## Abstract

A technique is proposed for measuring the linear birefringence and linear diattenuation of an optical sample using a polarimeter. In the proposed approach, the principal axis angle (α), phase retardance (β), diattenuation axis angle (θ* _{d}*), and diattenuation (

*D*) are derived using an analytical model based on the Mueller matrix formulation and the Stokes parameters. The dynamic measurement ranges of the four parameters are shown to be α = 0~180°, β = 0~180°, θ

*= 0~180°, and*

_{d}*D*= 0~1, respectively. Thus, full-range measurements are possible for all parameters other than β. In this study, the proposed methodology does not require the principal birefringence axes and diattenuation axes to be aligned. In addition, the linear birefringence and linear diattenuation properties are decoupled within the analytical model, and thus the birefringence properties of the sample can be solved directly without any prior knowledge of the diattenuation parameters. Also, the characteristic parameters in the baked polarizer with linear birefringence are successfully extracted from an optically equivalent model and proved by the respective simulation and experiment introduced in this study.

© 2009 OSA

## 1. Introduction

Diattenuation is the property of an optical system whereby the intensity transmittance of the exiting beam depends on the polarization state of the incident beam [1–3]. It has been shown that the diattenuation of an optical sample can be calculated directly from the elements of the corresponding Mueller matrix; namely a matrix containing information about all of the polarization properties of the system [3]. In 1990, Chenault and Chipman [4] measured the diattenuation and birefringence of IR samples using a spectropolarimeter. A few years later, the same authors used a rotating sample spectropolarimeter to measure the birefringence spectra of three ferroelectric liquid crystals, namely 764E, SCE4 and SCE9, in the infrared wavelength region from 2.5 to 16.5 μm [5]. In 1993, the same group presented a technique for measuring the diattenuation and birefringence spectra of a sample rotated between two stationary linear polarizers in an IR-spectrometer. The feasibility of the proposed approach was demonstrated by calibrating a quasi-achromatic retarder consisting of two plates of different birefringent materials for incident wavelengths in the range 3 ~14 μm [6]. In [4–6], the optical parameters were solved using an analytical approach based on the first three terms of the Fourier-Bessel expansion of optical intensity. However, in [4–6], an assumption was made that the principal birefringence and diattenuation axes were aligned. Furthermore, the linear birefringence and linear diattenuation parameters were coupled within the corresponding analytical models. Bueno and Artal [7,8] used a double-pass polarimeter to investigate the ocular diattenuation phenomenon. In their approach, the information contained within 16 double-pass retinal images was processed using a Mueller matrix formulation, and it was shown that the human eye has a diattenuation of around 0.10 on average. In 2004, Todorovic *et al*. [9] investigated the diattenuation of biological samples using an optical coherence tomography (OCT) technique and a Mueller matrix formulation. In 2003, Huang and Knighton [10] measured the diattenuation spectrum of the retinal nerve fiber layer (RNFL) using a multispectral imaging micropolarimeter. However, in [7–10], they only measured the diattenuation of material.

In recent years, several methodologies have been presented for measuring the linear birefringence and diattenuation properties of optical samples using near-field scanning optical microscopy (NSOM) techniques. For example, Higgins *et al*. [11] presented a method for measuring the diattenuation of mesoscopic crystals in which an electro-optic modulator and a quarter-wave plate were used to modulate the polarization of the light coupled into the near-field optical-fiber probe and a lock-in detection scheme based on the amplitude and phase of the detected signal was used to extract the sample parameters. Campillo and Hsu [12] measured the birefringence and diattenuation of SiN membranes using a SNOM in which the polarization of the input light was adjusted using a photo-elastic modulator (PEM) and the detected signal was processed using a Fourier analysis scheme. Fasolka *et al*. [13] and Goldner *et al*. [14–16] adopted a similar approach to measure the local optical properties of photonic block copolymers and isotactic polystyrene crystallites, respectively. In contrast to previous methods, the schemes presented in [13–16] assumed that the principal axes of birefringence and diattenuation were not necessarily aligned in the optical sample. However, as in the methods proposed in [4–6], the linear birefringence and linear diattenuation parameters were coupled within the corresponding analytical models. In addition, solving the sample parameters with a Fourier-based scheme is intended to simplify the measurements and data analysis is a bit more involved in the Fourier-transform case.

