Abstract

Liquid-crystal modal wave-front correctors provide much better wave-front correction than do piston correctors with the same number of actuators; moreover, use of additional degrees of freedom of the driving ac voltage signals may further improve device performance. Some practical aspects of the operation of liquid-crystal modal wave-front correctors are discussed. Special attention is paid to the interference of various contact responses and to the formation of required phase shapes through wider control of signal frequencies and electric phase shifts. The study is based on an analytic approach and numerical investigation; major theoretical conclusions are verified experimentally.

© 2004 Optical Society of America

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References

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  1. R. K. Tyson, Principles of Adaptive Optics, 2nd ed. (Academic, Boston, Mass., 1998).
  2. M. A. Vorontsov, V. I. Shmal’gauzen, Principles of Adaptive Optics (Nauka, Moscow, 1985).
  3. S. R. Restaino, S. W. Teare, eds., Proceedings of the 3rd International Workshop on Adaptive Optics for Industry and Medicine (Starline Printing, Albuquerque, N. Mex., 2002).
  4. Meadowlark Optics, http://www.meadowlark.com/catalog/SLMs/slm2.htm .
  5. D. Dayton, S. Browne, J. Gonglewski, S. Restaino, “Characterization and control of a multielement dual-frequency liquid-crystal device for high-speed adaptive optical wave-front correction,” Appl. Opt. 40, 2345–2355 (2001).
    [CrossRef]
  6. D. C. Dayton, S. L. Browne, S. P. Sandven, J. D. Gonglewski, A. V. Kudryashov, “Theory and laboratory demonstrations on the use of a nematic liquid-crystal phase modulator for controlled turbulence generation and adaptive optics,” Appl. Opt. 37, 5579–5589 (1998).
    [CrossRef]
  7. F. Vargas-Martin, P. Prieto, P. Artal, “Correction of the aberrations in the human eye with liquid crystal spatial light modulators: limits to the performance,” J. Opt. Soc. Am. A 15, 2552–2562 (1998).
    [CrossRef]
  8. M. Loktev, D. W. De Lima Monteiro, G. Vdovin, “Comparison study of the performance of piston, thin plate and membrane mirrors for correction of turbulence-induced phase distortions,” Opt. Commun. 192, 91–99 (2001).
    [CrossRef]
  9. A. F. Naumov, “Modal wavefront correctors,” Proc. P. N. Lebedev Phys. Inst. 217, 177–182 (1993).
  10. A. F. Naumov, G. V. Vdovin, “Multichannel liquid-crystal-based wave-front corrector with modal influence functions,” Opt. Lett. 23, 1550–1552 (1998).
    [CrossRef]
  11. S. P. Kotova, M. Y. Kvashnin, M. A. Rakhmatulin, O. A. Zayakin, I. R. Guralnik, N. A. Klimov, P. Clark, G. D. Love, A. F. Naumov, C. D. Saunter, M. Y. Loktev, G. V. Vdovin, L. V. Toporkova, “Modal liquid crystal wavefront corrector,” Opt. Express 10, 1258–1272 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-22-1258 .
    [CrossRef] [PubMed]
  12. G. V. Vdovin, I. R. Guralnik, S. P. Kotova, M. Yu. Loktev, A. F. Naumov, “Liquid crystal lenses with programmable focal distance. I. Theory; II. Experiment,” Quantum Electron. 29, 256–264 (1999).
    [CrossRef]
  13. I. R. Guralnik, V. N. Belopukhov, G. D. Love, A. F. Naumov, “Interdependence of the electrical and optical properties of liquid crystals for phase modulation applications,” J. Appl. Phys. 87, 4069–4074 (2000).
    [CrossRef]
  14. I. R. Guralnik, S. A. Samagin, “Electrophysics of the modal multi-channel liquid-crystal wavefront corrector,” Quantum Electron. 32, 362–366 (2002).
    [CrossRef]
  15. M. Y. Loktev, A. F. Naumov, I. R. Guralnik, “Static and dynamic models of liquid crystal wavefront correctors,” in Laser Optics 2000, Control of Laser Beam Characteristics and Nonlinear Methods for Wavefront Control, L. N. Soms, V. E. Sherstobitov, eds., Proc. SPIE4353, 9–16 (2000).
    [CrossRef]
  16. I. R. Guralnik, M. Y. Loktev, A. F. Naumov, “Electrophysics of adaptive LC lenses,” in International Conference on Atomic and Molecular Pulsed Lasers III, V. F. Tarasenko, G. V. Mayer, G. G. Petrash, eds., Proc. SPIE4071, 209–218 (2000).
    [CrossRef]
  17. M. B. Allen, E. I. Isaacson, Numerical Analysis for Applied Science (Wiley, New York, 1998).
  18. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1965).
  19. M. Y. Loktev, G. V. Vdovin, P. M. Sarro, “Modal corrector integrated in silicon: possibilities for implementation,” in Optics in Atmospheric Propagation and Adaptive Systems V, A. Kohnle, J. D. Gonglewski, eds., Proc. SPIE4884, 196–205 (2003).
    [CrossRef]
  20. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C++, 2nd ed. (Cambridge U. Press, Cambridge, 2002), pp. 413–417.
  21. L. M. Blinov, V. G. Chigrinov, Electrooptic Effects in Liquid Crystal Materials (Springer, New York, 1996).
  22. M. S. Bazaraa, H. D. Sherali, C. M. Shetty, Nonlinear Programming: Theory and Algorithms (Wiley, New York, 1993).
  23. G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1968).

