## Characterization on five effective parameters of anisotropic optical material using Stokes parameters-Demonstration by a fiber-type polarimeter

Optics Express, Vol. 18, Issue 9, pp. 9133-9150 (2010)

http://dx.doi.org/10.1364/OE.18.009133

Acrobat PDF (1271 KB)

### Abstract

An analytical technique based on the Mueller matrix method and the Stokes parameters is proposed for extracting five effective parameters on the principal axis angle, phase retardance, diattenuation axis angle, diattenuation and optical rotation angle of anisotropic optical materials. The linear birefringence (LB) / circular birefringence (CB) properties and linear diattenuation (LD) properties are decoupled within the analytical model. The analytical method is then integrated with a genetic algorithm to extract the optical properties of samples with linear birefringence property using a fiber-based polarimeter. The result demonstrates the feasibility of analytical model in characterizing five effective parameters of anisotropic optical material. Also, it confirms that the proposed fiber-based polarimeter provides a simple alternative to existing fiber-based probes for parameter measurement in the near field or the remote environment. A low birefringence fiber-based polarimeter based on effective parameters and genetic algorithm without using a fiber polarization controller is first proposed confirmatively.

© 2010 OSA

## 1. Introduction

2. D. B. Chenault and R. A. Chipman, “Measurements of linear diattenuation and linear retardation spectra with a rotating sample spectropolarimeter,” Appl. Opt. **32**(19), 3513–3519 (1993). [CrossRef] [PubMed]

*et al.*[4

4. M. J. Fasolka, L. S. Goldner, J. Hwang, A. M. Urbas, P. DeRege, T. Swager, and E. L. Thomas, “Measuring local optical properties: near-field polarimetry of photonic block copolymer morphology,” Phys. Rev. Lett. **90**(1), 016107 (2003). [CrossRef] [PubMed]

*et al.*[5

5. L. S. Goldner, M. J. Fasolka, S. Nougier, H. P. Nguyen, G. W. Bryant, J. Hwang, K. D. Weston, K. L. Beers, A. Urbas, and E. L. Thomas, “Fourier analysis near-field polarimetry for measurement of local optical properties of thin films,” Appl. Opt. **42**(19), 3864–3881 (2003). [CrossRef] [PubMed]

6. L. S. Goldner, M. J. Fasolka, and S. N. Goldie, “Measurement of the local diattenuation and retardance of thin polymer films using near-field polarimetry,” Appl. Scanned Probe Microscopy Polymers **897**, 65–84 (2005). [CrossRef]

7. A. L. Campillo and J. W. P. Hsu, “Near-field scanning optical microscope studies of the anisotropic stress variations in patterned SiN membranes,” J. Appl. Phys. **91**(2), 646–651 (2002). [CrossRef]

7. A. L. Campillo and J. W. P. Hsu, “Near-field scanning optical microscope studies of the anisotropic stress variations in patterned SiN membranes,” J. Appl. Phys. **91**(2), 646–651 (2002). [CrossRef]

4. M. J. Fasolka, L. S. Goldner, J. Hwang, A. M. Urbas, P. DeRege, T. Swager, and E. L. Thomas, “Measuring local optical properties: near-field polarimetry of photonic block copolymer morphology,” Phys. Rev. Lett. **90**(1), 016107 (2003). [CrossRef] [PubMed]

7. A. L. Campillo and J. W. P. Hsu, “Near-field scanning optical microscope studies of the anisotropic stress variations in patterned SiN membranes,” J. Appl. Phys. **91**(2), 646–651 (2002). [CrossRef]

*et al.*[8

8. D. B. Chenault, R. A. Chipman, and S. Y. Lu, “Electro-optic coefficient spectrum of cadmium telluride,” Appl. Opt. **33**(31), 7382–7389 (1994). [CrossRef] [PubMed]

9. E. A. Sornsin and R. A. Chipman, “Visible Mueller matrix spectropolarimetry,” SPIE **3121**, 156–160 (1997). [CrossRef]

*et al.*[10

10. 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 **17**(18), 15860–15884 (2009). [CrossRef] [PubMed]

## 2. Method of measuring five effective optical parameters of an anisotropic material

_{d}and diattenuation D has the form in Eq. (2)

*Ŝ*and

_{c}*S*are input and output Stokes vector, respectively. The CB component of the sample is in front of the LB and LD components.

