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Polarization calibration with large apertures in full field
of view for a full Stokes imaging polarimeter based
on liquid-crystal variable retarders
Ying Zhang, Huijie Zhao,* and Na Li
School of Instrumentation Science & Opto-electronics Engineering, Beihang University, No. 37,
Xueyuan Road, Haidian District, Beijing 100191, China
*Corresponding author: optoelectronicsbuaa@gmail.com
Received 31 October 2012; revised 10 January 2013; accepted 12 January 2013;
posted 15 January 2013 (Doc. ID 178768); published 18 February 2013
Currently, polarization calibration for full Stokes imaging polarimeters is limited by the apertures of the
retarders. In this paper, an improved polarization calibration with large apertures in full field of view for
full Stokes imaging polarimeters based on liquid-crystal variable retarders is proposed and investigated
theoretically and experimentally. The experimental precision of polarization calibration is 1.7% for linear
polarization states and 8.8% for circular ones for an imaging polarimeter with a 100 mm aperture and 10°
field of view. The feasibility for full Stokes polarization image is also confirmed in experiment for iden-
tifying objects due to degree of polarization and degree of circular polarization images. © 2013 Optical
Society of America
OCIS codes: 110.5405, 000.3110.
1. Introduction
Imaging polarimeters are used to map the state of
polarization across a scene of interest [1,2] and have
been proposed over the past three decades as power-
ful tools to enhance the information available in a
variety of applications, including remote sensing
[3–9], microscopy [10–12], medical monitoring [13],
and scientific research [14,15] for obtaining surface
features, shape, shading, and roughness of the scene
of interest.
Imaging polarimeters with different construction
principles have been proposed. These are based on
interferometric configurations [16], Fourier trans-
form [17], special birefringent prisms [18,19], polar-
ization gratings [20,21], micropolarizers [22–26], and
liquid-crystal variable retarders (LCVRs) [27–29].
The focus of the research in this field has been the
estimation of uncertainties, removing noise, perfor-
mance uniformity and error analysis, use of novel
devices, and optimizing design [30–40].
Generally, constructions of all polarimetric setups
tend to increase measurement accuracy and to auto-
matize the measurement process as well as design
small, compact devices without mechanically rotat-
ing elements. Stokes polarimeters based on LCVRs
are a good example of this [27–29].
Imaging polarimeters are typically not capable of
full Stokes polarization imaging, relying instead on
only a subset of the Stokes-vector components. The
first three components are for linear polarization
imaging, which can be used to improve the object’s
visibility in scattering media. The last component is
just for circular polarization imaging, which can be
used to improve the contrast of images formed by
circular or elliptical polarized light. In some applica-
tions, it is essential to measure all of the available
polarization information. Full Stokes imaging polari-
meters [41–43] have been proposed to enable the
simultaneous capture of both linear and circular
components of the Stokes vector [7,28].
In general for imaging polarimeters, it is necessary
to perform calibration [44–52] to account for polari-
zation aberrations. For each pixel and wavelength,
1559-128X/13/061284-09$15.00/0
© 2013 Optical Society of America
1284 APPLIED OPTICS / Vol. 52, No. 6 / 20 February 2013
calibration is performed to find the actual values of
the system matrix. A minimum of four independent
Stokes vectors must be generated that can form a
maximum volume polyhedron inscribed inside the
Poincaré sphere. Vedel et al. [42] present a polariza-
tion calibration by means of a polarization source
by a polarization state generator (PSG) with a white
light source. This proposed PSG is realized by a high-
quality linear polarizer and a quarter-wave plate.
Baba et al. [50] and Boulbry et al. [51] also use this
calibration for a full Stokes polarimeter, but the area
of this calibration is still limited by the aperture
of the retarder, which is usually smaller than two
inches at present. Pust and Shaw [29] propose a
model of the last column of the instrument matrix
according to the retardance and equivalent rotation
angle of each LCVR in the case of assuming they are
ideal pure retarders because the 2 in. (0.05 m) dia-
meter zero-order wave plates (for each wavelength)
are not available to produce a circular polarization
state across the full aperture. However, this model
still causes underestimation of the magnitude of
the circular Stokes parameter at 90%.