A three-dimensional stressed photoelastic medium can be reduced to an optically equivalent model in [17,18]. The term “optical equivalence” is used in the sense that, for a given incident light ellipse, the exit-light ellipse (characterized by its ellipticity and azimuth of its axis) will be the same for the given stressed medium and the equivalent model [17]. It can be shown that such an optically equivalent model consists of a linear retarder and a pure rotator [19]. The evaluation of two characteristic directions and one characteristic retardation constitutes the determination of the optically equivalent model. One of the experimental methods such as an automated phase-stepping method in [18] has been developed to enable the accurate and full-field determination of the three characteristic parameters for three-dimensional or integrated photoelasticity. In addition, it is known that lossless optical systems consist of waveplates and polarization rotators are represented by unitary matrices, and that for every unitary optical media, there is an optically equivalent model consisting of one waveplate followed by one polarization rotator [19]. In particular, the general twisted nematic liquid crystal (LC) layer is a unitary system and therefore can be represented by the combination of an equivalent waveplate and polarization rotators. Such characteristic parameters can provide a framework for analyzing LC mode, as well as providing a means to measure LC cell parameters, namely the twist angle and LC cell retardation. For example, Tang and Kwok [20] derived the equations relating the two LC cell parameters to the three equivalent characteristic parameters, and a Stokes parameter method with simple operation was also introduced to measure the characteristic parameters. Once the characteristic parameters are known, the LC cell twist angle and retardation can be solved. Hence, to adopt the concept of an optically equivalent model is noteworthy and expected to be useful.

Accordingly, this study presents a straightforward method for measuring the local birefringence and diattenuation properties of an optical sample using a polarimeter and a Mueller matrix formulation. In the proposed approach, we use Stokes parameters and change the state of polarization of a light to measure the optical sample. It is shown that the proposed methodology enables full range measurements of the principal axis angle, diattenuation axis angle and diattenuation, respectively. In addition, the retardance of the optical sample can be measured over the range 0° ~180°. As in [13–16], the proposed approach assumes that the principal axes of birefringence and diattenuation are not aligned. Furthermore, the linear birefringence and linear diattenuation parameters are decoupled in the analytical model. Thus, the birefringence parameters can be solved directly without any prior knowledge of the diattenuation parameters. The concept of characteristic parameters will be used to prove the derived algorithm for measuring the local effective birefringence and diattenuation properties in tested samples such as a baked polarizer and a composite sample that includes a baked polarizer in series with a quarter-wave plate. Two cases are discussed when considering the diattenuation axis of a baked polarizer is aligned or not aligned to the principal axis angle of a quarter-wave plate.

## 2. Basic measuring system and methods

#### 2.1 Basic Stokes parameter method

The state of polarization of a light beam can be completely described by the Stokes vector$S={\{{S}_{0},{S}_{1},{S}_{2},{S}_{3}\}}^{T}$, in which the four components are given by

*S*is the total light intensity;

_{0}*S*is the intensity difference between horizontally and vertically polarized components;

_{1}*S*is the intensity difference between +45° and −45° polarized components; and

_{2}*S*is the intensity difference between right- and left-circularly polarized components. In general, a polarizer oriented horizontally (x), vertically (y), or at an angle of ±45° transmits light with intensities of

_{3}*I*or

_{x/y}*I*, respectively, while a quarter-wave plate followed by a polarizer oriented at ±45° with respect to the fast axis of the wave plate transmits circularly polarized light with an intensity of

_{45°/-45°}*I*.

_{rcp/lcp}Any optical component which changes the polarization state of a light ray can be modeled by a 4 x 4 Mueller matrix. The inner product of this Mueller matrix *M* and the Stokes vector $\widehat{S}$ representing the polarization state of the incident light yields a Stokes vector *S* which describes the polarization state of the light exiting the optical sample [21]. According to the Stokes-Mueller formulation, the Stokes vector of the output light can be derived as follows:

In this study, the samples of interest are assumed to have both linear birefringence and linear diattenuation properties. According to [22], the Mueller matrix for a linear birefringent material such as a wave plate or retarder can be expressed as

*is the diattenuation axis angle and*

_{d}*u*and

*v*are the transmittances of the optical sample for light rays polarized parallel and perpendicular to the diattenuation axis, respectively. Since

*u*is taken to be the larger of the two transmittance values, the diattenuation axis therefore represents the polarization direction with higher transmission [14]. The diattenuation,

*D*, of the sample is given by

Assume that the sample of interest comprises an optical material with linear birefringence properties positioned in front of a second optical material with linear diattenuation properties [14]. As a result, the composite sample has four parameters of interest, namely the principal axis angle (α), the phase retardance (β), the diattenuation axis angle (θ* _{d}*), and the diattenuation (

*D*). The output Stokes vector

*S*of the composite sample can be obtained as

_{c}*c*indicates different input polarization states, e.g.

*S*represents a 0° linear polarization state,

_{0°}*S*represents a 45° linear polarization state,

_{45°}*S*represents a right-hand circular polarization state, and so on. The Mueller matrix elements for the two optical materials within the composite sample are expressed as

_{RHC}As described in the following sections, given a knowledge of the input polarization state and the measured values of the output Stokes parameters, Eqs. (7) ~ (22) enable the four parameters of interest to be inversely derived.

