Abstract

Three Monte Carlo programs were developed which keep track of the status of polarization of light traveling through mono-disperse solutions of micro-spheres. These programs were described in detail in our previous article [1]. This paper illustrates a series of Monte Carlo simulations that model common experiments of light transmission and reflection of scattering media. Furthermore the codes were expanded to model light propagating through poly-disperse solutions of micro-spheres of different radii distributions.

©2005 Optical Society of America

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References

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  1. J.C. Ramella-Roman , S.A. Prahl , and S.L. Jacques , “ Three Monte Carlo programs of polarized light transport into scattering media: part I ,” Opt. Express   13 , 4420 – 4438 , ( 2005 ), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-12-4420
    [Crossref] [PubMed]
  2. S. Bartel and A. H. Hielscher , “ Monte Carlo simulations of the diffuse backscattering Mueller matrix for highly scattering media ,” Appl. Opt.   39 , 1580 – 1588 , ( 2000 ).
    [Crossref]
  3. B. D. Cameron , M. J. Rakovic , M. Mehrubeoglu , G. W. Kattawar , S. Rastegar , L. V. Wang , and G. Cote , “ Measurement and calculation of the two-dimensional backscattering Mueller matrix of a turbid medium ,” Opt. Lett.   23 , 485 – 487 , 1998 .
    [Crossref]
  4. M. J. Rakovic , G. W. Kattawar , M. Mehrubeoglu , B. D. Cameron , L. V. Wang , S. Rastegar , and G. L. Cote , “ Light backscattering polarization patterns from turbid media: theory and experiment ,” Appl. Opt.   38 , 3399 – 3408 , ( 1999 ).
    [Crossref]
  5. Daniel CÔté and I. Alex Vitkin , “ Robust concentration determination of optically active molecules in turbid media with validated three-dimensional polarization sensitive Monte Carlo calculations ,” Opt. Express   13 , 148 – 163 , 2005 . http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-1-148
    [Crossref] [PubMed]
  6. M. Mehrubeoglu , N. Kehtarnavaz , S. Rastegar , and L. V. Wang , “ Effect of molecular concentrations in tissue simulating phantoms on images obtained using diffuse reflectance polarimetry ,” Opt. Express   3 , 286 – 297 , ( 1998 ). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-3-7-286
    [Crossref] [PubMed]
  7. M. Xu , “ Electric field Monte Carlo simulation of polarized light propagation in turbid media ”, Opt. Express   26 , 6530 – 6539 , ( 2004 ). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-26-6530
    [Crossref]
  8. K.F. Evans and G.L. Stephens , “ A new polarized atmospheric radiative transfer model ,” J.Quant.Spectrosc. Radiat. Transfer.   46 , 413 – 423 , ( 1991 ).
    [Crossref]
  9. The experimental data gently provided by Dr. Cameron.
  10. S. L. Jacques , R. J. Roman , and K. Lee , “ Imaging superficial tissues with polarized light ,” Lasers Surg. Med.   26 , 119 – 129 , ( 2000 ).
    [Crossref] [PubMed]
  11. G. Yao and L-H. Wang , “ Two-dimensional depth-resolved Muller matrix characterization of biological tissue by optical coherence tomography ,” Opt. Letters   24 , 537 – 539 , ( 1999 ).
    [Crossref]
  12. R. J. Allen and M. R. Platt , “ Lidar for multiple backscattering and depolarization observations ,” Appl. Opt.   16 ; 3193 – 3199 , ( 1977 ).
    [Crossref] [PubMed]
  13. G. Zaccanti , P. Bruscaglioni , M. Gurioli , and P. Sansoni , “ Laboratory simulations of lidar returns from clouds: experimental and numerical results ,” Appl. Opt.   32 , 1590 – 1597 , ( 1993 ).
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  14. P. Bruscaglioni , G. Zaccanti , and Q. Wei , “ Transmission of a pulsed polarized light beam through a thick turbid media: numerical results ,” Appl. Opt.   32 , 6142 – 6150 , ( 1993 ).
    [Crossref] [PubMed]
  15. J. He , A Karlsson , J. Swartling , and S. Anderson-Engles , “ Light scattering by multiple red blood cells ,” J. Opt. Soc. Am. A.   21 , 1953 – 1961 , ( 2004 ).
    [Crossref]
  16. S. V. Tsinopoulos , E. J. Sellountos , and D. Polyzos “ Light scattering by aggregated red blood cells ,” Appl. Opt.   41 , 1408 – 1417 , ( 2002 ).
    [Crossref] [PubMed]
  17. L. Whang , S.L. Jacques , and L. Zheng , “ MCML - Monte Carlo modeling of light transport in multi-layered tissues ,” Comput. Methods Programs Biomed.   47 , 131 – 46 , ( 1995 ).
    [Crossref]
  18. W. H. Press , S. A. Teukolsky , W. T. Vetterling , and B. P. Flannery , Numerical Recipes in C, the art of Scientific Computing , ( Cambridge University Press , 1992 ).

