## Mueller matrix approach for determination of optical rotation in chiral turbid media in backscattering geometry

Optics Express, Vol. 14, Issue 1, pp. 190-202 (2006)

http://dx.doi.org/10.1364/OPEX.14.000190

Acrobat PDF (360 KB)

### Abstract

For in vivo determination of optically active (chiral) substances in turbid media, like for example glucose in human tissue, the backscattering geometry is particularly convenient. However, recent polarimetric measurements performed in the backscattering geometry have shown that, in this geometry, the relatively small rotation of the polarization vector arising due to the optical activity of the medium is totally swamped by the much larger changes in the orientation angle of the polarization vector due to scattering. We show that the change in the orientation angle of the polarization vector arises due to the combined effect of linear diattenuation and linear retardance of light scattered at large angles and can be decoupled from the pure optical rotation component using polar decomposition of Mueller matrix. For this purpose, the method developed earlier for polar decomposition of Mueller matrix was extended to incorporate optical rotation in the medium. The validity of this approach for accurate determination of the degree of optical rotation using the Mueller matrix measured from the medium in both forward and backscattering geometry was tested by conducting studies on chiral turbid samples prepared using known concentration of scatterers and glucose molecules.

© 2006 Optical Society of America

## 1. Introduction

1. S.L. Jacques, R.J. Roman, and K. Lee, “Imaging skin pathology with polarized light,” J. Biomed. Opt. **7**, 329–340 (2002). [CrossRef] [PubMed]

04. V. Backman, R. Gurjar, K. Badizadegan, I. Itzkan, R.R. Dassari, L.T. Perelman, and M.S. Feld, “Polarized light scattering spectroscopy for quantitative measurement of epithelial cellular structure,” IEEE J. Sel. Top. Quantum Electron. **5**, 1019–1026 (1999). [CrossRef]

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

12. 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. 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]

17. 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]

19. R.R. Ansari, S. Bockle, and L. Rovati, “New optical scheme for a polarimetric-based glucose sensor,” J. Biomed. Opt. **9**, 103–115 (2004). [CrossRef] [PubMed]

22. Danial Cote and I.A. 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]

22. Danial Cote and I.A. 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]

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]

## 2. Theory

### 2.1 Polar decomposition process for separating out linear retardance and circular retardance:

**M**(component that causes different amplitude changes for its orthogonal eigen states), a retarder

_{D}**M**(component that causes dephasing of two eigen states) and a depolarizer

_{R}**M**(component that causes depolarization) has been described in details by Lu and Chipman [23

_{Δ}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]

**M**is defined as

_{D}_{ij}is matrix element of i

^{th}row and j

^{th}column of Mueller matrix

**M**.

**M**,

_{Δ}**M**and

_{R}**M**′ have the following form

**M**].

**′**is the sub matrix of

**M**′ and can be written as

**′**as

_{1}, λ

_{2}and λ

_{3}are eigen values of m

**′**(m

**′**) . The sign “+” or “-” in the right side of Eq. (8) is determined by the sign of determinant of m

**′**.

_{R}of the retardance matrix

**M**can be obtained as

_{R}**M**can further be used to determine the value of optical rotation. The total retardance (R) and the elements of the retardance vector

_{R}_{1}, r

_{2}, r

_{3}] can be written as

_{ijk}is Levi-civita permutation symbol.

**M**can be written as a combination of linear retardance matrix and optical rotation matrix

_{R}_{3}

^{2}) can be expressed as

_{3}

^{2}are function of linear retardance (δ) and optical rotation (Φ) and are independent of the orientation of the fast axis of the linear retarder (θ). Therefore, using Eqs. (13) and (14), linear retardance and optical rotation (Φ) can be the expressed as

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]

**M**. The total polar decomposition process is illustrated in the flow chart shown in Fig. 1.

_{R}24. 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]

24. 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]

### 2.2 Decomposition of single scattering Mueller matrix:

**M**), we first consider the case of scattering from spherical scatterers with known size and refractive index.

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

_{medium}= 1.33 respectively. The variation in the value for the orientation angle of the polarization vector (γ) as a function of scattering angle Θ of an incident beam polarized at an angle γ

_{0}= 45° after single scattering from the scatterer is shown in Fig. 2(a). Here, γ is defined as

*et al*[22

22. Danial Cote and I.A. 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]

_{11}(Θ)+S

_{12}(Θ)}] and perpendicular [0.5 × {S

_{11}(Θ)-S

_{12}(Θ)}] to the scattering plane. The change in orientation angle of linear polarization vector of light scattered from the achiral scatterer is therefore a combined effect of linear diattenuation and linear retardance and there is no contribution of circular retardance to it. In order to unambiguously detect optical rotation introduced by the presence of chiral substances in a turbid medium (i.e., due to circular retardance property of the medium), it would be necessary to filter out the contribution of the additional rotation of polarization vector arising due to the combined effect of linear diattenuation and linear retardance. To investigate the efficacy of polar decomposition of Mueller matrix to accomplish this objective, we constructed Mueller matrix for a chiral medium having scatterer with the same scattering parameters. The rotatory power of the chiral medium was taken to be 0.01745 radian / cm.

