Abstract

Optical excitation of fully leaky guided modes is used to characterize in detail the optical tensor profile in a liquid-crystal layer confined between two standard glass plates as for a conventional cell. The angle-dependent reflectivity and transmissivity of such a structure are explored analytically, numerically, and experimentally. By suitable choice of incident and detected polarizations it is shown possible to obtain a detailed characterization of the director profile in the cell even though the leaky optical modes lead to rather broad features in the recorded data. Using a two-prism coupling technique, matching the prisms to the glass of the cell with identical index-matching fluid, allows access to sets of both reflectivity data and transmissivity data over a sufficient range of in-plane photon wave vectors to yield unambiguous director profiles when the data are compared with modeling theory. The specific cell explored in this study contains a homogeneously aligned ferroelectric smectic material in which there is a cusped, chevron, director profile. The results presented for such a complex structure show that this powerful new form of quantified conoscopy is likely to provide the primary route forward for optical characterization of conventional cells of this nature.

© 1999 Optical Society of America

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References

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  1. K. R. Welford, J. R. Sambles, and M. G. Clark, “Guided modes and surface plasmon-polaritons observed with a nematic liquid crystal using attenuated total reflection,” Liq. Cryst. 2, 91 (1987).
    [CrossRef]
  2. S. J. Elston and J. R. Sambles, “The configuration in a ferroelectric liquid crystal cell in terms of a rigid chevron structure,” Mol. Cryst. Liq. Cryst. 200, 167 (1991).
    [CrossRef]
  3. S. Ito, F. Kremer, T. Fischer, and K. Knoll, “Guided optical waves in a ferroelectric liquid crystal layer: a birefringence analysis of molecular orientation on the switching process,” Mol. Cryst. Liq. Cryst. 264, 99 (1995).
    [CrossRef]
  4. C. R. Lavers and J. R. Sambles, “An examination of the optical dielectric tensor of a liquid crystal waveguide,” Ferroelectrics 113, 339 (1991).
    [CrossRef]
  5. Fuzi Yang and J. R. Sambles, “Optical characterization of liquid crystals by means of half-leaky guided modes,” J. Opt. Soc. Am. B 10, 858 (1993).
    [CrossRef]
  6. Fuzi Yang and J. R. Sambles, “The optical tensor configuration in a surface stabilized ferroelectric liquid crystal determined by using half-leaky guided modes,” Liq. Cryst. 13, 1 (1993).
    [CrossRef]
  7. Fuzi Yang and J. R. Sambles, “Optical characterization of a uniaxial material using the polarization-conversion reflectivity technique,” J. Opt. Soc. Am. B 11, 605 (1994).
    [CrossRef]
  8. D. Y. K. Ko and J. R. Sambles, “Scattering matrix method for propagation of radiation in stratified media: attenuated total reflection studies of liquid crystals,” J. Opt. Soc. Am. A 5, 1863 (1988).
    [CrossRef]
  9. D. J. Mikulin, D. A. Coley, and J. R. Sambles, “Fitting reflectivity data from liquid crystal cells using genetic algorithms,” Liq. Cryst. 22, 301 (1997).
    [CrossRef]
  10. N. Itoh, M. Koden, S. Miyoshi, and T. Wada, “Study of molecular orientational states of ferroelectric liquid crystals in a surface stabilized geometry,” Liq. Cryst. 15, 669 (1993).
    [CrossRef]

1997

D. J. Mikulin, D. A. Coley, and J. R. Sambles, “Fitting reflectivity data from liquid crystal cells using genetic algorithms,” Liq. Cryst. 22, 301 (1997).
[CrossRef]

1995

S. Ito, F. Kremer, T. Fischer, and K. Knoll, “Guided optical waves in a ferroelectric liquid crystal layer: a birefringence analysis of molecular orientation on the switching process,” Mol. Cryst. Liq. Cryst. 264, 99 (1995).
[CrossRef]

1994

1993

N. Itoh, M. Koden, S. Miyoshi, and T. Wada, “Study of molecular orientational states of ferroelectric liquid crystals in a surface stabilized geometry,” Liq. Cryst. 15, 669 (1993).
[CrossRef]

