Research Article

Vol. 57, No. 15 / 20 May 2018 / Applied Optics

4283

Stokes polarimeter performance: general noisemodel and analysis
NATHAN HAGEN*

AND YUKITOSHI OTANIUtsunomiya University, Department of Optical Engineering, Center for Optical Research and Engineering, 7-1-2 Yoto, Utsunomiya,Tochigi 321-0954, Japan*Corresponding author: nh@hagenlab.org
Received 17 November 2017; accepted 16 April 2018; posted 20 April 2018 (Doc. ID 313742); published 18 May 2018

We calculate the photometric Stokes parameter covariance matrices and SNRs estimated by polarimeters
exposed to general noise distributions, such as mixed Poisson–Gaussian (PG) noise. The measurement model
includes the effects of optical losses and detector quantum efficiency, enabling quantitative comparison of
instruments that have different photometric efficiencies. We demonstrate this capability by comparing the performance of many common polarimeter configurations, including diattenuator-based systems, such as Azzam’s
four-detector polarimeter [Opt. Lett. 10, 309 (1985)] and Kudenov’s stacked photovoltaic polarimeter [Opt.
Express 24, 14737 (2016)]. Working with the full covariance matrix under mixed PG noise, we also show that
instruments optimized under assumptions of Gaussian noise simultaneously exhibit optimal behavior under
Poisson noise. © 2018 Optical Society of America
OCIS codes: (260.5430) Polarization; (280.0280) Remote sensing and sensors; (230.5440) Polarization-selective devices;(030.4280) Noise in imaging systems.
https://doi.org/10.1364/AO.57.004283

1. INTRODUCTION
A number of researchers have worked on characterizing andoptimizing the performance of Stokes polarimeters [1–4].
Whereas the predominant performance metric has been thecondition number of the polarimetric measurement matrix,
or the equally weighted variance (EWV) of the Stokes parameters, more recent research has included the use of parametervariances and covariances. Most analyses have assumed independent identically distributed Gaussian noise (IG). Most
modern systems, however, operate in the shot-noise-limited
noise regime, and so a more appropriate model is Poissondistributed noise (PP, for “pure Poisson”). To incorporate bothregimes, we form a mixed Poisson–Gaussian (PG) noise model,
which can be taken as Gaussian or Poisson in the limit of smallor large detector noise. Thus, the three important noise regimes
for optical instruments can be written as IG, PG, or PP, inorder of increasing shot noise.
The focus on condition number and EWV as performance
metrics obscures important differences between systems, suchas detection efficiency, and ignores unequal variance values andcorrelations among the various Stokes parameter estimates.
Our analysis in the following fills in each of these three gaps,
presenting the full parameter covariance matrix in absolute
units for any noise regime. With this, we make quantitative
1559-128X/18/154283-14 Journal © 2018 Optical Society of America

comparisons among the many various linear and full Stokes
polarimeter designs, and in particular the less common designs
that incorporate diattenuating detectors, highlighting the factthat the covariance matrices of optimized polarimeters takeon a similar form in both IG and PP noise.
2. LINEAR STOKES POLARIMETERS
Linear Stokes polarimeters estimate only the first three parameters s0 , s 1 , s2  of the full Stokes vector s 0 , s1 , s 2 , s3 , i.e., thosestates corresponding to linear polarization. A division-of-time(DoT) linear Stokes polarimeter of this sort employs a single
rotating polarizer [Fig. 1(a)], whereas a typical division-offocal-plane polarimeter (DoFP) employs a tiled layout of micropolarizers placed on top of the detector array [Figs. 1(b)–1(e)][5]. These two system layouts produce identical measurement
matrices, and so we will not need to distinguish between thetwo in the following analysis. Rather, we will distinguish between the polarizer angles used to measure, starting with themost common choice of using the angles θ  0°, 45°, 90°,
and 135° [i.e., Fig. 1(b)].
Using Mueller calculus, we can model the intensity I measured at the detector for any given input polarization state s. Fora polarizer oriented at one of the orientation angles 0°, 45°, 60°,90°, 120°, or 135°, we find

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Vol. 57, No. 15 / 20 May 2018 / Applied Optics(a)the photoelectron count rather than the photon count, with theproportionality between the two being the external quantum
efficiency ϵ. Whereas the former (photon counts) is required
to compare instruments with differing quantum efficiencies,
the latter parameterizes the noise level. For IG noise, the measurement model can be more flexible because it is not signal
dependent, allowing s 0 to take on arbitrary units, as long as
the noise parameter shares the same choice of scaling.