2002 (2)

2001 (2)

D. Dayton, S. Browne, J. Gonglewski, S. Restaino, “Characterization and control of a multielement dual-frequency liquid-crystal device for high-speed adaptive optical wave-front correction,” Appl. Opt. 40, 2345–2355 (2001).
[CrossRef]

M. Loktev, D. W. De Lima Monteiro, G. Vdovin, “Comparison study of the performance of piston, thin plate and membrane mirrors for correction of turbulence-induced phase distortions,” Opt. Commun. 192, 91–99 (2001).
[CrossRef]

2000 (1)

I. R. Guralnik, V. N. Belopukhov, G. D. Love, A. F. Naumov, “Interdependence of the electrical and optical properties of liquid crystals for phase modulation applications,” J. Appl. Phys. 87, 4069–4074 (2000).
[CrossRef]

1999 (1)

G. V. Vdovin, I. R. Guralnik, S. P. Kotova, M. Yu. Loktev, A. F. Naumov, “Liquid crystal lenses with programmable focal distance. I. Theory; II. Experiment,” Quantum Electron. 29, 256–264 (1999).
[CrossRef]

1998 (3)

1993 (1)

A. F. Naumov, “Modal wavefront correctors,” Proc. P. N. Lebedev Phys. Inst. 217, 177–182 (1993).

Allen, M. B.

M. B. Allen, E. I. Isaacson, Numerical Analysis for Applied Science (Wiley, New York, 1998).

Artal, P.

Bazaraa, M. S.

M. S. Bazaraa, H. D. Sherali, C. M. Shetty, Nonlinear Programming: Theory and Algorithms (Wiley, New York, 1993).

Belopukhov, V. N.

I. R. Guralnik, V. N. Belopukhov, G. D. Love, A. F. Naumov, “Interdependence of the electrical and optical properties of liquid crystals for phase modulation applications,” J. Appl. Phys. 87, 4069–4074 (2000).
[CrossRef]

Blinov, L. M.

L. M. Blinov, V. G. Chigrinov, Electrooptic Effects in Liquid Crystal Materials (Springer, New York, 1996).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1965).

Browne, S.

Browne, S. L.

Chigrinov, V. G.

L. M. Blinov, V. G. Chigrinov, Electrooptic Effects in Liquid Crystal Materials (Springer, New York, 1996).

Clark, P.

Dayton, D.

Dayton, D. C.

De Lima Monteiro, D. W.

