## Abstract

We present a new real-time Stokes parameter measurement technique using three polarized beam splitters without mechanical motion or electrical tuning. This system can analyze the polarization state of light at 30 kHz, limited only by the speed of the detector analog to digital converters. The optical system is also compact ($52\times 30\times 25\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{mm}$) because it consists only of small volume optical devices. We show that the system can measure arbitrary polarization states with an accuracy of better than 0.006 in the normalized Stokes parameters. We also demonstrate the ability to measure fast dynamic polarization states by analyzing the state produced by a fast rotating quarter-wave plate and the time-dependent stress induced in a PMMA block by hitting the block with a hammer.

© 2019 Optical Society of America

## 1. INTRODUCTION

A real-time polarimeter is needed for measuring and monitoring dynamic polarization states, such as for monitoring thin-film thickness *in situ* during thin-film preparation. Compact systems are also needed for measuring in confined measurement spaces. Among the many techniques for measuring the Stokes parameters [1–4], most conventional polarimeters are relatively slow, requiring seconds to minutes for measurement, and use motors and other devices that can be difficult to make compact.

In 1982 and 1985, Azzam proposed two compact Stokes polarimeter designs, both of which use four detectors. The first polarimeter uses three beam splitters to separate the beam to be collected at the four detectors, allowing the system to measure the linear Stokes vector elements (${s}_{0}$, ${s}_{1}$, and ${s}_{2}$) [5,6]. The second polarimeter uses only the four detectors themselves, such that the reflection from the surface of each detector face induces a polarization change in the beam, while the fraction of light absorbed by the detector is used in measurement [7]. This system can measure the complete Stokes vector of the beam. Because both of these systems employ no moving parts or electronic scanning, their measurement speeds are limited only by the speed of the detectors and their analog to digital (A/D) converters. Disadvantages of these two approaches are that, for the first system, only the linear Stokes vector is measured, and for the second system, the alignment is difficult—due to the need to use complex 3D angles for each detector—and the calibration needs to be precise. That is, the calibration needs to accurately measure the polarization change on the beam induced by each detector element. In order to overcome the alignment difficulty, Kawabata proposed modifying the four-detector layout for transmission operation instead of reflection [8]. Kawabata’s polarimeter uses the polarization change of transmission and reflection on beam splitters, instead of Azzam’s use of the reflection on detectors. While this polarimeter is easier to align than Azzam’s, it too requires precise calibration.

In Section 2 below, we discuss the polarimeter’s principles of measurement. In order to demonstrate the system’s advantages, in Section 3 we show experimental results and evaluate the measurement accuracy and speed.

## 2. COMPACT AND HIGH-SPEED STOKES POLARIMETER USING THREE POLARIZED BEAM SPLITTERS

Figure 1 shows the optical setup for the high-speed measurement of input Stokes parameters [${s}_{0}$, ${s}_{1}$, ${s}_{2}$, and ${s}_{3}$]. The basic idea is to use each of three sets of polarizing-beam splitters plus detector-pair to analyze the ${s}_{1}$, ${s}_{2}$, and ${s}_{3}$ components of the Stokes vector directly. In order for this to work, we need to split the input light into three paths without changing the input polarization state. This splitting is accomplished with a special configuration of six normal beam splitters forming the three-way polarization-preserving splitters (3WPPS).

At first, we outline the construction of a new beam-splitting technique that separates a single input beam into three, such that the state of polarization (SOP) in the three outputs is the same as the input SOP. Figure 2 shows a related setup to separate a beam in two while maintaining the polarization state using three normal beam splitters. The input axis electric field amplitudes ${E}_{x}$ and ${E}_{y}$ are changed by transmission through or reflection by the first beam splitter due to its linear retardances ${\delta}_{1}$ and ${\delta}_{2}$, and linear diattenuations ${D}_{1}=({p}_{1}-{q}_{1})/({p}_{1}+{q}_{1})$ and ${D}_{2}=({p}_{2}-{q}_{2})/({p}_{2}+{q}_{2})$, where $p$ and $q$ are $x$ and $y$ axis transmittances, respectively. After passing the second beam splitter, which has approximately the same polarization properties as the first beam splitter (${p}_{1}\approx {p}_{3},{q}_{1}\approx {q}_{3}$, and ${\delta}_{1}\approx {\delta}_{3}$ for transmission mode, ${p}_{2}\approx {p}_{4}$, ${q}_{2}\approx {q}_{4}$, and ${\delta}_{2}\approx {\delta}_{4}$ for reflection mode), rotated by 90° relative to the first, the total linear diattenuations and linear retardances become ${D}_{1}-{D}_{3}=({p}_{3}{q}_{1}+{q}_{3}{p}_{1})/({p}_{3}{q}_{1}+{q}_{3}{p}_{1})\approx 0$ and ${\delta}_{1}-{\delta}_{2}\approx 0$ for transmission mode, and ${D}_{2}-{D}_{4}=({p}_{4}{q}_{2}-{q}_{4}{p}_{2})/({p}_{4}{q}_{2}+{q}_{4}{p}_{2})\approx 0$ and ${\delta}_{2}-{\delta}_{4}\approx 0$ for reflection mode, respectively. Therefore, the polarization effects induced by the first beam splitter are canceled by the second beam splitter if the second splitter is rotated by 90°. In a similar way, we can connect two of these in series to separate the input beam into three, where each of the three outputs still maintains the same SOP as the original input if all the beam splitters have the same polarization properties. We call this assembly of two sets of three beam splitters connected in series the 3WPPS.

