Abstract

This paper discusses an innovative low-cost deformable mirror made of a magnetic liquid (ferrofluid) whose surface is actuated by an hexagonal array of small current carrying coils. Predicted and experimental performances of a 37-actuator ferrofluid deformable mirror are presented along with wavefront correction examples. We show the validity of the model used to compute the actuators currents to obtain a desired wavefront shape. We demonstrate that the ferrofluid deformable mirror can correct a 11 µm low order aberrated wavefront to a residual RMS error of 0.05 µm corresponding to a Strehl ratio of 0.82.

© 2007 Optical Society of America

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References

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  1. H. W. Babcock, "Possibility of compensating astronomical seeing," Publ. Astron. Soc. Pac. 65, 229 (1953).
    [CrossRef]
  2. P. Hickson and M. K. Mulrooney, "University of British Columbia-NASA Multi-Narrowband Survey. I. Description and Photometric Properties of the Survey," Astrophys. J.Suppl. 115, 35-42 (1997).
    [CrossRef]
  3. R. Cabanac and E. F. Borra, "A search for peculiar objects with the NASA Orbital Debris Observatory 3-m Liquid Mirror Telescope," Astrophys. J. 509, 309-323 (1998).
    [CrossRef]
  4. R. Sica, S. Sargoytchev, E. F. Borra, L. Girard, S. Argall, C. T. Sarrow and S. Flatt. "Lidar measurements taken with a large aperture liquid mirror: 1. The Rayleigh-Scatter system," Appl. Opt. 34, 6925 (1995).
    [CrossRef] [PubMed]
  5. R. Wuerker, "Bistatic LMT Lidar Alignment," Opt. Eng. 36, 1421-1424 (1997).
    [CrossRef]
  6. R. J. Sica and T. Russell, "Measurements of the effects of gravity waves in the middle atmosphere using parametric models of density fluctuations. Part I," J. Atmos. Sci. 56, 1308-1329 (1999).
    [CrossRef]
  7. R. Ragazzoni and E. Marchetti, "A liquid adaptive mirror," Astron. Astrophys. 283, L17-L19 (1994).
  8. G. Vdovin, M. Loktev, and A. Simonov, "Low-cost deformable mirrors: technologies and goals," Proc. SPIE 5894, 1-10 (2005).
  9. R. E. Rosensweig, Ferrohydrodynamics. (Dover, 1997).
  10. W. L. H. Shutter and L. A. Whitehead, "A wide sky coverage ferrofluid mercury telescope," Astrophys. J. 424, L139-L141 (1994).
    [CrossRef]
  11. E. M. Vuelban, N. Bhattacharya and J. J. M. Braat, "Liquid deformable mirror for high-order wavefront correction," Opt. Lett. 31, 11, 1717-1719 (2006).
    [CrossRef] [PubMed]
  12. E. F. Borra, A. M. Ritcey, R. Bergamasco, P. Laird, J. Gingras, M. Dallaire, L. Da Silva and H. Yockell-Lelievre "Nanoengineered Astronomical Optics," Astron. Astrophys. 419, 777-782 (2004).
    [CrossRef]
  13. P. Laird, E. F. Borra, R. Bergamesco, J. Gingras, L. Truong and A. Ritcey, "Deformable mirrors based on magnetic liquids," Proc. SPIE 5490, 1493-1501 (2004).
    [CrossRef]
  14. E. F. Borra, D. Brousseau, and A. Vincent., "Large magnetic liquid mirrors," Astron. Astrophys. 446, 389, (2006).
    [CrossRef]
  15. D. Brousseau, E. F. Borra, H. J. Ruel, and J. Parent, "A magnetic liquid deformable mirror for high stroke and low order axially symmetrical aberrations," Opt. Express 14, 11486 (2006).
    [CrossRef] [PubMed]
  16. L. Thibos, R. A. Applegate, J. T. Schweigerling, and R. Webb, "Standards for reporting the optical aberrations of eyes," in OSA Trends in Optics and Photonics35, 232-244, (2000).
  17. C. Y. Matuo and A. M. Figueiredo Neto, "Time dependence of the magnetic grain concentration and secondary grain aggregation in ferronematic lyotropic liquid crystals subjected to magnetic field gradients," Phys. Rev E 60, 1815-1820 (1999).
    [CrossRef]
  18. V. S. Mendelev and A. O. Ivanov, "Ferrofluid aggregation in chains under the influence of a magnetic field," Phys. Rev. E 70, 051502 (2004).
    [CrossRef]
  19. E. F. Borra, R. Content, L. Girard, S. Szapiel, L. M. Tremblay and E. Boily, "Liquid mirrors: Optical shop tests and contributions to the technology," Astrophys. J. 393, 829-847 (1992).
    [CrossRef]

2006 (3)

2005 (1)

G. Vdovin, M. Loktev, and A. Simonov, "Low-cost deformable mirrors: technologies and goals," Proc. SPIE 5894, 1-10 (2005).

