Abstract

The conoscopic method for investigating the optical properties of a liquid crystal cell is studied with the aim of determining the effects of the approximations used in the calculation on the results. We confirm that the chiral liquid crystal cell forming a helical structure can be regarded as a single biaxial plate for analyzing the conoscopic image only if the helical pitch is less than several multiples of the wavelength of light. This approximation implies that the square of the refractive index along a direction is averaged over all the layers. An incorrectly chosen value for one of the principal refractive indices to be used in the analysis of the conoscopic data can lead to an incorrect conclusion, especially for the case when the wavelength dispersion of the refractive index is neglected. A thicker cell and a longer wavelength of the incident light can minimize these limitations of the conoscopic method. We propose a novel simulation method to find the molecular distribution in a liquid crystal cell based on the average-refractive-index approximation and the conoscopic data. This is shown to be a fast, more efficient, and useful method for estimating the director distributions.

© 2008 Optical Society of America

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  1. One of the examples is the Conoscope manufactured by Autronic-Melchers GmbH, Karlsruhe, Germany.
  2. Y. Galerne, “Refractive index measurements at a 2nd order smectic A to C phase transition,” J. Phys. (Paris) 39, 1311-1316 (1978).
    [CrossRef]
  3. E. Gorecka, A. D. L. Chandani, Y. Ouchi, H. Takezoe, and A. Fukuda, “Molecular orientational structures in ferroelectric, ferrielectric and antiferroelectric smectic liquid crystal phases as studied by conoscope observation,” Jpn. J. Appl. Phys. 29, 131-137 (1990).
    [CrossRef]
  4. T. Fujikawa, K. Hiraoka, T. Isozaki, K. Kajikawa, H. Takezoe, and A. Fukuda, “Construction of dynamic conoscope observation system using CCD camera and image processor,” Jpn. J. Appl. Phys., Part 1 32, 985-988 (1993).
    [CrossRef]
  5. J. K. Song, A. D. L. Chandani, O. E. Panarina, A. Fukuda, J. K. Vij, V. Goertz, and J. W. Goodby, “Study of the SmC* phase in the Tokyo mixture by conoscopy using tilted cell,” Ferroelectrics 344, 41-47 (2006).
    [CrossRef]
  6. J. K. Song, J. K. Vij, and I. Kobayashi, “Interlayer interactions and the dependence of biaxiality of the chiral smectic-C * phase on electric field in the helical unwinding process,” Phys. Rev. E 75, 051705 (2007).
    [CrossRef]
  7. S. Suwa, H. Hoshi, Y. Takanishi, K. Ishikawa, H. Takezoe, and B. Zeks, “Helix unwinding process in a short-pitch ferroelectric liquid crystal mixture studied by conoscopy,” Jpn. J. Appl. Phys., Part 1 42, 1335-1337 (2003).
    [CrossRef]
  8. S. Suwa, Y. Takanishi, H. Hoshi, K. Ishikawa, and H. Takezoe, “Helix unwinding process in the chiral smectic C phase of MHPOBC as observed by conoscopy,” Liq. Cryst. 30, 499-505 (2003).
    [CrossRef]
  9. A. Fukuda, Y. Takanishi, T. Isozaki, K. Ishikawa, and H. Takezoe, “Antiferroelectric chiral smectic liquid crystals,” J. Mater. Chem. 4, 997-1016 (1994).
    [CrossRef]
  10. J. K. Song, A. D. L. Chandani, A. Fukuda, J. K. Vij, I. Kobayashi, and A. V. Emelyanenko, “Temperature-induced sign reversal of biaxiality observed by conoscopy in some ferroelectric Sm-C * liquid crystals,” Phys. Rev. E 76, 011709 (2007).
    [CrossRef]
  11. M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999), Chap. 14.4.
  12. D. W. Berreman, “Optics in stratified and anisotropic media: 4×4-matrix formulation,” J. Opt. Soc. Am. 62, 502-510 (1972).
    [CrossRef]
  13. P. Yeh, “Electromagnetic propagation in birefringent layered media,” J. Opt. Soc. Am. 69, 742-756 (1979).
    [CrossRef]
  14. D. Dunmur and K. Toriyama, Handbook of Liquid Crystals (Wiley-VCH, 1998), Vol. 1, Chap. 7, Sec. 3.
  15. J. Li and S.-T. Wu, “Self-consistency of Vuks equations for liquid-crystal refractive indices,” J. Appl. Phys. 96, 6253-6258 (2004).
    [CrossRef]
  16. I.-C. Khoo and S.-T. Wu, Optics and Nonlinear Optics of Liquid Crystals (World Scientific, 1993), Chap. 2.
  17. J. K. Song, U. Manna, and J. K. Vij, “Mechanism of field induced unwinding of SmC* helix and bias field dependence of dielectric permittivity and macroscopic spontaneous polarization,” Europhys. Lett. 82, 26003 (2008).
    [CrossRef]
  18. I. Abdulhalim, “Simplified optical scatterometry for periodic nanoarrays in the near-quasi-static limit,” Appl. Opt. 46, 2219-2229 (2007).
    [CrossRef] [PubMed]
  19. A. R. MacGregor, “Method for computing homogeneous liquid-crystal conoscopic figures,” J. Opt. Soc. Am. A 7, 337-347 (1990).
    [CrossRef]
  20. I. Abdulhalim, “Analytic propagation matrix method for linear optics of arbitrary biaxial layered media,” J. Opt. A, Pure Appl. Opt. 1, 646-653 (1999).
    [CrossRef]

