Abstract

Optical excitation of half-leaky guided modes has been used, for the first time to our knowledge, to characterize in detail the optical tensor profile in a liquid-crystal layer. The thin liquid-crystal layer is sandwiched between a high-index pyramid, with an index greater than the maximum index of the liquid crystal, and a glass substrate with an index lower than the minimum index of the liquid crystal. Analysis of this geometry shows that over a small angle range encompassed by a pseudocritical angle and a critical angle associated with the pyramid–liquid crystal and the pyramid–glass-substrate boundaries, respectively, there may exist sharp resonant features in the angle-dependent optical reflectivity. More particularly, if the director is tilted and twisted out of the plane of incidence there is strong TE-to-TM optical conversion over this small angular window, which is sensitive to details of the director profile. A 90° twisted, homogeneously aligned nematic liquid-crystal layer has been studied with this technique at two different wavelengths, 632.8 and 514.5 nm. Greater than 60% conversion from TM incident radiation to TE output has been recorded from sharp resonances in the half-leaky guided-wave angle window. If one fits these resonances with predictions from multilayer optics theory, one obtains extraordinary detail of the director profile in the cell. Since no metallic coatings are used, and all that is required are two glass plates of high and low index, respectively, the technique offers the potential of extremely useful applications in the examination of detailed director profiles in commercially fabricated cells.

© 1993 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. C. R. Lavers and J. R. Sambles, “An examination of the optical dielectric tensor of a liquid crystal waveguide,” Ferroelectrics 113, 339 (1991).
    [Crossref]
  4. R. T. Kersten, “The prism-film coupler as a precision instrument, part 1: accuracy and capabilities of prism couplers as instruments,” Opt. Acta 22, 503 (1975).
    [Crossref]
  5. J. Xue, N. A. Clark, and M. R. Meadows, “Surface orientation transitions in surface stabilized ferroelectric liquid crystal structures,” Appl. Phys. Lett. 53, 2397 (1988).
    [Crossref]
  6. M. Born and E. Wolfe, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975).
  7. S. J. Elston and J. R. Sambles, “Examination of near surface alignment in tilted smectic liquid crystal filled cells,” Mol. Cryst. Liq. Cryst. 208, 1 (1991).
    [Crossref]
  8. J. Cognard, “Alignment of nematic liquid crystals and their mixtures,” Mol. Cryst. Liq. Cryst. Suppl. 78, 1 (1981).
    [Crossref]
  9. D. Y. K. Ko and J. R. Sambles, “Scattering matrix method for propagation of radiation in stratified media: attentuated total reflection studies of liquid crystal,” J. Opt. Soc. Am. A 5, 1863 (1988).
    [Crossref]

1991 (3)

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]

S. J. Elston and J. R. Sambles, “Examination of near surface alignment in tilted smectic liquid crystal filled cells,” Mol. Cryst. Liq. Cryst. 208, 1 (1991).
[Crossref]

1988 (2)

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

J. Xue, N. A. Clark, and M. R. Meadows, “Surface orientation transitions in surface stabilized ferroelectric liquid crystal structures,” Appl. Phys. Lett. 53, 2397 (1988).
[Crossref]

1987 (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]

1981 (1)

J. Cognard, “Alignment of nematic liquid crystals and their mixtures,” Mol. Cryst. Liq. Cryst. Suppl. 78, 1 (1981).
[Crossref]

1975 (1)

R. T. Kersten, “The prism-film coupler as a precision instrument, part 1: accuracy and capabilities of prism couplers as instruments,” Opt. Acta 22, 503 (1975).
[Crossref]

Born, M.

M. Born and E. Wolfe, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975).

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]

Clark, N. A.

J. Xue, N. A. Clark, and M. R. Meadows, “Surface orientation transitions in surface stabilized ferroelectric liquid crystal structures,” Appl. Phys. Lett. 53, 2397 (1988).
[Crossref]

Cognard, J.

J. Cognard, “Alignment of nematic liquid crystals and their mixtures,” Mol. Cryst. Liq. Cryst. Suppl. 78, 1 (1981).
[Crossref]

Elston, S. J.