2004 (3)

V. S. Mendelev and A. O. Ivanov, "Ferrofluid aggregation in chains under the influence of a magnetic field," Phys. Rev. E 70, 051502 (2004).
[CrossRef]

E. F. Borra, A. M. Ritcey, R. Bergamasco, P. Laird, J. Gingras, M. Dallaire, L. Da Silva and H. Yockell-Lelievre "Nanoengineered Astronomical Optics," Astron. Astrophys. 419, 777-782 (2004).
[CrossRef]

P. Laird, E. F. Borra, R. Bergamesco, J. Gingras, L. Truong and A. Ritcey, "Deformable mirrors based on magnetic liquids," Proc. SPIE 5490, 1493-1501 (2004).
[CrossRef]

1999 (2)

C. Y. Matuo and A. M. Figueiredo Neto, "Time dependence of the magnetic grain concentration and secondary grain aggregation in ferronematic lyotropic liquid crystals subjected to magnetic field gradients," Phys. Rev E 60, 1815-1820 (1999).
[CrossRef]

R. J. Sica and T. Russell, "Measurements of the effects of gravity waves in the middle atmosphere using parametric models of density fluctuations. Part I," J. Atmos. Sci. 56, 1308-1329 (1999).
[CrossRef]

1998 (1)

R. Cabanac and E. F. Borra, "A search for peculiar objects with the NASA Orbital Debris Observatory 3-m Liquid Mirror Telescope," Astrophys. J. 509, 309-323 (1998).
[CrossRef]

1997 (2)

R. Wuerker, "Bistatic LMT Lidar Alignment," Opt. Eng. 36, 1421-1424 (1997).
[CrossRef]

P. Hickson and M. K. Mulrooney, "University of British Columbia-NASA Multi-Narrowband Survey. I. Description and Photometric Properties of the Survey," Astrophys. J.Suppl. 115, 35-42 (1997).
[CrossRef]

1995 (1)

1994 (2)

R. Ragazzoni and E. Marchetti, "A liquid adaptive mirror," Astron. Astrophys. 283, L17-L19 (1994).

W. L. H. Shutter and L. A. Whitehead, "A wide sky coverage ferrofluid mercury telescope," Astrophys. J. 424, L139-L141 (1994).
[CrossRef]

1992 (1)

E. F. Borra, R. Content, L. Girard, S. Szapiel, L. M. Tremblay and E. Boily, "Liquid mirrors: Optical shop tests and contributions to the technology," Astrophys. J. 393, 829-847 (1992).
[CrossRef]

1953 (1)

H. W. Babcock, "Possibility of compensating astronomical seeing," Publ. Astron. Soc. Pac. 65, 229 (1953).
[CrossRef]

Argall, S.

Babcock, H. W.

H. W. Babcock, "Possibility of compensating astronomical seeing," Publ. Astron. Soc. Pac. 65, 229 (1953).
[CrossRef]

Bergamasco, R.

E. F. Borra, A. M. Ritcey, R. Bergamasco, P. Laird, J. Gingras, M. Dallaire, L. Da Silva and H. Yockell-Lelievre "Nanoengineered Astronomical Optics," Astron. Astrophys. 419, 777-782 (2004).
[CrossRef]

Bergamesco, R.

P. Laird, E. F. Borra, R. Bergamesco, J. Gingras, L. Truong and A. Ritcey, "Deformable mirrors based on magnetic liquids," Proc. SPIE 5490, 1493-1501 (2004).
[CrossRef]

Bhattacharya, N.

Boily, E.

E. F. Borra, R. Content, L. Girard, S. Szapiel, L. M. Tremblay and E. Boily, "Liquid mirrors: Optical shop tests and contributions to the technology," Astrophys. J. 393, 829-847 (1992).
[CrossRef]

Borra, E. F.