2008 (1)

J. K. Song, U. Manna, and J. K. Vij, “Mechanism of field induced unwinding of SmC* helix and bias field dependence of dielectric permittivity and macroscopic spontaneous polarization,” Europhys. Lett. 82, 26003 (2008).
[CrossRef]

2007 (3)

J. K. Song, A. D. L. Chandani, A. Fukuda, J. K. Vij, I. Kobayashi, and A. V. Emelyanenko, “Temperature-induced sign reversal of biaxiality observed by conoscopy in some ferroelectric Sm-C * liquid crystals,” Phys. Rev. E 76, 011709 (2007).
[CrossRef]

J. K. Song, J. K. Vij, and I. Kobayashi, “Interlayer interactions and the dependence of biaxiality of the chiral smectic-C * phase on electric field in the helical unwinding process,” Phys. Rev. E 75, 051705 (2007).
[CrossRef]

I. Abdulhalim, “Simplified optical scatterometry for periodic nanoarrays in the near-quasi-static limit,” Appl. Opt. 46, 2219-2229 (2007).
[CrossRef] [PubMed]

2006 (1)

J. K. Song, A. D. L. Chandani, O. E. Panarina, A. Fukuda, J. K. Vij, V. Goertz, and J. W. Goodby, “Study of the SmC* phase in the Tokyo mixture by conoscopy using tilted cell,” Ferroelectrics 344, 41-47 (2006).
[CrossRef]

2004 (1)

J. Li and S.-T. Wu, “Self-consistency of Vuks equations for liquid-crystal refractive indices,” J. Appl. Phys. 96, 6253-6258 (2004).
[CrossRef]

2003 (2)

S. Suwa, H. Hoshi, Y. Takanishi, K. Ishikawa, H. Takezoe, and B. Zeks, “Helix unwinding process in a short-pitch ferroelectric liquid crystal mixture studied by conoscopy,” Jpn. J. Appl. Phys., Part 1 42, 1335-1337 (2003).
[CrossRef]

S. Suwa, Y. Takanishi, H. Hoshi, K. Ishikawa, and H. Takezoe, “Helix unwinding process in the chiral smectic C phase of MHPOBC as observed by conoscopy,” Liq. Cryst. 30, 499-505 (2003).
[CrossRef]

1999 (2)

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999), Chap. 14.4.

I. Abdulhalim, “Analytic propagation matrix method for linear optics of arbitrary biaxial layered media,” J. Opt. A, Pure Appl. Opt. 1, 646-653 (1999).
[CrossRef]

1998 (1)

D. Dunmur and K. Toriyama, Handbook of Liquid Crystals (Wiley-VCH, 1998), Vol. 1, Chap. 7, Sec. 3.

1994 (1)

A. Fukuda, Y. Takanishi, T. Isozaki, K. Ishikawa, and H. Takezoe, “Antiferroelectric chiral smectic liquid crystals,” J. Mater. Chem. 4, 997-1016 (1994).
[CrossRef]

1993 (2)

T. Fujikawa, K. Hiraoka, T. Isozaki, K. Kajikawa, H. Takezoe, and A. Fukuda, “Construction of dynamic conoscope observation system using CCD camera and image processor,” Jpn. J. Appl. Phys., Part 1 32, 985-988 (1993).
[CrossRef]

I.-C. Khoo and S.-T. Wu, Optics and Nonlinear Optics of Liquid Crystals (World Scientific, 1993), Chap. 2.

