Abstract

Accurate modeling of a high-resolution, liquid-crystal-based, optical phased array (OPA) is demonstrated. The modeling method is extendable to cases where the array element size is close to the wavelength of light. This is accomplished through calculating an equilibrium liquid-crystal (LC) director field that takes into account the fringing electric fields in LC OPAs with small array elements and by calculating the light transmission with a finite-difference time-domain method that has been extended for use in birefringent materials. The diffraction efficiency for a test device is calculated and compared with the simulation.

© 2005 Optical Society of America

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  1. P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holz, S. Liberman, H. Q. Nguyen, D. P. Resler, R. C. Sharp, E. A. Watson, “Optical phased array technology,” Proc. IEEE 84, 268–298 (1996).
    [CrossRef]
  2. D. P. Resler, D. S. Hobbs, R. C. Sharp, L. J. Friedman, T. A. Dorschner, “High-efficiency liquid-crystal optical phased-array beam steering,” Opt. Lett. 21, 689–691 (1996).
    [CrossRef] [PubMed]
  3. R. M. Matic, “Blazed phase liquid crystal beam steering,” in Laser Beam Propagation and Control, H. Weichel, L. F. DeSandre, eds., Proc. SPIE2120, 194–205 (1994).
    [CrossRef]
  4. L. J. Friedman, D. S. Hobbs, S. Lieberman, D. L. Corkum, H. W. Nguyen, R. C. Sharp, T. A. Dorschner, “Spatially resolved phase imaging of a programmable liquid-crystal grating,” Appl. Opt. 35, 6236–6240 (1996).
    [CrossRef] [PubMed]
  5. V. G. Dominic, E. A. Watson, “Measurement and modeling of the angular dispersion in liquid crystal broadband beam steering devices,” Opt. Eng. (Bellingham) 35, 3371–3379 (1996).
    [CrossRef]
  6. X. Wang, D. Wilson, R. Muller, P. Maker, D. Psaltis, “Liquid-crystal blazed-grating beam deflector,” Appl. Opt. 39, 6545–6555 (2000).
    [CrossRef]
  7. M. T. Gruneisen, J. M. Wilkes, “Compensated imaging by real-time holography with optically addressed spatial light modulators,” in Spatial Light Modulators, G. Burdge, S. C. Esener, eds., Vol. 14 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1997).
  8. G. D. Love, “Wavefront correction of Zernike modes with a liquid-crystal spatial light modulator,” Appl. Opt. 36, 1517–1524 (1997).
    [CrossRef] [PubMed]
  9. M. T. Gruneisen, T. Martinez, D. V. Wick, J. M. Wilkes, J. T. Baker, I. Percheron, “Holographic compensation of severe dynamic aberrations in membrane-mirror based telescope systems,” in High-Resolution Wavefront Control: Methods, Devices, and Applications, J. D. Gonglewski, M. A. Vorontsov, eds., Proc. SPIE3760, 142–152 (1999).
    [CrossRef]
  10. C. M. Titus, P. J. Bos, J. R. Kelly, E. C. Gartland, “Comparison of analytical calculations to finite-difference time-domain simulations of one-dimensional spatially varying anisotropic liquid crystal structures,” Jpn. J. Appl. Phys., Part 1 38, 1488–1494 (1999).
    [CrossRef]
  11. C. M. Titus, J. R. Kelly, E. C. Gartland, S. V. Shiyanovskii, J. A. Anderson, P. J. Bos, “Asymmetric transmissive behavior of liquid-crystal diffraction gratings,” Opt. Lett. 26, 1188–1190 (2001).
    [CrossRef]
  12. L. M. Blinov, “Electro-optical effects in liquid crystal,” Sov. Phys. Usp. 17, 658–672 (1975).
    [CrossRef]
  13. P. G. De Gennes, J. Prost, The Physics of Liquid Crystals (Oxford U. Press, Oxford, UK, 1993).
  14. J. Anderson, P. Watson, P. Bos, Liquid Crystal Display 3-D Director Simulator Software and Technical Guide (Artech House, Norwood, Mass., 2001).
  15. K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. AP-14, 302–307 (1966).
  16. A. Taflove, S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 2nd ed. (Artech House, Norwood, Mass., 2000).
  17. E. E. Kriezis, S. J. Elston, “Light wave propagation in liquid crystal display by the 2-D finite-difference time-domain method,” Opt. Commun. 177, 69–77 (2000).
    [CrossRef]
  18. Bin Wang, Xinghua Wang, P. J. Bos, “Study of switchable liquid crystal polymer grating by finite-difference time-domain calculation,” in Liquid Crystals VII, I.-C. Khoo, ed., Proc. SPIE5213, 104–110 (2003).
    [CrossRef]
  19. G. F. Barrick, P. J. Bos, C. E. Titus, B. Winker, “Computing the liquid crystal director field in optical phased arrays,” Opt. Eng. (Bellingham) 43, 924–932 (2004).
    [CrossRef]
  20. A. Yefet, P. G. Petropoulos, “A staggered fourth-order accuracy explicit finite difference scheme for the time-domain Maxwell’s equations,” J. Comput. Phys. 168, 286–315 (2001).
    [CrossRef]
  21. J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
    [CrossRef]
  22. M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999).
  23. X. Wang, B. Wang, J. J. Pouch, F. A. Miranda, M. Fisch, J. E. Anderson, V. Sergan, P. J. Bos, “Performance evaluation of liquid crystal on silicon (LCOS) spatial light modulator,” Opt. Eng. (Bellingham) 43, 2769–2774 (2004).
    [CrossRef]
  24. X. Wang, B. Wang, J. J. Pouch, F. A. Miranda, J. E. Anderson, P. J. Bos, “Liquid crystal on silicon (LCOS) wavefront corrector and beam steerer,” in Advanced Wavefront Control: Methods, Devices, and Applications, J. D. Gonglewski, M. A. Vorontsov, M. T. Gruneisen, eds., Proc. SPIE5162, 139–146 (2003).
    [CrossRef]
  25. K. Creath, “Choosing a phase-measurement algorithm for measurement of coated LIGO optics,” in Laser Interferometry X: Techniques and Analysis, M. Kujawinska, R. J. Pryputniewicz, M. Takeda, eds., Proc. SPIE4101, 46–55 (2000).
    [CrossRef]
  26. B. Apter, U. Efron, E. Bahat-Treidel, “On the fringing-field effect in liquid-crystal beam-steering devices,” Appl. Opt. 43, 11–19 (2004).
    [CrossRef] [PubMed]

