## Electric field Monte Carlo simulation of polarized light propagation in turbid media

Optics Express, Vol. 12, Issue 26, pp. 6530-6539 (2004)

http://dx.doi.org/10.1364/OPEX.12.006530

Acrobat PDF (156 KB)

### Abstract

A Monte Carlo method based on tracing the multiply scattered electric field is presented to simulate the propagation of polarized light in turbid media. Multiple scattering of light comprises a series of updates of the parallel and perpendicular components of the complex electric field with respect to the scattering plane by the amplitude scattering matrix and rotations of the local coordinate system spanned by the unit vectors in the directions of the parallel and perpendicular electric field components and the propagation direction of light. The backscattering speckle pattern and the backscattering Mueller matrix of an aqueous suspension of polystyrene spheres in a slab geometry are computed using this Electric Field Monte Carlo (EMC) method. An efficient algorithm computing the Mueller matrix in the pure backscattering direction is detailed in the paper.

© 2004 Optical Society of America

## 1. Introduction

4. S. R. Arridge, “Optical tomography in medical imaging,” Inverse Problems **15**, R41–R93 (1999). [CrossRef]

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

8. A. D. Kim and J. B. Keller, “Light propagation in biological tissue,” J. Opt. Soc. Am. A **20**, 92–98 (2003). [CrossRef]

9. G. W. Kattawar and G. N. Plass, “Radiance and polarization of multiple scattered light from haze and clouds,” Appl. Opt. **7**, 1519–1527 (1968). [CrossRef] [PubMed]

18. X. Wang and L. V. Wang, “Propagation of polarized light in birefringent turbid media: A Monte Carlo study,” J. Biomed. Opt. **7**, 279–290 (2002). [CrossRef] [PubMed]

**I**=(

*I*,

*Q*,

*U*,

*V*)

^{T}in simulation where

*I*=〈|

*E*

_{l}|

^{2}+|

*E*

_{r}|

^{2}〉,

*Q*=〈|

*E*

_{l}|

^{2}-|

*E*

_{r}|

^{2}〉,

*U*=〈

*E*

_{r}+

*E*

_{l}

*V*=-

*i*〈

*E*

_{r}-

*E*

_{l}

*E*

_{l}and

*E*

_{r}are two orthogonal complex electric field components perpendicular to the propagation direction, the superscript “

*T*” denotes transpose, and 〈〉 means ensemble average. Light scattering involves a rotation of the Stokes vector to a local scattering reference frame and the multiplication of the Stokes vector by the 4×4 Mueller matrix which prescribes how polarized light is scattered by an isolated particle in that reference frame. The Stokes vector is summed up at the detector assuming the detected light is incoherent. Many implementations of the above Monte Carlo approach to simulate polarized light propagation in turbid media have appeared in the literature[9

9. G. W. Kattawar and G. N. Plass, “Radiance and polarization of multiple scattered light from haze and clouds,” Appl. Opt. **7**, 1519–1527 (1968). [CrossRef] [PubMed]

18. X. Wang and L. V. Wang, “Propagation of polarized light in birefringent turbid media: A Monte Carlo study,” J. Biomed. Opt. **7**, 279–290 (2002). [CrossRef] [PubMed]

13. M. J. Rakovic, G. W. Kattawar, M. Mehrbeolu, 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(15)**, 3399–3408 (1999). [CrossRef]

14. S. Bartel and A. H. Hielscher, “Monte Carlo Simulations of the Diffuse Backscattering Mueller Matrix for Highly Scattering Media,” Appl. Opt. **39(10)**, 1580–1588 (2000). [CrossRef]

16. H. H. Tynes, G. W. Kattawar, E. P. Zege, I. L. Katsev, A. S. Prikhach, and L. I. Chaikovskaya, “Monte Carlo and Multicomponent Approximation Methods for Vector Radiative Transfer by use of Effective Mueller Matrix Calculations,” Appl. Opt. **40(3)**, 400–412 (2001). [CrossRef]

18. X. Wang and L. V. Wang, “Propagation of polarized light in birefringent turbid media: A Monte Carlo study,” J. Biomed. Opt. **7**, 279–290 (2002). [CrossRef] [PubMed]

17. B. Kaplan, G. Ledanois, and B. villon, “Mueller Matrix of Dense Polystyrene Latex Sphere Suspensions: Measurements and Monte Carlo Simulation,” Appl. Opt. **40(16)**, 2769–2777 (2001). [CrossRef]