1990 (2)

A. R. MacGregor, “Method for computing homogeneous liquid-crystal conoscopic figures,” J. Opt. Soc. Am. A 7, 337-347 (1990).
[CrossRef]

E. Gorecka, A. D. L. Chandani, Y. Ouchi, H. Takezoe, and A. Fukuda, “Molecular orientational structures in ferroelectric, ferrielectric and antiferroelectric smectic liquid crystal phases as studied by conoscope observation,” Jpn. J. Appl. Phys. 29, 131-137 (1990).
[CrossRef]

1979 (1)

1978 (1)

Y. Galerne, “Refractive index measurements at a 2nd order smectic A to C phase transition,” J. Phys. (Paris) 39, 1311-1316 (1978).
[CrossRef]

1972 (1)

Abdulhalim, I.

I. Abdulhalim, “Simplified optical scatterometry for periodic nanoarrays in the near-quasi-static limit,” Appl. Opt. 46, 2219-2229 (2007).
[CrossRef] [PubMed]

I. Abdulhalim, “Analytic propagation matrix method for linear optics of arbitrary biaxial layered media,” J. Opt. A, Pure Appl. Opt. 1, 646-653 (1999).
[CrossRef]

Berreman, D. W.

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999), Chap. 14.4.

Chandani, A. D. L.

J. K. Song, A. D. L. Chandani, A. Fukuda, J. K. Vij, I. Kobayashi, and A. V. Emelyanenko, “Temperature-induced sign reversal of biaxiality observed by conoscopy in some ferroelectric Sm-C * liquid crystals,” Phys. Rev. E 76, 011709 (2007).
[CrossRef]

J. K. Song, A. D. L. Chandani, O. E. Panarina, A. Fukuda, J. K. Vij, V. Goertz, and J. W. Goodby, “Study of the SmC* phase in the Tokyo mixture by conoscopy using tilted cell,” Ferroelectrics 344, 41-47 (2006).
[CrossRef]

E. Gorecka, A. D. L. Chandani, Y. Ouchi, H. Takezoe, and A. Fukuda, “Molecular orientational structures in ferroelectric, ferrielectric and antiferroelectric smectic liquid crystal phases as studied by conoscope observation,” Jpn. J. Appl. Phys. 29, 131-137 (1990).
[CrossRef]

Dunmur, D.

D. Dunmur and K. Toriyama, Handbook of Liquid Crystals (Wiley-VCH, 1998), Vol. 1, Chap. 7, Sec. 3.

Emelyanenko, A. V.

J. K. Song, A. D. L. Chandani, A. Fukuda, J. K. Vij, I. Kobayashi, and A. V. Emelyanenko, “Temperature-induced sign reversal of biaxiality observed by conoscopy in some ferroelectric Sm-C * liquid crystals,” Phys. Rev. E 76, 011709 (2007).
[CrossRef]

Fujikawa, T.

T. Fujikawa, K. Hiraoka, T. Isozaki, K. Kajikawa, H. Takezoe, and A. Fukuda, “Construction of dynamic conoscope observation system using CCD camera and image processor,” Jpn. J. Appl. Phys., Part 1 32, 985-988 (1993).
[CrossRef]

Fukuda, A.

J. K. Song, A. D. L. Chandani, A. Fukuda, J. K. Vij, I. Kobayashi, and A. V. Emelyanenko, “Temperature-induced sign reversal of biaxiality observed by conoscopy in some ferroelectric Sm-C * liquid crystals,” Phys. Rev. E 76, 011709 (2007).
[CrossRef]

J. K. Song, A. D. L. Chandani, O. E. Panarina, A. Fukuda, J. K. Vij, V. Goertz, and J. W. Goodby, “Study of the SmC* phase in the Tokyo mixture by conoscopy using tilted cell,” Ferroelectrics 344, 41-47 (2006).
[CrossRef]

A. Fukuda, Y. Takanishi, T. Isozaki, K. Ishikawa, and H. Takezoe, “Antiferroelectric chiral smectic liquid crystals,” J. Mater. Chem. 4, 997-1016 (1994).
[CrossRef]

T. Fujikawa, K. Hiraoka, T. Isozaki, K. Kajikawa, H. Takezoe, and A. Fukuda, “Construction of dynamic conoscope observation system using CCD camera and image processor,” Jpn. J. Appl. Phys., Part 1 32, 985-988 (1993).
[CrossRef]

E. Gorecka, A. D. L. Chandani, Y. Ouchi, H. Takezoe, and A. Fukuda, “Molecular orientational structures in ferroelectric, ferrielectric and antiferroelectric smectic liquid crystal phases as studied by conoscope observation,” Jpn. J. Appl. Phys. 29, 131-137 (1990).
[CrossRef]

Galerne, Y.