(b)

A. 0-45-90-135 Configuration

In the case of the 0-45-90-135 configuration, we sample theinput light at the four angles θ  0°, 45°, 90°, and 135°, which
we can collect into matrix form as
(f)
(c)

(d)

(e)

Fig. 1. Two methods for linear Stokes polarimetry: (a) rotating analyzer, and (b)–(f ) tile of micropolarizer elements, arranged at orientations (b) 0-45-90-135, (c) 0-60-120, and (d) 0-45-90, and two
methods mixing measurements with and without polarizer elements:(e) N-0-45, and (f ) N-N-0-45.

ϵ
I 0  s0  s 1   n0 ,2ϵI 45  s0  s 2   n45 ,2
pﬃﬃﬃ 
3
ϵ1
I 60 
s  n60 ,s0 − s1 2 2
22
ϵI 90  s0 − s1   n90 ,2
pﬃﬃﬃ 
3
ϵ1
I 120 s  n120 ,s0 − s1 −2 2
22ϵI 135  s0 − s2   n135 ,(1)2
where each measurement I i is accompanied by a noise term ni ,and ϵ represents the polarimeter’s detection efficiency. For an
ideal lossless optical system, ϵ is equal to the detector quantum
efficiency, whereas for a real system, it can be described as
the “external quantum efficiency”—the ratio of detected photoelectrons per unit photon entering the optical system’s
entrance pupil.
Two of the configurations that we will consider later employ
a measurement with the polarizer removed, in which case themodel for the measured intensity is
I none  s0  nz :In the analysis to follow, we will assume all of the noise terms
to have zero mean, i.e., hI i i  I i and hni i  0. This causes nodifficulties for Poisson-distributed noise, because our definitionof the signal I and noise n results in the mean value of the
Poisson-distributed variable to be incorporated into I , whereas
n retains the zero-mean stochastic portion.
For photometric measurements, the most convenient unit
with which to model Poisson noise contributions is light “intensity” such that s 0 represents the number of photons incident
on the entrance pupil of the instrument during the measurement integration time. Poisson noise itself is parameterized by1
0
10 1 0 1
s0
1 1
0 0
n0
I0
C B s 1 C B n1 C
B I1 C ϵ B 1 0
10
CB C B C
B C B@ I 2 A 2 @ 1 −1 0 0 A @ s 2 A @ n2 A,
I3s3
n3
1 0 −1 0|ﬄﬄﬄ{zﬄﬄﬄ} |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} |ﬄﬄﬄ{zﬄﬄﬄ} |ﬄﬄﬄ{zﬄﬄﬄ}
0

I

s

W

n

or I  Ws  n. Here, W is generally called the measurement
matrix. By inspection, we can form estimators for the input
Stokes vector elements as1
I  I 45  I 90  I 135 ,
2ϵ 01ŝ1  I 0 − I 90 ,ϵ1ŝ2  I 45 − I 135 ,ϵŝ0 which we can rewrite in matrix form as
0
10 1
1
1 1 1
1
I0
ŝ0
B 2 0 −2 0 C B I 1 C
B ŝ1 C1CB C
B C B@ ŝ2 A 2ϵ @ 0 2 0 −2 A @ I 2 A
ŝ3
I3
0 0 0
0
|ﬄﬄﬄ{zﬄﬄﬄ} |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} |ﬄﬄﬄ{zﬄﬄﬄ}
0

ŝ

A

(2)

I

or ŝ  AI, where A is the analysis matrix—sometimes referred
to as the synthesis matrix or the polarimetric data reduction
matrix [6]. We use a hat (^) over any parameter that represents
an estimate of a quantity. Thus ŝi is a stochastic variable, as its
value depends on the measurements, and therefore also on thenoise, whereas si is deterministic (noise free).
Taking the expectation value of each estimator, we confirm
that they provide unbiased estimates of the Stokes vectors, i.e.,hŝ i i  si . For the parameter measurement statistics, we next
need to determine the variances, from which we can readily
calculate the SNR values. Using the standard formula for variance, we can write, for each si ,vars i   hŝ 2i i − hs i i2  hŝ2i i − s2i :(3)

We start with ŝ0 and solve for the second moment, substituting
in our definitions for the measurement terms I i and the input(deterministic) Stokes vector elements si into the estimator:

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Vol. 57, No. 15 / 20 May 2018 / Applied Optics


1
2
I

I

I

I

45
90
135
4ϵ2 0
1
 2 h2ϵs 0  n0  n45  n90  n135 2 i
4ϵ1
 2 h4ϵ2 s20  4ϵs0 n0  4ϵs0 n45  4ϵs 0 n90  4ϵs0 n135
4ϵ
 n20  n245  n290  n2135  2n0 n45  2n0 n90
hŝ20 i 

 2n0 n135  2n45 n90  2n45 n135  2n90 n135 i
1 2 4ϵ2 s20  hn20 i  hn245 i  hn290 i  hn2135 i
4ϵ 2hn0 n45 i  2hn0 n90 i  2hn0 n135 i
 2hn45 n90 i  2hn45 n135 i  2hn90 n135 i:(4)We show this derivation in detail to make clear that at this point,
we have made no assumptions on the statistical distribution of
the noise, other than the fact that the noise samples have welldefined mean and variance. Correlated noise causes no difficulty
up to this point, and any noise distributions (not just Poisson orGaussian) with finite mean and variance values are valid within
this analysis [7]. However, to simplify the analysis, we make ourfirst substantial assumptions about the characteristics of thenoise: (1) that the noise terms all share the same variance(hn2i i  hn2j i ≡ hn2 i), and (2) that the noise terms are independent of one another (hni nj i  hni ihnj i). This latter assumption
in particular is useful, in that all mixed moments, due to thezero-mean property of the noise, become zero:1
hŝ20 i  s20  2 hn2 i,ϵso that1
varŝ 0   2 v d ,ϵwhere we define v d  hn2 i as the variance of the measurement noise.
Performing a similar calculation for the second and third
Stokes parameters, we obtain similar expressions:22
varŝ 2   2 v d :varŝ1   2 v d ,ϵϵLikewise, calculating the covariances among the parameter
estimates follows along a similar path:covŝ0 , ŝ 1   hŝ0 − s0 ŝ1 − s1 i  hŝ 0 ŝ 1 i − s 0 s 1 :So we proceed to calculate the joint moment11I  I 45  I 90  I 135  · I 0  I 90 
hŝ0 ŝ1 i 
2ϵ 0ϵ1 s0 s1  2 hn2 i,ϵwhere we have skipped the intermediate steps this time. From
this we obtain the covariance:1
covŝ0 , ŝ 1   2 v d :ϵ
By symmetry, we can also conclude thatcovŝ0 , ŝ 2  

1
v :
ϵ2 d

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The remaining covariance element cannot be obtained bysymmetry arguments alone, but calculation gives hŝ1 , ŝ2 i  0,
so that we can form the full covariance matrix K of ourestimators:
10
covŝ0 , ŝ 1  covŝ 0 , ŝ2 varŝ 0 CBK IG  @ covŝ1 , ŝ0 varŝ1 
covŝ 1 , ŝ2  A
covŝ2 , ŝ0  covŝ2 , ŝ 1 
0
1
1 1 1
v B
C
 d2 @ 1 2 0 A:ϵ
1 0 2

varŝ2 

The EWV performance metric is obtained as the trace of K.
For measurements taken in the detector-limited noise
regime, such as IG noise, this covariance matrix tells us everything we can know about the second-order statistics of ourestimated parameters.
B. Mixed Gaussian–Poisson NoiseTo expand the preceding analysis to a more general noise regime,
we need to form a model for each of our noise terms. In the shotnoise-limited regime, each noise term will be signal dependent,such that the variance of a Poisson-distributed noise variableequals its mean value. For each of the intensities in the measurement model [Eq. (1)], we can thus write the noise variances as
9varn0   2ϵ s0  s 1   v d>>ϵ>varn45   2 s 0  s2   v d
>
>
>
ϵ
>
varn90   2 s 0 − s1   v d>=ϵvarn135   2 s 0 − s2  pvﬃﬃd:(5)>
varn60   2ϵ s 0 − 12 s 1  p23ﬃﬃ s2   v d >>>>>varn120   2ϵ s 0 − 12 s1 − 23 s2   v d >>;varnnone   ϵs0  v dHere, we maintain our definition of v d as the Gaussiandistributed (signal-independent) noise from the detector.
From this mixed PG noise model [8,9], we can take theshot-noise-limited regime (PP noise) by taking the limitv d → 0, or the detector-noise-limited regime (IG) by taking
v d ≫ everything else.
Calculating the variances and covariances under the PG
noise model involves returning to Eq. (4) and substituting
the appropriate variances [Eq. (5)] for each hn2i i. Note that
under PG noise, each independent realization of the noise isindependent, so that once again all of the cross terms cancel:
1varŝ0   2 hn20 i  hn245 i  hn290 i  hn2135 i
4ϵ 2hn0 n45 i  2hn0 n90 i  2hn0 n135 i
 2hn45 n90 i  2hn45 n135 i  2hn90 n135 i


