## Extraction of anisotropic parameters of turbid media using hybrid model comprising differential- and decomposition-based Mueller matrices |

Optics Express, Vol. 21, Issue 14, pp. 16831-16853 (2013)

http://dx.doi.org/10.1364/OE.21.016831

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### Abstract

A hybrid model comprising the differential Mueller matrix formalism and the Mueller matrix decomposition method is proposed for extracting the linear birefringence (LB), linear dichroism (LD), circular birefringence (CB), circular dichroism (CD), and depolarization properties (Dep) of turbid optical samples. In contrast to the differential-based Mueller matrix method, the proposed hybrid model provides full-range measurements of all the anisotropic properties of the optical sample. Furthermore, compared to the decomposition-based Mueller matrix method, the proposed model is insensitive to the multiplication order of the constituent basis matrices. The validity of the proposed method is confirmed by extracting the anisotropic properties of a compound chitosan-glucose-microsphere sample with LB/CB/Dep properties and two ferrofluidic samples with CB/CD/Dep and LB/LD/Dep properties, respectively. It is shown that the proposed hybrid model not only yields full-range measurements of all the anisotropic parameters, but is also more accurate and more stable than the decomposition method. Moreover, compared to the decomposition method, the proposed model more accurately reflects the dependency of the phase retardation angle and linear dichroism angle on the direction of the external magnetic field for ferrofluidic samples. Overall, the results presented in this study confirm that the proposed model has significant potential for extracting the optical parameters of real-world samples characterized by either single or multiple anisotropic properties.

© 2013 OSA

## 1. Introduction

*et al.*proposed a method for measuring the absorption coefficient, scattering coefficient and anisotropy factor of bovine muscle, human tissue and polyurethane using a single integrating sphere [1

1. P. R. Bargo, S. A. Prahl, T. T. Goodell, R. A. Sleven, G. Koval, G. Blair, and S. L. Jacques, “In vivo determination of optical properties of normal and tumor tissue with white light reflectance and an empirical light transport model during endoscopy,” J. Biomed. Opt. **10**(3), 034018 (2005). [CrossRef] [PubMed]

3. J. W. Pickering, S. A. Prahl, N. van Wieringen, J. F. Beek, H. J. C. M. Sterenborg, and M. J. C. van Gemert, “Double-Integrating-Sphere System for Measuring the Optical Properties of Tissue,” Appl. Opt. **32**(4), 399–410 (1993). [CrossRef] [PubMed]

4. A. A. Oraevsky, S. L. Jacques, and F. K. Tittel, “Measurement of tissue optical properties by time-resolved detection of laser-induced transient stress,” Appl. Opt. **36**(1), 402–415 (1997). [CrossRef] [PubMed]

7. D. Contini, A. Torricelli, A. Pifferi, L. Spinelli, F. Paglia, and R. Cubeddu, “Multi-channel time-resolved system for functional near infrared spectroscopy,” Opt. Express **14**(12), 5418–5432 (2006). [CrossRef] [PubMed]

8. S. Fantini, M.-A. Franceschini, J. S. Maier, S. A. Walker, B. B. Barbieri, and E. Gratton, “Frequency-domain multichannel optical detector for noninvasive tissue spectroscopy and oximetry,” Opt. Eng. **34**(1), 32–42 (1995). [CrossRef]

10. N. Shah, A. E. Cerussi, D. Jakubowski, D. Hsiang, J. Butler, and B. J. Tromberg, “Spatial variations in optical and physiological properties of healthy breast tissue,” J. Biomed. Opt. **9**(3), 534–540 (2004). [CrossRef] [PubMed]

11. S. J. Yeh, O. S. Khalil, C. F. Hanna, and S. Kantor, “Near-infrared thermo-optical response of the localized reflectance of intact diabetic and nondiabetic human skin,” J. Biomed. Opt. **8**(3), 534–544 (2003). [CrossRef] [PubMed]

