Abstract

A model is presented for computing the conoscopic figures produced by a uniformly oriented liquid crystal (LC) in a thin cell observed with a polarizing microscope. The method takes account of refraction and primary reflection at each interface within the cell but neglects multiple reflections. It is shown that the form of the isochromes in the figures is given qualitatively by the intersections of Bertin’s surfaces with the LC surface but that refraction also influences the form of the isochromes. Examples of calculated conoscopic figures are presented, including the symmetric flash figure and the optic-normal figure. The way in which these are altered by small changes in the optic-axis orientation is demonstrated. To complete the model, the polarizing effect of the microscope condenser and objective lens systems is estimated by calculating the Fresnel coefficients for model lens systems; the effect on the conoscopic figures is noted. Finally the director tilt in a 7-μm chiral smectic C LC cell is determined by comparing an experimentally measured figure with a set of calculated figures.

© 1990 Optical Society of America

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References

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  1. E. E. Wahlström, Optical Crystallography, 3rd ed. (Wiley, New York, 1962).
  2. P. G. de Gennes, The Physics of Liquid Crystals (Oxford U. Press,Oxford, 1975).
  3. D. W. Berreman, “Optics in smoothly varying anisotropic planar structures: application to liquid crystal twist cells,” J. Opt. Soc. Am. 63, 1374–1380 (1973).
    [CrossRef]
  4. H. Wöhler, G. Haas, M. Fritsch, D. A. Mlynski, “Faster 4 × 4 matrix method for uniaxial inhomogeneous media,” J. Opt. Soc. Am. A 5, 1554–1557 (1988).
    [CrossRef]
  5. P. Yeh, Optical Waves in Layered Media (Wiley, Chichester, UK, 1988).
  6. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).
  7. E. Hecht, A. Zajac, Optics (Addison-Wesley, Reading, Mass., 1974).
  8. R. Kingslake, Lens Design Fundamentals (Academic, New York, 1978).
  9. E. L. Hall, R. P. Kruger, S. J. Dwyer, D. L. Hall, R. W. McLaren, G. S. Lodwick, “A survey of processing and feature extraction techniques for radiographic images,” IEEE Trans. Comput. C-20, 1032–1044 (1971).
    [CrossRef]
  10. F. D. Bloss, An Introduction to the Methods of Optical Crystallography (Holt, Rinehart & Winston, New York, 1961).
  11. W. B. Kamb, “Isogyres in interference figures,” Am. Mineral 43, 1029–1067 (1958).
  12. A. R. MacGregor, “Method for computing the optical properties of a smectic C* liquid crystal cell,” J. Opt. Soc. Am. A 6, 1493–1503 (1989).
    [CrossRef]
  13. S. J. Elston, J. R. Sambles, M. G. Clark, “Determination of the director alignment in a ferroelectric liquid crystal device by observation of optical modes,” J. Mod. Opt. 36, 1019–1025 (1989).
    [CrossRef]
  14. H. Goldstein, Classical Mechanics, 2nd ed. (Addison-Wesley, Reading, Mass., 1980).
  15. H. Birecki, F. J. Kahn, “Effects of cell and material properties on multiplexing levels of twisted nematic liquid crystal displays,” in The Physics and Chemistry of Liquid Crystal Devices, G. J. Sprokel, ed. (Plenum, New York, 1980), pp. 125–142.

1989 (2)

A. R. MacGregor, “Method for computing the optical properties of a smectic C* liquid crystal cell,” J. Opt. Soc. Am. A 6, 1493–1503 (1989).
[CrossRef]

S. J. Elston, J. R. Sambles, M. G. Clark, “Determination of the director alignment in a ferroelectric liquid crystal device by observation of optical modes,” J. Mod. Opt. 36, 1019–1025 (1989).
[CrossRef]

1988 (1)

1973 (1)

1971 (1)

E. L. Hall, R. P. Kruger, S. J. Dwyer, D. L. Hall, R. W. McLaren, G. S. Lodwick, “A survey of processing and feature extraction techniques for radiographic images,” IEEE Trans. Comput. C-20, 1032–1044 (1971).
[CrossRef]

1958 (1)

W. B. Kamb, “Isogyres in interference figures,” Am. Mineral 43, 1029–1067 (1958).

Berreman, D. W.

