LetterVol. 44, No. 4 / 15 February 2019 / Optics Letters

891

Robust full Stokes imaging polarimeter withdynamic calibration
SHUHEI SHIBATA,1,2,* NATHAN HAGEN,1,2

AND YUKITOSHI OTANI

1,2

1

Faculty of Engineering, Utsunomiya University, 7-1-2 Yoto, Utsunomiya, Tochigi, Japan
Center for Optical Research and Education, Utsunomiya University, 7-1-2 Yoto, Utsunomiya, Tochigi, Japan*Corresponding author: shibata_s@opt.utsunomiya-u.ac.jp

2

Received 5 November 2018; revised 10 January 2019; accepted 11 January 2019; posted 14 January 2019 (Doc. ID 349490);
published 7 February 2019

We present a full Stokes imaging polarimeter using a rotating
retarder in combination with a polarization camera—a detector array on which a pixelated polarizer array is attached. By
itself, a polarization camera cannot capture the full Stokes
parameters, but we add a rotating retarder in front and show
how it can be used to provide full Stokes images. In addition,
we demonstrate the advantage that it can be recalibrated
dynamically while taking measurements, allowing accurate
measurements even in environments where the retardancein changing. © 2019 Optical Society of America
https://doi.org/10.1364/OL.44.000891

Polarization measurement is important in many applications, suchas visualizing stress within a material and determining an object’s
surface orientation and shape. The complete polarization state ofincoherent light can be expressed using the four Stokes parameterss 0 , s1 , s 2 , and s3 , indicating the overall light intensity component,
the difference between horizontal and vertical linear polarization,
the difference between 45° and 135° linear polarization, and thedifference between right and left circular polarization.
Many methods for Stokes parameter measurement havebeen proposed [1–7], among which one widely used method
employs a rotating retarder and a rotating analyzer [1]. Thismethod is capable of calibrating the waveplate retardance simultaneously with measurement of the Stokes parameters. It
is difficult to operate at high speed due to the trouble of accurately synchronizing dual continuously rotating motors together with the camera readout at high speed. As a result,typical dual rotating retarder and analyzer systems designed
for imaging generally require a few minutes for measurements.
Another polarization measurement method employs pairs ofliquid crystal retarders or photoelastic modulators [2,3],
allowing measurement without mechanical rotation. Whilethese two methods can measure quickly, they possess otherdrawbacks. And whereas a liquid crystal polarimeter can often
achieve images at up to 30 frames per second (fps), it is known
to have a lower accuracy than other methods due to nonuniformity among the liquid crystal elements and scattering.
Finally, a system based on photoelastic modulators can often
achieve a rate limited by the camera readout speed, but it
has a small aperture that limits light collection and field of view.
0146-9592/19/040891-04 Journal © 2019 Optical Society of America

Stokes imaging can also be performed in snapshot form (i.e.,
at a rate limited by the detector readout) by encoding thepolarization state into a channeled image, without electronic
or mechanical scanning [4]. Its resolution, however, is reducedby the need to perform low-pass filtering of the image in orderto reconstruct the Stokes images.
Polarization cameras—detector arrays to which have been attached pixelated micropolarizer filters (see Fig. 1)—have recently
become available, allowing the imaging of three Stokes parameters, s 0 , s1 , and s2 , in snapshot form [5,6]. However, these cameras cannot measure s3 directly because the filter array does notinclude a retarder element. A polarization camera that does employ a microretarder array attached to a micropolarizer array and
CCD sensor has been demonstrated [7]. This camera can measure the full Stokes parameters in real-time, but requires that theindividual retardances in the microretarder array be calibrated inadvance of a measurement. This can cause difficulty in the presence of temperature changes, which cause the retardances tochange as well. Imaging the Stokes parameters with a rotating
monolithic retarder and micropolarizer-equipped polarization
camera allows a user to remove this temperature sensitivity.
This paper demonstrates two new methods for measuring thecomplete spatially resolved Stokes parameters, including s3 ,either statically or dynamically, using a single rotating retarderelement placed in front of the polarization camera. Vaughn et al.
proposed a Mueller matrix polarimeter using rotating retarders
and a micropolarizer array but have not applied this approachto Stokes imaging and do not investigate its dynamic calibration
capabilities [8]. In method 1, the retardance of the rotating
retarder is calibrated in advance, and then Stokes parameters
are measured. In method 2, the retardance of the rotating

Fig. 1. Optical setup of rotating retarder and polarization camera.

