Abstract

An investigation is made of the vectorial and geometrical representation of the polarization of light propagating through a weakly inhomogeneous absorbing anisotropic and optically active medium. When the approximations of geometrical optics are used, Maxwell’s equations lead to the equation ∂G/∂x3 = (i/2)(Ω + iT)G, governing the behavior of polarized light propagating along the x3 axis in the medium, where x3 is the propagation distance along a light path, G is the complex amplitude of the electric vector, the vectors Ω = (0, Ω1, Ω2, Ω3) and T = (T0, T1, T2, T3), whose basis vectors are the unit matrix and the Pauli spin matrices, represent the optical properties of the medium. The two successive transformations of the resulting equation by the Stokes vector and the normalized polarization vector s yield a simple vector equation s/x3=Ω^×s+(T^×s)×s, where Ω^ = (Ω1, Ω2, Ω3) and T^ = (T1, T2, T3) are defined as the birefringent vector and the dichroic vector, respectively, representing the birefringence and the dichroism of the absorbing medium. The component Ω1 (or T1) shows the linear birefringence (or the dichroism) along the x1 and x2 axes, Ω2 (or T2) shows the linear birefringence (or dichroism) along the bisectors of the x1 and x2 axes, and Ω3 (or T3) shows the circular birefringence (or dichroism). The vector equation can represent clearly the geometrical behavior of the polarization of light in the inhomogeneous absorbing medium with the help of the Poincaré sphere.

© 1983 Optical Society of America

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References

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  1. H. Kubo and R. Nagata, “Stokes parameters representation of the light propagation equations in inhomogeneous anisotropic, optically active media,” Opt. Commun. 34, 306–308 (1980).
    [CrossRef]
  2. H. Kubo and R. Nagata, “Determination of dielectric tensor fields in weakly inhomogeneous anisotropic media. II,” J. Opt. Soc. Am. 71, 327–332 (1981).
    [CrossRef]
  3. R. C. Jones, “A new calculus for the treatment of optical system. VI. Properties of the N matrices,” J. Opt. Soc. Am. 38, 671–685 (1948).
    [CrossRef]
  4. W. H. MacMaster, “Matrix representation of polarization,” Rev. Mod. Phys. 33, 8–24 (1961).
    [CrossRef]
  5. R. M. A. Azzam, “Propagation of partially polarized light through anisotropic media with or without depolarization: a differential 4 × 4 matrix calculus,” J. Opt. Soc. Am. 68, 1756–1767 (1977).
    [CrossRef]
  6. R. Ulrich, “Representation of codirectional coupled waves,” Opt. Lett. 1, 109–111 (1977).
    [CrossRef] [PubMed]

1981 (1)

1980 (1)

H. Kubo and R. Nagata, “Stokes parameters representation of the light propagation equations in inhomogeneous anisotropic, optically active media,” Opt. Commun. 34, 306–308 (1980).
[CrossRef]

1977 (2)

1961 (1)

W. H. MacMaster, “Matrix representation of polarization,” Rev. Mod. Phys. 33, 8–24 (1961).
[CrossRef]

1948 (1)

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Figures (3)

Fig. 1
Fig. 1

When Ω ^ = constant and T ^ = 0, the end point of s rotates uniformly about the unit birefringent vector ω ^ = Ω ^ / Ω at the constant rate of frequency Ω along the circle C1 determined by the initial states s0 and ω ^ on the unit Poincaré sphere.

Fig. 2
Fig. 2

Locus of the end point of s when Ω ^ = 0 and T ^ = constant. P±, eigenpolarizations ± τ ^; P+ (or P), major (or minor) absorption point. The end point P moves toward P nonuniformly about the rotation axis T ^ × s along the arc P+P0P or the arc P + P 0 P - of a great circle P + P 0 P - P 0 , according to whether the initial state s0 is P0 or P 0 .

