Optical Engineering 58(10), 103103 (October 2019)

Video-rate full-Stokes imaging polarimeterusing two polarization cameras
Shuhei Shibata,a,* Masayuki Suzuki,b Nathan Hagen,b,c and Yukitoshi Otanib,caUtsunomiya University, Center for Industry-University Innovation Support, Utsunomiya, Japan
Utsunomiya University, Department of Optical Engineering, Utsunomiya, Japanc
Utsunomiya University, Center for Optical Research and Education, Utsunomiya, Japanb

Abstract. We show how to construct and calibrate a full-Stokes imaging polarimeter system by combining
the video data from two separate polarization cameras with a nonpolarizing beamsplitter and a waveplate.
As a result, this system can capture the full Stokes vector at each pixel for 3 megapixel images at up to 60 Hz.
To demonstrate some of the advantages of measuring the s 3 Stokes vector component that is normally notmeasured in polarization cameras, we show three experiments: viewing three-dimensional glasses, detecting
a scarab beetle in a natural environment via the circular dichroism of its shell, and mixing an optically active
liquid with a neutral liquid. © 2019 Society of Photo-Optical Instrumentation Engineers (SPIE) [DOI: 10.1117/1.OE.58.10.103103]
Keywords: polarization; image polarimetry; snapshot measurement.
Paper 190895 received Jul. 3, 2019; accepted for publication Sep. 24, 2019; published online Oct. 15, 2019.

1 Introduction
A polarization camera is a division of focal plane imaging
polarimeter employing an array of micropolarizers oriented
at 0 deg, 45 deg, 90 deg, and 135 deg attached to the pixels
of a detector array.1–3 Many researchers have taken advantage of the video-rate snapshot capabilities of polarization
cameras to improve measurement capabilities in interferometry,4,5 differential interference contrast microscopy,6
three-dimensional (3-D) profilometry,7 holography,8,9 andbirefringence measurement.10
A disadvantage of a micropolarizer polarization camerais that a single camera cannot measure the polarization
state ellipticity. To measure elliptically or circularly polarized light—the s3 Stokes vector component—it is necessary
to use microretarders on the pixels of the camera and not justmicropolarizers. One advantage of measuring the complete
Stokes vector (s0 through s3 ) and not just the linear components (s0 through s2 ) given by a standard polarization camerais that the degree of polarization (DOP) can be calculated,
separating the polarized component from the unpolarizedcomponent of the light. While we have previously shownhow to construct an imaging Stokes polarimeter using arotating retarder in front of a polarization camera,11 this canonly measure the Stokes vector at a rate of 10 Hz, requires
synchronization with a rotating motor, and is restricted tosmall aperture lenses due to the motor.
For video-rate snapshot imaging, we propose combining
two polarization cameras. In the discussion below, we first
explain how to measure the complete polarization state in asnapshot and show how to calibrate the (unwanted) residual
polarization properties of the beamsplitter and waveplate inour system. Finally, we demonstrate the advantage of capturing the complete state of polarization by imaging a pair of
3-D glasses, tracking the motion of a scarab beetle using
the circular dichroism of its shell, and viewing the mixing*Address all correspondence to Shuhei Shibata, E-mail: shibata_s@cc.utsunomiya-u.ac.jp

Optical Engineeringof an optically active liquid (sugar water) with an opticallyneutral liquid (salt water).
2 Measurement Procedure of Imaging Stokes
Polarimetry
To measure the complete polarization state image in a snapshot, the system combines a “nonpolarizing” beamsplitter,a quarter-wave plate oriented at 0 deg, and two polarization
cameras in the transmission/reflection arms of the beamsplitter, as shown in Fig. 1. For an ideal system, the input
polarization state for this ideal setup can be measured witheight intensities I i , where i ¼ 0 ∼ 3 indicates the four pixels
in the transmission-arm camera and i ¼ 4 ∼ 7 indicates thefour pixels in the reflection-arm camera. We can model theseeight intensities detected by two polarization cameras as afunction of the input Stokes parameters s ¼ ½s0 ; s1 ; s2 ; s3 T
via the Mueller calculus. In matrix form,12 we can write
I ¼ Ws;where W is the ideal measurement matrix
0

