OE Letters1
I 0 ¼ ðs0 þ s1 Þ þ n0 ;2
1
I 45 ¼ ðs0 þ s2 Þ þ n45 ;2
1
I 90 ¼ ðs0 − s1 Þ þ n90 ;2
1
I 135 ¼ ðs0 − s2 Þ þ n135 ;2

EQ-TARGET;temp:intralink-;e001;326;752

Generating highperformance polarization
measurements with lowperformance polarizers:
demonstration witha microgrid polarization
cameraNathan Hagen,a,* Shibata Shuhei,a,b and Yukitoshi Otania,bUtsunomiya University, Center for Optical Research andEngineering, Department of Optical Engineering,
Utsunomiya, Tochigi, Japanb
Utsunomiya University, Graduate School of Engineering,
Utsunomiya, Tochigi, Japana(1)

for random noise values nθ . In the analysis to follow,
we will assume all of the noise terms to have zero mean,i.e., hI i þ ni i ¼ I i and hni i ¼ 0. This causes no difficulties
for Poisson-distributed noise, since our definition of the signal I and noise n results in the mean value of the Poissondistributed variable to be incorporated into I while n retains
the zero-mean stochastic portion.
The four equations of (1) have the matrix form
0
1
1
1 1
0 0 0 s0 1 0 n0 1
I0
B I 45 C 1 B 1 0
1 0C
B
C B s1 C B n45 C
BC@ I 90 A ¼ 2 @ 1 −1 0 0 A @ s2 A þ@ n90 A
1 0 −1 0
I 135
s3
n135
|ﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄ} |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} |ﬄﬄﬄ{zﬄﬄﬄ} |ﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄ}
0

EQ-TARGET;temp:intralink-;e002;326;556

Abstract. Microgrid polarization cameras have historically experienced lower performance than is typical formonolithic polarization systems. Specifically, their polarizer elements have had a lower extinction ratio and alarger orientation error. We show how to use the calibrated
parameters of a nonideal polarizer to modify the polarization measurement model, effectively allowing one togenerate high-performance measurements from lowperformance elements. We demonstrate the effectivenessof this approach on a commercial polarization camera andestimate the signal-to-noise ratio penalty for using nonideal polarizers in this camera as being 1.25× versus a
system using ideal polarizers. © The Authors. Published by SPIE
under a Creative Commons Attribution 4.0 Unported License. Distribution or
reproduction of this work in whole or in part requires full attribution of the original
publication, including its DOI. [DOI: 10.1117/1.OE.58.8.080501]

Keywords: polarization; imaging polarimetry; noise.
Paper 190765L received Jun. 3, 2019; accepted for publication Aug.
5, 2019; published online Aug. 21, 2019.

I

s

W

(2)

n

or I ¼ Ws þ n, where W is the measurement matrix. By
inspection, we can obtain the input linear Stokes vector
elements as1
s^ 0 ¼ ðI 0 þ I 45 þ I 90 þ I 135 Þ;2s^ 1 ¼ I 0 − I 90 ;EQ-TARGET;temp:intralink-;e003;326;441s^ 2 ¼ I 45 − I 135 ;

(3)

or
0
1 0 I0 1
1
1 1 1
1
s^ 0
I C
@ s^ 1 A ¼ 1 @ 2 0 −2 0 A B@ 45 A;I 90
20 2 0 −2s^ 2
|ﬄﬄﬄ{zﬄﬄﬄ} |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} I 135|ﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄ}
A
s^
0

(4)

EQ-TARGET;temp:intralink-;e004;326;361

I

A polarizer can be used to estimate the orientation anddegree of linear polarization of light by rotating the polarizer
to the four angles: 0 deg, 45 deg, 90 deg, and 135 deg andmeasuring the light intensity transmitted at each angle. In apolarization camera, this can be done in a single snapshot by
combining the measurements of four adjacent pixels, each ofwhich have micropolarizers oriented at each of the fourorientation angles (Fig. 1).1 For incoherent light, we can use
the Stokes vector s ¼ ðs0 ; s1 ; s2 ; s3 ÞT to represent the input
polarization state and the Mueller calculus to model the four
intensities I θ measured at the detector:

K ij ¼

X
3

EQ-TARGET;temp:intralink-;e005;326;236

k¼0

sk

3
X
l¼0





Ail Ajl W lk þ vd

3
X
l¼0


Ail Ajl ;|ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ}|ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ}

Poisson noise partGaussian noise part(5)

where sk is the k’th Stokes vector element in units of photoelectrons and vd is the detector noise variance, also in units ofphotoelectrons. For an ideal polarizer, this set of four measurements produces variances in the Stokes vector components as3*Address all correspondence to Nathan Hagen, E-mail: nh@hagenlab.org