## 3. Experimental methods

**M**) we generated the required four incident polarization states (linear polarization at angles of 0°, 45°, 90° from the horizontal and right circular polarization) and recorded the intensity of the light transmitted through sample after it passed through the suitably oriented analyzers (linear polarization at angles of 0°, 45°, 90° from the horizontal and right circular polarization) [26,27,28

_{i}28. Justin 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]

**M**) was constructed using the relation [26]

_{s}_{s}) of the samples with known concentration of microspheres were calculated using Mie theory. In order to prepare the chiral turbid samples, known concentration of D (dextro-rotatory) glucose solution was added to the microspheres suspension. The values for μ

_{s}of the individual chiral turbid samples were measured separately in a spectrophotometer (GBC, Cintra 20, Australia).

## 4. Results and discussion

**M**) and the decomposed diattenuation (

**M**) and retardance (

_{D}**M**) matrices for the linear retarder. The value for δ obtained using the decomposed retardance matrix

_{R}**M**was found to be 1.23.

_{R}**M**and

_{D}**M**for the combination of the linear retarder and glucose solution. Using the matrix

_{R}**M**and following the procedure described in section 2.1, the values for the linear retardance and optical rotation were obtained to be δ = 1.16 and ψ = 0.068 radian respectively. These values are reasonably close to the corresponding values for δ and ψ of the linear retarder and pure glucose solution obtained from separate measurements (δ = 1.23, ψ = 0.064 radian).

_{R}_{s}= 1.5 mm

^{-1}, g = 0.91) in both forward and backscattering geometry. The polar decomposition based approach could successfully decouple the scattering induced linear diattenuation and linear retardance from the measured Mueller matrix for the achiral turbid sample and yield a value of ψ = 0.003 radian (data not shown here). The measured Mueller matrix (

**M**) and the decomposed diattenuation (

**M**), depolarization (

_{D}**M**) and retardance (

_{Δ}**M**) matrices of a chiral turbid sample (prepared using aqueous suspension of 2.0 μm diameter polystyrene microspheres and known concentration of glucose) for forward and backscattering geometry are displayed in Table 2(a) and 2(b) respectively. The concentration of glucose in the sample was 5M. The value for μ

_{R}_{s}of this chiral turbid sample was determined to be μ

_{s}= 0.6 mm

^{-1}. The values for the parameters, linear diattenuation (d), linear retardance (δ), optical rotation (ψ) and degree of linear polarization [defined here as P

_{L}= (Q

^{2}+ U

^{2})

^{1/2}/ I)] for this sample are summarized in Table 3.

14. I.A. Vitkin and E. Hoskinson, “Polarization studies in multiply scattering chiral media,” Opt. Eng. **39**, 353–362 (2000). [CrossRef]

18. 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]

**13**, 148–163 (2005).http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-1-148 [CrossRef] [PubMed]

_{L}obtained from the pure depolarizing matrix is considerably lower in the backscattering direction as compared to that obtained for the forward scattering direction, suggests that this might be the case here. This aspect is being investigated in more details by carrying out systematic studies on chiral scattering samples having different values of μ

_{s}and g.

_{s}(= 5 mm

^{-1}). The value for ψ recovered after polar decomposition of Mueller matrix detected in forward geometry was found to be ψ = 0.076 radian. This value of ψ is about 10 % higher than the corresponding value obtained from the turbid sample with μ

_{s}= 0.6 mm

^{-1}and is about 18 % higher than that obtained from the clear solution of glucose with the same concentration (5M). As noted earlier, this gradual increase of the value of ψ with increasing value of μ

_{s}of the chiral scattering sample arises due to the increase in path length of photons due to increased degree of multiple scattering of light in the medium. For the same reason, the value for P

_{L}for this sample was also found to be lower (P

_{L}= 0.249) as compared to that obtained for the sample with μ

_{s}= 0.6mm

^{-1}.

