## Polar decomposition of 3×3 Mueller matrix: a tool for quantitative tissue polarimetry

Optics Express, Vol. 14, Issue 20, pp. 9324-9337 (2006)

http://dx.doi.org/10.1364/OE.14.009324

Acrobat PDF (234 KB)

### Abstract

The polarization properties of any medium are completely described by the sixteen element Mueller matrix that relates the polarization parameters of the light incident on the medium to that emerging from it. Measurement of all the elements of the matrix requires a minimum of sixteen measurements involving both linear and circularly polarized light. However, for many diagnostic applications, it would be useful if the polarization parameters can be quantified with linear polarization measurements alone. In this paper, we present a method based on polar decomposition of Mueller matrix for quantification of the polarization parameters of a scattering medium using the nine element (3×3) Mueller matrix that requires linear polarization measurements only. The methodology for decomposition of the 3×3 Mueller matrix is based on the previously developed decomposition process for sixteen element (4×4) Mueller matrix but with an assumption that the depolarization of linearly polarized light due to scattering is independent of the orientation angle of the incident linear polarization vector. Studies conducted on various scattering samples demonstrated that this assumption is valid for a turbid medium like biological tissue where the depolarization of linearly polarized light primarily arises due to the randomization of the field vector’s direction as a result of multiple scattering. For such medium, polar decomposition of 3×3 Mueller matrix can be used to quantify the four independent polarization parameters namely, the linear retardance (δ), the circular retardance (ψ), the linear depolarization coefficient (Δ) and the linear diattenuation (d) with reasonable accuracy. Since this approach requires measurements using linear polarizers only, it considerably simplifies measurement procedure and might find useful applications in tissue diagnosis using the retrieved polarization parameters.

© 2006 Optical Society of America

## 1. Introduction

1. D. Bicout, C. Brosseau, A.S. Martinez, and J.M. Schmitt, “Depolarization of multiply scattered waves by spherical diffusers: Influence of the size parameter,” Phys. Rev. E **49**, 1767–1770 (1994). [CrossRef]

6. N. Ghosh, A. Pradhan, P.K. Gupta, S. Gupta, V. Jaiswal, and R.P. Singh, “Depolarization of light in a multiply scattering medium: effect of refractive index of scatterer,” Phys. Rev. E **70**, 066607 (2004). [CrossRef]

7. 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]

14. F. Boulvert, B. Boulbry, G Le Brun, S. Rivet, and J. Cariou, “Analysis of the depolarization properties of irradiated pig skin,” J. Opt. A: Pure Appl. Opt. **7**, 21–28 (2005). [CrossRef]

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

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

22. S. Manhas, M. K. Swami, P. Buddhiwant, N. Ghosh, P. K. Gupta, 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] [PubMed]

23. S. Yau Lu and R.A. Chipman, “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A **13**, 1106–1113 (1996). [CrossRef]

22. S. Manhas, M. K. Swami, P. Buddhiwant, N. Ghosh, P. K. Gupta, 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] [PubMed]

23. S. Yau Lu and R.A. Chipman, “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A **13**, 1106–1113 (1996). [CrossRef]

25. B.L. Boulesteix, A. De Martino, B. Dre villon, and L. Schwartz, “Mueller polarimetric imaging system with liquid crystal,” Appl. Opt. **43**, 2824–2832 (2004). [CrossRef]

27. E. Gar cia-Caurel, A. De Martino, and B. Drevillon, “Spectroscopic Mueller polarimeter based on liquid crystal devices,” Thin Solid Films **455–456**, 120–123 (2004). [CrossRef]

## 2. Theory

### 2.1 Polar decomposition of nine element (3×3) Mueller matrix

_{D}), a retarder (M

_{R}) and a depolarizer (M

_{Δ}) has been described in reference [23

23. S. Yau Lu and R.A. Chipman, “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A **13**, 1106–1113 (1996). [CrossRef]

_{D}). The inverse of M

_{D}is multiplied with M to remove the diattenuation and the remaining matrix (M

^{/}) consists of retardance and depolarization.

