1. INTRODUCTION

The ability to measure birefringence distributions is important for various applications. These include the analysis of inner stress by means of photoelasticity in optically anisotropic solids composed of transparent materials such as crystals, polymers, or glasses. Besides these established applications, other potential use cases for birefringence imaging techniques are being explored. Examples include biomedical research [1–4], material inspection systems [5], microfluidics [6], and polarization-sensitive optical diffraction tomography [7]. In many applications, a two-dimensional (2D) birefringence measurement system is desirable. The system requirements are to identify the relative retardation and the position of the refractive index axes. Several researchers have proposed and discussed such measurement systems. Reviews on relevant techniques used in photoelasticity can be found in the literature [8–14]. Their utility can also be evaluated in the broader context of general two-dimensional birefringence. A common method is the phase-shifting technique [15–18], applied in precise and fast 2D analysis systems. Using circularly polarized light and a polarization image sensor, it allows determination of the position of the fast axis and phase differences of up to ${\pi \mathord{/ {\vphantom {\pi 2}}} 2}$ rad [18]. Stokes vector and Mueller matrix polarimetry have been used to characterize materials and measure the state of polarization [19–21]. The literature offers several other studies dealing with photoelasticity, birefringence, and polarimetry, e.g.,  [22–29]. Approaches differ in experimental design and evaluation techniques. Most use rotating optical polarization components or variable retarders and determine birefringence by analyzing the resulting changes in light intensity. Technological advances in imaging and measurement systems continue to facilitate new approaches. The recent introduction of polarization-sensitive cameras represents such an advancement, and they are increasingly used in research studies, e.g.,  [30–32]. Currently, most techniques are complex, require expensive hardware, or are only able to measure phase differences of up to ${\pi \mathord{/ {\vphantom {\pi 2}}} 2}$, corresponding to a quarter of the applied wavelength. This range may be expanded by considering other information, for example, conducting the measurements with two or more wavelengths [12], but the periodic nature of the relative retardation, a fundamental challenge to all methods [29], may require the maximum measurement range of the phase difference in particular cases. We introduce a measurement system that consists of a single rotatable linear polarizer and a polarization camera. The measurement range of the phase difference is up to $\pi$ rad (half of the applied wavelength), and the relative position of the refractive index axes can be determined. However, fast and slow axis cannot be distinguished. The overall idea and concept are based on a previous study [33], in which we analyzed birefringence of a fluid in a Taylor–Couette flow. There, the circular geometry of the experiment naturally lead to a rotation of the sample, and thus polarizer and polarization camera could remain at fixed positions. The approach described in the present study could be more broadly applicable.

 

Fig. 1. Principal optical setup.

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Fig. 2. Schematic structure of a division-of-focal-plane polarization image sensor.

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2. THEORY

The basic optical measurement setup is shown in Fig. 1. Light from a source passes a linear polarizer $P(\phi)$. A global coordinate system $x-y$ is given as a reference. The polarizer is rotated by an angle $\phi$. A transparent sample ${X_{\Delta ,\theta}}$ with an unknown 2D distribution of the birefringent properties $\Delta$ and $\theta$ is to be examined, where $\Delta$ is the relative phase difference and $\theta$ the angle of the refractive index axis ${n_1}$. We define this axis to be the fast (${n_1} \lt {n_2}$) vibration direction of the light. This definition sets $\theta$ to the range [0, $\pi$]. If the optical properties are assumed to be constant throughout the thickness of the sample, the phase shift $\Delta$ is given by

Here, $\lambda$ is the wavelength of the light, $L$ the thickness of the sample, and $\delta n = | {{n_2} - {n_1}} |$ the difference of the refractive indexes. $\delta n \cdot L$ is referred to as optical retardation. A polarization camera detects the light that passes through the sample. The camera used in this study is based on a division-of-focal-plane polarization image sensor (Sony IMX 250 MZR [34]). The basic structure of such a polarization image sensor is shown in Fig. 2.