#### 2.2 Determination of linear birefringence properties

In the method proposed in this study, the linear birefringence properties of the composite sample, i.e., α and β, are extracted from the output Stokes vectors corresponding to three different input polarization states, namely two linear polarization states, i.e., ${\widehat{S}}_{{0}^{\circ}}=\left[1,1,0,0\right]$ and ${\widehat{S}}_{{45}^{\circ}}=\left[1,\text{0},1,0\right]$, and one right-hand circular polarization state, i.e., ${\widehat{S}}_{RHC}=\left[1,\text{0},\text{0},1\right]$. The output Stokes vectors are obtained from Eq. (6) as

*RHC*denote the three input polarization states, respectively. From Eqs. (23) and (24), it follows that

Therefore, the principal axis angle of the linear birefringent material can be obtained as

It should be noted here that the transmittance parameters, *u* and *v*, have positive values and the range of the phase retardation is deliberately limited to β = 0 ~180° such that sin(β) also has a positive value in Eq. (26). Thus, the appropriate quadrant in Eq. (27) can be correctly determined and the corresponding value of α extracted. For example, if $-{S}_{{0}^{\circ}}\left({S}_{3}\right)>0$ and${S}_{{45}^{\circ}}\left({S}_{3}\right)>0$, then 2α is located in quadrant I. Similarly, if $-{S}_{{0}^{\circ}}\left({S}_{3}\right)>0$ and${S}_{{45}^{\circ}}\left({S}_{3}\right)<0$, then 2α is located in quadrant II. Likewise, if $-{S}_{{0}^{\circ}}\left({S}_{3}\right)<0$ and${S}_{{45}^{\circ}}\left({S}_{3}\right)<0$, then 2α is located in quadrant III. Finally, if $-{S}_{{0}^{\circ}}\left({S}_{3}\right)<0$ and${S}_{{45}^{\circ}}\left({S}_{3}\right)>0$, then 2α is located in quadrant IV. In other words, the measurable range of 2α extends from 0 ~360°, i.e., the range of α extends from 0 ~ 180°. Consequently, the proposed method enables the principal axis angle of the composite sample to be measured over the full range.

From Eqs. (24) and (25), it follows that

Therefore, the retardance can be obtained as

It should be noted here that while the proposed methodology yields a full-range measurement of the principal axis angle, the measurable range of β is limited to 0 ~ 180° in order to facilitate the implementation of Eq. (27).

#### 2.3 Determination of linear diattenuation properties

When determining the linear diattenuation properties of the composite sample, two additional linearly polarized input lights are chosen, namely ${\widehat{S}}_{{90}^{\circ}}=\left[1,\text{-}1,0,0\right]$and${\widehat{S}}_{{135}^{\circ}}=\left[1,\text{0},\text{-}1,0\right]$, respectively. The corresponding output Stokes vectors are expressed as

From Eqs. (23) and (30), it can be shown that

Meanwhile, from Eqs. (24) and (31), it follows that

Combining Eqs. (33) and (34) yields the following formulation for the diattenuation axis angle:

In Eqs. (33) and (34), *u* is bigger than *v*, and thus the (*u*-*v*) term has a positive value. Therefore, the quadrant determination method described in the previous section can also be applied to Eq. (35) to determine the correct value of θ* _{d}*. As with the principal axis angle measurement α, the measurement range of 2θ

*extends from 0 ~ 360°, and thus a full-range measurement of θ*

_{d}*, i.e., 0 ~ 180°, is obtained. Having determined the value of the diattenuation axis θ*

_{d}*, the diattenuation*

_{d}*D*can be obtained in one of two different ways. For example,

*D*can be obtained directly from Eqs. (32) and (33) as

Alternatively, *D* can be derived from Eqs. (32) and (34) as

Naturally, Eqs. (36) and (37) should provide the same solution. Thus, a comparison of the results obtained using these two equations provide a useful means of verifying the accuracy of the experimental measurements. On the other hand, the accuracy of extracted parameters also can be verified by the other equations derived from Muller matrix.

## 3. Analytical results and error analysis

This section commences by performing simulations to confirm the ability of the proposed analytical model to extract the four parameters of interest over the measurement ranges defined in the previous sections. Thereafter, simulations are performed to evaluate the accuracy of the results obtained from the proposed method for composite samples with varying degrees of linear 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

This sub-section utilizes an analytical approach to demonstrate the feasibility of the proposed measurement methodology. The theoretical values of the output Stokes parameters, namely *S _{0°}*,

*S*,

_{45°}*S*,

_{90°}*S*, and

_{135°}*S*, are derived using the Jones matrix formulation given known values of the sample parameters and a knowledge of the input Stokes vectors. The theoretical Stokes values are then inserted into the analytical model derived in Section 2 and the optical parameters are inversely derived. The extracted values of the optical parameters are then compared with the input values used in the Jones matrix formulation.