1
ϵ
ϵ
s0  s1   v d  s0  s2   v d
 2
4ϵ
2
2



ϵ
ϵ
 s0 − s 1   v d  s0 − s 2   v d2
2
1
1
 s0  2 vd :
2ϵ
ϵ

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Solving for the remaining terms, we obtain the covariance matrix under the PG noise:
0
1
1
1
1
1
2 s0  ϵ vd
2 s1
2 s2
C
1B 1
C,
s0  2ϵ v d0
K PG  B
2 s1
@
A
ϵ
1
2
s
0
s

v
0
2 2
ϵ d
from which we can clearly see the dependence of the measurements on the input state of polarization (SOP).
We can also rewrite the form of the covariance matrix for thelimit of PP noise, giving
10
1 s˜1 s˜2
s0 @s˜1 2 0 A,
K PP 
2ϵs˜2 0 2where the tilde (∼) represents a normalized parameter:s˜i ≡ si ∕s0 . The matrix diagonal, and thus the EWV, has exactly
the same form here as it did for IG noise, with the only difference being the scale factor in front.
From the estimated variances, we can calculate the SNR ofeach of the Stokes parameters:
SNRŝ0  s0varŝ0 1∕2

s0
,
1∕2
1
1
s

v
2
0
d
2ϵ
ϵ

which in the IG and PP noise regimes becomesSNR IG ŝ 0   ϵs0 ∕σ d ,and

SNR PP ŝ0  

SNR IG ŝ2   ϵs 2 ∕pﬃﬃﬃ
2σ d ,

pﬃﬃﬃﬃﬃﬃﬃﬃ
2ϵs 0 ,

pﬃﬃﬃﬃﬃﬃ
SNR PP ŝ2   s˜2 ϵs0 :and

C. Matrix Approach for Calculating the CovarianceMatrix

Although we have derived the variances and SNRs using thedefinition of the variance [Eq. (3)], we can calculate the covariance matrix directly from the measurement and analysis matrices. The covariance matrix K of Stokes vector elements can be
written in terms of the analysis matrix A as
K  AΓA T ,(6)where Γ is the covariance matrix of the raw detector measurements I i . Under IG, PG, and PP noise, Γ is diagonal of theform
01
varn0 
0
…
0
B 0
0 C
varn1  …
BCΓ  B ..
C,
..
..
..
@ .
A
.
.
.
0

0

…

or, under IG noise, simplyΓIG  v d I,

varnM 

k0

l0

l0

where L is the number of measurements (the number of rows inW). An advantage of the indicial form is that we can write theresult directly in terms of the measurement matrix W without
having to first form Γ. The result, Eq. (7), differs from thatof P
Goudail [10] by the addition of the Gaussian term
v d Ail Ajl .
Something that must be kept in mind when working withthe covariance matrices is that for singular A, the covariancematrix is in general not singular. Rather, one will typicallysee that two of the parameters will be fully correlated with
one another, with covariance elements equal in magnitude
to the variances along the diagonal. Whereas the 2-norm condition number κ2 of a singular matrix goes to infinity, giving aclear indication of a problem, this behavior in the covariancematrix is more subtle, requiring that one look carefully at thestructure of the matrix to spot the singular structure. Thus, κ 2 isoften a convenient supplement to covariance analysis.
D. 0-60-120 Configuration

where σ d is the detector noise standard deviation, i.e.,
pﬃﬃﬃﬃﬃ
σ d  v d . In accordance with what we expect, the SNR shows
an increase with higher detection efficiency ϵ and with more
light s 0 .
Likewise, calculating the SNRs for ŝ1 and ŝ2 gives
pﬃﬃﬃ
pﬃﬃﬃﬃﬃﬃ
SNR IG ŝ 1   ϵs1 ∕ 2σ d , and SNR PP ŝ 1   s˜1 ϵs0
andfor M × M identity matrix I. In indicial form, Eq. (6)becomes [10]
X
 X
3
L−1
L−1
X
K ij sk
Ail Ajl W lk  v d
Ail Ajl ,(7)