D. Brousseau, E. F. Borra, H. J. Ruel, and J. Parent, "A magnetic liquid deformable mirror for high stroke and low order axially symmetrical aberrations," Opt. Express 14, 11486 (2006).
[CrossRef] [PubMed]

E. F. Borra, D. Brousseau, and A. Vincent., "Large magnetic liquid mirrors," Astron. Astrophys. 446, 389, (2006).
[CrossRef]

P. Laird, E. F. Borra, R. Bergamesco, J. Gingras, L. Truong and A. Ritcey, "Deformable mirrors based on magnetic liquids," Proc. SPIE 5490, 1493-1501 (2004).
[CrossRef]

E. F. Borra, A. M. Ritcey, R. Bergamasco, P. Laird, J. Gingras, M. Dallaire, L. Da Silva and H. Yockell-Lelievre "Nanoengineered Astronomical Optics," Astron. Astrophys. 419, 777-782 (2004).
[CrossRef]

R. Cabanac and E. F. Borra, "A search for peculiar objects with the NASA Orbital Debris Observatory 3-m Liquid Mirror Telescope," Astrophys. J. 509, 309-323 (1998).
[CrossRef]

R. Sica, S. Sargoytchev, E. F. Borra, L. Girard, S. Argall, C. T. Sarrow and S. Flatt. "Lidar measurements taken with a large aperture liquid mirror: 1. The Rayleigh-Scatter system," Appl. Opt. 34, 6925 (1995).
[CrossRef] [PubMed]

E. F. Borra, R. Content, L. Girard, S. Szapiel, L. M. Tremblay and E. Boily, "Liquid mirrors: Optical shop tests and contributions to the technology," Astrophys. J. 393, 829-847 (1992).
[CrossRef]

Braat, J. J. M.

Brousseau, D.

Cabanac, R.

R. Cabanac and E. F. Borra, "A search for peculiar objects with the NASA Orbital Debris Observatory 3-m Liquid Mirror Telescope," Astrophys. J. 509, 309-323 (1998).
[CrossRef]

Content, R.

E. F. Borra, R. Content, L. Girard, S. Szapiel, L. M. Tremblay and E. Boily, "Liquid mirrors: Optical shop tests and contributions to the technology," Astrophys. J. 393, 829-847 (1992).
[CrossRef]

Da Silva, L.

E. F. Borra, A. M. Ritcey, R. Bergamasco, P. Laird, J. Gingras, M. Dallaire, L. Da Silva and H. Yockell-Lelievre "Nanoengineered Astronomical Optics," Astron. Astrophys. 419, 777-782 (2004).
[CrossRef]

Dallaire, M.

E. F. Borra, A. M. Ritcey, R. Bergamasco, P. Laird, J. Gingras, M. Dallaire, L. Da Silva and H. Yockell-Lelievre "Nanoengineered Astronomical Optics," Astron. Astrophys. 419, 777-782 (2004).
[CrossRef]

Figueiredo Neto, A. M.

C. Y. Matuo and A. M. Figueiredo Neto, "Time dependence of the magnetic grain concentration and secondary grain aggregation in ferronematic lyotropic liquid crystals subjected to magnetic field gradients," Phys. Rev E 60, 1815-1820 (1999).
[CrossRef]

Flatt, S.

Gingras, J.

P. Laird, E. F. Borra, R. Bergamesco, J. Gingras, L. Truong and A. Ritcey, "Deformable mirrors based on magnetic liquids," Proc. SPIE 5490, 1493-1501 (2004).
[CrossRef]

E. F. Borra, A. M. Ritcey, R. Bergamasco, P. Laird, J. Gingras, M. Dallaire, L. Da Silva and H. Yockell-Lelievre "Nanoengineered Astronomical Optics," Astron. Astrophys. 419, 777-782 (2004).
[CrossRef]

Girard, L.

R. Sica, S. Sargoytchev, E. F. Borra, L. Girard, S. Argall, C. T. Sarrow and S. Flatt. "Lidar measurements taken with a large aperture liquid mirror: 1. The Rayleigh-Scatter system," Appl. Opt. 34, 6925 (1995).
[CrossRef] [PubMed]

E. F. Borra, R. Content, L. Girard, S. Szapiel, L. M. Tremblay and E. Boily, "Liquid mirrors: Optical shop tests and contributions to the technology," Astrophys. J. 393, 829-847 (1992).
[CrossRef]

Hickson, P.