Optical Engineering

i.e., s ¼ AI, where A is the analysis matrix. To evaluate theperformance of a polarization measurement, we can use
the measurement and analysis matrix elements to calculate
the Stokes vector covariance matrix K as2,3080501-1

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August 2019 • Vol. 58(8)OE Letters
EQ-TARGET;temp:intralink-;;326;752

cosð2α0 Þ ¼ cosð2ϵ0 Þ → 1;
cosð2α1 Þ ¼ cosð90° þ 2ϵ1 Þ → −2ϵ1 ;cosð2α2 Þ ¼ cosð180° þ 2ϵ2 Þ → −1;
cosð2α3 Þ ¼ cosð270° þ 2ϵ3 Þ → 2ϵ3 ;andFig. 1 A polarization camera showing a 2 × 2 pixels group from thedetector array and their polarization filter orientations.

EQ-TARGET;temp:intralink-;;326;667

sinð2α0 Þ ¼ sinð2ϵ0 Þ → 2ϵ0 ;sinð2α1 Þ ¼ sinð90° þ 2ϵ1 Þ → 1;1varð^s0 Þ ¼ s0 þ vd ;2varð^s1 Þ ¼ s0 þ 2vd ;EQ-TARGET;temp:intralink-;e006;63;627

sinð2α2 Þ ¼ sinð180° þ 2ϵ2 Þ → −2ϵ2 ;sinð2α3 Þ ¼ sinð270° þ 2ϵ3 Þ → −1;varð^s2 Þ ¼ s0 þ 2vd :(6)The equally weighted variance (EWV) is often used tosummarize the measurement performance, such that in thiscase EWV ¼ 52 s0 þ 5vd .3
A nonideal linear polarizer is a diattenuator defined byits diattenuation D, its orientation α, and its efficiency η(the average transmission of the diattenuator for all input
polarization angles):40

EQ-TARGET;temp:intralink-;e007;63;484

Dc

Ds

0

1 − Ds2

1
2η Dcs0

1
2η Dcs

1 − Dc2

0

0

0

1−D

1

B Dc
B
Mld ¼ ηB
B
@ Ds
0

1
C
C
C;
C
A

where ϵn represents the angular error about the nominal orientation. Likewise, each diattenuation can be represented as
Di ¼ 1 − Δi , where the diattenuation errors Δi are assumed
to be small. For this first-order approximation regime, theanalysis matrix A in the Stokes vector estimation:
0 1
1
0
1 I0
A00 A01 A02 A03
s^00
B I1 CB 0 C 1@C
A10 A11 A12 A13 A B@ s^ 1 A ¼
@I A
4
2
0
A20 A21 A22 A23s^ 2|ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ}
I3|ﬄﬄﬄ{zﬄﬄﬄ}|ﬄﬄﬄ{zﬄﬄﬄ}
A
s^ 0
0

EQ-TARGET;temp:intralink-;e009;326;528

(9)

I

(7)

simplifies to
A00 ¼ 2 þ Δ0 − Δ2 þ 2ϵ1 − 2ϵ3 ;EQ-TARGET;temp:intralink-;e010;326;419

for c ¼ cosð2αÞ and s ¼ sinð2αÞ. An ideal polarizer has
D ¼ 1, η ¼ 1, and orientation α. The polarizer’s extinction
ratio X is derived from the diattenuation as X ¼
ð1 þ DÞ∕ð1 − DÞ.
Modeling each pixel of a polarization camera as a nonideal polarizer, we can calibrate the diattenuation Dn and orientation αn parameters at each pixel n, then use the Mueller
calculus to estimate the polarization state by combining measurements I i from each set of four neighboring pixels. This
produces the measurement model5
0I 1

01

D0 cosð2α0 Þ D0 sinð2α0 Þ 1
s0 !
B I 2 C 1 B 1 D1 cosð2α1 Þ D1 sinð2α1 Þ C
B C¼ BC@ I A 2 @ 1 D cosð2α Þ D sinð2α Þ A s1 :
3
2
2
2
2
s2
I4
1 D3 cosð2α3 Þ D3 sinð2α3 Þ |ﬄﬄﬄ{zﬄﬄﬄ}|ﬄﬄﬄ{zﬄﬄﬄ}|ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ
ﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} s0