Birecki, H.

H. Birecki, F. J. Kahn, “Effects of cell and material properties on multiplexing levels of twisted nematic liquid crystal displays,” in The Physics and Chemistry of Liquid Crystal Devices, G. J. Sprokel, ed. (Plenum, New York, 1980), pp. 125–142.

Bloss, F. D.

F. D. Bloss, An Introduction to the Methods of Optical Crystallography (Holt, Rinehart & Winston, New York, 1961).

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

Clark, M. G.

S. J. Elston, J. R. Sambles, M. G. Clark, “Determination of the director alignment in a ferroelectric liquid crystal device by observation of optical modes,” J. Mod. Opt. 36, 1019–1025 (1989).
[CrossRef]

de Gennes, P. G.

P. G. de Gennes, The Physics of Liquid Crystals (Oxford U. Press,Oxford, 1975).

Dwyer, S. J.

E. L. Hall, R. P. Kruger, S. J. Dwyer, D. L. Hall, R. W. McLaren, G. S. Lodwick, “A survey of processing and feature extraction techniques for radiographic images,” IEEE Trans. Comput. C-20, 1032–1044 (1971).
[CrossRef]

Elston, S. J.

S. J. Elston, J. R. Sambles, M. G. Clark, “Determination of the director alignment in a ferroelectric liquid crystal device by observation of optical modes,” J. Mod. Opt. 36, 1019–1025 (1989).
[CrossRef]

Fritsch, M.

Goldstein, H.

H. Goldstein, Classical Mechanics, 2nd ed. (Addison-Wesley, Reading, Mass., 1980).

Haas, G.

Hall, D. L.

E. L. Hall, R. P. Kruger, S. J. Dwyer, D. L. Hall, R. W. McLaren, G. S. Lodwick, “A survey of processing and feature extraction techniques for radiographic images,” IEEE Trans. Comput. C-20, 1032–1044 (1971).
[CrossRef]

Hall, E. L.

E. L. Hall, R. P. Kruger, S. J. Dwyer, D. L. Hall, R. W. McLaren, G. S. Lodwick, “A survey of processing and feature extraction techniques for radiographic images,” IEEE Trans. Comput. C-20, 1032–1044 (1971).
[CrossRef]

Hecht, E.

E. Hecht, A. Zajac, Optics (Addison-Wesley, Reading, Mass., 1974).

Kahn, F. J.

H. Birecki, F. J. Kahn, “Effects of cell and material properties on multiplexing levels of twisted nematic liquid crystal displays,” in The Physics and Chemistry of Liquid Crystal Devices, G. J. Sprokel, ed. (Plenum, New York, 1980), pp. 125–142.

Kamb, W. B.

W. B. Kamb, “Isogyres in interference figures,” Am. Mineral 43, 1029–1067 (1958).

Kingslake, R.

R. Kingslake, Lens Design Fundamentals (Academic, New York, 1978).

Kruger, R. P.

E. L. Hall, R. P. Kruger, S. J. Dwyer, D. L. Hall, R. W. McLaren, G. S. Lodwick, “A survey of processing and feature extraction techniques for radiographic images,” IEEE Trans. Comput. C-20, 1032–1044 (1971).
[CrossRef]

Lodwick, G. S.

E. L. Hall, R. P. Kruger, S. J. Dwyer, D. L. Hall, R. W. McLaren, G. S. Lodwick, “A survey of processing and feature extraction techniques for radiographic images,” IEEE Trans. Comput. C-20, 1032–1044 (1971).
[CrossRef]

MacGregor, A. R.

McLaren, R. W.