Small polarizers are aligned in four defined directions, forming a polarizer filter array. The direction depends on the position of the pixel. Arrays of these polarized pixel combinations enable the camera to simultaneously measure the intensities passing through the specified directions, which we define as ${I_0},{I_{45}},{I_{90}},{I_{135}}$. Four neighboring pixels with polarizers oriented at 0°, 45°, 90°, and 135° form a so-called superpixel, as depicted in Fig. 2. In our setting, direction ${I_0}$ is in line with the $x$ axis. A polarization camera introduces errors such as field-of-view errors resulting from the spatial arrangement of the polarizer filter array [35] and errors introduced by optical imperfections of the polarizer filter array [36,37]. Alternatively, instead of the polarization camera, a second rotatable linear polarizer mounted in front of a monochrome camera could be used. The second polarizer would then be rotated to the corresponding positions: 0°, 45°, 90°, and 135°. An advantage of such a setup would be the mitigation of the errors introduced by a polarization camera and to have the opportunity to apply polarizers of the highest possible quality. The advantage of the polarization camera, however, is the concurrent measurement of the four directions of polarization.

With the definition of the Stokes parameters being

Stokes parameters and Mueller matrices are used to describe the state of polarization and its changes, respectively. The following vectors and matrixes refer to the fixed $x-y$ coordinate system and can be found in the literature, for example, in [38,39]. We define ${\vec S_{\text{in}}}$ as the corresponding Stokes vector of an unpolarized light source and $P(\phi)$ as the Mueller matrix of a linear polarizer oriented at $\phi$,

A linear retarder ${X_{\Delta ,\theta}}$ inducing a phase shift $\Delta$ with an orientation angle of the fast axis $\theta$ can be expressed as

The relation between incident and outgoing Stokes vector is then given by

Using the normalized representation of the Stokes parameters ${\vec S_N} = {1 \mathord{/ {\vphantom {1 {{S_0}}}}} {{S_0}}} \cdot \vec S$ (note the difference to a regular vector normalization), the expressions for ${S_{{1_N}}}$ and ${S_{{2_N}}}$ that are independent of the light source intensity are derived,

We can see that

This is in line with the definition of ${S_{{1_N}}}$ and ${S_{{2_N}}}$, as their corresponding intensities (${I_0},{I_{90}}$ and ${I_{45}},{I_{135}}$) are rotated by $\pi/4$. Due to the range for the degree of linear polarization (DOLP),

${S_{{1_N}}}$ and ${S_{{2_N}}}$ must fulfil

The position of the linear polarizer is described by the known angular coordinate $\phi$. This reduces Eq. (6) to two equations with two unknowns ($\Delta ,\theta$) and two measurable parameters (${S_{{1_N}}},{S_{{2_N}}}$). For a specific position of the linear polarizer, we can therefore derive explicit relations for $\Delta ,\theta$, here done for the case $\phi = 0$,