Y. Galerne, “Refractive index measurements at a 2nd order smectic A to C phase transition,” J. Phys. (Paris) 39, 1311-1316 (1978).
[CrossRef]

Goertz, V.

J. K. Song, A. D. L. Chandani, O. E. Panarina, A. Fukuda, J. K. Vij, V. Goertz, and J. W. Goodby, “Study of the SmC* phase in the Tokyo mixture by conoscopy using tilted cell,” Ferroelectrics 344, 41-47 (2006).
[CrossRef]

Goodby, J. W.

J. K. Song, A. D. L. Chandani, O. E. Panarina, A. Fukuda, J. K. Vij, V. Goertz, and J. W. Goodby, “Study of the SmC* phase in the Tokyo mixture by conoscopy using tilted cell,” Ferroelectrics 344, 41-47 (2006).
[CrossRef]

Gorecka, E.

E. Gorecka, A. D. L. Chandani, Y. Ouchi, H. Takezoe, and A. Fukuda, “Molecular orientational structures in ferroelectric, ferrielectric and antiferroelectric smectic liquid crystal phases as studied by conoscope observation,” Jpn. J. Appl. Phys. 29, 131-137 (1990).
[CrossRef]

Hiraoka, K.

T. Fujikawa, K. Hiraoka, T. Isozaki, K. Kajikawa, H. Takezoe, and A. Fukuda, “Construction of dynamic conoscope observation system using CCD camera and image processor,” Jpn. J. Appl. Phys., Part 1 32, 985-988 (1993).
[CrossRef]

Hoshi, H.

S. Suwa, Y. Takanishi, H. Hoshi, K. Ishikawa, and H. Takezoe, “Helix unwinding process in the chiral smectic C phase of MHPOBC as observed by conoscopy,” Liq. Cryst. 30, 499-505 (2003).
[CrossRef]

S. Suwa, H. Hoshi, Y. Takanishi, K. Ishikawa, H. Takezoe, and B. Zeks, “Helix unwinding process in a short-pitch ferroelectric liquid crystal mixture studied by conoscopy,” Jpn. J. Appl. Phys., Part 1 42, 1335-1337 (2003).
[CrossRef]

Ishikawa, K.

S. Suwa, H. Hoshi, Y. Takanishi, K. Ishikawa, H. Takezoe, and B. Zeks, “Helix unwinding process in a short-pitch ferroelectric liquid crystal mixture studied by conoscopy,” Jpn. J. Appl. Phys., Part 1 42, 1335-1337 (2003).
[CrossRef]

S. Suwa, Y. Takanishi, H. Hoshi, K. Ishikawa, and H. Takezoe, “Helix unwinding process in the chiral smectic C phase of MHPOBC as observed by conoscopy,” Liq. Cryst. 30, 499-505 (2003).
[CrossRef]

A. Fukuda, Y. Takanishi, T. Isozaki, K. Ishikawa, and H. Takezoe, “Antiferroelectric chiral smectic liquid crystals,” J. Mater. Chem. 4, 997-1016 (1994).
[CrossRef]

Isozaki, T.

A. Fukuda, Y. Takanishi, T. Isozaki, K. Ishikawa, and H. Takezoe, “Antiferroelectric chiral smectic liquid crystals,” J. Mater. Chem. 4, 997-1016 (1994).
[CrossRef]

T. Fujikawa, K. Hiraoka, T. Isozaki, K. Kajikawa, H. Takezoe, and A. Fukuda, “Construction of dynamic conoscope observation system using CCD camera and image processor,” Jpn. J. Appl. Phys., Part 1 32, 985-988 (1993).
[CrossRef]

Kajikawa, K.

