Abstract

The analog switching mode in ferroelectric liquid crystals, sometimes referred to as “V-shaped switching,” has, thanks to its submillisecond switching capability, attracted much interest for future fast electro-optic displays where it is to be used for amplitude modulation. We have studied this mode for analog phase-only modulation. As V-shaped switching is based on a conical motion of the index ellipsoid this presents a challenging problem since both the orientation of the slow and fast axes, as well as the amount of birefringence varies in the switching process. We show theoretically, partly by means of Poincaré sphere analysis, that it is in fact possible to obtain near-lossless analog phase modulation between zero and π radians in an ideal V-shaped switching cell through careful tuning of the polarization state of the input light. Furthermore, we were able to demonstrate this experimentally in a fabricated cell. Although this cell deviated slightly from the ideal conditions, e.g., the tilt cone half-angle was 38° instead of the desired 45°, we still obtained a continuous phase modulation between zero and 0.78π rad with less than 2% modulation of the amplitude; the measured values agree very well with our numerical simulations of the real device.

© 2006 Optical Society of America

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  1. N. A. Clark and S. T. Lagerwall, "Submicrosecond bistable electro-optic switching in liquid crystals," Appl. Phys. Lett. 36, 899-901 (1980).
    [CrossRef]
  2. M. O. Freeman, T. A. Brown, and D. M. Walba, "Quantized complex ferroelectric liquid crystal spatial light modulators," Appl. Opt. 31, 3917-3929 (1992).
    [CrossRef] [PubMed]
  3. S. E. Broomfield, M. A. A. Neil, and E. G. S. Paige, "Programmable multiple-level phase modulation that uses ferroelectric liquid-crystal spatial light modulators," Appl. Opt. 34, 6652-6665 (1995).
    [CrossRef] [PubMed]
  4. D. Engström, S. Hård, P. Rudquist, K. D'havé, T. Matuszczyk, M. Skeren, and B. Löfving, "Beam steering experiment with two cascaded ferroelectric liquid-crystal spatial light modulators," Appl. Opt. 43, 1559-1569 (2004).
    [CrossRef] [PubMed]
  5. K. D'havé, P. Rudquist, M. Matuszczyk, S. T. Lagerwall, H. Pauwels, and R. S. Dabrowski, "Antiferroelectric liquid crystals with 45 degrees tilt: new electro-optic effects in liquid crystals," in Liquid Crystal Materials, Devices, and Flat Panel Displays, R. Shashidhar and B. Gnade, eds., Proc. SPIE 3955,33-44 (2000).
  6. S. Garoff and R. B. Meyer, "Electroclinic effect at the A-C phase change in a chiral smectic liquid crystal," Phys. Rev. Lett. 38, 848-851 (1977).
    [CrossRef]
  7. G. Andersson, I. Dahl, L. Komitov, S. T. Lagerwall, K. Skarp, and B. Stebler, "Device physics of the soft-mode electro-optic effect," J. Appl. Phys. 66, 4983-4995 (1989).
    [CrossRef]
  8. J. E. Stockley, "Chiral smectic liquid crystal beam deflector with large analog phase modulation," Ph.D. dissertation (University of Colorado, 1996).
  9. J. E. Stockley, G. D. Sharp, S. A. Serati, and K. M. Johnson, "Analog optical phase modulator based on chiral smectic and polymer cholesteric liquid crystals," Opt. Lett. 20, 2441-2443 (1995).
    [CrossRef] [PubMed]
  10. J. Stockley, X. Xia, T. Ewing, and S. Serati, "Liquid crystal optical phase modulators for beam steering," in Advances in Liquid Crystalline Materials and Technologies (Materials Research Society, 2002), Vol. 709, pp. 55-66.
  11. A. Fukuda, "Pretransitional effect in AF-F switching: to suppress it or to enhance it, that is my question about AFLCDs," Asia Display'95: Proceedings of the 15th International Display Research Conference (1995), pp. 61-64.
  12. S. Inui, N. Iimura, T. Suzuki, H. Wane, K. Miyachi, Y. Takanishi, and A. Fukuda, "Threshold-less antiferroelectricity in liquid crystals and its application to displays," J. Mater. Chem. 6, 671-673 (1996).
    [CrossRef]
  13. P. Rudquist, J. P. F. Lagerwall, M. Buivydas, F. Gouda, S. T. Lagerwall, N. A. Clark, J. E. Maclennan, R. Shao, D. A. Coleman, S. Bardon, T. Bellini, D. R. Link, G. Natale, M. A. Glaser, D. M. Walba, M. D. Wand, and X.-H. Chen, "The case of thresholdless antiferroelectricity: polarization-stabilized twisted SmC* liquid crystals give V-shaped electro-optic response," J. Mater. Chem. 9, 1257-1261 (1999).
    [CrossRef]
  14. N. A. Clark, D. Coleman, and J. E. Maclennan, "Electrostatics and the behaviour of chiral smectic C: 'block' polarization screening of applied voltage and 'V-shaped' switching," Liq. Crys. 20, 985-990 (2000).
    [CrossRef]
  15. R. B. Meyer, L. Liebert, L. Strzelecki, and P. Keller, "Ferroelectric liquid crystals," J. Phys. (Paris), Lett. 36, 69-71 (1975).
  16. M. Copic, J. E. Maclennan, and N. A. Clark, "Structure and dynamics of ferroelectric liquid crystal cells exhibiting thresholdless switching," Phys. Rev. E 65, 021708 (2002).
  17. M. J. O'Callaghan, "Switching dynamics and surface forces in thresholdless 'V-shaped' switching ferroelectric liquid crystal," Phys. Rev. E 67, 011710 (2003).
  18. D. Coleman, D. Mueller, N. A. Clark, J. E. Maclennan, R.-F. Shao, S. Bardon, and D. M. Walba, "Control of molecular orientation in electrostatically stabilized ferroelectric liquid crystals," Phys. Rev. Lett. 91, 175505 (2003).
    [CrossRef] [PubMed]
  19. M. J. O'Callaghan, M. D. Wand, C. M. Walker, and M. Nakata, "Charge controlled, fixed optic axis analog ('V-shaped') switching of a bent core ferroelectric liquid crystal," Appl. Phys. Lett. 85, 6344-6346 (2004).
    [CrossRef]
  20. A. Hammarquist, K. D'havé, M. Matuszczyk, and P. Rudquist "V-shaped ferroelectric liquid crystal structure stabilized by dielectric surface layers," Europhys. Lett. (to be published).
  21. See, for example, D. Goldstein, Polarized Light (Dekker, 2003), pp. 241-273.
  22. P. Yeh and C. Gu, Optics of Liquid Crystal Displays(Wiley, 1999), pp. 103-152.
  23. J. Nicolás, J. Campos, and M. J. Yzuel, "Phase and amplitude modulation of elliptic polarization states by nonabsorbing anisotropic elements: application to liquid-crystal devices," J. Opt. Soc. Am. A 19, 1013-1020 (2002).
    [CrossRef]
  24. The cells used in this work were made in house in the Nanofabrication Laboratory at the Department of Microtechnology and Nanoscience at Chalmers University of Technology.
  25. T. Rieker, N. A. Clark, G. S. Smith, D. S. Parmar, and E. B. Sirota, "'Chevron' local layer structure in surface-stabilized ferroelectric smectic-C cells," Phys. Rev. Lett. 59, 2658-2661 (1987).
    [CrossRef] [PubMed]