_{Δ}) can be written as

_{R}) consists of both linear and circular retardance and can be written as

_{Δ}from the matrix M

^{/}and find out the retardance matrix M

_{R}. In order to obtain the values for the depolarization coefficient, we construct a matrix M

_{DR}as

_{DR}is always unity for a normalized matrix. If no depolarization were present (a=b=1), then the other two Eigen values would be 1 and cos

^{2}δ respectively. In presence of depolarization, the Eigen values would be scaled by the square of the value of the linear depolarization coefficient. If the depolarization of linearly polarized light can be assumed to be independent of the orientation angle of the incident linear polarization vector, that is if a=b=Δ, then the remaining two Eigen values of the matrix M

_{DR}would be Δ

^{2}and Δ

^{2}cos

^{2}δ respectively. Hence, the larger of the remaining two Eigen values would be equal to the square of the linear depolarization coefficient (Δ) and thus this Eigen value can be used to find out the value of Δ. Once the value of Δ is obtained from the Eigen values of M

_{DR}, one can readily construct the depolarization matrix as

_{Δ}is thereafter multiplied with M

^{/}to obtain the retardance matrix M

_{R}

_{R}to estimate the values for δ and ψ. The orientation angle of the axis of the linear retarder (θ) can also be obtained using the value of δ and ψ in Eq. (3).

22. S. Manhas, M. K. Swami, P. Buddhiwant, N. Ghosh, P. K. Gupta, 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] [PubMed]

**13**, 1106–1113 (1996). [CrossRef]

_{DR}is constructed here (using the matrix M

^{/}), Eigen values of which are utilized to decouple the depolarization matrix (M

_{Δ}) and the retardance matrix (M

_{R}) with the assumption that the depolarization of linearly polarized light is independent of the orientation angle of the incident linear polarization vector. The accuracy of this approach for decomposing the retardance and the depolarization matrices in a depolarizing medium would thus be determined by the validity of this assumption. It should also be noted that the decomposition of Mueller matrix also depends upon the order in which the diattenuator, depolarizer and retarder matrices are multiplied. Based on the order of these matrices, six possible decompositions can be performed. Among these, the group in which the diattenuator matrix comes ahead of the retardance and the depolarization matrix [M=M

_{Δ}M

_{R}M

_{D}] always lead to a physically realizable Mueller matrix [28

28. J. Morio and F. Goudail, “Influence of the order of diattenuator, retarder, and polarizer in polar decomposition of Mueller matrices,” Opt. Lett. **29**, 2234–2236 (2004). [CrossRef] [PubMed]

## 3. Experimental methods

^{0}, 45

^{0}and 90

^{0}from the horizontal).

_{i}), we generated the required three incident polarization states (linear polarization at angles of 0

^{0}, 45

^{0}and 90

^{0}from the horizontal) and recorded the intensity of the light transmitted (or backscattered) through samples after it passed through the suitably oriented analyzers (linear polarization at angles of 0

^{0}, 45

^{0}and 90

^{0}from the horizontal). We define polarization state generator (PSG) and polarization state analyzer (PSA) matrix as [24]

_{S}

_{vec}is the sample Mueller matrix written in a 1×9 vector form.

_{s}) of the samples were varied by dilution. The samples were kept inside quartz cuvette having path length of either 5 mm or 10 mm while taking measurements. The collagen samples used in this study were extracted from eggshell membrane. Eggshell membrane is known to contain mainly Type I and Type V collagen. The Type I collagen from the inner shell membrane was extracted by acid-pepsin digestion method [29

29. M. Wong, M.J. Hendrix, K. Von der Mark, C. Little, and R. Stern, “Collagen in the egg shell membranes of the hen,” Dev. Biol. **104 (1)**, 28–36 (1984). [CrossRef]

## 4. Results and discussion

_{medium}) was taken to be n=1.59 and n

_{medium}=1.33 respectively. The computed 3×3 non depolarizing Mueller matrix was decomposed into diattenuation (M

_{D}) and retardance (M

_{R}) matrices following the procedure described in Section 2 and the values for diattenuation (d), linear retardance (δ) and optical rotation (ψ) of scattered light was calculated using Eqs. (4) and (5). The variation of d, δ and ψ as a function of scattering angle Θ is shown in Fig. 2.