Using the same methods, we can calculate the covariances forother polarimeter configurations. Bernard and Wehner, e.g.,determined that the optimal set of three linear analyzer angles
for estimating the linear Stokes vector is f0°, 60°, 120°g [shownin Fig. 1(c)] [11,12]. For this “0-60-120” configuration, thethree measurements take the formϵ
I 0  s0  s 1   n0 ,2
pﬃﬃﬃ 
3
ϵ1
I 60 
s  n60 , ands − s 2 2
2 0 2 1
p
ﬃﬃ
ﬃ


3
ϵ
1
s2  n120 ,s0 − s1 −I 120 2
2
2
so that the measurement matrix W, analysis matrix A, and PG
noise covariance matrix K are obtained as
01 1
1 1
1
0
3
3
3
1 1
0
0
C
B2
1
ﬃﬃ
p
C
− 13 C
ϵB
2B
3
1
3 −3
C
C,
B
B
0
1
−
A B
W @
2
2
A,
2
ϵ @ 0 p1ﬃﬃ − p1ﬃﬃ C
pﬃﬃ
A
3
3
1 − 12 − 23 0
0 0
0
and

0

s 2v
2B 0 ϵ d
K PG  @ s1
3ϵ
s2

s1
2s0  s 1  4ϵ v d
−s2

1
s2
C
−s2
A:
4
2s 0 − s1  ϵ v dHere, we see that whereas the variances of ŝ1 and ŝ2 in the
0-45-90-135 configuration are independent of the input SOP,
the variances for the 0-60-120 setup have the curious property
that they depend on the input state of s1 but not of s 2 . This is afeature of the 0-60-120 that we have also observed experimentally in our lab.

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E. N-0-45 and N-N-0-45 Configurations

With a DoFP polarimeter in mind, Perkins and Gruev argued
that an optimal four-measurement configuration can beachieved with one fully transmissive filter (i.e., no polarizer)
plus just two polarizer angles of 0° and 45° [Fig. 1(e)] [13].
This setup can also be used to estimate the linear Stokes vector
using only three measurements, and so we start with the threemeasurement form and continue on to Perkins’s suggested
four-measurement system. For this “N-0-45” configuration[Fig. 1(e)], the three measurements take the form
I N  ϵs0  nz ,ϵ
I 0  s0  s1   n0 , and2
ϵI 45  s0  s2   n45 ,
2

and
0

s0  1ϵ v d

1B
K PG  @ −s0 − 1ϵ v d
ϵ−s0 − 1ϵ v d−s0 − 1ϵ v d−s 0 − 1ϵ v ds 0  1ϵ v d

3s0  2s2  5ϵ vd

3s0  2s 1  5ϵ v ds0  1ϵ vd

1CA:

Perkins’s original four-measurement system [Fig. 1(f )] uses
two samples without a polarizer, producing the measurement
equation and covariance matrix1−s0 − 1ϵ v d−s 0 − 1ϵ v ds0  1ϵ v d1BCs0  1ϵ v d
K PG  @ −s0 − 1ϵ v d 5s0  2s1  9ϵ v dA,2ϵ−s0 − 1ϵ v ds 0  1ϵ v d
5s0  2s2  9ϵ v d0Table 1.

where we see that the variance of ŝ0 and all of the off-diagonalterms have been halved, but the variances of ŝ 1 and ŝ 2 have been
reduced by somewhat less than that.
Finally, it is interesting to see the consequences of getting by
with only three of the original four polarizers in a conventionalpolarization camera, leaving only the 0°, 45°, and 90° polarizers[Fig. 1(d)]. The measurement, analysis, and covariance matrices are obtained as
0 1 0 1 1!
1 1 0 0ϵ1 B 1 0 −1 C
1 0 1 0 ,
A @
W
A,
2
ϵ −1 2 −1
1 −1 0 0
0 0 0
andso that the measurement, analysis, and covariance matrices are
obtained as
0
1
1
0
1 0 0
1 0 0 0B
CC
B1 1
−1 2 0 C
1B
C
B
B
C,
A B
W  ε@ 2 2 0 0 A,
ε @ −1 0 2 C
A
1
0 12 0
2
0 0 0