2004

M. J. O'Callaghan, M. D. Wand, C. M. Walker, and M. Nakata, "Charge controlled, fixed optic axis analog ('V-shaped') switching of a bent core ferroelectric liquid crystal," Appl. Phys. Lett. 85, 6344-6346 (2004).
[CrossRef]

D. Engström, S. Hård, P. Rudquist, K. D'havé, T. Matuszczyk, M. Skeren, and B. Löfving, "Beam steering experiment with two cascaded ferroelectric liquid-crystal spatial light modulators," Appl. Opt. 43, 1559-1569 (2004).
[CrossRef] [PubMed]

2003

M. J. O'Callaghan, "Switching dynamics and surface forces in thresholdless 'V-shaped' switching ferroelectric liquid crystal," Phys. Rev. E 67, 011710 (2003).

D. Coleman, D. Mueller, N. A. Clark, J. E. Maclennan, R.-F. Shao, S. Bardon, and D. M. Walba, "Control of molecular orientation in electrostatically stabilized ferroelectric liquid crystals," Phys. Rev. Lett. 91, 175505 (2003).
[CrossRef] [PubMed]

2002

J. Nicolás, J. Campos, and M. J. Yzuel, "Phase and amplitude modulation of elliptic polarization states by nonabsorbing anisotropic elements: application to liquid-crystal devices," J. Opt. Soc. Am. A 19, 1013-1020 (2002).
[CrossRef]

M. Copic, J. E. Maclennan, and N. A. Clark, "Structure and dynamics of ferroelectric liquid crystal cells exhibiting thresholdless switching," Phys. Rev. E 65, 021708 (2002).

2000

N. A. Clark, D. Coleman, and J. E. Maclennan, "Electrostatics and the behaviour of chiral smectic C: 'block' polarization screening of applied voltage and 'V-shaped' switching," Liq. Crys. 20, 985-990 (2000).
[CrossRef]

1999

P. Rudquist, J. P. F. Lagerwall, M. Buivydas, F. Gouda, S. T. Lagerwall, N. A. Clark, J. E. Maclennan, R. Shao, D. A. Coleman, S. Bardon, T. Bellini, D. R. Link, G. Natale, M. A. Glaser, D. M. Walba, M. D. Wand, and X.-H. Chen, "The case of thresholdless antiferroelectricity: polarization-stabilized twisted SmC* liquid crystals give V-shaped electro-optic response," J. Mater. Chem. 9, 1257-1261 (1999).
[CrossRef]

1996

S. Inui, N. Iimura, T. Suzuki, H. Wane, K. Miyachi, Y. Takanishi, and A. Fukuda, "Threshold-less antiferroelectricity in liquid crystals and its application to displays," J. Mater. Chem. 6, 671-673 (1996).
[CrossRef]

1995

1992

1989

G. Andersson, I. Dahl, L. Komitov, S. T. Lagerwall, K. Skarp, and B. Stebler, "Device physics of the soft-mode electro-optic effect," J. Appl. Phys. 66, 4983-4995 (1989).
[CrossRef]

1987

T. Rieker, N. A. Clark, G. S. Smith, D. S. Parmar, and E. B. Sirota, "'Chevron' local layer structure in surface-stabilized ferroelectric smectic-C cells," Phys. Rev. Lett. 59, 2658-2661 (1987).
[CrossRef] [PubMed]

1980

N. A. Clark and S. T. Lagerwall, "Submicrosecond bistable electro-optic switching in liquid crystals," Appl. Phys. Lett. 36, 899-901 (1980).
[CrossRef]

1977

S. Garoff and R. B. Meyer, "Electroclinic effect at the A-C phase change in a chiral smectic liquid crystal," Phys. Rev. Lett. 38, 848-851 (1977).
[CrossRef]

1975

R. B. Meyer, L. Liebert, L. Strzelecki, and P. Keller, "Ferroelectric liquid crystals," J. Phys. (Paris), Lett. 36, 69-71 (1975).

Andersson, G.

G. Andersson, I. Dahl, L. Komitov, S. T. Lagerwall, K. Skarp, and B. Stebler, "Device physics of the soft-mode electro-optic effect," J. Appl. Phys. 66, 4983-4995 (1989).
[CrossRef]

Bardon, S.

D. Coleman, D. Mueller, N. A. Clark, J. E. Maclennan, R.-F. Shao, S. Bardon, and D. M. Walba, "Control of molecular orientation in electrostatically stabilized ferroelectric liquid crystals," Phys. Rev. Lett. 91, 175505 (2003).
[CrossRef] [PubMed]

P. Rudquist, J. P. F. Lagerwall, M. Buivydas, F. Gouda, S. T. Lagerwall, N. A. Clark, J. E. Maclennan, R. Shao, D. A. Coleman, S. Bardon, T. Bellini, D. R. Link, G. Natale, M. A. Glaser, D. M. Walba, M. D. Wand, and X.-H. Chen, "The case of thresholdless antiferroelectricity: polarization-stabilized twisted SmC* liquid crystals give V-shaped electro-optic response," J. Mater. Chem. 9, 1257-1261 (1999).
[CrossRef]

Bellini, T.

P. Rudquist, J. P. F. Lagerwall, M. Buivydas, F. Gouda, S. T. Lagerwall, N. A. Clark, J. E. Maclennan, R. Shao, D. A. Coleman, S. Bardon, T. Bellini, D. R. Link, G. Natale, M. A. Glaser, D. M. Walba, M. D. Wand, and X.-H. Chen, "The case of thresholdless antiferroelectricity: polarization-stabilized twisted SmC* liquid crystals give V-shaped electro-optic response," J. Mater. Chem. 9, 1257-1261 (1999).
[CrossRef]

Broomfield, S. E.

Brown, T. A.

Buivydas, M.

P. Rudquist, J. P. F. Lagerwall, M. Buivydas, F. Gouda, S. T. Lagerwall, N. A. Clark, J. E. Maclennan, R. Shao, D. A. Coleman, S. Bardon, T. Bellini, D. R. Link, G. Natale, M. A. Glaser, D. M. Walba, M. D. Wand, and X.-H. Chen, "The case of thresholdless antiferroelectricity: polarization-stabilized twisted SmC* liquid crystals give V-shaped electro-optic response," J. Mater. Chem. 9, 1257-1261 (1999).
[CrossRef]

Campos, J.