Fuzi Yang and J. R. Sambles, “The optical tensor configuration in a surface stabilized ferroelectric liquid crystal determined by using half-leaky guided modes,” Liq. Cryst. 13, 1 (1993).
[CrossRef]

Fuzi Yang and J. R. Sambles, “Optical characterization of liquid crystals by means of half-leaky guided modes,” J. Opt. Soc. Am. B 10, 858 (1993).
[CrossRef]

1991

C. R. Lavers and J. R. Sambles, “An examination of the optical dielectric tensor of a liquid crystal waveguide,” Ferroelectrics 113, 339 (1991).
[CrossRef]

S. J. Elston and J. R. Sambles, “The configuration in a ferroelectric liquid crystal cell in terms of a rigid chevron structure,” Mol. Cryst. Liq. Cryst. 200, 167 (1991).
[CrossRef]

1988

1987

K. R. Welford, J. R. Sambles, and M. G. Clark, “Guided modes and surface plasmon-polaritons observed with a nematic liquid crystal using attenuated total reflection,” Liq. Cryst. 2, 91 (1987).
[CrossRef]

Clark, M. G.

K. R. Welford, J. R. Sambles, and M. G. Clark, “Guided modes and surface plasmon-polaritons observed with a nematic liquid crystal using attenuated total reflection,” Liq. Cryst. 2, 91 (1987).
[CrossRef]

Coley, D. A.

D. J. Mikulin, D. A. Coley, and J. R. Sambles, “Fitting reflectivity data from liquid crystal cells using genetic algorithms,” Liq. Cryst. 22, 301 (1997).
[CrossRef]

Elston, S. J.

S. J. Elston and J. R. Sambles, “The configuration in a ferroelectric liquid crystal cell in terms of a rigid chevron structure,” Mol. Cryst. Liq. Cryst. 200, 167 (1991).
[CrossRef]

Fischer, T.

S. Ito, F. Kremer, T. Fischer, and K. Knoll, “Guided optical waves in a ferroelectric liquid crystal layer: a birefringence analysis of molecular orientation on the switching process,” Mol. Cryst. Liq. Cryst. 264, 99 (1995).
[CrossRef]

Ito, S.

S. Ito, F. Kremer, T. Fischer, and K. Knoll, “Guided optical waves in a ferroelectric liquid crystal layer: a birefringence analysis of molecular orientation on the switching process,” Mol. Cryst. Liq. Cryst. 264, 99 (1995).
[CrossRef]

Itoh, N.

N. Itoh, M. Koden, S. Miyoshi, and T. Wada, “Study of molecular orientational states of ferroelectric liquid crystals in a surface stabilized geometry,” Liq. Cryst. 15, 669 (1993).
[CrossRef]

Knoll, K.

S. Ito, F. Kremer, T. Fischer, and K. Knoll, “Guided optical waves in a ferroelectric liquid crystal layer: a birefringence analysis of molecular orientation on the switching process,” Mol. Cryst. Liq. Cryst. 264, 99 (1995).
[CrossRef]

Ko, D. Y. K.

Koden, M.

N. Itoh, M. Koden, S. Miyoshi, and T. Wada, “Study of molecular orientational states of ferroelectric liquid crystals in a surface stabilized geometry,” Liq. Cryst. 15, 669 (1993).
[CrossRef]

Kremer, F.

S. Ito, F. Kremer, T. Fischer, and K. Knoll, “Guided optical waves in a ferroelectric liquid crystal layer: a birefringence analysis of molecular orientation on the switching process,” Mol. Cryst. Liq. Cryst. 264, 99 (1995).
[CrossRef]

Lavers, C. R.

C. R. Lavers and J. R. Sambles, “An examination of the optical dielectric tensor of a liquid crystal waveguide,” Ferroelectrics 113, 339 (1991).
[CrossRef]

Mikulin, D. J.

D. J. Mikulin, D. A. Coley, and J. R. Sambles, “Fitting reflectivity data from liquid crystal cells using genetic algorithms,” Liq. Cryst. 22, 301 (1997).
[CrossRef]

Miyoshi, S.