892

Vol. 44, No. 4 / 15 February 2019 / Optics Letters

Letterretarder is calibrated dynamically, at the same time as the Stokes
parameter measurements. The latter method is particularly usefulin environments where the retardance cannot be controlled, suchas when measuring in the presence of changes in the temperatureor the observation wavelength. Figure 1 shows the setup for
Stokes parameter measurement, in which the micropolarizer
elements have orientations φ  0°, 45°, 90°, and 135° [5].
The Mueller calculus representing the polarized light propagation through the system is given by
S out  A · R · S in ,(1)

where S in is the input polarization state of the light, S out thestate after the polarizer elements and incident on CCD sensor,
and R and A the Mueller matrices of the rotating retarder andanalyzer. From this equation, we can obtain the detected intensity I at the CCD sensor as1
I  s0  s1 f1 − 1 − cos δsin2 2ξg2
 s2 1 − cos δ sin 2ξ cos 2ξ − s3 sin δ sin 2ξ cos 2φ s 1 1 − cos δ sin 2ξ cos 2ξ  s 2 f1 − 1 − cos δcos2 2ξg s3 sin δ cos 2ξ sin 2φ,(2)where si , δ, and ξ in Eq. (2) are elements of the input polarization state S in , the retardance, and the azimuthal angle of therotating retarder.
In the first algorithm, the unknown Stokes parameters aremeasured by setting the direction of the rotating retarder at two
orthogonal orientations 0° and 90°, or at 45° and 135°. For
these angles, the corresponding intensities I 0° ∼ I 135° are1
I 0°  fs0  s1 cos 2φ  s2 cos δ  s 3 sin δ sin 2φg, (3)21
I 45°  fs0  s1 cos δ − s 3 sin δ cos 2φ  s 2 sin 2φg,
2

(4)

1
I 90°  fs0  s1 cos 2φ  s 2 cos δ − s3 sin δ sin 2φg,
2

(5)

1
I 135°  fs0  s 1 cos δ  s 3 sin δ cos 2φ  s 2 sin 2φg: (6)2
The intensities obtained by the four different micropolarizer
element orientation angles 0°, 45°, 90°, and 135° for each of
the four retarder orientations indicated in Eqs. (3)–(6) are1
I 0  s0  s1 ,
(7)
2
1
(8)
I 1  fs0  s2 cos δ  s3 sin δg,2
1
(9)I 2  s0 − s1 ,21
I 3  fs 0 − s2 cos δ  s3 sin δg,
(10)
2
1
(11)
I 4  fs 0  s1 cos δ − s3 sin δg,2
1
(12)
I 5  s0  s2 ,2
1(13)
I 6  fs0 − s1 cos δ − s 3 sin δg,2

1
I 7  s0 − s 2 ,
2

(14)

1
I 8  s 0  s1 ,
2

(15)

1
I 9  fs0  s2 cos δ − s 3 sin δg,
2

(16)

1
I 10  s0 − s 1 ,
2

(17)

1
I 11  fs 0 − s2 cos δ − s3 sin δg,
2

(18)

1
I 12  fs 0  s1 cos δ  s3 sin δg,
2

(19)

1
I 13  s0  s2 ,
2

(20)

1
I 14  fs0 − s1 cos δ  s3 sin δg,
2

(21)

1
I 15  s0 − s 2 ,
(22)
2
where I 0 ∼ I 3 indicate the intensities obtained for the rotating
retarder set to 0° for each pixel polarizer element oriented at 0°,45°, 90°, and 135°. I 4 ∼ I 7 are the four pixel intensities obtained for the retarder oriented at 45°. I 8 ∼ I 11 are likewise
for the retarder at 90°, and I 12 ∼ I 15 for the retarder at
135°. Finally, the unknown input normalized Stokes parameters can be estimated from these 16 measurements by
3 2
3
2
1
1
2 3
s0
7 6 I 4 −I 6 I 12 −I 14 7
6 I 0 −I 2
7 6 2I 5 I 7  cos δ 7
I 0 I 2
6 s1 7 676
6
6 7  6 I −I I −I 7
6
(23)
7:
I 5 −I 7
4 s2 5 6 1 3 9 11 777
6
4 2I 0 I 2  cos δ 5 4 I 5 I 7 5s3
I 1 −I 3 −I 9 I 11
I 4 −I 6 −I 12 I 14
2I 0 I 2  sin δ2I 5 I 7  sin δ

The I 8 , I 10 , I 13 , and I 15 intensities are not used in Eq. (23)because they are the same intensities as other intensities. The
retardance δ of the rotating retarder can be measured from thepolarizer elements oriented at 0°, 45°, and 90° with the relation−1 I 1 − I 3  I 9 − I 11:(24)
δ  cos2I 5 − I 7 
The main drawback of this approach, which we call the “two-shot
algorithm,” is that the retardance of the rotating retarder must be
calibrated in advance and not dynamically because the Stokes
parameters measurement use two images taken at ξ  0° and
90° or 45° and 135°; however, for retardance measurement it
is necessary to capture three images at ξ  0°, 45°, and 90°.
When the incident polarization has s2  0, the retardance cannot
be calculated by Eq. (24). Note that if the rotating retarder employed in the system is a half-wave or a quarter-wave plate, theStokes parameters cannot be calibrated at all via this algorithm
because the Stokes parameters become undefined in that case.
In the second algorithm, the retardance of the rotating
retarder can be calibrated by setting the orientation of the rotating retarder to three positions, 0°, 45°, and 90°, during measurement. In this case, the normalized Stokes parameters of the
input light are obtained from