Chen, X.-H.

P. Rudquist, J. P. F. Lagerwall, M. Buivydas, F. Gouda, S. T. Lagerwall, N. A. Clark, J. E. Maclennan, R. Shao, D. A. Coleman, S. Bardon, T. Bellini, D. R. Link, G. Natale, M. A. Glaser, D. M. Walba, M. D. Wand, and X.-H. Chen, "The case of thresholdless antiferroelectricity: polarization-stabilized twisted SmC* liquid crystals give V-shaped electro-optic response," J. Mater. Chem. 9, 1257-1261 (1999).
[CrossRef]

Clark, N. A.

D. Coleman, D. Mueller, N. A. Clark, J. E. Maclennan, R.-F. Shao, S. Bardon, and D. M. Walba, "Control of molecular orientation in electrostatically stabilized ferroelectric liquid crystals," Phys. Rev. Lett. 91, 175505 (2003).
[CrossRef] [PubMed]

M. Copic, J. E. Maclennan, and N. A. Clark, "Structure and dynamics of ferroelectric liquid crystal cells exhibiting thresholdless switching," Phys. Rev. E 65, 021708 (2002).

N. A. Clark, D. Coleman, and J. E. Maclennan, "Electrostatics and the behaviour of chiral smectic C: 'block' polarization screening of applied voltage and 'V-shaped' switching," Liq. Crys. 20, 985-990 (2000).
[CrossRef]

P. Rudquist, J. P. F. Lagerwall, M. Buivydas, F. Gouda, S. T. Lagerwall, N. A. Clark, J. E. Maclennan, R. Shao, D. A. Coleman, S. Bardon, T. Bellini, D. R. Link, G. Natale, M. A. Glaser, D. M. Walba, M. D. Wand, and X.-H. Chen, "The case of thresholdless antiferroelectricity: polarization-stabilized twisted SmC* liquid crystals give V-shaped electro-optic response," J. Mater. Chem. 9, 1257-1261 (1999).
[CrossRef]

T. Rieker, N. A. Clark, G. S. Smith, D. S. Parmar, and E. B. Sirota, "'Chevron' local layer structure in surface-stabilized ferroelectric smectic-C cells," Phys. Rev. Lett. 59, 2658-2661 (1987).
[CrossRef] [PubMed]

N. A. Clark and S. T. Lagerwall, "Submicrosecond bistable electro-optic switching in liquid crystals," Appl. Phys. Lett. 36, 899-901 (1980).
[CrossRef]

Coleman, D.

D. Coleman, D. Mueller, N. A. Clark, J. E. Maclennan, R.-F. Shao, S. Bardon, and D. M. Walba, "Control of molecular orientation in electrostatically stabilized ferroelectric liquid crystals," Phys. Rev. Lett. 91, 175505 (2003).
[CrossRef] [PubMed]

N. A. Clark, D. Coleman, and J. E. Maclennan, "Electrostatics and the behaviour of chiral smectic C: 'block' polarization screening of applied voltage and 'V-shaped' switching," Liq. Crys. 20, 985-990 (2000).
[CrossRef]

Coleman, D. A.

P. Rudquist, J. P. F. Lagerwall, M. Buivydas, F. Gouda, S. T. Lagerwall, N. A. Clark, J. E. Maclennan, R. Shao, D. A. Coleman, S. Bardon, T. Bellini, D. R. Link, G. Natale, M. A. Glaser, D. M. Walba, M. D. Wand, and X.-H. Chen, "The case of thresholdless antiferroelectricity: polarization-stabilized twisted SmC* liquid crystals give V-shaped electro-optic response," J. Mater. Chem. 9, 1257-1261 (1999).
[CrossRef]

Copic, M.

M. Copic, J. E. Maclennan, and N. A. Clark, "Structure and dynamics of ferroelectric liquid crystal cells exhibiting thresholdless switching," Phys. Rev. E 65, 021708 (2002).

Dabrowski, R. S.

K. D'havé, P. Rudquist, M. Matuszczyk, S. T. Lagerwall, H. Pauwels, and R. S. Dabrowski, "Antiferroelectric liquid crystals with 45 degrees tilt: new electro-optic effects in liquid crystals," in Liquid Crystal Materials, Devices, and Flat Panel Displays, R. Shashidhar and B. Gnade, eds., Proc. SPIE 3955,33-44 (2000).

Dahl, I.

G. Andersson, I. Dahl, L. Komitov, S. T. Lagerwall, K. Skarp, and B. Stebler, "Device physics of the soft-mode electro-optic effect," J. Appl. Phys. 66, 4983-4995 (1989).
[CrossRef]

D'havé, K.

D. Engström, S. Hård, P. Rudquist, K. D'havé, T. Matuszczyk, M. Skeren, and B. Löfving, "Beam steering experiment with two cascaded ferroelectric liquid-crystal spatial light modulators," Appl. Opt. 43, 1559-1569 (2004).
[CrossRef] [PubMed]

K. D'havé, P. Rudquist, M. Matuszczyk, S. T. Lagerwall, H. Pauwels, and R. S. Dabrowski, "Antiferroelectric liquid crystals with 45 degrees tilt: new electro-optic effects in liquid crystals," in Liquid Crystal Materials, Devices, and Flat Panel Displays, R. Shashidhar and B. Gnade, eds., Proc. SPIE 3955,33-44 (2000).

A. Hammarquist, K. D'havé, M. Matuszczyk, and P. Rudquist "V-shaped ferroelectric liquid crystal structure stabilized by dielectric surface layers," Europhys. Lett. (to be published).

Engström, D.

Ewing, T.

J. Stockley, X. Xia, T. Ewing, and S. Serati, "Liquid crystal optical phase modulators for beam steering," in Advances in Liquid Crystalline Materials and Technologies (Materials Research Society, 2002), Vol. 709, pp. 55-66.

Freeman, M. O.

Fukuda, A.

S. Inui, N. Iimura, T. Suzuki, H. Wane, K. Miyachi, Y. Takanishi, and A. Fukuda, "Threshold-less antiferroelectricity in liquid crystals and its application to displays," J. Mater. Chem. 6, 671-673 (1996).
[CrossRef]

A. Fukuda, "Pretransitional effect in AF-F switching: to suppress it or to enhance it, that is my question about AFLCDs," Asia Display'95: Proceedings of the 15th International Display Research Conference (1995), pp. 61-64.

Garoff, S.

S. Garoff and R. B. Meyer, "Electroclinic effect at the A-C phase change in a chiral smectic liquid crystal," Phys. Rev. Lett. 38, 848-851 (1977).
[CrossRef]

Glaser, M. A.