Next, Fig. 3 shows how the three polarizing beam splitters (PBSs) and six detector elements—one set after each of the three beams split from the 3WPPS—are used to measure the input SOP.

If we assume that the transmittance and reflectance are equal for each PBS, and that the gain of each detector is the same, then the normalized Stokes parameters can be measured by

In a realistic model, the PBS will have a different transmittance ${\alpha}_{i}$ and reflectance ${\beta}_{i}$, and the gains ${k}_{i}$ and ${k}_{i}^{\prime}$ of each detector will be different ($i=1$ for ${\tilde{s}}_{1}$, $i=2$ for ${\tilde{s}}_{2}$, and $i=3$ for ${\tilde{s}}_{3}$), so that it is necessary to calibrate them. If we combine the ${\alpha}_{i}$, ${\beta}_{i}$, ${k}_{i}$, and ${k}_{i}^{\prime}$ into one equation, then we can define three total calibration parameters ${g}_{i}$ that need to be determined:

In order to measure ${g}_{i}$, we can pass linear polarization oriented at 22.5°, $\tilde{\mathit{s}}=(1,1/\sqrt{2},1/\sqrt{2},0)$, into the Stokes polarimeter. The ${g}_{i}$ parameters are then determined by## 3. EXPERIMENTAL RESULTS OF THE STOKES POLARIMETER

We characterize the polarization characteristics of the 3WPPS to determine how close it is to achieving the ideal (zero) linear retardance and ideal (zero) linear diattenuation required to maintain the input polarization state. Figure 4 shows the spectroscopic measurement of linear retardance $\delta (\lambda )$ and linear diattenuation $D(\lambda )$ of the 3WPPS by an Axoscan Mueller matrix polarimeter [9], indicating that the $\delta (\lambda )$ and $D(\lambda )$ values are small but not quite zero. Thus, the six normal beam splitters used here do not quite share the same polarization properties. While errors such as these can also be produced by misalignment, the fact that we measured negligible circular retardance and circular diattenuation implies that errors in azimuthal angle alignment (rotation about the optical axis) are not to blame.

In experiments introduced below, we use a $\lambda =633\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{nm}$ light source. Taking the retardance and diattenuation from the measurements of Fig. 4, we can simulate the influence that the nonideal behavior in $\delta $ and $D$ have on the polarization state transmitted by the 3WPPS (Fig. 5). For all input SOPs, we find that the normalized Stokes parameter measurement error is less than 0.033 for ${\tilde{s}}_{1}$, 0.046 for ${\tilde{s}}_{2}$, and 0.042 for ${\tilde{s}}_{3}$. These errors can be further reduced by calibrating the residual linear diattenuation and linear retardance.

In order to calibrate these residual values, we can substitute these values from Fig. 4 in the Stokes parameter measurement model. If we assume that the residual linear diattenuation ${D}_{(i)}$ and residual linear retardance ${\delta}_{(i)}$ produced by transmission/reflection before each detector’s polarizing beam splitter (the three splitters shown in Fig. 3) are small, then we have ${D}_{(i)}^{2}\approx 0$, sin ${\delta}_{(i)}\approx {\delta}_{(i)}$, and cos ${\delta}_{(i)}\approx 1$. In this regime, the initial normalized Stokes parameter estimates ${\tilde{s}}_{1}$, ${\tilde{s}}_{2}$, and ${\tilde{s}}_{3}$ obtained from Eq. (4) can be corrected to ${\tilde{s}}_{1}^{\prime}$, ${\tilde{s}}_{2}^{\prime}$, and ${\tilde{s}}_{3}^{\prime}$ using