M. Loktev, D. W. De Lima Monteiro, G. Vdovin, “Comparison study of the performance of piston, thin plate and membrane mirrors for correction of turbulence-induced phase distortions,” Opt. Commun. 192, 91–99 (2001).
[CrossRef]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C++, 2nd ed. (Cambridge U. Press, Cambridge, 2002), pp. 413–417.

Gonglewski, J.

Gonglewski, J. D.

Guralnik, I. R.

I. R. Guralnik, S. A. Samagin, “Electrophysics of the modal multi-channel liquid-crystal wavefront corrector,” Quantum Electron. 32, 362–366 (2002).
[CrossRef]

S. P. Kotova, M. Y. Kvashnin, M. A. Rakhmatulin, O. A. Zayakin, I. R. Guralnik, N. A. Klimov, P. Clark, G. D. Love, A. F. Naumov, C. D. Saunter, M. Y. Loktev, G. V. Vdovin, L. V. Toporkova, “Modal liquid crystal wavefront corrector,” Opt. Express 10, 1258–1272 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-22-1258 .
[CrossRef] [PubMed]

I. R. Guralnik, V. N. Belopukhov, G. D. Love, A. F. Naumov, “Interdependence of the electrical and optical properties of liquid crystals for phase modulation applications,” J. Appl. Phys. 87, 4069–4074 (2000).
[CrossRef]

G. V. Vdovin, I. R. Guralnik, S. P. Kotova, M. Yu. Loktev, A. F. Naumov, “Liquid crystal lenses with programmable focal distance. I. Theory; II. Experiment,” Quantum Electron. 29, 256–264 (1999).
[CrossRef]

M. Y. Loktev, A. F. Naumov, I. R. Guralnik, “Static and dynamic models of liquid crystal wavefront correctors,” in Laser Optics 2000, Control of Laser Beam Characteristics and Nonlinear Methods for Wavefront Control, L. N. Soms, V. E. Sherstobitov, eds., Proc. SPIE4353, 9–16 (2000).
[CrossRef]

I. R. Guralnik, M. Y. Loktev, A. F. Naumov, “Electrophysics of adaptive LC lenses,” in International Conference on Atomic and Molecular Pulsed Lasers III, V. F. Tarasenko, G. V. Mayer, G. G. Petrash, eds., Proc. SPIE4071, 209–218 (2000).
[CrossRef]

Isaacson, E. I.

M. B. Allen, E. I. Isaacson, Numerical Analysis for Applied Science (Wiley, New York, 1998).

Klimov, N. A.

Korn, G. A.

G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1968).

Korn, T. M.

G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1968).

Kotova, S. P.

Kudryashov, A. V.

Kvashnin, M. Y.

Loktev, M.

M. Loktev, D. W. De Lima Monteiro, G. Vdovin, “Comparison study of the performance of piston, thin plate and membrane mirrors for correction of turbulence-induced phase distortions,” Opt. Commun. 192, 91–99 (2001).
[CrossRef]

Loktev, M. Y.

S. P. Kotova, M. Y. Kvashnin, M. A. Rakhmatulin, O. A. Zayakin, I. R. Guralnik, N. A. Klimov, P. Clark, G. D. Love, A. F. Naumov, C. D. Saunter, M. Y. Loktev, G. V. Vdovin, L. V. Toporkova, “Modal liquid crystal wavefront corrector,” Opt. Express 10, 1258–1272 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-22-1258 .
[CrossRef] [PubMed]

M. Y. Loktev, A. F. Naumov, I. R. Guralnik, “Static and dynamic models of liquid crystal wavefront correctors,” in Laser Optics 2000, Control of Laser Beam Characteristics and Nonlinear Methods for Wavefront Control, L. N. Soms, V. E. Sherstobitov, eds., Proc. SPIE4353, 9–16 (2000).
[CrossRef]

I. R. Guralnik, M. Y. Loktev, A. F. Naumov, “Electrophysics of adaptive LC lenses,” in International Conference on Atomic and Molecular Pulsed Lasers III, V. F. Tarasenko, G. V. Mayer, G. G. Petrash, eds., Proc. SPIE4071, 209–218 (2000).
[CrossRef]

M. Y. Loktev, G. V. Vdovin, P. M. Sarro, “Modal corrector integrated in silicon: possibilities for implementation,” in Optics in Atmospheric Propagation and Adaptive Systems V, A. Kohnle, J. D. Gonglewski, eds., Proc. SPIE4884, 196–205 (2003).
[CrossRef]

Loktev, M. Yu.