In this paper, an improved polarization calibration
for a full Stokes imaging polarimeter based on
LCVRs is proposed to solve the problem of small area
of polarization source and realize the calibration in
full field of view.
2. Full Stokes Imaging Polarimeter Based on LCVRs
Our proposed full Stokes imaging polarimeter based
on LCVRs (as shown in Fig. 1)[27] is utilized to
measure the full Stokes vector of each pixel in a scene
to recognize different objects in a complicated back-
ground by both of polarization and spectral detection
by a single imager. The light is focused onto a field
lens at the focal plane by a telescope lens with a
100 mm aperture and a 10° field of view. Then, by
passing through the LCVRs (Meadowlark LRC-200),
polarizer, and filters, transmitted light is collected
by the imaging lens, and the imaging is detected
by CCD.
The LCVRs use nematic liquid-crystal materials
to electrically control polarization. They can provide
tunable retardation by changing the effective bire-
fringence of the material with applied voltage, thus
altering the transmitted light to some polarization
forms. The retardance due to LCVRs is a function of
both of wavelength and the applied voltage [53] and
is experimentally measured, as shown in Fig. 2.It
can be found that the retardance is accurate for only
the narrowband; therefore, a filter wheel is used in
our imaging polarimeter to lock the working wave-
length band, which contains five 10 nm band filters
(China Daheng Group, GCC-2020) centered at 476,
530, 568, 647, and 676 nm as well as an opaque filter
for dark measurement. Therefore, the Stokes vector
of detected light can be obtained monochromatically,
and full-color images can be realized due to several
monochromatic images.
A fixed linear polarizer (Meadowlark DP-100-VIS)
is used as a polarization analyzer with extinction
ratio larger than 40 dB in the wavelength band from
476 to 676 nm as shown in Fig. 3.
Four intensity images for one wavelength band can
be collected when four sets of retardance values of
the LCVRs are set respectively in a response time
shorter than 50 ms. Then a set of linear equations
is obtained, and the four units of the Stokes vector
including S0,S1,S2, and S3can be calculated out. If
only one LCVR is used to modulate the polarization
state of the light, the linear equations will be rele-
vant and only part units of Stokes vector can be
calculated. In addition, image acquisition is realized
by a Qimaging Exi-1394, which exhibits 12 bit data
and frame speeds up to 10 frames∕s. A photograph of
our proposed imaging polarimeter is shown in Fig. 4.
3. Principle of Polarization Calibration
The polarization state of the incident light can be
defined by a Stokes vector,
Sin 0
B
B
B
@
S0
S1
S2
S3
1
C
C
C
A
0
B
B
B
@
IHIV
IH−IV
I45°−I135°
IR−IL
1
C
C
C
A
;(1)
where IH,IV,I45, and I135 are the intensity in the
horizontal, vertical, 45°, and 135° linearly polarized
states, respectively, and IRand ILare the intensity
in the right and left circularly polarized states,
Fig. 1. (Color online) Schematic of the imaging polarimeter with LCVRs.