Figure 6 shows the equipment layout used to evaluate the proposed Stokes polarimeter, composed of an He–Ne laser of wavelength 632.8 nm, and a circular polarizer built by combining a polarizer and a quarter-wave plate (QWP) and a sample. These generate the input polarization state, which is then measured by the proposed Stokes polarimeter. Inside the polarimeter proper, the actual retardance of the QWP is 89.9° as measured by the Axoscan polarimeter. The influence on the normalized Stokes parameter measurement by the small retardance error 0.1° in the QWP inside the polarimeter is less than 0.0005, and is therefore smaller than the errors generated elsewhere in the system, so that we can ignore them in the analysis below. The measurement speed of the polarimeter is determined by the speed of the A/D converters (16-bit, $3\mathrm{\times}{10}^{4}$ samples/s) used in the six silicon PIN photodiodes.

In order to evaluate the performance of the proposed polarimeter, we measure the Stokes parameters for an input polarization state produced by a rotating QWP [Fig. 7(a)], and by a rotating polarizer [Fig. 7(c)]. In this measurement, the calibration parameters ${g}_{1}=1.25$, ${g}_{2}=0.89$, and ${g}_{3}=1.07$ are obtained by Eq. (3) before measurement. The results show small deviations [see Figs. 7(b) and 7(d)] from the known input state, caused by the nonzero residual linear retardance and linear diattenuation of the 3WPPS. Once we calibrate these residual values and apply the correction using Eq. (5), the normalized Stokes parameter errors are reduced by an order of magnitude to $<0.006$ [Figs. 7(b) and 7(d)].

In order to demonstrate high-speed measurement, we perform two experiments. In the first experiment, we measure the Stokes parameters of a QWP rotated at high speed ($\omega =25\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{rps}$) by a servomotor [Fig. 8(a)]. Even for the resulting 100 Hz frequency modulations, produced from ${\tilde{s}}_{1}=\mathrm{cos}(4\omega \mathrm{t})$ and ${\tilde{s}}_{2}=\mathrm{cos}(4\omega \mathrm{t}+\pi /2)$, and 50 Hz frequency modulations produced by ${\tilde{s}}_{3}=\mathrm{sin}(2\omega \mathrm{t})$, the system can easily follow the dynamic changes in the polarization state. For measurement speed, the system samples at 30 kHz [Fig. 8(b)], while the delay time required for Stokes parameter calculations is only 3 μs. Thus, the system is easily adapted to interactive use or for other real-time applications.

In the second experiment, we measure the rapid polarization change induced in a PMMA (acrylic polymer) block ($50\times 20\times 100\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{mm}$) by hitting it with a hammer. As the hammer impacts the plastic, the resulting stresses in the material induce stress-induced birefringence via the photoelastic effect, producing the dynamic Stokes parameter measurement shown in Fig. 9(a). For this measurement, we use right-handed circular polarization incident on the cube, so that we can observe birefringence in any orientation within the material. From the resulting measurements, we calculate the induced linear retardance ${\delta}_{\mathrm{PMMA}}$ and the azimuthal angle of birefringence $\varphi $, from which we obtain the average azimuthal angle of the principal stress difference $\mathrm{\Delta}\sigma $ within the PMMA block [Fig. 9(b)]:

## 4. CONCLUSION

A real-time Stokes polarimeter with three PBSs and a 3WPPS has been proposed. The 3WSPPS is a method of nulling the unwanted polarization effect in beam splitters by taking advantage of their symmetry for canceling their polarization properties. While the cancellation is not perfect, we have demonstrated that the residual linear retardance can be reduced to less than 1.5°, and that the residual linear diattenuation can be reduced to less than 0.03, across a spectral range spanning almost the entire visible spectrum. Using the instrument, we have shown that even without calibration we can measure the normalized Stokes parameters to an absolute accuracy of better than 0.05, and to an absolute accuracy of better than 0.006 with calibration.

Using the proposed system, we have shown that it is possible to make a high-accuracy polarimeter with a temporal sampling speed of 30 μs, and a real-time calculation delay of only 3 μs, while the optical system is only $52\times 30\times 25\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{mm}$ in size. This system is straightforward to assemble and align, and consists of readily available inexpensive components, and yet is capable of achieving a temporal resolution that is faster than that of photoelastic modulator (PEM)-based systems, though its field of view is more restrictive.

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