G. V. Vdovin, I. R. Guralnik, S. P. Kotova, M. Yu. Loktev, A. F. Naumov, “Liquid crystal lenses with programmable focal distance. I. Theory; II. Experiment,” Quantum Electron. 29, 256–264 (1999).
[CrossRef]

Love, G. D.

Naumov, A. F.

S. P. Kotova, M. Y. Kvashnin, M. A. Rakhmatulin, O. A. Zayakin, I. R. Guralnik, N. A. Klimov, P. Clark, G. D. Love, A. F. Naumov, C. D. Saunter, M. Y. Loktev, G. V. Vdovin, L. V. Toporkova, “Modal liquid crystal wavefront corrector,” Opt. Express 10, 1258–1272 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-22-1258 .
[CrossRef] [PubMed]

I. R. Guralnik, V. N. Belopukhov, G. D. Love, A. F. Naumov, “Interdependence of the electrical and optical properties of liquid crystals for phase modulation applications,” J. Appl. Phys. 87, 4069–4074 (2000).
[CrossRef]

G. V. Vdovin, I. R. Guralnik, S. P. Kotova, M. Yu. Loktev, A. F. Naumov, “Liquid crystal lenses with programmable focal distance. I. Theory; II. Experiment,” Quantum Electron. 29, 256–264 (1999).
[CrossRef]

A. F. Naumov, G. V. Vdovin, “Multichannel liquid-crystal-based wave-front corrector with modal influence functions,” Opt. Lett. 23, 1550–1552 (1998).
[CrossRef]

A. F. Naumov, “Modal wavefront correctors,” Proc. P. N. Lebedev Phys. Inst. 217, 177–182 (1993).

I. R. Guralnik, M. Y. Loktev, A. F. Naumov, “Electrophysics of adaptive LC lenses,” in International Conference on Atomic and Molecular Pulsed Lasers III, V. F. Tarasenko, G. V. Mayer, G. G. Petrash, eds., Proc. SPIE4071, 209–218 (2000).
[CrossRef]

M. Y. Loktev, A. F. Naumov, I. R. Guralnik, “Static and dynamic models of liquid crystal wavefront correctors,” in Laser Optics 2000, Control of Laser Beam Characteristics and Nonlinear Methods for Wavefront Control, L. N. Soms, V. E. Sherstobitov, eds., Proc. SPIE4353, 9–16 (2000).
[CrossRef]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C++, 2nd ed. (Cambridge U. Press, Cambridge, 2002), pp. 413–417.

Prieto, P.

Rakhmatulin, M. A.

Restaino, S.

Samagin, S. A.

I. R. Guralnik, S. A. Samagin, “Electrophysics of the modal multi-channel liquid-crystal wavefront corrector,” Quantum Electron. 32, 362–366 (2002).
[CrossRef]

Sandven, S. P.

Sarro, P. M.

M. Y. Loktev, G. V. Vdovin, P. M. Sarro, “Modal corrector integrated in silicon: possibilities for implementation,” in Optics in Atmospheric Propagation and Adaptive Systems V, A. Kohnle, J. D. Gonglewski, eds., Proc. SPIE4884, 196–205 (2003).
[CrossRef]

Saunter, C. D.

Sherali, H. D.

M. S. Bazaraa, H. D. Sherali, C. M. Shetty, Nonlinear Programming: Theory and Algorithms (Wiley, New York, 1993).

Shetty, C. M.

M. S. Bazaraa, H. D. Sherali, C. M. Shetty, Nonlinear Programming: Theory and Algorithms (Wiley, New York, 1993).

Shmal’gauzen, V. I.