## 5. Conclusion

## References

1. | S.L. Jacques, R.J. Roman, and K. Lee, “Imaging skin pathology with polarized light,” J. Biomed. Opt. |

02. | S. P. Morgan and I. M. Stockford, “Surface-reflection elimination in polarization imaging of superficial tissue,” Opt. Lett. |

03. | 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. |

04. | V. Backman, R. Gurjar, K. Badizadegan, I. Itzkan, R.R. Dassari, L.T. Perelman, and M.S. Feld, “Polarized light scattering spectroscopy for quantitative measurement of epithelial cellular structure,” IEEE J. Sel. Top. Quantum Electron. |

05. | M.I. Mischenko, J.W. Hovenier, and L.D. Travis, “Light scattering by nonspherical particles” Academic Press, San Diego, 1999. |

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

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

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

09. | 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. |

10. | 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. | 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 |

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

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

14. | I.A. Vitkin and E. Hoskinson, “Polarization studies in multiply scattering chiral media,” Opt. Eng. |

15. | 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. |

16. | I.A. 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 |

17. | 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. | 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. |

19. | R.R. Ansari, S. Bockle, and L. Rovati, “New optical scheme for a polarimetric-based glucose sensor,” J. Biomed. Opt. |

20. | M.P. Silverman, W. Strange, J. Badoz, and I.A. Vitkin, “Enhanced optical rotation and diminished depolarization in diffusive scattering from a chiral liquid,” Opt. Commun. |

21. | D. Cote and I.A. Vitkin, “Balanced detection for low-noise precision polarimetric measurements of optically active, multiply scattering tissue phantoms,” J. Biomed. Opt. |

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

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

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

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

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

27. | E. Collette, “Polarized Light: Fundamentals and Applications,” Marcel Dekker Inc. New York (1990). |

28. | Justin 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. |

**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

**Virtual Issues**

Vol. 1, Iss. 2 *Virtual Journal for Biomedical Optics*

**Citation**

S. Manhas, M. K. Swami, P. Buddhiwant, N. Ghosh, P. K. Gupta, and J. Singh, "Mueller matrix approach for determination of optical rotation in chiral turbid media in backscattering geometry," Opt. Express **14**, 190-202 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-1-190

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### References

- S.L. Jacques, R.J. Roman and K. Lee, "Imaging skin pathology with polarized light," J. Biomed. Opt. 7, 329 -340 (2002). [CrossRef] [PubMed]
- S. P. Morgan and I. M. Stockford, "Surface-reflection elimination in polarization imaging of superficial tissue," Opt. Lett. 28, 114-116 (2003). [CrossRef] [PubMed]
- 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, 6535-6546 (1992). [CrossRef] [PubMed]
- V. Backman, R. Gurjar, K. Badizadegan, I. Itzkan, R.R. Dassari, L.T. Perelman, and M.S. Feld, "Polarized light scattering spectroscopy for quantitative measurement of epithelial cellular structure," IEEE J. Sel. Top. Quantum Electron. 5, 1019 - 1026 (1999). [CrossRef]
- M.I. Mischenko, J.W. Hovenier, L.D. Travis, "Light scattering by nonspherical particles" Academic Press, San Diego, 1999.
- D. Bicout, C. Brosseau, A.S. Martinez , J.M. Schmitt "Depolarization of multiply scattered waves by spherical diffsers :Influence of size parameter," Phys. Rev. E 49, 1767-1770 (1994). [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]
- A.D. Kim, M. Moscoso, "Influence of the refractive index on the depolarization of multiply scattered waves," Phys. Rev. E 64, 026612, 1 -4 (2001). [CrossRef]
- 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, P.K. Gupta, "Depolarization of light in tissue phantoms - effect of a distribution in the size of scatterers," Opt. Express 11, 2198 -2205 (2003). <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-18-2198">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-18-2198</a> [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]
- R. J. McNichols, 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 - 1227 (1997). [CrossRef] [PubMed]
- I.A.Vitkin, E. Hoskinson, "Polarization studies in multiply scattering chiral media," Opt. Eng. 39, 353-362 (2000). [CrossRef]
- K.C.Hadley, 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.A. Vitkin, R.D. Laszlo, 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). <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-4-222">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-4-222</a> [PubMed]
- 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]
- 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]
- R.R. Ansari, S. Bockle, and L. Rovati, "New optical scheme for a polarimetric-based glucose sensor," J. Biomed. Opt. 9, 103 - 115 (2004). [CrossRef] [PubMed]
- M.P.Silverman, W. Strange, J. Badoz, I.A. Vitkin, "Enhanced optical rotation and diminished depolarization in diffusive scattering from a chiral liquid," Opt. Commun.132, 410-416 (1996). [CrossRef]
- D. Cote and I.A. Vitkin, "Balanced detection for low-noise precision polarimetric measurements of optically active, multiply scattering tissue phantoms," J. Biomed. Opt. 9, 213 - 220 (2004). [CrossRef] [PubMed]
- Danial Cote and I.A. 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). <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-1-148">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-1-148</a> [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]
- 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]
- C.F. Bohren, D.R. Huffman, "Absorption and scattering of light by small particles," Wiley, New York (1983).
- R.A.Chipman " Hand book of optics (polarimetry)," OSA / McGraw-Hill, 22.1-22.35, (1994).
- E. Collette, "Polarized Light: Fundamentals and Applications," Marcel Dekker Inc. New York (1990).
- Justin 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]

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