P. Rudquist, J. P. F. Lagerwall, M. Buivydas, F. Gouda, S. T. Lagerwall, N. A. Clark, J. E. Maclennan, R. Shao, D. A. Coleman, S. Bardon, T. Bellini, D. R. Link, G. Natale, M. A. Glaser, D. M. Walba, M. D. Wand, and X.-H. Chen, "The case of thresholdless antiferroelectricity: polarization-stabilized twisted SmC* liquid crystals give V-shaped electro-optic response," J. Mater. Chem. 9, 1257-1261 (1999).
[CrossRef]

Goldstein, D.

See, for example, D. Goldstein, Polarized Light (Dekker, 2003), pp. 241-273.

Gouda, F.

P. Rudquist, J. P. F. Lagerwall, M. Buivydas, F. Gouda, S. T. Lagerwall, N. A. Clark, J. E. Maclennan, R. Shao, D. A. Coleman, S. Bardon, T. Bellini, D. R. Link, G. Natale, M. A. Glaser, D. M. Walba, M. D. Wand, and X.-H. Chen, "The case of thresholdless antiferroelectricity: polarization-stabilized twisted SmC* liquid crystals give V-shaped electro-optic response," J. Mater. Chem. 9, 1257-1261 (1999).
[CrossRef]

Gu, C.

P. Yeh and C. Gu, Optics of Liquid Crystal Displays(Wiley, 1999), pp. 103-152.

Hammarquist, A.

A. Hammarquist, K. D'havé, M. Matuszczyk, and P. Rudquist "V-shaped ferroelectric liquid crystal structure stabilized by dielectric surface layers," Europhys. Lett. (to be published).

Hård, S.

Iimura, N.

S. Inui, N. Iimura, T. Suzuki, H. Wane, K. Miyachi, Y. Takanishi, and A. Fukuda, "Threshold-less antiferroelectricity in liquid crystals and its application to displays," J. Mater. Chem. 6, 671-673 (1996).
[CrossRef]

Inui, S.

S. Inui, N. Iimura, T. Suzuki, H. Wane, K. Miyachi, Y. Takanishi, and A. Fukuda, "Threshold-less antiferroelectricity in liquid crystals and its application to displays," J. Mater. Chem. 6, 671-673 (1996).
[CrossRef]

Johnson, K. M.

Keller, P.

R. B. Meyer, L. Liebert, L. Strzelecki, and P. Keller, "Ferroelectric liquid crystals," J. Phys. (Paris), Lett. 36, 69-71 (1975).

Komitov, L.

G. Andersson, I. Dahl, L. Komitov, S. T. Lagerwall, K. Skarp, and B. Stebler, "Device physics of the soft-mode electro-optic effect," J. Appl. Phys. 66, 4983-4995 (1989).
[CrossRef]

Lagerwall, J. P. F.

P. Rudquist, J. P. F. Lagerwall, M. Buivydas, F. Gouda, S. T. Lagerwall, N. A. Clark, J. E. Maclennan, R. Shao, D. A. Coleman, S. Bardon, T. Bellini, D. R. Link, G. Natale, M. A. Glaser, D. M. Walba, M. D. Wand, and X.-H. Chen, "The case of thresholdless antiferroelectricity: polarization-stabilized twisted SmC* liquid crystals give V-shaped electro-optic response," J. Mater. Chem. 9, 1257-1261 (1999).
[CrossRef]

Lagerwall, S. T.

P. Rudquist, J. P. F. Lagerwall, M. Buivydas, F. Gouda, S. T. Lagerwall, N. A. Clark, J. E. Maclennan, R. Shao, D. A. Coleman, S. Bardon, T. Bellini, D. R. Link, G. Natale, M. A. Glaser, D. M. Walba, M. D. Wand, and X.-H. Chen, "The case of thresholdless antiferroelectricity: polarization-stabilized twisted SmC* liquid crystals give V-shaped electro-optic response," J. Mater. Chem. 9, 1257-1261 (1999).
[CrossRef]

G. Andersson, I. Dahl, L. Komitov, S. T. Lagerwall, K. Skarp, and B. Stebler, "Device physics of the soft-mode electro-optic effect," J. Appl. Phys. 66, 4983-4995 (1989).
[CrossRef]

N. A. Clark and S. T. Lagerwall, "Submicrosecond bistable electro-optic switching in liquid crystals," Appl. Phys. Lett. 36, 899-901 (1980).
[CrossRef]

K. D'havé, P. Rudquist, M. Matuszczyk, S. T. Lagerwall, H. Pauwels, and R. S. Dabrowski, "Antiferroelectric liquid crystals with 45 degrees tilt: new electro-optic effects in liquid crystals," in Liquid Crystal Materials, Devices, and Flat Panel Displays, R. Shashidhar and B. Gnade, eds., Proc. SPIE 3955,33-44 (2000).

Liebert, L.

R. B. Meyer, L. Liebert, L. Strzelecki, and P. Keller, "Ferroelectric liquid crystals," J. Phys. (Paris), Lett. 36, 69-71 (1975).

Link, D. R.

P. Rudquist, J. P. F. Lagerwall, M. Buivydas, F. Gouda, S. T. Lagerwall, N. A. Clark, J. E. Maclennan, R. Shao, D. A. Coleman, S. Bardon, T. Bellini, D. R. Link, G. Natale, M. A. Glaser, D. M. Walba, M. D. Wand, and X.-H. Chen, "The case of thresholdless antiferroelectricity: polarization-stabilized twisted SmC* liquid crystals give V-shaped electro-optic response," J. Mater. Chem. 9, 1257-1261 (1999).
[CrossRef]

Löfving, B.

Maclennan, J. E.

D. Coleman, D. Mueller, N. A. Clark, J. E. Maclennan, R.-F. Shao, S. Bardon, and D. M. Walba, "Control of molecular orientation in electrostatically stabilized ferroelectric liquid crystals," Phys. Rev. Lett. 91, 175505 (2003).
[CrossRef] [PubMed]

M. Copic, J. E. Maclennan, and N. A. Clark, "Structure and dynamics of ferroelectric liquid crystal cells exhibiting thresholdless switching," Phys. Rev. E 65, 021708 (2002).

N. A. Clark, D. Coleman, and J. E. Maclennan, "Electrostatics and the behaviour of chiral smectic C: 'block' polarization screening of applied voltage and 'V-shaped' switching," Liq. Crys. 20, 985-990 (2000).
[CrossRef]

P. Rudquist, J. P. F. Lagerwall, M. Buivydas, F. Gouda, S. T. Lagerwall, N. A. Clark, J. E. Maclennan, R. Shao, D. A. Coleman, S. Bardon, T. Bellini, D. R. Link, G. Natale, M. A. Glaser, D. M. Walba, M. D. Wand, and X.-H. Chen, "The case of thresholdless antiferroelectricity: polarization-stabilized twisted SmC* liquid crystals give V-shaped electro-optic response," J. Mater. Chem. 9, 1257-1261 (1999).
[CrossRef]

Matuszczyk, M.

