Abstract

An exact representation of the transfer matrix for stratified homogeneous uniaxial media is derived. It can be used to calculate optical quantities such as reflectance and transmittance by means of Berreman’s 4 × 4 matrix method, permitting calculations for thick homogeneous slabs such as polarizers in one single step without the commonly used truncated series expansion. When the dielectric tensor of an inhomogeneous medium varies continuously with the normal to the plane of stratification, the medium is divided into thin slabs. The transfer matrix of the whole medium is then obtained by multiplying the transfer matrices of all slabs. Treating each slab as homogeneous gives satisfactory results, as shown in an example of a periodic structure for which an analytic solution is known.

© 1988 Optical Society of America

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References

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  1. S. Teitler, B. Henvis, “Refraction is stratified anisotropic media,” J. Opt. Soc. Am. 60, 830–834 (1970).
    [CrossRef]
  2. W. Berreman, “Optics in stratified and anisotropic media: 4 × 4 matrix formulation,” J. Opt. Soc. Am. 62, 502–510 (1972).
    [CrossRef]
  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. R. J. Gagnon, “Liquid-crystal twist-cell optics,” J. Opt. Soc. Am. 71, 348–353 (1981).
    [CrossRef]
  5. I. Abdulhalim, L. Benguigui, R. Weil, “Selective reflection by helicoidal liquid crystals. Results of an exact calculation using the 4 × 4 characteristic matrix method,” J. Phys. (Paris) 46, 815–825 (1985).
    [CrossRef]
  6. C. W. Oseen, “Beiträge zur Theorie der anisotropen Flüssigkeiten,” Ark. Mat. Astron. Fys. A 21, 14–35 (1925).
  7. E. G. Sauter, “Light propagation in twisted dielectric media,” Appl. Phys. B 27, 137–139 (1982).
    [CrossRef]
  8. O. Parodi, “Light propagation along the helical axis in chiral smectics C,” J. Phys. (Paris) Colloq. 36, 22–23 (1975).
    [CrossRef]
  9. See, for example, R. A. Gabel, R. A. Roberts, Signals and Linear Systems (Wiley, New York, 1987), pp. 158–169.
  10. H. Kahn, F. J. Birecki, in The Physics and Chemistry of Liquid Crystal Devices, J. Sprokel, ed. (Plenum, New York, 1980), pp. 125–142.
  11. J. Nehring, “Light propagation and reflection by absorbing cholesteric layers,” J. Chem. Phys. 75, 4326–4337 (1981).
    [CrossRef]

1985 (1)

I. Abdulhalim, L. Benguigui, R. Weil, “Selective reflection by helicoidal liquid crystals. Results of an exact calculation using the 4 × 4 characteristic matrix method,” J. Phys. (Paris) 46, 815–825 (1985).
[CrossRef]

1982 (1)

E. G. Sauter, “Light propagation in twisted dielectric media,” Appl. Phys. B 27, 137–139 (1982).
[CrossRef]

1981 (2)

J. Nehring, “Light propagation and reflection by absorbing cholesteric layers,” J. Chem. Phys. 75, 4326–4337 (1981).
[CrossRef]

R. J. Gagnon, “Liquid-crystal twist-cell optics,” J. Opt. Soc. Am. 71, 348–353 (1981).
[CrossRef]

1975 (1)

O. Parodi, “Light propagation along the helical axis in chiral smectics C,” J. Phys. (Paris) Colloq. 36, 22–23 (1975).
[CrossRef]

1973 (1)

1972 (1)

1970 (1)

1925 (1)

C. W. Oseen, “Beiträge zur Theorie der anisotropen Flüssigkeiten,” Ark. Mat. Astron. Fys. A 21, 14–35 (1925).

Abdulhalim, I.

I. Abdulhalim, L. Benguigui, R. Weil, “Selective reflection by helicoidal liquid crystals. Results of an exact calculation using the 4 × 4 characteristic matrix method,” J. Phys. (Paris) 46, 815–825 (1985).
[CrossRef]

Benguigui, L.

I. Abdulhalim, L. Benguigui, R. Weil, “Selective reflection by helicoidal liquid crystals. Results of an exact calculation using the 4 × 4 characteristic matrix method,” J. Phys. (Paris) 46, 815–825 (1985).
[CrossRef]

Berreman, D. W.