17. B. Kaplan, G. Ledanois, and B. villon, “Mueller Matrix of Dense Polystyrene Latex Sphere Suspensions: Measurements and Monte Carlo Simulation,” Appl. Opt. **40(16)**, 2769–2777 (2001). [CrossRef]

14. S. Bartel and A. H. Hielscher, “Monte Carlo Simulations of the Diffuse Backscattering Mueller Matrix for Highly Scattering Media,” Appl. Opt. **39(10)**, 1580–1588 (2000). [CrossRef]

16. H. H. Tynes, G. W. Kattawar, E. P. Zege, I. L. Katsev, A. S. Prikhach, and L. I. Chaikovskaya, “Monte Carlo and Multicomponent Approximation Methods for Vector Radiative Transfer by use of Effective Mueller Matrix Calculations,” Appl. Opt. **40(3)**, 400–412 (2001). [CrossRef]

## 2. Theoretical formalism

*θ*due to the symmetry[22]. The parallel and perpendicular components of the electric field with respect to the scattering plane are scattered according to

*S*

_{j}(

*θ*) where

*j*=2, 1, respectively[21].

**m**,

**n**,

**s**) where

**m**and

**n**are the unit vectors in the directions of the parallel and perpendicular components of the electric field with respect to the scattering plane of the previous scattering event and s is the photon’s propagation direction prior to the current scattering [see Fig. 1]. The propagation direction

*s′*of the photon after the current scattering is given by:

*θ*and

*ϕ*are the scattering and azimuthal angles of the current scattering respectively.

**s**and

**s′**. The unit vectors in the directions of the parallel and perpendicular electric fields with respect to the current scattering plane are given by

**e′**

_{2}=

**e**

_{2}

**m**,

**n**,

**s**) is rotated to (

**m′**,

**n′**,

**s′**) whose

**m′**=

**e′**

_{1}and

**n′**=

**e′**

_{2}after scattering. The incident electric field

**E**=

*E*

_{1}

**m**+

*E*

_{2}

**n**is scattered to

**E′**=

*E′*

_{1}

**m′**+

*E′*

_{2}

**n′**whose parallel and perpendicular components are given by

*E′*

_{1}=

*S*

_{2}

**E**·

**e**

_{1}and

*E′*

_{2}=

*S*

_{1}

**E**·

**e**

_{2}, respectively.

*θ*and the azimuthal angle

*ϕ*, the local coordinate system is updated by

*F*(

*θ,ϕ*), the scattered wave intensity propagating along the (

*θ,ϕ*) direction, given by

*E*

_{1}|

^{2}+|

*E*

_{2}|

^{2}=1 is assumed unity in Eq. (8). The scattered light intensity |

*E′*

_{1}|

^{2}+|

*E′*

_{2}|

^{2}is then conserved (equal to unity) in the series of scattering events. Absorption of light, if any, is accounted for by adjusting the photon weight as in the conventional Monte Carlo simulations[23

23. R. Y. Rubinstein, *Simulation and the Monte Carlo method* (John Wiley & Sons, 1981). [CrossRef]

*A*rotates the local coordinate system (

**m**,

**n**,

**s**) each time the photon is scattered by a scatterer. The electric field is updated simultaneously by the matrix

*B*. The free path between consecutive scattering events is sampled in the same fashion as that in a conventional Monte Carlo method. The state of a multiply scattered photon is specified by the final local coordinate system (

**m**

^{(n)},

**n**

^{(n)},

**s**

^{(n)}) after consecutive rotations, the final complex electric field components

*l*inside the host medium where

*n*denotes the number of scattering events the photon have experienced before being detected. The detected electric field is given by

**E**

_{d}=[

**m**

^{(n)}+

**n**

^{(n)}]exp(

*ikl*) where

*k*=2

*π*/

*λ*is the wave number,

*λ*is the wavelength of light in the host medium, and we have assumed a convention that the temporal dependence of a plane wave of frequency

*ω*is exp(-

*iωt*). The phase of the detected photon accrues from phase delays due to the interaction of light with scatterers and propagation through the host medium. The photon weight

*w*, unity at incidence, is multiplied by the albedo of the scatterer at each scattering event. Once the photon hits a detector, the electric field at the detector is increased by

*w*

^{1/2}

**E**

_{d}and the Stokes vector is increased by the Stokes vector computed from the electric field

*w*

^{1/2}

**E**

_{d}.