2004 (3)

G. F. Barrick, P. J. Bos, C. E. Titus, B. Winker, “Computing the liquid crystal director field in optical phased arrays,” Opt. Eng. (Bellingham) 43, 924–932 (2004).
[CrossRef]

X. Wang, B. Wang, J. J. Pouch, F. A. Miranda, M. Fisch, J. E. Anderson, V. Sergan, P. J. Bos, “Performance evaluation of liquid crystal on silicon (LCOS) spatial light modulator,” Opt. Eng. (Bellingham) 43, 2769–2774 (2004).
[CrossRef]

B. Apter, U. Efron, E. Bahat-Treidel, “On the fringing-field effect in liquid-crystal beam-steering devices,” Appl. Opt. 43, 11–19 (2004).
[CrossRef] [PubMed]

2001 (2)

A. Yefet, P. G. Petropoulos, “A staggered fourth-order accuracy explicit finite difference scheme for the time-domain Maxwell’s equations,” J. Comput. Phys. 168, 286–315 (2001).
[CrossRef]

C. M. Titus, J. R. Kelly, E. C. Gartland, S. V. Shiyanovskii, J. A. Anderson, P. J. Bos, “Asymmetric transmissive behavior of liquid-crystal diffraction gratings,” Opt. Lett. 26, 1188–1190 (2001).
[CrossRef]

2000 (2)

X. Wang, D. Wilson, R. Muller, P. Maker, D. Psaltis, “Liquid-crystal blazed-grating beam deflector,” Appl. Opt. 39, 6545–6555 (2000).
[CrossRef]

E. E. Kriezis, S. J. Elston, “Light wave propagation in liquid crystal display by the 2-D finite-difference time-domain method,” Opt. Commun. 177, 69–77 (2000).
[CrossRef]

1999 (1)

C. M. Titus, P. J. Bos, J. R. Kelly, E. C. Gartland, “Comparison of analytical calculations to finite-difference time-domain simulations of one-dimensional spatially varying anisotropic liquid crystal structures,” Jpn. J. Appl. Phys., Part 1 38, 1488–1494 (1999).
[CrossRef]

1997 (1)

1996 (4)