_{11}(Θ)+S

_{12}(Θ)}] and perpendicular [0.5×{S

_{11}(Θ)-S

_{12}(Θ)}] to the scattering plane [22

**14**, 190–202 (2006). [CrossRef] [PubMed]

**14**, 190–202 (2006). [CrossRef] [PubMed]

_{s}=2 mm

^{-1}). The sample was kept inside a quartz cuvette with path length of 10 mm. The value for the linear depolarization coefficient (Δ) obtained from the decomposition was Δ=0.74 (where Δ=1 corresponds to completely polarized light). It is important to note here that the values for the linear depolarization coefficients for incident horizontally polarized light and light polarized at 45

^{0}from horizontal direction are comparable for this sample (the value for the elements M

_{22}and M

_{33}of the depolarization matrix are nearly equal). Measurements were also conducted on samples with different other concentration of the scatterers. The value for M

_{22}was always found to be nearly equal to the value for M

_{33}for all the turbid scattering samples (µ

_{s}=1-6 mm

^{-1}). These results confirmed that the assumption, the depolarization of linearly polarized light due to scattering is independent of the orientation angle of the incident linear polarization vector, is valid for a turbid medium. This should be expected also because in a turbid medium, depolarization of linearly polarized light takes place mainly because of the randomization of the filed vector’s direction by a random sequence of scattering at arbitrary scattering and azimuthal angles. This can be treated to be an incoherent addition of Mueller matrices of isotropic depolarizers and thus the linear depolarization coefficients of the resultant depolarizing Mueller matrix will be the same for the two different incident linear polarization states.

_{Δ}, M

_{R}and M

_{D}are also displayed in the table. Following the procedure described in Section 2, the value for linear retardance (δ), orientation of the linear retarder (θ) and linear depolarization (Δ) were estimated to be δ=1.57, θ=5.7° and Δ=0.75 respectively. These values are reasonably close to the corresponding values obtained from separate measurements on the linear retarder and the pure depolarizer (δ=1.56, θ=5.5° and Δ=0.74).

_{Δ}, M

_{R}and M

_{D}for this combination of linear retarder and scattering samples are displayed in Table 2. The values δ=1.56, θ=10.3° and Δ=0.79 obtained from the polar decomposition approach are again found to be reasonably close to that expected for this combination.

8. D.J. Maitland and J.T. Walsh Jr., “Quantitative measurement of linear birefringence during heating of native collagen,” Lasers Surg. Med. **20**, 310–318 (1997). [CrossRef] [PubMed]

10. G.L. Liu, Y. Li, and B.D. Cameron, “Polarization based optical imaging and processing techniques with application to the cancer diagnostics,” Proceedings SPIE **4617**, 208–220 (2002). [CrossRef]

12. J. Zhang, S. Guo, W. Jung, J.S. Nelson, and Z. Chen, “Determination of birefringence and absolute optic axis orientation using polarization-sensitive optical coherence tomography with PM fibers,” Opt. Express **11**, 3262–3270 (2003). [CrossRef] [PubMed]

14. F. Boulvert, B. Boulbry, G Le Brun, S. Rivet, and J. Cariou, “Analysis of the depolarization properties of irradiated pig skin,” J. Opt. A: Pure Appl. Opt. **7**, 21–28 (2005). [CrossRef]

^{0}in the plane normal to the direction of propagation of the incident light. The magnitude of Δ (λ), δ (λ) and d (λ) was not found to change significantly with altered physical orientation of the sample. However, as expected, the orientation angle of the linear retarder θ, obtained from the decomposed retardance matrix M

_{R}was observed to change with altered physical orientation of the sample.

## 6. Conclusion

_{44}and M

_{14}elements of the 4×4 Mueller matrix and these therefore cannot be estimated using the decomposition of 3×3 Mueller matrix. Further, the decomposition of the 3×3 Mueller matrix is based on the assumption that the depolarization of linearly polarized light due to scattering is independent of the orientation angle of the incident linear polarization vector which is valid for a turbid and multiply scattering medium like tissue. The applicability of the approach was tested by carrying out studies on various samples having known scattering and polarization properties. Since the 3×3 Mueller matrix requires measurements using linear polarizers only, it considerably simplifies measurement procedure and also allows one to quantify the wavelength dependence of the useful polarization parameters of a turbid medium like biological tissue with relative ease. This approach may thus find useful diagnostic applications.

