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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References

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Appl. Opt.

Biomed. Opt.

K.C.Hadley, I.A. Vitkin, "Optical rotation and linear and circular depolarization rates in diffusively scattered light from chiral, racimic and achiral turbid media," J. Biomed. Opt. 7, 291-299 (2002).
[CrossRef] [PubMed]

IEEE J. Sel. Top. Quantum Electron.

V. Backman, R. Gurjar, K. Badizadegan, I. Itzkan, R.R. Dassari, L.T. Perelman, and M.S. Feld, "Polarized light scattering spectroscopy for quantitative measurement of epithelial cellular structure," IEEE J. Sel. Top. Quantum Electron. 5, 1019 - 1026 (1999).
[CrossRef]

IEEE Trans. Biomed. Eng.

B.D.Cameron and G.L. Cote, "Noninvasive glucose sensing utilizing a digital closed loop polarimetric approach," IEEE Trans. Biomed. Eng. 44, 1221 - 1227 (1997).
[CrossRef] [PubMed]

J. Biomed. Opt.

R. J. McNichols, G.L. Cote, "Optical glucose sensing in biological fluids: an overview," J. Biomed. Opt. 5, 5 - 16 (2000).
[CrossRef] [PubMed]

R.R. Ansari, S. Bockle, and L. Rovati, "New optical scheme for a polarimetric-based glucose sensor," J. Biomed. Opt. 9, 103 - 115 (2004).
[CrossRef] [PubMed]

S.L. Jacques, R.J. Roman and K. Lee, "Imaging skin pathology with polarized light," J. Biomed. Opt. 7, 329 -340 (2002).
[CrossRef] [PubMed]

V. Sankaran, J. T. Walsh, Jr., and D. J. Maitland, "Comparative study of polarized light propagation in biological tissues," J. Biomed. Opt. 7, 300 - 306 (2002).
[CrossRef] [PubMed]

D. Cote and I.A. Vitkin, "Balanced detection for low-noise precision polarimetric measurements of optically active, multiply scattering tissue phantoms," J. Biomed. Opt. 9, 213 - 220 (2004).
[CrossRef] [PubMed]

Justin S. Baba, J.R. Chung, A.H. DeLaughter, B.D. Cameron, G.L. Cote," Development and calibration of an automated Mueller matrix polarization imaging system," J. Biomed. Opt. 7, 341 - 348 (2002).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A

Opt. Commun.

A. Vitkin and R.C.N Studinski, "Polarization preservation in diffusive scattering from in-vivo turbid biological media: Effects of tissue optical absorption in the exact backscattering direction," Opt. Commun. 190, 37 - 43 (2001).
[CrossRef]

N. Ghosh, P.K. Gupta, H.S. Patel, B. Jain and B.N. Singh, "Depolarization of light in tissue phantoms - effect of collection geometry," Opt. Commun. 222, 93 -100 (2003).
[CrossRef]

M.P.Silverman, W. Strange, J. Badoz, I.A. Vitkin, "Enhanced optical rotation and diminished depolarization in diffusive scattering from a chiral liquid," Opt. Commun.132, 410-416 (1996).
[CrossRef]

Opt. Eng.

I.A.Vitkin, E. Hoskinson, "Polarization studies in multiply scattering chiral media," Opt. Eng. 39, 353-362 (2000).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev.

N. Ghosh, A. Pradhan, P. K. Gupta, S. Gupta, V. Jaiswal and R. P. Singh, "Depolarization of light in a multiply scattering medium: effect of refractive index of scatterer," Phys. Rev. E 70, 066607 (2004).
[CrossRef]

Phys. Rev. E

A.D. Kim, M. Moscoso, "Influence of the refractive index on the depolarization of multiply scattered waves," Phys. Rev. E 64, 026612, 1 -4 (2001).
[CrossRef]

D. Bicout, C. Brosseau, A.S. Martinez , J.M. Schmitt "Depolarization of multiply scattered waves by spherical diffsers :Influence of size parameter," Phys. Rev. E 49, 1767-1770 (1994).
[CrossRef]

Other

M.I. Mischenko, J.W. Hovenier, L.D. Travis, "Light scattering by nonspherical particles" Academic Press, San Diego, 1999.

C.F. Bohren, D.R. Huffman, "Absorption and scattering of light by small particles," Wiley, New York (1983).

R.A.Chipman " Hand book of optics (polarimetry)," OSA / McGraw-Hill, 22.1-22.35, (1994).

E. Collette, "Polarized Light: Fundamentals and Applications," Marcel Dekker Inc. New York (1990).

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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)

Tables Icon

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

Tables Icon

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

Tables Icon

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

Tables Icon

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.

Tables Icon

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)

Equations on this page are rendered with MathJax. Learn more.

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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