P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holz, S. Liberman, H. Q. Nguyen, D. P. Resler, R. C. Sharp, E. A. Watson, “Optical phased array technology,” Proc. IEEE 84, 268–298 (1996).
[CrossRef]

D. P. Resler, D. S. Hobbs, R. C. Sharp, L. J. Friedman, T. A. Dorschner, “High-efficiency liquid-crystal optical phased-array beam steering,” Opt. Lett. 21, 689–691 (1996).
[CrossRef] [PubMed]

L. J. Friedman, D. S. Hobbs, S. Lieberman, D. L. Corkum, H. W. Nguyen, R. C. Sharp, T. A. Dorschner, “Spatially resolved phase imaging of a programmable liquid-crystal grating,” Appl. Opt. 35, 6236–6240 (1996).
[CrossRef] [PubMed]

V. G. Dominic, E. A. Watson, “Measurement and modeling of the angular dispersion in liquid crystal broadband beam steering devices,” Opt. Eng. (Bellingham) 35, 3371–3379 (1996).
[CrossRef]

1994 (1)

J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

1975 (1)

L. M. Blinov, “Electro-optical effects in liquid crystal,” Sov. Phys. Usp. 17, 658–672 (1975).
[CrossRef]

1966 (1)

K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. AP-14, 302–307 (1966).

Anderson, J.

J. Anderson, P. Watson, P. Bos, Liquid Crystal Display 3-D Director Simulator Software and Technical Guide (Artech House, Norwood, Mass., 2001).

Anderson, J. A.

Anderson, J. E.

X. Wang, B. Wang, J. J. Pouch, F. A. Miranda, M. Fisch, J. E. Anderson, V. Sergan, P. J. Bos, “Performance evaluation of liquid crystal on silicon (LCOS) spatial light modulator,” Opt. Eng. (Bellingham) 43, 2769–2774 (2004).
[CrossRef]

X. Wang, B. Wang, J. J. Pouch, F. A. Miranda, J. E. Anderson, P. J. Bos, “Liquid crystal on silicon (LCOS) wavefront corrector and beam steerer,” in Advanced Wavefront Control: Methods, Devices, and Applications, J. D. Gonglewski, M. A. Vorontsov, M. T. Gruneisen, eds., Proc. SPIE5162, 139–146 (2003).
[CrossRef]

Apter, B.

Bahat-Treidel, E.

Baker, J. T.

M. T. Gruneisen, T. Martinez, D. V. Wick, J. M. Wilkes, J. T. Baker, I. Percheron, “Holographic compensation of severe dynamic aberrations in membrane-mirror based telescope systems,” in High-Resolution Wavefront Control: Methods, Devices, and Applications, J. D. Gonglewski, M. A. Vorontsov, eds., Proc. SPIE3760, 142–152 (1999).
[CrossRef]

Barrick, G. F.

G. F. Barrick, P. J. Bos, C. E. Titus, B. Winker, “Computing the liquid crystal director field in optical phased arrays,” Opt. Eng. (Bellingham) 43, 924–932 (2004).
[CrossRef]

Berenger, J.-P.

J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

Blinov, L. M.

L. M. Blinov, “Electro-optical effects in liquid crystal,” Sov. Phys. Usp. 17, 658–672 (1975).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999).

Bos, P.

J. Anderson, P. Watson, P. Bos, Liquid Crystal Display 3-D Director Simulator Software and Technical Guide (Artech House, Norwood, Mass., 2001).

Bos, P. J.

G. F. Barrick, P. J. Bos, C. E. Titus, B. Winker, “Computing the liquid crystal director field in optical phased arrays,” Opt. Eng. (Bellingham) 43, 924–932 (2004).
[CrossRef]

X. Wang, B. Wang, J. J. Pouch, F. A. Miranda, M. Fisch, J. E. Anderson, V. Sergan, P. J. Bos, “Performance evaluation of liquid crystal on silicon (LCOS) spatial light modulator,” Opt. Eng. (Bellingham) 43, 2769–2774 (2004).
[CrossRef]

C. M. Titus, J. R. Kelly, E. C. Gartland, S. V. Shiyanovskii, J. A. Anderson, P. J. Bos, “Asymmetric transmissive behavior of liquid-crystal diffraction gratings,” Opt. Lett. 26, 1188–1190 (2001).
[CrossRef]