Vol. 44, No. 4 / 15 February 2019 / Optics Letters

Letter
3

2

2

3

1

s0
6 I 0 −I 2 7
6 s1 7 6 I 0 I 2 7
7 6
7S in  6
4 s2 5  6 I 5 −I 7 7,
4 I 5 I 7 5s3
I 1 −I 3 −I 9 I 11(25)

2I 0 I 2  sin δwithδ  cos−1I 1 − I 3  I 9 − I 11
2I 5 − I 7 
I 4 − I 6  I 12 − I 14
:
(26)
 cos
2I 1 − I 3 
We refer to this as the “three-shot algorithm.” Note that thewave plate retardance must be something other than ahalf-wave for this calibration method.
To demonstrate both methods, we use the experimental setup
shown in Fig. 2, which shows a polarization state generator illuminating a sample in transmission mode, form which light ispassed into the polarization camera of a photonic crystal micropolarizer array. The extinction ratio of the micropolarizers is about
10, the spectral range is 510 to 550 nm, the pixel size is
4.65 × 4.65 μm, and the overall number of pixels is 1120 × 868.
Prior to measurement, we first calibrate the retardance of therotating retarder using the two algorithms described above andcompare the results to a measurement obtained by a commercial
Mueller matrix polarimeter (Axometrics AxoScan). These results
are compared in Table 1. The AxoScan measurements consist of
an average of 10 measurements taken at a wavelength of 520 nm,
giving the retardance of the rotating retarder as 136.6°. The proposed two-shot and three-shot calibration algorithm retardance
estimates were obtained by taking the mean of nine pixels in thecenter of the measurement image and averaging over 10 measurements. The estimated retardance values were 135.3° and
136.7° for the two-shot and three-shot algorithms, respectively.
These calibrated retardances are used in both algorithms for the
Stokes parameter measurements given in the experiments below.
Next, we test the accuracy of our proposed algorithms formeasuring the Stokes parameters when the polarization state−1893

generator is set to produce various polarization states. The resulting Stokes parameter estimates are shown in Table 2, obtained by extracting the spatial average of the image and atemporal average over 10 measurements. In the measurement
we consider the linear diattenuation values Dφ of each micropolarizer; because the extinction ratio is low (∼10) and varies
from pixel to pixel, the intensity of Eq. (2) can be rewritten as1
I  s0  Dφ s1 f1 − 1 − cos δsin2 2ξg2
 s2 1 − cos δ sin 2ξ cos 2ξ − s 3 sin δ sin 2ξ cos 2φ
× Dφ s1 1 − cos δ sin 2ξ cos 2ξ
 s2 f1 − 1 − cos δcos2 2ξg  s3 sin δ cos 2ξ sin 2φ:(27)

The linear diattenuation Dφ influences the amplitude in the intensity equation [Eq. (27)]. At first, we estimated them by
Fourier analysis using input rotating linear polarization images
captured by only the polarization camera before measurement.
Then we substitute them into Eq. (27). It indicates an absoluteaccuracy of 0.01 and relative precision of 0.008, results that arecomparable to conventional nonimaging (single point) Stokes
polarimeter measurements [1]. The validity of the Stokes parameters measurement was confirmed in both calibration methods.
To demonstrate the advantage of dynamic calibration, weshow two measurement results in which the retardance ischanged by the wavelength of the incident light or by the external temperature around the polarimeter. Figure 3 shows retardance values in regard to change in wavelength from 510 to
550 nm. The photonic crystal polarization camera allows only
this narrow wavelength range. For this experiment, the incident
polarization is 22.5° linear polarization. As expected the retardances exhibit a linear decrease with wavelength.
The second measurement (Fig. 4) shows retardance changing
with temperature from 20°C to 30°C. By the dynamic calibration, the calibrated retardances show a linear variation over thetemperature range. The results of Stokes parameters measurement
and retardance between the two-shot (static) and the three-shot(dynamic) methods for right circular polarization is shown in
Table 2. Accuracy of Results for Stokes Parameter
Measurements by Rotating Retarder and Polarization
CamerasStokes Parameters
Polarization
State

Fig. 2. Experimental setup of the imaging Stokes polarimeter.