E. L. Hall, R. P. Kruger, S. J. Dwyer, D. L. Hall, R. W. McLaren, G. S. Lodwick, “A survey of processing and feature extraction techniques for radiographic images,” IEEE Trans. Comput. C-20, 1032–1044 (1971).
[CrossRef]

Mlynski, D. A.

Sambles, J. R.

S. J. Elston, J. R. Sambles, M. G. Clark, “Determination of the director alignment in a ferroelectric liquid crystal device by observation of optical modes,” J. Mod. Opt. 36, 1019–1025 (1989).
[CrossRef]

Wahlström, E. E.

E. E. Wahlström, Optical Crystallography, 3rd ed. (Wiley, New York, 1962).

Wöhler, H.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

Yeh, P.

P. Yeh, Optical Waves in Layered Media (Wiley, Chichester, UK, 1988).

Zajac, A.

E. Hecht, A. Zajac, Optics (Addison-Wesley, Reading, Mass., 1974).

Am. Mineral (1)

W. B. Kamb, “Isogyres in interference figures,” Am. Mineral 43, 1029–1067 (1958).

IEEE Trans. Comput. (1)

E. L. Hall, R. P. Kruger, S. J. Dwyer, D. L. Hall, R. W. McLaren, G. S. Lodwick, “A survey of processing and feature extraction techniques for radiographic images,” IEEE Trans. Comput. C-20, 1032–1044 (1971).
[CrossRef]

J. Mod. Opt. (1)

S. J. Elston, J. R. Sambles, M. G. Clark, “Determination of the director alignment in a ferroelectric liquid crystal device by observation of optical modes,” J. Mod. Opt. 36, 1019–1025 (1989).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Other (9)

H. Goldstein, Classical Mechanics, 2nd ed. (Addison-Wesley, Reading, Mass., 1980).

H. Birecki, F. J. Kahn, “Effects of cell and material properties on multiplexing levels of twisted nematic liquid crystal displays,” in The Physics and Chemistry of Liquid Crystal Devices, G. J. Sprokel, ed. (Plenum, New York, 1980), pp. 125–142.

P. Yeh, Optical Waves in Layered Media (Wiley, Chichester, UK, 1988).

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

E. Hecht, A. Zajac, Optics (Addison-Wesley, Reading, Mass., 1974).

R. Kingslake, Lens Design Fundamentals (Academic, New York, 1978).

F. D. Bloss, An Introduction to the Methods of Optical Crystallography (Holt, Rinehart & Winston, New York, 1961).

E. E. Wahlström, Optical Crystallography, 3rd ed. (Wiley, New York, 1962).

P. G. de Gennes, The Physics of Liquid Crystals (Oxford U. Press,Oxford, 1975).

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Figures (12)

Fig. 1
Fig. 1

(a) Reflection and refraction of light incident upon a uniaxial slab within an isotropic medium. The propagation directions of the incident, reflected, forward-propagating ordinary and extraordinary, and backward-propagating ordinary and extraordinary waves are defined by the wave vectors K, K′, Ko, Ke, Ko, and Ke, respectively. ŝ is a unit vector perpendicular to the plane of incidence, and p ̂ and p ̂ are unit vectors parallel to the plane of incidence. (b) Definition of the tilt (θ) and twist (ϕ) angles of the LC director n.

Fig. 2
Fig. 2

Schematic diagram of a LC cell, showing the rubbed-nylon alignment layers, transparent indium tin oxide (ITO) electrodes, and glass substrates. The z axis is parallel to the principal optic axis of the microscope.

Fig. 3
Fig. 3

Coordinate system used in the conoscopic calculations. The x′ and y′ axes are parallel to the polarizer and analyzer axes, respectively. Linearly polarized light is incident upon the LC cell in the yz plane (the z axis points into the page) with electric field components Ap and As parallel and perpendicular, respectively, to the plane of incidence. Only the components of the output electric field vectors Ap and As parallel to the y′ axis are transmitted by the analyzer. P and Q are the conjugate points in the input and output planes.