1
B1
B
B1
B
1B
1
W¼ B
B
2B1
B1
B
@1
1

EQ-TARGET;temp:intralink-;e002;326;265

1
0
−1
0
1
0
−1
0

0
0
0
0
0
1
0
−1

1
0
1 C
C
0 C
C
−1 C
C:
0 C
C
0 C
C
0 A
0

(2)

The Stokes vector s of the input light is estimated bys ¼ AI;(3)

EQ-TARGET;temp:intralink-;e003;326;134

where the ideal analysis matrix A is0091-3286/2019/$28.00 © 2019 SPIE

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EQ-TARGET;temp:intralink-;e001;326;307

October 2019 • Vol. 58(10)Shibata et al.: Video-rate full-Stokes imaging polarimeter using two polarization cameras

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
ðs00 þ s21 þ s22 Þ, (−1 ≦ e ≦ 1). The orientation angle θ ofthe ellipse major axis is given by θ ¼ 0.5 tan−1 ðs2 ∕s1 Þ,(−90 deg ≦ θ ≦ 90 deg).13

Fig. 1 The optical layout of the snapshot full-Stokes imager.

0

1
1B
2
A¼ B
4@0
0

EQ-TARGET;temp:intralink-;e004;63;515

1
0
0
4

1
−2
0
0

1
0
0
−4

1
2
0
0

1
0
−4
0

1
−2
0
0

1
1
0C
C:
4A
0

(4)

Once we have determined the Stokes parameters at eachpixel in the scene, we can calculate any desired polarization
properties
them. For example, the DOP is obtained as
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
pfrom
2
DOP ¼ s1 þ s22 þ s23 ∕s0 , (0 ≦ DOP ≦ 1). One advantage
is that we can show the pure-polarization Stokes vector s 0
without nonpolarized light by s 0 ¼ ½DOPs0 ; s1 ; s2 ; s3 T.
Using s 0 , we can calculate the ellipticity e as e ¼ s3 ∕3 Experimental Equipment
Our system shown in Fig. 2(a) uses two of the same polarization camera models produced by SONY, each having a 2064 ×
1560 grid of 2.5-μm pixels, a visible-near infrared detection
range, 60 fps, and 12 bit depth. The micropolarizers are oriented at 0 deg, 45 deg, 90 deg, and 135 deg related to 2 ×2 pixels. The nonpolarized beamsplitter is designed to operate
in the visible spectrum range and is 20 × 20 mm in size. The
quartz quarter-wave plate is designed to operate at 632.8 nm.
When constructing this system, there are several details thatneed to be taken into account. First, the presence of the beamsplitter and optical mounts means that the back focal distanceof the objective lens has to be at least 50 mm for the image
plane to occur at the face of the detector. Thus, a standard single lens reflex objective lens, with typical back focal distancesof 42 mm or so, are not compatible. To obtain a long backfocus, we use a Fujinon EX 75 projection lens, which has afocal length f ¼ 75 mm and a back focal distance of 62.1 mm.
Second, one must also take care to prevent glancing
reflections off the interior faces of the beamsplitter. For
example, if the camera is assembled without an aperture stop
designed for this purpose, then the detector array will likely
see multiple subimages overlapping the main image, as
shown in Fig. 2. Rays producing the unwanted images resulting from glancing reflections are shown with red and blue
dashed lines, while the rays producing the main image areshown as black solid lines. To remove these unwanted rays,
we insert an aperture stop with diameter 7 mm set between
the imaging lens and the beamsplitter, as shown in Fig. 2(a).

