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Compact and high-speed Stokes polarimeterusing three-way polarization-preserving
beam splitters
SHUHEI SHIBATA,1,* NATHAN HAGEN,2,3

SHUICHI KAWABATA,4 AND YUKITOSHI OTANI2,3

1

Center for Industry-University Innovation Support, Utsunomiya University, 7-1-2 Yoto, Utsunomiya, Tochigi, Japan
Department of Optical Engineering, Utsunomiya University, 7-1-2 Yoto, Utsunomiya, Tochigi, Japan
3
Center for Optical Research and Education, Utsunomiya University, 7-1-2 Yoto, Utsunomiya, Tochigi, Japan
4
Faculty of Engineering, Tokyo Polytechnic University, 1583 Iiyama, Atsugi, Kanagawa, Japan*Corresponding author: shibata_s@cc.utsunomiya-u.ac.jp
2

Received 22 May 2019; revised 25 June 2019; accepted 26 June 2019; posted 27 June 2019 (Doc. ID 368239); published 17 July 2019

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 × 30 × 25 mm) because it consists only of small volume optical devices. We show that the system can measurearbitrary 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 witha hammer. © 2019 Optical Society of America
https://doi.org/10.1364/AO.58.005644

1. INTRODUCTION
A real-time polarimeter is needed for measuring and monitoring dynamic polarization states, such as for monitoring thinfilm 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 makecompact.
In 1982 and 1985, Azzam proposed two compact Stokes
polarimeter designs, both of which use four detectors. The firstpolarimeter uses three beam splitters to separate the beam to becollected at the four detectors, allowing the system to measure
the linear Stokes vector elements (s 0 , s1 , and s2 ) [5,6]. The second polarimeter uses only the four detectors themselves, suchthat the reflection from the surface of each detector face induces
a polarization change in the beam, while the fraction of lightabsorbed by the detector is used in measurement [7]. Thissystem 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 forthe second system, the alignment is difficult—due to the need1559-128X/19/215644-06 Journal © 2019 Optical Society of America

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 inducedby 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 ofmeasurement. In order to demonstrate the system’s advantages,
in Section 3 we show experimental results and evaluate themeasurement accuracy and speed.
2. COMPACT AND HIGH-SPEED STOKESPOLARIMETER USING THREE POLARIZEDBEAM SPLITTERS
Figure 1 shows the optical setup for the high-speed measurement of input Stokes parameters [s0 , s1 , s2 , 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 , s2 , 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

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Fig. 1. Optical setup of Stokes polarimeter using 3WPPS.

configuration of six normal beam splitters forming the threeway 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 thatthe state of polarization (SOP) in the three outputs is the same as
the input SOP. Figure 2 shows a related setup to separate a beamin 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 δ1 andδ2 , and linear diattenuations D1  p1 − q1 ∕p1  q1  andD2  p2 − q 2 ∕p2  q2 , 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 (p1 ≈ p3 , q1 ≈ q3 , and δ1 ≈ δ3 for transmission mode, p2 ≈ p4 , q2 ≈ q 4 , and δ2 ≈ δ4 for reflectionmode), rotated by 90° relative to the first, the total linear diattenuations and linear retardances become D1 − D3  p3 q1 
q3 p1 ∕p3 q1  q3 p1  ≈ 0 and δ1 − δ2 ≈ 0 for transmissionmode, and D2 − D4  p4 q2 − q4 p2 ∕p4 q2  q4 p2  ≈ 0 andδ2 − δ4 ≈ 0 for reflection mode, respectively. Therefore, thepolarization effects induced by the first beam splitter are canceled by the second beam splitter if the second splitter is rotatedby 90°. In a similar way, we can connect two of these in series toseparate 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 thisassembly 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 thethree 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

Fig. 2. Concept for maintaining the input polarization while dividing the light into two beams.

s̃1 

I 0° − I 90°,
I 0°  I 90°s̃2 

I 45° − I −45°,
I 45°  I −45°s̃3 

I RC − I LC,
I RC  I LC
(1)

Fig. 3. Optical setup for Stokes parameter measurement using a PBS and detector pair oriented at (a) 0° to split horizontal and verticalpolarizations for measuring s1 , (b) 45° to split the two states needed for measuring s2 , and (c) 0° in combination with a QWP oriented at
45° for measuring s3 .