12. A. Dimofte, J. C. Finlay, and T. C. Zhu, “A method for determination of the absorption and scattering properties interstitially in turbid media,” Phys. Med. Biol. **50**(10), 2291–2311 (2005). [CrossRef] [PubMed]

13. R. O. Esenaliev, Y. Y. Petrov, O. Hartrumpf, D. J. Deyo, and D. S. Prough, “Continuous, noninvasive monitoring of total hemoglobin concentration by an optoacoustic technique,” Appl. Opt. **43**(17), 3401–3407 (2004). [CrossRef] [PubMed]

14. C. L. Darling, G. D. Huynh, and D. Fried, “Light scattering properties of natural and artificially demineralized dental enamel at 1310 nm,” J. Biomed. Opt. **11**(3), 034023 (2006). [CrossRef] [PubMed]

*et al.*[15

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

*et al*. [18

18. X. Wang, G. Yao, and L. V. Wang, “Monte Carlo Model and Single-Scattering Approximation of the Propagation of Polarized Light in Turbid Media Containing Glucose,” Appl. Opt. **41**(4), 792–801 (2002). [CrossRef] [PubMed]

19. X. Wang and L. V. Wang, “Propagation of polarized light in birefringent turbid media: A Monte Carlo study,” J. Biomed. Opt. **7**(3), 279–290 (2002). [CrossRef] [PubMed]

*et al.*[20

20. N. Ghosh, M. F. G. Wood, S. H. Li, R. D. Weisel, B. C. Wilson, R. K. Li, and I. A. Vitkin, “Mueller matrix decomposition for polarized light assessment of biological tissues,” J Biophotonics **2**(3), 145–156 (2009). [CrossRef] [PubMed]

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

24. T.-T.-H. Pham and Y.-L. Lo, “Extraction of effective parameters of turbid media utilizing Mueller matrix approach -A study of glucose sensing,” J. Biomed. Opt. **17**(9), 097002 (2012). [CrossRef]

25. R. M. A. Azzam, “Propagation of partially polarized light through anisotropic media with or without depolarization: A differential 4 × 4 matrix calculus,” J. Opt. Soc. Am. **68**(12), 1756–1767 (1978). [CrossRef]

26. R. Ossikovski, “Differential matrix formalism for depolarizing anisotropic media,” Opt. Lett. **36**(12), 2330–2332 (2011). [CrossRef] [PubMed]

27. N. Ortega-Quijano and J. L. Arce-Diego, “Depolarizing differential Mueller matrices,” Opt. Lett. **36**(13), 2429–2431 (2011). [CrossRef] [PubMed]

28. N. Ortega-Quijano and J. L. Arce-Diego, “Mueller matrix differential decomposition for direction reversal: application to samples measured in reflection and backscattering,” Opt. Express **19**(15), 14348–14353 (2011). [CrossRef] [PubMed]

## 2. Mueller-stokes parameter measurement method

**M**is the Mueller matrix of the sample and

_{sample}**S**is the input Stokes vector. In the present study, the sample is illuminated with four different input polarization lights, namely three linear polarization lights (i.e.,

_{input}**M**(i.e.,

_{sample}*M*

_{11}~

*M*

_{44}) are then obtained as

## 3. Differential Mueller matrix

26. R. Ossikovski, “Differential matrix formalism for depolarizing anisotropic media,” Opt. Lett. **36**(12), 2330–2332 (2011). [CrossRef] [PubMed]

*α*, phase retardation

*β*, optical rotation angle

*γ*, orientation angle

*θ*, linear dichroism

_{d}*D*, and circular dichroism

*R*can be obtained respectively as Similarly, the differential Mueller matrix describing the depolarization effect can be obtained as

25. R. M. A. Azzam, “Propagation of partially polarized light through anisotropic media with or without depolarization: A differential 4 × 4 matrix calculus,” J. Opt. Soc. Am. **68**(12), 1756–1767 (1978). [CrossRef]