_{RHC}In evaluating the ability of the proposed method to extract the principal axis angle of the composite sample, the input parameters were specified as follows: phase retardance β = 60°, diattenuation axis angle θ* _{d}* = 35°, and transmittances

*u*= 0.81 and

*v*= 0.04 (

*D*= 0.9059). Figure 1 compares the value of the principal axis angle extracted using Eq. (27) with the input value over the range 0 ~ 180°. It is observed that a good agreement is obtained between the two values at all 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 composite sample, the principal axis angle was specified as α = 50°, the diattenuation axis angle was given as θ* _{d}* =35°, and the transmittances were assigned values of

*u*= 0.81 and

*v*= 0.04 (

*D*= 0.9059). Figure 2 compares the values of the phase retardance extracted using Eq. (29) with the input values 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°. In other words, as expected, the proposed method does not support a full-range measurement of the phase retardance parameter.

In extracting the diattenuation axis angle, θ* _{d}*, using Eq. (35), the parameters of the composite sample were specified as follows: principal axis angle α = 50°, retardance β = 60°, and transmittances

*u*= 0.81 and

*v*= 0.04 (

*D*= 0.9059). The results presented in Fig. 3 confirm the ability of the analytical model to obtain measurements of the diattenuation axis angle over the full range of 0 ~ 180°.

Finally, in extracting the diattenuation *D* of the composite sample, the principal axis angle was specified as α = 50°, the phase retardance was given as β = 60°, and the diattenuation axis angle was set as θ* _{d}* = 35°. Figure 4
plots the value of the diattenuation obtained from Eq. (36) with respect to the input value. A good agreement is observed between the two sets of results for all values of

*D*in the range 0 ~ 1. Consequently, the ability of the proposed method to obtain full-range measurements of the diattenuation is confirmed.

Overall, the results presented in Figs. 1 ~ 4 demonstrate that the proposed method yields full range measurements of all the linear birefringence / linear diattenuation parameters of interest other than the phase retardance, which is limited to the range 0 ~ 180°. Nonetheless, the proposed method enables the complete characterization of optical samples such as thin films, in which the phase retardance is limited to this restricted range due to their very thin thickness.

#### 3.2 Error analysis of proposed measurement methodology

To evaluate the robustness of the proposed analytical model, this section commences by using the Jones matrix formulation to derive the theoretical output Stokes parameters *S _{0°}*,

*S*,

_{45°}*S*,

_{90°}*S*, and

_{135°}*S*for a composite sample with known 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

_{RHC}*S*,

_{0°}*S*,

_{45°}*S*,

_{90°}*S*, and

_{135°}*S*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 birefringence and diattentuation properties of the composite sample. Finally, the extracted values of the birefringence and diattentuation are compared with the given values used in the Jones matrix formulation. The simulations consider three different optical materials, namely a material with proper birefringence and diattenuation, a material with slight diattenuation (e.g. a wave plate), and a material with slight birefringence (e.g. a polarizer).

_{RHC}### 3.2.1 Materials with proper birefringence and diattenuation

In deriving the theoretical values of the output Stokes parameters, the properties of the composite sample were assigned as follows: principal axis angle α = 50°, retardance β = 60°, diattenuation axis angle θ* _{d}* = 35°, and transmittances

*u*= 0.81 and

*v*= 0.04 (

*D*= 0.9059). The values of α, β, θ

*, and*

_{d}*D*were then extracted from Eqs. (27), (29), (35), and (36), respectively. Figure 5 compares the extracted values of the sample parameters (α’, β’, θ

*, and*

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

*, and*

_{d}*D*) 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 α, β, θ

*, and*

_{d}*D*are found to have values of ±0.0235°, ±0.1083°, ±0.0451°, and ±0.0039, respectively. Thus, it is inferred that the analytical model is robust toward experimental errors in the output Stokes parameters when applied to optical samples with relatively high values of retardance (β = 60°) and diattentuation (

*D*= 0.9059), respectively.

### 3.2.2 Materials with slight diattenuation

To verify the performance of the proposed analytical model in measuring the properties of optical components with a low degree of diattenuation such as wave plates, the sample parameters were specified as follows: principal axis angle α = 50°, retardance β = 60°, diattenuation axis angle θ* _{d}* = 35°, and transmittances

*u*= 0.99 and

*v*= 0.986 (

*D*= 0.0020). Figure 6 compares the extracted values of the four parameters with the corresponding input values for assumed errors in the output Stokes parameters of −0.5% ~ +0.5%. The error bars for α, β, θ

*, and*

_{d}*D*are found to be ±0.0231°, ±0.1119°, −35° ~ 145°, and ±0.0024, respectively. Thus, it is evident that for samples with low diattenuation, the results obtained for the diattenuation axis angle and diattenuation, respectively, are highly sensitive to errors in the output Stokes parameters. However, it can be seen that the extracted values of the birefringence properties deviate only slightly from the input values. In other words, the de-coupling of the birefringence and diattentuation parameters in the analytical model is beneficial in maintaining the accuracy of the birefringence results.

