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Mueller matrix approach for determination of optical rotation in chiral turbid media in backscattering geometry

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Abstract

For in vivo determination of optically active (chiral) substances in turbid media, like for example glucose in human tissue, the backscattering geometry is particularly convenient. However, recent polarimetric measurements performed in the backscattering geometry have shown that, in this geometry, the relatively small rotation of the polarization vector arising due to the optical activity of the medium is totally swamped by the much larger changes in the orientation angle of the polarization vector due to scattering. We show that the change in the orientation angle of the polarization vector arises due to the combined effect of linear diattenuation and linear retardance of light scattered at large angles and can be decoupled from the pure optical rotation component using polar decomposition of Mueller matrix. For this purpose, the method developed earlier for polar decomposition of Mueller matrix was extended to incorporate optical rotation in the medium. The validity of this approach for accurate determination of the degree of optical rotation using the Mueller matrix measured from the medium in both forward and backscattering geometry was tested by conducting studies on chiral turbid samples prepared using known concentration of scatterers and glucose molecules.

©2006 Optical Society of America

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Figures (3)

Fig. 1.
Fig. 1. Flow chart for polar decomposition of an experimentally obtained Mueller Matrix
Fig. 2.
Fig. 2. (a) The variation of the orientation angle (γ) as a function of scattering angle Θ. (b) The values for linear retardance δ (solid line), diattenuation d (dotted line) and optical rotation ψ (dash dotted line) obtained from polar decomposition of single scattering Mueller matrix (δ and ψ are in radian).
Fig. 3.
Fig. 3. (a) Depolarization (represented by degree of polarization P) (b) diattenuation (d), (c) linear retardance (δ) and (d) Optical rotation (ψ) map of the chiral spherical scatterer obtained from polar decomposition of Mueller matrix (X and Y are in cm, δ and ψ are in radian).

Tables (6)

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Table 1.(a) Measured Mueller matrix and the decomposed components for the linear retarder.

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Table 1.(b) Measured Mueller matrix and the decomposed components for the combination of the linear retarder and glucose (5M) solution.

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Table 2.(a) Measured Mueller matrix and the decomposed components for the chiral turbid sample (μs = 0.6 mm-1, glucose 5M) in forward scattering geometry

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Table 2.(b) Measured Mueller matrix and the decomposed components for the chiral turbid sample (μs = 0.6 mm-1, glucose 5M) in backscattering geometry.

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Table 3. Comparison between the different polarization parameters for the chiral turbid sample (μs = 0.6 mm-1, glucose 5M) in forward and backscattering geometry.

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Table 4. Measured Mueller matrix and the decomposed components for the chiral turbid sample (μs = 5 mm-1, glucose 5M) in forward scattering geometry.

Equations (21)

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M = M Δ M R M D
M D = ( 1 D T D m D )
m D = 1 D 2 I + ( 1 1 D 2 ) D ̂ D ̂ T
D = 1 m 00 [ m 01 m 02 m 03 ] T , D ̂ = D D
M Δ M R = M = MM D 1
M Δ = ( 1 0 P Δ m Δ )
M R = ( 1 0 0 m R )
M = ( 1 0 P Δ m ' )
m = m Δ m R
m Δ = ± [ m ( m ) T + ( λ 1 λ 2 + λ 2 λ 3 + λ 3 λ 1 ) I ] 1 × [ ( λ 1 + λ 2 + λ 3 ) m ( m ) T + λ 1 λ 2 λ 3 I ]
m R = m Δ 1 m
r i = 1 2 sin R j , k = 1 3 ε ijk ( m R ) jk
R = cos 1 { tr ( M R ) 2 1 }
M R = ( 1 0 0 0 0 cos 2 2 θ + sin 2 2 θ cos δ sin 2 θ cos 2 θ ( 1 cos δ ) sin 2 θ sin δ 0 θ cos θ ( cos δ ) sin 2 2 θ + cos 2 2 θ cos δ cos 2 θ sin δ 0 sin 2 θ sin δ cos 2 θ sin δ cos δ ) × ( 1 0 0 0 0 cos 2 Ψ sin2Ψ 0 0 sin 2 Ψ cos 2 Ψ 0 0 0 0 1 )
R = cos 1 { 2 cos 1 ( Ψ ) cos 2 ( 2 δ ) 1 }
r 3 2 = sin 2 Ψ cos 2 ( δ 2 ) 1 cos 2 ( Ψ ) cos 2 ( δ 2 )
δ = 2 cos 1 { r 3 2 ( 1 cos 2 ( R 2 ) ) + cos 2 ( R 2 ) }
Ψ = cos 1 { cos ( R 2 ) cos ( δ 2 ) }
θ = 1 2 tan 1 { r 3 r 2 }
γ = 0.5 × tan 1 ( U Q )
M i = PSA M S PSG
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