Fig. 3
Fig. 3

(a) The end point of s draws a spiral curve when Ω ^ and T ^ are parallel, constant, and an example for ( Ω ^ , T ^ ) > 0. The spiral curve is wound densely as s gets nearer P±. (b) The locus of the end point of s in (a) is composed of two circular motions, as shown in Figs. 1 and 2. Point P moves toward the direction of resultant vector Ω ^ × s + ( T ^ × s ) × s at the rate of |∂s/∂x3|. The resultant vector is tangent to the sphere at point P.

Equations (29)

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× H = 1 c D t ,             × E = - 1 c B t ,
· D = 0 ,             · B = 0 ,
D = [ ] E ,             B = H ,
[ ¯ ] = [ 11 12 21 22 ] = [ 11 s + i 11 s 12 s + i 12 a + i 12 s - 12 a 21 s + i 21 a + i 21 s - 21 a 22 s + i 22 s ] ,
E j = G j exp [ i ( k 0 Φ - ω t ) ] , Φ = [ ( ½ ) ( 11 s + 22 s ) ] 1 / 2 d x 3
2 G j + [ 2 i k 0 ( grad Φ ) · + i k 0 ( 2 Φ ) - k 0 2 ( grad Φ ) 2 ] G j - ( x j + i k 0 Φ x j ) div G - ( i k 0 x j - k 0 2 Φ x j ) [ ( grad Φ ) · G ] + k 0 2 k = 1 3 j k G k = 0             ( j = 1 , 2 , 3 ) ,
2 i k 0 Φ x 3 G j x 3 ,             k 0 2 ( Φ x 3 ) 2 G j ,             k 0 2 1 j G j ,             k 0 2 2 j G j             ( for j = 1 , 2 ) ,
2 G k x j 2 ,             2 G k x j x k ,             i k 0 2 Φ x j 2 G k ,             i k 0 2 Φ x j x k G k , i k 0 Φ x j G k x k ,             i k 0 Φ x k G k x j ,             k 0 2 Φ x j Φ x k G k             ( for j , k = 1 , 2 , 3 and j k ) ; 2 i k 0 Φ x 1 G k x 1 δ 1 k ,             2 i k 0 Φ x 2 G k x 2 δ 2 k , k 0 2 ( Φ x 1 ) 2 G k δ 1 k ,             k 0 2 ( Φ x 2 ) 2 G k δ 2 k k 0 2 k 3 G 3 δ 3 k ,             k 0 2 3 k G k ,             ( for k = 1 , 2 , 3 ) ,
i k 0 j k Φ x j G k ,             j k G k x j ,             j k x j G k             ( for j , k = 1 , 2 , 3 ) .
G 1 x 3 = i k 0 2 ( 0 s ) 1 / 2 [ ( 11 - 11 s + 22 s 2 ) G 1 + 12 G 2 ] , G 2 x 3 = i k 0 2 ( 0 s ) 1 / 2 [ 21 G 1 + ( 22 - 11 s + 22 s 2 ) G 2 ] ,
x 3 [ G 1 G 2 ] = i 2 [ Ω 1 + i T 0 + i T 1 Ω 2 + i Ω 3 + i T 2 - T 3 Ω 2 - i Ω 3 + i T 2 + T 3 - Ω 1 + i T 0 - i T 1 ] [ G 1 G 2 ] ,
