Abstract

The concentration determination of optically active species in moderately turbid suspensions is studied both experimentally and with a validated three-dimensional polarization-sensitive Monte Carlo model. It is shown that the orientation of the polarization of the scattered light exhibits a strong dependence on exit position in the side or backscattered directions, but not in the forward direction. In addition, it is shown that the increased path length of photons due to multiple scattering in a 1 cm cuvette filled with forward-peaked scatterers (anisotropy around 0.93) increases the optical rotation by up to 15%, but only for scattering coefficients under 30 cm-1, after which it decreases again. It is concluded that in order to avoid systematic errors in concentration determination of optically-active molecular species in turbid samples, the scattered light in the forward direction should be used.

© 2005 Optical Society of America

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    [CrossRef]
  28. N.  Ghosh, P. K.  Gupta, H. S.  Patel, B.  Jain, B. N.  Singh, “Depolarization of light in tissue phantoms -effect of collection geometry,” Opt. Comm. 222, 93–100 (2003).
    [CrossRef]
  29. V.  Sankaran, K.  Schönenberger, J. T.  Walsh, D. J.  Maitland, “Polarization discrimination of coherently propagating light in turbid media,” Appl. Opt. 38, 4252–4261 (1999).
    [CrossRef]
  30. M. P.  Silverman, W.  Strange, J.  Badoz, I. A.  Vitkin, “Enhanced optical rotation and diminished depolarization in diffusive scattering from a chiral liquid,” Opt. Comm. 132, 410–416 (1996).
    [CrossRef]
  31. K. C.  Hadley, I. A.  Vitkin, “Optical rotation and linear and circular depolarization rates in diffusively scattered light from chiral, racemic, and achiral turbid media,” J. Biomed. Opt. 7, 291–299 (2002).
    [CrossRef] [PubMed]
  32. C. F.  Bohren, D. R.  Huffman, Absorption and scattering of light by small particles (Wiley, New York, 1983), chap. 2, pp. 46–56.

2004 (2)

D.  Côté, 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]

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

2003 (2)

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

F.  Jaillon, H.  Saint-Jalmes, “Description and time reduction of a Monte Carlo code to simulate propagation of polarized light through scattering media,” Appl. Opt. 42, 3290–3296 (2003).
[CrossRef] [PubMed]

2002 (3)

X.  Wang, G.  Yao, L. V.  Wang, “Monte Carlo model and single-scattering approximation of the propagation of polarized light in turbid media containing glucose,” Appl. Opt. 41, 792–801 (2002).
[CrossRef] [PubMed]

S. L.  Jacques, J. C.  Ramella-Roman, K.  Lee, “Imaging skin pathology with polarized light,” J. Biomed. Opt. 7, 329–340 (2002).
[CrossRef] [PubMed]

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

2001 (3)

2000 (4)

1999 (2)

V.  Sankaran, K.  Schönenberger, J. T.  Walsh, D. J.  Maitland, “Polarization discrimination of coherently propagating light in turbid media,” Appl. Opt. 38, 4252–4261 (1999).
[CrossRef]

A. N.  Yaroslavsky, I. V.  Yaroslavsky, T.  Goldbach, H.-J.  Schwarsmaier, “Influence of the scattering phase function approximation on the optical properties of blood determined from the integrating sphere measurements,” J. Biomed. Opt. 4, 47–53 (1999).
[CrossRef]

1998 (1)

1997 (1)

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

1996 (1)

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

1995 (1)

L.  Wang, S. L.  Jacques, L.  Zheng, “MCML-Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Meth. Prog. Biomed. 47, 131–146 (1995).
[CrossRef]

1994 (1)

1992 (1)

1990 (1)

M. S.  Patterson, B. C.  Wilson, D. R.  Wyman, “The propagation of optical radiation in tissue I. Models of radiation transport and their application,” Lasers in Medical Science 6, 155–166 (1990).
[CrossRef]

1988 (1)

