Abstract

Optical transmittance and reflectance of continuously varying anisotropic planar media, such as nematic liquid crystals in Schadt–Helfrich twist cells or cholesterics between parallel rubbed surfaces, have previously been computed with a 4 × 4 matrix method by considering the medium as broken up into many thin parallel layers and treating each as if it had homogeneous anisotropic optical parameters. A matrix multiplication was done for each layer, and unless each layer was much less than one wavelength thick, several more multiplications were done within each layer. Here we show how to do numerical computations with equal accuracy using much thicker layers. We use a truncated power series to approximate the variation of optical parameters through each layer. We also show two ways to obtain fast convergence of numerical computations with layers of homogeneous anisotropic material that are several wavelengths thick. We use the method to get a better understanding of the optical properties of twist cells, particularly for oblique rays. The possibility of measuring elastic constants by comparing measured with computed transmittance of twist cells is suggested.

© 1973 Optical Society of America

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References

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  1. J. Billard, Thesis, University of Paris (1966).
  2. S. Teitler and B. Henvis, J. Opt. Soc. Am. 60, 830 (1970).
    [Crossref]
  3. D. W. Berreman and T. J. Scheffer, Phys. Rev. Lett. 25, 577 (1970).
    [Crossref]
  4. D. W. Berreman, J. Opt. Soc. Am. 62, 502 (1972).
    [Crossref]
  5. M. Schadt and W. Helfrich, Appl. Phys. Lett. 18, 127 (1971).
    [Crossref]
  6. C. W. Oseen, Ark. Mat. Astron. Fys. 19, 1 (1925).
  7. F. C. Frank, Discuss. Faraday Soc. 25, 19 (1958).
    [Crossref]
  8. I. Haller, J. Chem. Phys. 57, 1400 (1972).
    [Crossref]
  9. D. W. Berreman, 28, 1683 (1972).
  10. F. J. Kahn, Appl. Phys. Lett. 22, 386 (1973).
    [Crossref]
  11. A. Penz (private communication).
  12. H. Gruler (private communication).
  13. J. T. Jenkins, J. Fluid Mech. 45, 465 (1971).
    [Crossref]
  14. J. Nehring and A. Saupe, J. Chem. Phys. 54, 337 (1971).
    [Crossref]
  15. W. Helfrich, Appl. Phys. Lett. 17, 531 (1970).
    [Crossref]

1973 (1)

F. J. Kahn, Appl. Phys. Lett. 22, 386 (1973).
[Crossref]

1972 (2)

1971 (3)

M. Schadt and W. Helfrich, Appl. Phys. Lett. 18, 127 (1971).
[Crossref]

J. T. Jenkins, J. Fluid Mech. 45, 465 (1971).
[Crossref]

J. Nehring and A. Saupe, J. Chem. Phys. 54, 337 (1971).
[Crossref]

1970 (3)

W. Helfrich, Appl. Phys. Lett. 17, 531 (1970).
[Crossref]

S. Teitler and B. Henvis, J. Opt. Soc. Am. 60, 830 (1970).
[Crossref]

D. W. Berreman and T. J. Scheffer, Phys. Rev. Lett. 25, 577 (1970).
[Crossref]

1958 (1)

F. C. Frank, Discuss. Faraday Soc. 25, 19 (1958).
[Crossref]

1925 (1)

C. W. Oseen, Ark. Mat. Astron. Fys. 19, 1 (1925).

Berreman, D. W.

D. W. Berreman, J. Opt. Soc. Am. 62, 502 (1972).
[Crossref]

D. W. Berreman and T. J. Scheffer, Phys. Rev. Lett. 25, 577 (1970).
[Crossref]

D. W. Berreman, 28, 1683 (1972).

Billard, J.

J. Billard, Thesis, University of Paris (1966).

Frank, F. C.

F. C. Frank, Discuss. Faraday Soc. 25, 19 (1958).
[Crossref]

Gruler, H.

H. Gruler (private communication).

Haller, I.

I. Haller, J. Chem. Phys. 57, 1400 (1972).
[Crossref]

Helfrich, W.

M. Schadt and W. Helfrich, Appl. Phys. Lett. 18, 127 (1971).
[Crossref]

W. Helfrich, Appl. Phys. Lett. 17, 531 (1970).
[Crossref]

Henvis, B.

Jenkins, J. T.