N. Itoh, M. Koden, S. Miyoshi, and T. Wada, “Study of molecular orientational states of ferroelectric liquid crystals in a surface stabilized geometry,” Liq. Cryst. 15, 669 (1993).
[CrossRef]

Sambles, J. R.

D. J. Mikulin, D. A. Coley, and J. R. Sambles, “Fitting reflectivity data from liquid crystal cells using genetic algorithms,” Liq. Cryst. 22, 301 (1997).
[CrossRef]

Fuzi Yang and J. R. Sambles, “Optical characterization of a uniaxial material using the polarization-conversion reflectivity technique,” J. Opt. Soc. Am. B 11, 605 (1994).
[CrossRef]

Fuzi Yang and J. R. Sambles, “The optical tensor configuration in a surface stabilized ferroelectric liquid crystal determined by using half-leaky guided modes,” Liq. Cryst. 13, 1 (1993).
[CrossRef]

Fuzi Yang and J. R. Sambles, “Optical characterization of liquid crystals by means of half-leaky guided modes,” J. Opt. Soc. Am. B 10, 858 (1993).
[CrossRef]

S. J. Elston and J. R. Sambles, “The configuration in a ferroelectric liquid crystal cell in terms of a rigid chevron structure,” Mol. Cryst. Liq. Cryst. 200, 167 (1991).
[CrossRef]

C. R. Lavers and J. R. Sambles, “An examination of the optical dielectric tensor of a liquid crystal waveguide,” Ferroelectrics 113, 339 (1991).
[CrossRef]

D. Y. K. Ko and J. R. Sambles, “Scattering matrix method for propagation of radiation in stratified media: attenuated total reflection studies of liquid crystals,” J. Opt. Soc. Am. A 5, 1863 (1988).
[CrossRef]

K. R. Welford, J. R. Sambles, and M. G. Clark, “Guided modes and surface plasmon-polaritons observed with a nematic liquid crystal using attenuated total reflection,” Liq. Cryst. 2, 91 (1987).
[CrossRef]

Wada, T.

N. Itoh, M. Koden, S. Miyoshi, and T. Wada, “Study of molecular orientational states of ferroelectric liquid crystals in a surface stabilized geometry,” Liq. Cryst. 15, 669 (1993).
[CrossRef]

Welford, K. R.

K. R. Welford, J. R. Sambles, and M. G. Clark, “Guided modes and surface plasmon-polaritons observed with a nematic liquid crystal using attenuated total reflection,” Liq. Cryst. 2, 91 (1987).
[CrossRef]

Yang, Fuzi

Ferroelectrics

C. R. Lavers and J. R. Sambles, “An examination of the optical dielectric tensor of a liquid crystal waveguide,” Ferroelectrics 113, 339 (1991).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Liq. Cryst.

K. R. Welford, J. R. Sambles, and M. G. Clark, “Guided modes and surface plasmon-polaritons observed with a nematic liquid crystal using attenuated total reflection,” Liq. Cryst. 2, 91 (1987).
[CrossRef]

Fuzi Yang and J. R. Sambles, “The optical tensor configuration in a surface stabilized ferroelectric liquid crystal determined by using half-leaky guided modes,” Liq. Cryst. 13, 1 (1993).
[CrossRef]

D. J. Mikulin, D. A. Coley, and J. R. Sambles, “Fitting reflectivity data from liquid crystal cells using genetic algorithms,” Liq. Cryst. 22, 301 (1997).
[CrossRef]

N. Itoh, M. Koden, S. Miyoshi, and T. Wada, “Study of molecular orientational states of ferroelectric liquid crystals in a surface stabilized geometry,” Liq. Cryst. 15, 669 (1993).
[CrossRef]

Mol. Cryst. Liq. Cryst.