C. M. Titus, P. J. Bos, J. R. Kelly, E. C. Gartland, “Comparison of analytical calculations to finite-difference time-domain simulations of one-dimensional spatially varying anisotropic liquid crystal structures,” Jpn. J. Appl. Phys., Part 1 38, 1488–1494 (1999).
[CrossRef]

Bin Wang, Xinghua Wang, P. J. Bos, “Study of switchable liquid crystal polymer grating by finite-difference time-domain calculation,” in Liquid Crystals VII, I.-C. Khoo, ed., Proc. SPIE5213, 104–110 (2003).
[CrossRef]

X. Wang, B. Wang, J. J. Pouch, F. A. Miranda, J. E. Anderson, P. J. Bos, “Liquid crystal on silicon (LCOS) wavefront corrector and beam steerer,” in Advanced Wavefront Control: Methods, Devices, and Applications, J. D. Gonglewski, M. A. Vorontsov, M. T. Gruneisen, eds., Proc. SPIE5162, 139–146 (2003).
[CrossRef]

Corkum, D. L.

P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holz, S. Liberman, H. Q. Nguyen, D. P. Resler, R. C. Sharp, E. A. Watson, “Optical phased array technology,” Proc. IEEE 84, 268–298 (1996).
[CrossRef]

L. J. Friedman, D. S. Hobbs, S. Lieberman, D. L. Corkum, H. W. Nguyen, R. C. Sharp, T. A. Dorschner, “Spatially resolved phase imaging of a programmable liquid-crystal grating,” Appl. Opt. 35, 6236–6240 (1996).
[CrossRef] [PubMed]

Creath, K.

K. Creath, “Choosing a phase-measurement algorithm for measurement of coated LIGO optics,” in Laser Interferometry X: Techniques and Analysis, M. Kujawinska, R. J. Pryputniewicz, M. Takeda, eds., Proc. SPIE4101, 46–55 (2000).
[CrossRef]

De Gennes, P. G.

P. G. De Gennes, J. Prost, The Physics of Liquid Crystals (Oxford U. Press, Oxford, UK, 1993).

Dominic, V. G.

V. G. Dominic, E. A. Watson, “Measurement and modeling of the angular dispersion in liquid crystal broadband beam steering devices,” Opt. Eng. (Bellingham) 35, 3371–3379 (1996).
[CrossRef]

Dorschner, T. A.

Efron, U.

Elston, S. J.

E. E. Kriezis, S. J. Elston, “Light wave propagation in liquid crystal display by the 2-D finite-difference time-domain method,” Opt. Commun. 177, 69–77 (2000).
[CrossRef]

Fisch, M.

X. Wang, B. Wang, J. J. Pouch, F. A. Miranda, M. Fisch, J. E. Anderson, V. Sergan, P. J. Bos, “Performance evaluation of liquid crystal on silicon (LCOS) spatial light modulator,” Opt. Eng. (Bellingham) 43, 2769–2774 (2004).
[CrossRef]

Friedman, L. J.

Gartland, E. C.

C. M. Titus, J. R. Kelly, E. C. Gartland, S. V. Shiyanovskii, J. A. Anderson, P. J. Bos, “Asymmetric transmissive behavior of liquid-crystal diffraction gratings,” Opt. Lett. 26, 1188–1190 (2001).
[CrossRef]

C. M. Titus, P. J. Bos, J. R. Kelly, E. C. Gartland, “Comparison of analytical calculations to finite-difference time-domain simulations of one-dimensional spatially varying anisotropic liquid crystal structures,” Jpn. J. Appl. Phys., Part 1 38, 1488–1494 (1999).
[CrossRef]

Gruneisen, M. T.

M. T. Gruneisen, J. M. Wilkes, “Compensated imaging by real-time holography with optically addressed spatial light modulators,” in Spatial Light Modulators, G. Burdge, S. C. Esener, eds., Vol. 14 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1997).

M. T. Gruneisen, T. Martinez, D. V. Wick, J. M. Wilkes, J. T. Baker, I. Percheron, “Holographic compensation of severe dynamic aberrations in membrane-mirror based telescope systems,” in High-Resolution Wavefront Control: Methods, Devices, and Applications, J. D. Gonglewski, M. A. Vorontsov, eds., Proc. SPIE3760, 142–152 (1999).
[CrossRef]

Hagness, S. C.

A. Taflove, S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 2nd ed. (Artech House, Norwood, Mass., 2000).

Hobbs, D. S.