S. J. Elston and J. R. Sambles, “Examination of near surface alignment in tilted smectic liquid crystal filled cells,” Mol. Cryst. Liq. Cryst. 208, 1 (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]

Kersten, R. T.

R. T. Kersten, “The prism-film coupler as a precision instrument, part 1: accuracy and capabilities of prism couplers as instruments,” Opt. Acta 22, 503 (1975).
[Crossref]

Ko, D. Y. K.

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]

Meadows, M. R.

J. Xue, N. A. Clark, and M. R. Meadows, “Surface orientation transitions in surface stabilized ferroelectric liquid crystal structures,” Appl. Phys. Lett. 53, 2397 (1988).
[Crossref]

Sambles, J. R.

S. J. Elston and J. R. Sambles, “Examination of near surface alignment in tilted smectic liquid crystal filled cells,” Mol. Cryst. Liq. Cryst. 208, 1 (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: attentuated total reflection studies of liquid crystal,” 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]

Sambles, J.R.

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]

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]

Wolfe, E.

M. Born and E. Wolfe, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975).

Xue, J.

J. Xue, N. A. Clark, and M. R. Meadows, “Surface orientation transitions in surface stabilized ferroelectric liquid crystal structures,” Appl. Phys. Lett. 53, 2397 (1988).
[Crossref]

Appl. Phys. Lett. (1)

J. Xue, N. A. Clark, and M. R. Meadows, “Surface orientation transitions in surface stabilized ferroelectric liquid crystal structures,” Appl. Phys. Lett. 53, 2397 (1988).
[Crossref]

Ferroelectrics (1)

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 (1)

Liq. Cryst. (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]

Mol. Cryst. Liq. Cryst. (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]

S. J. Elston and J. R. Sambles, “Examination of near surface alignment in tilted smectic liquid crystal filled cells,” Mol. Cryst. Liq. Cryst. 208, 1 (1991).
[Crossref]

Mol. Cryst. Liq. Cryst. Suppl. (1)

J. Cognard, “Alignment of nematic liquid crystals and their mixtures,” Mol. Cryst. Liq. Cryst. Suppl. 78, 1 (1981).
[Crossref]

Opt. Acta (1)

R. T. Kersten, “The prism-film coupler as a precision instrument, part 1: accuracy and capabilities of prism couplers as instruments,” Opt. Acta 22, 503 (1975).
[Crossref]

Other (1)

M. Born and E. Wolfe, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975).

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Figures (12)

Fig. 1
Fig. 1

Geometry for analyzing the HLGM method.

Fig. 2
Fig. 2

Field distribution for (a) an s-polarized and (b) a p-polarized incident beam in the geometry shown at an internal angle of 55.32°. The parameters of the geometry are λ = 632.8 nm; pyramid: = 3.2400; ITO: = 3.6400 + i0.1000, thickness 40 nm; polyimide: = 3.0980 + i0.0400, thickness 50 nm; liquid crystal: xx = yy = 2.2490 + i0.0005, zz = 2.8610 + i0.0010, thickness 3.49 μm, twist angle, 77°, tilt angle, 90°; substrate: = 2.1228.

Fig. 3
Fig. 3

Reflectivities, (a) Rps, and (b) Rsp, versus internal angles of incidence for the HLGM geometry with the parameters the same as given in the caption of Fig. 2.

Fig. 4
Fig. 4

Reflectivities versus internal angle of incidence for (a) the HLGM geometry, Rps, and (b) the metal-clad waveguide geometry, Rpp. The parameters of the geometry in (a) are as for Fig. 2 with a twist angle of 77° (solid curve) and 78° (dashed curve) and a tilt angle of 90°. The parameters of the geometry in (b) are different only insofar as the ITO and polyimide are replaced by silver and SiOx, respectively. Silver coatings: = −17.62 + i0.62; thicknesses 46.5 and 150 nm for top and bottom layers. SiOx: = 2.4500 + i0.0010; thickness, 20 nm.

Fig. 5
Fig. 5

Reflectivities versus internal angle of incidence for (a) the HLGM geometry, Rps, and (b) the metal-clad waveguide geometry, Rpp. The parameters of both geometries are the same as shown in Fig. 4 except for a twist angle of 77° and a tilt angle of 90° (solid curve) and 91° (dashed curve).

