## 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 Optical Society of America

## 1. Introduction

Anisotropic optical materials are used for a diverse range of applications [1], and thus many methods have been proposed for measuring their fundamental optical properties including the linear birefringence (LB), circular birefringence (CB), linear diattenuation (LD), and circular diattenuation (CD). In [2,3], a technique to measure linear diattenuation and retardance spectra of infrared materials in transmission was proposed. The intensity modulation that resulted from the rotation of the sample was Fourier analyzed, and the linear diattenuation and linear retardance of the sample were calculated from the Fourier series coefficients for each wavelength. However, in extracting the sample parameters, an assumption was made that the principal birefringence and diattenuation axes were aligned.

Several methodologies have been proposed for measuring the linear birefringence and diattenuation properties of optical samples using near-field scanning optical microscopy (NSOM) with a near-field optical-fiber probe. Fasolka *et al.* [4] and Goldner *et al.* [5, 6] used a photo-elastic modulator (PEM) and a Fourier analysis scheme to measure the principal axis angle, retardance, diattenuation axis angle, and diattenuation of the local optical properties of photonic block copolymers and polystyrene-b-polyisoprene block copolymers, respectively. Also, Campillo and Hsu [7] measured the birefringence and diattenuation properties of SiN membranes using a NSOM technique in which the polarization of the input light was adjusted using a PEM and the detected signal was processed using a Fourier analysis scheme. To minimize polarization effects due to asymmetries of the optical fiber probe, the extinction ratios were measured for two orthogonal polarizations and only optical fiber probes with minimal difference in extinction ratios were used for this experiment. Also, a fiber polarization controller with the three paddles allows any retardance added by the fiber to be cancelled out [7]. It is found that without knowing properties of a near-field optical fiber probe and compensating for its polarization effects, the signal will be easily contaminated. However, the algorithms developed in [4–7] were overly complex for practical use.

Chenault *et al.* [8] proposed one method using an infrared Mueller matrix spectropolarimeter to measure a retardance spectrum for the electro-optic coefficient of cadmium telluride. Retardance spectra were calculated from Mueller matrix spectra, and then the electro-optic coefficient was calculated at each wavelength by a least-square fit to the resulting retardance as a function of voltage. Also, a Mueller matrix spectropolarimeter was used to measure an achromatic retarder in transmission, a reflective beamsplitter, and the electro-optic dispersion of a spatial light modulator by Sornsin and Chipman [9]. In a recent study, Chen *et al.* [10] proposed a technique for measuring the LB and LD of an optical sample using a polarimeter based on the Mueller matrix formulation and the Stokes parameters. Unlike the existing methods introduced above, the LB and LD parameters were decoupled within the analytical model. However, certain anisotropic materials have not only the LB and LD, but also CB.

The present study proposes a method based on the Mueller matrix formulation and the Stokes parameters for determining the effective optical parameters of an anisotropic material on the LB, LD, and CB. The feasibility of the proposed method is demonstrated by extracting the five effective optical parameters of an optical fiber. Also, the feasibility of the proposed low-birefringence fiber-based polarimeter is demonstrated by measuring the LB property of a quarter-wave plate.

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

This section proposes an analytical method for determining the effective LB, LD and CB properties of an anisotropic optical material utilizing the Mueller matrix formulation and the Stokes parameters. In developing the optically equivalent model of the anisotropic material, the CB component of the sample is assumed to be in front of the LB and LD components. According to [11], the Mueller matrix for a LB material such as a waveplate or retarder with a slow axis principal angle α and a retardance β can be expressed in Eq. (1)

_{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}The output Stokes vector *S _{c}* in Fig. 1 can be calculated as

