Abstract

Berreman’s 4 × 4 matrix method is used to study the optical properties of special configurations. Under certain conditions the expression for the characteristic matrix of a homogeneous anisotropic slab can be greatly simplified. If these conditions are also fulfilled for each slab of a multilayered system, relations between the transmission and the reflection coefficients can be derived.

© 1991 Optical Society of America

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References

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  1. D. W. Berreman, “Optics in stratified and anisotropic media,”J. Opt. Soc. Am. 62, 502–510 (1972).
    [CrossRef]
  2. H. Wöhler, G. Haas, M. Fritsch, D. A. Mlynski, “Faster 4 × 4 matrix method for inhomogeneous uniaxial media,” J. Opt. Soc. Am. A 5, 1554–1557 (1988).
    [CrossRef]
  3. 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.
  4. K. Eidner, “Light propagation in stratified anisotropic media: orthogonality and symmetry properties of the 4 × 4 matrix formalisms,” J. Opt. Soc. Am. A 6, 1657–1660 (1989).
    [CrossRef]
  5. M. Foresti, “Plane-wave propagation in a plane-oriented nematic liquid-crystal multilayer: the case of a wave incident upon the plane containing the liquid-crystal directors,” J. Opt. Soc. Am. A 6, 1254–1259 (1989).
    [CrossRef]

1989

1988

1972

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.

Eidner, K.

Foresti, M.

Fritsch, M.

Haas, G.

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.

Mlynski, D. A.

Wöhler, H.

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Figures (2)

Fig. 1
Fig. 1

Conoscope of a 60° twisted nematic cell with parallel polarizers.

Fig. 2
Fig. 2

(a) Incident, reflected, and transmitted waves at a multilayered medium with ϕ = 0. (b) Multilayer and surrounding media are rotated around the x axis by a half-turn.

Equations (41)