*θ,ϕ*) which distribute according to the normalized phase function

*p*(

*θ,ϕ*)=

*F*(

*θ,ϕ*)/

*πQ*

_{sca}

*x*

^{2}where

*Q*

_{sca}is the scattering efficiency,

*x*=

*ka*is the size parameter, and

*a*is the radius of the particle. The rejection technique[24, 25] has been widely used to sample such a distribution function. The procedure is to choose a doublet (

*µ*≡cos

*θ,ϕ*) where

*µ*and

*ϕ*are uniformly distributed over [-1,1] and [0,2

*π*] respectively and a random number

*f*uniformly distributed over [0,max

_{θ,ϕ}

*p*(

*θ,ϕ*)], the doublet (

*µ,ϕ*) is accepted as the scattering and azimuthal angles if

*f*<

*p*(arccos

*µ,ϕ*), otherwise generate a new doublet (

*µ,ϕ*), a new

*f*and start over. Each trial involves one Mie calculation of the amplitude scattering matrix at that trial scattering angle. The number of rejections per scattering event is large (on the order of one for Rayleigh particles and can be much more significant for larger particles). This limits the efficiency of the Monte Carlo simulation. Mie calculations of the amplitude scattering matrix are hence usually performed on a fixed set of scattering angles in advance to generate a table of scattering matrices and reduce the computation load.

*θ*by drawing of one random number uniformly distributed over [0,1] and looking up an inverse table of the marginal distribution function

*ϕ*is then obtained by the rejection method for the conditional probability

*p*(

*ϕ*|

*θ*)=

*p*(

*θ*,

*ϕ*)/

*p*(

*θ*). One Mie calculation for the looked up scattering angle is required if the table of scattering matrices is not generated in advance for each scattering event.

*θ*according to

*p*(

*θ*) while to sample

*ϕ*uniformly distributed over [0,2

*π*]. At the same time, we modify

*F*(

*θ,ϕ*) in the update rule of the electric field (6) to

*F′*(

*θ*)=|

*S*

_{2}|

^{2}+|

*S*

_{1}|

^{2}/2 such that the light intensity is no longer conserved in the series of scattering events. This strategy saves the rejection sampling of the angle

*ϕ*but the simulation result is prone to be contaminated by hotspots and has a larger variance compared to the first strategy because it is unfavorable to the statistical variance when photons of varying weights, rather than equal weights, are accumulated by the detector. The second sampling strategy is not used in simulation.

## 3. Results and Discussion

### 3.1. Backscattering speckle pattern

*α*=

*x,y, z*component of the outgoing light intensity

*I*

_{α}(

*θ,ϕ*) in direction (

*θ,ϕ*) on the boundary of the medium comprises a multitude of independently-phased additive complex electric fields in that direction (outside the coherent backscattering cone[26

26. P.-E. Wolf and G. Maret, “Weak Localization and Coherent Backscattering of Photons in Disordered Media,” Phys. Rev. Lett. **55(24)**, 2696–2699 (1985). [CrossRef]

28. E. Akkermans, P. E. Wolf, and R. Maynard, “Coherent backscattering of light by disordered media: analysis of the peak line shape,” Phys. Rev. Lett. **56(14)**, 1471–1474 (1986). [CrossRef]

29. J. W. Goodman, “Statistical properties of laser speckle patterns,” in *Laser speckle and related phenomena* ,J. C. Dainty, ed., pp. 9–75 (Springer-Verlag, Berlin, 1975). [CrossRef]

*I*

_{α}〉 is the mean intensity. The normalized light intensity

*η*=

*I*

_{α}/〈

*I*

_{α}〉 follows the exponential distribution exp(-

*η*).

*x*direction and propagates in the

*z*direction (normal to the surface of the slab). The size of the polystyrene sphere is 0.46

*µ*m. The wavelength of the incident wave is

*λ*

_{vac}=0.52

*µ*m in vaccum. The refractive indices of the polystyrene sphere and water is

*n*

_{sct}=1.59 and

*n*

_{bg}=1.33, respectively. The mean scattering length of light inside the solution is

*l*

_{s}=2.80

*µ*m with 2

*πn*

_{bg}

*l*

_{s}/

*λ*

_{vac}=45. The thickness of the slab is

*L*

_{z}=4

*l*

_{s}. Total 5×10

^{8}photons are launched into the medium. Both backscattering electric field and Stokes vector are recorded versus the direction (

*θ,ϕ*) of the backscattered light where

*θ*is the angle away from the surface normal (0≤

*θ*≤

*π*/2) and

*ϕ*is the azimuthal angle (0≤

*ϕ*≤2

*π*). The ranges of the angles

*θ*and

*ϕ*are uniformly divided into 1125 and 360 bins in simulation, respectively.