Figures 3(a) and 3(b) show Eq. (6) for the case $\phi = 0$ and Figs. 3(e) and 3(f) the corresponding inverse relationships from Eq. (10). From Eqs. (6) and (10) and Fig. 3 it is clear that the Stokes parameters ${S_{{1_N}}}$ and ${S_{{2_N}}}$ are periodic with period ${\pi \mathord{/ {\vphantom {\pi 2}}} 2}$ in $\theta$ (the range [${-}\pi/4$, 0] corresponding to [$\pi/4$, ${\pi \mathord{/ {\vphantom {\pi 2}}} 2}$]). Figure 3(f) shows the orientation angle for all possible combinations of ${S_{{1_N}}}$ and ${S_{{2_N}}}$ (with $S_{{1_N}}^2 + S_{{2_N}}^2 \le 1$). It stays in the range [0, ${\pi \mathord{/ {\vphantom {\pi 2}}} 2}$]. This means that no distinction between fast and slow vibration axis can be made, as otherwise the period of $\theta$ would have to be $\pi$ and the range of values [0, $\pi$]. The periodicity of $\Delta$ is $2\pi$. When the retardation $\delta n \cdot L$ is a multiple of $\lambda$, the phase difference $\Delta$ is a multiple of $2\pi$ and the state of polarization will be linear again, orientated at $\phi$ equal to the state and orientation of the linear polarizer $P(\phi)$. Both relations in Eq. (10) are not defined for ${S_{{1_N}}} = 1$ and ${S_{{2_N}}} = 0$. Such a Stokes vector can be anywhere on an isoclinic or isochromatic [blue dashed lines and black solid lines in Figs. 3(a) and 3(b)], with corresponding values for $\Delta$ and $\theta$. Looking at Fig. 3(e), we can see that $\Delta = \pi$ applies for a circle defined by $S_{{1_N}}^2 + S_{{2_N}}^2 = 1$. In this case, the light beam is fully linear-polarized (${\rm DOLP} = 1$). Fully linear polarization is only possible if retardation $\delta n \cdot L$ is an odd multiple of $\lambda /2$(besides the discussed and trivial case of ${S_{{1_N}}} = 1$ and ${S_{{2_N}}} = 0$). This means that the relative phase between the two oscillating and perpendicular electromagnetic vibrations has shifted by an odd multiple of $\pi$, the location of maximum (relative) phase difference. Relation (10) also give insight into the direct measurement ranges of $\Delta$ and $\theta$. Unless further information is supplied, how many periods of retardation have occurred remain undefined. If the only available information is based on ${S_{{1_N}}}$ and ${S_{{2_N}}}$, the period number, also known as order, remains unclear. Moreover, retardation could also be in the “second half” of the corresponding period number, meaning that retardation could be between either [0, $\lambda /2$] or [$\lambda /2$, $\lambda$], which corresponds to relative phase differences of [0, $\pi$] and [$\pi, 2\pi$]. Summarizing, the orientation of the refractive index axes can only be measured in the range [0, ${\pi \mathord{/ {\vphantom {\pi 2}}} 2}$], and it is not possible to distinguish between the fast and slow axis. Phase differences $\Delta$ are measurable in the range [0, $\pi$]. However, for any measured (relative) phase difference $\Delta$, the following (absolute) phase differences ${\Delta ^A}$ are possible:

As the relation (10) is not defined for the case ${S_{{1_N}}} = 1,{S_{{2_N}}} = 0$, we suggest multiple measurements at different angles $\phi$ to determine the optical properties. Rotating the linear polarizer by $\phi$ will lead to distributions of ${S_{{1_N}}}(\phi)$ and ${S_{{2_N}}}(\phi)$. Figures 3(c) and 3(d) show distributions for the case $\theta = 0$. From such distributions, we will be able to clearly identify the optical parameters within the discussed ranges.

 

Fig. 3. Stokes parameters ${S_{{1_N}}}$ and ${S_{{2_N}}}$ in relation to the phase difference $\Delta$ and angle $\theta$. (a) and (b) Eq. (6) for the case $\phi = 0$. Blue dashed lines indicate isoclinics (no change in polarization as linear polarization is in line with refractive index axes). Black solid lines indicate isochromatics (retardation being a multiple of $\lambda$). (c) and (d) Eq. (6) for the case $\theta = 0$; (e) and (f) Eq. (10), being the inverse relationships of Eq. (6) for the case $\phi = 0$. The phase difference $\Delta$ in (e) is not defined for the case ${S_{{1_N}}} = 1$. The surface in (f) was created using atan2(), the two-argument arctangent.

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3. MATERIAL AND METHODS

A. Applied Materials

A halogen incandescent lamp was applied as the light source and a white optical diffuser was placed between the lamp and the linear polarizer to illuminate the specimen as evenly as possible. The linear polarizer had an extinction ratio of 10,000:1 and was manually rotated using a continuous rotation mount (Thorlabs #RSP2/M). The test sample was an achromatic polymer quarter-wave plate (inducing a phase difference of ${\pi \mathord{/ {\vphantom {\pi 2}}} 2}$ rad) with a wavelength range of 450–600 nm, retardance tolerance of 10 nm, and diameter of 25 mm (Edmund Optics Inc., article number #88-198). Placed before the specimen was a green bandpass filter (CWL 526 nm, FWHM 53 nm; Edmund Optics Inc., #46-053). The camera was a monochrome polarization camera (Phoenix PHX050S-P from Lucid Vision Labs [40] based on a Sony IMX250MZR CMOS (mono) sensor [34]) equipped with a Schneider xenon 50 mm/0.95 lens. The f-number was set to f/0.95. The camera resolution is 2448 × 2048 (5 MP) pixels with a pixel size of 3.45 µm × 3.45 µm. As indicated in Fig. 2, four neighboring single pixels, each having a different linear polarizer, form a superpixel. This leads to a spatial resolution for Stokes parameter measurements of 1224 × 1024 (1.25 MP) superpixels.