### 3.2.3 Materials with low birefringence

Optical components such as polarizers are characterized by a low degree of phase retardance. In evaluating the performance of the proposed analytical model in extracting the birefringence and diattenuation parameters of such components, the principal axis angle was specified as α = 50°, the retardance was assigned a value of β = 0.0001°, the diattenuation axis angle was given as θ* _{d}* = 35°, and the transmittances were set as

*u*= 0.81 and

*v*= 0.04, respectively (D = 0.9059). Figure 7 compares the extracted values of the polarizer parameters with the input values given assumed errors of −0.5% ~ +0.5% in the experimentally-derived values of the output Stokes parameters. From inspection, the error bars are determined to have values of just α= ±0.0240°, β= ±4.7736×10

^{−7}°, θ

*= ±0.0517°, and*

_{d}*D*= ±0.0035. Therefore, the ability of the proposed method to extract the optical parameters of samples with a low degree of birefringence is confirmed.

Overall, the results presented in Figs. 5 ~7 demonstrate the robustness of the proposed analytical model to errors in the output Stokes parameters for all optical samples other than those with a low degree of diattentuation such as wave plates. For such samples, the 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.

## 4. Experimental measurements and results

#### 4.1 Experimental setup

Figure 8 presents a schematic illustration of the experimental setup used in this study to evaluate the practical feasibility of the proposed measurement method. Note that in this figure, P is a polarizer (GTH5M, Thorlabs Co.) and Q is a quarter-wave plate (QWP0-633-04-4-R10, CVI Co.). As shown, the polarizer and quarter-wave plate were used to produce linearly polarized light orientated at 0°, 45°, 90°, and 135° to the horizontal plane, respectively, and right-handed circularly polarized light. In addition, the slow axis or the diattenuation axis of the sample was set to various positions (i.e., 0°, 30°, 60°, 90°, 120°, or 150°) during the measurement process using a rotary stage. In the experiments, the illuminating light was provided by a frequency-stable He-Ne laser (SL 02/2, SIOS Co.) with a central wavelength of 632.8 nm. Finally, the output Stokes parameters were determined in accordance with the intensity measurements obtained using a commercial Stokes polarimeter (PAX5710, Thorlabs Co.).

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. (6). Therefore, there is no need to ensure that the five input lights (i.e., four linearly polarized lights and one right-hand circularly polarized light) 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 five input lights has an identical intensity. In the present experiments, this is achieved by using a Neutral Density Filter (NDC-100C-2, ONSET Co.) and a power meter detector (8842A, OPHIT Co.) positioned between the polarizer and the sample (see Fig. 8).

_{14}The Stokes polarimeter shown in Fig. 8 yields four intensity measurements, namely the total intensity *S _{0}*,

*S*,

_{1n}*S*, and

_{2n}*S*, respectively, where the subscript

_{3n}*n*denotes normalization. However, the analytical equations used to solve the diattenuation axis angle and diattenuation, i.e. Equations (35) and (36), respectively, use the

*S*term to solve the optical parameters, and thus the normalized values of the Stokes parameters obtained from the Stokes polarimeter cannot be used. To resolve this problem, the total intensity

_{0}*S*is multiplied to un-normalized the Stokes parameters prior to their substitution into the analytical model. The sampling rate of polarimeter used in the present experiments of measuring the output Stokes parameters is 30 samples per second. Therefore, we used one hundred data points to calculate the standard deviation and the average data in experiments.

_{0}The validity of the proposed measurement method was evaluated using four different optical samples, namely a quarter-wave plate (QWP0-633-04-4-R10, CVI Co.), a polarizer (GTH5M, Thorlabs Co.), a second polarizer (LLC2-82-18S, OPTIMAX Co.) baked in an oven at a temperature of 150°C for 80 minutes, and a composite sample comprising of the quarter-wave plate and the baked polarizer, respectively. The quarter-wave plate and polarizer were chosen specifically to evaluate the performance of the proposed method in measuring the parameters of samples with low diattenuation and low birefringence, respectively. It is noted that the polarizers used in the tested samples for baking are polymer polarizers. Meanwhile, the baked polarizer and composite sample were chosen to evaluate the performance of the proposed measurement system in measuring the optical parameters of samples with a leaky diattenuation axis and with both birefringence and diattenuation properties, respectively.