*K*

_{22}and

*K*

_{33}are degrees of linear depolarization and

*K*

_{44}is the degree of circular depolarization. In general, the degree of depolarization is quantified by the depolarization index, Δ, which has a value of 0 for a non-depolarizing sample and 1 for an ideal depolarizer [32

32. J. J. Gil and E. Bernabeu, “Depolarization and polarization indices of an optical system,” Opt. Acta (Lond.) **33**(2), 185–189 (1986). [CrossRef]

*K*

_{22},

*K*

_{33}and

*K*

_{44}, the depolarization index can be obtained as [33

33. R. A. Chipman, “Depolarization index and the average degree of polarization,” Appl. Opt. **44**(13), 2490–2495 (2005). [CrossRef] [PubMed]

34. B. J. DeBoo, J. M. Sasian, and R. A. Chipman, “Depolarization of diffusely reflecting man-made objects,” Appl. Opt. **44**(26), 5434–5445 (2005). [CrossRef] [PubMed]

*β*) and rotation angle (

*γ*) given in Eqs. (10) and (11), respectively, are restricted due to an ambiguity in the value of the angle in the corresponding trigonometric functions. Thus, as described in the following subsection, the present study proposes a hybrid model in which the differential Mueller matrix formalism is combined with the Mueller matrix decomposition method.

## 4. Hybrid Mueller matrix model

### 4.1 LB/CB/Dep composite sample

**M**is the macroscopic matrix describing the LB and CB properties of the sample and

_{B}**M**is the macroscopic Mueller matrix describing the depolarization properties. It is noted in Fig. 2 that two different combinations of the Dep and LB/CB properties are considered in order to check which solution of Eq. (18) is closer to that obtained from the differential model given in Eq. (8). In Eq. (18), the macroscopic depolarization matrix,

_{Δ}**M**, can be obtained directly from Eq. (16). Thus, matrix

_{Δ}**M**describing the LB and CB properties of the sample can be calculated from the measured matrix

_{B}**M**as follows:Alternatively,

_{BΔ}**M**can be derived from Eq. (6) via a process of inverse differential calculation and formulated as

_{B}*β*sin(2

*α*) and

*β*cos(2

*α*) can be obtained respectively as From Eqs. (22) and (23), the orientation angle (

*α*) and phase retardation (

*β*) of the composite sample can be obtained as It is noted that in the decomposition method used within the proposed hybrid model, the parameters given in Eqs. (21), (24) and (25) are extracted for both cases shown in Fig. 2 (i.e.,

*α*,

*β*and γ in Eqs. (21), (24) and (25) are equal to 0<

*α*<180°, 0<

*β*<360° and 0<

*γ*<180°, respectively. In other words, the proposed hybrid model enables parameters

*α*,

*β*and γ to be measured over the full range.

### 4.2 CB/CD/Dep composite sample

**M**is the macroscopic matrix describing the CB and CD properties of the sample and

_{C}**M**is the macroscopic Mueller matrix describing the depolarization properties. In practice,

_{Δ}**M**can be determined experimentally from the Stokes polarimeter measurements, while

_{CΔ}**M**can be calculated from Eq. (16). Matrix

_{Δ}**M**can thus be obtained asAlternatively, matrix

_{C}**M**can be derived from the differential matrix given in Eq. (6) as follows:Equating Eqs. (27) and (28), the circular birefringence (γ) and circular dichroism (

_{C}*R*) can be obtained respectively as As for the previous sample, the values of γ and

*R*are calculated for each configuration shown in Fig. 3. The extracted parameters are then used to establish the corresponding differential Mueller matrix given in Eq. (8) in order to check which solutions are closer to those determined from the measured differential Mueller matrix in Eq. (5). It is noted that Eqs. (29) and (30) yield values of

*γ*and

*R*in the ranges 0<

*γ*<180° and −1<

*R*<1, respectively. In other words, the proposed hybrid model provides full range measurements of both the CB properties and the CD properties of the composite sample.

