## Propagation of polarized light in birefringent turbid media: time-resolved simulations

Optics Express, Vol. 9, Issue 5, pp. 254-259 (2001)

http://dx.doi.org/10.1364/OE.9.000254

Acrobat PDF (316 KB)

### Abstract

A Monte Carlo model was used to analyze the propagation of polarized light in linearly birefringent turbid media, such as fibrous tissues. Linearly and circularly polarized light sources were used to demonstrate the change of polarizations in turbid media with different birefringent parameters. Videos of spatially distributed polarization states of light backscattered from or propagating in birefringent media are presented.

© Optical Society of America

## 1. Introduction

1. A. Ambirajan and D. C. Look, “A backward Monte Carlo study of the multiple scattering of a polarized laser beam,” J. Quant. Spectrosc. Transfer **58**, 171–192(1997). [CrossRef]

*et al*. [2

2. 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]

*et al*. [3

3. 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]

4. G. Yao and L. V. Wang, “Propagation of polarized light in turbid media: simulated animation sequences,” Opt. Express **7**, 198–203 (2000). http://www.opticsexpress.org/oearchive/source/23140.htm. [CrossRef] [PubMed]

5. J. F. de Boer, T. E. Milner, M. J. C. van Gemert, and J. S. Nelson, “Two-dimensional birefringence imaging in biological tissue by polarization-sensitive optical coherence tomography,” Opt. Lett. **22**, 934–936(1997). [CrossRef] [PubMed]

8. S. L. Jiao, G. Yao, and L. V. Wang, “Depth-resolved two-dimensional Stokes vectors of backscattered light and Mueller matrices of biological tissue by optical coherence tomography,” Appl. Opt. **39**, 6318–6324 (2000). [CrossRef]

## 2. Monte Carlo algorithm for birefringent turbid media

2. 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]

4. G. Yao and L. V. Wang, “Propagation of polarized light in turbid media: simulated animation sequences,” Opt. Express **7**, 198–203 (2000). http://www.opticsexpress.org/oearchive/source/23140.htm. [CrossRef] [PubMed]

*n*times in a birefringent turbid medium as

*µ*

_{s},

*µ*

_{a}are the scattering and absorption coefficients, respectively;

*δ*is the linear birefringence value; (

*x*′,

*y*′) is the detection point on the upper surface of the turbid medium in the laboratory coordinate. S

_{0}and

*µ*

_{s}/(

*µ*

_{a}+

*µ*

_{s})]

^{n}expresses the remaining energy after the photon has been scattered

*n*times.

**R**(

*ϕ*) is the rotation matrix that connects the two Stokes vectors that describe the same polarization state but with respect to the two reference planes such that one reference plane coincides with the other after a counterclockwise rotation by angle

*ϕ*around the direction of light propagation [2

2. 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]

**M**(Θ) is the matrix for each single scattering event based on Mie theory [9], where Θ is the scattering angle.

**T**(Δ,

*β*) describes the birefringent effect on the photon packet in each free path, which can be expressed as

*β*is the azimuthal angle of the birefringent slow axis at the

*x-y*plane in the local coordinate of the propagating photon. Δ is the phase retardation, which can be obtained by

*s*is the step length;

*λ*is the wavelength of the light

*in vacuo*; and Δ

*n*is the difference between the maximum and minimum refractive indices in the plane perpendicular to the propagation orientation of the photon packet. When the angle

*α*between the slow axis of the medium and the propagation orientation of the photon packet is known, Δ

*n*can be shown as

*n*

_{s}and

*n*

_{f}are the refractive indices of the birefringent medium along the slow axis and the fast axis, respectively; the linear birefringence value

*δ*is

*n*

_{s}

*-n*

_{f}.

## 3. Results

*λ*is 594 nm. The radius of the spherical scattering particles in the media is 350 nm. The refractive index of the scattering particles is 1.57.

*n*

_{s}is 1.33. The anisotropic factor g is calculated to be 0.9.

*µ*

_{s}and

*µ*

_{a}are 90 cm

^{-1}and 1 cm

^{-1}, respectively.

### 3.1 Polarization of light backscattered from birefringent turbid media

*δ*is 1.33/1000. Every picture in Figs. 2 and 3 is centered at the point of incidence. The actual size of each picture is 1×1 mm

^{2}.

*x-y*plane, where the incident light is linearly polarized along the

*x*-axis. The linear birefringence value

*δ*is set to be 1.33/1000 and 1.33×2/1000, respectively. For right-circularly polarized incident light, the time-resolved DOCP of backscattered light from the birefringent turbid media with the birefringent slow axes oriented along the cross section of the z-axis is shown in Fig. 5(b). The period of the oscillations in Fig. 5 is inversely proportional to δ. In isotropic scattering media, the DOLP and DOCP of backscattered light decrease when the time of propagation increases, because multiply scattered photons usually have greater path length than weakly scattered photons. However, in birefringent scattering media, the DOLP and DOCP present oscillations as a function of time because of the periodic phase retardation caused by the birefringence, a similar phenomenon was observed in optical-coherence tomography [10

10. J. F. de Boer, T. E. Milner, and J. S. Nelson, “Determination of the depth-resolved Stokes parameters of light backscattered from turbid media by use of polarization-sensitive optical coherence tomography,” Opt. Lett. **24**, 300–302 (1999). [CrossRef]

^{2}. These videos demonstrate how the polarization states of the diffusely reflected light depend on the propagation time as well as the birefringence in the turbid media.

