Abstract

A deformable mirror based on the principle of total internal reflection of light from an electrostatically deformed liquid–air interface was realized and used to perform closed-loop adaptive optical (AO) correction on a collimated laser beam aberrated by a rotating phase disk. Equations describing the resonant and oscillatory behavior of the liquid system were obtained and applied to the system under consideration. Characterization of the mirror included open- and closed-loop frequency responses, determination of rise times, the damping times of the liquid, and the influence of liquid surface motion in the absence of external optical aberrations. The performance of the AO system was determined for static and dynamic aberrations for various sets of system parameters. The predictions of the general expressions were compared to the results of the experimental realization and were found to be in good agreement.

© 2012 Optical Society of America

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References

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  1. R. W. Wood, “The mercury paraboloid as a reflecting telescope,” Astrophys. J. 29, 164–176 (1909).
    [CrossRef]
  2. E. F. Borra, M. Beauchemin, and R. Lalande, “Liquid mirror telescopes: observations with a 1 meter diameter prototype and scaling-up considerations,” Astrophys. J. 297, 846–851 (1985).
    [CrossRef]
  3. P. Hickson, E. F. Borra, R. Cabanac, R. Content, B. K. Gibson, and G. A. H. Walker, “UBC/Laval 2.7 meter liquid mirror telescope,” Astrophys. J. 436, L201–L204 (1994).
    [CrossRef]
  4. D. Brousseau, E. F. Borra, M. Rochette, and D. B. Landry, “Linearization of the response of a 91-actuator magnetic liquid deformable mirror,” Opt. Express 18, 8239–8250 (2010).
    [CrossRef]
  5. G. Vdovin, “Closed-loop adaptive optical system with a liquid mirror,” Opt. Lett. 34, 524–526 (2009).
    [CrossRef]
  6. R. A. Ibrahim, Liquid Sloshing Dynamics, Theory and Applications (Cambridge University, 2005).
  7. H. F. Bauer, “Tables of zeros of cross product Bessel functions,” J. Math. Comput. 18, 128–135 (1964).
  8. L. Landau and E. Lifshitz, Fluid Mechanics, 2nd ed., Vol. 6 of Course of Theoretical Physics (Butterworth Heinemann, 2009).
  9. J. Dong, J. Qi, and R. Miao, “Measurement of the damping of liquid surface wave by diffraction method,” Braz. J. Phys. 37, 1129–1133 (2007).
    [CrossRef]
  10. J. Miles, “Surface-wave damping in closed basins,” Proc. R. Soc. Lond. A 297, 459–475 (1967).
    [CrossRef]
  11. C. Mei and L. Liu, “The damping of surface gravity waves in a bounded liquid,” J. Fluid. Mech. 59, 239–256 (1973).
    [CrossRef]
  12. H. Lamb, Hydrodynamics,6th ed. (Cambridge University, 1975).
  13. J. J. DiStefano, A. R. Stubberud, and I. J. Williams, Outline of Theory and Problems of Feedback and Control Systems, 2nd ed., Schaum’s Outline Series (McGraw-Hill, 1990).

2010 (1)

2009 (1)

2007 (1)

J. Dong, J. Qi, and R. Miao, “Measurement of the damping of liquid surface wave by diffraction method,” Braz. J. Phys. 37, 1129–1133 (2007).
[CrossRef]

1994 (1)

P. Hickson, E. F. Borra, R. Cabanac, R. Content, B. K. Gibson, and G. A. H. Walker, “UBC/Laval 2.7 meter liquid mirror telescope,” Astrophys. J. 436, L201–L204 (1994).
[CrossRef]

1985 (1)

E. F. Borra, M. Beauchemin, and R. Lalande, “Liquid mirror telescopes: observations with a 1 meter diameter prototype and scaling-up considerations,” Astrophys. J. 297, 846–851 (1985).
[CrossRef]

1973 (1)