4287

0

s0  2ϵ v d
1B
s1
K PG  @ϵ−s 0 − 2ϵ v ds1s0  2ϵ v d−s 1

1−s0 − 2ϵ v d
C
−s1
A:
6
3s0  2s2  ϵ v d

3. COMPARISONS OF LINEAR STOKESPOLARIMETERS
The Stokes parameter covariance matrix represents the complete statistical information we have about the parameter estimates up to second order, providing an ideal metric to evaluate
and compare system performance. In particular, the advantages
that the covariance matrix has over the widely used matrix condition number κ, or the EWV, are the following.
1. κ is invariant to detection efficiency ϵ, so that acomparison of systems having different efficiencies must incorporate an additional metric.
2. κ alone cannot be scaled to give the SNR of a polarimetric measurement.
3. κ does not provide information on the dependence of
SNR on the input SOP.
4. Both κ and EWV combine all of the Stokes vector estimation characteristics into one metric, obscuring informationon the differences in variance among the parameters as well as
correlations between estimates.

Compiled Variance Results for the Five Linear Stokes Polarimeter Configurationsa

Configurationvarŝ 0 

varŝ 1 

varŝ 2 

EWV

κ1

κ2

κ∞

κF

5
20/3
10
11
19/2

3.00
3.73
4.50
6.00
6.00

1.41
1.41
2.41
3.19
3.61

2.00
3.15
4.00
3.00
3.00

3.16
3.16
3.87
4.69
5.34

IG Regime: □ vd ∕ϵ

20-45-90-135
0-60-1200-45-90N-0-45
N-N-0-45

1
4/3
2
1
1/2

2
8/3
2
5
9/2

0-45-90-135
0-60-120b
0-45-90N-0-45
N-N-0-45

1
2
2
3

1
4
2
3  3 s˜1
1
3

s˜1
2
5

s˜1
4

2
8/3
6
5
9/2

PP Regime: □ s 0 ∕ϵ

a

1
1
1
4

1
4
2
3  3 s˜1

3  2˜s2
32  s˜2
51
4  2 s˜252
104
3  3 s˜1

5  2˜s2
4  s˜1  s˜2
11
1
4  s˜1  2 s˜2Each of the IG regime entries (□) share a v d ∕ϵ2 factor, whereas each of the PP regime entries (□) share a s 0 ∕ϵ factor.
The dependence on s˜1 in the varŝ 2  column is not a typo.

b

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The variances for the linear Stokes polarimeter configurations are listed in Table 1 for easy comparison. From the table,
we see that the smallest variances are achieved for the conventional 0-45-90-135 configuration and for the N-N-0-45 system. However, because these two use four measurements tothe other systems’ three, they can be considered as unfairly
rewarded and so should be scaled by a factor of 4/3 to apply
to working with the same number of photons. Applying thisscale factor, we find that all of the variances of the 0-45-90135 and the 0-60-120 configurations are identical under IG
noise. They do not, however, have the same covariances (theoff-diagonal elements).
Farther down in Table 1, we see differences among the configurations showing up in the PP regime. Thus, in PP noise,
whereas the 0-45-90-135 and 0-60-120 configurations perform
equivalently when s˜1  0 (when scaled to compensate for thedifferences in the number of measurements), the 0-60-120 system’s performance worsens as s˜1 increases from zero toward 1 andimproves as s˜1 decreases from zero toward −1. Note that thedependence on s˜1 in the varŝ2  is not a typo. Whereas the performances of the two systems remain the same on average, they are
not the same for any specific measurement except when s˜1  0.
For the other polarimeter configurations shown in Table 1,
we see that the N-0-45 and N-N-0-45 configurations havegood performance for estimating s0 , which should be no surprise, but that they make a heavy sacrifice in performance for
estimating s1 and s2 , producing variances that are more than
twice as high. In addition, we see that the 0-45-90 configuration, by dropping the fourth measurement, has doubled thevariance in ŝ 0 and tripled it in ŝ 2 . More complex relationships
develop under PP noise, but the basic trend remains.
Whereas the polarimeters considered here use single-channel polarization detection, so that half of the input light is lost,
DoT (but not DoFP) implementations can improve the efficiency by a factor of 2 by using a polarization beam splitterand two detection channels. This can be used to either double
the amount of light collected, effectively removing the factor of
1/2 at the front of each measurement matrix, or halving thenumber of measurements required. Doubling the number ofmeasurements also causes a doubling of the detector noise v d .
The equivalent performance of the 0-60-120 and 0-45-90135 configurations supports the following conclusion reachedby several researchers: if the configuration is optimal, there isno noise benefit to spreading the measurements to sample themeasurement space more evenly [1,3,14]. This property is confirmed by the covariance matrices in Table 1, where we also seethat the equivalence holds true in general for IG noise, but it
holds in PP noise only when averaging the performance acrossall polarization states. Running a simulation over all configurations that sample the half-circle evenly at up to 360 angles, we
confirmed that the equivalence holds for all such configurations, once normalized by the number of measurements taken.
Whereas Tyo concurred with this result [3], he also pointed
out that instrument robustness to calibration error and perturbation improves with use of more sampling angles, so that there
is some trade-off between noise performance and robustness.
The SNR values corresponding to the parameter variances
are listed in Table 2.