### 3.2 Polarization propagation in single-layer birefringent turbid media

*x-z*plane in the laboratory coordinate. The actual size of each video sequence is 1×2 mm

^{2}. The polarization states of the photons at the expanding edges are retained well in the isotropic turbid medium because the photons there experience very few scattering events. However, in the birefringent turbid media, the polarization states at the expanding edges depend on the geometric positions of the edges in the media, the orientation of the birefringence in the media and the incident polarization state, though the DOP of the light at the expanding edges is retained well. The alternate and periodic changes between the circular and linear polarization states of the light at the expanding edges are caused by the periodic phase retardation in the birefringent media.

### 3.3 Polarization propagation in two-layer birefringent turbid media

*δ*in each layer has a combined effect on the polarization propagation of light in the medium.

*y*-axis and

*x*-axis, respectively. The two layers have the same birefringent value

*δ*that equals 1.33/1000. The thicknesses of the upper and lower layers are 0.4 mm and 0.6 mm, respectively. Videos of DOP, DOLP and DOCP in this two-layer birefringent turbid medium for different incident polarizations (the linear polarization oriented at x-axis and the right-circular polarization) are shown in Figs. 10 and 11, respectively. The gray dot lines in pictures show the interface of the two layers. The actual size of each video sequence is 1×2 mm

^{2}. It is noticeable that the polarization patterns are different at the upper and lower sides of the two-layer interface because of the different orientations of the birefringence in the two layers.

## 4. Conclusion

*δ*were compared and discussed.

## References and links

1. | A. Ambirajan and D. C. Look, “A backward Monte Carlo study of the multiple scattering of a polarized laser beam,” J. Quant. Spectrosc. Transfer |

2. | 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. |

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

4. | G. Yao and L. V. Wang, “Propagation of polarized light in turbid media: simulated animation sequences,” Opt. Express |

5. | J. F. de Boer, T. E. Milner, M. J. C. van Gemert, and J. S. Nelson, “Two-dimensional birefringence imaging in biological tissue by polarization-sensitive optical coherence tomography,” Opt. Lett. |

6. | M. J. Everett, K. Schoenenberger, B. W. Colston Jr., and L. B. Da Silva, “Birefringence characterization of biological tissue by use of optical coherence tomography,” Opt. Lett. |

7. | G. Yao and L. V. Wang, “Two-dimensional depth-resolved Mueller matrix characterization of biological tissue by optical coherence tomography,” Opt. Lett. |

8. | S. L. Jiao, G. Yao, and L. V. Wang, “Depth-resolved two-dimensional Stokes vectors of backscattered light and Mueller matrices of biological tissue by optical coherence tomography,” Appl. Opt. |

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

10. | J. F. de Boer, T. E. Milner, and J. S. Nelson, “Determination of the depth-resolved Stokes parameters of light backscattered from turbid media by use of polarization-sensitive optical coherence tomography,” Opt. Lett. |

**OCIS Codes**

(170.5280) Medical optics and biotechnology : Photon migration

(260.5430) Physical optics : Polarization

(290.4020) Scattering : Mie theory

(290.4210) Scattering : Multiple scattering

(290.7050) Scattering : Turbid media

**ToC Category:**

Research Papers

**History**

Original Manuscript: July 12, 2001

Published: August 27, 2001

**Citation**

Xueding Wang and Lihong Wang, "Propagation of polarized light in birefringent turbid media: time-resolved simulations," Opt. Express **9**, 254-259 (2001)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-9-5-254

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### References

- A. Ambirajan and D. C. Look, "A backward Monte Carlo study of the multiple scattering of a polarized laser beam," J. Quant. Spectrosc. Transfer 58, 171-192(1997). [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]
- 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]
- G. Yao and L. V. Wang, "Propagation of polarized light in turbid media: simulated animation sequences," Opt. Express 7, 198-203 (2000). http://www.opticsexpress.org/oearchive/source/23140.htm. [CrossRef] [PubMed]
- J. F. deBoer, T. E. Milner, M. J. C. van Gemert and J. S. Nelson, "Two-dimensional birefringence imaging in biological tissue by polarization-sensitive optical coherence tomography," Opt. Lett. 22, 934-936(1997). [CrossRef] [PubMed]
- M. J. Everett, K. Schoenenberger, B. W. Colston, Jr. and L. B. Da Silva, "Birefringence characterization of biological tissue by use of optical coherence tomography," Opt. Lett. 23, 228-230(1998). [CrossRef]
- G. Yao and L. V. Wang, "Two-dimensional depth-resolved Mueller matrix characterization of biological tissue by optical coherence tomography," Opt. Lett. 24, 537-539 (1999). [CrossRef]
- S. L. Jiao, G. Yao and L. V. Wang, "Depth-resolved two-dimensional Stokes vectors of backscattered light and Mueller matrices of biological tissue by optical coherence tomography," Appl. Opt. 39, 6318-6324 (2000). [CrossRef]
- H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).
- J. F. de Boer, T. E. Milner, and J. S. Nelson, "Determination of the depth-resolved Stokes parameters of light backscattered from turbid media by use of polarization-sensitive optical coherence tomography," Opt. Lett. 24, 300-302 (1999). [CrossRef]

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