B. Measurement Procedure

The linear polarizer is rotated by changing $\phi$. Looking at relation (10) and Fig. 3, we see that besides the periodicities in $\Delta$ and $\theta$, the Stokes parameters ${S_{{1_N}}}$ and ${S_{{2_N}}}$ are also periodic with period $\pi$ in $\phi$. To get full resolution along one period of $\phi$, the polarizer will hence have to be rotated up to $\pi$ rad.

For every polarizer position $\phi$, the first three Stokes parameters are measured,

Formally, the first Stokes parameter can be derived as ${S_0} = {I_0} + {I_{90}} = {I_{45}} + {I_{135}}$. We therefore average both measurements. The normalized form results following ${\vec S _N} = {1 \mathord{/ {\vphantom {1 {{S_0}}}}} {{S_0}}} \cdot \vec S$. Ideally, the degree of polarization (DOP) should equal one: ${{\rm DOP}} = 1$. The DOP is defined as (note the difference from the DOLP):

However, the measured Stokes vector will only be partially polarized (${{\rm DOP}} \lt 1$). Noisy unpolarized light intensities arising from the environment and from imperfections of the optical components are unavoidable. This is particularly the case when a distributed light source instead of a laser is applied, as is done in this study. Partially polarized Stokes vectors can be considered as a superposition of a fully polarized Stokes vector ${\vec S _P}$ and an unpolarized Stokes vector ${\vec S _U}$ [38],

The normalized representation of the fully polarized part of the Stokes vector measured at a polarizer position $\phi$ can be written as

Rotating the linear polarizer $P(\phi)$ and taking images at different positions $\phi$ results in measured distributions of ${S_{{1_N}}}(\phi)$ and ${S_{{2_N}}}(\phi)$. Fitting Eq. (16) to these measurements gives the parameters $\Delta ,\,\,\theta ,\,\,{\rm DOP},\,\,d\phi$. This can be simultaneously done throughout an area, resulting in a 2D birefringence measurement.

C. Camera Calibration

The camera was calibrated following the superpixel calibration described in [36,37]. Test images for the training data were taken at three different intensity values (achieved by varying the exposure time) and at 10 different polarizer positions $P(\phi)$, each having a step size of 20° between them. The training data were acquired using the same optical setup as shown in Fig. 1 but without a specimen ${X_{\Delta ,\theta}}$. At first, aligning imperfections between polarizer $P(\phi)$ and polarization camera were corrected with $d\phi$. Therefore, the idealized distributions of the Stokes parameters ${S_{{1_N}}}(\phi)$ and ${S_{{2_N}}}(\phi)$ were fitted to the measurements,

These equations result when setting $\Delta = 0$ and ${{\rm DOP}}=100\%$ in Eq. (16). Fitting Eq. (17) to the measurements gives $d\phi$. The resulting distributions [Eq. (17)] (including $d\phi$) are then used as training data for the superpixel calibration, which gives a calibration function as follows:

Here $\vec I = {[{\begin{array}{*{20}{c}}{{I_0}}&{{I_{45}}}&{{I_{90}}}&{{I_{135}}}\end{array}}]^T}$ are the measured intensities, $\vec d$ is the dark noise offset, and $\underline G$ the gain correction of the superpixel. Each analyzed superpixel was calibrated separately. However, variation among the superpixels was negligible, and we hence averaged the calibration function over all involved superpixels. Figure 4 summarizes the calibration.