J. M.  Steinke, A. P.  Shepherd, “Diffusion model of the optical absorbance of whole blood,” J. Opt. Soc. Am. B 5, 813–822 (1988).
[CrossRef]

1987 (1)

A. J.  Welch, G.  Yoon, M. J.  van Gemert, “Practical models for light distribution in laser-irradiated tissue,” Lasers Surg Med 6, 488–493 (1987).
[CrossRef] [PubMed]

1982 (1)

W. F.  March, B.  Rabinovitch, R. L.  Adams, “Noninvasive glucose monitoring of the aqueous humor of the eye: Part II. Animal studies and the scleral lens,” Diabetes Care 5, 259–265 (1982).
[CrossRef] [PubMed]

1980 (1)

Adams, R. L.

W. F.  March, B.  Rabinovitch, R. L.  Adams, “Noninvasive glucose monitoring of the aqueous humor of the eye: Part II. Animal studies and the scleral lens,” Diabetes Care 5, 259–265 (1982).
[CrossRef] [PubMed]

Ansari, R. R.

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

Badoz, J.

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

Bartel, S.

Bockle, S.

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

Bohren, C. F.

C. F.  Bohren, D. R.  Huffman, Absorption and scattering of light by small particles (Wiley, New York, 1983), chap. 2, pp. 46–56.

Bonner, R. F.

Cameron, B. D.

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

Coté, G. L.

R. J.  McNichols, G. L.  Coté, “Optical glucose sensing in biological fluids: an overview,” J. Biomed. Opt. 5, 5–16 (2000).
[CrossRef] [PubMed]

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

Côté, D.

D.  Côté, 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]

D.  Côté, I. A.  Vitkin, “Pol-MC: a three-dimensional polarization-sensitive Monte Carlo implementation for light propagation in tissue,” Available at http://www.novajo.ca/ont-canc-inst-biophotonics/.

Drévillon, B.

Fantini, S.

Franceschini, M. A.

Freyer, J. P.

Gandjbakhche, A. H.

Germer, T. A.

T. A.  Germer, “SCATMECH: Polarized Light Scattering C++ Class Library,” Available at http://physics.nist.gov/scatmech.

Ghosh, N.

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

Goldbach, T.

A. N.  Yaroslavsky, I. V.  Yaroslavsky, T.  Goldbach, H.-J.  Schwarsmaier, “Influence of the scattering phase function approximation on the optical properties of blood determined from the integrating sphere measurements,” J. Biomed. Opt. 4, 47–53 (1999).
[CrossRef]

Gratton, E.

Gupta, P. K.

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

Hadley, K. C.

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

Hielscher, A. H.

Hoskinson, E.

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

Huffman, D. R.

C. F.  Bohren, D. R.  Huffman, Absorption and scattering of light by small particles (Wiley, New York, 1983), chap. 2, pp. 46–56.

Jacques, S. L.

S. L.  Jacques, J. C.  Ramella-Roman, K.  Lee, “Imaging skin pathology with polarized light,” J. Biomed. Opt. 7, 329–340 (2002).
[CrossRef] [PubMed]

L.  Wang, S. L.  Jacques, L.  Zheng, “MCML-Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Meth. Prog. Biomed. 47, 131–146 (1995).
[CrossRef]

Jaillon, F.

Jain, B.

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

Johnson, T. M.

Kaplan, B.

Kehtarnavaz, N.

Keller, J. B.

Ledanois, G.

Lee, K.

S. L.  Jacques, J. C.  Ramella-Roman, K.  Lee, “Imaging skin pathology with polarized light,” J. Biomed. Opt. 7, 329–340 (2002).
[CrossRef] [PubMed]

Maier, J. S.

Maitland, D. J.

March, W. F.

W. F.  March, B.  Rabinovitch, R. L.  Adams, “Noninvasive glucose monitoring of the aqueous humor of the eye: Part II. Animal studies and the scleral lens,” Diabetes Care 5, 259–265 (1982).
[CrossRef] [PubMed]

McNichols, R. J.

