## Characterization of backscattering Mueller matrix patterns of highly scattering media with triple scattering assumption

Optics Express, Vol. 15, Issue 15, pp. 9672-9680 (2007)

http://dx.doi.org/10.1364/OE.15.009672

Acrobat PDF (1252 KB)

### Abstract

We report on the use of an effective Mueller matrix to characterize the spatially-resolved diffuse backscattering Mueller matrix patterns of highly scattering media. The matrix expressions are based on assuming that the photon trajectories include only three scattering events. The numerically determined effective Mueller matrix elements are compared with the Monte Carlo simulated diffuse backscattering Mueller matrix for the polystyrene sphere suspensions. The results show that the two-dimensional intensity pattern maps of the effective Mueller matrix elements have good agreements with Monte Carlo simulations in azimuthal structure symmetry and radial dependence. It is demonstrated that this effective Mueller matrix can be used to quantitatively predict and interpret an experimentally-determined diffuse backscattering Mueller matrix from highly scattering media.

© 2007 Optical Society of America

## 1. Introduction

1. W.S. Bickel and W.M. Bailey, “Stokes vectors, Mueller matrices, and polarized light scattering,” Am. J. Phys. **53**, 468–478 (1985). [CrossRef]

*et al*. used the steady state backscattering Mueller matrix patterns to differentiate cancerous from noncancerous cell suspensions [4

4. A.H. Hielscher, A.A. Eick, J.R. Mourant, D. Shen, J.P. Freyer, and I.J. Bigio, “Diffuse backscattering Mueller matrices of highly scattering media,” Opt. Express **1**, 441–454 (1997). [CrossRef] [PubMed]

*et al*. measured the Mueller matrix of hard or soft biological tissues by a double-beam polarization-sensitive optical coherence tomography imaging technique, and extracted the polarization parameters, such as magnitude and orientation of birefringence and diattenuation [5

5. G. Yao and L.H. Wang, “Two dimensional depth resolved Mueller matrix measurement in biological tissue with optical coherence tomography,” Opt. Lett. **24**, 537–539 (1999). [CrossRef]

6. S.L. Jiao and L.H. Wang, “Two-dimensional depth-resoved Mueller matrix of biological tissue measured with double-beam polarization-senstive optical coherence tomography,” Opt. Lett. **27**, 101–103 (2002). [CrossRef]

*et al*. used the measured backscattering Mueller matrix to determine the optical rotation in chirl turbid media by a polar decomposition approach [7

7. S. Manhas 1, M.K. Swami, P. Buddhiwant, N. Ghosh, P.K. Gupta1, and K. Singh, “Mueller matrix approach for determination of optical rotation in chiral turbid media in backscattering geometry,” Opt. Express **14**, 190–202 (2006). [CrossRef]

*et al*. presented a complete spatial-temporal polarization pattern description of a Mueller matrix of scattering media and the polarization patterns were used to separate the polarimetric contributions of different scattering paths [8

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

*et al*. reported a novel optical polarization-imaging system which can be used to generate the full 16-element Mueller matrix in less than 70-sec [9

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

*et al*., Yao

*et al*., and Kaplan

*et al*. developed a Monte Carlo model based on the Stokes-Mueller formulation for a multiple scattering Mueller matrix [11

11. G. Yao and L.H. Wang, “Propagation of polarizes light in turbid media: simulated animation sequences,” Opt. Express **7**, 198–203 (2000). [CrossRef] [PubMed]

13. 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**, 10392 (2005). [CrossRef] [PubMed]

*et al*. presented a single-scattering model and compared it with a Monte Carlo model. Agreement was satisfactorily achieved between the two methods in the non-diffusing regime, but the single-scattering model becomes inaccurate in the diffuse regime [14

14. X. Wang, G. Yao, and L.H. 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]

*et al*. was the first to present a double scattering method to explain the diffuse backscattering Mueller matrix in turbid media. They were able to reproduce the azimuthal symmetry in all 16 matrix elements, but obtained only poor agreement in the radial dependence [15