A. Hammarquist, K. D'havé, M. Matuszczyk, and P. Rudquist "V-shaped ferroelectric liquid crystal structure stabilized by dielectric surface layers," Europhys. Lett. (to be published).

K. D'havé, P. Rudquist, M. Matuszczyk, S. T. Lagerwall, H. Pauwels, and R. S. Dabrowski, "Antiferroelectric liquid crystals with 45 degrees tilt: new electro-optic effects in liquid crystals," in Liquid Crystal Materials, Devices, and Flat Panel Displays, R. Shashidhar and B. Gnade, eds., Proc. SPIE 3955,33-44 (2000).

Matuszczyk, T.

Meyer, R. B.

S. Garoff and R. B. Meyer, "Electroclinic effect at the A-C phase change in a chiral smectic liquid crystal," Phys. Rev. Lett. 38, 848-851 (1977).
[CrossRef]

R. B. Meyer, L. Liebert, L. Strzelecki, and P. Keller, "Ferroelectric liquid crystals," J. Phys. (Paris), Lett. 36, 69-71 (1975).

Miyachi, K.

S. Inui, N. Iimura, T. Suzuki, H. Wane, K. Miyachi, Y. Takanishi, and A. Fukuda, "Threshold-less antiferroelectricity in liquid crystals and its application to displays," J. Mater. Chem. 6, 671-673 (1996).
[CrossRef]

Mueller, D.

D. Coleman, D. Mueller, N. A. Clark, J. E. Maclennan, R.-F. Shao, S. Bardon, and D. M. Walba, "Control of molecular orientation in electrostatically stabilized ferroelectric liquid crystals," Phys. Rev. Lett. 91, 175505 (2003).
[CrossRef] [PubMed]

Nakata, M.

M. J. O'Callaghan, M. D. Wand, C. M. Walker, and M. Nakata, "Charge controlled, fixed optic axis analog ('V-shaped') switching of a bent core ferroelectric liquid crystal," Appl. Phys. Lett. 85, 6344-6346 (2004).
[CrossRef]

Natale, G.

P. Rudquist, J. P. F. Lagerwall, M. Buivydas, F. Gouda, S. T. Lagerwall, N. A. Clark, J. E. Maclennan, R. Shao, D. A. Coleman, S. Bardon, T. Bellini, D. R. Link, G. Natale, M. A. Glaser, D. M. Walba, M. D. Wand, and X.-H. Chen, "The case of thresholdless antiferroelectricity: polarization-stabilized twisted SmC* liquid crystals give V-shaped electro-optic response," J. Mater. Chem. 9, 1257-1261 (1999).
[CrossRef]

Neil, M. A. A.

Nicolás, J.

O'Callaghan, M. J.

M. J. O'Callaghan, M. D. Wand, C. M. Walker, and M. Nakata, "Charge controlled, fixed optic axis analog ('V-shaped') switching of a bent core ferroelectric liquid crystal," Appl. Phys. Lett. 85, 6344-6346 (2004).
[CrossRef]

M. J. O'Callaghan, "Switching dynamics and surface forces in thresholdless 'V-shaped' switching ferroelectric liquid crystal," Phys. Rev. E 67, 011710 (2003).

Paige, E. G. S.

Parmar, D. S.

T. Rieker, N. A. Clark, G. S. Smith, D. S. Parmar, and E. B. Sirota, "'Chevron' local layer structure in surface-stabilized ferroelectric smectic-C cells," Phys. Rev. Lett. 59, 2658-2661 (1987).
[CrossRef] [PubMed]

Pauwels, H.

K. D'havé, P. Rudquist, M. Matuszczyk, S. T. Lagerwall, H. Pauwels, and R. S. Dabrowski, "Antiferroelectric liquid crystals with 45 degrees tilt: new electro-optic effects in liquid crystals," in Liquid Crystal Materials, Devices, and Flat Panel Displays, R. Shashidhar and B. Gnade, eds., Proc. SPIE 3955,33-44 (2000).

Rieker, T.

T. Rieker, N. A. Clark, G. S. Smith, D. S. Parmar, and E. B. Sirota, "'Chevron' local layer structure in surface-stabilized ferroelectric smectic-C cells," Phys. Rev. Lett. 59, 2658-2661 (1987).
[CrossRef] [PubMed]

Rudquist, P.

D. Engström, S. Hård, P. Rudquist, K. D'havé, T. Matuszczyk, M. Skeren, and B. Löfving, "Beam steering experiment with two cascaded ferroelectric liquid-crystal spatial light modulators," Appl. Opt. 43, 1559-1569 (2004).
[CrossRef] [PubMed]

P. Rudquist, J. P. F. Lagerwall, M. Buivydas, F. Gouda, S. T. Lagerwall, N. A. Clark, J. E. Maclennan, R. Shao, D. A. Coleman, S. Bardon, T. Bellini, D. R. Link, G. Natale, M. A. Glaser, D. M. Walba, M. D. Wand, and X.-H. Chen, "The case of thresholdless antiferroelectricity: polarization-stabilized twisted SmC* liquid crystals give V-shaped electro-optic response," J. Mater. Chem. 9, 1257-1261 (1999).
[CrossRef]

K. D'havé, P. Rudquist, M. Matuszczyk, S. T. Lagerwall, H. Pauwels, and R. S. Dabrowski, "Antiferroelectric liquid crystals with 45 degrees tilt: new electro-optic effects in liquid crystals," in Liquid Crystal Materials, Devices, and Flat Panel Displays, R. Shashidhar and B. Gnade, eds., Proc. SPIE 3955,33-44 (2000).

A. Hammarquist, K. D'havé, M. Matuszczyk, and P. Rudquist "V-shaped ferroelectric liquid crystal structure stabilized by dielectric surface layers," Europhys. Lett. (to be published).

Serati, S.

J. Stockley, X. Xia, T. Ewing, and S. Serati, "Liquid crystal optical phase modulators for beam steering," in Advances in Liquid Crystalline Materials and Technologies (Materials Research Society, 2002), Vol. 709, pp. 55-66.

Serati, S. A.

Shao, R.

P. Rudquist, J. P. F. Lagerwall, M. Buivydas, F. Gouda, S. T. Lagerwall, N. A. Clark, J. E. Maclennan, R. Shao, D. A. Coleman, S. Bardon, T. Bellini, D. R. Link, G. Natale, M. A. Glaser, D. M. Walba, M. D. Wand, and X.-H. Chen, "The case of thresholdless antiferroelectricity: polarization-stabilized twisted SmC* liquid crystals give V-shaped electro-optic response," J. Mater. Chem. 9, 1257-1261 (1999).
[CrossRef]

Shao, R.-F.