*x*component

*I*

_{x}/〈

*I*

_{x}〉 where

*I*

_{x}=|

*E*

_{x}|

^{2}and 〈

*I*

_{x}〉 is estimated by the mean of the first and second Stokes vectors. The appearance of speckles is obvious in Fig. 2(a). The first order statistics about

*I*

_{x}/〈

*I*

_{x}〉 can be computed from its histogram. This yields a negative exponential distribution exp(-

*η*) as expected [see Fig. 2(b)]. The other two

*y*and

*z*components of light behave similarly and not displayed.

### 3.2. Backscattering Mueller matrix

*µ*m. The wavelength of the incident wave is

*λ*

_{vac}=0.63

*µ*m in vacuum. The refractive indices of the polystyrene sphere and water is

*n*

_{sct}=1.59 and

*n*

_{bg}=1.33, respectively. The thickness of the slab is

*L*

_{z}=4

*l*

_{s}. Total 3×10

^{8}photons are launched into the medium.

17. B. Kaplan, G. Ledanois, and B. villon, “Mueller Matrix of Dense Polystyrene Latex Sphere Suspensions: Measurements and Monte Carlo Simulation,” Appl. Opt. **40(16)**, 2769–2777 (2001). [CrossRef]

**40(16)**, 2769–2777 (2001). [CrossRef]

^{8}photons launched using one Pentium III 1GHz CPU. In computation of the Stokes vector in Monte Carlo simulations, one should note that the Stokes vector upon which Mueller matrix is defined depends on the reference system used. The backscattered Stokes vector is defined in the reference system whose

*x′y′z′*axes coincide with -

*x*,

*y*,-

*z*axes of the reference system of the incident Stokes vector respectively.

*M*

^{bs}(

*ρ,ϕ*) of the medium shall satisfy the following relation[19, 30

30. D. S. Saxon, “Tensor Scattering Matrix for the Electromagnetic Field,” Phys. Rev. **100(6)**, 1771–1775 (1955). [CrossRef]

*ρ,ϕ*is the polar coordinate of the position of the outgoing beam and

*R*(

*ϕ*) is the rotation matrix

*ρ*)≡

*M*

^{bs}(

*ρ,ϕ*=0) relates

**I′**

_{i,o}are the incident and outgoing Stokes vectors with respect to the

*ϕ*=0 plane.

**E**

_{i}=cos

*αe*

^{-iβ}

*x̂*+sin

*αe*

^{iβ}

*ŷ*where

*α*∊(0,

*π*/2) and

*β*∊(0,

*π*) are uniform random numbers. The corresponding incident Stokes vector with respect to the

*xy*axes (i.e., the

*y*=0 plane) is

**I**

_{i}=(1,cos2

*α*, sin2

*α*cos2

*β*, sin2

*α*sin2

*β*)

^{T}. The incident Stokes vector with respect to the

*ϕ*=0 plane is given by

**I′**

_{i}=

*R*(

*ϕ*)

**I**

_{i}. The ensemble average of

**I′**

_{i}(

**I′**

_{i})

^{T}over the random variates

*α*and

*β*yields

**I′**

_{i}(

**I′**

_{i})

^{T}〉 is given by

*l*

_{s}×20

*l*

_{s}in size, with the laser being incident in the center. The displayed Mueller matrix has been normalized by the maximum light intensity of the (1,1) element of

*M*

^{bs}. Only 7 elements of the Mueller matrix are independent. The symmetrical relation between different elements of the Mueller matrix has been considered by Kattawa et. al.[19, 31

31. B. D. Cameron, M. J. Rakovic, M. Mehrbeoglu, G. W. Kattawar, S. Rastegar, L. V. Wang, and G. L. Cot, “Measurement and calculationof the two-dimensional backscattering Mueller matrix of a turbid medium,” Opt. Lett. **23(7)**, 485–487 (1998). [CrossRef]