15. M.J. Raković and G.W. Kattawar, “Theoretical analysis of polarization patterns from incoherent backscattering of light,” Appl. Opt. **37**, 3333–3338 (1998). [CrossRef]

17. M.J. Rakovic, G.W. Kattawar, M. Mehrubeoglu, B.D. Cameron, L.H. Wang, S. Rastegar, and G.L. Coté, “Light backscattering polarization patterns from turbid media: theory and experiments,” Appl. Opt. **38**, 3399–3408 (1999). [CrossRef]

18. Yong Deng, Qiang Lu, Qingming Luo, and Shaoqun Zeng, “A third-order scattering model for the diffuse backscattering intensity patterns of polarized light from a turbid medium,” Appl. Phys. Lett. **90**, 153902 (2007). [CrossRef]

## 2. Theory

_{0}, S

_{1}, S

_{2}, S

_{3})

^{T}, which is a mathematical representation of the polarization state, where

*S*

_{0}=〈|

*E*|

_{x}^{2}+|

*E*|

_{y}^{2}〉,

*S*

_{1}=〈|

*E*|

_{x}^{2}-|

*E*|

_{y}^{2}〉,

*S*

_{2}=〈2

*E*

_{x}*E*cos

_{y}*δ*〉,

*S*

_{3}=〈2

*E*sin

_{x}E_{y}*δ*〉,

*E*and

_{x}*E*are two orthogonal complex electric field components perpendicular to the propagation direction, δ denotes the phase difference with respect to

_{y}*Ex*and

*Ey*. And the brackets <> refers to ensemble average. Some of the Stokes components have an obvious interpretation. For example,

*S*

_{0}is equal to the total intensity.

*S*be the Stokes vector of the incident laser beam that is injected into a highly scattering medium surface, and

_{in}*S*be the Stokes vector of diffuse backscattering light that escapes from the upper surface of the medium. As light is described by a four-component vector, this interaction with the scattering medium can be described as a multiplication of the Stokes vector with a 4x4 matrix:

_{out}*S*=M

_{out}*S*. This sixteen-element matrix is called the Mueller matrix of the scattering medium, as follows:

_{in}18. Yong Deng, Qiang Lu, Qingming Luo, and Shaoqun Zeng, “A third-order scattering model for the diffuse backscattering intensity patterns of polarized light from a turbid medium,” Appl. Phys. Lett. **90**, 153902 (2007). [CrossRef]

*µ*and

_{t}*µ*are the extinction coefficient and the scattering coefficient, respectively;

_{s}*M*(

*θ*) is a single scattering matrix [2,19]. In this study we consider only isotropically distributed spherical particles, which reduces the Mueller matrix to four independent elements:

*m*are derived [19].

_{ij}*R*(

*Φ*) is a 4×4 rotation matrix, which transforms between the scattering plane and the reference plane;

*Φ*is defined as positive for clockwise rotation [2]:

## 3. Results

_{0}=1.33 and of the polystyrene spheres scatterers n=1.59 at a wavelength of λ=0.6328µm. And Mie theory can be used to calculate the scattering cross section C

_{s}of the individual particles. Given the number density N

_{s}, that is, the particle concentration, the scattering coefficients of turbid media can be determined as µ

_{s}=N

_{s}C

_{s}. For statistical results, large quantities of photon packets (10

^{8}) are traced in the Monte Carlo simulations. The thickness of the aqueous suspensions of polystyrene spheres is 2.5cm. Firstly, for the aqueous suspension of polystyrene spheres of particle diameter 0.486 µm with scattering coefficients 14.8 cm

^{-1}, the numerical calculations of triple scattering model are given as two-dimensional images of the surface in Fig. 1(a), and the calculations of double scattering model are shown in Fig. 1(b). As a comparison, Monte Carlo simulated results for the same aqueous suspension of polystyrene spheres are also shown in Fig. 1(c).