Table 2. Compiled SNR Results for the Five Linear
Stokes Polarimeter Configurationsa
SNRŝ 0 

Configuration

SNRŝ 1 

SNRŝ 2 

IG Regime: □ ϵs 0 ∕σd0-45-90-135
0-60-1200-45-90N-0-45
N-N-0-45

1
0.866
0.707
1
1.414

0.707˜s1
0.612˜s1
0.707˜s1
0.447˜s1
0.471˜s1
PP Regime: □0-45-90-135
0-60-1200-45-90

1.414
0.866
1

N-0-45

1

N-N-0-45

2

pﬃﬃﬃﬃﬃﬃﬃﬃﬃ
ϵ∕s 0

s1
1.732
p
ﬃﬃﬃﬃﬃﬃﬃ s12˜s1
s1
p1.414
ﬃﬃﬃﬃﬃﬃﬃﬃﬃ s1
32˜s1
2
pﬃﬃﬃﬃﬃﬃﬃﬃﬃ
s1
54˜s10.707˜s2
0.612˜s2
0.408˜s2
0.447˜s2
0.471˜s2
s2
1.732
p
ﬃﬃﬃﬃﬃﬃﬃ s2
2˜s1
1
pﬃﬃﬃﬃﬃﬃﬃﬃﬃs
32˜s 2 2
1.414
pﬃﬃﬃﬃﬃﬃﬃﬃﬃ s 2
32˜s 2
2
pﬃﬃﬃﬃﬃﬃﬃﬃﬃs
52˜s 2 2aEach of the IG regime entriesp(□)
ﬃﬃﬃﬃﬃﬃﬃﬃshare a ϵs 0 ∕σ d factor, whereas each of the
PP regime entries (□) share a ϵ∕s 0 factor.

4. ROTATING-RETARDER FULL STOKESPOLARIMETERS
Whereas the discussion up to this point has focused on linear
Stokes polarimetry, full Stokes polarimeters, estimating all fourof the Stokes vector elements, occupy perhaps an even more
important place in research work. The most widely used
implementation for this is probably the rotating-retarder fixedpolarizer (RRFP) configuration, as shown in Fig. 2(a).

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 2. Instrument layouts for (a) RRFP [15], (b) DVRFP [2],(c) DRRFP [16], (d) RRPC, (e) Azzam’s four-detector polarimeter[17], and (f ) diattenuator photovoltaic detector stack [18]. In eachsubfigure, R i is the ith retarder, Dj the jth detector, and P the polarizer
element.

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Table 3. Instrument Configurations for Three RP
Polarimeters, Where θ Indicates the Retarder Angle and ϕ
the Polarizer Angle
Configuration

Measurement #

δ

θ

ϕ

1
2
3
4
1
2
3
4
1
2
3
4
5
6
7
8

90°
90°
90°
90°
131.8°
131.8°
131.8°
131.8°
90°
90°
90°
90°
90°
90°
90°
90°

−45°
0°
30°
60°
−51.69°−15.12°
15.12°
51.69°
22.5°
22.5°
22.5°
22.5°
67.5°
67.5°
67.5°
67.5°