[ 11 - ( k k 0 ) 2 ] E 1 + 12 E 2 = 0 , 21 E 1 + [ 22 - ( k k 0 ) 2 ] E 2 = 0 ,
k ± 2 = k 0 2 2 { 11 + 22 ± [ ( 11 - 22 ) 2 + 4 12 21 ] 1 / 2 } .
E = { E 10 + [ 1 ρ ] exp ( i K + x 3 ) + E 10 - [ 1 - 21 12 ρ ] exp ( i K - x 3 ) } exp ( - i ω t ) = G a exp [ i ( K + x 3 - ω t ) ] ,
x 3 [ G 1 a G 2 a ] = i k 0 2 4 K + [ 11 - 22 2 12 2 21 - ( 11 - 22 ) ] ( G 1 a G 2 a ) .
x 3 ( G 1 c G 2 c ) = i k 0 2 4 C K + ( Ω 1 + i T 0 + i T 1 Ω 2 + i Ω 3 + i T 2 - T 3 Ω 2 - i Ω 3 + i T 2 + T 3 - Ω 1 + i T 0 - i T 1 ) ( G 1 c G 2 c ) ,
E = G c exp { i [ K + x 3 - i ( k 0 2 / 4 K + ) ( 11 s + 22 s ) x 3 - ω t ] } ,
S = E σ E = G σ G ,
σ ^ 0 = [ 1 0 0 1 ] ,             σ ^ 1 = [ 1 0 0 - 1 ] ,             σ ^ 2 = ( 0 1 1 0 ) ,             σ ^ 3 = ( 0 i - i 0 ) .
G x 3 = ( i / 2 ) [ i T 0 σ ^ 0 + ( Ω 1 + i T 1 ) σ ^ 1 + ( Ω 2 + i T 2 ) σ ^ 2 + ( Ω 3 + i T 3 ) σ ^ 3 ] G = ( i / 2 ) ( Ω + i T ) G ,
S x 3 = - ( i / 2 ) G [ ( Ω + i T ) σ - σ ( Ω + i T ) ] G .
S x 3 = [ ω ] S = { [ ω ] s + [ ω ] a } S , [ ω ] s = - [ T 0 T 1 T 2 T 3 T 1 T 0 0 0 T 2 0 T 0 0 T 3 0 0 T 0 ] , [ ω ] a = [ 0 0 0 0 0 0 - Ω 3 Ω 2 0 Ω 3 0 - Ω 1 0 - Ω 2 Ω 1 0 ] ,
S c x 3 = [ ω ] c S c = { [ ω ] s c + [ ω ] a c } S c , [ ω ] s c = - ρ + ρ * 2 [ T 0 ( ρ ) T 1 - i ρ 0 Ω 1 T 2 - i ρ 0 Ω 2 T 3 - i ρ 0 Ω 3 T 1 - i ρ 0 Ω 1 T 0 ( ρ ) 0 0 T 2 - i ρ 0 Ω 2 0 T 0 ( ρ ) 0 T 3 - i ρ 0 Ω 3 0 0 T 0 ( ρ ) ] , [ ω ] a c = ρ + ρ * 2 [ 0 0 0 0 0 0 - Ω 3 - i ρ 0 T 3 Ω 2 + i ρ 0 T 2 0 Ω 3 + i ρ 0 T 3 0 - Ω 1 - i ρ 0 T 1 0 - Ω 2 - i ρ 0 T 2 Ω 1 + i ρ 0 T 1 0 ] , ρ = k 0 2 C K + ,             ρ 0 = ρ - ρ * ρ + ρ * ,             T 0 ( ρ ) = i 2 ρ 0 ρ ρ * k 0 ( 0 s ) 1 / 2 ,
s x 3 = Ω ^ × s + ( T ^ × s ) × s = Ω ω ^ × s + T ( τ ^ × s ) × s ,
s x 3 = Ω ^ × s ,
s x 3 = ( T ^ × s ) × s
d s ϕ / d x 3 = T sin s ϕ ,
s = 1 cosh T x 3 - α α ( ω ^ , s 0 ) sinh T x 3 × { - α α ω ^ { sinh T x 3 - α α ( ω ^ , s 0 ) cosh T x 3 } + ( ω ^ × s 0 ) × ω ^ cos Ω x 3 + ω ^ × s 0 sin Ω x 3 } ,
s = ω ^ ( ω ^ , s 0 ) + ( ω ^ × s 0 ) × ω ^ cos Ω x 3 + ω ^ × s 0 sin Ω x 3 ,