#### 4.2 Experimental results

### 4.2.1 Quarter-wave plate as a sample

As a result, Figs. 9 and 10 illustrate the experimental results obtained for the linear birefringence and linear diattenuation properties of the quarter-wave plate. In Fig. 9, the average standard deviations of the principal axis angle and phase retardance are found to be 0.03° and 0.04°, respectively. In Fig. 10, it is observed that the extracted value of the diattenutation axis angle varies non-linearly with changes in the slow axis angle of the quarter-wave plate. Moreover, it can be seen that the value of the extracted diattentuation fluctuates notably with changes in the rotational position of the sample. Therefore, as discussed in Sub-section 3.2.2, the proposed method fails to obtain reliable measurements of the diattenuation parameters of samples with low diattenuation.

### 4.2.2 Polarizer as a sample

Figures 11
and 12
present the experimental results obtained for the birefringence and diattentuation parameters of the polarizer. As expected, Fig. 12 shows that the diattenuation of the polarizer has a value equal to approximately 1. Moreover, the average standard deviations of θ* _{d}* and

*D*are found to be around 0.03° and 5.38×10

^{−5}, respectively. In contrast to the simulation results presented in Sub-section 3.2.3, it can be seen that the proposed measurement system performs poorly in extracting the birefringence properties of the optical sample. In the analytical model proposed in this study, the principal axis angle and phase retardance are both determined using Stokes parameter

*S*(see Eqs. (27) and (29), respectively). According to Eq. (1), this parameter should have a value of zero for light emitted from a polarizer. However, in practice, the experimental value of

_{3}*S*determined by the Stokes polarimeter may have an error of as much as 0.5% due to the presence of noise. Furthermore, when the values of

_{3}*S*associated with the different input polarization states are inserted into Eqs. (27) and (29), respectively, this error is amplified via the effect of the arctangent function. Consequently, the extracted values of both birefringence parameters deviate significantly from the expected values for optical samples with low birefringence.

_{3}### 4.2.3 Baked polarizer as a sample

Figures 13
and 14
illustrate the experimental results obtained for the effective linear birefringence and linear diattenuation properties of the baked polarizer. The average measured values of the four optical parameters of baked polarizer with different diattenuation axis angles from 0° to 180° in increments of 30° are summarized in Table 1
. As expected, the diattenuation has a value of less than 1 since the prolonged exposure of the polarizer to a high-temperature environment causes the optical intensity of the input light to leak through one of the diattenuation axes. The average value of *D* is determined to be 0.952, and the measured *D* correlated with different diattenuation axis angles are shown in Fig. 14. Figure 13 shows that the baked polarizer as mentioned before causes the polarizer to develop distinct birefringence properties. However, it is observed that the retardance has a non-isotropic characteristic. Interestingly, it is also seen that the principal axis angle and the diattenuation axis angle do not coincide exactly with one another. From inspection, the average standard deviations of the experimental measurements of α, β, θ* _{d}*, and

*D*are found to be around 0.03°, 0.03°, 0.01°, and 4.16×10

^{−4}, respectively. The results show that the sample has both birefringence and diattenuation. Thus, the ability of the measurement system to extract the characteristic parameters of samples with both birefringence and diattenuation is confirmed.

From above figures, it is found that some errors appear to be systematic changes with rotation of the sample, and this could be caused by some factors. Indeed, the analytical models about systematic errors caused by imperfect components and misalignment [24] and effect of input states of polarization [23] in measuring Stokes parameters were discussed.

To confirm the extracted characteristic parameters of baked polarizer as shown in Table 1, we adopt different input lights polarized linearly at 0°, 20°, 40°, 60°, 80°, 100°, 120°, 140°, 160°, and 180° and make experiments when the diattenuation axis of baked polarizer placed at 0°, 30°, 60°, 90°, 120°, and 150°, respectively. On the other hand, the normalized Stokes parameters are simulated from the Mueller matrix of sample by the different input polarization Stokes vectors. The Mueller matrix of sample is obtained by substituting the four extracted characteristic parameters, as illustrated in Table 1, into the optically equivalent model of a linearly birefringent material in series with a linear diattenuation material, as shown in Eq. (6). As compared with each other, it is found that the measured normalized Stokes parameters are well fitted to the simulated normalized Stokes parameters. Among those experimental results, only two results when the diattenuation axis of baked polarizer is placed at 120° or 150° are summarized and illustrated to demonstrate the feasibility in an optically equivalent model for the baked polarizer. Tables 2 and 3 show the values of the measured and simulated normalized Stokes parameters, respectively. The relation between the measured and simulated normalized Stokes parameter is clearly illustrated in Figs. 15 and 16 .From Figs. 15(c) and 16(c), it is found that the deviation between the measured and simulated Stokes parameters occurs when the diattenuation angle is perpendicular to the incident polarization angle. It is due to the output light from the sample is of little intensity. The errors arising from environmental perturbations, imperfect components, and misalignment [23] do exist in the measurement of output Stokes parameters. In addition, a slightly over-polarized Stokes vector (the degree of polarization $P=\sqrt{{S}_{1}^{2}+{S}_{2}^{2}+{S}_{3}^{2}}/{S}_{0}=1.009>1$) occurred in Figs. 15(a) and 16(a) when the linear polarization light is incident in an angle of 20° and 180°, respectively. It is owing to the fluctuations of the laser output power and the high noise of the room temperature in the pyroelectric detector [24]. Therefore the physical consistency of the measured Stokes vector must be always satisfied before the data analysis for obtaining good experimental results. Also, the experimental error on the measured Stokes vector induced by the effect of input states of polarization is in need of consideration [23]. From Figs. 15 and 16, the effective birefringence and diattenuation in the baked polarizer are successfully obtained. Then the optically equivalent model of baked polarizer is proven to be reasonable and feasible.