20 February 2013 / Vol. 52, No. 6 / APPLIED OPTICS 1285
respectively. Then some terms can be expressed as
follows. The degree of linear polarization (DOLP) can
be expressed as
DOLP
S2
1S2
2
qS0
;(2)
The degree of circular polarization (DOCP) can be
defined by
DOCP jS3j
S0
;(3)
The degree of polarization (DOP) is
DOP
S2
1S2
2S2
3
qS0
;(4)
and the polarization state of transmitted light is
Sout S0
0S0
1S0
2S0
3TMMUE ×Sin;(5)
where MMUE is the Mueller matrix of the imaging
polarimeter,
MMUE 0
B
B
@
m00 m01 m02 m03
m10 m11 m12 m13
m20 m21 m22 m23
m30 m31 m32 m33
1
C
C
A
:(6)
In an ideal case, MMUE can be presented as
MMUE Y
1
iN;−1
Mi;(7)
where Miis the Mueller matrix of the ith optical
element of the imaging polarimeter, so MMUE can
be expressed as
MMUE Mlens2×Mfilter ×MLP ×MLCVR2
×MLCVR1×Mlens1;(8)
where Mlens2,Mfilter,MLP ,MLCVR2,MLCVR1, and
Mlens1are the ideal Mueller matrixes of the imaging
lens, filter, polarizer, LCVRs, and front lenses,
respectively. When considering the influences of
instrumental polarizations, the field angle, and the
wavelength of the elements, Miis different from its
real value. For different retardance, the Mueller
matrix of the LCVR is also changed, so the instru-
ment matrix must also be changed. Each group of
retardance value is corresponding to the Mueller
matrix MMUE and one intensity value I. Here Iis
the first term of the Stokes vector, S0
0, and can be
obtained from Eqs. (5)and(6):
S0
0Im00 m01 m02 m03 ×Sin:(9)
Four groups of retardance values can be obtained
by changing the voltage applied on the two LCVRs. A
4×4matrix can be obtained by extracting the first
row from the four MMUE, which is defined as the
instrument matrix and can be expressed as
Fig. 2. Experimental results of LCVR retardance and applied
voltage for different wavelength.
Fig. 3. (Color online) Transmittance of linear polarizer.
1286 APPLIED OPTICS / Vol. 52, No. 6 / 20 February 2013
MINS 0
B
B
@
a00 a01 a02 a03
a10 a11 a12 a13
a20 a21 a22 a23
a30 a31 a32 a33
1
C
C
A
;(10)
where a00 a01 a02 a03 ,a10 a11 a12 a13 ,
a20 a21 a22 a23 , and a30 a31 a32 a33 are
the first row of the four MMUE corresponding to
the four groups of retardance values, respectively.
Therefore, there is a relationship,
IOUT 0
B
B
@
I0
I1
I2
I3
1
C
C
A
MINS ×0
B
B
@
S0
S1
S2
S3
1
C
C
A
0
B
B
@
a00 a01 a02 a03
a10 a11 a12 a13
a20 a21 a22 a23
a30 a31 a32 a33
1
C
C
A
×0
B
B
@
S0
S1
S2
S3
1
C
C
A
;(11)
where I0,I1,I2, and I3are the four intensity values
from the detector. To reduce the amplification of
image-exposure errors to Stokes-vector errors during
Stokes-vector retrieval, we followed the work of Tyo
[34], and the retardances of the LCVRs for each
image have been chosen to minimize the condition
number of the system matrix.
The polarization state of the incident light can be
expressed as
Sin M−1
INS ×Iout:(12)
As a result, it is necessary to calibrate the MINS.
The 16 elements of the instrument matrix MINS can
be calculated by four groups of the known incident
light S0S1S2S3Tand the detected intensity
values I0I1I2I3T.
Our proposed polarization calibration for an ima-
ging polarimeter based on an LCVR is shown in Fig. 5.
An integrating sphere light source is collimated and
the emitted light passes through an aperture stop. An
eigenstate generator (EGS) including a linear polari-
zer (Meadowlark DP-200-VIS) and a precision achro-
matic retarder (Meadowlark AQM-100-545) is used to
produce six polarization eigenstates: linear polarized
light at angles of 0°, 90°, 45°, and −45°aswellas
circular right-handed and left-handed polarized light.
The retardance of the achromatic retarder is shown
in Fig. 6.
Then the polarized beam is diffused by a micro-
lens-array-based diffuser (Thorlabs ED1-C50), which
is used to diffuse a collimated polarized beam. Each
microlens has the same focal length, and the diffused
angle of the light is determined by the focal length
of the microlens. The change in polarization state
caused by this diffuser is less than 0.1% [43]. The
diffuser is operated on the focal plane of a collimating
lens, so the diffused polarized light is collimated by
the lens again. As a result, the diffuser, in conjunc-
tion with the collimating lens, changes the colli-
mated polarized beam with a small diameter in 0°
field of view into a collimated beam with a larger
diameter, and the field of view is from 0° to ω0. The
diameter of the collimated polarized light after the
lens should be larger than the one of the polarimeter
to be calibrated, which can be expressed as
D02f0tan θ
2
D0>D ;13
where D0is the diameter of the collimated polarized
light, f0is the focal length of the collimating lens, θis
the diffused angle of the diffuser, and Dis the diam-
eter of the imaging polarimeter to be calibrated.