R. J.  McNichols, G. L.  Coté, “Optical glucose sensing in biological fluids: an overview,” J. Biomed. Opt. 5, 5–16 (2000).
[CrossRef] [PubMed]

Mehrübeoglu, M.

Moscoso, M.

Mourant, J. R.

Papanicolaou, G.

Patel, H. S.

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

Patterson, M. S.

M. S.  Patterson, B. C.  Wilson, D. R.  Wyman, “The propagation of optical radiation in tissue I. Models of radiation transport and their application,” Lasers in Medical Science 6, 155–166 (1990).
[CrossRef]

Rabinovitch, B.

W. F.  March, B.  Rabinovitch, R. L.  Adams, “Noninvasive glucose monitoring of the aqueous humor of the eye: Part II. Animal studies and the scleral lens,” Diabetes Care 5, 259–265 (1982).
[CrossRef] [PubMed]

Ramella-Roman, J. C.

S. L.  Jacques, J. C.  Ramella-Roman, K.  Lee, “Imaging skin pathology with polarized light,” J. Biomed. Opt. 7, 329–340 (2002).
[CrossRef] [PubMed]

Rastegar, S.

Rovati, L.

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

Saint-Jalmes, H.

Sankaran, V.

Schmitt, J. M.

Schönenberger, K.

Schwarsmaier, H.-J.

A. N.  Yaroslavsky, I. V.  Yaroslavsky, T.  Goldbach, H.-J.  Schwarsmaier, “Influence of the scattering phase function approximation on the optical properties of blood determined from the integrating sphere measurements,” J. Biomed. Opt. 4, 47–53 (1999).
[CrossRef]

Shepherd, A. P.

J. M.  Steinke, A. P.  Shepherd, “Diffusion model of the optical absorbance of whole blood,” J. Opt. Soc. Am. B 5, 813–822 (1988).
[CrossRef]

Silverman, M. P.

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

Singh, B. N.

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

Steinke, J. M.

J. M.  Steinke, A. P.  Shepherd, “Diffusion model of the optical absorbance of whole blood,” J. Opt. Soc. Am. B 5, 813–822 (1988).
[CrossRef]

Strange, W.

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

van de Hulst, H. C.

H. C.  van de Hulst, Light scattering by small particles (Dover, New York, 1981).

van Gemert, M. J.

A. J.  Welch, G.  Yoon, M. J.  van Gemert, “Practical models for light distribution in laser-irradiated tissue,” Lasers Surg Med 6, 488–493 (1987).
[CrossRef] [PubMed]

Vitkin, I. A.

D.  Côté, 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]

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

I. A.  Vitkin, E.  Hoskinson, “Polarization studies in multiply scattering chiral media,” Opt. Eng. 39, 353–362 (2000).
[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. Comm. 132, 410–416 (1996).
[CrossRef]

D.  Côté, I. A.  Vitkin, “Pol-MC: a three-dimensional polarization-sensitive Monte Carlo implementation for light propagation in tissue,” Available at http://www.novajo.ca/ont-canc-inst-biophotonics/.

Walker, S. A.

Walsh, J. T.

Wang, L.

L.  Wang, S. L.  Jacques, L.  Zheng, “MCML-Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Meth. Prog. Biomed. 47, 131–146 (1995).
[CrossRef]

Wang, L. V.

Wang, X.

Welch, A. J.

A. J.  Welch, G.  Yoon, M. J.  van Gemert, “Practical models for light distribution in laser-irradiated tissue,” Lasers Surg Med 6, 488–493 (1987).
[CrossRef] [PubMed]

Wilson, B. C.

M. S.  Patterson, B. C.  Wilson, D. R.  Wyman, “The propagation of optical radiation in tissue I. Models of radiation transport and their application,” Lasers in Medical Science 6, 155–166 (1990).
[CrossRef]

Wiscombe, W. J.

Wyman, D. R.

M. S.  Patterson, B. C.  Wilson, D. R.  Wyman, “The propagation of optical radiation in tissue I. Models of radiation transport and their application,” Lasers in Medical Science 6, 155–166 (1990).
[CrossRef]

Yao, G.