T. Fujikawa, K. Hiraoka, T. Isozaki, K. Kajikawa, H. Takezoe, and A. Fukuda, “Construction of dynamic conoscope observation system using CCD camera and image processor,” Jpn. J. Appl. Phys., Part 1 32, 985-988 (1993).
[CrossRef]

Khoo, I.-C.

I.-C. Khoo and S.-T. Wu, Optics and Nonlinear Optics of Liquid Crystals (World Scientific, 1993), Chap. 2.

Kobayashi, I.

J. K. Song, A. D. L. Chandani, A. Fukuda, J. K. Vij, I. Kobayashi, and A. V. Emelyanenko, “Temperature-induced sign reversal of biaxiality observed by conoscopy in some ferroelectric Sm-C * liquid crystals,” Phys. Rev. E 76, 011709 (2007).
[CrossRef]

J. K. Song, J. K. Vij, and I. Kobayashi, “Interlayer interactions and the dependence of biaxiality of the chiral smectic-C * phase on electric field in the helical unwinding process,” Phys. Rev. E 75, 051705 (2007).
[CrossRef]

Li, J.

J. Li and S.-T. Wu, “Self-consistency of Vuks equations for liquid-crystal refractive indices,” J. Appl. Phys. 96, 6253-6258 (2004).
[CrossRef]

MacGregor, A. R.

Manna, U.

J. K. Song, U. Manna, and J. K. Vij, “Mechanism of field induced unwinding of SmC* helix and bias field dependence of dielectric permittivity and macroscopic spontaneous polarization,” Europhys. Lett. 82, 26003 (2008).
[CrossRef]

Ouchi, Y.

E. Gorecka, A. D. L. Chandani, Y. Ouchi, H. Takezoe, and A. Fukuda, “Molecular orientational structures in ferroelectric, ferrielectric and antiferroelectric smectic liquid crystal phases as studied by conoscope observation,” Jpn. J. Appl. Phys. 29, 131-137 (1990).
[CrossRef]

Panarina, O. E.

J. K. Song, A. D. L. Chandani, O. E. Panarina, A. Fukuda, J. K. Vij, V. Goertz, and J. W. Goodby, “Study of the SmC* phase in the Tokyo mixture by conoscopy using tilted cell,” Ferroelectrics 344, 41-47 (2006).
[CrossRef]

Song, J. K.

J. K. Song, U. Manna, and J. K. Vij, “Mechanism of field induced unwinding of SmC* helix and bias field dependence of dielectric permittivity and macroscopic spontaneous polarization,” Europhys. Lett. 82, 26003 (2008).
[CrossRef]

J. K. Song, A. D. L. Chandani, A. Fukuda, J. K. Vij, I. Kobayashi, and A. V. Emelyanenko, “Temperature-induced sign reversal of biaxiality observed by conoscopy in some ferroelectric Sm-C * liquid crystals,” Phys. Rev. E 76, 011709 (2007).
[CrossRef]

J. K. Song, J. K. Vij, and I. Kobayashi, “Interlayer interactions and the dependence of biaxiality of the chiral smectic-C * phase on electric field in the helical unwinding process,” Phys. Rev. E 75, 051705 (2007).
[CrossRef]

J. K. Song, A. D. L. Chandani, O. E. Panarina, A. Fukuda, J. K. Vij, V. Goertz, and J. W. Goodby, “Study of the SmC* phase in the Tokyo mixture by conoscopy using tilted cell,” Ferroelectrics 344, 41-47 (2006).
[CrossRef]

Suwa, S.

S. Suwa, H. Hoshi, Y. Takanishi, K. Ishikawa, H. Takezoe, and B. Zeks, “Helix unwinding process in a short-pitch ferroelectric liquid crystal mixture studied by conoscopy,” Jpn. J. Appl. Phys., Part 1 42, 1335-1337 (2003).
[CrossRef]

S. Suwa, Y. Takanishi, H. Hoshi, K. Ishikawa, and H. Takezoe, “Helix unwinding process in the chiral smectic C phase of MHPOBC as observed by conoscopy,” Liq. Cryst. 30, 499-505 (2003).
[CrossRef]

Takanishi, Y.