S. J. Elston and J. R. Sambles, “The configuration in a ferroelectric liquid crystal cell in terms of a rigid chevron structure,” Mol. Cryst. Liq. Cryst. 200, 167 (1991).
[CrossRef]

S. Ito, F. Kremer, T. Fischer, and K. Knoll, “Guided optical waves in a ferroelectric liquid crystal layer: a birefringence analysis of molecular orientation on the switching process,” Mol. Cryst. Liq. Cryst. 264, 99 (1995).
[CrossRef]

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Figures (12)

Fig. 1
Fig. 1

Geometry for analyzing the FLGM method.

Fig. 2
Fig. 2

Schematic of a conventional liquid-crystal cell.

Fig. 3
Fig. 3

Model polarization-conversion reflectivity, Rsp, for a uniformly twisted slab of uniaxial material with 90° director tilt, having a thickness of 3 µm, with =2.7850+i0.0005 and =2.200+i0.0005. The solid curve is for a twist of 77°; the dashed curve is for a twist of 76°.

Fig. 4
Fig. 4

Model polarization-conversion reflectivity, Rsp, for the same structure as for Fig. 3 with a twist angle of 77°; for the continuous curve the tilt is 90°, while for the dashed curve the tilt is 89°.

Fig. 5
Fig. 5

Model reflectivities for a cell similar to that modeled in Figs. 3 and 4 but with a tilt of 90°. The continuous curves are for a fixed twist angle of 13° throughout the cell, while the dashed curves are for the same structure except for modified thin (100-nm) regions at both cell walls where the director twists linearly from the alignment direction, 0°, out to 13°: (a) Rpp and (b) Rsp.

Fig. 6
Fig. 6

Model polarization-conversion reflectivities Rps and Rsp for a cell similar to that modeled in Figs. 3 and 4 for different tilts: (a) 88°, (b) 90°.

Fig. 7
Fig. 7

Model reflectivities and transmissivities for a 90° twisted nematic cell with different tilt angles, 90° (solid curve), 89° (short dashed curve), and 91° (longer dashed curve): (a) Rpp, (b) Rss, (c) Rsp and Tsp, (d) Rps and Tps, and (e) Tpp and (f) Tss.

Fig. 8
Fig. 8

Model polarization-conversion transmissivity, Tsp, for a cell similar to that modeled in Fig. 5 with a tilt of 90°. The continuous curve is for a fixed twist angle of 13° throughout the cell, while the dashed curve is for the same structure except for modified thin (100-nm) regions at both cell walls where the director twists linearly from the alignment direction, 0°, out to 13°.

Fig. 9
Fig. 9

Schematic of the experimental geometry.

Fig. 10
Fig. 10

Prism-coupled cell geometry.

Fig. 11
Fig. 11

Experimental data (crosses) and fitted theory (curves) for a SCE8* cell: (a) Rpp and Tpp, (b) Rss and Tss, (c) Rps and Rsp, (d) Tsp and (e) Tps.

Fig. 12
Fig. 12

Fitted profile of the director in the SCE8* layer: (a) Twist angle measured from the incident plane xz, (b) tilt angle measured from the y axis, and (c) layer tilt angle calculated with a cone angle of 19.50° together with the twist/tilt profiles.

Equations (18)

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1/2 sin β=no sin αo,
1/2 sin β=ne(αe)sin αe,
cos ψ=cos θ cos αe-sin θ sin αe sin ϕ,
ne=+Δ cos2 ψ1/2,
Δ=-.
cos γ=sin α cos θ+sin θ cos α sin φ[1-(cos θ cos α-sin θ sin α sin ϕ)2]1/2.
rsp=2()1/2(sin θ cos ϕ cos β)(q cos θ-q sin θ sin ϕ)ΔD,
rps=2()1/2(sin θ cos ϕ cos β)(q cos θ+q sin θ sin ϕ)ΔD,
D=1/2(d1+d2)+cos β(d3+d4),
d1= cos2 β(l+mq),
d2=q(lq-q2+p),
d3=(lq-q2+33),
d4=(lq-mq2+),
l=[(33-mq2)]1/2,
33=+Δ cos2 θ=-Δ sin2 θ,
m=-Δ sin2 θ cos2 ϕ,
p=-Δ sin2 θ sin2 ϕ.
sin ϕ cos δ sin θ+sin δ cos θ=cos χ.

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