C. Mei and L. Liu, “The damping of surface gravity waves in a bounded liquid,” J. Fluid. Mech. 59, 239–256 (1973).
[CrossRef]

1967 (1)

J. Miles, “Surface-wave damping in closed basins,” Proc. R. Soc. Lond. A 297, 459–475 (1967).
[CrossRef]

1964 (1)

H. F. Bauer, “Tables of zeros of cross product Bessel functions,” J. Math. Comput. 18, 128–135 (1964).

1909 (1)

R. W. Wood, “The mercury paraboloid as a reflecting telescope,” Astrophys. J. 29, 164–176 (1909).
[CrossRef]

Bauer, H. F.

H. F. Bauer, “Tables of zeros of cross product Bessel functions,” J. Math. Comput. 18, 128–135 (1964).

Beauchemin, M.

E. F. Borra, M. Beauchemin, and R. Lalande, “Liquid mirror telescopes: observations with a 1 meter diameter prototype and scaling-up considerations,” Astrophys. J. 297, 846–851 (1985).
[CrossRef]

Borra, E. F.

D. Brousseau, E. F. Borra, M. Rochette, and D. B. Landry, “Linearization of the response of a 91-actuator magnetic liquid deformable mirror,” Opt. Express 18, 8239–8250 (2010).
[CrossRef]

P. Hickson, E. F. Borra, R. Cabanac, R. Content, B. K. Gibson, and G. A. H. Walker, “UBC/Laval 2.7 meter liquid mirror telescope,” Astrophys. J. 436, L201–L204 (1994).
[CrossRef]

E. F. Borra, M. Beauchemin, and R. Lalande, “Liquid mirror telescopes: observations with a 1 meter diameter prototype and scaling-up considerations,” Astrophys. J. 297, 846–851 (1985).
[CrossRef]

Brousseau, D.

Cabanac, R.

P. Hickson, E. F. Borra, R. Cabanac, R. Content, B. K. Gibson, and G. A. H. Walker, “UBC/Laval 2.7 meter liquid mirror telescope,” Astrophys. J. 436, L201–L204 (1994).
[CrossRef]

Content, R.

P. Hickson, E. F. Borra, R. Cabanac, R. Content, B. K. Gibson, and G. A. H. Walker, “UBC/Laval 2.7 meter liquid mirror telescope,” Astrophys. J. 436, L201–L204 (1994).
[CrossRef]

DiStefano, J. J.

J. J. DiStefano, A. R. Stubberud, and I. J. Williams, Outline of Theory and Problems of Feedback and Control Systems, 2nd ed., Schaum’s Outline Series (McGraw-Hill, 1990).

Dong, J.

J. Dong, J. Qi, and R. Miao, “Measurement of the damping of liquid surface wave by diffraction method,” Braz. J. Phys. 37, 1129–1133 (2007).
[CrossRef]

Gibson, B. K.

P. Hickson, E. F. Borra, R. Cabanac, R. Content, B. K. Gibson, and G. A. H. Walker, “UBC/Laval 2.7 meter liquid mirror telescope,” Astrophys. J. 436, L201–L204 (1994).
[CrossRef]

Hickson, P.

P. Hickson, E. F. Borra, R. Cabanac, R. Content, B. K. Gibson, and G. A. H. Walker, “UBC/Laval 2.7 meter liquid mirror telescope,” Astrophys. J. 436, L201–L204 (1994).
[CrossRef]

Ibrahim, R. A.

R. A. Ibrahim, Liquid Sloshing Dynamics, Theory and Applications (Cambridge University, 2005).

Lalande, R.

E. F. Borra, M. Beauchemin, and R. Lalande, “Liquid mirror telescopes: observations with a 1 meter diameter prototype and scaling-up considerations,” Astrophys. J. 297, 846–851 (1985).
[CrossRef]

Lamb, H.

H. Lamb, Hydrodynamics,6th ed. (Cambridge University, 1975).

Landau, L.