## References

1. | D. Bicout, C. Brosseau, A.S. Martinez, and J.M. Schmitt, “Depolarization of multiply scattered waves by spherical diffusers: Influence of the size parameter,” Phys. Rev. E |

2. | A.D. Kim and M. Moscoso, “Influence of the refractive index on the depolarization of multiply scattered waves,” Phys. Rev. E |

3. | V. Sankaran, J.T. Walsh Jr., and D.J. Maitland, “Comparative study of polarized light propagation in biological tissues,” J. Biomed. Opt. |

4. | N. Ghosh, P.K. Gupta, H.S. Patel, B. Jain, and B.N. Singh, “Depolarization of light in tissue phantoms - effect of collection geometry,” Opt. Commun. |

5. | N. Ghosh, H.S. Patel, and P.K. Gupta, “Depolarization of light in tissue phantoms - effect of a distribution in the size of scatterers,” Opt. Express |

6. | N. Ghosh, A. Pradhan, P.K. Gupta, S. Gupta, V. Jaiswal, and R.P. Singh, “Depolarization of light in a multiply scattering medium: effect of refractive index of scatterer,” Phys. Rev. E |

7. | 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. |

8. | D.J. Maitland and J.T. Walsh Jr., “Quantitative measurement of linear birefringence during heating of native collagen,” Lasers Surg. Med. |

9. | M.H. Smith, “Interpreting Mueller matrix images of tissues,” Proceedings SPIE |

10. | G.L. Liu, Y. Li, and B.D. Cameron, “Polarization based optical imaging and processing techniques with application to the cancer diagnostics,” Proceedings SPIE |

11. | C.W. Sun, L.S. Lu, C.C. Yang, Y.W. Kiang, and M.J. Su, “Myocardial tissue characterization based on the time-resolved Stokes-Mueller formalism,” Opt. Express |

12. | J. Zhang, S. Guo, W. Jung, J.S. Nelson, and Z. Chen, “Determination of birefringence and absolute optic axis orientation using polarization-sensitive optical coherence tomography with PM fibers,” Opt. Express |

13. | O. Kostyuk and R.A. Brown, “Novel Spectroscopic Technique for In Situ Monitoring of Collagen Fibril Alignment in Gels,” Biophys. J. |

14. | F. Boulvert, B. Boulbry, G Le Brun, S. Rivet, and J. Cariou, “Analysis of the depolarization properties of irradiated pig skin,” J. Opt. A: Pure Appl. Opt. |

15. | R.J. McNichols and G.L. Cote, “Optical glucose sensing in biological fluids: an overview,” J. Biomed. Opt. |

16. | B.D. Cameron and G.L. Cote, “Noninvasive glucose sensing utilizing a digital closed loop polarimetric approach,” IEEE Trans. Biomed. Eng. |

17. | I.A. Vitkin and R.C.N Studinski, “Polarization preservation in diffusive scattering from in-vivo turbid biological media: Effects of tissue optical absorption in the exact backscattering direction,” Opt. Commun. |

18. | K.C. Hadley and I.A. Vitkin, “Optical rotation and linear and circular depolarization rates in diffusively scattered light from chiral, racimic and achiral turbid media,” J. Biomed. Opt. |

19. | I. Vitkin, R.D. Laszlo, and C.L. Whyman, “Effects of molecular asymmetry of optically active molecules on the polarization properties of multiply scattered light,” Opt. Express |

20. | X. Wang, G. Yao, and L.V. Yang, “Monte Carlo model and single scattering approx. Of the propagation of polarized light in turbid media containing glucose,” Appl. Opt. |

21. | D. Cote and I. Vitkin, “Robust concentration determination of optically active molecule in turbid media with validated three dimensional polarization sensitive Monte Carlo calculation,” Opt. Express |

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

23. | S. Yau Lu and R.A. Chipman, “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A |

24. | R.A. Chipman, “Hand book of optics (polarimetry),” OSA/McGraw-Hill, 22.1–22.35, (1994). |