13. M. J. Rakovic, G. W. Kattawar, M. Mehrbeolu, 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(15)**, 3399–3408 (1999). [CrossRef]

31. B. D. Cameron, M. J. Rakovic, M. Mehrbeoglu, G. W. Kattawar, S. Rastegar, L. V. Wang, and G. L. Cot, “Measurement and calculationof the two-dimensional backscattering Mueller matrix of a turbid medium,” Opt. Lett. **23(7)**, 485–487 (1998). [CrossRef]

32. I. Berezhnyy and A. Dogariu, “Time-resolved Mueller matrix imaging polarimetry,” Opt. Exp. **12(19)**, 4635–4649 (2004). [CrossRef]

13. M. J. Rakovic, G. W. Kattawar, M. Mehrbeolu, 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(15)**, 3399–3408 (1999). [CrossRef]

*ρ*). Each element of the reduced Mueller matrix is given as a one-dimensional curve versus the distance

*ρ*from the origin in Fig. (4). The reduced backscattering Mueller matrix is found to be 2×2 block diagonal. The first two elements of the Stokes vector

*I′*

_{1,2}are then decoupled from the rest two elements of the Stokes vector

*I′*

_{3,4}. This means, for example, the backscattered light due to a normally incident light linearly polarized in the

*ϕ*=0 plane (

**I′**

_{i}=(1,1,0,0)

^{T}) has no circular polarization component (the fourth element of Stokes vector

**I′**

_{o}=

*ρ*)

**I′**

_{i}is always zero) with respect to the

*ϕ*=0 plane. The circular polarization component of this backscattered light is no longer zero with respect to a different reference frame rather than the

*ϕ*=0 plane.

## 4. Conclusion

33. EMC is available at http://www.sci.ccny.cuny.edu/minxu.

## Acknowledgments

## References and links

1. | A. Ishimaru, |

2. | A. Yodh and B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today |

3. | S. K. Gayen and R. R. Alfano, “Emerging optical biomedical imaging techniques,” Opt. Photon. News |

4. | S. R. Arridge, “Optical tomography in medical imaging,” Inverse Problems |

5. | S. Chandrasekhar, |

6. | K. F. Evans and G. L. Stephens, “A new polarized atmospheric radiative transfer model,” J. Quant. Spectrosc. Radiat. Transfer |

7. | A. D. Kim and M. Moscoso, “Chebyshev Spectral methods for radiative transfer,” SIAM J. Sci. Comput. |

8. | A. D. Kim and J. B. Keller, “Light propagation in biological tissue,” J. Opt. Soc. Am. A |

9. | G. W. Kattawar and G. N. Plass, “Radiance and polarization of multiple scattered light from haze and clouds,” Appl. Opt. |

10. | I. Lux and L. Koblinger, |

11. | J. M. Schmitt, A. H. Gandjbakhche, and R. F. Bonner, “Use of polarized light to discriminate short-path photons in a multiply scattering medium,” Appl. Opt. |

12. | P. Bruscaglioni, G. Zaccanti, and Q. Wei, “Transmission of a pulsed polarized light beam through thick turbid media: numerical results,” Appl. Opt. |

13. | M. J. Rakovic, G. W. Kattawar, M. Mehrbeolu, B. D. Cameron, L. V. Wang, S. Rastegar, and G. L. Cote, “Light Backscattering Polarization Patterns from Turbid Media: Theory and Experiment,” Appl. Opt. |

14. | S. Bartel and A. H. Hielscher, “Monte Carlo Simulations of the Diffuse Backscattering Mueller Matrix for Highly Scattering Media,” Appl. Opt. |

15. | M. Moscoso, J. B. Keller, and G. Papanicolaou, “Depolarization and blurring of optical images by biological tissue,” J. Opt. Soc. Am. A |

16. | H. H. Tynes, G. W. Kattawar, E. P. Zege, I. L. Katsev, A. S. Prikhach, and L. I. Chaikovskaya, “Monte Carlo and Multicomponent Approximation Methods for Vector Radiative Transfer by use of Effective Mueller Matrix Calculations,” Appl. Opt. |

17. | B. Kaplan, G. Ledanois, and B. villon, “Mueller Matrix of Dense Polystyrene Latex Sphere Suspensions: Measurements and Monte Carlo Simulation,” Appl. Opt. |

18. | X. Wang and L. V. Wang, “Propagation of polarized light in birefringent turbid media: A Monte Carlo study,” J. Biomed. Opt. |