D. Coleman, D. Mueller, N. A. Clark, J. E. Maclennan, R.-F. Shao, S. Bardon, and D. M. Walba, "Control of molecular orientation in electrostatically stabilized ferroelectric liquid crystals," Phys. Rev. Lett. 91, 175505 (2003).
[CrossRef] [PubMed]

Sharp, G. D.

Sirota, E. B.

T. Rieker, N. A. Clark, G. S. Smith, D. S. Parmar, and E. B. Sirota, "'Chevron' local layer structure in surface-stabilized ferroelectric smectic-C cells," Phys. Rev. Lett. 59, 2658-2661 (1987).
[CrossRef] [PubMed]

Skarp, K.

G. Andersson, I. Dahl, L. Komitov, S. T. Lagerwall, K. Skarp, and B. Stebler, "Device physics of the soft-mode electro-optic effect," J. Appl. Phys. 66, 4983-4995 (1989).
[CrossRef]

Skeren, M.

Smith, G. S.

T. Rieker, N. A. Clark, G. S. Smith, D. S. Parmar, and E. B. Sirota, "'Chevron' local layer structure in surface-stabilized ferroelectric smectic-C cells," Phys. Rev. Lett. 59, 2658-2661 (1987).
[CrossRef] [PubMed]

Stebler, B.

G. Andersson, I. Dahl, L. Komitov, S. T. Lagerwall, K. Skarp, and B. Stebler, "Device physics of the soft-mode electro-optic effect," J. Appl. Phys. 66, 4983-4995 (1989).
[CrossRef]

Stockley, J.

J. Stockley, X. Xia, T. Ewing, and S. Serati, "Liquid crystal optical phase modulators for beam steering," in Advances in Liquid Crystalline Materials and Technologies (Materials Research Society, 2002), Vol. 709, pp. 55-66.

Stockley, J. E.

J. E. Stockley, G. D. Sharp, S. A. Serati, and K. M. Johnson, "Analog optical phase modulator based on chiral smectic and polymer cholesteric liquid crystals," Opt. Lett. 20, 2441-2443 (1995).
[CrossRef] [PubMed]

J. E. Stockley, "Chiral smectic liquid crystal beam deflector with large analog phase modulation," Ph.D. dissertation (University of Colorado, 1996).

Strzelecki, L.

R. B. Meyer, L. Liebert, L. Strzelecki, and P. Keller, "Ferroelectric liquid crystals," J. Phys. (Paris), Lett. 36, 69-71 (1975).

Suzuki, T.

S. Inui, N. Iimura, T. Suzuki, H. Wane, K. Miyachi, Y. Takanishi, and A. Fukuda, "Threshold-less antiferroelectricity in liquid crystals and its application to displays," J. Mater. Chem. 6, 671-673 (1996).
[CrossRef]

Takanishi, Y.

S. Inui, N. Iimura, T. Suzuki, H. Wane, K. Miyachi, Y. Takanishi, and A. Fukuda, "Threshold-less antiferroelectricity in liquid crystals and its application to displays," J. Mater. Chem. 6, 671-673 (1996).
[CrossRef]

Walba, D. M.

D. Coleman, D. Mueller, N. A. Clark, J. E. Maclennan, R.-F. Shao, S. Bardon, and D. M. Walba, "Control of molecular orientation in electrostatically stabilized ferroelectric liquid crystals," Phys. Rev. Lett. 91, 175505 (2003).
[CrossRef] [PubMed]

P. Rudquist, J. P. F. Lagerwall, M. Buivydas, F. Gouda, S. T. Lagerwall, N. A. Clark, J. E. Maclennan, R. Shao, D. A. Coleman, S. Bardon, T. Bellini, D. R. Link, G. Natale, M. A. Glaser, D. M. Walba, M. D. Wand, and X.-H. Chen, "The case of thresholdless antiferroelectricity: polarization-stabilized twisted SmC* liquid crystals give V-shaped electro-optic response," J. Mater. Chem. 9, 1257-1261 (1999).
[CrossRef]

M. O. Freeman, T. A. Brown, and D. M. Walba, "Quantized complex ferroelectric liquid crystal spatial light modulators," Appl. Opt. 31, 3917-3929 (1992).
[CrossRef] [PubMed]

Walker, C. M.

M. J. O'Callaghan, M. D. Wand, C. M. Walker, and M. Nakata, "Charge controlled, fixed optic axis analog ('V-shaped') switching of a bent core ferroelectric liquid crystal," Appl. Phys. Lett. 85, 6344-6346 (2004).
[CrossRef]

Wand, M. D.

M. J. O'Callaghan, M. D. Wand, C. M. Walker, and M. Nakata, "Charge controlled, fixed optic axis analog ('V-shaped') switching of a bent core ferroelectric liquid crystal," Appl. Phys. Lett. 85, 6344-6346 (2004).
[CrossRef]

P. Rudquist, J. P. F. Lagerwall, M. Buivydas, F. Gouda, S. T. Lagerwall, N. A. Clark, J. E. Maclennan, R. Shao, D. A. Coleman, S. Bardon, T. Bellini, D. R. Link, G. Natale, M. A. Glaser, D. M. Walba, M. D. Wand, and X.-H. Chen, "The case of thresholdless antiferroelectricity: polarization-stabilized twisted SmC* liquid crystals give V-shaped electro-optic response," J. Mater. Chem. 9, 1257-1261 (1999).
[CrossRef]

Wane, H.

S. Inui, N. Iimura, T. Suzuki, H. Wane, K. Miyachi, Y. Takanishi, and A. Fukuda, "Threshold-less antiferroelectricity in liquid crystals and its application to displays," J. Mater. Chem. 6, 671-673 (1996).
[CrossRef]

Xia, X.

J. Stockley, X. Xia, T. Ewing, and S. Serati, "Liquid crystal optical phase modulators for beam steering," in Advances in Liquid Crystalline Materials and Technologies (Materials Research Society, 2002), Vol. 709, pp. 55-66.

Yeh, P.

P. Yeh and C. Gu, Optics of Liquid Crystal Displays(Wiley, 1999), pp. 103-152.

Yzuel, M. J.

Appl. Opt.

Appl. Phys. Lett.