_{c}*S*in Fig. 1 can be calculated as

_{c}*],[M*

_{ld}*], and [M*

_{lb}*] represent the effective Mueller matrices corresponding to the LD, LB and CB properties of an anisotropic material, respectively. The non-zero Mueller matrix elements in Eq. (4) are expressed respectively as As described in the following sections, given a knowledge of the input polarization state (i.e. linearly polarized, circularly polarized) and the measured values of the output Stokes parameters, Eqs. (5) ~(13) provide the means to solve the five effective optical parameters of the anisotropic material. It should be noted here that the complex terms in elements*

_{cb}*m*,

_{12}*m*,

_{13}*m*,

_{22}*m*,

_{23}*m*, and

_{32}*m*are not required to solve the effective optical parameters of the anisotropic material. In the methodology proposed in this study, the five effective optical parameters of the anisotropic material, i.e. α, β, θ

_{33}*, D, and γ, are extracted using six input polarization lights, namely four linear polarization lights (i.e.*

_{d}*, can be determined from Eq. (27) using a quadrant determination method. Note that the range of 2θ*

_{d}*is defined as 0 ~ 360°, and thus θ*

_{d}*is obtained within the range 0 ~ 180°. Once the diattenuation axis angle, θ*

_{d}*, has been obtained, the diattenuation, D, can be solved using one of two different methods. In the first method, D is obtained from Eqs. (24) and (25) asIn the second method, D is obtained from Eqs. (24) and (26) asNote that Eqs. (28) and (29) yield the same theoretical solution, and thus the equality (or otherwise) of Eqs. (28) and (29) provides the means to check the correctness of the experimental results. Once the diattenuation axis angle, diattenuation, and retardance are known, Eqs. (18) and (19) can be used to calculate the principal axis angle α as where Therefore, using Eqs. (30)~(34), 2α can be obtained as*

_{d}_{d}), diattenuation (D), and optical rotation (γ) can be extracted using Eqs. (35), (23), (27), (28), and (36), respectively. It is noted that the proposed methodology does not require the principal birefringence axes and diattenuation axes to be aligned. Moreover, while the LB and CB properties are coupled within the analytical model, the LB/CB and LD properties are decoupled. Thus, the LB/CB properties of the sample can be solved directly without any prior knowledge of the LD properties.

## 3. Analytical simulations and error analysis

### 3.1 Analytical simulations

_{0°}, S

_{45°}, S

_{90°}, S

_{135°}, S

_{RHC}, and S

_{LHC}, were obtained for a hypothetical anisotropic sample using the Jones matrix formulation based on known values of the sample parameters and a knowledge of the input Stokes vectors. The theoretical Stokes values were inserted into the analytical model derived in Section 2 and the effective optical parameters were then inversely derived. Finally, the extracted values of the effective optical parameters were compared with the input values used in the Jones matrix formulation.

_{d}= 35°, diattenuation D= 0.03, and optical rotation angle γ = 15°. Figure 2 plots the value of the principal axis angle extracted using Eq. (35) against the input value of α over the range 0 ~ 180°. A good agreement is observed between the two values of α, and thus the ability of the proposed method to obtain full-range measurements of the principal axis angle is confirmed.

_{d}= 35°, the diattenuation was set as D= 0.03, and the optical rotation angle was assigned a value of γ = 15°. Figure 3 compares the value of the phase retardance extracted using Eq. (23) with the input values of the phase retardance over the full range of 0 ~360°. It can be seen that the extracted values are only consistent with the input values over the range 0 ~180°. Thus, as expected, the proposed method only enables the phase retardance to be extracted over the limited range of 0 ~180°.

_{d}, using Eq. (27), the effective parameters of the anisotropic material were specified as follows: principal axis angle α = 50°, retardance β = 60°, diattenuation D= 0.03, and optical rotation angle γ = 15°. The results presented in Fig. 4 confirm the ability of the analytical model to extract the diattenuation axis angle over the full range of 0 ~180°.

_{d}= 35° and the optical rotation angle was assigned a value of γ = 15°. Figure 5 plots the value of the diattenuation obtained from Eq. (28) against the input value of D. It is observed that a good agreement exists between the two values of D over the range 0 ~1. Thus, the ability of the proposed method to obtain full-range measurements of the diattenuation is confirmed.