### 4.2.4 Composite sample comprising quarter-wave plate and baked polarizer

To further confirm the validity of the above conclusion in sub-section 4.2.3, the composite sample composed of a quarter-wave plate in series a baked polarizer is examined. Two cases as the diattenuation axis of baked polarizer is aligned to or not aligned to the principal axis of quarter-wave plate are discussed. The optical parameters of the composite sample were measured using the following three-step procedure. Initially, the quarter-wave plate was inserted into the measurement system shown in Fig. 8 and rotated such that its principal axis angle was orientated at 90° to the horizontal. The principal axis angle, phase retardation, diattenuation axis angle and diattenuation were then extracted using the analytical model derived in Section 2. In the second step, the quarter-wave plate was removed from the experimental system and the baked polarizer was introduced at a position immediately behind that vacated by the wave plate. The polarizer was adjusted such that its principal axis angle was orientated at 90° and its birefringence and diattenuation properties were then measured. Finally, the quarter-wave plate was reintroduced into the experimental measurement system in its original position, and the characteristic parameters of the composite sample were measured. Table 4
summarizes the experimental results obtained at each stage of the measurement procedure (i.e., θ* _{d}* =α=90° in the first case). The results show that when the principal birefringence axes of the quarter-wave plate and the baked polarizer are aligned, the total phase retardation of the composite sample is equal simply to the sum of the retardance values of the individual components within the sample. Therefore the retardance indeed exists in the baked polarizer. In addition, it can be seen that the total diattenuation of the composite sample is dominated by that of the baked polarizer and has a value of less than 1.

Next, consider the case when the diattenuation axis of baked polarizer is not aligned to the principal axis of quarter-wave plate (i.e., θ* _{d}* =0° and α =−8° in the second case). All the experimental procedures are same as those in the first case. Table 5
summarizes the experimental results obtained at each stage of the measurement procedure. The results show that when the principal birefringence axes of the quarter-wave plate and baked polarizer are aligned, the total phase retardation of the composite sample is equal simply to the sum of the retardance values of the individual components within the sample. Therefore the retardance indeed exists in the baked polarizer and it is found that when the diattenuation axis θ

*of the baked polarizer is rotated to*

_{d}**~**0°, its extracted principal axis angle α

**~**−8° does not coincide exactly with the diattenuation axis. As a result, the feasibility of the new proposed algorithm is demonstrated by Tables 4 and 5.

## 5. Conclusions and discussion

This study has proposed a technique based on a polarimeter and the Mueller matrix formulation for measuring the linear birefringence and linear diattenuation properties of an optical sample. The validity of the proposed approach has been demonstrated by measuring the principal axis angle (α), retardance (β), diattenuation axis angle (θ* _{d}*), and diattenuation (

*D*) of a quarter-wave plate, a polarizer, a baked polarizer, and a composite sample comprising a quarter-wave plate and a baked polarizer, respectively. It has been shown that the proposed methodology enables the full-range measurement of the principal axis angle, diattenuation axis angle, and diattenuation, respectively. However, the dynamic measurement range of the retardance is limited to 0 ~180°. Nonetheless, the proposed method still enables the full characterization of samples such as thin films, in which the retardance is restricted to the range 0 ~180° due to their very thin thickness. The experimental results have shown that the extracted values of the principal axis angle, retardance, diattenuation axis angle, and diattenuation of baked polarizer have average standard deviations of α = 0.03°, β = 0.03°, θ

*= 0.01°, and*

_{d}*D*= 4.16×10

^{−4}, respectively. Meanwhile, the average standard deviations of quarter-wave plate with low diattenuation are α = 0.03° and β = 0.04°, and polarizer with low birefringence are θ

*= 0.03° and*

_{d}*D*= 5.38×10

^{−5}. Unreliable measurement of a quarter-wave plate in the diattenuation parameters or a polarizer in the birefringence parameters is found in this study. Aside from the limited measurement range of the retardance, the proposed technique also has a limit if the values of birefringence or diattenuation are very low. In this case, the serious deviation is caused by the numerical round-off problem in mathematical models, and it relies on the processing minimum digit value from a polarimeter in the whole measurement system.