### 4.3 LB/LD/Dep composite sample

**M**is the macroscopic Mueller matrix describing the LB and LD properties of the sample, while

_{L}**M**is the macroscopic Mueller matrix describing the depolarization properties.

_{Δ}**M**can be obtained directly from Eq. (16). Matrix

_{Δ}**M**can then be determined from the measured matrix

_{L}**M**as follows:Alternatively,

_{LΔ}**M**can be inversely derived from Eq. (6) as

_{L}**M**is highly complicated and is therefore presented in the appendix to this study. In the present study, the unknown parameters in Eq. (33) are extracted by means of a Genetic Algorithm [35–37

_{L}37. T. C. Yu and Y. L. Lo, “A novel heterodyne polarimeter for the multiple-parameter measurements of twisted nematic liquid crystal cell using a genetic algorithm approach,” J. Lightwave Technol. **25**(3), 946–951 (2007). [CrossRef]

**M**given in Eq. (32) as the target function. It is again noted that the GA yields full range measurements of all of the LB and LD properties.

_{L}### 4.4 LB/CB/LD/CD/Dep composite sample

*α*,

*β*,

*γ*,

*θ*,

_{d}*D*,

*R*and Δ can be obtained. It is found that

*α*,

*θ*,

_{d}*D*, and

*R*are measured over the full range, but

*β*and

*γ*are measured over the restricted range. Subsequently, in order to improve the restricted ranges of

*β*and

*γ*, the hybrid model is employed. The extracted

*β*and

*γ*obtained by the differential Mueller matrix formalism are used to re-build a macroscopic LB/CB Mueller matrix by the inverse differential calculation. Finally, according to the macroscopic LB/CB Mueller matrix,

*β*and

*γ*then can be extracted by employing the hybrid model for the LB/CB case and using Eqs. (21), (22), (23), and (25). The problem in the restricted ranges of

*β*and

*γ*thus can be improved.

*β*and

*γ*are out of range. Thus, it affects the correction in deriving the macroscopic LB/CB Mueller matrix by the inverse differential calculation. Therefore, the modified algorithm proposed here has some limitation but still provides a good extension of the measurement range than just only using the differential Mueller matrix formalism. The robustness of this modified algorithm can be confirmed in the simulation.

## 5. Simulation results

*α*: 0~180°,

*θ*: 0~180°;

_{d}*γ*: 0~180°;

*β*: 0~360°;

*D*: 0~1; and R: −1~1).

### 5.1 LB/CB/Dep composite sample

*α*,

*β*and

*γ*for an anisotropic sample with known LB, CB and Dep properties and then comparing the extracted values with the theoretical input values. Figure 6(a) compares the extracted value of the principal axis of phase retardation (

*α*) with the input value of

*α*over the full measurement range, i.e.,

*α*: 0~180°. Note that the remaining sample parameters have values of

*β*= 60°,

*γ*= 15° and Δ = 0.4. Figures 6(b) and 6(c) present the equivalent results for the phase retardation (

*β*) and orientation angle (

*γ*), respectively. Note that parameters

*α*,

*β*and

*γ*are extracted using Eqs. (21), (24) and (25) in the hybrid model, Eqs. (9)-(11) in the differential Mueller matrix method, and the method proposed in [24

24. T.-T.-H. Pham and Y.-L. Lo, “Extraction of effective parameters of turbid media utilizing Mueller matrix approach -A study of glucose sensing,” J. Biomed. Opt. **17**(9), 097002 (2012). [CrossRef]

*α*to be measured over the full range. However, it is seen in Figs. 6(b) and 6(c) that the differential Mueller matrix method is unable to obtain full-range measurements of

*β*and

*γ*. By contrast, the proposed hybrid model enables both parameters to be measured over the full range. In general, it is noted that the parameter values extracted by the decomposition method all deviate from the input values for the composite sample. Finally, it is noted that the depolarization is correctly extracted as Δ = 0.4 in every case.