Yaroslavsky, A. N.

A. N.  Yaroslavsky, I. V.  Yaroslavsky, T.  Goldbach, H.-J.  Schwarsmaier, “Influence of the scattering phase function approximation on the optical properties of blood determined from the integrating sphere measurements,” J. Biomed. Opt. 4, 47–53 (1999).
[CrossRef]

Yaroslavsky, I. V.

A. N.  Yaroslavsky, I. V.  Yaroslavsky, T.  Goldbach, H.-J.  Schwarsmaier, “Influence of the scattering phase function approximation on the optical properties of blood determined from the integrating sphere measurements,” J. Biomed. Opt. 4, 47–53 (1999).
[CrossRef]

Yoon, G.

A. J.  Welch, G.  Yoon, M. J.  van Gemert, “Practical models for light distribution in laser-irradiated tissue,” Lasers Surg Med 6, 488–493 (1987).
[CrossRef] [PubMed]

Zheng, L.

L.  Wang, S. L.  Jacques, L.  Zheng, “MCML-Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Meth. Prog. Biomed. 47, 131–146 (1995).
[CrossRef]

Appl. Opt. (8)

W. J.  Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt. 19, 1505–1509 (1980).
[CrossRef] [PubMed]

J. M.  Schmitt, A. H.  Gandjbakhche, R. F.  Bonner, “Use of polarized light to discriminate short-path photons in a multiply scattering medium,” Appl. Opt. 31, 6535–6546 (1992).
[CrossRef] [PubMed]

S.  Bartel, A. H.  Hielscher, “Monte Carlo simulations of the diffuse backscattering Mueller matrix for highly scattering media,” Appl. Opt. 39, 1580–1588 (2000).
[CrossRef]

V.  Sankaran, K.  Schönenberger, J. T.  Walsh, D. J.  Maitland, “Polarization discrimination of coherently propagating light in turbid media,” Appl. Opt. 38, 4252–4261 (1999).
[CrossRef]

B.  Kaplan, G.  Ledanois, B.  Drévillon, “Mueller Matrix of dense polystyrene latex sphere supsensions: measurements and Monte Carlo simulations,” Appl. Opt. 40, 2769–2777 (2001).
[CrossRef]

J. R.  Mourant, T. M.  Johnson, J. P.  Freyer, “Characterizing mammalian cells and cell phantoms by polarized backscattering fiber-optic measurements,” Appl. Opt. 40, 5114–5123 (2001).
[CrossRef]

X.  Wang, G.  Yao, L. V.  Wang, “Monte Carlo model and single-scattering approximation of the propagation of polarized light in turbid media containing glucose,” Appl. Opt. 41, 792–801 (2002).
[CrossRef] [PubMed]

F.  Jaillon, H.  Saint-Jalmes, “Description and time reduction of a Monte Carlo code to simulate propagation of polarized light through scattering media,” Appl. Opt. 42, 3290–3296 (2003).
[CrossRef] [PubMed]

Comput. Meth. Prog. Biomed. (1)

L.  Wang, S. L.  Jacques, L.  Zheng, “MCML-Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Meth. Prog. Biomed. 47, 131–146 (1995).
[CrossRef]

Diabetes Care (1)

W. F.  March, B.  Rabinovitch, R. L.  Adams, “Noninvasive glucose monitoring of the aqueous humor of the eye: Part II. Animal studies and the scleral lens,” Diabetes Care 5, 259–265 (1982).
[CrossRef] [PubMed]

IEEE Trans. Biomed. Eng. (1)

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

J. Biomed. Opt. (6)

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

D.  Côté, 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]

R. J.  McNichols, G. L.  Coté, “Optical glucose sensing in biological fluids: an overview,” J. Biomed. Opt. 5, 5–16 (2000).
[CrossRef] [PubMed]