_{d}= 35°, and the diattenuation was specified as D = 0.03. Figure 6 shows that the value obtained from Eq. (36) for the optical rotation angle is consistent with the input value of γ over the range 0° ~180°. Thus, the results confirm the ability of the proposed method to obtain full-range measurements of the optical rotation angle.

10. 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 **17**(18), 15860–15884 (2009). [CrossRef] [PubMed]

### 3.2 Error analysis of proposed measurement methodology

_{0°}, S

_{45°}, S

_{90°}, S

_{135°}, S

_{RHC}and S

_{LHC}for a composite sample with known LB/CB birefringence and diattentuation properties and known input polarization states. To simulate the error in the values of the output Stokes parameters obtained in a typical experimental measurement procedure, 500 sets of theoretical values of S

_{0°}, S

_{45°}, S

_{90°}, S

_{135°}, S

_{RHC}and S

_{LHC}with random perturbations between −0.5% ~ + 0.5% are deliberately introduced. These perturbed values are then inserted into the analytical model in order to derive the corresponding LB/CB birefringence and diattentuation properties of the composite sample. Finally, the extracted values of the LB/CB birefringence and diattentuation are compared with the given values used in the Jones matrix formulation.

_{d}= 102.440°, diattenuation D = 0.040, and optical rotation angle γ = −24.356°. It is noted that those data in above are based on the experimental data in Table 2 . The values of α, β, θ

_{d}, D, and γ were then extracted from Eqs. (35), (23), (27), (28) and (36), respectively. Figure 7(a) - (e) compares the extracted values of the sample parameters (α’, β’, θ

_{d}’, D’, and γ’) with the input values (α, β, θ

_{d}, D, and γ) subject to the assumption of errors in the range −0.5% ~ + 0.5% in the values of the output Stokes parameters. From inspection, the error bars of parameters in α, β, θ

_{d}, D, and γ are found to have values of ±0.002°, ±0.091°, ±0.893°, ±0.002, and ±0.040°, respectively. Thus, it is inferred that the analytical model is robust toward experimental errors in the output Stokes parameters when applied to an optical fiber in this study.

10. 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 **17**(18), 15860–15884 (2009). [CrossRef] [PubMed]

## 4. Experimental setup and results for measuring effective parameters of optical fiber

*S*since the terms of

_{C}/S_{0}*m*,

_{12}*m*, and

_{13}*m*are not zero in Eq. (4). Therefore, there is no need to ensure that the six input lights have an identical optical intensity before passing through the sample. However, if the sample has a diattenuation property, the output Stokes parameters cannot be normalized, and thus additional steps must be taken to ensure that each of the input lights has an identical intensity.

_{14}*, and D were determined to be 0.98°, 0.95°, 2.12°, and 0.01°, respectively. It is noted that the sampling rate of polarimeter used in the present experiments of measuring the output Stokes parameters is 30 samples per second. Therefore, one hundred data points are used to calculate the standard deviation and the average data in experiments.*

_{d}*, D, and γ measurements were found to be 1.37°, 1.01°, 13.76°, 0.02, and 1.80°, respectively, based on a total of one hundred data points of measuring the output Stokes parameters in experiments. From inspection, the standard deviations of five effective optical parameters in experiment are one order larger than their error bars analyzed in Sub-section 3.2. Obviously, it is explained that only an assumption of errors ranging from −0.5% ~ +0.5% in the values of the output Stokes parameters from a commercial polarimeter is considered in an analytical model. The other experimental errors caused by the factors in misalignments, non-perfect optical components, and variations of light intensity in the whole system are also possibly occurred.*

_{d}## 5. Extraction of LB sample parameters using fiber-type polarimeter and genetic algorithm

### 5.1 Experimental setup for measuring parameters of LB sample using fiber-type polarimeter and genetic algorithm

*), the diattenuation (D), and the optical rotation angle (γ), the output Stokes vector*

_{d}*Q*in Fig. 9 has the form

_{c}*], [M*

_{ld}*], and [M*

_{lb}*] are the effective Mueller matrices of the optical fiber and [M*

_{cb}*] is the effective Mueller matrix of the LB sample. It is noted that the case where the optical fiber is assumed to have just LB and LD parameters can be obtained simply by replacing the Mueller matrix used to simulate the optical rotation of the fiber, i.e. [M*