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where I is the light intensity measured by a photodetector, as
indicated in Fig. 3.
In a realistic model, the PBS will have a different transmittance αi and reflectance βi , and the gains ki and ki0 of eachdetector will be different (i  1 for s̃1, i  2 for s̃2, and i 
3 for s̃3 ), so that it is necessary to calibrate them. If we combine
the αi , βi , ki , and ki0 into one equation, then we can define three
total calibration parameters g i that need to be determined:gi 

k i αi:ki0 βi(2)

In order to measurep
g iﬃﬃ,ﬃ we canpﬃﬃﬃ pass linear polarization oriented
at 22.5°, s̃  1, 1∕ 2, 1∕ 2, 0, into the Stokes polarimeter.
The g i parameters are then determined bypﬃﬃﬃ
2 − 1 I 0°
g 1  pﬃﬃﬃ,·
2  1 I 90°

pﬃﬃﬃ
2 − 1 I 45°
g 2  pﬃﬃﬃ,·
2  1 I −45°Ig 3  RC :
I LC
(3)

Finally, the calibrated Stokes parameters can be estimated withthe calibration parameters g i as
I 0° − g 1 I 90°,
I 0°  g 1 I 90°
I − g 3 I LC:s̃3  RCI RC  g 3 I LC

s̃1 

s̃2 

I 45° − g 2 I −45°,
I 45°  g 2 I −45°
(4)

3. EXPERIMENTAL RESULTS OF THE STOKESPOLARIMETERWe 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 tomaintain the input polarization state. Figure 4 shows the
spectroscopic measurement of linear retardance δλ and linear
diattenuation Dλ of the 3WPPS by an Axoscan Mueller matrix polarimeter [9], indicating that the δλ and Dλ values
are small but not quite zero. Thus, the six normal beam splittersused 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 retardanceand circular diattenuation implies that errors in azimuthal anglealignment (rotation about the optical axis) are not to blame.
In experiments introduced below, we use a λ  633 nm
light source. Taking the retardance and diattenuation fromthe measurements of Fig. 4, we can simulate the influence thatthe nonideal behavior in δ and D have on the polarization statetransmitted by the 3WPPS (Fig. 5). For all input SOPs, we findthat the normalized Stokes parameter measurement error is lessthan 0.033 for s̃1, 0.046 for s̃2, and 0.042 for s̃3. These errors
can be further reduced by calibrating the residual linear diattenuation and linear retardance.
In order to calibrate these residual values, we can substitutethese values from Fig. 4 in the Stokes parameter measurement
model. If we assume that the residual linear diattenuation Diand residual linear retardance δi produced by transmission/reflection before each detector’s polarizing beam splitter

Fig. 4. Measurements of beam splitter polarization properties by the Axoscan polarimeter. In (a)–(d), the black dashed curves indicate the values
after the first beam splitter, while the red curves indicate the values after the combination of two beam splitters. Panels (e) and (f ) show the residual
values produced before each detector’s polarizing beam splitter, i.e., exiting the 3WPPS at points (1), (2), and (3), as shown.

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Fig. 5. Simulation of the impact that the residual retardance and diattenuation of the 3WPPS will have on the measurement of the Stokes
parameters at λ  633 nm. Plus/minus ellipticity indicates right/left-handedness. The image gray scale indicates the error in the normalized
Stokes parameters.

Fig. 6. Measurement setup.

(the three splitters shown in Fig. 3) are small, then we haveD2i ≈ 0, sin δi ≈ δi , and cos δi ≈ 1. In this regime, theinitial normalized Stokes parameter estimates s̃1 , s̃2 , and s̃3 obtained from Eq. (4) can be corrected to s̃10 , s̃20 , and s̃30 usings̃10 

s̃1 − D1,
1 − s̃1 D1s̃30  s̃3 

s̃20  s̃2 D3 s̃1 s̃3
 δ3 s̃2 :1 − s̃1 D1

D2 s̃1 s̃2− δ s̃ ,1 − s̃1 D1 2 3(5)

Residual linear diattenuations D1 , D2 , and D3 are shownfor the light exiting the 3WPPS at locations (1), (2), and (3),respectively, as indicated in Fig. 4(e). The corresponding
residual linear retardances δ1 , δ2 , and δ3 are likewise shownin Fig. 4(f ). This correction step is only needed for those applications where the imperfect cancellation of the beam splitterpolarization properties causes an unacceptable level of error. If
the six normal beam splitters of the 3WPPS have entirely thesame polarization properties, this step is not necessary.