 

Fig. 4. Nominal, measured, and calibrated data for an exemplary measurement point with $d\phi = - 0.29^\circ$.

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D. Performance Tests

In total, we conducted a series of six tests. For every test, a region of 200 pixels × 160 pixels (100 superpixels × 80 superpixels) was analyzed. This gives a sufficient number of measurements (8000) without having to consider the computational effort fitting Eq. (16). The position of the linear polarizer $P(\phi)$ was varied between 0° and 180°. At every position $\phi$, one image was taken with the polarization camera. Starting at $\phi = 0^\circ$ and increasing the angle by 1° for every step gave 181 evaluated polarizer positions. The intensities ${I_0},{I_{45}},{I_{90}},{I_{135}}$ of the analyzed pixels were calibrated with Eq. (18). Following Eq. (12), the normalized Stokes parameters ${S_{{1_N}}}$ and ${S_{{2_N}}}$ were obtained for every evaluated polarizer position. This gives distributions along $\phi$ for every analyzed superpixel. Fitting Eq. (16) to the measurements ${S_{{1_N}}}(\phi)$ and ${S_{{2_N}}}(\phi)$ leads to estimates for $\Delta ,\theta ,{\rm DOP},d\phi$. Repeating this for every superpixel generates a 2D distribution. Parameter $d\phi$ was fitted in Section 3.C for the camera calibration. However, it is fitted again for every measurement, as the results of Section 3.C did not always match the results we achieved when fitting parameter $d\phi$ concurrently with Eq. (16). One reason for this could be that the quarter-wave plate rotates the incident linear polarized light. The second reason is that we assume our manual positioning of the polarizer $P(\phi)$ to be slightly inconsistent from measurement to measurement.

 

Fig. 5. Exemplary measurement results for a quarter-wave plate (the results correspond to Test 2 in Table 1). (a) Measured and fitted distributions for one superpixel; (b) variation of $\Delta$ for the analyzed area of 100 superpixels × 80 superpixels.

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Table 1. Average (av) and Standard Deviation (std) of the 144 Measurement Points Spaced over the 2D Area, Conducted with (QW) and without (-) the Quarter-Wave Plate

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4. RESULTS AND DISCUSSION

Figure 5 shows the results for a test carried out with the quarter-wave plate. The results correspond to Test 2 in Table 1. Figure 5(a) shows an example of measured and fitted distributions of normalized Stokes parameters ${S_{{1_N}}}(\phi)$ and ${S_{{2_N}}}(\phi)$ for a superpixel. The data were fitted with a nonlinear least-squares algorithm. Figure 5(b) shows the variation of $\Delta$ for all measured 100 superpixels × 80 superpixels. All phase differences are in the range [1.584, 1.609]. In the following subsections, we analyze the 2D measurement capability, compare the performed tests, and discuss the number of polarizer positions $P(\phi)$ necessary for satisfactory results. Finally, an exemplary test case is given to demonstrate the measurement system.

A. 2D Measurements

As each superpixel is evaluated separately, a 2D analysis is obtained. Table 1 summarizes the results for the six conducted tests. Average (av) and standard deviation (std) were calculated for the analyzed superpixels. Looking at the average values of $\theta$, our test scheme for the quarter-wave plate becomes obvious: 0°, 10°, 20°, 30°, and 40°. Measurements of angle $\theta$ describe the relative position of one of the refractive index axes in the range [0°, 90°] without distinguishing between slow and fast axis (compare Section 2). Results for Test 2 are depicted in Fig. 5. The mean measured phase difference $\Delta$ throughout the wave plate is 1.597. This is reasonably close to the ideal phase difference of ${\pi \mathord{/ {\vphantom {\pi 2}}} 2} \approx 1.571$ that a quarter-wave plate is supposed to induce. The difference of about 0.026 is well within the retardance tolerance of the quarter-wave plate,

The DOP is close to 100%, indicating low ambient light influence. However, the conditions of these measurements were similar to the conditions of the calibration measurements, where the DOP was set to 100%, and thus the calibration function will have eliminated most ambient conditions. Rather surprising are the results for the correction angle $d\phi$ of about 2.5°. In Fig. 4 and during the camera calibration, the misalignment between camera and initial polarizer position $P(\phi = 0)$ was found to be $d\phi \lt \pm 0.5{{^\circ}}$. We therefore assume that the quarter-wave plate rotates the incident polarized light. Test 5 was conducted with strong ambient light. This influenced the DOP but was found to have little to no effect on the other parameters. Therefore, by introducing parameter DOP in relation (16), this approach can be applied when ambient light is present.