_{lbS}*], by a unit matrix. In other words, Eq. (37) is equally applicable to the case in which the optical fiber is assumed to have either four effective optical parameters (i.e. α, β, θ*

_{cb}*, and*

_{d}*D*) or five effective optical parameters (i.e. α, β, θ

*, D, and γ). In Eq. (37), the Mueller matrix of the LB sample, [M*

_{d}*], is expressed as*

_{lbS}*and β*

_{S}*are the principal axis angle and retardance of the LB sample, respectively. As described in the following section, the values of α*

_{S}*and β*

_{S}*of the LB sample can be determined by substituting the effective optical parameters of the optical fiber and the three sets of Stokes parameters associated with the input polarization lights (i.e.*

_{S}*Q*,

_{0°}*Q*, and

_{45°}*Q*) into a GA.

_{RHC}### 5.2 Genetic algorithm (GA) for extracting sample parameters

13. 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. Light. Tech. **25**, (2007). [CrossRef]

*P*and

_{1}*P*are the parents,

_{2}*P*and

_{1}’*P*are the offspring, and δ is a uniformly distributed random variable with a small real value.

_{2}’*P”*is the new offspring after mutation and random_value is a small uniformly distributed random variable applied to the original solution.

*ϕ*and the experimental values. The experimental setup shown in Fig. 9 yields three sets of Stokes parameters for each input polarization light, i.e.

*Q*(

_{0°}*S*),

_{1n}*Q*(

_{0°}*S*),

_{2n}*Q*(

_{0°}*S*),

_{3n}*Q*(

_{45°}*S*),

_{1n}*Q*(

_{45°}*S*),

_{2n}*Q*(

_{45°}*S*),

_{3n}*Q*(

_{RHC}*S*),

_{1n}*Q*(

_{RHC}*S*), and

_{2n}*Q*(

_{RHC}*S*), respectively, where the subscript

_{3n}*n*indicates that the Stokes parameters are normalized. Thus, the error function can be defined as [12]where

*ϕ*

_{i}_{,Experiment}represents the experimental values of the nine Stokes parameters associated with the three sets output Stokes parameters and

*ϕ*

_{i}_{,Compute}represents the corresponding values of the Stokes parameters computed using the Mueller matrix method based upon the estimated values of the sample parameters. In other words, the objective of the GA is to determine the LB parameters of the optical sample (α

*, β*

_{S}*) which minimize the distance between the computed values of the Stokes parameters and the experimental values.*

_{S}*, β*

_{S}*). The Mueller matrix method is then used to calculate the corresponding set of Stokes parameters*

_{S}*ϕ*(α

*, β*

_{S}*) for the three input lights. The calculated values of the Stokes parameters are then substituted into Eq. (42) to determine the corresponding error value. If the distance between the experimental Stokes parameter values (*

_{S}*ϕ*

_{i}_{,Experiment}) and the computed values (

*ϕ*

_{i}_{,Compute}) falls within an allowable range, the optimization program is terminated. However, if the error value exceeds the permissible range, the selection, crossover and mutation operators are applied to generate a new population of candidate solutions. This process is repeated iteratively until the error distance falls within the acceptable range and the specified number of iteration loops has been completed. In performing the present simulations, the maximum number of iteration loops was set as 500, and the initial range of the candidate solutions for α

_{S}and β

_{S}were specified in the ranges 0° ≤ α

_{S}≤ 180° and 0° ≤ β

_{S}≤ 180°, respectively. It is noted that different numbers of iteration loops for testing the convergence of GA are investigated in the simulation program. The error value would converge after about 400~500 iteration loops. Therefore the loop number 500 was chosen in the proposed method.

### 5.3 Experimental results for LB sample parameters based on four effective parameters of optical fiber

*, and*

_{d}*D*. Having determined the four effective parameters of the optical fiber (the extracted parameters in Table 1), the experimental setup shown in Fig. 9 was used to extract the parameters of an LB sample using the GA optimization procedure. Note that the polarizer, quarter-wave plate, neutral density filter and power meter detector were the same as those used to measure the effective properties of the optical fiber.