Fig. 7. Measured normalized Stokes parameters of (a) a rotating analyzer and (c) a rotating QWP. The dashed curves indicate the known polarization state, while the squares indicate the measurements. (b) and (d) show differences between the measured and ideal normalized Stokes parameters. Filled dots show the measurements corrected using Eq. (5), while the open squares show the uncorrected measurements obtained directly from
Eq. (4).

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Fig. 8. Measured normalized Stokes parameters of the QWP rotatedby the servomotor in 25 rps during (a) 250 ms, (b) 1 ms.

Figure 6 shows the equipment layout used to evaluate theproposed Stokes polarimeter, composed of an He–Ne laser ofwavelength 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 thenormalized 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 theanalysis below. The measurement speed of the polarimeter is

Research Article
determined by the speed of the A/D converters (16-bit,
3×104 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 arotating 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(ω  25 rps) by a servomotor [Fig. 8(a)]. Even for the resulting 100 Hz frequency modulations, produced from s̃1 cos4ωt and s̃2  cos4ωt  π∕2, and 50 Hz frequency
modulations produced by s̃3  sin2ωt, 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 × 20 × 100 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 shownin Fig. 9(a). For this measurement, we use right-handed circularpolarization incident on the cube, so that we can observe birefringence in any orientation within the material. From theFig. 9. Demonstration of high-speed measurement of the polarization state after passing through an acrylic block due to impact by a hammer.
(a) Time-resolved Stokes parameters; (b) linear retardance δPMMA and azimuthal angle ϕ calculated by Eq. (6) using the Stokes parameters of (a);(c) principal stress difference Δσ calculated by Eq. (7) using the linear retardance of (b).

Research Article
resulting measurements, we calculate the induced linear retardance δPMMA and the azimuthal angle of birefringence ϕ, fromwhich we obtain the average azimuthal angle of the principal
stress difference Δσ within the PMMA block [Fig. 9(b)]:pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
01
s 102  s202
−1 s 2−1
δPMMA  tan: (6)ϕ  tan 0 ,s1
2s30
If we assume that the PMMA block is thin enough to be treatedunder the principles of two-dimensional photoelasticity, we canestimate the average principal stress difference Δσ along thepath of the light, using the measured retardance δPMMA given
by Eq. (6), the path length through the block d  20 mm,
the light wavelength λ  632.8 nm, and the stress-optic coefficient G  6.0 × 10−12 Pa−1 of the acrylic (PMMA),
λ
δ
:
(7)
Δσ 
2πGd PMMA
The time period during which we can observe stress changes in
the material is about 1 ms after impact, with the maximum
observed principal stress difference being 3.09 MPa. If we takethe time derivative of the measurements, we find that the measured stress Δσ increases slowly over the first 100 μs or so, thenincreases in a roughly linear fashion over the next 500 μs, after
which the stress reaches a peak and begins to relax. During theperiod when the stress builds, we see that the orientation ϕ ofthe path-averaged stress rotates from a starting point near −40°
(the static stress orientation of the material) to about −80°, after
which it undergoes an almost instantaneous 150° rotation toabout 70°. After this point, the stress orientation changes
are smoother, showing that the angle slowly relaxes back to
its original static orientation. The sudden shift by 150° is clearly
not a wrapped phase phenomenon because the orientation relaxes smoothly back to its preimpact state.
4. CONCLUSION
A real-time Stokes polarimeter with three PBSs and a 3WPPS
has been proposed. The 3WSPPS is a method of nullingthe unwanted polarization effect in beam splitters by taking

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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 toless 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 haveshown 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 × 30 × 25 mm in size. Thissystem is straightforward to assemble and align, and consists ofreadily available inexpensive components, and yet is capable ofachieving 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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