 

Fig. 6. Optimization results for one superpixel. The number # on the $x$ axis corresponds to the number of polarizer positions considered in the fitting algorithm and the angle [°] to the relative difference of the positions. (a) Phase difference and (b) relative position of the refractive index axis.

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Table 2. Intensity Measurements 1–8 of the Equivalent Polarizer (P) and Analyzer (A) Positions for a Plane Polariscope Phase-Stepping Method of the #2-45° Configuration in Comparison with the Procedure Described by Nurse [28]^a

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B. Number of Required Polarizer Positions

For each test in Table 1, we analyzed 181 polarizer positions $P(\phi)$ from $\phi = 0^\circ$ up to $\phi = 180^\circ$. This represents a rather large experimental effort, so this section investigates the possibility of decreasing the number of polarizer positions. The question of how many different positions are necessary to achieve reasonable results is studied in Fig. 6 using an exemplary superpixel. The parameters $\Delta ,\theta ,{\rm DOP},d\phi$ were fitted applying only a limited number of the available polarizer positions $P(\phi)$. At first, only one position was used. This is similar to Eq. (10), which calculates $\Delta ,\theta$ from single values for ${S_{{1_N}}},{S_{{2_N}}}$. When only one position is considered, no results for the DOP and $d\phi$ are available, as our fitting algorithm requires at least the same number of equations as parameters. The DOP is then set to 100% and $d\phi = 0^\circ$. We analyzed all 180 different positions, ranging from $\phi = 0^\circ {-} 179^\circ$. The results are shown in Fig. 6 and are labeled #1. We then plotted the results considering two, three, four, six, and nine polarizer positions. The angles between the positions are shown in the $x$ axis. By assuming periodic continuation of the polarizer positions from $0 - \pi$ to $\pi - 2\pi$ (meaning position $\phi = 180{{^\circ + x}}$ is equal to position $\phi = x$), we analyzed all 180 possible combinations. For example, in the case of three positions, we considered positions ${\phi _1} = x,\,\,{\phi _2} = x + 60^\circ ,\,\,{\phi _3} = x + 2 \cdot 60^\circ$ and varied $x$ from 0°–179°. The starting position for the fitting algorithm was ${\Delta _0} = \pi ,\,\,{\theta _0} = 0^\circ ,\,{{\rm DOP}_0} = 100\% ,\,\,d\phi = 0^\circ$.

Figure 6 indicates that configurations #1 and #2-90° are not suitable for birefringence measurements. Acceptable results are available for #2-45°, #3-60°, and thereafter. However, compared with Table 1, accuracy in the area close to $\Delta = 0$ is decreased. The general procedure is relatable to a phase-stepping method for a plane polariscope composed of two rotatable linear polarizers. One polarizes the incident light (polarizer, orientation $\phi$) and the other analyzes the transmitted light (analyzer). The polarization camera hereby concurrently acts as four analyzer positions (0°, 45°, 90°, 135°). The #2-45° configuration can be interpreted as a phase-stepping method that gives eight intensity measurements for the governing phase-stepping equations [8,9]. This is summarized in Table 2, together with the method conducted by Nurse [28] for comparison. As Nurse is using a three-wavelength approach, we suggest the use of a color polarization camera in this context.