Holz, M.

P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holz, S. Liberman, H. Q. Nguyen, D. P. Resler, R. C. Sharp, E. A. Watson, “Optical phased array technology,” Proc. IEEE 84, 268–298 (1996).
[CrossRef]

Kelly, J. R.

C. M. Titus, J. R. Kelly, E. C. Gartland, S. V. Shiyanovskii, J. A. Anderson, P. J. Bos, “Asymmetric transmissive behavior of liquid-crystal diffraction gratings,” Opt. Lett. 26, 1188–1190 (2001).
[CrossRef]

C. M. Titus, P. J. Bos, J. R. Kelly, E. C. Gartland, “Comparison of analytical calculations to finite-difference time-domain simulations of one-dimensional spatially varying anisotropic liquid crystal structures,” Jpn. J. Appl. Phys., Part 1 38, 1488–1494 (1999).
[CrossRef]

Kriezis, E. E.

E. E. Kriezis, S. J. Elston, “Light wave propagation in liquid crystal display by the 2-D finite-difference time-domain method,” Opt. Commun. 177, 69–77 (2000).
[CrossRef]

Liberman, S.

P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holz, S. Liberman, H. Q. Nguyen, D. P. Resler, R. C. Sharp, E. A. Watson, “Optical phased array technology,” Proc. IEEE 84, 268–298 (1996).
[CrossRef]

Lieberman, S.

Love, G. D.

Maker, P.

Martinez, T.

M. T. Gruneisen, T. Martinez, D. V. Wick, J. M. Wilkes, J. T. Baker, I. Percheron, “Holographic compensation of severe dynamic aberrations in membrane-mirror based telescope systems,” in High-Resolution Wavefront Control: Methods, Devices, and Applications, J. D. Gonglewski, M. A. Vorontsov, eds., Proc. SPIE3760, 142–152 (1999).
[CrossRef]

Matic, R. M.

R. M. Matic, “Blazed phase liquid crystal beam steering,” in Laser Beam Propagation and Control, H. Weichel, L. F. DeSandre, eds., Proc. SPIE2120, 194–205 (1994).
[CrossRef]

McManamon, P. F.

P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holz, S. Liberman, H. Q. Nguyen, D. P. Resler, R. C. Sharp, E. A. Watson, “Optical phased array technology,” Proc. IEEE 84, 268–298 (1996).
[CrossRef]

Miranda, F. A.

X. Wang, B. Wang, J. J. Pouch, F. A. Miranda, M. Fisch, J. E. Anderson, V. Sergan, P. J. Bos, “Performance evaluation of liquid crystal on silicon (LCOS) spatial light modulator,” Opt. Eng. (Bellingham) 43, 2769–2774 (2004).
[CrossRef]

X. Wang, B. Wang, J. J. Pouch, F. A. Miranda, J. E. Anderson, P. J. Bos, “Liquid crystal on silicon (LCOS) wavefront corrector and beam steerer,” in Advanced Wavefront Control: Methods, Devices, and Applications, J. D. Gonglewski, M. A. Vorontsov, M. T. Gruneisen, eds., Proc. SPIE5162, 139–146 (2003).
[CrossRef]

Muller, R.

Nguyen, H. Q.

P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holz, S. Liberman, H. Q. Nguyen, D. P. Resler, R. C. Sharp, E. A. Watson, “Optical phased array technology,” Proc. IEEE 84, 268–298 (1996).
[CrossRef]

Nguyen, H. W.

Percheron, I.

M. T. Gruneisen, T. Martinez, D. V. Wick, J. M. Wilkes, J. T. Baker, I. Percheron, “Holographic compensation of severe dynamic aberrations in membrane-mirror based telescope systems,” in High-Resolution Wavefront Control: Methods, Devices, and Applications, J. D. Gonglewski, M. A. Vorontsov, eds., Proc. SPIE3760, 142–152 (1999).
[CrossRef]

Petropoulos, P. G.

A. Yefet, P. G. Petropoulos, “A staggered fourth-order accuracy explicit finite difference scheme for the time-domain Maxwell’s equations,” J. Comput. Phys. 168, 286–315 (2001).
[CrossRef]

Pouch, J. J.