A03 ¼ 2 − Δ1 þ Δ3 þ 2ϵ0 − 2ϵ2 ;A10 ¼ 4 þ Δ0 þ 3Δ2 þ 2ϵ1 − 2ϵ3 ;A11 ¼ Δ0 − Δ2 − 4ϵ0 þ 2ϵ1 þ 4ϵ2 − 2ϵ3 ;A12 ¼ −4 − 3Δ0 − Δ2 þ 2ϵ1 − 2ϵ3 ;A13 ¼ Δ0 − Δ2 þ 4ϵ0 − 2ϵ1 þ 4ϵ2 þ 2ϵ3 ;A21 ¼ 4 þ Δ1 þ 3Δ3 − 2ϵ0 þ 2ϵ2
A22 ¼ Δ1 − Δ3 þ 2ϵ0 − 4ϵ1 − 2ϵ2 − 4ϵ3 ;A23 ¼ −4 − 3Δ1 − Δ3 − 2ϵ0 þ 2ϵ2 :W(8)The polarizer efficiency η does not appear here because it
typically cannot be separated from the pixel’s internal quantum efficiency qint —the two appear together as the single
factor qext ¼ ηqint as the “external” quantum efficiency.
Because we do not calibrate the camera’s quantum efficiency
here, the equations below assume qext ¼ 1, so that the Stokes
parameters are represented in units of photoelectrons rather
than photons.
The general analytical form of A is quite complex, but if
we assume that the orientation errors are small then we canuse a Taylor expansion about their nominal values:
Optical Engineering

A02 ¼ 2 − Δ0 þ Δ2 − 2ϵ1 þ 2ϵ3 ;A20 ¼ Δ1 − Δ3 þ 2ϵ0 þ 4ϵ1 − 2ϵ2 þ 4ϵ3 ;EQ-TARGET;temp:intralink-;e008;63;292

I

A01 ¼ 2 þ Δ1 − Δ3 − 2ϵ0 þ 2ϵ2 ;Here, s^ 10 indicates the calibration-corrected Stokes estimate,while s^ i without a prime indicates the estimate assuming an
ideal polarizer. In Eqs. (9) and (10), note that it is generally
not useful to define α0 as anything but zero, since the systemreference axis is a degree of freedom that we have to choose
in most situations. In this case, we can set ϵ0 ¼ 0.
To demonstrate this technique, we use a commerciallyavailable polarization camera to measure the Stokes vector
elements of the light transmitted through a high-extinction-ratio Glan–Thompson polarizer as the polarizer isrotated through a set of angles from 0 deg to 180 deg.
The polarization camera is a photonic lattice PI-110, having

080501-2

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August 2019 • Vol. 58(8)OE Lettersincreases linearly with Δ0 and Δ2 , and varð^s20 Þ with Δ1 andΔ3 , respectively. The Stokes parameters are often expressedin normalized form s̃i ¼ si ∕s0 , for which the expressions in(11) need to be modified. The normalized parameter variances are related to their unnormalized counterparts by6varðs̃i0 Þ ¼

EQ-TARGET;temp:intralink-;e012;326;697

1
hs 0 i2varðsi0 Þ þ i0 4 varðs00 Þ;
0 2
hs0 i
hs0 i

(12)

which together with Eq. (11) gives
vd
ð2½1 þ Δ0 þ Δ2  þ s̃21 Þ;s20vvarðs̃20 Þ ¼ 2d ð2½1 þ Δ1 þ Δ3  þ s̃22 Þ;
s0

EQ-TARGET;temp:intralink-;e013;326;641

Fig. 2 The measured uncorrected linear Stokes parameters s^ 1 and s^ 2
and corrected parameters s^ 10 and s^ 20 obtained while rotating the input
angle of polarization from 0 deg to 180 deg. Note that the Stokes
parameters, here, are shown in their normalized form: ~s 1 ¼ s1 ∕s 0 ands̃ 2 ¼ s2 ∕s 0 . Each curve is obtained as an average of the results fromall pixels of the camera, while the grayed regions astride each curve
give the standard deviation of the result, also calculated using all pixels on the camera. The dashed curves indicate the measurement
result using the ideal polarization analysis matrix [Eq. (4)] and thesolid curves give the measurement with the corrected analysis matrix[Eq. (10)].