Fig. 4
Fig. 4

Lens triplet, consisting of a sphero-ellipsoidal (Lens 1) and two aplanatic (Lenses 2 and 3) lenses, used to model the microscope condenser lens system. The same triplet was reversed to model the objective lens system.

Fig. 5
Fig. 5

Center sections of the first three Bertin surfaces corresponding to phase differences of one, two, and three wavelengths. The surfaces connect points of equal phase difference between ordinary and extraordinary waves emanating from the point O.

Fig. 6
Fig. 6

The symmetric flash figure produced by a cell containing homogeneously aligned LC’s with no director tilt (θ = 0) and (a) ϕ′ = 0 and (b) ϕ′ = 3°. In (a) the isogyres form a dark cross, which breaks into two hyperbolic isogyres, as shown in (b), when the sample is rotated. (c) A cross section of the LC layer showing the cone of light and the Bertin surfaces. The intersection of the 2λ surface with the LC surface, the 2λ isochrome, causes the broadening of the vertical arms of the isogyres relative to the horizontal arms, but no other part of the isochrome is visible. (d) Rotation of the polarization of light by refraction at the surfaces of an isotropic slab on the microscope stage. The polarization rotation (away from the x′ axis) has been exaggerated by a factor of 4 and was calculated for a slab with a refractive index of 1.6. It is this effect that obscures the isochrome in (a) and (b).

Fig. 7
Fig. 7

Conoscopic figures produced by a LC cell with θ = 5°and (a) ϕ′ = 0 and (b) ϕ′ = 3°. The 2λ Bertin’s surface shown in (c) produces the isochrome in the top parts of (a) and (b). When the LC cell is rotated, the isogyre breaks in two, but the isochrome rotates unchanged with the cell.

Fig. 8
Fig. 8

(a) The optic-normal (θ = 90°) conoscopic figure and (b) the corresponding orientation of the Bertin surfaces. No isochromes are visible because none of the Bertin surfaces intersects the LC surface within the field of view.

Fig. 9
Fig. 9

Conoscopic figures produced by a LC cell with θ = 85°and (a) ϕ′ = 0 and (b) ϕ′ = 45°. The center of the isogyre cross lies directly over the LC optic axis in both figures, and the cross retains its orientation as the LC cell is rotated.

Fig. 10
Fig. 10

Conoscopic figures produced by a LC cell with (a) θ = 10°, (b) θ = 20°, (c) θ = 30°, (d) θ = 40°, (e) θ = 50°, (f) θ = 60°, (g) θ = 70°, and (h) θ = 80°. As the tilt angle increases, the 1λ isochrome and then the optic axis enter the field of view, and finally the 2λ isochrome leaves the field of view.

Fig. 11
Fig. 11

(a) The isotropic cross produced by a pair of the triplet lens systems shown in Fig. 4 when the LC cell is removed. This figure is a result of the lens-induced polarization rotation shown (exaggerated by a factor of 4) in (b). When the lens-induced polarization is taken into account, the conoscopic figures are altered as shown in (c) for θ = 5°[cf. Figs. 7(a) and 7(d)] for θ = 10°[cf. Fig. 10(a)].

Fig. 12
Fig. 12

(a) Conoscopic figure produced by a 7-μm-thick layer of SmC* LC. This was compared with calculated conoscopic figures with tilt angles (θ) of 0°to 5°in 1°steps. The closest match was with the θ = 2° figure shown (b) as image processed and (c) as calculated.