16. B.D. Cameron, M.J. Raković, M. Mehrubeoglu, G.W. Kattawar, S. Rastegar, L.H. Wang, and G. Coté, “Measurement and calculation of the two-dimensional backscattering Mueller matrix of a turbid medium,” Opt. Lett. **23**, 485–487 (1998). [CrossRef]

^{-1}. We found that, as the scaterer sizes increase, the mean values of M44 increase. The calculations have a good agreement with Monte Carlo simulated results. The results demonstrate that the pattern variations of the effective Mueller matrix can accurately reveal the variations of the scaterer sizes. Furthermore, notwithstanding the changes in scattering coefficients or anisotropy factor g of the suspensions, the pattern features of the effective Mueller matrix coincide with Monte Carlo simulations well.

## 4. Conclusions

## Acknowledgments

## References and links

1. | W.S. Bickel and W.M. Bailey, “Stokes vectors, Mueller matrices, and polarized light scattering,” Am. J. Phys. |

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

3. | W.S. Bickel, A.J. Watkins, and G. Videen, “The light-scattering Mueller matrix elements for Rayleigh, Rayleigh-Gans, and Mie spheres,” Am. J. Phys. 55, 559–561 (1987). |

4. | A.H. Hielscher, A.A. Eick, J.R. Mourant, D. Shen, J.P. Freyer, and I.J. Bigio, “Diffuse backscattering Mueller matrices of highly scattering media,” Opt. Express |

5. | G. Yao and L.H. Wang, “Two dimensional depth resolved Mueller matrix measurement in biological tissue with optical coherence tomography,” Opt. Lett. |

6. | S.L. Jiao and L.H. Wang, “Two-dimensional depth-resoved Mueller matrix of biological tissue measured with double-beam polarization-senstive optical coherence tomography,” Opt. Lett. |

7. | S. Manhas 1, M.K. Swami, P. Buddhiwant, N. Ghosh, P.K. Gupta1, and K. Singh, “Mueller matrix approach for determination of optical rotation in chiral turbid media in backscattering geometry,” Opt. Express |

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

9. | J.S. Baba, J.R. Chung, A.H. DeLaughter, B.D. Cameron, and G.L. Cote,“Development and calibration of an automated Mueller matrix polarization imaging system,” J. Biomed. Opt. |

10. | S. Bartel and A.H. Hielscher, “Monte Carlo simulation of the diffuse backscattering Mueller matrix for highly scattering media,” Appl. Opt. |

11. | G. Yao and L.H. Wang, “Propagation of polarizes light in turbid media: simulated animation sequences,” Opt. Express |

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

13. | 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 |

14. | X. Wang, G. Yao, and L.H. Wang, “Monte Carlo model and single scattering approximation of the propagation of polarized light in turbid media containing glucose,” Appl. Opt. |

15. | M.J. Raković and G.W. Kattawar, “Theoretical analysis of polarization patterns from incoherent backscattering of light,” Appl. Opt. |

16. | B.D. Cameron, M.J. Raković, M. Mehrubeoglu, G.W. Kattawar, S. Rastegar, L.H. Wang, and G. Coté, “Measurement and calculation of the two-dimensional backscattering Mueller matrix of a turbid medium,” Opt. Lett. |

17. | M.J. Rakovic, G.W. Kattawar, M. Mehrubeoglu, B.D. Cameron, L.H. Wang, S. Rastegar, and G.L. Coté, “Light backscattering polarization patterns from turbid media: theory and experiments,” Appl. Opt. |

18. | Yong Deng, Qiang Lu, Qingming Luo, and Shaoqun Zeng, “A third-order scattering model for the diffuse backscattering intensity patterns of polarized light from a turbid medium,” Appl. Phys. Lett. |

19. | G. Bohren and D. Hoffman, |

**OCIS Codes**

(170.5280) Medical optics and biotechnology : Photon migration

(260.5430) Physical optics : Polarization

(290.0290) Scattering : Scattering

(290.1350) Scattering : Backscattering

(290.4210) Scattering : Multiple scattering

(290.7050) Scattering : Turbid media

**ToC Category:**

Scattering

**History**

Original Manuscript: May 17, 2007

Revised Manuscript: July 2, 2007

Manuscript Accepted: July 11, 2007

Published: July 19, 2007

**Virtual Issues**

Vol. 2, Iss. 8 *Virtual Journal for Biomedical Optics*

**Citation**

Yong Deng, Shaoqun Zeng, Qiang Lu, Dan zhu, and Qingming Luo, "Characterization of backscattering Mueller matrix patterns of highly scattering media with triple scattering assumption," Opt. Express **15**, 9672-9680 (2007)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-15-15-9672