*],[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*

_{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. ${\widehat{S}}_{{0}^{0}}={[\begin{array}{cccc}1,& 1,& 0,& 0\end{array}]}^{T}$,${\widehat{S}}_{{45}^{0}}={[\begin{array}{cccc}1,& 0,& 1,& 0\end{array}]}^{T}$, ${\widehat{S}}_{{90}^{0}}={[\begin{array}{cccc}1,& -1,& 0,& 0\end{array}]}^{T}$, and ${\widehat{S}}_{{135}^{0}}={[\begin{array}{cccc}1,& 0,& -1,& 0\end{array}]}^{T}$) and two circular polarization lights (i.e. right-handed ${\widehat{S}}_{RHC}={[\begin{array}{cccc}1,& 0,& 0,& 1\end{array}]}^{T}$ and left-handed ${\widehat{S}}_{LHC}={[\begin{array}{cccc}1,& 0,& 0,& 1\end{array}]}^{T}$). Therefore, the corresponding output Stokes vectors can be obtained from Eq. (4) as*

_{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) 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

In this section, the ability of the proposed analytical model to extract the five effective optical parameters of interest over the measurement ranges defined in the previous section is verified using a simulation technique. Thereafter, simulations are performed to evaluate the accuracy of the results obtained from the proposed method for composite samples with varying degrees of linear/circular birefringence and linear diattenuation given an assumption of errors ranging from −0.5% ~ +0.5% in the values of the output Stokes parameters. For the sake of simplicity, it should be noted that the error simulated in this study is according to the precision of a commercial polarimeter used in the whole experimental system.

#### 3.1 Analytical simulations

In performing the simulations, the theoretical values of the output Stokes parameters for the six input lights, namely S_{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.

In evaluating the ability of the proposed method to extract the principal axis angle of an anisotropic sample, the input parameters were specified as follows: phase retardance β = 60°, diattenuation axis angle θ_{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.

In assessing the ability of the proposed method to extract the phase retardance of the anisotropic material, the principal axis angle was specified as α = 50°, the diattenuation axis angle was given as θ_{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°.

In extracting the diattenuation axis angle, θ_{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°.

In extracting the diattenuation D of the anisotropic sample, the principal axis angle was specified as α = 50°, the phase retardance was given as β = 60°, the diattenuation axis angle was set as θ_{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.

Finally, in extracting the optical rotation angle γ of the anisotropic sample, the principal axis angle was specified as α = 50°, the phase retardance was given as β = 60°, the diattenuation axis angle was set as θ_{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.

Overall, the results presented in Figs. 2 - 6 demonstrate that the proposed analytical method yields full range measurements of all the optical parameters of interest other than the phase retardance, which is limited to the range 0 ~ 180°. In other words, in contrast to the method presented in [10] in which only the LB and LD parameters can be obtained, the method proposed in this study enables both the LB / LD parameters and the CB properties to be obtained.

In the following section, the analytical model introduced in Section 2 is used to extract effective parameters in LB, LD and CB of an optical fiber. Then, the parameters of an optical sample are extracted successfully using a fiber-type polarimeter and a genetic algorithm for demonstration

#### 3.2 Error analysis of proposed measurement methodology

This section evaluates the robustness of the proposed analytical model by using the Jones matrix formulation to derive the theoretical output Stokes parameters S_{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.

In deriving the theoretical values of the output Stokes parameters, the effective properties of the optical sample (i.e. an optical fiber in this study) were assigned as follows: the principal axis angle α = 123.780°, retardance β = 22.554°, diattenuation axis angle θ_{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.

Overall, the results presented in Fig. 7(a) - (e) demonstrates the robustness of the proposed analytical model to errors in the output Stokes parameters for optical samples such as an optical fiber used in this study. It is noted that the LB/CB birefringence properties can be extracted with a high degree of precision, but the values obtained for the diattenuation axis angle and diattenuation, respectively, are highly sensitive to experimental errors, and it has similarities in [10].