 

Fig. 7. 2D birefringence measurement of an exemplary test case with three plastic cuvettes (first three from the left) and a glass cuvette (right). (a) Parallel polarizers ($\phi = 0\,,\,{I_0}$); (b) crossed polarizers ($\phi = 0\,,\,{I_{90}}$). ${I_0}$ and ${I_{90}}$ refer to the corresponding images of the polarization camera. (c) Vector plot with the length of the vector corresponding to the measured phase difference $\Delta$ and the orientation to the relative position of the refractive index axes; (d) filled 2D contour plot with seven isolines separating eight levels of birefringence. Plot (c) and (d) show that the three plastic cuvettes to the left have a similar birefringence distribution, resulting from internal stresses, presumably due to the same manufacturing process. The glass cuvette to the right does not show any birefringence above $\Delta \gt \pi /4$, but the edges induce phase differences between $\pi /8 \lt \Delta \lt \pi /4$.

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C. Error Analysis

This subsection investigates the effect of inevitable measurement errors on the quality of results. Assuming an equal measurement uncertainty for ${S_{{1_N}}}$ and ${S_{{2_N}}}$ of $\delta {S_N}$, the error propagation on phase difference and extinction angle are modeled from the ideal relations as

We can see that measurement errors for the phase difference $\Delta$ are larger when $\Delta$ is close to zero or multiples of $\pi$. This can be seen in the $\Delta$ results of Table 1. For all superpixels, the same calibration function [Eq. (18)] was applied, as differences between the calibration functions were negligible. Because of the similar superpixel performance, the standard deviations of Tests 1–5 are small (std $\Delta = 0.005$). However, in agreement with Eq. (20), these small differences are intensified in Test 6, causing the resulting nonzero average value of 0.04 and the standard deviation to be nearly 10 times higher (std $\Delta = 0.04$). Equation (20) also explains the slower convergence to the mean value of Test 6 in Fig. 6(a). Measurement results for $\theta$ tend to be inaccurate if $\Delta$ is zero (in this case, $\theta$ does not exist) or a multiple of $2\pi$. The term $\sin [2(\phi - \theta)]$ in relation (20) explains the poorly performing cases of configurations #1 and #2-90° in Section 4.B. If the polarizer positions $\phi$ happen to inconveniently lead to $\sin [2(\phi - \theta)] = 0$, measurements will be corrupted.

D. Exemplary Test Case

The images in Fig. 7 show an exemplary test case with three plastic cuvettes and one glass cuvette as specimens. Internal stresses in the plastic cuvettes induce birefringence, whereas little to no birefringence is seen with the glass cuvette. The visible result is the cumulated birefringence of both surfaces. The optical materials and measurement method of Section 3 were used and four polarizer positions were taken into account: $\phi = 0^\circ ,45^\circ ,90^\circ ,135^\circ$. Any desired spatial resolution is possible, with the actual camera resolution being the limiting factor. In this case, we analyzed every eighth superpixel position. Each intensity ${I_0},{I_{45}},{I_{90}},{I_{135}}$ was derived by averaging the corresponding 37 pixels in a circular neighborhood with a diameter of eight superpixels. Figure 7(c) is a vector plot where the length corresponds to the measured phase difference $\Delta$ and the orientation to the relative position of the refractive index axes between [${-}{\pi \mathord{/ {\vphantom {\pi 4}}} 4},\pi/4$]. Figure 7(d) is a 2D contour plot visualizing coherent areas. We can see that phase differences range from 0 to $\pi$; therefore, some areas will most likely exceed the measurement limit of $\pi$ rad. However, these cases exceeding the limit are few and occur in close proximity.

5. CONCLUSION

The described technique presents a practical approach to measure 2D birefringence. The proposed procedure is able to identify optical retardations of up to half the wavelength of light (phase differences $\pi$ rad) and gives the relative positions of the refractive index axes. In comparison to methods based on circularly polarized light, the use of linearly polarized light increases the measurement range for retardation but cannot distinguish between the fast and the slow axis. It may represent an attractive alternative to previous methods, especially in the case of phase differences larger than ${\pi \mathord{/ {\vphantom {\pi 2}}} 2}$. The experimental setup and data analysis are comparatively simple and enable the rapid evaluation of a large sample area. The use of a polarization-sensitive camera in this measurement context is promising and suggests that further research in the field should be undertaken.