Tables (1)

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Table 1 Optical Constants Used in the Conoscopic Calculations

Equations (46)

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E i ( z = 0 ) = ( A s ŝ + A p p ̂ ) exp ( i ω t ) ,
E r ( z = 0 ) = ( B s ŝ + B p p ̂ ) exp ( i ω t ) ,
E t ( z = 0 ) = ( C o ô + C e ê ) exp ( i ω t ) ,
p ̂ = K × ŝ / | K | ,
p ̂ = K × ŝ / | K | .
K = β y + K z z ,
K = β y K z z ,
K o = β y + K o z z ,
K e = β y + K e z z .
β = ( ω / c ) n sin χ ,
K z = ( ω / c ) n cos χ .
χ o = arctan ( β / K o z ) ,
χ e = arctan ( β / K e z ) .
n e ( χ e ) = K e z ( ω / c ) cos χ e .
ô = N o ( β cos ϕ , β sin θ sin ϕ K o z cos θ , 0 ) T ,
ê = N e [ K e z cos θ β sin θ sin ϕ n e 2 ( χ e ) n o 2 , β cos ϕ n e 2 ( χ e ) n o 2 , K e z sin θ + β cos θ sin ϕ n e 2 ( χ e ) n e 2 ] T ,
E i ( z = d ) [ C o ô exp ( i K o z d ) + C e ê exp ( i K e z d ) ] exp ( i ω t ) ,
E r ( z = d ) = ( C o ô + ( C e ê ) exp ( i ω t ) ,
E t ( z = d ) = ( A s ŝ + A p p ̂ ) exp ( i ω t ) ,
K o = β y + K o z z ,
K e = β y + K e z z .
T p = E p ( 2 n 1 cos χ 1 n 2 cos χ 1 + n 1 cos χ 2 ) ,
T s = E s ( 2 n 1 cos χ 1 n 1 cos χ 1 + n 2 cos χ 2 ) ,
tan χ = r tan χ max r max .
A s = E cos ρ ,
A p = E sin ρ .
T = A s sin ρ + A p cos ρ ,
χ 2 = arcsin ( n sin χ 3 n g l ) ,
θ t 1 = arctan ( n sin χ 1 n g l n cos χ 1 ) .
Δ = ( K e z K o z ) d .
= [ sin θ cos ϕ sin ϕ cos θ cos ϕ sin θ sin ϕ cos ϕ cos θ sin ϕ cos θ 0 sin θ ] .
( K o ) ell = T K o = ( K o z cos θ β sin θ sin ϕ , β cos ϕ , K o z sin θ + β cos θ sin ϕ ) T ,
( K e ) ell = T K e = ( K e z cos θ β sin θ sin ϕ , β cos ϕ , K e z sin θ + β cos θ sin ϕ ) T .
K o 2 n o 2 = ω 2 c 2 ,
K e a 2 + K e b 2 n e 2 + K e c 2 n o 2 = ω 2 c 2 ,
K o z = ± [ ( n o ω c ) 2 β 2 ] 1 / 2 ,
K e z = β 13 33 ± ( ) 1 / 2 33 [ 33 ( ω c ) 2 β 2 ( 1 Δ cos 2 θ cos 2 ϕ ) ] ,
13 = Δ sin θ cos θ sin ϕ , 33 = + Δ sin 2 θ , Δ = , = n e 2 , = n o 2 .
ŝ o = ( K o ) ell / | K o | = ( s o a , s o b , s o c ) T ,
ŝ e = ( K e ) ell / | K e | = ( s e a , s e b , s e c ) T .
( ô ) ell = ( s o b , s o a , 0 ) T ,
( ê ) ell = [ s e a n e 2 ( χ e ) n o 2 , s e b n e 2 ( χ e ) n o 2 , s e c n e 2 ( χ e ) n e 2 ] T ,
( ô ) ell = N o ( β cos ϕ , β sin θ sin ϕ K o z cos θ , 0 ) T ,
( ê ) ell = N e [ K e z cos θ β sin θ sin ϕ n e 2 ( χ e ) n o 2 , β cos ϕ n e 2 ( χ e n o 2 ) , K e z sin θ + β cos θ sin ϕ n e 2 ( χ e ) n e 2 ] T ,
ô = ( ô ) ell ,
ê = ( ê ) ell .

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