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

Figure 8
presents a schematic illustration of the experimental setup proposed in this study for characterizing the LB, LD and CB properties of an optically anisotropic material such as an optical fiber. Initially, the LB and LD properties of a single-mode optical fiber (630HP, Thorlabs Co.) with a length of 47 cm were measured using a commercial Stokes polarimeter (PAX5710, Thorlabs Co.). The transmitted light is provided by a frequency-stable He-Ne laser (SL 02/2, SIOS Co.) with a central wavelength of 632.8 nm. Note that in Fig. 8, P is a polarizer (GTH5M, Thorlabs Co.) and Q is a quarter-wave plate (QWP0-633-04-4-R10, CVI Co.), which are used to produce linear polarization lights using 0°, 45°, 90°, 135°, right handed circular, and left handed circular polarization light. As shown, the polarized light is coupled into an optical fiber by a fiber coupler. Note that the neutral density filter (NDC-100-2, ONSET Co.) and power meter detector (8842A, OPHIT Co.) shown in Fig. 8 are used to ensure that each of the input polarization lights has an identical intensity. For a sample with no diattenuation, the output Stokes parameters can be normalized by *S _{C} /S_{0}* since the terms of

*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}Here, the experiments are conducted in two different cases of four and five effective parameters being assumed in an optical fiber. Tables 1 and 2 illustrate the extracted four and five effective optical parameters of the optical fiber, respectively. By doing this way, it is understood that which assumption on the four or five parameters can be effectively characterized in the optical fiber, and this can be demonstrated in the following Section 5.

Table 1 shows the values of the four effective optical parameters of the optical fiber extracted using Eqs. (21), (23), (27), and (28) in Section 2. (Note that γ is set equal to 0 since the optical fiber is assumed to have LB and LD parameters only). In this case, the optical fiber was illuminated by five input lights (i.e. four linear polarization lights on 0°, 45°, 90°, 135° and one right-hand circular polarization lights, respectively). The standard deviations of the measurements of α, β, θ* _{d}*, 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.

Table 2 shows the values of the five effective optical parameters of the optical fiber extracted using Eqs. (35), (23), (27), (28), and (36) in Section 2, respectively. In this case, it is noted that the optical fiber was illuminated by six input lights (i.e. four linear polarization lights on 0°, 45°, 90°, 135° and two right- and left-hand circular polarization lights, respectively) rather than five. The standard deviations of the α, β, θ* _{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.

## 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

Figure 9 presents a schematic diagram of the setup used to measure the parameters of a LB sample using a fiber-type polarimeter, a genetic algorithm (GA) and the Stokes method. As shown, the quarter-wave plate produces RHC polarization light, while the polarizer produces linear polarization light with an orientation of 0° or 45° to the horizontal plane. The polarized light is passed through a neutral density filter and power meter detector and is then coupled into an optical fiber. The light emerging from the fiber is passed through the LB sample and is then coupled into a Stokes polarimeter. During the experimental procedure, the slow axis of the LB sample is rotated to 0°, 30°, 60°, 90°, 120°, 150° or 180°, and the corresponding values of the principal axis angle and phase retardance are extracted using a GA. For comparison purposes, the parameters of the LB sample are extracted using two different treatments of the optical fiber parameters, namely (1) the optical fiber is assumed to have LB and LD parameters (the extracted parameters in Table 1) only, and (2) the optical fiber is assumed to have both LB / LD and CB parameters (the extracted parameter in Table 2).

Given the assumption that the optical fiber has five effective optical parameters, i.e. the principal axis angle (α), the retardance (β), the diattenuation axis angle (θ* _{d}*), the diattenuation (D), and the optical rotation angle (γ), the output Stokes vector

*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

A genetic algorithm (GA) is a search technique used in computing to find exact or approximate solutions to optimization and search problems. For example, Cheng and Lo [12] described the use of a genetic algorithm and two thermally modulated fiber Bragg grating reflection intensity spectra to perform the inverse extraction of multiple physical parameters of arbitrary FBGs, including the grating period, grating position, grating length, chirped direction, and refractive-index modulation. Yu and Lo [13] presented a new heterodyne polarimeter for measuring the multiple parameters of a twisted nematic liquid crystal (TNLC) by applying a genetic algorithm. Lin et al. [14] proposed an approach for measuring the cell thickness, twist angle, pretilt angle, and azimuth angle of twisted nematic liquid crystal cells utilizing the Stokes parameter method and a genetic algorithm.