S. L.  Jacques, J. C.  Ramella-Roman, K.  Lee, “Imaging skin pathology with polarized light,” J. Biomed. Opt. 7, 329–340 (2002).
[CrossRef] [PubMed]

A. N.  Yaroslavsky, I. V.  Yaroslavsky, T.  Goldbach, H.-J.  Schwarsmaier, “Influence of the scattering phase function approximation on the optical properties of blood determined from the integrating sphere measurements,” J. Biomed. Opt. 4, 47–53 (1999).
[CrossRef]

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

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[CrossRef]

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[CrossRef]

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[CrossRef] [PubMed]

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

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Figures (7)

Fig. 1.
Fig. 1.

Schematic diagram of the experimental setup used for the measurements. Chop.-mechanical chopper; P1: polarizer; PEM- photo-elastic modulator; A1,2: apertures; PC: polarizing beam splitter cube; L x,y - lenses; D x,y - photodetectors. fc and fp are the modulation frequencies for the mechanical chopper and PEM respectively.

Fig. 2.
Fig. 2.

a) Experimental measurements (squares) and calculation (line) of surviving linear polarization fraction for the forward-scattered beam in 1 µm diameter microspheres suspensions of various scattering coefficients, with no absorption (phase function with anisotropy g of 0.917). The error bars (shown on graph) are smaller than the symbols for most measurements. The cuvette has a thickness of 1 cm, the imaged area is approximately 5 mm in diameter with an acceptance angle of 15° for the apertures and lenses used. No adjustable parameters were used. b) Experimental measurements (symbols, no error bars provided) from Fig. 2a and 2c in Ghosh et al.[28] for surviving linear polarization fraction of the forward-scattered beam in cuvettes for thicknesses 0.5 cm (blue line) and 1 cm (red line) with 1.072 µm diameter (g=0.923) microspheres suspensions of various scattering coefficients, and corresponding calculations (lines, same colors as symbols) with the model presented here.

Fig. 3.
Fig. 3.

Experimental measurements (squares) and Monte Carlo predictions of the optical rotation (solid black line) as a function of scattering coefficient µs , in the transmission direction through a turbid 1 cm cuvette for solutions containing 1.2 M of glucose and 1.4 µm diameter microspheres (phase function with anisotropy of 0.943, no adjustable parameters were used in the calculation). The absorption coefficient µa is negligible. The rotation as predicted from the average path length if Eq. (1) held is also shown on the graph (blue line), and can be seen to increase with scattering coefficient, unlike the rotation which reaches a plateau.

Fig. 4.
Fig. 4.

Monte Carlo calculations of the beam orientation as a function of exit position for the (a) forward, (b) side and (c) back scattering geometry. The scattering coefficient is 20 cm-1 and the scatterer diameters are 1.0 (blue line), 1.4 (red) and 1.8 (green) µm, absorption is negligible. The scatterers have different phase functions and different anisotropies (0.917, 0.930 and 0.922 respectively) because of the different sizes. The beam is incident with a 45° polarization and all three solutions contain 1 M of glucose (effect of glucose on background index not considered). Regardless of the scatterer size, the rotation on axis in the forward direction is the same. Notice the sensitivity to exit position in directions other than the forward direction, with a sharp orientation change in the back direction. Another solution with no glucose and 1.8 µm scatterer diameter (black line) demonstrates that this change of orientation is solely due to scattering since the sharp change in orientation is also observed in that case.

Fig. 5.
Fig. 5.

Orientation of the polarization of a 45° incident beam after single scattering from a spherical scatterer (polystyrene spherical scatterer of 1.4 µm diameter in water, g=0.93). The polarization changes from its original 45° in the forward direction to 135° in the back scattering direction. Inset: Diagram illustrating the orientation of polarization in two limiting cases (forward and backward scattering).

Fig. 6.
Fig. 6.

Schematic diagram of the important points and vectors used in calculating the intersection between the photon path and a triangular surface element.

Fig. 7.
Fig. 7.