### 5.2 CB/CD/Dep composite sample

*γ*) and circular dichroism (

*R*), respectively. Figure 7(a) compares the extracted value of

*γ*with the input value over the full range of

*γ*: 0~180° given

*R*= 0.2 and Δ = 0.4. Similarly, Fig. 7(b) compares the extracted value of

*R*with the input value of

*R*over the full range of

*R*: −1 ~1 given

*γ*= 15° and Δ = 0.4. Note that in the hybrid model,

*γ*and R are extracted using Eqs. (29) and (30), respectively, while in the differential Mueller matrix method,

*γ*and R are extracted using Eqs. (11) and (14), respectively.

*γ*to be measured only over the range of

*γ*: 0~90°. Figure 7(b) shows that the hybrid model and the differential calculation method both enable the circular dichroism to be measured over the full range. However, the values of

*R*extracted by the decomposition method deviate from the theoretical input values since the composite sample is assumed.

### 5.3 LB/LD/Dep composite sample

*α*,

*β*,

*θ*, and

_{d}*D*and then comparing the extracted values with the known input values. Figure 8(a) presents the extracted values of

*α*for input values of

*α*over the full range (i.e.,

*α*: 0~180°) given parameter settings of

*β*= 60°,

*θ*= 35°,

_{d}*D*= 0.4, and Δ = 0.4. Figures 8(b)-8(d) present the equivalent results for parameters

*β*,

*θ*and

_{d}*D*, respectively. Note that in the hybrid model, parameters

*α*,

*β*,

*θ*, and

_{d}*D*are extracted using a GA, while in the differential calculation method,

*α*,

*β*,

*θ*, and

_{d}*D*are extracted using Eqs. (9)-(10) and (12)-(13).

*α*,

*θ*, and

_{d}*D*to be measured over the full range. However,

*β*can only be measured over the half-range (i.e.,

*β*: 0~180°). It is noted that the decomposition method enables all four parameters to be measured over the full range. However, as for the two previous composite samples, the extracted parameter values deviate from the input values.

### 5.4 LB/CB/LD/CD/Dep composite sample

*α*for input value of

*α*over the full range (i.e.,

*α*: 0~180°) given parameter settings of

*β*= 60°,

*θ*= 35°,

_{d}*D*= 0.4, and Δ = 0.4. Figures 9(b)-9(f) present the equivalent results for parameters

*θ*,

_{d}*D*,

*R*,

*β*, and

*γ*, respectively. Note that

*α*,

*θ*,

_{d}*D*, and

*R*are extracted using differential Mueller matrix formalism in Eqs. (9), (12), (13), and (14), and

*β*and

*γ*are extracted by the modified algorithm as described in Subsection 4.4.

*α*,

*θ*,

_{d}*D*, and

*R*to be measured over the full range. However,

*β*and

*γ*can only be measured over the half-range by the differential Mueller matrix formalism. The limitation is then improved by the hybrid model, and it enables

*β*and

*γ*to be measured over the near full-range (i.e.,

*β*: 0~350° and

*γ*: 0~170°)). It can be seen in Figs. 9(e) and 9(f) in black dots.

*α*= 60°,

*β*= 30°,

*γ*= 30°,

*θ*= 70°, D = 0.2,

_{d}*R*= 0.6, Δ = 0.3; (2)

*α*= 120°,

*β*= 15°,

*γ*= 45°,

*θ*= 105°, D = 0.6,

_{d}*R*= 0.4, Δ = 0.5) are simulated and the similar results are observed. While

*β*and

*γ*are out of range (i.e.,

*β*> 350° and

*γ*>170°), the deviations in the extracted values of

*β*and

*γ*occur. In the proposed modified algorithm, the hybrid model can provide the exact quadrant determination but cannot eliminate the existing errors induced by using the differential Mueller matrix formalism.