Table 1. Comparing Retardance Estimates by the Two
Proposed Algorithms and the AxoScan

Horizontal
linearpolarization−45° linear
polarization

Retardance [°]Methods

Mean

Repeatability Standard Deviation

AxoScanTwo-shotThree-shot

136.6
135.3
136.7

0.06
0.34
0.24

Right circularpolarization

Theoretical
Value
2 3
1
617
6 7
405
0
2 3
1
6 0 7
6 7
4 −1 5
0
2 3
1
607
6 7
405
1

Two-Shot2

Three-Shot

3 2
3
1
1
6 0.99  0.007 7 6 0.99  0.007 76
7 6
7
4 0.01  0.008 5 4 0.01  0.008 5
0.00  0.005
0.00  0.005
2
3 2
3
1
1
6 0.01  0.001 7 6 0.00  0.001 767 674 −0.99  0.008 5 4 −0.99  0.007 5
0.01  0.007
0.00  0.008
3 2
2
3
1
1
6 0.00  0.007 7 6 0.00  0.007 77 6
6
7
4 0.01  0.002 5 4 −0.01  0.007 5
0.99  0.006
0.99  0.004

894

Vol. 44, No. 4 / 15 February 2019 / Optics Letters

LetterFig. 3. Retardance of the rotating retarder changing wavelength.
The black line gives the theoretical retardance, while the red circles
are measurements from our proposed system.

Fig. 5. Dynamic Stokes parameter images of stressed U-shaped
PET plastic film (Visualization 1).

Fig. 4. Retardance of the rotating retarder changing temperature.
A black line is measurement by the Axoscan, the red points are measurements by our instrument.

Table 3. Comparison of Polarization Measurements
Between the Two-Shot and the Three-Shot Methods
Room Temperature [°C]
Fixed retardance [°]Two-shotStokes parameters

Retardance [°]Three-shot
Stokes parameters

20
2

3
1
6 −0.06 76
7
4 0.04 5
0.96
137.2
3
2
1
6 0.02 77
6
4 0.05 5
1.01

25
2

135.3

3
1
6 −0.03 77
6
4 0.04 5
0.99

30
3
1
6 −0.08 76
7
4 0.03 5
0.93
2

135.7
2

3
1
6 −0.01 77
6
4 0.03 5
0.99

132.1
2

3
1
6 −0.02 76
7
4 0.01 5
1.02

Table 3. The two-shot method exhibits errors because it cannot
recalibrate the retardance. However, the three-shot method is not
influenced by retardance error caused by the temperature change.
The strong point of working with a polarization camera is its
ability to image dynamic variation of polarization, and our calibration method adapts this instrument so that it can easily measure circular polarization and not just linear polarization. To
demonstrate our method’s ability to measure dynamic phenomena, we present a third experiment, showing the polarization

change due to time-varying stress-induced birefringence. A finger
presses down on a U-shaped film of polyethylene terephthalate(PET) plastic, as shown in Fig. 5. Using the two-shot algorithm,
we measured the dynamic Stokes parameters of the light transmitted by the U-shaped film. In this case, the Stokes parameters
were measured at a rate of 10 Hz by synchronizing the rotating
retarder with the polarization camera readout. For this measurement, the incident polarization state is right circular polarization
to allow for birefringence measurement that is independent oforientation. The resulting time-resolved Stokes parameters areshown in Fig. 5, where we can see the dynamic changes in
the Stokes parameters due to stress induced in a U-shaped film
from pressing down on an edge with a finger.
In conclusion, we have shown two methods to calibrate andperform complete Stokes polarimetry with a rotating retarder infront of a polarization camera employing a pixelated micropolarizer. While both methods allow dynamic capture of full
Stokes parameter images, one method requires calibration inadvance while another allows dynamic recalibration during
measurement. The dynamic recalibration method can be usefulfor measuring outdoors over long time periods This capability
can be of particular advantage for imagers such as microbolometer infrared cameras that experience rapid changes in detector
calibration due to drift and pixel response nonuniformity.
REFERENCES
1. M. Tanaka, Y. Nakajima, H. Amamiya, M. Chujo, and Y. Otani, Proc.
SPIE 7461, 74610O (2009).
2. N. J. Pust and J. A. Shaw, Appl. Opt. 45, 5470 (2006).
3. D. J. Diner, A. Davis, B. Hancock, G. Gutt, R. A. Chipman, and B.
Cairns, Appl. Opt. 46, 8428 (2007).
4. M. W. Kudenov, M. J. Escuti, E. Dereniak, and K. Oka, Appl. Opt. 50,
2283 (2011).
5. L. Fabre, Y. Inoue, T. Aoki, and S. Kawakami, Appl. Opt. 48, 1347 (2009).
6. V. Gruev, R. Perkins, and T. York, Opt. Express 18, 19087 (2010).
7. S. Yoneyama, H. Kikuta, and K. Moriwaki, Exp. Mech. 45, 451(2005).
8. I. J. Vaughn, O. G. Rodríguez-Herrera, M. Xu, and J. S. Tyo, Proc. SPIE
9613, 961312 (2015).