*Q*,

_{0°}*Q*, and

_{45°}*Q*) were inserted into the GA in order to extract the principal axis angle and retardance of the LB sample. Figure 11 shows the corresponding results for the case in which the slow axis of the sample was set to various positions (i.e. 0°, 30°, 60°, 90°, 120°, 150°, or 180°) during the measurement process using a precision rotary stage. As shown, a relatively poor agreement is obtained between the extracted values of the LB sample parameters and the known values. Specifically, the experimentally-derived values of the principal axis parameter have a linear correlation with the true values, but are shifted slightly compared to the actual values, while the experimental values of the retardance have a significant non-linear correlation with the true values. The standard deviations of the measured α

_{RHC}*and β*

_{S}*values were found to be 10.49° and 6.04°, respectively. In other words, the results suggest that the use of just four effective parameters is insufficient to fully characterize the optical performance of the fiber used in the polarimeter shown in Fig. 11.*

_{S}### 5.4 Experimental results for LB sample parameters based on five effective parameters of optical fiber

*and D (LD), and γ (CB). The five effective optical parameters of the fiber (the extracted parameters in Table 2) and the three sets of normalized Stokes parameters (*

_{d}*Q*,

_{0°}*Q*, and

_{45°}*Q*) obtained using the setup shown in Fig. 9 were inserted into the GA in order to extract the principal axis angle and retardance of the LB sample.

_{RHC}*and β*

_{S}*measurements were determined to be 2.55° and 0.69°, respectively.*

_{S}*and β*

_{S}*measurements shown in Fig. 12 are notably smaller than those of the measurements shown in Fig. 11, in which the CB properties of the optical fiber are ignored. Thus, the results imply that in utilizing the GA optimization method to extract the sample parameters, the Mueller matrix of the optical fiber used to calculate the Stokes parameters should be based on all five effective optical parameters, namely the principal axis angle (α), the retardance (β), the diattenuation axis angle (θ*

_{S}_{d}), the diattenuation (D), and the optical rotation (γ). Moreover, the experimental results of proposed method are compared with the previous experimental results [10] which were extracted without using an optical fiber probe. Obviously, the standard deviations of α

*and β*

_{S}*in the proposed method are larger than those in [10], and it is because an optical fiber probe induces additional experimental errors. Also, the GA optimization method results in intrinsic errors in calculation.*

_{S}## 6. Conclusions

*= 13.76°, D = 0.02, and γ = 1.80°, respectively.*

_{d}## Acknowledgements

## References and links

1. | G. F. Smith, |

2. | D. B. Chenault and R. A. Chipman, “Measurements of linear diattenuation and linear retardation spectra with a rotating sample spectropolarimeter,” Appl. Opt. |

3. | D. B. Chenault and R. A. Chipman, “Infrared birefringence spectra for cadmium-sulfide and cadmium selenide,” Opt. Lett. |

4. | M. J. Fasolka, L. S. Goldner, J. Hwang, A. M. Urbas, P. DeRege, T. Swager, and E. L. Thomas, “Measuring local optical properties: near-field polarimetry of photonic block copolymer morphology,” Phys. Rev. Lett. |

5. | L. S. Goldner, M. J. Fasolka, S. Nougier, H. P. Nguyen, G. W. Bryant, J. Hwang, K. D. Weston, K. L. Beers, A. Urbas, and E. L. Thomas, “Fourier analysis near-field polarimetry for measurement of local optical properties of thin films,” Appl. Opt. |

6. | L. S. Goldner, M. J. Fasolka, and S. N. Goldie, “Measurement of the local diattenuation and retardance of thin polymer films using near-field polarimetry,” Appl. Scanned Probe Microscopy Polymers |

7. | A. L. Campillo and J. W. P. Hsu, “Near-field scanning optical microscope studies of the anisotropic stress variations in patterned SiN membranes,” J. Appl. Phys. |

8. | D. B. Chenault, R. A. Chipman, and S. Y. Lu, “Electro-optic coefficient spectrum of cadmium telluride,” Appl. Opt. |

9. | E. A. Sornsin and R. A. Chipman, “Visible Mueller matrix spectropolarimetry,” SPIE |