2005 (2)

2004 (2)

M. Xu , “ Electric field Monte Carlo simulation of polarized light propagation in turbid media ”, Opt. Express   26 , 6530 – 6539 , ( 2004 ). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-26-6530
[Crossref]

J. He , A Karlsson , J. Swartling , and S. Anderson-Engles , “ Light scattering by multiple red blood cells ,” J. Opt. Soc. Am. A.   21 , 1953 – 1961 , ( 2004 ).
[Crossref]

2002 (1)

2000 (2)

S. L. Jacques , R. J. Roman , and K. Lee , “ Imaging superficial tissues with polarized light ,” Lasers Surg. Med.   26 , 119 – 129 , ( 2000 ).
[Crossref] [PubMed]

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

1999 (2)

G. Yao and L-H. Wang , “ Two-dimensional depth-resolved Muller matrix characterization of biological tissue by optical coherence tomography ,” Opt. Letters   24 , 537 – 539 , ( 1999 ).
[Crossref]

M. J. Rakovic , G. W. Kattawar , M. Mehrubeoglu , B. D. Cameron , L. V. Wang , S. Rastegar , and G. L. Cote , “ Light backscattering polarization patterns from turbid media: theory and experiment ,” Appl. Opt.   38 , 3399 – 3408 , ( 1999 ).
[Crossref]

1998 (2)

1995 (1)

L. Whang , S.L. Jacques , and L. Zheng , “ MCML - Monte Carlo modeling of light transport in multi-layered tissues ,” Comput. Methods Programs Biomed.   47 , 131 – 46 , ( 1995 ).
[Crossref]

1993 (2)

1991 (1)

K.F. Evans and G.L. Stephens , “ A new polarized atmospheric radiative transfer model ,” J.Quant.Spectrosc. Radiat. Transfer.   46 , 413 – 423 , ( 1991 ).
[Crossref]

1977 (1)

Alex Vitkin, I.

Allen, R. J.

Anderson-Engles, S.

J. He , A Karlsson , J. Swartling , and S. Anderson-Engles , “ Light scattering by multiple red blood cells ,” J. Opt. Soc. Am. A.   21 , 1953 – 1961 , ( 2004 ).
[Crossref]

Bartel, S.

Bruscaglioni, P.

Cameron, B. D.

Cote, G.

Cote, G. L.

CÔté, Daniel

Evans, K.F.

K.F. Evans and G.L. Stephens , “ A new polarized atmospheric radiative transfer model ,” J.Quant.Spectrosc. Radiat. Transfer.   46 , 413 – 423 , ( 1991 ).
[Crossref]

Flannery, B. P.

W. H. Press , S. A. Teukolsky , W. T. Vetterling , and B. P. Flannery , Numerical Recipes in C, the art of Scientific Computing , ( Cambridge University Press , 1992 ).

Gurioli, M.

He, J.

J. He , A Karlsson , J. Swartling , and S. Anderson-Engles , “ Light scattering by multiple red blood cells ,” J. Opt. Soc. Am. A.   21 , 1953 – 1961 , ( 2004 ).
[Crossref]

Hielscher, A. H.

Jacques, S. L.

S. L. Jacques , R. J. Roman , and K. Lee , “ Imaging superficial tissues with polarized light ,” Lasers Surg. Med.   26 , 119 – 129 , ( 2000 ).
[Crossref] [PubMed]

Jacques, S.L.

Karlsson, A

J. He , A Karlsson , J. Swartling , and S. Anderson-Engles , “ Light scattering by multiple red blood cells ,” J. Opt. Soc. Am. A.   21 , 1953 – 1961 , ( 2004 ).
[Crossref]

Kattawar, G. W.