Fig. 2 (a) The system layout including the lens and imaging rays. The aperture stop iris is stopped down
to cut the innermost rays (the red and blue dashed lines appearing closest to the optical axis) that canglance off the interior faces of the beamsplitter. (b) The detected pair of images with the aperture left wide
open (showing several overlapping subimages). (c) The detected pair of images with the iris stopped
down such that only the main image is visible.

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October 2019 • Vol. 58(10)Shibata et al.: Video-rate full-Stokes imaging polarimeter using two polarization cameras

In general, the two images captured by the two cameras
will not be aligned to one another, as in Fig. 3(a). [The image
in the beamsplitter’s reflected path of Fig. 3(a) has been
flipped horizontally to present the same chirality.] To alignthe two images, we estimate the relative image translation via
the phase-correlation method14 for the two polarization cameras. First we calculate the intensity (s0 ) images from eachcamera, I T ðn; mÞ and I R ðn; mÞ as
I T ðn;mÞ ¼ I 0 þI 1 þ I 2 þ I 3 I R ðn;mÞ ¼ I 4 þ I 5 þI 6 þI 7 ;EQ-TARGET;temp:intralink-;e005;63;664(5)where ðm; nÞ are the pixel coordinates. Subscripts “R” and“T” represent the reflection and transmission arms after thebeamsplitter. If we take I T as the reference image, then theimage I R is shifted relative to I T by ðm0 ; n0 Þ:
I R ðm; nÞ ¼ I T ðm − m0 ; n − n0 Þ:

EQ-TARGET;temp:intralink-;e006;63;589(6)

The Fourier transform of Eq. (6) is given as
FR ðξ; ηÞ ¼ FT ðξ; ηÞ exp½−i2πðm0 ξ þ n0 ηÞ;

EQ-TARGET;temp:intralink-;e007;63;547(7)where ðξ; ηÞ are the Fourier-domain conjugates of ðx; yÞ.
Since a linear shift of the image in the canonical domain
produces a linear phase in the spatial frequency domain,
the phase correlation function Δðm; nÞ can be estimatedby applying an inverse Fourier transform of the ratio
FR ðξ; ηÞ∕FT ðξ; ηÞ. This method, however, is sensitive todifferences in average intensity between the two images, andachieving equal intensity can be difficult for our system dueto the differences in beamsplitter transmission and reflection.
To get around this problem, we modify the calculation to use
the normalized cross-power spectrum

EQ-TARGET;temp:intralink-;e008;63;395

FR ðξ; ηÞFT ðξ; ηÞ¼ exp½−i2πðm0 ξ þ n0 ηÞ;
jFR ðξ; ηÞFT ðξ; ηÞj

where * represents a complex conjugate. The phase correlation function now contains little dependence on different DC
values between two images and can be obtained by applying
the inverse Fourier transform to the result of Eq. (8), giving
the result shown in Fig. 3(b). The peak position ðm0 ; n0 Þ ofthe phase correlation indicates the shift amount between thereflection image I R and the transmission image I T . Using thispeak position ðm0 ; n0 Þ, we can match the two images, as
shown in Fig. 3(c).

(8)

4 Calibration Method
To estimate the Stokes image via Eq. (3), we have assumed
the beamsplitter to have no polarization dependence and thequarter-wave plate to be ideal. To measure the actual polarization properties of the beamsplitter, we measure its spectroscopic Mueller matrix (400 to 800 nm) near the opticalaxis by an AxoScan Mueller matrix polarimeter.15 From themeasurements, we find that the nominally nonpolarizing
beamsplitter has small linear diattenuation values DT and
DR , and large linear retardance values δT and δR , along thetransmission and reflection arms, as shown in Fig. 4. Other
polarization properties such as circular retardance and depolarization (among others) are measured to be approximately
zero; therefore, we ignore them.
To calibrate the linear diattenuations and linear retardances, we align the orientation 0 deg of the beamsplitter retardance and the quarter-wave plate fast axis, allowing for a
simple correction. With a shared orientation axis, the retardances of the beamsplitter and waveplate sum to become
one total retardance δtotal ¼ δT þ δQWP that acts like a single
retarder, where δQWP is the retardance of the quarter-wave
plate alone. We can rewrite the Mueller matrices of the beamsplitter in transmission and reflection as MT and MR , where
each matrix represents a retarder oriented along the horizontal and vertical directions, respectively:

Fig. 3 (a) Addition of the two intensity images without correcting for translation between the two. (b) The
phase-correlation function ðm 0 ; n 0 Þ is calculated by applying an inverse Fourier transform of Eq. (8).
(c) Addition of the two intensity images after correcting for translation ðm 0 ; n 0 Þ.
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October 2019 • Vol. 58(10)Shibata et al.: Video-rate full-Stokes imaging polarimeter using two polarization cameras

Fig. 4 The spectroscopic (a) linear diattenuation and (b) linear retardance measurements for the transmission (black line) and reflection (red dashed line) directions of the beamsplitter, as measured by
the AxoScan Mueller polarimeter.

0

1
B DT
MT ¼ B@ 0
0

EQ-TARGET;temp:intralink-;e009;63;549

0

1
B DR
MR ¼ B@ 0
0

EQ-TARGET;temp:intralink-;e010;63;488

DT
1
0
0

0
0
ð1 − DT Þ cosðδtotal Þ−ð1 − DT Þ sinðδtotal Þ

DR
1
0
0

0
0
−ð1 − DR Þ cosðδR Þ
ð1 − DR Þ sinðδR Þ

1
0
C
0
C;
ð1 − DT Þ sinðδtotal Þ A
ð1 − DT Þ cosðδtotal Þ

(9)

1

0
0

C
C:
−ð1 − DR Þ sinðδR Þ A−ð1 − DR Þ cosðδR Þ(10)

The measurement matrix W 0 can be rewritten using MT and MR as0

1 þ DT
B 1
B
B 1 − DT
B
B 1
1
0
W ¼ B
2B
B 1 þ DR
B 1
B
@ 1 − DR
1

EQ-TARGET;temp:intralink-;e011;63;429

1 þ DTDT−1 þ DTDT
1 þ DRDR−1 þ DRDR

100
ð1 − DT Þ cosðδtotal Þð1 − DT Þ sinðδtotal Þ C
C
C
0
0
C
−ð1 − DT Þ cosðδtotal Þ −ð1 − DT Þ sinðδtotal Þ C
C:
C
0
0
C
−ð1 − DR Þ cosðδR Þ−ð1 − DR Þ sinðδR Þ CCA00ð1 − DR Þ cosðδR Þ
ð1 − DR Þ sinðδR Þ(11)

The analysis matrix A 0 is given by the pseudoinverse of W 0
0

a00
B
1
a10
A0 ¼ B
16 @ a20
a30

EQ-TARGET;temp:intralink-;e012;63;292

a01
a11
a21
a31

a02
a12
a22
a32

a03
a13
a23
a33

a04
a14
a24
a34

a05
a15
a25
a35

a06
a16
a26
a36

1
a07
a17 C
C;
a27 A
a37

(12)

the elements of which are given asa00 ¼ −3DR − DT þ 2

EQ-TARGET;temp:intralink-;e013;63;213

a01 ¼ a03 ¼ a05 ¼ a07 ¼ 2

a02 ¼ 3DR þ DT þ 2

a06 ¼ DR þ 3DT þ 2 a10 ¼ −3DR þ DT þ 4

a11 ¼ a13 ¼ −3DR þ DT

a14 ¼ DR − 3DT þ 4

a16 ¼ DR − 3DT − 4

a15 ¼ a17 ¼ DR − 3DT

8 sin δtotal
ðDT − 1Þ sinðδR − δtotal Þ
8 cos δtotal;
a35 ¼ −a37 ¼
ðDT − 1Þ sinðδR − δtotal Þ
Optical Engineering