N. A. Clark and S. T. Lagerwall, "Submicrosecond bistable electro-optic switching in liquid crystals," Appl. Phys. Lett. 36, 899-901 (1980).
[CrossRef]

M. J. O'Callaghan, M. D. Wand, C. M. Walker, and M. Nakata, "Charge controlled, fixed optic axis analog ('V-shaped') switching of a bent core ferroelectric liquid crystal," Appl. Phys. Lett. 85, 6344-6346 (2004).
[CrossRef]

J. Appl. Phys.

G. Andersson, I. Dahl, L. Komitov, S. T. Lagerwall, K. Skarp, and B. Stebler, "Device physics of the soft-mode electro-optic effect," J. Appl. Phys. 66, 4983-4995 (1989).
[CrossRef]

J. Mater. Chem.

S. Inui, N. Iimura, T. Suzuki, H. Wane, K. Miyachi, Y. Takanishi, and A. Fukuda, "Threshold-less antiferroelectricity in liquid crystals and its application to displays," J. Mater. Chem. 6, 671-673 (1996).
[CrossRef]

P. Rudquist, J. P. F. Lagerwall, M. Buivydas, F. Gouda, S. T. Lagerwall, N. A. Clark, J. E. Maclennan, R. Shao, D. A. Coleman, S. Bardon, T. Bellini, D. R. Link, G. Natale, M. A. Glaser, D. M. Walba, M. D. Wand, and X.-H. Chen, "The case of thresholdless antiferroelectricity: polarization-stabilized twisted SmC* liquid crystals give V-shaped electro-optic response," J. Mater. Chem. 9, 1257-1261 (1999).
[CrossRef]

J. Opt. Soc. Am. A

J. Phys.

R. B. Meyer, L. Liebert, L. Strzelecki, and P. Keller, "Ferroelectric liquid crystals," J. Phys. (Paris), Lett. 36, 69-71 (1975).

Liq. Crys.

N. A. Clark, D. Coleman, and J. E. Maclennan, "Electrostatics and the behaviour of chiral smectic C: 'block' polarization screening of applied voltage and 'V-shaped' switching," Liq. Crys. 20, 985-990 (2000).
[CrossRef]

Opt. Lett.

Phys. Rev. E

M. Copic, J. E. Maclennan, and N. A. Clark, "Structure and dynamics of ferroelectric liquid crystal cells exhibiting thresholdless switching," Phys. Rev. E 65, 021708 (2002).

M. J. O'Callaghan, "Switching dynamics and surface forces in thresholdless 'V-shaped' switching ferroelectric liquid crystal," Phys. Rev. E 67, 011710 (2003).

Phys. Rev. Lett.

D. Coleman, D. Mueller, N. A. Clark, J. E. Maclennan, R.-F. Shao, S. Bardon, and D. M. Walba, "Control of molecular orientation in electrostatically stabilized ferroelectric liquid crystals," Phys. Rev. Lett. 91, 175505 (2003).
[CrossRef] [PubMed]

S. Garoff and R. B. Meyer, "Electroclinic effect at the A-C phase change in a chiral smectic liquid crystal," Phys. Rev. Lett. 38, 848-851 (1977).
[CrossRef]

T. Rieker, N. A. Clark, G. S. Smith, D. S. Parmar, and E. B. Sirota, "'Chevron' local layer structure in surface-stabilized ferroelectric smectic-C cells," Phys. Rev. Lett. 59, 2658-2661 (1987).
[CrossRef] [PubMed]

Other

A. Hammarquist, K. D'havé, M. Matuszczyk, and P. Rudquist "V-shaped ferroelectric liquid crystal structure stabilized by dielectric surface layers," Europhys. Lett. (to be published).

See, for example, D. Goldstein, Polarized Light (Dekker, 2003), pp. 241-273.

P. Yeh and C. Gu, Optics of Liquid Crystal Displays(Wiley, 1999), pp. 103-152.

The cells used in this work were made in house in the Nanofabrication Laboratory at the Department of Microtechnology and Nanoscience at Chalmers University of Technology.

K. D'havé, P. Rudquist, M. Matuszczyk, S. T. Lagerwall, H. Pauwels, and R. S. Dabrowski, "Antiferroelectric liquid crystals with 45 degrees tilt: new electro-optic effects in liquid crystals," in Liquid Crystal Materials, Devices, and Flat Panel Displays, R. Shashidhar and B. Gnade, eds., Proc. SPIE 3955,33-44 (2000).

J. E. Stockley, "Chiral smectic liquid crystal beam deflector with large analog phase modulation," Ph.D. dissertation (University of Colorado, 1996).

J. Stockley, X. Xia, T. Ewing, and S. Serati, "Liquid crystal optical phase modulators for beam steering," in Advances in Liquid Crystalline Materials and Technologies (Materials Research Society, 2002), Vol. 709, pp. 55-66.

A. Fukuda, "Pretransitional effect in AF-F switching: to suppress it or to enhance it, that is my question about AFLCDs," Asia Display'95: Proceedings of the 15th International Display Research Conference (1995), pp. 61-64.

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Figures (18)

Fig. 1
Fig. 1

Schematic of the V-FLC structure. The material is in the bookshelf geometry, i.e., the smectic layers of the smectic C* phase are homogeneously aligned perpendicular to the glass plates of the cell. The director can be switched into any position on the tilt cone, having an angle of 2θ. The spontaneous polarization P s is tangential to the tilt cone surface.

Fig. 2
Fig. 2

Orientation of the director for (a) no applied electric field, (b) a medium-strength electric field, and (c) a strong electric field.

Fig. 3
Fig. 3

Schematic of the geometry of the V-FLC cell, with the director n, the tilt half-cone on which the director is confined, and the polarization vector Ps. Light travels in the positive z direction and is polarized in the (x, y) plane. The glass substrates, which confine the LC material, are parallel with the (x, y) plane. The voltage over the LC cell causes an electric field along the z axis that rotates the director along the cone.

Fig. 4
Fig. 4

Effective refractive indices for V-FLC materials with no = 1.50 and ne = 1.65 as function of the angle α. Shown are the ordinary and extraordinary refractive indices (dashed-dotted and dashed lines, respectively) and the rotation-angle dependent effective extraordinary indices for materials with θ = 45° and θ = 38° (solid and dotted curves, respectively).

Fig. 5
Fig. 5

Poincaré sphere representation of the angles χ and φ describing a general polarization state E.

Fig. 6
Fig. 6

Outgoing polarization states plotted on the Poincaré sphere, when varying the director in-plane angle α. (a) For a half-wave plate (a single position in the middle of the drawn square at the south pole of the sphere), (b) for a V-FLC that works as a half-wave plate for the outermost director positions (dashed curve), and (c) for a V-FLC that works as a half-wave plate for the central director position (dashed-dotted curve). In all cases, the incident light was circularly polarized (denoted by the circle at the sphere's north pole). Note that both curves are plotted as dotted curves on the rear side of the sphere.

Fig. 7
Fig. 7

(a) Relative intensity and (b) the phase retardation of the x and y components of the output fields as functions of the director in-plane angle α for a half-wave plate (solid line), a V-FLC cell that works as a half-wave plate for the extreme director positions (dashed curve), and a V-FLC cell that works as a half-wave plate for the central director position (dashed-dotted curve). For the half-wave plate, the lines for the x and y components coincide.

Fig. 8
Fig. 8

An illustration of the optimization of the polarization state of E in. When switching the V-FLC cell, thus changing α, any input polarization state E in yields a continuous curve E out(α), which is limited by a circle with a radius rl .