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### References

- W.S. Bickel and W.M. Bailey, "Stokes vectors, Mueller matrices, and polarized light scattering," Am. J. Phys. 53, 468-478 (1985). [CrossRef]
- H.C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).
- <jrn>. W.S. Bickel, A.J. Watkins, and G. Videen, "The light-scattering Mueller matrix elements for Rayleigh -Gans, and Mie spheres," Am. J. Phys. 55, 559-561 (1987).</jrn
- A.H. Hielscher, A.A. Eick, J.R. Mourant, D. Shen, J.P. Freyer, and I.J. Bigio, "Diffuse backscattering Mueller matrices of highly scattering media," Opt. Express 1, 441-454 (1997). [CrossRef] [PubMed]
- G. Yao and L.H. Wang, "Two dimensional depth resolved Mueller matrix measurement in biological tissue with optical coherence tomography," Opt. Lett. 24, 537-539 (1999). [CrossRef]
- S.L. Jiao, L.H. Wang, "Two-dimensional depth-resoved Mueller matrix of biological tissue measured with double-beam polarization-senstive optical coherence tomography," Opt. Lett. 27, 101-103 (2002). [CrossRef]
- S. Manhas1, M.K. Swami, P. Buddhiwant, N. Ghosh, P.K. Gupta1 and K. Singh, "Mueller matrix approach for determination of optical rotation in chiral turbid media in backscattering geometry," Opt. Express 14, 190-202 (2006). [CrossRef]
- I. Berezhnyy and A. Dogariu, "Time-resolved Mueller matrix imaging polarimetry," Opt. Express 12, 4635-4649 (2004). [CrossRef] [PubMed]
- J.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]
- S. Bartel and A.H. Hielscher, "Monte Carlo simulation of the diffuse backscattering Mueller matrix for highly scattering media," Appl. Opt. 39, 1580-1588(2000). [CrossRef]
- G. Yao, L.H. Wang, "Propagation of polarizes light in turbid media: simulated animation sequences," Opt. Express 7, 198-203 (2000). [CrossRef] [PubMed]
- B. Kaplan, G. Ledanois, B. Villon, "Mueller Matrix of Dense Polystyrene Latex Sphere Suspensions: Measurements and Monte Carlo Simulation," Appl. Opt. 40, 2769-2777 (2001). [CrossRef]
- J.C. Ramella-Roman, S.A. Prahl, S.L. Jacques, "Three Monte Carlo programs of polarized light transport into scattering media: part I," Opt. Express 13, 10392 (2005). [CrossRef] [PubMed]
- X. Wang, G. Yao and L.H. 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]
- M.J. Rakovic and G.W. Kattawar, "Theoretical analysis of polarization patterns from incoherent backscattering of light," Appl. Opt. 37, 3333-3338 (1998). [CrossRef]
- B.D. Cameron, M.J. Rakovic, M. Mehrubeoglu, G.W. Kattawar, S. Rastegar, L.H. 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]
- M.J. Rakovic, G.W. Kattawar, M. Mehrubeoglu, B.D. Cameron, L.H. Wang, S. Rastegar, and G.L. Coté, "Light backscattering polarization patterns from turbid media: theory and experiments," Appl. Opt. 38, 3399-3408 (1999). [CrossRef]
- Yong Deng, Qiang Lu, Qingming Luo, Shaoqun Zeng, "A third-order scattering model for the diffuse backscattering intensity patterns of polarized light from a turbid medium," Appl. Phys. Lett. 90, 153902 (2007). [CrossRef]
- G. Bohren and D. Hoffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

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