J. T. Jenkins, J. Fluid Mech. 45, 465 (1971).
[Crossref]

Kahn, F. J.

F. J. Kahn, Appl. Phys. Lett. 22, 386 (1973).
[Crossref]

Nehring, J.

J. Nehring and A. Saupe, J. Chem. Phys. 54, 337 (1971).
[Crossref]

Oseen, C. W.

C. W. Oseen, Ark. Mat. Astron. Fys. 19, 1 (1925).

Penz, A.

A. Penz (private communication).

Saupe, A.

J. Nehring and A. Saupe, J. Chem. Phys. 54, 337 (1971).
[Crossref]

Schadt, M.

M. Schadt and W. Helfrich, Appl. Phys. Lett. 18, 127 (1971).
[Crossref]

Scheffer, T. J.

D. W. Berreman and T. J. Scheffer, Phys. Rev. Lett. 25, 577 (1970).
[Crossref]

Teitler, S.

Appl. Phys. Lett. (3)

M. Schadt and W. Helfrich, Appl. Phys. Lett. 18, 127 (1971).
[Crossref]

F. J. Kahn, Appl. Phys. Lett. 22, 386 (1973).
[Crossref]

W. Helfrich, Appl. Phys. Lett. 17, 531 (1970).
[Crossref]

Ark. Mat. Astron. Fys. (1)

C. W. Oseen, Ark. Mat. Astron. Fys. 19, 1 (1925).

Discuss. Faraday Soc. (1)

F. C. Frank, Discuss. Faraday Soc. 25, 19 (1958).
[Crossref]

J. Chem. Phys. (2)

I. Haller, J. Chem. Phys. 57, 1400 (1972).
[Crossref]

J. Nehring and A. Saupe, J. Chem. Phys. 54, 337 (1971).
[Crossref]

J. Fluid Mech. (1)

J. T. Jenkins, J. Fluid Mech. 45, 465 (1971).
[Crossref]

J. Opt. Soc. Am. (2)

Phys. Rev. Lett. (1)

D. W. Berreman and T. J. Scheffer, Phys. Rev. Lett. 25, 577 (1970).
[Crossref]

Other (4)

J. Billard, Thesis, University of Paris (1966).

A. Penz (private communication).

H. Gruler (private communication).

D. W. Berreman, 28, 1683 (1972).

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Figures (6)

Fig. 1
Fig. 1

Euler angles of optical dielectric tensor with respect to the surface (x,y) plane and the normal. The incident and reflected rays are in the x, z plane.

Fig. 2
Fig. 2

Orientation of molecules and optical dielectric tensors between two surfaces of a 90° twist cell when (a) less than the critical field is applied, (b) about twice the critical field is applied, and (c) a very strong field is applied. The angle of tilt is α and the bounding values of Z are Z1 and Z2.

Fig. 3
Fig. 3

Projected view of the x, y, z coordinate system showing both tilt angle α and azimuthal or turn angle β of the director, L, which coincides with the optic axis. L does not vary in the x, y plane. Electric or magnetic fields D or B are applied in the z direction.

Fig. 4
Fig. 4

Tilt and turn angles α and β as a function of Z in a quarter-turn twist cell 10μm thick containing MBBA doped to give positive dielectric anisotropy, for seven values of V/Vc. An arbitrary angle may be added to β if β(Z1) ≠ 0. Values of V/Vc for numbered curves are (1) ≤1, (2) 1.083, (3) 1.295, (4) 1.69, (5) 2.56, (6) 3.42, (7) 4.12.

Fig. 5
Fig. 5

Transmittance as a function of angle of incidence in the x, z plane when β(Z1) = 0, the polarizer transmits optic electric vectors perpendicular to β(Z1) and the analyzer is parallel with the polarizer. Curves are for the seven values of V/Vc given in Fig. 4.

Fig. 6
Fig. 6

Transmittance as in Fig. 5 except that β(Z1) = 45°; or the incident beam is turned −45° about the Z axis with respect to the configuration for Fig. 5.