M. A. Vorontsov, V. I. Shmal’gauzen, Principles of Adaptive Optics (Nauka, Moscow, 1985).

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C++, 2nd ed. (Cambridge U. Press, Cambridge, 2002), pp. 413–417.

Toporkova, L. V.

Tyson, R. K.

R. K. Tyson, Principles of Adaptive Optics, 2nd ed. (Academic, Boston, Mass., 1998).

Vargas-Martin, F.

Vdovin, G.

M. Loktev, D. W. De Lima Monteiro, G. Vdovin, “Comparison study of the performance of piston, thin plate and membrane mirrors for correction of turbulence-induced phase distortions,” Opt. Commun. 192, 91–99 (2001).
[CrossRef]

Vdovin, G. V.

S. P. Kotova, M. Y. Kvashnin, M. A. Rakhmatulin, O. A. Zayakin, I. R. Guralnik, N. A. Klimov, P. Clark, G. D. Love, A. F. Naumov, C. D. Saunter, M. Y. Loktev, G. V. Vdovin, L. V. Toporkova, “Modal liquid crystal wavefront corrector,” Opt. Express 10, 1258–1272 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-22-1258 .
[CrossRef] [PubMed]

G. V. Vdovin, I. R. Guralnik, S. P. Kotova, M. Yu. Loktev, A. F. Naumov, “Liquid crystal lenses with programmable focal distance. I. Theory; II. Experiment,” Quantum Electron. 29, 256–264 (1999).
[CrossRef]

A. F. Naumov, G. V. Vdovin, “Multichannel liquid-crystal-based wave-front corrector with modal influence functions,” Opt. Lett. 23, 1550–1552 (1998).
[CrossRef]

M. Y. Loktev, G. V. Vdovin, P. M. Sarro, “Modal corrector integrated in silicon: possibilities for implementation,” in Optics in Atmospheric Propagation and Adaptive Systems V, A. Kohnle, J. D. Gonglewski, eds., Proc. SPIE4884, 196–205 (2003).
[CrossRef]

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C++, 2nd ed. (Cambridge U. Press, Cambridge, 2002), pp. 413–417.

Vorontsov, M. A.

M. A. Vorontsov, V. I. Shmal’gauzen, Principles of Adaptive Optics (Nauka, Moscow, 1985).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1965).

Zayakin, O. A.

Appl. Opt. (2)

J. Appl. Phys. (1)

I. R. Guralnik, V. N. Belopukhov, G. D. Love, A. F. Naumov, “Interdependence of the electrical and optical properties of liquid crystals for phase modulation applications,” J. Appl. Phys. 87, 4069–4074 (2000).
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Commun. (1)

M. Loktev, D. W. De Lima Monteiro, G. Vdovin, “Comparison study of the performance of piston, thin plate and membrane mirrors for correction of turbulence-induced phase distortions,” Opt. Commun. 192, 91–99 (2001).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Proc. P. N. Lebedev Phys. Inst. (1)

A. F. Naumov, “Modal wavefront correctors,” Proc. P. N. Lebedev Phys. Inst. 217, 177–182 (1993).

Quantum Electron. (2)

G. V. Vdovin, I. R. Guralnik, S. P. Kotova, M. Yu. Loktev, A. F. Naumov, “Liquid crystal lenses with programmable focal distance. I. Theory; II. Experiment,” Quantum Electron. 29, 256–264 (1999).
[CrossRef]

I. R. Guralnik, S. A. Samagin, “Electrophysics of the modal multi-channel liquid-crystal wavefront corrector,” Quantum Electron. 32, 362–366 (2002).
[CrossRef]

Other (13)

M. Y. Loktev, A. F. Naumov, I. R. Guralnik, “Static and dynamic models of liquid crystal wavefront correctors,” in Laser Optics 2000, Control of Laser Beam Characteristics and Nonlinear Methods for Wavefront Control, L. N. Soms, V. E. Sherstobitov, eds., Proc. SPIE4353, 9–16 (2000).
[CrossRef]

I. R. Guralnik, M. Y. Loktev, A. F. Naumov, “Electrophysics of adaptive LC lenses,” in International Conference on Atomic and Molecular Pulsed Lasers III, V. F. Tarasenko, G. V. Mayer, G. G. Petrash, eds., Proc. SPIE4071, 209–218 (2000).
[CrossRef]

M. B. Allen, E. I. Isaacson, Numerical Analysis for Applied Science (Wiley, New York, 1998).

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1965).