Fig. 4. Our proposed imaging polarimeter.
Integrating
sphere light
source
Collimator
Stop
EGS Diffuser Collimating lens
Polarimeter
detector
Intensity
detector
Calibrated
Imaging
Polarimeter
Fig. 5. (Color online) Schematic of the proposed polarization
calibration.
0.18
0.20
0.22
0.24
0.26
0.28
0.30
300 400 500 600 700 800 900 1000
Retardance i n Waves
Wavelength in nm
Fig. 6. (Color online) Retardance of achromatic retarder
(AQM-100-545).
20 February 2013 / Vol. 52, No. 6 / APPLIED OPTICS 1287
The field of view of the collimated light after the
collimated lens should also be larger than the one
of the polarimeter to be calibrated, which can be
expressed as
ω0arctan d
2f0
ω0>ω;14
where ω0is the field of view of the collimated light after
the collimated lens, dis the diameter of the collimated
light before the engineered diffuser, and ωis the field
of view of the polarimeter to be calibrated. Then the
whole aperture of the optical system can be satisfied
in full field of view by our proposed polarization cali-
bration using a light wave with different polarization
states. Therefore, the full field-of-view calibration can
be achieved by using our proposed method one time
with higher precision and a simpler process.
The Stokes parameters of the collimated light are
measured by a polarimeter detector (Meadowlark
PMI-VIS) with a resolution of 0.001 of a Stokes para-
meter. The intensity of the light is detected by an
intensity detector as a referred intensity during
the calibration.
The linear polarized light generated by the EGS is
used to calibrate the first three columns of MINS,and
the circular polarized light is used to calibrate the
fourth column. The data of the two calibration parts
can compose one matrix to be processed using the
least-squares method [39]. Each polarization state
is used five times repeatedly, and the coefficient of
Stokes can be obtained, expressed as
A
0
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
@
11 0 0
.
.
..
.
..
.
..
.
.
1−10 0
.
.
..
.
..
.
..
.
.
10 1 0
.
.
..
.
..
.
..
.
.
10−10
.
.
..
.
..
.
..
.
.
1cos π
2·λ0
λ0sin π
2·λ0
λ
.
.
..
.
..
.
..
.
.
1−cos π
2·λ0
λ0−sin π
2·λ0
λ
.
.
..
.
..
.
..
.
.
1
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
A30×4
;
(15)
where λ0is the central wavelength of the wave plate
in the EGS and λis the central wavelength of
the filter. The first row of Ais the Stokes vector of
the incident light with 0° linear polarization state.
Accordingly, the first five rows are equal to each
other because each polarization state is used five
times repeatedly. The sixth row is the Stokes vector
of the incident light with 90° linear polarization
state. Then the sixth through tenth rows are the
same, too. Analogously, the 11th, 16th, 21st, and
26th rows are for 45° and −45° linear polarization
and left-handed and right-handed circular polariza-
tion, respectively.
The corresponding detected intensity can form a
matrix Lexpressed as
LI
0
B
B
B
B
B
@
I0;1I1;1I2;1I3;1
I0;2I1;2I2;2I3;2
I0;3I1;3I2;3I3;3
.
.
..
.
..
.
..
.
.
I0;30 I1;30 I2;30 I3;30
1
C
C
C
C
C
A30×4
:(16)
The errors would be different since the polariza-
tion states of the input light to the system are diff-
erent, so it is an unequal precision measurement.