25. | B.L. Boulesteix, A. De Martino, B. Dre villon, and L. Schwartz, “Mueller polarimetric imaging system with liquid crystal,” Appl. Opt. |

26. | E. Collett and V. Gazerro, “Polarization measurements in a spectrofluorophotometer,” Opt. Commun. |

27. | E. Gar cia-Caurel, A. De Martino, and B. Drevillon, “Spectroscopic Mueller polarimeter based on liquid crystal devices,” Thin Solid Films |

28. | J. Morio and F. Goudail, “Influence of the order of diattenuator, retarder, and polarizer in polar decomposition of Mueller matrices,” Opt. Lett. |

29. | M. Wong, M.J. Hendrix, K. Von der Mark, C. Little, and R. Stern, “Collagen in the egg shell membranes of the hen,” Dev. Biol. |

30. | C.F. Bohren and D.R. Huffman, “Absorption and scattering of light by small particles,” Wiley, New York (1983). |

**OCIS Codes**

(110.7050) Imaging systems : Turbid media

(120.5410) Instrumentation, measurement, and metrology : Polarimetry

(170.0170) Medical optics and biotechnology : Medical optics and biotechnology

(170.4580) Medical optics and biotechnology : Optical diagnostics for medicine

(290.4210) Scattering : Multiple scattering

**ToC Category:**

Medical Optics and Biotechnology

**History**

Original Manuscript: July 7, 2006

Revised Manuscript: September 7, 2006

Manuscript Accepted: September 12, 2006

Published: October 2, 2006

**Virtual Issues**

Vol. 1, Iss. 11 *Virtual Journal for Biomedical Optics*

**Citation**

M. K. Swami, S. Manhas, P. Buddhiwant, N. Ghosh, A. Uppal, and P. K. Gupta, "Polar decomposition of 3 x 3 Mueller matrix: a tool for quantitative tissue polarimetry," Opt. Express **14**, 9324-9337 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-20-9324