M. Y. Loktev, G. V. Vdovin, P. M. Sarro, “Modal corrector integrated in silicon: possibilities for implementation,” in Optics in Atmospheric Propagation and Adaptive Systems V, A. Kohnle, J. D. Gonglewski, eds., Proc. SPIE4884, 196–205 (2003).
[CrossRef]

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C++, 2nd ed. (Cambridge U. Press, Cambridge, 2002), pp. 413–417.

L. M. Blinov, V. G. Chigrinov, Electrooptic Effects in Liquid Crystal Materials (Springer, New York, 1996).

M. S. Bazaraa, H. D. Sherali, C. M. Shetty, Nonlinear Programming: Theory and Algorithms (Wiley, New York, 1993).

G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1968).

R. K. Tyson, Principles of Adaptive Optics, 2nd ed. (Academic, Boston, Mass., 1998).

M. A. Vorontsov, V. I. Shmal’gauzen, Principles of Adaptive Optics (Nauka, Moscow, 1985).

S. R. Restaino, S. W. Teare, eds., Proceedings of the 3rd International Workshop on Adaptive Optics for Industry and Medicine (Starline Printing, Albuquerque, N. Mex., 2002).

Meadowlark Optics, http://www.meadowlark.com/catalog/SLMs/slm2.htm .

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Figures (13)

Fig. 1
Fig. 1

(a) Cross section and (b) contacts layout of the LC MWC.

Fig. 2
Fig. 2

Equivalent electrical circuits of the LC MWC and the control unit.

Fig. 3
Fig. 3

Vector layout: O, origin; P, observation point. Vectors starting at O are denoted r; those that originate at the contacts are denoted R.

Fig. 4
Fig. 4

(a)–(c) Calculated influence functions of the central contact in a 37-channel LC MWC for various CU resistances and all active contacts. (d) Influence function for a single active contact, shown for comparison. Corrector parameters: diameter of pointlike contacts, 0.5 mm; aperture diameter, 30 mm; intercontact distance, 3.3 mm; sheet resistance, ρ s = 500 kΩ; contact impedance, Z c = 691 kΩ; intercontact impedance, Z ic = 878 kΩ; frequency, 500 Hz.

Fig. 5
Fig. 5

Cross section of the influence functions of the central contact for various CU resistances and all active contacts; intercontact distance, 3.3 mm; other corrector parameters identical to those for Fig. 4. The influence function for a single active contact is shown for comparison.

Fig. 6
Fig. 6

Cross sections of the influence functions for various frequencies. (a), (c) Amplitude and (b), (d) phase components of the influence functions are presented. CU resistance: (a), (b) 1 kΩ and (c), (d) 100 MΩ; intercontact distance, 3.3 mm; other parameters identical to those for Fig. 4.

Fig. 7
Fig. 7

Approximation of (a), (b) negative and (c), (d) positive defocus in a 38-channel modal LC corrector; optimization by voltage amplitudes for frequencies (a), (c) 1 kHz and (b), (d) 100 kHz.

Fig. 8
Fig. 8

Cross section of the voltage distribution for two active contacts operating on the same frequency and different phase shifts; results of numerical simulation.

Fig. 9
Fig. 9

Approximation of (a)–(c) negative defocus and (d)–(f) negative spherical aberration in a LC MWC. Results of optimization (a), (d) by voltage amplitudes and (b), (e) by both amplitudes and phase shifts. (c) Ideal defocus and (f) spherical aberration are shown for comparison. Corrector parameters are similar to those given in Table 2; signal frequency, 1 kHz; residual aberrations are given in Table 3.