## 6. Experimental measurement of anisotropic parameters of turbid media

### 6.1 Composite chitosan-glucose-microsphere sample

24. T.-T.-H. Pham and Y.-L. Lo, “Extraction of effective parameters of turbid media utilizing Mueller matrix approach -A study of glucose sensing,” J. Biomed. Opt. **17**(9), 097002 (2012). [CrossRef]

*β*) obtained using the hybrid method vary approximately linearly with the chitosan concentration. Moreover, the optical rotation angle (γ) and depolarization have approximately constant values since the different samples contain equal amounts of glucose and microparticles. In addition, it is seen that the measured values and tendencies of

*β*and γ are similar in both types of chitosan / glucose sample (i.e., with and without suspended particles, respectively. From inspection, the standard deviations of the extracted values of

*α*,

*β*and γ are found to be just 0.43°, 0.04° and 0.01°, respectively. By contrast, the standard deviations of

*α*,

*β*and γ extracted using the decomposition method are 0.85°, 0.05° and 0.08°, respectively. In other words, the measurement performance of the hybrid model is more stable than that of the decomposition method.

*β*) and depolarization have approximately constant values since all of the samples contain the same amount of chitosan and microspheres. In addition, it is observed that for both types of sample, the measurement results obtained using the hybrid model are more stable than those obtained using the decomposition method.

### 6.2 Ferrofluidic samples with CB/CD/Dep and LB/LD/Dep properties

_{3}O

_{4}) nanoparticles were synthesized using the thermal decomposition method described in [38

38. S. Sun, H. Zeng, D. B. Robinson, S. Raoux, P. M. Rice, S. X. Wang, and G. Li, “Monodisperse MFe2O4 (M = Fe, Co, Mn) nanoparticles,” J. Am. Chem. Soc. **126**(1), 273–279 (2004). [CrossRef] [PubMed]

_{3}O

_{4}nanoparticles were dispersed in hexane to create stock solutions with concentrations of 0.01 M and 0.025 M, respectively. Figure 13 shows the image of Fe

_{3}O

_{4}nanoparticles obtained from the transmission electron microscopy (JEM-2010 Electron Microscope, JEOL Co. 200 KV) by our group. From inspection, the average diameter of the Fe

_{3}O

_{4}nanoparticles is 6 nm. The ferrofluidic suspensions were placed in square quartz containers with external side-lengths of 12.5 mm and internal side-lengths of 10 mm.

#### 6.2.1 Ferrofluidic sample with CB/CD/Dep properties

*R*) increases as the intensity of the external magnetic field increases. However, the depolarization of the ferrofluidic sample remains approximately constant for all values of the magnetic field intensity since the sample contains a fixed concentration of Fe

_{3}O

_{4}nanoparticles. From inspection, the standard deviations of the values of γ and R obtained using the hybrid model are found to be 0.02° and 5.45 × 10

^{−4}, respectively. Thus, the experimental stability of the proposed hybrid model is confirmed. Also, from inspection the experimental results of the measured R and Dep obtained by the decomposition method and the hybrid model, the hybrid model is more stable than that of the decomposition method.

#### 6.2.2 Ferrofluidic sample with LB/LD/Dep properties

_{3}O

_{4}nanoparticle concentration of 0.025 M given external magnetic field intensities ranging from 0 to 450 G. For both methods, a good correlation exists between the measured values of the phase retardation (

*β*) and the intensity of the external magnetic field. Moreover, the linear dichroism (

*D*) increases with an increasing magnetic field intensity. The standard deviations of

*α*,

*β*,

*θ*and D extracted using the hybrid model are 0.05°, 0.04°, 2.03° and 9.5 × 10

_{d,}^{−4}, respectively. As expected, the extracted value of the depolarization is around twice that of the sample with a Fe

_{3}O

_{4}concentration of 0.01 M. It is seen that the orientation angle of the phase retardation (

*α*) is more closely aligned with the direction of the magnetic field as the intensity of the magnetic field is increased. Notably, the orientation angle of the linear dichroism (

*θ*) extracted by the hybrid model also converges toward the direction of the magnetic field as the intensity of the magnetic field increases. However, such a tendency is not apparent in the results extracted using the decomposition method. In other words, the superior measurement performance of the proposed hybrid model is once again confirmed.