L. Landau and E. Lifshitz, Fluid Mechanics, 2nd ed., Vol. 6 of Course of Theoretical Physics (Butterworth Heinemann, 2009).

Landry, D. B.

Lifshitz, E.

L. Landau and E. Lifshitz, Fluid Mechanics, 2nd ed., Vol. 6 of Course of Theoretical Physics (Butterworth Heinemann, 2009).

Liu, L.

C. Mei and L. Liu, “The damping of surface gravity waves in a bounded liquid,” J. Fluid. Mech. 59, 239–256 (1973).
[CrossRef]

Mei, C.

C. Mei and L. Liu, “The damping of surface gravity waves in a bounded liquid,” J. Fluid. Mech. 59, 239–256 (1973).
[CrossRef]

Miao, R.

J. Dong, J. Qi, and R. Miao, “Measurement of the damping of liquid surface wave by diffraction method,” Braz. J. Phys. 37, 1129–1133 (2007).
[CrossRef]

Miles, J.

J. Miles, “Surface-wave damping in closed basins,” Proc. R. Soc. Lond. A 297, 459–475 (1967).
[CrossRef]

Qi, J.

J. Dong, J. Qi, and R. Miao, “Measurement of the damping of liquid surface wave by diffraction method,” Braz. J. Phys. 37, 1129–1133 (2007).
[CrossRef]

Rochette, M.

Stubberud, A. R.

J. J. DiStefano, A. R. Stubberud, and I. J. Williams, Outline of Theory and Problems of Feedback and Control Systems, 2nd ed., Schaum’s Outline Series (McGraw-Hill, 1990).

Vdovin, G.

Walker, G. A. H.

P. Hickson, E. F. Borra, R. Cabanac, R. Content, B. K. Gibson, and G. A. H. Walker, “UBC/Laval 2.7 meter liquid mirror telescope,” Astrophys. J. 436, L201–L204 (1994).
[CrossRef]

Williams, I. J.

J. J. DiStefano, A. R. Stubberud, and I. J. Williams, Outline of Theory and Problems of Feedback and Control Systems, 2nd ed., Schaum’s Outline Series (McGraw-Hill, 1990).

Wood, R. W.

R. W. Wood, “The mercury paraboloid as a reflecting telescope,” Astrophys. J. 29, 164–176 (1909).
[CrossRef]

Astrophys. J. (3)

R. W. Wood, “The mercury paraboloid as a reflecting telescope,” Astrophys. J. 29, 164–176 (1909).
[CrossRef]

E. F. Borra, M. Beauchemin, and R. Lalande, “Liquid mirror telescopes: observations with a 1 meter diameter prototype and scaling-up considerations,” Astrophys. J. 297, 846–851 (1985).
[CrossRef]

P. Hickson, E. F. Borra, R. Cabanac, R. Content, B. K. Gibson, and G. A. H. Walker, “UBC/Laval 2.7 meter liquid mirror telescope,” Astrophys. J. 436, L201–L204 (1994).
[CrossRef]

Braz. J. Phys. (1)

J. Dong, J. Qi, and R. Miao, “Measurement of the damping of liquid surface wave by diffraction method,” Braz. J. Phys. 37, 1129–1133 (2007).
[CrossRef]

J. Fluid. Mech. (1)

C. Mei and L. Liu, “The damping of surface gravity waves in a bounded liquid,” J. Fluid. Mech. 59, 239–256 (1973).
[CrossRef]

J. Math. Comput. (1)

H. F. Bauer, “Tables of zeros of cross product Bessel functions,” J. Math. Comput. 18, 128–135 (1964).

Opt. Express (1)

Opt. Lett. (1)

Proc. R. Soc. Lond. A (1)

J. Miles, “Surface-wave damping in closed basins,” Proc. R. Soc. Lond. A 297, 459–475 (1967).
[CrossRef]

Other (4)

L. Landau and E. Lifshitz, Fluid Mechanics, 2nd ed., Vol. 6 of Course of Theoretical Physics (Butterworth Heinemann, 2009).