Kehtarnavaz, N.

Lee, K.

S. L. Jacques , R. J. Roman , and K. Lee , “ Imaging superficial tissues with polarized light ,” Lasers Surg. Med.   26 , 119 – 129 , ( 2000 ).
[Crossref] [PubMed]

Mehrubeoglu, M.

Platt, M. R.

Polyzos, D.

Prahl, S.A.

Press, W. H.

W. H. Press , S. A. Teukolsky , W. T. Vetterling , and B. P. Flannery , Numerical Recipes in C, the art of Scientific Computing , ( Cambridge University Press , 1992 ).

Rakovic, M. J.

Ramella-Roman, J.C.

Rastegar, S.

Roman, R. J.

S. L. Jacques , R. J. Roman , and K. Lee , “ Imaging superficial tissues with polarized light ,” Lasers Surg. Med.   26 , 119 – 129 , ( 2000 ).
[Crossref] [PubMed]

Sansoni, P.

Sellountos, E. J.

Stephens, G.L.

K.F. Evans and G.L. Stephens , “ A new polarized atmospheric radiative transfer model ,” J.Quant.Spectrosc. Radiat. Transfer.   46 , 413 – 423 , ( 1991 ).
[Crossref]

Swartling, J.

J. He , A Karlsson , J. Swartling , and S. Anderson-Engles , “ Light scattering by multiple red blood cells ,” J. Opt. Soc. Am. A.   21 , 1953 – 1961 , ( 2004 ).
[Crossref]

Teukolsky, S. A.

W. H. Press , S. A. Teukolsky , W. T. Vetterling , and B. P. Flannery , Numerical Recipes in C, the art of Scientific Computing , ( Cambridge University Press , 1992 ).

Tsinopoulos, S. V.

Vetterling, W. T.

W. H. Press , S. A. Teukolsky , W. T. Vetterling , and B. P. Flannery , Numerical Recipes in C, the art of Scientific Computing , ( Cambridge University Press , 1992 ).

Wang, L. V.

Wang, L-H.

G. Yao and L-H. Wang , “ Two-dimensional depth-resolved Muller matrix characterization of biological tissue by optical coherence tomography ,” Opt. Letters   24 , 537 – 539 , ( 1999 ).
[Crossref]

Wei, Q.

Whang, L.

L. Whang , S.L. Jacques , and L. Zheng , “ MCML - Monte Carlo modeling of light transport in multi-layered tissues ,” Comput. Methods Programs Biomed.   47 , 131 – 46 , ( 1995 ).
[Crossref]

Xu, M.

M. Xu , “ Electric field Monte Carlo simulation of polarized light propagation in turbid media ”, Opt. Express   26 , 6530 – 6539 , ( 2004 ). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-26-6530
[Crossref]

Yao, G.

G. Yao and L-H. Wang , “ Two-dimensional depth-resolved Muller matrix characterization of biological tissue by optical coherence tomography ,” Opt. Letters   24 , 537 – 539 , ( 1999 ).
[Crossref]

Zaccanti, G.

Zheng, L.

L. Whang , S.L. Jacques , and L. Zheng , “ MCML - Monte Carlo modeling of light transport in multi-layered tissues ,” Comput. Methods Programs Biomed.   47 , 131 – 46 , ( 1995 ).
[Crossref]

Appl. Opt. (6)

Comput. Methods Programs Biomed. (1)

L. Whang , S.L. Jacques , and L. Zheng , “ MCML - Monte Carlo modeling of light transport in multi-layered tissues ,” Comput. Methods Programs Biomed.   47 , 131 – 46 , ( 1995 ).
[Crossref]

J. Opt. Soc. Am. A. (1)

J. He , A Karlsson , J. Swartling , and S. Anderson-Engles , “ Light scattering by multiple red blood cells ,” J. Opt. Soc. Am. A.   21 , 1953 – 1961 , ( 2004 ).
[Crossref]

J.Quant.Spectrosc. Radiat. Transfer. (1)

K.F. Evans and G.L. Stephens , “ A new polarized atmospheric radiative transfer model ,” J.Quant.Spectrosc. Radiat. Transfer.   46 , 413 – 423 , ( 1991 ).
[Crossref]

Lasers Surg. Med. (1)

S. L. Jacques , R. J. Roman , and K. Lee , “ Imaging superficial tissues with polarized light ,” Lasers Surg. Med.   26 , 119 – 129 , ( 2000 ).
[Crossref] [PubMed]

Opt. Express (4)

Opt. Lett. (1)

Opt. Letters (1)

G. Yao and L-H. Wang , “ Two-dimensional depth-resolved Muller matrix characterization of biological tissue by optical coherence tomography ,” Opt. Letters   24 , 537 – 539 , ( 1999 ).
[Crossref]

Other (2)

The experimental data gently provided by Dr. Cameron.