Equations (33)

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[ E x H y E y H x ] Z = Z 2 = P [ E x H y E y H x ] Z = Z 1
ψ ( Z 2 ) = P ψ ( Z 1 ) ,
ψ z = i v Δ ( Z ) ψ D ( Z ) ψ ,
Δ = [ Δ 11 Δ 12 Δ 13 0 Δ 21 Δ 11 Δ 23 0 0 0 0 Δ 34 Δ 23 Δ 13 Δ 43 0 ] ,
11 = a cos 2 ψ + b sin 2 ψ , 12 = ( a b ) sin ψ cos ψ cos θ , 13 = ( a b ) sin ψ cos ψ sin θ , 22 = ( a sin 2 ψ + b cos 2 ψ ) cos 2 θ + c sin 2 θ , 23 = ( a sin 2 ψ + b cos 2 ψ c ) sin θ cos θ , 33 = ( a sin 2 ψ + b cos 2 ψ ) sin 2 θ + c cos 2 θ ,
Δ 11 = X ( 13 cos φ 23 sin φ ) / 33 , Δ 13 = X ( 13 sin φ + 23 cos φ ) / 33 , Δ 12 = 1 ( X 2 / 33 ) , Δ 21 = [ ( 11 22 2 13 2 23 2 2 33 ) cos 2 φ ( 12 13 23 33 ) sin 2 φ ] + ( 11 + 22 2 13 2 + 23 2 2 33 ) , Δ 43 = [ ( 11 22 2 13 2 23 2 2 33 ) cos 2 φ ( 12 13 23 33 ) sin 2 φ ] + ( 11 + 22 2 13 2 + 23 2 2 33 ) X 2 , Δ 23 = [ ( 11 22 2 13 2 23 2 2 33 ) sin 2 φ + ( 12 13 23 33 ) cos 2 φ ] ,
Δ 34 = 1.
P ( Z 1 , Z 2 ) = exp ( Z 1 Z 2 D ( z ) d z ) .
P ( z + h , z ) = P 0 + h P + h 2 2 ! P + + h n n ! P N + ,
P ( z ) = D ( z ) : P ( z ) ,
P 0 = 1 , P = D : 1 , P = D : P + D : 1 , P = D : P + 2 D : P + D : 1 , P = D : P + 3 D : P + 3 D : P + D : 1.
P = U + V ,
U = n = 0 h n n ! D n = exp ( h D )
V = n = 0 h n n ! V n ,
V 0 = 0 ,
V 1 = 0 ,
V 2 = D : 1 ,
V 3 = D : V 2 + 2 D : D + 2 · 1 2 ! D : 1 ,
V 4 = D : V 3 + 3 D : ( D 2 + V 2 ) + 3 · 2 2 ! D : D + 0 ( D ) ,
V n = D : V n 1 + ( n 1 ) D : ( D n 2 + V n 2 ) + ( n 1 ) ( n 2 ) 2 ! D : ( D n 3 + V n 3 ) + 0 ( D ) .
Δ 0 = [ 0 Δ 12 0 0 Δ 21 0 0 0 0 0 0 1 0 0 Δ 43 0 ] .
Q = exp ( i ν h Δ 0 ) .
Q 11 = Q 22 = cos [ ( Δ 12 · Δ 21 ) 1 2 ν d ] , Q 12 = i ( Δ 12 / Δ 21 ) 1 2 sin [ ( Δ 12 · Δ 21 ) 1 2 ν d ] , Q 21 = Q 12 · Δ 21 / Δ 12 , Q 33 = Q 44 = cos ( Δ 43 1 2 ν d ) , Q 34 = i ( Δ 43 ) 1 2 sin ( Δ 43 1 2 v d ) ,
Q 43 = Q 34 Δ 43 .
U = Q + ( U Q ) .
( U Q ) = i ν h ( Δ Δ 0 ) + ( i ν h ) 2 2 ! ( Δ 2 Δ 0 2 ) + ( i ν h ) 3 3 ! ( Δ 3 Δ 0 3 ) + .
Δ n Δ 0 n = [ ( Δ Δ 0 ) ( Δ n 1 + Δ 0 n 1 ) + ( Δ + Δ 0 ) ( Δ n 1 Δ 0 n 1 ) ] / 2 ,
Δ n 1 + Δ 0 n 1 = ( Δ n 1 Δ 0 n 1 ) + 2 Δ 0 n 1
U ( h ) = [ U ( h / n ) ] n .
n 0 = 1.54 ( a , b = 2.37 ) n e = 1.75 ( c = 3.06 ) ,
n g = 1.54 ( = 2.37 ) .
k 11 / k 33 = 0.79
k 22 / k 33 = 0.48.