Berreman, W.

Birecki, F. J.

H. Kahn, F. J. Birecki, in The Physics and Chemistry of Liquid Crystal Devices, J. Sprokel, ed. (Plenum, New York, 1980), pp. 125–142.

Gabel, R. A.

See, for example, R. A. Gabel, R. A. Roberts, Signals and Linear Systems (Wiley, New York, 1987), pp. 158–169.

Gagnon, R. J.

Henvis, B.

Kahn, H.

H. Kahn, F. J. Birecki, in The Physics and Chemistry of Liquid Crystal Devices, J. Sprokel, ed. (Plenum, New York, 1980), pp. 125–142.

Nehring, J.

J. Nehring, “Light propagation and reflection by absorbing cholesteric layers,” J. Chem. Phys. 75, 4326–4337 (1981).
[CrossRef]

Oseen, C. W.

C. W. Oseen, “Beiträge zur Theorie der anisotropen Flüssigkeiten,” Ark. Mat. Astron. Fys. A 21, 14–35 (1925).

Parodi, O.

O. Parodi, “Light propagation along the helical axis in chiral smectics C,” J. Phys. (Paris) Colloq. 36, 22–23 (1975).
[CrossRef]

Roberts, R. A.

See, for example, R. A. Gabel, R. A. Roberts, Signals and Linear Systems (Wiley, New York, 1987), pp. 158–169.

Sauter, E. G.

E. G. Sauter, “Light propagation in twisted dielectric media,” Appl. Phys. B 27, 137–139 (1982).
[CrossRef]

Teitler, S.

Weil, R.

I. Abdulhalim, L. Benguigui, R. Weil, “Selective reflection by helicoidal liquid crystals. Results of an exact calculation using the 4 × 4 characteristic matrix method,” J. Phys. (Paris) 46, 815–825 (1985).
[CrossRef]

Appl. Phys. B (1)

E. G. Sauter, “Light propagation in twisted dielectric media,” Appl. Phys. B 27, 137–139 (1982).
[CrossRef]

Ark. Mat. Astron. Fys. A (1)

C. W. Oseen, “Beiträge zur Theorie der anisotropen Flüssigkeiten,” Ark. Mat. Astron. Fys. A 21, 14–35 (1925).

J. Chem. Phys. (1)

J. Nehring, “Light propagation and reflection by absorbing cholesteric layers,” J. Chem. Phys. 75, 4326–4337 (1981).
[CrossRef]

J. Opt. Soc. Am. (4)

J. Phys. (Paris) (1)

I. Abdulhalim, L. Benguigui, R. Weil, “Selective reflection by helicoidal liquid crystals. Results of an exact calculation using the 4 × 4 characteristic matrix method,” J. Phys. (Paris) 46, 815–825 (1985).
[CrossRef]

J. Phys. (Paris) Colloq. (1)

O. Parodi, “Light propagation along the helical axis in chiral smectics C,” J. Phys. (Paris) Colloq. 36, 22–23 (1975).
[CrossRef]

Other (2)

See, for example, R. A. Gabel, R. A. Roberts, Signals and Linear Systems (Wiley, New York, 1987), pp. 158–169.

H. Kahn, F. J. Birecki, in The Physics and Chemistry of Liquid Crystal Devices, J. Sprokel, ed. (Plenum, New York, 1980), pp. 125–142.

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Figures (2)

Fig. 1
Fig. 1

Rπ,π, Rσ,σ (solid line), Rπ,σ, Rσ,π (dashed line) versus λ/p of a cholesteric layer. d/p = 10, = 2, = 3, 0 = 1.

Fig. 2
Fig. 2

Maximum difference of Rπ,π, Rπ,σ, Rσ,π, and Rσ,σ between the exact solution and the approximation by homogeneous slabs versus the number of slabs per full turn (1 ≤ λ/p ≤ 2).