The error matrix corresponding to Lcan be ex-
pressed as
Pdiag1
σ2
1
;…;1
σ2
2
;…;1
σ2
3
;…;1
σ2
4
;…;1
σ2
5
;…;1
σ2
6
;…30×30
;
(17)
where σiis the error corresponding to the six differ-
ent polarization states. Pcan be considered as the
weight matrix of L, and 1∕σ2
iand σ2
iare the weight
and variance of the ith row of L, respectively. Because
the six different polarization states are used five
times repeatedly, the variance for each polarization
state is the same. Therefore, every five terms at
the diagonal line of Pare the weight 1∕σ2
ifor one
polarization state.
According to Eqs. (11), (15), and (16), the
relationship
IT
iI0;i I1;i I2;i I3;i TMINS ×AT
i;(18)
ITMINS ×AT(19)
Table 1. Experimental Results for STDV of the Polarization Calibration (λ670 nm)
STDV in 50% Area of
CCD around the Center
STDV in Whole
Area of CCD
Polarized State of Incident Light σDOLP σDOCP σDOP σDOLP σDOCP σDOP
Linear polarized light generated by a 30° linear polarizer 1.01% 0.35% 1.07% 1.57% 0.52% 1.65%
Linear polarized light generated by a 120° linear polarizer 1.03% 0.36% 1.09% 1.59% 0.55% 1.68%
Elliptical polarized light generated by a 30° linear polarizer and a quarter-wave plate 4.28% 2.64% 5.03% 7.52% 4.49% 8.76%
Elliptical polarized light generated by a 120° linear polarizer and a quarter-wave plate 4.41% 2.54% 5.09% 7.45% 4.51% 8.71%
1288 APPLIED OPTICS / Vol. 52, No. 6 / 20 February 2013
can be achieved, where Iiis the ith row of matrix I
and Aiis the ith row of matrix A.
Then, MINS can be calculated out by using the con-
dition of the least-squares method for the unequal
precision measurement [39],
MINS ATPA−1ATPIT:(20)
Therefore, the polarization calibration can be
achieved by obtaining MINS due to Eq. (20).
4. Experimental Results and Discussion
In order to evaluate the feasibility and performances
of the proposed polarization calibration, the experi-
ments for standard deviation (STDV) measurement
for proposed polarization calibration and perfor-
mances of imaging polarimeter using proposed polar-
ization calibration are executed.
A. STDV Measurement for Proposed Polarization
Calibration
It is worthily noted that the STDVof the DOP is used
to describe the performance of the proposed polariza-
tion calibration here. Then the STDV of the polariza-
tion calibration for four types of polarized light with
λ670 nm is executed into the imaging polarimeter
and is shown in Table 1.
What is more, the STDV of the DOP in five differ-
ent wavelength bands is shown in Fig. 7. It is found
that the STDVof the DOP is 1.1% for linear polarized
light and 5.2% for elliptical one in the 50% area of
the CCD around the center. Also, it is increased to
1.7% for linear polarized light and 8.8% for elliptical
one in the whole area of the CCD. The reason is that
the light in a smaller field of view is focused around
the center of the CCD and the corresponding incident
angles in the calibration system are smaller, so the
polarization aberrations [52] are smaller; therefore,
higher polarization detection precision can be rea-
lized. At the same time, the light in the larger field
of view is focused near the edge of the CCD, and the
incident angles are larger, so the polarization aberra-
tions are larger, and the polarization detection preci-
sion is lower.
It can be also seen that the polarization detection
precisions for linear polarized light in the five bands
are nearly equal, while different for circular ones.
This is because there is uncertainty in the exact
Stokes vector obtained with the precision achromatic
retarder, as the retardance value is dependent upon
the incidence angle of light and the wavelength, and
Fig. 7. (Color online) STDV of DOP in five different wavelength
bands.
(a) Image of spectral intensity in 476 nm (b) Image of DOP in 476 nm
(c) Image of spectral intensity in 530 nm (d) Image of DOP in 530 nm
(e) Image of spectral intensity in 568 nm (f) Image of DOP in 568 nm
(g) Image of spectral intensity in 647 nm (h) Image of DOP in 647 nm
(i) Image of spectral intensity in 676 nm (j) Image of DOP in 676 nm
Fig. 8. (Color online) Experimental images of spectral intensity
and DOP in different wavelength.