Fig. 9
Fig. 9

Limiting radius rl , shown as a gray scale, as a function of the polarization state input parameters (χin, φin).

Fig. 10
Fig. 10

Minimum simulated rl for all possible input polarization states as a function of the effective cell thickness, d, for a material with a tilt cone half-angle of θ = 45° (solid) or θ = 38° (dashed).

Fig. 11
Fig. 11

(a) Output polarization states of the two optimal input polarization states, plotted on the Poincaré sphere with α as the parameter. (b) The settings of the polarizer and quarter-wave plate, the optimal input field vector and the resulting output field vector shown in the (x, y) plane for the two optima found for a cell with θ = 45° and an effective thickness of 1.85 μm. Also shown is the projection onto the (x, y) plane of the tilt cone. The polarizer and quarter-wave plate angles are αPol (1) = 112.5°, αλ∕4 (1) = 135.0°, αPol (2) = 22.5°, and αλ∕4 (2) = 135.0°. The input and output states of each optimum have the same orientation of the polarization ellipse, αell (1) = 45.0° and αell (2) = 135.0°, respectively, but they differ in handedness.

Fig. 12
Fig. 12

(a) Simulated relative intensity and (b) phase retardation of the x and y components of the output field as functions of the director in-plane angle α for one of the two optimal input polarization states for V-FLC cells with tilt cone half-angles θ = 45° and θ = 38°, respectively.

Fig. 13
Fig. 13

Characterization of the wavelength response with the V-FLC cell inserted between polarizers. (a) Measured intensity modulation depth for the voltage extremes and the simulated behavior of a half-wave plate with an optimal wavelength at 520 nm. (b) Measured intensity modulation depth for zero voltage and the simulated behavior of a quarter-wave plate with an optimal wavelength at 550 nm.

Fig. 14
Fig. 14

Measured intensity after the second (crossed) polarizer for the cell being rotated such that the first polarizer has a transmission direction making an angle of 0 and ±θ with the y axis as defined by Fig. 3. The curves correspond to the simulations for a cell with an effective cell thickness of 1.40 μm.

Fig. 15
Fig. 15

Measured variation of the output polarization state during analog switching of the V-FLC cell as a function of the polarization state input parameters (χin, φin). The intensity represented by the gray scale is Δ I x     2 + Δ I y     2 where ΔIx is the maximum intensity change in the x-polarized intensity as the cell voltage is changed over its full range, and likewise for ΔIy.

Fig. 16
Fig. 16

Mach–Zehnder interferometer setup.

Fig. 17
Fig. 17

(a) Measured relative intensity and (b) phase retardation of the x and y components of the output field as functions of the drive voltage for one of the optimal settings of the input polarization state.

Fig. 18
Fig. 18

Measured range of phase modulation of the x and y components of the output field over the entire modulation cycle of the triangular drive voltage as functions of the repetition rate of the voltage modulation.

Equations (167)

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

0.78 π
2 %
n
( 10   μs )
± 45 °
( ± 28 ° )
1.24 π rad
( ± 45 °
2 π
10 % 20 %
100   μs
π  rad
ε 3
ε 2
ε 1
P s
( T = P s × E )
2 θ
P s
P s
P s
P s
P s
n e
n o
n = ( x d , y d , z d ) = ( sin θ cos ψ , cos θ , sin θ sin ψ ) ,
0 ψ < π
tan α = x d y d = sin θ cos ψ cos θ .
sin β = z d | n | = sin θ sin ψ .
n e ( β )
n e ,
n e ( β ) = n e n o n e     2 cos 2 β + n e     2 sin 2 β ,
n o
θ = 45 °
θ = 38 °
n o = 1.50
n e = 1.65
P s
Γ = 2 π λ [ n e ( α ) n o ] d ,
E opt
E opt = [ E x E y ] = [ cos χ sin χ exp ( i φ ) ] ,
0 χ < π / 2
0 φ < 2 π
s 1 = | E x | 2 | E y | 2 = cos 2 χ sin 2 χ ,
s 2 = 2 | E x | | E y | cos ( arg E y arg E x ) = 2 cos χ sin χ cos φ ,
s 3 = 2 | E x | | E y | sin ( arg E y arg E x ) = 2 cos χ sin χ sin φ ,
( s 1 , s 2 , s 3 )
E in
E out = R ( α ) W WP ( Γ ) R ( α ) E in ,
W WP ( Γ )
( ψ = 0
ψ = π
( ψ = π / 2 )
E in
E out ( α )
r l
E in
r l
( χ in , φ in )
E in
E in
E out ( α )
r l ( χ in , φ in )
r l ( χ in , φ in )
d = 1.55   μm
n o = 1.50
n e = 1.65
λ = 543   nm
( χ in       ( 1 ) , φ in       ( 1 ) ) = ( 0.25 π , 0.75 π )
( χ in       ( 2 ) , φ in       ( 2 ) ) = ( 0.25 π , 1.75 π )
r l
r l
r l ( χ in , φ in )
( χ in , φ in )
θ = 45 °
θ = 38 °
15   μm
d 3   μm
r l
1.85   μm
1.55   μm
θ = 38 °
θ = 45 °
1.55   μm
α = 0
86 %
α ± 38 °
θ = 45 °
1.85   μm
θ = 38 °
E in
E out ( α )
( χ in , φ in )
± 6 %
± 2 %
θ = 45 °
θ = 38 °
θ = 45 °
θ = 38 °
100   nm
SiO 2
SiO 2
100   nm
300 ° C
3   in . × 3   in .
100 ° C
200   Hz
100   μs
80 ° C
( ψ = 0
ψ = π )
( ψ = π / 2 )
550   nm
π / 4   rad
α = 0
0.86 × π / 2
α = ± 38 °
543.5   nm
π / 4   rad
0.96 × π / 2   rad
60   V
30   Hz
d = 1.40   μm
n e
n o
α = ± θ
r l
r l
α Pol
α λ / 4
χ in
φ in
χ in
φ in
( χ in       ( 1 ) , φ in       ( 1 ) ) = ( 0.266 π , 0.768 π )
( χ in       ( 2 ) , φ in       ( 2 ) ) = ( 0.237 π , 1.774 π )
( χ in       ( 1 ) , φ in       ( 1 ) ) = ( 0.25 π , 0.75 π )
( χ in     ( 2 ) , φ in       ( 2 ) ) = ( 0.25 π , 1.75 π )
P 1
P 2
P 3
D 1
D 2
D 1
D 2
D 1
D 2
P 2
± 30   V
30   Hz
( α Pol = 69.5 °
α λ / 4 = 139.5 ° )
0.5 ± 0.025
( d = 1.55   μm )
d = 1.40   μm
0.8 π
0.73 π
0.78 π
30   Hz
± 30   V
1000   Hz
50   Hz
0.8 π
0.3 π
50 1000   Hz
38 °
Δ I x     2 + Δ I y     2

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