R. A. Ibrahim, Liquid Sloshing Dynamics, Theory and Applications (Cambridge University, 2005).

H. Lamb, Hydrodynamics,6th ed. (Cambridge University, 1975).

J. J. DiStefano, A. R. Stubberud, and I. J. Williams, Outline of Theory and Problems of Feedback and Control Systems, 2nd ed., Schaum’s Outline Series (McGraw-Hill, 1990).

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Figures (12)

Fig. 1.
Fig. 1.

Liquid mirror device.

Fig. 2.
Fig. 2.

Setup for the measurement of the frequency response and the time constants.

Fig. 3.
Fig. 3.

Frequency response of the liquid mirror device: (a) Bode plot and (b) Nichol’s chart.

Fig. 4.
Fig. 4.

Open-loop response of the liquid mirror to ambient excitations (a) in the time domain and (b) as a Fourier spectrum in the frequency domain with the frequencies of the excited modes of Table 2.

Fig. 5.
Fig. 5.

Rise and relaxation time constants of the liquid mirror device as a function of (a) the number of actuators and (b) the frequency.

Fig. 6.
Fig. 6.

Wavefronts of the (a) aberrated, (b) corrected, and (c) undisturbed beam. The image diameter is 5.2 mm, and the peak-to-valley phase differences are 2.49, 0.40, and 0.30 μm, respectively.

Fig. 7.
Fig. 7.

Far-field images of the beam—top row: reconstruction from the wavefront sensor data, bottom row: measurement with the CCD camera behind the microscope objective. The images show the aberrated (left), corrected (middle), and undisturbed beams (right).

Fig. 8.
Fig. 8.

Measurements of the wavefront rms error for the dynamic [(a) with the rotating phase disk] and static [(b) without phase disk] regimes; the passages with the activated feedback are shown with the gray background color. During the dynamic measurements, the feedback was switched on and off during the passage of the cut out part of the disk when the rms error was especially low.

Fig. 9.
Fig. 9.

Dynamic correction performance of the liquid mirror setup in terms of improvement, the ratio of the measured wavefront error induced by the rotating phase disk with and without closing the feedback loop of the liquid mirror system, as a function of (a) feedback gain, (b) feedback frequency, (c) rotation speed of the disk, and (d) number of modes.

Fig. 10.
Fig. 10.

Measurement of the beam deflection (a) without and (b) with closing the feedback loop. Analysis of the tip/tilt measurements shows that the liquid mirror system reduces the (c) amplitude and the (d) directional variance.

Fig. 11.
Fig. 11.

Measurement of the beam deflection (a) without and (b) with closing the feedback loop. The axis of deflection oriented along 75°–255° coincides with the direction in which the cross section of the beam is elongated on the liquid mirror and thus with its direction of incidence on the liquid surface. Analysis of the tip/tilt measurements shows that the liquid mirror system (c) reduces the amplitude and (d) distributes the deflection symmetrically around the center.

Fig. 12.
Fig. 12.

Fourier spectra of the liquid mirror response with a free surface (open-loop) and with closed-loop operation of the system. The arrows show the positions of the normal modes of the liquid container (compare Table 2).

Tables (3)

Tables Icon

Table 1. Frequency, f, Wavelength, λ, and Wave-Velocity, c, of the First 10 Modes

Tables Icon

Table 2. Measured, fmeas, and Calculated, fcalc, Resonances of the Fourier Spectrum of the Open-Loop Liquid Mirror Frequency Response (Compare Also Fig. 12)

Tables Icon

Table 3. Residual Errors and Strehl Ratios of an Aberrated, Corrected, and Undisturbed Beam

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

ωij2=gξijR(1+γξij2ρgR2)tanh(ξijhR)
ωij2=gkij+γkij3ρ
ωij2λij32πgλij2=8π3γρ.
c=ω·λ2π
δ=2υωh
τ=λ28π2υ
τ=a*nb

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