X. Wang, B. Wang, J. J. Pouch, F. A. Miranda, M. Fisch, J. E. Anderson, V. Sergan, P. J. Bos, “Performance evaluation of liquid crystal on silicon (LCOS) spatial light modulator,” Opt. Eng. (Bellingham) 43, 2769–2774 (2004).
[CrossRef]

X. Wang, B. Wang, J. J. Pouch, F. A. Miranda, J. E. Anderson, P. J. Bos, “Liquid crystal on silicon (LCOS) wavefront corrector and beam steerer,” in Advanced Wavefront Control: Methods, Devices, and Applications, J. D. Gonglewski, M. A. Vorontsov, M. T. Gruneisen, eds., Proc. SPIE5162, 139–146 (2003).
[CrossRef]

Prost, J.

P. G. De Gennes, J. Prost, The Physics of Liquid Crystals (Oxford U. Press, Oxford, UK, 1993).

Psaltis, D.

Resler, D. P.

P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holz, S. Liberman, H. Q. Nguyen, D. P. Resler, R. C. Sharp, E. A. Watson, “Optical phased array technology,” Proc. IEEE 84, 268–298 (1996).
[CrossRef]

D. P. Resler, D. S. Hobbs, R. C. Sharp, L. J. Friedman, T. A. Dorschner, “High-efficiency liquid-crystal optical phased-array beam steering,” Opt. Lett. 21, 689–691 (1996).
[CrossRef] [PubMed]

Sergan, V.

X. Wang, B. Wang, J. J. Pouch, F. A. Miranda, M. Fisch, J. E. Anderson, V. Sergan, P. J. Bos, “Performance evaluation of liquid crystal on silicon (LCOS) spatial light modulator,” Opt. Eng. (Bellingham) 43, 2769–2774 (2004).
[CrossRef]

Sharp, R. C.

Shiyanovskii, S. V.

Taflove, A.

A. Taflove, S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 2nd ed. (Artech House, Norwood, Mass., 2000).

Titus, C. E.

G. F. Barrick, P. J. Bos, C. E. Titus, B. Winker, “Computing the liquid crystal director field in optical phased arrays,” Opt. Eng. (Bellingham) 43, 924–932 (2004).
[CrossRef]

Titus, C. M.

C. M. Titus, J. R. Kelly, E. C. Gartland, S. V. Shiyanovskii, J. A. Anderson, P. J. Bos, “Asymmetric transmissive behavior of liquid-crystal diffraction gratings,” Opt. Lett. 26, 1188–1190 (2001).
[CrossRef]

C. M. Titus, P. J. Bos, J. R. Kelly, E. C. Gartland, “Comparison of analytical calculations to finite-difference time-domain simulations of one-dimensional spatially varying anisotropic liquid crystal structures,” Jpn. J. Appl. Phys., Part 1 38, 1488–1494 (1999).
[CrossRef]

Wang, B.

X. Wang, B. Wang, J. J. Pouch, F. A. Miranda, M. Fisch, J. E. Anderson, V. Sergan, P. J. Bos, “Performance evaluation of liquid crystal on silicon (LCOS) spatial light modulator,” Opt. Eng. (Bellingham) 43, 2769–2774 (2004).
[CrossRef]

X. Wang, B. Wang, J. J. Pouch, F. A. Miranda, J. E. Anderson, P. J. Bos, “Liquid crystal on silicon (LCOS) wavefront corrector and beam steerer,” in Advanced Wavefront Control: Methods, Devices, and Applications, J. D. Gonglewski, M. A. Vorontsov, M. T. Gruneisen, eds., Proc. SPIE5162, 139–146 (2003).
[CrossRef]

Wang, Bin

Bin Wang, Xinghua Wang, P. J. Bos, “Study of switchable liquid crystal polymer grating by finite-difference time-domain calculation,” in Liquid Crystals VII, I.-C. Khoo, ed., Proc. SPIE5213, 104–110 (2003).
[CrossRef]

Wang, X.