10. | 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 |

11. | I. C. Khoo, and F. Simoni, |

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

13. | 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. Light. Tech. |

14. | W. L. Lin, T. C. Yu, Y. L. Lo, and J. F. Lin, “A hybrid approach for measuring the parameters of twisted-nematic liquid crystal cells utilizing the stokes parameter method and a genetic algorithm,” J. Light. Tech. |

15. | Z. Michalewicz, |

**OCIS Codes**

(060.2270) Fiber optics and optical communications : Fiber characterization

(060.2300) Fiber optics and optical communications : Fiber measurements

(120.0120) Instrumentation, measurement, and metrology : Instrumentation, measurement, and metrology

(120.5410) Instrumentation, measurement, and metrology : Polarimetry

(160.1190) Materials : Anisotropic optical materials

(160.4760) Materials : Optical properties

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: January 20, 2010

Revised Manuscript: March 29, 2010

Manuscript Accepted: April 6, 2010

Published: April 16, 2010

**Citation**

Yu-Lung Lo, Thi-Thu-Hien Pham, and Po-Chun Chen, "Characterization on five effective parameters of anisotropic optical material using Stokes parameters—Demonstration by a fiber-type polarimeter," Opt. Express **18**, 9133-9150 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-9-9133

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### References

- G. F. Smith, Constitutive equations for anisotropic and isotropic materials, (North-Holland, 1994).
- D. B. Chenault and R. A. Chipman, “Measurements of linear diattenuation and linear retardation spectra with a rotating sample spectropolarimeter,” Appl. Opt. 32(19), 3513–3519 (1993). [CrossRef] [PubMed]
- D. B. Chenault and R. A. Chipman, “Infrared birefringence spectra for cadmium-sulfide and cadmium selenide,” Opt. Lett. 17, 4223–4227 (1992).
- M. J. Fasolka, L. S. Goldner, J. Hwang, A. M. Urbas, P. DeRege, T. Swager, and E. L. Thomas, “Measuring local optical properties: near-field polarimetry of photonic block copolymer morphology,” Phys. Rev. Lett. 90(1), 016107 (2003). [CrossRef] [PubMed]
- L. S. Goldner, M. J. Fasolka, S. Nougier, H. P. Nguyen, G. W. Bryant, J. Hwang, K. D. Weston, K. L. Beers, A. Urbas, and E. L. Thomas, “Fourier analysis near-field polarimetry for measurement of local optical properties of thin films,” Appl. Opt. 42(19), 3864–3881 (2003). [CrossRef] [PubMed]
- L. S. Goldner, M. J. Fasolka, and S. N. Goldie, “Measurement of the local diattenuation and retardance of thin polymer films using near-field polarimetry,” Appl. Scanned Probe Microscopy Polymers 897, 65–84 (2005). [CrossRef]
- A. L. Campillo and J. W. P. Hsu, “Near-field scanning optical microscope studies of the anisotropic stress variations in patterned SiN membranes,” J. Appl. Phys. 91(2), 646–651 (2002). [CrossRef]
- D. B. Chenault, R. A. Chipman, and S. Y. Lu, “Electro-optic coefficient spectrum of cadmium telluride,” Appl. Opt. 33(31), 7382–7389 (1994). [CrossRef] [PubMed]
- E. A. Sornsin and R. A. Chipman, “Visible Mueller matrix spectropolarimetry,” SPIE 3121, 156–160 (1997). [CrossRef]
- 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 17(18), 15860–15884 (2009). [CrossRef] [PubMed]
- I. C. Khoo, and F. Simoni, Physics of Liquid Crystalline Materials, (Gorden and Breach Science Publishers, 1991), Chap. 13.
- H.C. Cheng and Y. L. Lo, “The synthesis of multiple parameters of arbitrary FBGs via a genetic algorithm and two thermally modulated intensity spectra,” J. Light. Tech. 23, (2005).
- 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. Light. Tech. 25, (2007). [CrossRef]
- W. L. Lin, T. C. Yu, Y. L. Lo, and J. F. Lin, “A hybrid approach for measuring the parameters of twisted-nematic liquid crystal cells utilizing the stokes parameter method and a genetic algorithm,” J. Light. Tech. 27, (2009).
- Z. Michalewicz, Genetic Algorithm+ Data Structure = Evolution Programs, (Springer-Verlag, New York, 1994).

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