W. H. Press , S. A. Teukolsky , W. T. Vetterling , and B. P. Flannery , Numerical Recipes in C, the art of Scientific Computing , ( Cambridge University Press , 1992 ).

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Figures (14)

Fig. 1.
Fig. 1. Experimental set-up for back-reflected Mueller matrix measurements.

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Fig. 2.
Fig. 2. Comparison of Mueller matrix results for light backscattered from solutions of micro-spheres. The image on the left is an experimental result of Cameron et al. the image on the right is the Mueller matrix obtained with our Monte Carlo programs. (The left-hand image was gently provided by Dr. Cameron.)

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Fig. 3.
Fig. 3. Experimental apparatus. We studied backscattering from a solution of micro-spheres (diameter 2μm) when the polarized illumination is at an angle of 30°.

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Fig. 4.
Fig. 4. Experimental results (left) and Monte Carlo simulation (right) of the first 9 of elements of the 16 element of Mueller matrix. The micro-spheres diameter was 2μm and index of refraction 1.59. The micro-spheres were in a water media with index of refraction equal to 1.33. Each image is 4x4 cm. Light was incident at an angle θ from the normal. m1 1 values are in between 0 and 0.1 all other element were in between -.01 and 0.1; only positive values are shown in these images to enhance contrast.

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Fig. 5.
Fig. 5. Experimental setup, top view. Polarized images of light scattered by a micro-spheres solution were acquired with a digital camera. Laser light (632.8nm) was polarized and the scattered light was passed through an analyzer whose orientation was either parallel or perpendicular to the incident beam.

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Fig. 6.
Fig. 6. Top image is the experimental results obtained with polarizer and analyzer oriented parallel to each other, the bottom image is the Monte Carlo simulation.

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Fig. 7.
Fig. 7. Comparison of exittance for experiment (line) and Monte Carlo results (void circles) for parallel image. This intensity profile corresponds to the centerline of the axis of irradiation.

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Fig. 8.
Fig. 8. Top image is the experimental results obtained with the polarizer oriented parallel to the optical table and analyzer oriented perpendicular to the optical table, the bottom image is the corresponding Monte Carlo simulation. Both images were normalized by the maximum value of the parallel image.

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Fig. 9.
Fig. 9. Transmission experiment. The incident beam is polarized parallel to the optical table and an analyzer in front of the detector selects the status of polarization of light transmitted through the cuvette.

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Fig. 10.
Fig. 10. Experimental results and Monte Carlo simulations (black circles and squares) for a mono-disperse solution of micro-spheres. Diameter of micro-spheres was 482 nm, light wavelength was 543 nm. The back line is the heuristic model applied to the Monte Carlo data

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Fig. 11.
Fig. 11. Experimental results and Monte Carlo simulations (black circles and squares) for a mono-disperse solution of micro-spheres. Diameter was 0.308 μm, light wavelength was 0.543 μm.

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Fig. 12.
Fig. 12. A photon scattering in a poly-disperse solution of micro-spheres

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Fig. 13.
Fig. 13. Total transmittance and reflectance geometry; the reference plane for the incident polarized beam was the plane of the x and y unit vectors.

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Fig. 14.
Fig. 14. Degree of polarization for poly-disperse solutions of micro-spheres and mono-disperse solutions of identical anisotropy.

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Tables (2)

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Table 1. Results of total reflectance from a solution of poly-disperse modified gamma distribution of spheres. Comparison of Adding Doubling code by Evans and Monte Carlo simulations

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Table 2. Results of total transmission through a solution of poly-disperse modified gamma distribution of spheres. Comparison of Adding Doubling code by Evans and Monte Carlo simulations

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Equations (1)

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N ( r ) = a ∙ r α exp ( − b ∙ r γ )

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