_{d}## 7. Conclusions and discussions

*α*) and linear dichroism (

*θ*) of the ferrofluidic sample under high magnetic field intensities along the direction of the magnetic field.

_{d}## Appendix

## Acknowledgments

## References and links

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22. | X. Guo, M. F. G. Wood, N. Ghosh, and I. A. Vitkin, “Depolarization of light in turbid media: a scattering event resolved Monte Carlo study,” Appl. Opt. |

23. | N. Ghosh and I.A. Vitkin, “Tissue polarimetry: concepts, challenges, applications, and outlook ,” J. Biomed. Opt. |

24. | T.-T.-H. Pham and Y.-L. Lo, “Extraction of effective parameters of turbid media utilizing Mueller matrix approach -A study of glucose sensing,” J. Biomed. Opt. |

25. | R. M. A. Azzam, “Propagation of partially polarized light through anisotropic media with or without depolarization: A differential 4 × 4 matrix calculus,” J. Opt. Soc. Am. |

26. | R. Ossikovski, “Differential matrix formalism for depolarizing anisotropic media,” Opt. Lett. |

27. | N. Ortega-Quijano and J. L. Arce-Diego, “Depolarizing differential Mueller matrices,” Opt. Lett. |

28. | N. Ortega-Quijano and J. L. Arce-Diego, “Mueller matrix differential decomposition for direction reversal: application to samples measured in reflection and backscattering,” Opt. Express |

29. | D. S. Kliger, J. W. Lewis, and C. E. Randall, |

30. | P. C. Chen, Y. L. Lo, T. C. Yu, J. F. Lin, and T. T. Yang, “Measurement of linear birefringence and diattenuation properties of optical samples using polarimeter and Stokes parameters,” Opt. Express |

31. | Y. L. Lo, T. T. H. Pham, and P. C. Chen, “Characterization on five effective parameters of anisotropic optical material using Stokes parameters-Demonstration by a fiber-type polarimeter,” Opt. Express |

32. | J. J. Gil and E. Bernabeu, “Depolarization and polarization indices of an optical system,” Opt. Acta (Lond.) |

33. | R. A. Chipman, “Depolarization index and the average degree of polarization,” Appl. Opt. |

34. | B. J. DeBoo, J. M. Sasian, and R. A. Chipman, “Depolarization of diffusely reflecting man-made objects,” Appl. Opt. |

35. | Z. Michalewicz, |

36. | H. C. Cheng and Y. L. Lo, “The synthesis of multiple parameters of arbitrary FBGs via a genetic algorithm, and two thermally modulated intensity,” J. Lightwave Technol. |

37. | T. C. Yu and Y. L. Lo, “A novel heterodyne polarimeter for the multiple-parameter measurements of twisted nematic liquid crystal cell using a genetic algorithm approach,” J. Lightwave Technol. |

38. | S. Sun, H. Zeng, D. B. Robinson, S. Raoux, P. M. Rice, S. X. Wang, and G. Li, “Monodisperse MFe2O4 (M = Fe, Co, Mn) nanoparticles,” J. Am. Chem. Soc. |

**OCIS Codes**

(120.5410) Instrumentation, measurement, and metrology : Polarimetry

(160.1190) Materials : Anisotropic optical materials

(160.4760) Materials : Optical properties

(290.7050) Scattering : Turbid media

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: April 24, 2013

Revised Manuscript: June 5, 2013

Manuscript Accepted: June 30, 2013

Published: July 5, 2013

**Virtual Issues**

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

**Citation**

Chia-Chi Liao and Yu-Lung Lo, "Extraction of anisotropic parameters of turbid media using hybrid model comprising differential- and decomposition-based Mueller matrices," Opt. Express **21**, 16831-16853 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-14-16831

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