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ψ / z = - i k 0 Δ · 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 ] ,
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 1 sin φ 1 ,
ψ ( z 2 ) = P ( z 2 , z 1 ) · ψ ( z 1 ) .
P ( h ) = β 0 I + β 1 Δ + β 2 Δ 2 + β 3 Δ 3 ,
λ 1 , 2 = ± ( - X 2 ) 1 / 2 , λ 3 , 4 = - Δ sin θ cos θ sin ϕ 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 , i j k l .
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 ^ = P · U ,
U = [ 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0 ] .
P 11 = 1 Y ( cos 2 ϕ cos x 1 + λ 1 2 sin 2 ϕ cos x 3 ) , P 12 = - i λ 1 Y ( cos 2 ϕ sin x 1 + λ 1 3 λ 3 sin 2 ϕ sin x 3 ) , P 13 = λ 1 2 sin ϕ cos ϕ Y ( cos x 1 - cos x 3 ) , P 14 = - i sin ϕ cos ϕ Y ( λ 1 sin x 1 - λ 1 2 λ 3 sin x 3 ) , P 21 = - i Y ( λ 1 cos 2 ϕ sin x 1 + λ 3 sin 2 ϕ sin x 3 ) , P 23 = - i sin ϕ cos ϕ Y ( λ 1 sin x 1 - λ 3 sin x 3 ) , P 24 = sin ϕ cos ϕ Y ( cos x 1 - cos x 3 ) , P 33 = 1 Y ( λ 1 2 sin 2 ϕ cos x 1 + cos 2 ϕ cos x 3 ) , P 34 = - i Y ( λ 1 sin 2 ϕ sin x 1 + λ 3 cos 2 ϕ sin x 3 ) , P 43 = - i Y ( λ 1 3 sin 2 ϕ sin x 1 + λ 3 cos 2 ϕ sin x 3 ) ,
Y = - X 2 sin 2 ϕ . x k = k 0 λ k h .
P - 1 = A · P · A ,             A = diag ( 1 , - 1 , 1 , - 1 ) .
P = P ( n ) · P ( n - 1 ) P ( 3 ) · P ( 2 ) · P ( 1 ) .
P - 1 = A · P ( 1 ) · P ( 2 ) · P ( 3 ) P ( n - 1 ) · P ( n ) · A ,
P - 1 = A · U · P T · U · A ,
P - 1 = [ P 22 - P 12 P 42 - P 32 - P 21 P 11 - P 41 P 31 P 24 - P 14 P 44 - P 34 - P 23 P 13 - P 43 P 33 ] ,
r π σ = 1 2 n 1 D [ cos φ 2 ( Q 31 Q 43 - Q 33 Q 41 ) + n 2 cos 2 φ 2 ( Q 31 Q 44 - Q 34 Q 41 ) + n 2 ( Q 32 Q 43 - Q 33 Q 42 ) + n 2 2 cos φ 2 ( Q 32 Q 44 - Q 34 Q 42 ) ] , r σ π = 1 2 n 1 D [ cos φ 2 ( Q 13 Q 21 - Q 11 Q 23 ) + n 2 cos 2 φ 2 ( Q 14 Q 21 - Q 11 Q 24 ) + n 2 ( Q 13 Q 22 - Q 12 Q 23 ) + n 2 2 cos φ 2 ( Q 14 Q 22 - Q 12 Q 24 ) ] ,
Q - 1 · Q = I ,
Q 13 Q 21 - Q 11 Q 23 = Q 31 Q 43 - Q 33 Q 41 , Q 14 Q 21 - Q 11 Q 24 = Q 31 Q 44 - Q 34 Q 41 , Q 13 Q 22 - Q 12 Q 23 = Q 32 Q 43 - Q 33 Q 42 , Q 14 Q 22 - Q 12 Q 24 = Q 32 Q 44 - Q 34 Q 42 ,
r π σ = r σ π .
P 11 = 1 Y ( sin 2 θ cos x 1 + X 2 cos 2 θ cos x 3 ) , P 12 = - i Y ( λ 1 sin 2 θ sin x 1 + λ 3 X 2 cos 2 θ sin x 3 ) , P 13 = i sin θ cos θ X Y ( λ 1 sin x 1 - λ 3 sin x 3 ) , P 14 = - sin θ cos θ X Y ( cos x 1 - cos x 3 ) , P 21 = - i Y ( λ 1 sin 2 θ sin x 1 + X 2 λ 3 cos 2 θ sin x 3 ) , P 23 = - sin θ cos θ X Y ( cos x 1 - cos x 3 ) , P 24 = i sin θ cos θ X Y ( sin x 1 λ 1 - sin x 3 λ 3 ) , P 33 = 1 Y ( X 2 cos 2 θ cos x 1 + sin 2 θ cos x 3 ) , P 34 = - i Y ( X 2 λ 1 cos 2 θ sin x 1 + λ 3 sin 2 θ sin x 3 ) , P 43 = - i Y ( X 2 λ 1 cos 2 θ sin x 1 + λ 3 sin 2 θ sin x 3 ) ,
Y = - ( - X 2 ) cos 2 θ .
P - 1 = B · P · B ,             B = diag ( 1 , - 1 , - 1 , 1 ) .
P - 1 = B · U · P T · U · B ,
P - 1 = [ P 22 - P 12 - P 42 P 32 - P 21 P 11 P 41 - P 31 - P 24 P 14 P 44 - P 34 P 23 - P 13 - P 43 P 33 ] ,
r π σ = - r σ π .
P ˜ = P ( 1 ) · P ( 2 ) P ( n - 1 ) · P ( n )
P ˜ = U · P T · U .
t ˜ π π = n 2 cos φ 2 n 1 cos φ 1 t π π , t ˜ π σ = n 2 cos φ 2 n 1 cos φ 1 t σ π , t ˜ σ π = n 2 cos φ 2 n 1 cos φ 1 t π σ , t ˜ σ σ = n 2 cos φ 2 n 1 cos φ 1 t σ σ ,
T ˜ π π = T π π , T ˜ π σ = T σ π , T ˜ σ π = T π σ , T ˜ σ σ = T σ σ ,
P ¯ = [ P 11 P 12 - P 13 - P 14 P 21 P 22 - P 23 - P 24 - P 31 - P 32 P 33 P 34 - P 41 - P 42 P 43 P 44 ] ,
P = [ P 11 P 12 0 0 P 21 P 11 0 0 0 0 P 33 P 34 0 0 P 43 P 33 ] ,
P 11 = exp ( - i k 0 Δ 11 h ) cos x 3 , P 12 = - i ( Δ 12 / Δ 21 ) 1 / 2 exp ( - i k 0 Δ 11 h ) sin x 3 , P 21 = - i ( Δ 21 / Δ 12 ) 1 / 2 exp ( - i k 0 Δ 11 h ) sin x 3 , P 33 = cos x 1 , P 34 = - i ( Δ 34 / Δ 43 ) 1 / 2 sin x 1 , P 43 = - i ( Δ 43 / Δ 34 ) 1 / 2 sin x 1 ,
x 1 = k 0 ( Δ 12 Δ 21 ) 1 / 2 h , x 3 = k 0 ( Δ 34 Δ 43 ) 1 / 2 h .
ψ t = P · ( ψ i + ψ r )
Q · ψ t = ψ i + ψ r ,
r π π = ( h 11 - h 21 ) ( h 32 + h 42 ) - ( h 12 - h 22 ) ( h 31 + h 41 ) D , r π σ = 2 h 31 h 42 - h 32 h 41 D cos φ 1 , r σ π = - 2 h 11 h 22 - h 12 h 21 D cos φ 1 , r σ σ = - ( h 31 - h 41 ) ( h 12 + h 22 ) - ( h 32 - h 42 ) ( h 11 + h 21 ) D , t π π = h 32 - h 42 D cos φ 1 , t π σ = - h 31 + h 41 D cos φ 1 , t σ π = - h 12 + h 22 D , t σ σ = h 11 + h 21 D ,
D = ( h 11 + h 21 ) ( h 32 + h 42 ) - ( h 12 + h 22 ) ( h 31 + h 41 )
h 11 ½ ( cos φ 2 Q 11 + n 2 Q 12 ) , h 12 ½ ( Q 13 + n 2 cos φ 2 Q 14 ) , h 21 1 2 ( cos φ 2 Q 21 + n 2 Q 22 ) cos φ 1 n 1 , h 22 1 2 ( Q 23 + n 2 cos φ 2 Q 24 ) cos φ 1 n 1 , h 31 ½ ( cos φ 2 Q 31 + n 2 Q 32 ) , h 32 ½ ( Q 33 + n 2 cos φ 2 Q 34 ) , h 41 1 2 ( cos φ 2 Q 41 + n 2 Q 42 ) 1 n 1 cos φ 1 , h 42 1 2 ( Q 43 + n 2 cos φ 2 Q 44 ) 1 n 1 cos φ 1 .

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