P. Hickson and M. K. Mulrooney, "University of British Columbia-NASA Multi-Narrowband Survey. I. Description and Photometric Properties of the Survey," Astrophys. J.Suppl. 115, 35-42 (1997).
[CrossRef]

Ivanov, A. O.

V. S. Mendelev and A. O. Ivanov, "Ferrofluid aggregation in chains under the influence of a magnetic field," Phys. Rev. E 70, 051502 (2004).
[CrossRef]

Laird, P.

E. F. Borra, A. M. Ritcey, R. Bergamasco, P. Laird, J. Gingras, M. Dallaire, L. Da Silva and H. Yockell-Lelievre "Nanoengineered Astronomical Optics," Astron. Astrophys. 419, 777-782 (2004).
[CrossRef]

P. Laird, E. F. Borra, R. Bergamesco, J. Gingras, L. Truong and A. Ritcey, "Deformable mirrors based on magnetic liquids," Proc. SPIE 5490, 1493-1501 (2004).
[CrossRef]

Loktev, M.

G. Vdovin, M. Loktev, and A. Simonov, "Low-cost deformable mirrors: technologies and goals," Proc. SPIE 5894, 1-10 (2005).

Marchetti, E.

R. Ragazzoni and E. Marchetti, "A liquid adaptive mirror," Astron. Astrophys. 283, L17-L19 (1994).

Matuo, C. Y.

C. Y. Matuo and A. M. Figueiredo Neto, "Time dependence of the magnetic grain concentration and secondary grain aggregation in ferronematic lyotropic liquid crystals subjected to magnetic field gradients," Phys. Rev E 60, 1815-1820 (1999).
[CrossRef]

Mendelev, V. S.

V. S. Mendelev and A. O. Ivanov, "Ferrofluid aggregation in chains under the influence of a magnetic field," Phys. Rev. E 70, 051502 (2004).
[CrossRef]

Mulrooney, M. K.

P. Hickson and M. K. Mulrooney, "University of British Columbia-NASA Multi-Narrowband Survey. I. Description and Photometric Properties of the Survey," Astrophys. J.Suppl. 115, 35-42 (1997).
[CrossRef]

Parent, J.

Ragazzoni, R.

R. Ragazzoni and E. Marchetti, "A liquid adaptive mirror," Astron. Astrophys. 283, L17-L19 (1994).

Ritcey, A.

P. Laird, E. F. Borra, R. Bergamesco, J. Gingras, L. Truong and A. Ritcey, "Deformable mirrors based on magnetic liquids," Proc. SPIE 5490, 1493-1501 (2004).
[CrossRef]

Ritcey, A. M.

E. F. Borra, A. M. Ritcey, R. Bergamasco, P. Laird, J. Gingras, M. Dallaire, L. Da Silva and H. Yockell-Lelievre "Nanoengineered Astronomical Optics," Astron. Astrophys. 419, 777-782 (2004).
[CrossRef]

Ruel, H. J.

Russell, T.

R. J. Sica and T. Russell, "Measurements of the effects of gravity waves in the middle atmosphere using parametric models of density fluctuations. Part I," J. Atmos. Sci. 56, 1308-1329 (1999).
[CrossRef]

Sargoytchev, S.

Sarrow, C. T.

Shutter, W. L. H.

W. L. H. Shutter and L. A. Whitehead, "A wide sky coverage ferrofluid mercury telescope," Astrophys. J. 424, L139-L141 (1994).
[CrossRef]

Sica, R.

Sica, R. J.

R. J. Sica and T. Russell, "Measurements of the effects of gravity waves in the middle atmosphere using parametric models of density fluctuations. Part I," J. Atmos. Sci. 56, 1308-1329 (1999).
[CrossRef]

Simonov, A.

G. Vdovin, M. Loktev, and A. Simonov, "Low-cost deformable mirrors: technologies and goals," Proc. SPIE 5894, 1-10 (2005).

Szapiel, S.

E. F. Borra, R. Content, L. Girard, S. Szapiel, L. M. Tremblay and E. Boily, "Liquid mirrors: Optical shop tests and contributions to the technology," Astrophys. J. 393, 829-847 (1992).
[CrossRef]

Tremblay, L. M.