Fig. 6
Fig. 6

Reflectivities versus internal angle of incidence for (a) the HLGM geometry, Rps, and (b) the metal-clad waveguide geometry, Rpp. The parameters of both geometries are the same as shown in Fig. 4 except for the liquid crystal: xx = yy = 2.2490 + i0.0005, zz = 2.8610 + i0.0010 (solid curve) and xx = 2.2490 + i0.0005, yy = 2.2540 + i0.0005, zz = 2.8610 + i0.0010 (dashed curve), and a twist angle of 77° and a rotation angle of 0°.

Fig. 7
Fig. 7

Reflectivities versus internal angle of incidence for (a) the HLGM geometry, Rps, and (b) the metal-clad waveguide geometry, Rpp. The parameters of both geometries are the same as shown in Fig. 4 except for the twist angle, which is 13° across the whole of the 3.49 μm cell (solid curve) and which varies linearly from 0° to 13° in the upper 0.2 μm of the cell, is constant at 13° across the central 3.09 μm, and then varies linearly from 13° to 26° in the lower 0.2 μm (dashed curve).

Fig. 8
Fig. 8

Sample geometry for examining cells fabricated with commercial alignment.

Fig. 9
Fig. 9

Sample geometry used in the experiment.

Fig. 10
Fig. 10

Experimental data (crosses) and theoretically fitted results (solid curves) for a wavelength λ = 632.8 nm for (a) Rpp and (b) Rps. The fitted parameters of the geometry are pyramid: = 3.2400; SiOx: = 2.4500 + i0.0010, thickness 20 nm; liquid crystal: xx = yy = 2.3140 + i0.0005, zz = 2.9850 + i0.0008, thickness 1.965 μm, twist angle 86.8°–6.8° ± 0.7°, tilt angle 90°; substrate: = 2.1495.

Fig. 11
Fig. 11

Experimental data (crosses) and theoretically fitted results (solid curves) for a wavelength of λ = 514.5 nm for (a) Rpp and (b) Rps. The fitted parameters of the geometry are pyramid: = 3.3133; SiOx: = 2.5500 + i0.0010, thickness 20 nm; liquid crystal: xx = yy = 2.3505 + i0.0005, zz = 3.0900 + i0.0012, thickness 1.965 μm, twist angle 86.4°–6.4° ± 0.8°, tilt angle 90°; substrate: = 2.1625.

Fig. 12
Fig. 12

Experimental reflectivity (Rps) versus internal angle of incidence for two different positions of the beam spot on the bottom surface of the pyramid (solid and dashed curves), for (a) λ = 632.8 nm and (b) λ = 514.5 nm.

Equations (15)

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β cl = sin - 1 ( 3 / 1 ) 1 / 2 .
β c 2 = sin - 1 ( / 1 ) 1 / 2 ,
( 1 ) 1 / 2 sin β = n e ( α ) sin α .
cos ψ = cos θ cos α - sin θ sin α sin ϕ ,
( n e cos ψ ) 2 + ( n e sin ψ ) 2 = 1 ,
n e = ( + Δ cos 2 ψ ) 1 / 2 ,
Δ = - .
( 1 ) 1 / 2 sin β = [ Δ ( cos θ cot α - sin θ sin ϕ ) 2 + ( 1 + cot 2 α ) ] 1 / 2 .
Δ ( cos θ cot α - sin θ sin ϕ ) 2 + ( 1 + cot 2 α )
cot α = Δ cos θ sin θ sin ϕ + Δ cos 2 θ .
β c 3 = sin - 1 [ 1 + ( - ) cos 2 θ + ( - ) ( cos 2 θ + sin 2 θ sin 2 ϕ ) ] 1 / 2 ,
( β c 3 ) max = sin - 1 ( / 1 ) 1 / 2 ( θ = 0 °             or             ϕ = 0 ° ) ,
( β c 3 ) min = sin - 1 ( / 1 ) 1 / 2 ( θ = 90 ° ,             ϕ = 90 ° ) .
β c l < β < β c 3 ( β c 2 ) .
cos γ = sin α cos θ + sin θ cos ϕ sin ϕ [ 1 - ( cos θ cos α - sin θ sin α sin ϕ ) 2 ] 1 / 2 .

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