20 February 2013 / Vol. 52, No. 6 / APPLIED OPTICS 1289
the exact position of the fast axis changes with
wavelength.
B. Performances of Imaging Polarimeter Using Proposed
Polarization Calibration
The performances of the full Stokes-imaging-
polarimeter-based LCVRs using proposed polariza-
tion calibration are evaluated in the experiment.
The imaging polarimeter is operated on a platform
rotating from 0° to 180° horizontally and −60° to 60°
vertically to observe the objects in different direc-
tions. Spectral intensity images and the correspond-
ing DOP images in five wavelength bands are
obtained, as shown in Fig. 8.
It can be seen that the contrastof the words against
the walls in the area A of the building, as shown in
Fig. 8(a), in the spectral intensity images are lower
than the DOP images. The reason is that the words
and wall are materials with different roughness; as
a result, DOP images can present better contrast.
Analogously, in area B of the building, the limiting
lines of different parts are obscured in the spectral
intensity images; however, they are obvious in DOP
images. The reason is that the two planes beside the
limiting line always are made of different materials
and with different plane directions. In area C of the
building, the glasses in the window almost cannot be
figured out in spectral intensity images, but they are
very clear in DOP images, especially in the 476 nm, so
it seems that the same material has different polari-
zation properties for different wavelength bands. In
conclusion, it is found that, due to our proposed polar-
ization calibration, the LCVR-based imaging polari-
meter is viable for obtaining surface features and
roughness of building.
In order to evaluate the feasibility of our proposed
polarization calibration for a full Stokes polarization
image, experimental images for artificial grass in
shadow of a tree and on a lawn are obtained as shown
in Fig. 9. The artificial grass in the shadow and the
lawn background can be identified in the Fig. 9(a)
but not in Fig. 9(b). The contrast of the metallic
sculpture and the other nonmetallic substance can
be improved in Fig. 9(c). The contrast of the objects,
artificial grass in our experiment, and the back-
ground can be increased and the effect of shadow
can be removed in the DOP image. Furthermore, the
metallic sculpture can even be detected in the DOCP
image. In addition, the object (artificial grass) iden-
tifying results are shown in Fig. 10. It is found that
the artificial grass in the shadow and in the lawn
background can be identified effectively using the
probability statistical method in the image of the
DOP as shown in Fig. 10(b), but not in the spectral
intensity image. When considering an application,
the false alarm rate of identifying artificial grass in
the lawn background by the spectral intensity image
is high; as a result, the road is confused due to the
wrong identified result.
5. Conclusions
An improved polarization calibration for a full Stokes
imaging polarimeter based on LCVRs is proposed
and investigated to solve the problem of a small area
of polarization source and to realize the calibration
of polarimeters with large apertures in full field of
view. The feasibility and performances of the pro-
posed polarization calibration are evaluated in ex-
periment. The precision of polarization calibration is
1.7% for linear polarization states and 8.8% for circu-
lar ones in the case of an imaging polarimeter with a
100 mm aperture and 10° field of view. The feasibility
for a full Stokes polarization image is then tested in
experiments for identifying artificial grass and me-
tallic sculpture in a lawn background due to DOP
and DOCP images, respectively. Similar instruments
(a) DOP image (b) spectral intensity image (c) DOCP image
Fig. 9. (Color online) Experimental images for identifying artificial grass in the 476 nm band.
(a) (b) (c) (d)
Fig. 10. Experimental results for processed artificial grass image from Fig. 9: (a) original DOP image (b) processed DOP image, (c) original
spectral intensity image, and (d) processed spectral intensity image.
1290 APPLIED OPTICS / Vol. 52, No. 6 / 20 February 2013
aiming for higher polarization calibration precision,
will, however, need to take some other problems into
account.
This work was supported by the National Natural
Science Foundation of China (no. 61107013), by the
National Natural Science Foundation of China
(no. 61177008), and by the Program for Changjiang
Scholars and Innovative Research Team in Univer-
sity (IRT0705). The authors are grateful for all of the
valuable suggestions received during the course of
this research.
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