E. F. Borra, R. Content, L. Girard, S. Szapiel, L. M. Tremblay and E. Boily, "Liquid mirrors: Optical shop tests and contributions to the technology," Astrophys. J. 393, 829-847 (1992).
[CrossRef]

Truong, L.

P. Laird, E. F. Borra, R. Bergamesco, J. Gingras, L. Truong and A. Ritcey, "Deformable mirrors based on magnetic liquids," Proc. SPIE 5490, 1493-1501 (2004).
[CrossRef]

Vdovin, G.

G. Vdovin, M. Loktev, and A. Simonov, "Low-cost deformable mirrors: technologies and goals," Proc. SPIE 5894, 1-10 (2005).

Vincent, A.

E. F. Borra, D. Brousseau, and A. Vincent., "Large magnetic liquid mirrors," Astron. Astrophys. 446, 389, (2006).
[CrossRef]

Vuelban, E. M.

Whitehead, L. A.

W. L. H. Shutter and L. A. Whitehead, "A wide sky coverage ferrofluid mercury telescope," Astrophys. J. 424, L139-L141 (1994).
[CrossRef]

Wuerker, R.

R. Wuerker, "Bistatic LMT Lidar Alignment," Opt. Eng. 36, 1421-1424 (1997).
[CrossRef]

Yockell-Lelievre, H.

E. F. Borra, A. M. Ritcey, R. Bergamasco, P. Laird, J. Gingras, M. Dallaire, L. Da Silva and H. Yockell-Lelievre "Nanoengineered Astronomical Optics," Astron. Astrophys. 419, 777-782 (2004).
[CrossRef]

Appl. Opt. (1)

Astron. Astrophys. (3)

R. Ragazzoni and E. Marchetti, "A liquid adaptive mirror," Astron. Astrophys. 283, L17-L19 (1994).

E. F. Borra, A. M. Ritcey, R. Bergamasco, P. Laird, J. Gingras, M. Dallaire, L. Da Silva and H. Yockell-Lelievre "Nanoengineered Astronomical Optics," Astron. Astrophys. 419, 777-782 (2004).
[CrossRef]

E. F. Borra, D. Brousseau, and A. Vincent., "Large magnetic liquid mirrors," Astron. Astrophys. 446, 389, (2006).
[CrossRef]

Astrophys. J. (4)

W. L. H. Shutter and L. A. Whitehead, "A wide sky coverage ferrofluid mercury telescope," Astrophys. J. 424, L139-L141 (1994).
[CrossRef]

P. Hickson and M. K. Mulrooney, "University of British Columbia-NASA Multi-Narrowband Survey. I. Description and Photometric Properties of the Survey," Astrophys. J.Suppl. 115, 35-42 (1997).
[CrossRef]

R. Cabanac and E. F. Borra, "A search for peculiar objects with the NASA Orbital Debris Observatory 3-m Liquid Mirror Telescope," Astrophys. J. 509, 309-323 (1998).
[CrossRef]

E. F. Borra, R. Content, L. Girard, S. Szapiel, L. M. Tremblay and E. Boily, "Liquid mirrors: Optical shop tests and contributions to the technology," Astrophys. J. 393, 829-847 (1992).
[CrossRef]

J. Atmos. Sci. (1)

R. J. Sica and T. Russell, "Measurements of the effects of gravity waves in the middle atmosphere using parametric models of density fluctuations. Part I," J. Atmos. Sci. 56, 1308-1329 (1999).
[CrossRef]

Opt. Eng. (1)

R. Wuerker, "Bistatic LMT Lidar Alignment," Opt. Eng. 36, 1421-1424 (1997).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev E (1)

C. Y. Matuo and A. M. Figueiredo Neto, "Time dependence of the magnetic grain concentration and secondary grain aggregation in ferronematic lyotropic liquid crystals subjected to magnetic field gradients," Phys. Rev E 60, 1815-1820 (1999).
[CrossRef]

Phys. Rev. E (1)

V. S. Mendelev and A. O. Ivanov, "Ferrofluid aggregation in chains under the influence of a magnetic field," Phys. Rev. E 70, 051502 (2004).
[CrossRef]

Proc. SPIE (2)

P. Laird, E. F. Borra, R. Bergamesco, J. Gingras, L. Truong and A. Ritcey, "Deformable mirrors based on magnetic liquids," Proc. SPIE 5490, 1493-1501 (2004).
[CrossRef]

G. Vdovin, M. Loktev, and A. Simonov, "Low-cost deformable mirrors: technologies and goals," Proc. SPIE 5894, 1-10 (2005).