Equations (14)

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d ψ d z = i k 0 Δ ˜ ( z ) ψ ,
ψ = ( E x , H y , E y , H x ) T , k 0 = ω / c ( c is the velocity of light in vacuum ) , 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 ] , Δ 11 = Δ sin θ cos θ sin ϕ 33 X , Δ 12 = 1 X 2 33 , Δ 13 = Δ sin θ cos θ cos ϕ 33 X , Δ 21 = Δ sin 2 θ cos 2 ϕ 33 , Δ 23 = Δ sin 2 θ sin ϕ cos ϕ 33 , Δ 34 = 1 , Δ 43 = Δ sin 2 θ sin 2 ϕ 33 X 2 , 33 = + Δ cos 2 θ , X = n 0 sin A .
ψ ( z 2 ) = P ˜ ( z 2 , z 1 ) ψ ( z 1 ) .
P ˜ = exp ( i k 0 Δ ˜ h ) = I ˜ i k 0 h 1 ! Δ ˜ ( k 0 h ) 2 2 ! Δ ˜ 2 + ,
P ˜ = β 0 I ˜ + β 1 Δ ˜ + β 2 Δ ˜ 2 + β 3 Δ ˜ 3 .
exp ( i k 0 λ i h ) = β 0 + β 1 λ i + β 2 λ i 2 + β 3 λ i 3 , i = 1 , , 4 ,
λ 1 , 2 = ± ( X 2 ) 1 / 2 , λ 3 , 4 = 13 33 X ± ( ) 1 / 2 33 [ 33 ( 1 Δ sin 2 θ cos 2 ϕ ) X 2 ] 1 / 2 ,
β 0 = i = 1 4 λ j λ k λ l f i λ i j λ i k λ i l , β 1 = i = 1 4 ( λ j λ k + λ j λ l + λ k λ l ) f i λ i j λ i k λ i l , β 2 = i = 1 4 ( λ j + λ k + λ l ) f i λ i j λ i k λ i l , β 3 = i = 1 4 f i λ i j λ i k λ i l ,
λ i j = λ i λ j , f i = exp ( i k 0 λ i h ) , i , j , k , l = 1 , , 4.
P 22 = P 11 , P 31 = P 24 , P 32 = P 14 , P 41 = P 23 , P 42 = P 13 , P 44 = P 33 .
P 11 = 1 Y ( λ 1 2 sin 2 ϕ cos α 3 + cos 2 ϕ cos α 1 ) , P 12 = i λ 1 2 Y ( λ 1 2 λ 3 1 sin 2 ϕ sin α 3 + 1 λ 1 cos 2 ϕ sin α 1 ) , P 13 = λ 1 2 sin ϕ cos ϕ Y ( cos α 3 cos α 1 ) , P 14 = i sin ϕ cos ϕ Y ( λ 1 2 λ 3 sin α 3 λ 1 sin α 1 ) , P 21 = i Y ( λ 3 sin 2 ϕ sin α 3 + λ 1 cos 2 ϕ sin α 1 ) , P 23 = i sin ϕ cos ϕ Y ( λ 3 sin α 3 λ 1 sin α 1 ) , P 24 = sin ϕ cos ϕ Y ( cos α 3 cos α 1 ) , P 33 = 1 Y ( cos 2 ϕ cos α 3 + λ 1 2 sin 2 ϕ cos α 1 ) , P 34 = i 1 Y ( λ 3 cos 2 ϕ sin α 3 + λ 1 sin 2 ϕ sin α 1 ) , P 43 = i 1 Y ( λ 3 cos 2 ϕ sin α 3 + λ 1 3 sin 2 ϕ sin α 1 ) , Y = X 2 sin 2 ϕ , α i = k 0 λ i h .
ψ k = [ sin γ + i x k cos γ p k sin γ + i q k cos γ cos γ + i x k sin γ p k cos γ + i q k sin γ ] exp ( i k 0 m k z ) ,
m k = ± { ( λ / p ) 2 + + ± [ 4 ( λ / p ) 2 + + 2 ] 1 / 2 } 1 / 2 , + = ( + ) / 2 , = ( ) / 2 , x k = m k 2 + ( λ / p ) 2 2 λ / p m k , p k = m k + λ p x k , q k = λ p + m k x k .
P ˜ = ψ ˜ ( z 2 ) ψ ˜ 1 ( z 1 ) .

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