As compared to many of the optical parameter measurement schemes presented in the literature, the methodology proposed in this study does not require the birefringence and diattenuation axes of the sample to coincide. In addition, the linear birefringence and linear diattenuation parameters are decoupled in the analytical model. Thus, the birefringence of the optical sample can be evaluated without any prior knowledge of the diattenuation characteristics.

## Acknowledgements

The authors gratefully acknowledge the financial support provided to this study by the National Science Council of Taiwan under grant NSC96-2628-E-006-005-MY3.

## References and links

**1. **R. A. Chipman, “Polarization analysis of optical systems,” Opt. Eng. **28**, 90–99 (1989).

**2. **J. P. McGuire, Jr., “Image formation and analysis in optical systems with polarization aberrations,” Ph.D. thesis, University of Alabama in Huntsville, (1990).

**3. **R. A. Chipman, *Handbook of Optics* (McGraw-Hill, 1995), Vol. 2, Chap. 22.

**4. **D. B. Chenault and R. A. Chipman, ““Linear diattenuation and retardation measurements in an ir-spectropolarimeter,” Polarimetry: Radar, Infrared, Visible, Ultraviolet, And X-Ray **1317**, 263–278 (1990).

**5. **D. B. Chenault and R. A. Chipman, “Infrared birefringence spectra for cadmium-sulfide and cadmium selenide,” Opt. Lett. **17**, 4223–4227 (1992).

**6. **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]

**7. **J. M. Bueno and P. Artal, “Diattenuation of the human eye at the fovea,” Invest. Ophthalmol. Vis. Sci. **46**, ••• (2005).

**8. **J. M. Bueno and P. Artal, “Average double-pass ocular diattenuation using foveal fixation,” J. Mod. Opt. **55**(4), 849–859 (2008). [CrossRef]

**9. **M. Todorović, S. L. Jiao, L. V. Wang, and G. Stoica, “Determination of local polarization properties of biological samples in the presence of diattenuation by use of Mueller optical coherence tomography,” Opt. Lett. **29**(20), 2402–2404 (2004). [CrossRef] [PubMed]

**10. **X. R. Huang and R. W. Knighton, “Diattenuation and polarization preservation of retinal nerve fiber layer reflectance,” Appl. Opt. **42**(28), 5737–5743 (2003). [CrossRef] [PubMed]

**11. **D. A. Higgins, D. A. VandenBout, J. Kerimo, and P. F. Barbara, “Polarization-modulation near-field scanning optical microscopy of mesostructured materials,” J. Phys. Chem. **100**(32), 13794–13803 (1996). [CrossRef]

**12. **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]

**13. **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]

**14. **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]

**15. **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,” Applications of Scanned Probe Microscopy to Polymers **897**, 65–84 (2005). [CrossRef]

**16. **L. S. Goldner, S. N. Goldie, M. J. Fasolka, F. Renaldo, J. Hwang, and J. F. Douglas, “Near-field polarimetric characterization of polymer crystallites,” Appl. Phys. Lett. **85**(8), 1338–1340 (2004). [CrossRef]

**17. **L. S. Srinath and A. V. S. S. S. R. Sarma, “Determination of the optically equivalent model in three-dimensional photoelasticity,” Exp. Mech. **14**(3), 118–122 (1974). [CrossRef]

**18. **R. A. Tomlinson and E. A. Patterson, “The use of phase-stepping for the measurement of characteristic parameters in integrated photoelasticity,” Exp. Mech. **42**(1), 43–50 (2002). [CrossRef]

**19. **H. Hurwitz and R. C. Jones, “A new calculus for the treatment of optical systems-II: proof of three general equivalent theorems,” J. Opt. Soc. Am. **31**, 493–499 (1941).

**20. **S. T. Tang and H. S. Kwok, “Characteristic parameters of liquid crystal cells and their measurements,” J. Disp. Techno. **2**(1), 26–31 (2006). [CrossRef]

**21. **http://www.meadowlark.com>.

**22. **I. C. Khoo, and F. Simoni, *Physics of Liquid Crystalline Materials* (Gorden and Breach Science Publishers, 1991), Chap. 13.

**23. **H. Dong, Y. D. Gong, V. Paulose, P. Shum, and M. Olivo, “Effect of input states of polarization on the measurement error of Mueller matrix in a system having small polarization-dependent loss or gain,” Opt. Express **17**(15), 13017–13028 (2009). [CrossRef] [PubMed]

**24. **L. Giudicotti and M. Brombin, “Data analysis for a rotating quarter-wave, far-infrared Stokes polarimeter,” Appl. Opt. **46**(14), 2638–2648 (2007). [CrossRef] [PubMed]