S. Suwa, Y. Takanishi, H. Hoshi, K. Ishikawa, and H. Takezoe, “Helix unwinding process in the chiral smectic C phase of MHPOBC as observed by conoscopy,” Liq. Cryst. 30, 499-505 (2003).
[CrossRef]

S. Suwa, H. Hoshi, Y. Takanishi, K. Ishikawa, H. Takezoe, and B. Zeks, “Helix unwinding process in a short-pitch ferroelectric liquid crystal mixture studied by conoscopy,” Jpn. J. Appl. Phys., Part 1 42, 1335-1337 (2003).
[CrossRef]

A. Fukuda, Y. Takanishi, T. Isozaki, K. Ishikawa, and H. Takezoe, “Antiferroelectric chiral smectic liquid crystals,” J. Mater. Chem. 4, 997-1016 (1994).
[CrossRef]

Takezoe, H.

S. Suwa, Y. Takanishi, H. Hoshi, K. Ishikawa, and H. Takezoe, “Helix unwinding process in the chiral smectic C phase of MHPOBC as observed by conoscopy,” Liq. Cryst. 30, 499-505 (2003).
[CrossRef]

S. Suwa, H. Hoshi, Y. Takanishi, K. Ishikawa, H. Takezoe, and B. Zeks, “Helix unwinding process in a short-pitch ferroelectric liquid crystal mixture studied by conoscopy,” Jpn. J. Appl. Phys., Part 1 42, 1335-1337 (2003).
[CrossRef]

A. Fukuda, Y. Takanishi, T. Isozaki, K. Ishikawa, and H. Takezoe, “Antiferroelectric chiral smectic liquid crystals,” J. Mater. Chem. 4, 997-1016 (1994).
[CrossRef]

T. Fujikawa, K. Hiraoka, T. Isozaki, K. Kajikawa, H. Takezoe, and A. Fukuda, “Construction of dynamic conoscope observation system using CCD camera and image processor,” Jpn. J. Appl. Phys., Part 1 32, 985-988 (1993).
[CrossRef]

E. Gorecka, A. D. L. Chandani, Y. Ouchi, H. Takezoe, and A. Fukuda, “Molecular orientational structures in ferroelectric, ferrielectric and antiferroelectric smectic liquid crystal phases as studied by conoscope observation,” Jpn. J. Appl. Phys. 29, 131-137 (1990).
[CrossRef]

Toriyama, K.

D. Dunmur and K. Toriyama, Handbook of Liquid Crystals (Wiley-VCH, 1998), Vol. 1, Chap. 7, Sec. 3.

Vij, J. K.

J. K. Song, U. Manna, and J. K. Vij, “Mechanism of field induced unwinding of SmC* helix and bias field dependence of dielectric permittivity and macroscopic spontaneous polarization,” Europhys. Lett. 82, 26003 (2008).
[CrossRef]

J. K. Song, A. D. L. Chandani, A. Fukuda, J. K. Vij, I. Kobayashi, and A. V. Emelyanenko, “Temperature-induced sign reversal of biaxiality observed by conoscopy in some ferroelectric Sm-C * liquid crystals,” Phys. Rev. E 76, 011709 (2007).
[CrossRef]

J. K. Song, J. K. Vij, and I. Kobayashi, “Interlayer interactions and the dependence of biaxiality of the chiral smectic-C * phase on electric field in the helical unwinding process,” Phys. Rev. E 75, 051705 (2007).
[CrossRef]

J. K. Song, A. D. L. Chandani, O. E. Panarina, A. Fukuda, J. K. Vij, V. Goertz, and J. W. Goodby, “Study of the SmC* phase in the Tokyo mixture by conoscopy using tilted cell,” Ferroelectrics 344, 41-47 (2006).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999), Chap. 14.4.

Wu, S.-T.

J. Li and S.-T. Wu, “Self-consistency of Vuks equations for liquid-crystal refractive indices,” J. Appl. Phys. 96, 6253-6258 (2004).
[CrossRef]

I.-C. Khoo and S.-T. Wu, Optics and Nonlinear Optics of Liquid Crystals (World Scientific, 1993), Chap. 2.

Yeh, P.

Zeks, B.