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### References

- D. Bicout, C. Brosseau, A.S. Martinez, and J.M. Schmitt, "Depolarization of multiply scattered waves by spherical diffusers: Influence of the size parameter," Phys. Rev. E 49, 1767-1770 (1994). [CrossRef]
- A.D. Kim and M. Moscoso, "Influence of the refractive index on the depolarization of multiply scattered waves," Phys. Rev. E 64, 026612, 1-4 (2001). [CrossRef]
- V. Sankaran, J.T. Walsh, Jr., and D.J. Maitland, "Comparative study of polarized light propagation in biological tissues," J. Biomed. Opt. 7, 300-306 (2002). [CrossRef] [PubMed]
- N. Ghosh, P.K. Gupta, H.S. Patel, B. Jain, and B.N. Singh, "Depolarization of light in tissue phantoms - effect of collection geometry," Opt. Commun. 222, 93-100 (2003). [CrossRef]
- N. Ghosh, H.S. Patel, and P.K. Gupta, "Depolarization of light in tissue phantoms - effect of a distribution in the size of scatterers," Opt. Express 11, 2198-2205 (2003). [CrossRef] [PubMed]
- N. Ghosh, A. Pradhan, P.K. Gupta, S. Gupta, V. Jaiswal, and R.P. Singh, "Depolarization of light in a multiply scattering medium: effect of refractive index of scatterer," Phys. Rev. E 70, 066607 (2004). [CrossRef]
- 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]
- D.J. Maitland and J.T. WalshJr., "Quantitative measurement of linear birefringence during heating of native collagen," Lasers Surg. Med. 20, 310-318 (1997). [CrossRef] [PubMed]
- M.H. Smith, "Interpreting Mueller matrix images of tissues," Proceedings SPIE 4257, 82 - 89 (2001). [CrossRef]
- G.L. Liu, Y. Li, and B.D. Cameron, "Polarization based optical imaging and processing techniques with application to the cancer diagnostics," Proceedings SPIE 4617, 208 - 220 (2002). [CrossRef]
- C.W. Sun, L.S. Lu, C.C. Yang, Y.W. Kiang, and M.J. Su, "Myocardial tissue characterization based on the time-resolved Stokes-Mueller formalism," Opt. Express 10, 1347 - 1353 (2002). [PubMed]
- J. Zhang, S. Guo, W. Jung, J.S. Nelson, and Z. Chen, " Determination of birefringence and absolute optic axis orientation using polarization-sensitive optical coherence tomography with PM fibers," Opt. Express 11, 3262 - 3270 (2003). [CrossRef] [PubMed]
- O. Kostyuk and R.A. Brown, "Novel Spectroscopic Technique for In Situ Monitoring of Collagen Fibril Alignment in Gels," Biophys. J. 87, 648-655 (2004). [CrossRef] [PubMed]
- F. Boulvert, B. Boulbry, G Le Brun, S. Rivet, and J. Cariou, "Analysis of the depolarization properties of irradiated pig skin," J. Opt. A: Pure Appl. Opt. 7, 21 - 28 (2005). [CrossRef]
- R.J. McNichols and G.L. Cote, "Optical glucose sensing in biological fluids: an overview," J. Biomed. Opt. 5, 5 - 16 (2000). [CrossRef] [PubMed]
- B.D. Cameron and G.L. Cote, "Noninvasive glucose sensing utilizing a digital closed loop polarimetric approach," IEEE Trans. Biomed. Eng. 44, 1221-227 (1997). [CrossRef] [PubMed]
- I.A. Vitkin and R.C.N Studinski, "Polarization preservation in diffusive scattering from in-vivo turbid biological media: Effects of tissue optical absorption in the exact backscattering direction," Opt. Commun. 190, 37-43 (2001). [CrossRef]
- K.C. Hadley and I.A. Vitkin, "Optical rotation and linear and circular depolarization rates in diffusively scattered light from chiral, racimic and achiral turbid media," J. Biomed. Opt. 7, 291-299 (2002). [CrossRef] [PubMed]
- I. Vitkin, R.D. Laszlo, and C.L. Whyman, "Effects of molecular asymmetry of optically active molecules on the polarization properties of multiply scattered light," Opt. Express 10, 222 - 229 (2002). [PubMed]
- X. Wang, G. Yao, and L.V. Yang, "Monte Carlo model and single scattering approx. Of the propagation of polarized light in turbid media containing glucose," Appl. Opt. 41, 792 - 801, (2002). [CrossRef] [PubMed]
- D. Cote and I. Vitkin, "Robust concentration determination of optically active molecule in turbid media with validated three dimensional polarization sensitive Monte Carlo calculation," Opt. Express 13, 148-163 (2005). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-1-148 [CrossRef] [PubMed]
- S. Manhas, M. K. Swami, P. Buddhiwant, N. Ghosh, P. K. Gupta, 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] [PubMed]
- S. Yau Lu and R.A. Chipman, "Interpretation of Mueller matrices based on polar decomposition," J. Opt. Soc. Am. A 13, 1106-1113 (1996). [CrossRef]
- R.A. Chipman, "Hand book of optics (polarimetry)," OSA/McGraw-Hill, 22.1-22.35, (1994).
- B.L. Boulesteix, A. De Martino, B. Dre villon, and L. Schwartz, "Mueller polarimetric imaging system with liquid crystal," Appl. Opt. 43, 2824-2832 (2004). [CrossRef]
- E. Collett and V. Gazerro, "Polarization measurements in a spectrofluorophotometer," Opt. Commun. 129, 229-236 (1996). [CrossRef]
- E . Gar cia-Caurel, A. De Martino, and B. Drevillon, "Spectroscopic Mueller polarimeter based on liquid crystal devices," Thin Solid Films 455-456, 120-123 (2004). [CrossRef]
- <jrn>. J. Morio and F. Goudail, "Influence of the order of diattenuator, retarder, and polarizer in polar decomposition of Mueller matrices," Opt. Lett. 29, 2234-2236 (2004).</jrn> [CrossRef] [PubMed]
- M. Wong, M.J. Hendrix, K. Von der Mark, C. Little and R. Stern, "Collagen in the egg shell membranes of the hen," Dev. Biol. 104 (1), 28-36 (1984). [CrossRef]
- C.F. Bohren and D.R. Huffman, "Absorption and scattering of light by small particles," Wiley, New York (1983).

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