a12 ¼ −3DR þ DT − 4

8 sin δR
ðDT − 1Þ sinðδR − δtotal Þ
8 cos δR
a31 ¼ −a33 ¼
ðDT − 1Þ sinðδR − δtotal Þ

a20 ¼ a22 ¼ a24 ¼ a26 ¼ a30 ¼ a32 ¼ a34 ¼ a36 ¼ 0
a25 ¼ −a27 ¼ −a04 ¼ −DR − 3DT þ 2

a21 ¼ −a23 ¼ −(13)103103-4

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where DT and DR are assumed to be small so that we approximate DjT ≈ 0, DjR ≈ 0, for j ∈ ð2; 3; 4Þ.

To use analysis matrix A 0 for Stokes parameter measurement, we need to calibrate the values of DT , DR , δtotal , and δR
at each pixel ðm; nÞ of the respective camera. The calibration
values can be obtained by imaging four states of polarization:
0 deg, 45 deg, and 90 deg linear polarization, plus rightcircular polarization. We label detected intensities incident onthese polarization as I HLi , I 45°Li , I VLi , and I RCi , respectively.
The following equations give the calibration procedure:I HL0 − I VL2
I HL0 þ I VL2I− I VL6
DR ¼ HL4
I HL4 þ I VL6ðI− I RC3 ÞðI 45°L1 þ I 45°L3 Þδtotal ¼ tan−1 RC1
ðI RC1 þ I RC3 ÞðI 45°L1 − I 45°L3 Þ

EQ-TARGET;temp:intralink-;e014;63;642

DT ¼

δR ¼ tan−1

ðI RC5 − I RC7 ÞðI 45°L5 þ I 45°L7 Þ:ðI RC5 þ I RC7 ÞðI 45°L5 − I 45°L7 Þ(14)

To confirm the above calibration method of our polarimeter at 550  5 nm using bandpass filter set in front of
the beamsplitter, we measure the input polarization statesobtained by passing light through a fixed polarizer oriented
at 0 deg followed by a rotating quarter-wave plate. The influence of wavelength dispersion is ignored because thechange of the beamsplitter and linear retardance and linear
diattenuation is small in narrow spectrum range. Figure 5

shows the normalized Stokes parameters with and without
the calibrated measurements (i.e., using A 0 or A), where eachmeasurement point is obtained by averaging 10 images andtaking the mean value of all pixels. We can see that the calibration pushes the estimated states closer to the known input
states. The RMS errors between the estimated Stokes parameter values and the known input values improve from 0.138
to 0.033 for s̃1, from 0.082 to 0.015 for s̃2, and from 0.109 to
0.016 for s̃3.
The calibration procedure of Eqs. (12)–(14) can be used
independently at each pixel of the two cameras, so any fielddependence to the parameters is corrected for. In addition,
each pixel of the polarization cameras can have its polarization characteristics vary from its nominal values, so somecameras may require a pixel-dependent calibration of thepolarizer filter microgrid as well.16
5 Experimental Results
To show our polarimeter’s advantages of capturing videorate full-Stokes vectors, we demonstrate with three experiments. In the first demonstration, we measure the polarization state passing through a pair of 3-D glasses as the glasses
are turned over by hand from the white light flat source with
nonpolarization. The “lens elements” of each pair of 3-D
glasses are composed of one right and one left circular polarizer (one at each eye). If the glasses are viewed in reverse, thetwo “lens elements” become linear polarizers oriented at thesame angle. In Video 1 (Fig. 6), we can see the difference inthe right and left circularly polarized light in the ellipticity
image, as e ≈ þ1 or e ≈ −1. After the glasses are turned over,Fig. 5 The normalized Stokes parameters for 0 deg polarized light passed through a rotating quarterwave plate. The solid lines indicate the known input state while the squares show estimates obtained
without the calibration procedure (i.e., using analysis matrix A), while the filled circles show the estimatesobtained with the calibration procedure (i.e., A 0 ).