1164 × 874 pixels of 4.65-μm size, 20-Hz frame rate, 12-bit
depth, and 520  20-nm wavelength range. Figure 2 shows
the resulting Stokes parameter estimates before (^s1 and s^ 2 )and after (^s10 and s^ 20 ) applying the correction, corresponding
to using the ideal polarization analysis matrix [Eq. (4)] or acorrected analysis matrix [Eq. (10)]. We can see that the corrected curves display a higher contrast than the uncorrectedcurves, so that the mean values of the Stokes parameter(averaged across all camera pixels) at each angle of the input
polarizer are closer to the correct values. Looking closely at
the curves, we can also see that the standard deviation of thecorrected result is slightly larger than for the uncorrected
measurement.
To see why the corrected result has higher noise, we canuse Eqs. (5), (8), and (10) in their analytical form to calculate
the general equation for the variances. To keep the expressions as simple as possible, and because our experimental
data show that the diattenuation errors are much larger than
the angular errors in our system, we set the angular errors
to zero (ϵi ≈ 0). Thus, in the detector-limited noise regime(uniform Gaussian noise), the variances becomevd½8 þ Δ20 þ Δ21 þ Δ22 þ Δ23 − 2Δ0 Δ2 − 2Δ1 Δ3 
8≈ vd ;v
varð^s10 Þ ¼ d ½8 þ 3Δ20 þ 3Δ22 þ 2Δ0 ðΔ2 þ 4Þ þ 8Δ2 
4≈ 2vd ½1 þ Δ0 þ Δ2 ;vvarð^s20 Þ ¼ d ½8 þ 3Δ21 þ 3Δ23 þ 2Δ1 ðΔ3 þ 4Þ þ 8Δ3 
4≈ 2vd ½1 þ Δ1 þ Δ3 ;(11)

EQ-TARGET;temp:intralink-;e011;63;254varð^s00 Þ ¼

where the variances are the diagonal elements of the Stokes
vector covariance matrix, varð^s00 Þ ¼ K 00 , etc. The approximations in Eq. (11) assume small errors so that only the
first-order terms are retained. In this approximation, varð^s00 Þ
is independent of the diattenuation errors Δi , while varð^s10 Þ
Optical Engineeringvarðs̃10 Þ ¼

and where we have replaced the corrected parameter means
with the true parameter values: hs00 i → s0 , etc.
To see how these variance equations behave quantitatively, we can look at an example group of 4 pixels on thepolarization camera. For this group, we obtain calibrated diattenuations of
EQ-TARGET;temp:intralink-;;326;506

D0 ¼ 0.8277; D1 ¼ 0.7435; D2 ¼ 0.8133; D3 ¼ 0.7338;

and calibrated orientation angles of
EQ-TARGET;temp:intralink-;;326;463

α0 ¼ 0 deg;

α1 ¼ 49.650 deg;

α2 ¼ 88.694 deg;

α3 ¼ 134.399 deg;

so that the differences from the nominal angles are 0 deg,
4.650 deg, −1.306 deg, and −0.601 deg. Converting theangular errors to radians for the ϵi values, we obtain the error
parameters:
Δ0 ¼ 0.1723; Δ1 ¼ 0.2565; Δ2 ¼ 0.1867; Δ3 ¼ 0.2662;

EQ-TARGET;temp:intralink-;e014;326;369

ϵ0 ¼ 0; ϵ1 ¼ 0.0811; ϵ2 ¼ −0.0228; ϵ3 ¼ −0.0105:(14)With these parameters, the variances (11) in the detector-limited noise regime (uniform Gaussian noise) become, via
Eq. (5),varð^s00 Þ ¼ 1.00vd ;EQ-TARGET;temp:intralink-;e015;326;269

varð^s10 Þ ¼ 3.23vd ;varð^s20 Þ ¼ 3.01vd ;(15)

for detector noise variance vd . In comparison to Eq. (6), we
see that the corrected nonideal measurement for this pixel
group suffers an increase in the variances for measuring
s1 and s2 by factors of 1.62 and 1.51, respectively, while thevariance for s0 is basically unaffected. These correspond to aloss in signal-to-noise ratio (SNR) by factors of 1.27 in s1
and 1.23 in s2 .
Using the same Eqs. (2), (5), and (10), we can obtain similar results for the Poisson noise regime, for which the variance equations become

080501-3

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August 2019 • Vol. 58(8)OE LettersFig. 3 The increase (multiplying factor) in the normalized Stokes
parameter standard deviations stdðs^ 10 Þ and stdðs^ 20 Þ with respect tothe input polarization angle, where the increased noise is caused
by applying the correction of Eq. (9). The dashed lines indicate theaverage standard deviation ratio taken across all angles.

