the red dot circle. The resistors are separated by a dead space of 100 μm. (b) Photograph of the prototype array.

Fig. 3.
Fig. 3.

Schematic of the adaptive optical setup of TDM characterization. The continuous wave (CW) laser beam is directly sent to the TDM, and the phase deformations occurring in the pupil area are analyzed by a wavefront sensor.

Fig. 4.
Fig. 4.

Three experimental actuator responses of the TDM (colorscale in nanometers) at 5 mA with the resistor values and their corresponding position in the prototype array in red.

Fig. 5.
Fig. 5.

Experimental phase deformation created by the central actuator dissipating a power of 27 mW (blue) compared with results from finite element analysis simulation for an absorbed power of 9.4 mW (red crosses). The simulation is fitted to have the same area under the curve as the experimental data (same total power).

Fig. 6.
Fig. 6.

RMS of the phase deformation as a function of power dissipated by the central resistor in the substrate. The circles are the experimental measurement values. The blue curve corresponds to the best linear fit of the experimental data. The slope is indicated by the red diagonal line and the offset by the green horizontal line; it corresponds to the noise level of the wavefront sensor, which is lowered by the average of experimental data.

Fig. 7.
Fig. 7.

RMS of the residual image in a closed-loop control for the target mode Z33 at 50 nm PtV. The first iteration corresponds to the residual RMS, which may be obtained in an open-loop control.

Fig. 8.
Fig. 8.

(a) Target Zernike mode. (b) Zernike mode Z33 generated by the TDM over a pupil of 7 mm. An RoC of 350 m and a horizontal tilt of 5 μrad have been numerically subtracted. (c) The difference between the two modes is the residual phase image (colorscales are in nm).

Fig. 9.
Fig. 9.

Characterization of the modes generated by the TDM: for each mode the efficiency is given in the green bars (right portions of the bars) and the accuracy is given in the blue bars (left portions). All the Zernike target modes have an RMS of 10 nm.

Fig. 10.
Fig. 10.

Full scan of the Zernike modes from the second- and third-order: (a) RMS value of the Zernike produced by the mirror and (b) RMS value of the residual image.

Tables (1)

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Table 1. Correction of the Mode Matching by Extrapolation of the Experimental Results

Equations (7)

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ψr=ψtMa,
ϵ=ψtMa22.
liaiuiwithi=161,
ψ=i=1NαiZi,
RMSψ=αi2.
E=1|αt7α7|αt7.
A=α7i=120αi2.

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