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Fig. 6 The polarization images of the pair of 3-D glasses, captured at 30 Hz (Video 1, MPEG-4, 5.2 MB[URL: https://doi.org/10.1117/1.OE.58.10.103103.1]). The ellipticity image describes the degree of right(+1) and left (-1) circularly polarized light. From the figure, we see that one of the glasses is right- and theother left-circularly polarized. After the glasses are turned over, we can see that the ellipticity of bothglasses becomes zero. When viewed in reverse the glasses become linear polarizers.

Fig. 7 The polarization video of a scarab beetle walking on green leaves, captured at 30 Hz (Video 2,
MPEG-4, 6.9 MB [URL: https://doi.org/10.1117/1.OE.58.10.103103.2]). The beetle and leaves are similar
in color and are therefore difficult to distinguish. However, the beetle is easily detected in the ellipticity
image, where the circular diattenuation of its shell makes it stand out.

we can see that the two elements change into linear polarizers both oriented at 0 deg.
In our second demonstration, we observe a living
Japanese green scarab beetle (Anomala daimiana) under theroom light (the fluorescent light). It is well known that scarabbeetles’ shells exhibit circular diattenuation, as first discovered by Albert Michelson in 1911.17 Figure 7 shows theintensity image and ellipticity image of the scarab beetle
walking around a bed of green leaves, similar to what one
may find in natural conditions. In Video 2, we can see thatthe beetle and leaves are similar in color and are therefore
difficult to distinguish. However, the beetle is easily detectedin the ellipticity image, where the circular diattenuation ofits shell makes it stand out. Thus, a full Stokes imaging
polarimeter can be used as a scarab beetle detector, which
may have utility for farmers because scarab beetles arecommon vegetable-eating pests.
In our third demonstration, we measure the polarization
image obtained by passing 22.5 deg linearly polarized lightthrough a transparent liquid (a salt solution) and pouring asugar (sucrose) solution into the container. To reduce lightscatter as much as possible, we match the refractive indexes(n ¼ 1.376) of both solutions by taking the concentrations ofsugar as 30% by weight and that of salt as 24.18%. Because
Optical Engineeringsugar water is an optically active material, the polarization
angle rotates as it passes through the sugar solution. Salt
solution is optically neutral and produces no polarization
rotation. In Video 3 (Fig. 8), we show the DOP and thepolarization angle θ images, from which we can see thatin the intensity video the sugar water is indistinguishable
from the salt solution in the intensity images, but can beclearly distinguished and quantified by analyzing the polarization angle video. If this phenomenon is measured by asingle polarization camera, we cannot differentiate between
polarization change and scattering; however, our polarimeter,
through the full Stokes measurement, can show that scattering is negligible because DOP ≈ 1 during mixing (Fig. 8).
6 Conclusion
We have detailed the construction methods and calibration
for a video-rate Stokes imaging polarimeter constructed withtwo polarization cameras. The “nonpolarizing” beamsplitterused in this system is not an ideal element, but we havesucceeded in measuring the normalized Stokes parameterto an accuracy of better than 3.5% with the described calibration procedure. Finally, we demonstrated the advantagesof having access to the full Stokes vector in a video format
by collecting videos of a pair of 3-D glasses, a scarab beetle

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Fig. 8 The polarization video showing a sugar solution being poured into a salt solution tank, captured at
60 Hz (Video 3, MPEG-4, 11 MB [URL: https://doi.org/10.1117/1.OE.58.10.103103.3]). The backgroundlight source behind the tank is linearly polarized. The sugar water is indistinguishable from the salt
solution in the intensity images but can be clearly distinguished and quantified by analyzing the polarization angle images.

walking through a bed of leaves, and the mixing of a sugar
solution into a salt solution. The system currently captures
videos at a rate of 60 Hz, the frame rate of the two cameras.
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