S. Suwa, H. Hoshi, Y. Takanishi, K. Ishikawa, H. Takezoe, and B. Zeks, “Helix unwinding process in a short-pitch ferroelectric liquid crystal mixture studied by conoscopy,” Jpn. J. Appl. Phys., Part 1 42, 1335-1337 (2003).
[CrossRef]

Appl. Opt. (1)

Europhys. Lett. (1)

J. K. Song, U. Manna, and J. K. Vij, “Mechanism of field induced unwinding of SmC* helix and bias field dependence of dielectric permittivity and macroscopic spontaneous polarization,” Europhys. Lett. 82, 26003 (2008).
[CrossRef]

Ferroelectrics (1)

J. K. Song, A. D. L. Chandani, O. E. Panarina, A. Fukuda, J. K. Vij, V. Goertz, and J. W. Goodby, “Study of the SmC* phase in the Tokyo mixture by conoscopy using tilted cell,” Ferroelectrics 344, 41-47 (2006).
[CrossRef]

J. Appl. Phys. (1)

J. Li and S.-T. Wu, “Self-consistency of Vuks equations for liquid-crystal refractive indices,” J. Appl. Phys. 96, 6253-6258 (2004).
[CrossRef]

J. Mater. Chem. (1)

A. Fukuda, Y. Takanishi, T. Isozaki, K. Ishikawa, and H. Takezoe, “Antiferroelectric chiral smectic liquid crystals,” J. Mater. Chem. 4, 997-1016 (1994).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (1)

I. Abdulhalim, “Analytic propagation matrix method for linear optics of arbitrary biaxial layered media,” J. Opt. A, Pure Appl. Opt. 1, 646-653 (1999).
[CrossRef]

J. Opt. Soc. Am. (2)

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

J. Phys. (Paris) (1)

Y. Galerne, “Refractive index measurements at a 2nd order smectic A to C phase transition,” J. Phys. (Paris) 39, 1311-1316 (1978).
[CrossRef]

Jpn. J. Appl. Phys. (1)

E. Gorecka, A. D. L. Chandani, Y. Ouchi, H. Takezoe, and A. Fukuda, “Molecular orientational structures in ferroelectric, ferrielectric and antiferroelectric smectic liquid crystal phases as studied by conoscope observation,” Jpn. J. Appl. Phys. 29, 131-137 (1990).
[CrossRef]

Jpn. J. Appl. Phys., Part 1 (2)

T. Fujikawa, K. Hiraoka, T. Isozaki, K. Kajikawa, H. Takezoe, and A. Fukuda, “Construction of dynamic conoscope observation system using CCD camera and image processor,” Jpn. J. Appl. Phys., Part 1 32, 985-988 (1993).
[CrossRef]

S. Suwa, H. Hoshi, Y. Takanishi, K. Ishikawa, H. Takezoe, and B. Zeks, “Helix unwinding process in a short-pitch ferroelectric liquid crystal mixture studied by conoscopy,” Jpn. J. Appl. Phys., Part 1 42, 1335-1337 (2003).
[CrossRef]

Liq. Cryst. (1)

S. Suwa, Y. Takanishi, H. Hoshi, K. Ishikawa, and H. Takezoe, “Helix unwinding process in the chiral smectic C phase of MHPOBC as observed by conoscopy,” Liq. Cryst. 30, 499-505 (2003).
[CrossRef]

Phys. Rev. E (2)

J. K. Song, J. K. Vij, and I. Kobayashi, “Interlayer interactions and the dependence of biaxiality of the chiral smectic-C * phase on electric field in the helical unwinding process,” Phys. Rev. E 75, 051705 (2007).
[CrossRef]

J. K. Song, A. D. L. Chandani, A. Fukuda, J. K. Vij, I. Kobayashi, and A. V. Emelyanenko, “Temperature-induced sign reversal of biaxiality observed by conoscopy in some ferroelectric Sm-C * liquid crystals,” Phys. Rev. E 76, 011709 (2007).
[CrossRef]

Other (4)

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999), Chap. 14.4.

D. Dunmur and K. Toriyama, Handbook of Liquid Crystals (Wiley-VCH, 1998), Vol. 1, Chap. 7, Sec. 3.

I.-C. Khoo and S.-T. Wu, Optics and Nonlinear Optics of Liquid Crystals (World Scientific, 1993), Chap. 2.

One of the examples is the Conoscope manufactured by Autronic-Melchers GmbH, Karlsruhe, Germany.