Publ. Astron. Soc. Pac. (1)

H. W. Babcock, "Possibility of compensating astronomical seeing," Publ. Astron. Soc. Pac. 65, 229 (1953).
[CrossRef]

Other (2)

R. E. Rosensweig, Ferrohydrodynamics. (Dover, 1997).

L. Thibos, R. A. Applegate, J. T. Schweigerling, and R. Webb, "Standards for reporting the optical aberrations of eyes," in OSA Trends in Optics and Photonics35, 232-244, (2000).

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Figures (11)

Fig. 1.
Fig. 1.

(a). The 37-actuator ferrofluid deformable mirror with its support. Wires going to the current source can be seen at the left of the mirror. b) Closer view of the mirror showing the hexagonal array of actuators. Each actuator is 5 mm in diameter and the array covers a diameter of 35 mm. The support is made of aluminum to minimize ferromagnetic effects.

Fig. 2.
Fig. 2.

Influence function of a typical actuator. Both theoretical and experimental profiles are well described by a Gaussian. A Gaussian fit of theoretical data gives a FWHM of 8.4 mm and 8.9 mm for the experimental data. Amplitude in arbitrary units.

Fig. 3.
Fig. 3.

Computed wavefronts of the first 12 Zernike polynomials produced by a 37-actuator ferrofluid deformable mirror. Printthrough caused by the hexagonal array of the actuators can be seen on the wavefronts. Zernike polynomials numbering follows the OSA convention. Pupil size is 30 mm and amplitude is measured in µm.

Fig. 4.
Fig. 4.

Example of computed currents (for actuators without ferrite cores) to produce trefoil Z3 3 (left) and spherical aberration Z0 4 (right). The geometrical distribution of currents is not entirely symmetrical because the algorithm considers the presence of a 0.18 gauss y-oriented magnetic field component due to Earth’s magnetic field. Currents are measured in amperes.

Fig. 5.
Fig. 5.

Histogram of predicted residual wavefront RMS errors for the first 27 Zernike polynomials with a 37-actuator ferrofluid deformable mirror. RMS error is given in fraction of the targeted RMS amplitude of the Zernike term.

Fig. 6.
Fig. 6.

Shack-Hartmann wavefront sensor setup used to measure the wavefronts produced by the ferrofluid deformable mirror. Lenses L1 and L2 have a focal length of 100 mm while L3 has a focal length of 500 mm giving a magnification factor of 5 between the DM and the HASO WFS. The laser diode wavelength is 659.5 nm.

Fig. 7.
Fig. 7.

Experimental wavefronts for the first 16 Zernike polynomials produced by a 37-actuator ferrofluid deformable mirror. Amplitudes are measured in µm.

Fig. 8.
Fig. 8.

Histogram of wavefront residual RMS errors for the first 20 Zernike polynomials experimentally obtained with the 37-actuator ferrofluid deformable mirror. RMS error is given in fraction of the targeted RMS amplitude of the Zernike term.

Fig. 9.
Fig. 9.

Histograms of wavefront RMS amplitudes of the Zernike polynomials coefficients for optical misalignment of lens L2 and L3 in the setup of Fig. 6, before (left) and after (right) correction with the 37-actuator ferrofluid deformable mirror.

Fig. 10.
Fig. 10.

Computed PSFs (logarithmic scale) before (left) and after (right) correction with the 37-actuator ferrofluid deformable mirror. Strehl ratio in b) is 0.82. Wavelength used is 659.5 nm.

Fig. 11.
Fig. 11.

Histograms of wavefront RMS amplitudes of the Zernike polynomials coefficients produced by the insertion of two Petri dish in the beam between the ferrofluid deformable mirror and lens L3 before (left) and after (right) correction with the 37-actuator ferrofluid deformable mirror. High order terms are contributing the most to the residual error. Tip-tilt terms were not considered.

Equations (2)

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h ( x , y ) = ( μ r 1 ) 2 μ r μ 0 ρ g [ B n 2 ( x , y ) + μ r B t 2 ( x , y ) ]
B r = μ 0 I 2 a F t ( r a , z a , z r ) , B z = μ 0 I 2 a F n ( r a , z a , z r )

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