Fig. 10
Fig. 10

Function f that describes the mismatch frequency dependence on voltage.

Fig. 11
Fig. 11

(a) Amplitude of the first harmonics of the photodiode output versus amplitude of voltage at the second contact; (b) amplitude of phase delay oscillations in the LC-MWC versus frequency. Both dependencies are experimental.

Fig. 12
Fig. 12

Two-contact response of a 38-channel 80-mm LC-MWC for (a) 0.15 Ω and (b) 453 kΩ CU series resistance. EMF values of both control channels were adjusted to yield 7.07-V rms voltages at the corresponding contacts. CU-generated sine wave with 500 Hz frequency for the first channel and 600 Hz for the second channel.

Fig. 13
Fig. 13

(c) Approximation of target voltage distribution for astigmatism under FMC by use of (a) voltage-addressing and (b) current-addressing modes; base frequency, f base = 1 kHz.

Tables (4)

Tables Icon

Table 1 Experimental Investigation of the Screening Effect in a 38-Channel 80-mm LC MWC for Several CU Series Resistances Ra

Tables Icon

Table 2 Approximation of Zernike Aberrations in a 38-Channel Modal LC Corrector by Use of SF-VACa

Tables Icon

Table 3 Approximation of Zernike Aberrations in a LC MWCa

Tables Icon

Table 4 Approximation of Zernike Aberrations in a LC MWC by Use of MFC and Current-Addressing Modea

Equations (36)

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

2ur, t=ρc ur, tt+ρgu r, t,
2ur=χ2ur,
ur, tLk=εk-IkR,
Ik=Lki, Nkdl=- 1ρLku r, t, Nkdl,
ur, t|Lk- RρLku r, t, Nkdl=εk,k=1  N.
εkt=α=1M Ekα expiωαt,
ur, t=k=1N ukr, t,
ukr, t=α=1M ukαrexpiωαt.
Δukαr=χα2ukαr,
umαr|Lm- RρLmumαr, Nmdl=Ekαδmk,m=1  N,
2ukr=χk2ukr,
umr|Lm- RρLmumr, Nmdl=Ekδmk,m=1  N.
ukr= EkΔm=1N YmkH0iχk|r-rm|,
uαr |Lk=Ekα-IkαR, k=1  N.
Ikα= 1ρLkuαr,Nkdl.
Δ= δ12δ0-δ1,
εkt=Ek expiωt.
ur|Lk- RρLkur, Nkdl=Ek,k=1  N.
Uk=UkE1,  , EN, k=1  N.
ur|Lk=Uk, k=1  N.
ur, t=k=1Nα=1M ukαrexpiωαt,
τ= γd2π2K11,
urms2r= |ur, t|2= 1τtt+τ ur, tu*r, tdt.
urms2r= k,m=1Nα=1M ukαumα*+ k,m=1Nα,β=1M×ukαrumβ* rFωαβexpiωαβt+τ/2,
urms2r=|u1r|2+|u2r|2+2|u1ru2r|Fω12×cosω12t+τ/2+φ12r,
It= I0/21+cosΦ0+Φ1 cos ω12t.
C1=a1+b1=I0| sin Φ0J1Φ1|,
C2= a2+b2= I0cos Φ02J1Φ1Φ1 -J0Φ1,
urms2r =k=1N |ukr|2.
εkt=Ekα=1M Aα expiωαt=α=1M Ekα expiωαt,
uar|Lk- RρLkuαr, Nkdl=Ekα,k=1  N.
urms2r =α=1K |uαr|2.
Vr=k=1N XkH0iχk|r-rk|,
k=1N XkH0iχkdkm=Vm, m=1, 2  N,
Vkr|Ck=Ekδik.
H=H0iχ1aH0iχ1R12  H0iχ1R1NH0iχ2R12H0iχ2a  H0iχ2R2N  H0iχNR1NH0iχNR2N  H0iχNa,

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