Polarized photon incident on detecting interface, where and represent the orientation of linear polarizers in the laboratory. The reference frame of the Stokes vector is rotated such that ê is parallel to ×ê 3. The intensity that is detected is obtained by I in that reference frame.

Tables (2)

Tables Icon

Table 1. Table comparing Pol-MC results with MCML. The MCML simulation is done with three layers (air, tissue, air) of indices of (1.0,1.33,1.0), the Henyey-Greenstein phase function with anisotropy parameter g=0.93, and µs =10 cm-1, µa =0.1 cm-1 in the tissue layer. The Pol-MC calculation is done using two different phase functions: the Henyey-Greenstein phase function with anisotropy parameter g=0.93 (no polarization considered in this case), and the phase function from Mie theory for spherical scatterers of 1.4 µm-diameter (with an anisotropy g of 0.93). In all three cases, there is a 2% specular reflection at the first interface because of the index mismatch between air and tissue.

Tables Icon

Table 2. Table showing the effect of finite size on diffuse reflectance (back scattering) and transmittance (forward scattering) for calculations done with three layers (air, tissue, air) of indices of (1.0,1.33,1.0) with µs =10 cm-1 and µa =0.1 cm-1 with spherical polystyrene scatterers of 1.4 µm-diameter (which have a Mie scattering phase function with an anisotropy g of 0.93). As the lateral extent of the sample goes from infinite to finite, the transmittance and reflectance go down since a larger fraction of the light is lost through the sides of the sample.

Equations (33)

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C = α [ α ] λ T L ,
tan 2 γ = U Q ,
α = 1 2 tan 1 ( Q 2 f p Q f c π J 2 ( δ ) ( 2 f p ) ( f c ) Q 2 f p Q f c ) ,
S = ( I Q U V ) ,
I E E * + E E * ,
Q E E * E E * ,
U E E * + E * E ,
V i ( E E * E * E ) ,
I = ( 1 0 0 0 ) ,
Q = ( 0 1 0 0 ) ,
U = ( 0 0 1 0 ) .
e ̂ = BR 3 ( ϕ ) ( 1 0 0 ) T ,
e ̂ = BR 3 ( ϕ ) ( 0 1 0 ) T ,
S = R S ( ϕ ) S .
R 3 ( ϕ ) = ( cos ϕ sin ϕ 0 sin ϕ cos ϕ 0 0 0 1 ) ,
R S ( ϕ ) = ( 1 0 0 0 0 cos 2 ϕ sin 2 ϕ 0 0 sin 2 ϕ cos 2 ϕ 0 0 0 0 1 ) ,
B = ( e ̂ · x ̂ e ̂ · x ̂ e ̂ 3 · x ̂ e ̂ · y ̂ e ̂ · y ̂ e ̂ 3 · y ̂ e ̂ · z ̂ e ̂ · z ̂ e ̂ 3 · z ̂ ) .
e ̂ = BR ( θ ) ( 0 1 0 ) T ,
e ̂ 3 = BR ( θ ) ( 0 0 1 ) T ,
S = M S ( θ ) S .
R ( θ ) = ( 1 0 0 0 cos θ sin θ 0 sin θ cos θ ) ,
S + t d = C ,
OC ¯ · n ̂ = 0 ,
t = OS ¯ · n ̂ d · n ̂ .
u = OC ¯ · a a ,
u = OC ¯ · b b .
e ̂ = BR ( θ i θ t ) ( 0 1 0 ) T ,
e ̂ 3 = BR ( θ i θ t ) ( 0 1 0 ) T ,
S = 𝓣 ( θ i ) S ,
𝓣 ( θ i ) = 1 2 ( t p 2 + t s 2 t p 2 t s 2 0 0 t p 2 t s 2 t p 2 + t s 2 0 0 0 0 2 t p t s 0 0 0 0 2 t p t s ) ,
e ̂ = BR ( 2 θ i π ) ( 0 1 0 ) T ,
e ̂ 3 = BR ( 2 θ i π ) ( 0 0 1 ) T ,
S = ( θ i ) S ,

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