111varð^s00 Þ ≈ s0 þ s1 ðΔ0 − Δ2 Þ þ s2 ðΔ1 − Δ3 Þ;
288varð^s10 Þ ≈ s0 ð1 þ Δ0 þ Δ2 Þ þ s1 ðΔ2 − Δ0 Þ;

EQ-TARGET;temp:intralink-;e017;63;551

varð^s20 Þ ≈ s0 ð1 þ Δ1 þ Δ3 Þ þ s2 ðΔ3 − Δ1 Þ;(16)

where only the first-order terms are retained. As we expect,
the variances of Eqs. (11) and (16) approach those of theideal polarizer system3 as the diattenuation errors approachzero (Δi → 0).
For mixed Gaussian–Poisson noise, the Gaussian andPoisson expressions (11) and (16) are simply added together
to get the total variance. Using the example diattenuation
errors of Eq. (14), the Stokes parameter variances under
Poisson noise becomevarð^s00 Þ ¼ 0.500s0 − 0.001s1 − 0.002s2 ;EQ-TARGET;temp:intralink-;e017;63;390varð^s10 Þ ¼ 1.542s0 þ 0.092s1 þ 0s2 ;varð^s20 Þ ¼ 1.381s0 þ 0s1 þ 0.014s2 :(17)

The results of Eqs. (15) and (17) and Fig. 3 demonstratean SNR penalty due to reduced diattenuation that is similar tothat calculated by Tyo and Wei,7 where the authors concluded via numerical simulation that polarizer extinction
ratios as low as 3 give a 3.5 dB (i.e., a factor of 2.2) penalty
in SNR. Although the measurement system analyzed by Tyo
and Wei is not the same as the one we consider, here, the
SNR penalty is similar in magnitude. Our results also agreewith those of Roussel et al.,8 who analyzed microgrid polarization cameras and concluded that diattenuations of 0.75 to
0.85 will increase the variance in the estimated degree ofpolarization by 1.6 to 3. What is different from Roussel et al.

Optical Engineeringis that the analytic dependence of the Stokes parameter variance is expressed here in terms of the calibration parameters
directly, allowing us to see more clearly the effect of misalignment or low diattenuation on the measurements.
In Fig. 2, we can see that the mean values of the correctedcurves come close to the ideal behavior of a cosine curve
oscillating between þ1 and −1. The shaded regions surrounding each curve describe the values within one standard
deviation about the mean, where we can expect to find thedata most of the time. The noise fluctuations result primarily
from the shot noise of the measurements, but are also influenced by any error in the calibration parameters. As we can
expect for any linear estimation procedure, as the estimate
approaches the ideal cosine curve, noise fluctuations causesome of the data at the peaks and valleys to extend beyond
the physically valid region between þ1 and −1.
From these results, we can see that most polarimeters cantolerate a surprisingly low diattenuation if calibrated well. As
often is the case for indirect measurements, it is the accuracy
with which one can calibrate the system measurement modelthat limits the measurement performance more than theinherent accuracy of the components themselves. If we workwith low-extinction-ratio polarizers, we can still achieve
high-accuracy measurements, but only if we take extreme
care in the calibration and in ensuring the accuracy of ourreconstruction.9 For example, we should keep in mind thatthe diattenuation value of a pixel is a spectrally dependent
quantity, so that the measurement spectrum must be similar
to the spectrum used during calibration in order for the
results to be accurate.10
References
1. Y. Maruyama et al., “3.2-MP back-illuminated polarization image sensor with four-directional air-gap wire grid and 2.5-μm pixels,” IEEE
Trans. Electron Devices 65, 2544–2551 (2018).
2. F. Goudail, “Optimization of the contrast in active Stokes images,” Opt.
Lett. 34, 121–123 (2009).
3. N. Hagen and Y. Otani, “Stokes polarimeter performance: general noisemodel and analysis,” Appl. Opt. 57, 4283–4296 (2018).
4. R. A. Chipman, “Polarimetry,” Chapter 22 in Handbook of Optics,
M. Bass, Ed., Vol. 2, 2nd ed., pp. 22.1–22.37, McGraw-Hill, New York(1995).
5. N. Hagen, S. Shibata, and Y. Otani, “Calibration and performanceassessment of microgrid polarization cameras,” Opt. Eng. 58, 082408(2019).
6. N. Hagen, “Statistics of normalized Stokes polarization parameters,”
Appl. Opt. 57, 5356–5363 (2018).
7. J. S. Tyo and H. Wei, “Optimizing imaging polarimeters constructedwith imperfect optics,” Appl. Opt. 45, 5497–5503 (2006).
8. S. Roussel, M. Boffety, and F. Goudail, “Polarimetric precision ofa micropolarizer grid-based camera in the presence of additive andPoisson shot noise,” Opt. Express 26, 29968–29982 (2018).
9. J. S. Tyo, “Design of optimal polarimeters: maximization of signal-tonoise ratio and minimization of systematic error,” Appl. Opt. 41, 619–
630 (2002).
10. N. Hagen, “Flatfield correction errors due to spectral mismatching,”
Opt. Eng. 53(12), 123107 (2014).

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