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Figures (6)

Fig. 1
Fig. 1

Schematic for conoscopy: (a) setup, (b) a cell, and (c) the chemical structure of liquid crystals used. The x y plane is the screen plane, and the x, y, and z coordinate system is for the cell. The directions of two optic axes thick lines (blue online), a dark line dashed lines (red online) and the tilt direction of the refractive index ellipsoid (black thin line) are required for calculating the biaxiality and the apparent tilt angle Θ. Homeotropically aligned smectic C liquid crystals have a layered structure. In each smectic layer, the average direction of molecules is tilted from the normal to the cell by θ, called the tilt angle, and the tilting direction, ϕ, varies vertically from layer to layer and forms a helical structure. α and β are the angles of the optic axes and the Nth dark ring from the major axis of the index ellipsoid, respectively (N can be arbitrararily chosen, and here, N = 2 ). The directional vector ( ϑ , φ ) in the cell corresponds to the point ( x 0 , y 0 ) on the screen. ϑ and φ are the polar and azimuthal angles of the direction of the light in the cell. The chemical name of the laboratory named compound (RRI, Bangalore) 3.b.5 is [2s,3s] 4-(2-Chloro3-methlypentyloxycarbonyl) phenyl trans 4-n-decyloxycinnamate.

Fig. 2
Fig. 2

Simulated transmittances as a function of pitch or the cell thickness in TN mode by using the 4 × 4 matrix method ( T ( 4 × 4 ) ) and average-refractive-index method ( T ( average ) ) . Simulation condition: 90 ° twist nematic, n o = 1.5 , and n e = 1.7 .

Fig. 3
Fig. 3

(a), (b) Simulated conoscopic images and (c) experimentally obtained examples. The first images indexed by (ave.) in (a) and (b) were obtained by the average-refractive method, and all the others were from the 4 × 4 matrix method. The numbers in parentheses are the length of the pitch for the simulation condition of the 4 × 4 matrix method. The first two images in (c) were taken by using pure ( R ) MHPOOCBC ( p 0.4 μ m ) and a racemic mixture ( p 2 μ m ) of ( R ) MHPOOCBC 65% and ( S ) MHPOOCBC 35%, respectively, and the last one was taken by using Felix018 ( p 25 μ m ) .

Fig. 4
Fig. 4

Simulated transmittance as a function of the angle of light with respect to n e along the line passing the two optic axes in the simulated conoscopic images in Fig. 3 [see the inset image in (a)]. The pitches in the insets are the simulation condition for the 4 × 4 matrix method, and (ave.) indicates the average-refractive-index method. Points marked with arrows (blue online) are optic axes.

Fig. 5
Fig. 5

Transmittance along the curve passing between the two optic axes at wavelengths (red and blue online) obtained by using 3.b.5 (experimental conditions: E = 200 V mm , 40 ° C ). The chemical name of the laboratory named compound (RRI, Bangalore) 3.b.5 is [2s,3s] 4-(2-Chloro3-methyl- pentyloxycarbonyl) phenyl trans 4-n-decyloxycinnamate.

Fig. 6
Fig. 6

(a) Director distribution estimated from the conoscopic image using a racemic mixture of MHPOOCBC (right inset image), and the reproduced conoscopic image from the calculated director distribution (left inset image). The experimental conoscopic image was obtained by using the tilted method, so the center point is shifted. The circle in the simulated conoscopic image denotes the observable range in the actual experiment. (b) Helix distortion by the applied field in the cell. Initial uniform helix (dashed line, blue online) changes to the distorted helical structure (solid line, red online) by the field.

Equations (9)

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n o 1 = [ ( cos ζ n e ) 2 + ( sin ζ n o 1 ) 2 ] 1 2 ,
n o 1 l o 1 n o 2 l o 2 = m λ ( m = 0 , 1 , 2 ) ,
d [ n o 1 2 n o 1 2 sin 2 ϑ n o 2 2 n o 2 2 sin 2 ϑ ] = m λ ( m = 0 , 1 , 2 ) .
n i 2 1 n 2 + 2 = 4 π N α i i 3 ,
n i 2 = 1 m j = 1 , 2 , , m [ 4 π 3 N α i i j ( n 2 + 2 ) + 1 ] = 4 π N ( n 2 + 2 ) 3 [ 1 m j = 1 , 2 , , m ( α i i j ) ] + 1 = 4 π N α i i ( n 2 + 2 ) 3 + 1 .
n i j 2 4 π 3 N α i i j ( n 2 + 2 ) + 1 ,
n φ ̂ 2 n φ ̂ j 2 j = 1 , 2 , , m .
ϕ ( z ) q z + ϕ 0 sin q z + ϕ 1 sin ( 2 q z ) + ,
ϑ = sin 1 [ sin ( ϑ ) n 0 ] ,

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