In the GA used in the present study, the most suitable birefringence parameters of the LB sample have a greater probability of surviving to the next generation. Once superior solutions have been selected, a crossover operation is performed in which two solutions (parents) are selected at random and their bit information exchanged in order to generate two further solutions (offspring). In practice, the crossover operation is expected to improve the quality of the offspring generation (i.e. the quality of the birefringence parameter solutions) by progressively accumulating the exceptional bit information of the previous generations. In the present study, the crossover operation is performed using the real-valued crossover process [15], i.e.

*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}’In addition to the crossover operation, the GA also utilizes a mutation process to modify the existing individuals (solutions) in order to introduce an additional variability within the population, thereby improving the quality of the final solution. In the mutation process, a species string is picked at random, a mutation point is randomly selected within the string, and the corresponding bit information is then changed. In the GA used in the present study, the mutation operation is performed using the following real-valued mutation process [15]

where*P”*is the new offspring after mutation and random_value is a small uniformly distributed random variable applied to the original solution.

In the present study, the quality of the optical parameters obtained using the GA is evaluated using an error function (fitness function) defined in terms of the distance between the computed values of the Stokes parameters *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]

*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}Figure 10
presents a flow chart showing the major steps in the GA optimization procedure. As shown, the GA commences by generating an initial population of random solutions (α* _{S}*, β

*). 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

The LB parameters of the sample (a quarter-wave plate: QWP0-633-04-4-R10, CVI Co.) shown in Fig. 9 were initially extracted using the GA optimization procedure described in the previous section subject to the assumption that the optical fiber was characterized by just four effective optical parameters, namely α, β, θ* _{d}*, and

*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.

The four effective optical parameters of the fiber and the three sets of normalized Stokes parameters (*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

For comparison purposes, the optical parameters of the LB sample were also extracted for the case in which the optical fiber was assumed to have five effective optical parameters, namely α and β (LB), θ* _{d}* 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 (

*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}Figure 12
plots the extracted values of the principal axis angle and retardance of the quarter-wave plate at various values of the known principal axis angle. It is observed that a good correlation exists between the experimental values of the sample parameters and the true values. From inspection, the standard deviations of the α* _{S}* and β

*measurements were determined to be 2.55° and 0.69°, respectively.*

_{S}The standard deviations of the α* _{S}* and β

*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

This study has proposed an analytical method based on the Mueller matrix method and the Stokes parameters for extracting the linear birefringence, linear diattenuation, and circular birefringence effective properties of an anisotropic optical material. The proposed method enables all of the effective parameters other than the phase retardance to be extracted over the full range. Moreover, the LB/CB and LD properties are decoupled within the analytical model. The experimental values obtained using the proposed method for the principal axis angle, retardance, diattenuation axis angle, diattenuation, and optical rotation angle of a single-mode fiber have standard deviations of α = 1.37°, β = 1.01°, θ* _{d}* = 13.76°, D = 0.02, and γ = 1.80°, respectively.

It has been shown that the analytical model can be integrated with a genetic algorithm to construct a scheme for extracting the optical parameters of birefringent samples using a fiber-type polarimeter. Without using a fiber polarization controller and selecting only optical fiber probes with minimal difference in extinction ratios of two orthogonal polarizations, in general, the results have confirmed that the proposed polarimeter design provides a straightforward and practical alternative to existing optical-fiber based probes for optical parameter measurement in the near field or the remote environment.

## Acknowledgements

The authors gratefully acknowledge the financial support provided to this study by the National Science Council of Taiwan under grant no. NSC96-2628-E-006-005-MY3.

## References and links

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