X. Wang, B. Wang, J. J. Pouch, F. A. Miranda, M. Fisch, J. E. Anderson, V. Sergan, P. J. Bos, “Performance evaluation of liquid crystal on silicon (LCOS) spatial light modulator,” Opt. Eng. (Bellingham) 43, 2769–2774 (2004).
[CrossRef]

X. Wang, D. Wilson, R. Muller, P. Maker, D. Psaltis, “Liquid-crystal blazed-grating beam deflector,” Appl. Opt. 39, 6545–6555 (2000).
[CrossRef]

X. Wang, B. Wang, J. J. Pouch, F. A. Miranda, J. E. Anderson, P. J. Bos, “Liquid crystal on silicon (LCOS) wavefront corrector and beam steerer,” in Advanced Wavefront Control: Methods, Devices, and Applications, J. D. Gonglewski, M. A. Vorontsov, M. T. Gruneisen, eds., Proc. SPIE5162, 139–146 (2003).
[CrossRef]

Wang, Xinghua

Bin Wang, Xinghua Wang, P. J. Bos, “Study of switchable liquid crystal polymer grating by finite-difference time-domain calculation,” in Liquid Crystals VII, I.-C. Khoo, ed., Proc. SPIE5213, 104–110 (2003).
[CrossRef]

Watson, E. A.

V. G. Dominic, E. A. Watson, “Measurement and modeling of the angular dispersion in liquid crystal broadband beam steering devices,” Opt. Eng. (Bellingham) 35, 3371–3379 (1996).
[CrossRef]

P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holz, S. Liberman, H. Q. Nguyen, D. P. Resler, R. C. Sharp, E. A. Watson, “Optical phased array technology,” Proc. IEEE 84, 268–298 (1996).
[CrossRef]

Watson, P.

J. Anderson, P. Watson, P. Bos, Liquid Crystal Display 3-D Director Simulator Software and Technical Guide (Artech House, Norwood, Mass., 2001).

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M. T. Gruneisen, T. Martinez, D. V. Wick, J. M. Wilkes, J. T. Baker, I. Percheron, “Holographic compensation of severe dynamic aberrations in membrane-mirror based telescope systems,” in High-Resolution Wavefront Control: Methods, Devices, and Applications, J. D. Gonglewski, M. A. Vorontsov, eds., Proc. SPIE3760, 142–152 (1999).
[CrossRef]

Wilkes, J. M.

M. T. Gruneisen, T. Martinez, D. V. Wick, J. M. Wilkes, J. T. Baker, I. Percheron, “Holographic compensation of severe dynamic aberrations in membrane-mirror based telescope systems,” in High-Resolution Wavefront Control: Methods, Devices, and Applications, J. D. Gonglewski, M. A. Vorontsov, eds., Proc. SPIE3760, 142–152 (1999).
[CrossRef]

M. T. Gruneisen, J. M. Wilkes, “Compensated imaging by real-time holography with optically addressed spatial light modulators,” in Spatial Light Modulators, G. Burdge, S. C. Esener, eds., Vol. 14 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1997).

Wilson, D.

Winker, B.

G. F. Barrick, P. J. Bos, C. E. Titus, B. Winker, “Computing the liquid crystal director field in optical phased arrays,” Opt. Eng. (Bellingham) 43, 924–932 (2004).
[CrossRef]

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Appl. Opt. (4)

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K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. AP-14, 302–307 (1966).

J. Comput. Phys. (2)

A. Yefet, P. G. Petropoulos, “A staggered fourth-order accuracy explicit finite difference scheme for the time-domain Maxwell’s equations,” J. Comput. Phys. 168, 286–315 (2001).
[CrossRef]

J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
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C. M. Titus, P. J. Bos, J. R. Kelly, E. C. Gartland, “Comparison of analytical calculations to finite-difference time-domain simulations of one-dimensional spatially varying anisotropic liquid crystal structures,” Jpn. J. Appl. Phys., Part 1 38, 1488–1494 (1999).
[CrossRef]

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E. E. Kriezis, S. J. Elston, “Light wave propagation in liquid crystal display by the 2-D finite-difference time-domain method,” Opt. Commun. 177, 69–77 (2000).
[CrossRef]

Opt. Eng. (Bellingham) (3)

X. Wang, B. Wang, J. J. Pouch, F. A. Miranda, M. Fisch, J. E. Anderson, V. Sergan, P. J. Bos, “Performance evaluation of liquid crystal on silicon (LCOS) spatial light modulator,” Opt. Eng. (Bellingham) 43, 2769–2774 (2004).
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X. Wang, B. Wang, J. J. Pouch, F. A. Miranda, J. E. Anderson, P. J. Bos, “Liquid crystal on silicon (LCOS) wavefront corrector and beam steerer,” in Advanced Wavefront Control: Methods, Devices, and Applications, J. D. Gonglewski, M. A. Vorontsov, M. T. Gruneisen, eds., Proc. SPIE5162, 139–146 (2003).
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Figures (10)

Fig. 1
Fig. 1

Electro-optical (E-O) response of the LCOS device and the simplified voltage/phase pair for each of eight electrodes. The LC material used here is MLC6080, and K11=14.1×10-12 N (N stands for Newton, the unit of force), K22=7.1×10-12 N, K33=19.1×10-12 N, =11.10, =3.90, d=6 µm, no=1.5076, ne=1.7100, and q0=0.