19. | G. W. Kattawar, M. J. Raković, and B. D. Cameron, “Laser backscattering polarization patterns from turbid media: theory and experiments,” in |

20. | J. C. Ramella-Roman, “Imaging skin pathologies with polarized light: empirical and theoretical studies,” Ph.D. thesis, OGI School of Science & Engineering at Oregon Health & Science University (2004). |

21. | H. C. van de Hulst, |

22. | C. F. Bohren and D. R. Huffman, |

23. | R. Y. Rubinstein, |

24. | J. von Neumann, “Various techniques used in connection with random digits,” J. Res. Natl. Bur. Stand. |

25. | W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, |

26. | P.-E. Wolf and G. Maret, “Weak Localization and Coherent Backscattering of Photons in Disordered Media,” Phys. Rev. Lett. |

27. | M. P. V. Albada and A. Lagendijk, “Observation of Weak Localization of Light in a Random Medium,” Phys. Rev. Lett. |

28. | E. Akkermans, P. E. Wolf, and R. Maynard, “Coherent backscattering of light by disordered media: analysis of the peak line shape,” Phys. Rev. Lett. |

29. | J. W. Goodman, “Statistical properties of laser speckle patterns,” in |

30. | D. S. Saxon, “Tensor Scattering Matrix for the Electromagnetic Field,” Phys. Rev. |

31. | B. D. Cameron, M. J. Rakovic, M. Mehrbeoglu, G. W. Kattawar, S. Rastegar, L. V. Wang, and G. L. Cot, “Measurement and calculationof the two-dimensional backscattering Mueller matrix of a turbid medium,” Opt. Lett. |

32. | I. Berezhnyy and A. Dogariu, “Time-resolved Mueller matrix imaging polarimetry,” Opt. Exp. |

33. | EMC is available at http://www.sci.ccny.cuny.edu/minxu. |

**OCIS Codes**

(030.5620) Coherence and statistical optics : Radiative transfer

(030.6140) Coherence and statistical optics : Speckle

(170.5280) Medical optics and biotechnology : Photon migration

(290.1350) Scattering : Backscattering

(290.4210) Scattering : Multiple scattering

(290.7050) Scattering : Turbid media

**ToC Category:**

Research Papers

**History**

Original Manuscript: November 15, 2004

Revised Manuscript: December 13, 2004

Published: December 27, 2004

**Citation**

Min Xu, "Electric field Monte Carlo simulation of polarized light propagation in turbid media," Opt. Express **12**, 6530-6539 (2004)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-26-6530

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### References

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- A. Yodh and B. Chance, �??Spectroscopy and imaging with diffusing light,�?? Phys. Today 48 3, 38�??40 (1995).
- S. K. Gayen and R. R. Alfano, �??Emerging optical biomedical imaging techniques,�?? Opt. Photon. News 7 (3), 17�??22 (1996).
- S. R. Arridge, �??Optical tomography in medical imaging,�?? Inverse Problems 15, R41�??R93 (1999). [CrossRef]
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- A. D. Kim and M. Moscoso, �??Chebyshev Spectral methods for radiative transfer,�?? SIAM J. Sci. Comput. 23, 2074�??2094 (2002). [CrossRef]
- A. D. Kim and J. B. Keller, �??Light propagation in biological tissue,�?? J. Opt. Soc. Am. A 20, 92�??98 (2003). [CrossRef]
- G. W. Kattawar and G. N. Plass, �??Radiance and polarization of multiple scattered light from haze and clouds,�?? Appl. Opt. 7, 1519�??1527 (1968). [CrossRef] [PubMed]
- I. Lux and L. Koblinger, Monte Carlo Particle Transport Methods: Neutron and Photon Calculations (CRC Press, Boca Raton, Fla., 1991).
- J. M. Schmitt, A. H. Gandjbakhche, and R. F. Bonner, �??Use of polarized light to discriminate short-path photons in a multiply scattering medium,�?? Appl. Opt. 31(30), 6535 (1992). [CrossRef]
- P. Bruscaglioni, G. Zaccanti, and Q. Wei, �??Transmission of a pulsed polarized light beam through thick turbid media: numerical results,�?? Appl. Opt. 32(30), 6142�??6150 (1993). [CrossRef]
- M. J. Rakovic, G. W. Kattawar, M. Mehrbeolu, 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(15), 3399�??3408 (1999). [CrossRef]
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