Fig. 2
Fig. 2

Simulated director configuration in one period of an LC blazed grating for an eight-electrode/reset scheme. Electrode size is 19.0 µm, interpixel gap is 0.4 µm, and cell thickness is d=4.0 µm.

Fig. 3
Fig. 3

Simulated phase profile of an 8 pixel per reset maximum steering grating versus an ideal eight-step stairlike blazed grating for the visible version of LCOS. The elastic constant of the LC material is the same as that in Fig. 1. The dielectric constants are =12.10 and =4.10; d=4 µm, no=1.5035, ne=1.6742, the pixel size is 19 µm, the interpixel gap is 0.4 µm, and q0=0.

Fig. 4
Fig. 4

Near-field intensity distribution at the final time step (t=6.59×10-14 s) of beam propagation in a reflective LC blazed grating obtained by 2-D FDTD simulation. λ=632.8 nm, grid spacing Δx=Δy=1/20λ, and Δt=Δx/2c, where c is the speed of light. The input beam is a plane wave with Guassian beam intensity distribution with 1/e2 diameter=140 µm.

Fig. 5
Fig. 5

Far-field diffraction peak simulated by near-field-to-far-field transformation of the FDTD simulated near field. The far-field distance is 100 km. The inset is the magnified view of the first-order diffraction peak.

Fig. 6
Fig. 6

One frame of an interferogram of a beam reflected from LCOS: (a) noncompensated beam with aberration present, (b) compensated beam with aberration removed, (c) compensated beam with aberration removed and linear tip–tilt added to the phase ramp on LCOS to steer the beam to 0.6 mrad.

Fig. 7
Fig. 7

Far-field point-spread function measured at the focus of a low-numerical-aperture lens: (a) wave-front compensation off, (b) wave-front compensation on, (c) wave-front compensation+maximum steered beam.

Fig. 8
Fig. 8

Far-field beam profile: (a) aberration-free laser beam with intensity reduced by 80% to take into account the reflectivity loss of LCOS, (b) wave-front compensated but nonsteered laser beam with Strehl ratio=0.82 using (a) as a reference, (c) wave-front compensated and maximum steered beam with Strehl ratio=0.846 using (b) as a reference.

Fig. 9
Fig. 9

Comparison of DE as a function of steering angle for experimental results, FDTD simulation, and simple model.

Fig. 10
Fig. 10

Phase profile and corresponding far field of an LC blazed grating. The elastic constant of the LC material is the same as that in Fig. 1. The dielectric constants are =12.10 and =4.10, d=4 µm, no=1.5035, ne=1.6742, the pixel size is 19 µm, the interpixel gap is 0.4 µm, and q0=0. The steering angle is 2.04 mrad for (a) and (d), 8.16 mrad for (b) and (e), and 12.24 mrad for (c) and (f).

Tables (2)

Tables Icon

Table 1 Zernike Modes of the Transmitted Wave-Front Aberration Introduced by Surface Deformation of the Silicon Backplane

Tables Icon

Table 2 Comparison of Diffraction Efficiency and Steering Angle for Simple Model, Simulation, and Measurement at Maximum Steering Angle

Equations (10)

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

fg=12{K11(n)2+K22[n(×n)-q0]2+K33[n×(×n)]2-DE}.
D=0.
nit+1=nit-Δtγ1[fg]ni,i=x, y, z.
[fg]ni=fgni-ddxfg(dni/dx)-ddyfg(dni/dy).
E(r)t=-1(r)[×H(r)],
-H(r)t=μ0-1(r)[×E(r)].
Ψfar(r)=14πSn[Ψnear(r)G-GΨnear(r)]ds,
Ψfar(xfar, yfar)=exp-[(iπ)/4]8πk -ddexp(-ikR)R×yΨnear(x, y0)+ik(yfar-y0)Ψnear(x, y0)Rdx,
η=sin(π/q)π/q2,
ηtotal=sin(π/q)π/q21-ΛFΛ2,

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