0°
0°
0°
0°
0°
0°
0°
0°
0°
45°
90°
135°
0°
45°
90°
135°

Ambirajan

Sabatke

Shibata

Ambirajan and Look analyzed the best choice of azimuth
angles for use with the RRFP system employing a quarter-wave
retarder and found that the set of four angles −90°, −45°, 30°,
and 60° are a good choice for minimizing the measurement
matrix condition number [15]. Sabatke et al. considered simultaneously optimizing over angles and wave plate retardance δ,finding that the widely used choice of δ  90° is in fact not
optimal and that by choosing δ  132°, one can improve performance by about 50% [1].
A related configuration employs a rotating retarder in frontof a 0-45-90-135 DoFP polarization camera. This system takes
advantage of the four polarization angles simultaneously
sampled at the detector array to collect the full Stokes vector
image from only two retarder angles [19]. To our knowledge,
the performance of this system has not yet been analyzed,
despite its simplicity.
The configurations for the three RR polarimeters aretabulated in Table 3.
A. Ambirajan’s RRFP

Applying Mueller calculus to Ambirajan’s suggested instrument
setup [15], we obtain the measured intensities asϵ
I 0  s 0  s1   n0 ,2ϵ
I 1  s 0  s3   n1 ,2
pﬃﬃﬃ
pﬃﬃﬃ 

3
3
ϵ
1
s2 −
s  n2 , ands0  s1 
I2 
4
2 3
2
4
pﬃﬃﬃ
pﬃﬃﬃ 

3
3
ϵ
1
s −
s  n3s  s −I3 
2 0 4 1 4 2 2 3
giving
1
0
1
1
0
0
C
B2 2
1 C
B1 0
0
B2
2 C
pﬃﬃ
pﬃﬃ C:W  ϵB
B1 1
3
3C
−
B2 8
8
4 C
@
pﬃﬃ
pﬃﬃ A
1
1
− 83 − 43
2
8

4289

The full covariance matrix under PG noise is given in Table 7,
which in the limit of IG noise becomes
0
1
2.01 −2.63
0
0.13
B
C
−2.63 7.25
00.49 Cvd B
BC:K IG  2 B
ϵ @ 0
0
10.67
0 C
A
0.13
0.49
0
1.72
B. Sabatke’s RRFPThe RRFP configuration optimized by Sabatke et al. [1],employing a δ  132° wave plate at four azimuth angles, givesa measurement matrix of01
0.5 0.2888
0.3628 −0.1869
BC
B 0.5 0.2888 −0.3628 0.1869 CBC:W  ϵBC@ 0.5 −0.2897 −0.1881 −0.3615 A
0.5 −0.2897 0.1881 0.3615
The full covariance matrix under PG noise is given in Table 8,
which in the limit of IG noise becomes01
1 0 0 0v B0 3 0 0C
C:
K IG  2d B
ϵ @0 0 3 0A
0 0 0 3We see that the covariances disappear under IG noise, and thes1 –s 3 parameter variances become uniform. In the limit of PP
noise, the covariance matrix Table 8 becomes
0
1
1
s˜1
s˜2
s˜3
pﬃﬃﬃ
pﬃﬃﬃ
B
C
s˜1
3
s˜2 − 2s˜3 − 2s˜2 − s˜3 C
s0 B
B
C,
K PP  B
pﬃﬃﬃ
pﬃﬃﬃ
2ϵ @ s˜2 s˜2 − 2s˜33
− 2s˜1 C
A
pﬃﬃﬃ
pﬃﬃﬃ
s˜3
− 2s˜2− 2s˜1
3from which we see that the covariance matrix is no longer
diagonal but that the matrix trace is identical to that of theIG case, up to a scalar factor. In fact, this pattern of1∶3∶3∶3 ratios of variances along the diagonal is retained in
the full PG matrix, a feature shared by all of the optimized
polarimeter designs, as we will see in the following.
C. Shibata’s Rotating-Retarder Polarization Camera

Shibata et al. adapted a standard 0-45-90-135 configuration
DoFP linear Stokes polarimeter to full Stokes polarimetry byinserting a rotating quarter-wave plate in front and rotating
to two azimuth angle (θ) positions [19]. It is useful to look into
the performance of this system in comparison to others due toits simplicity and to the increasing availability of these DoFP
cameras. Unfortunately, the polarization camera cannot be converted into a snapshot full Stokes polarimeter with the additionof a fixed retarder in front, because this configuration can only
generate a 2D subset of states on the Poincaré sphere, so that s 1and s 2 cannot be separated from one another. Using a pair ofretarder azimuth angles generates a set of eight measurements,
which for θ  22.5° and 67.5° gives the measurement matrix

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0

1

B
B1
B
B
B1
B
1B
B1
W B
4B1
B
B
B1
B
B1
@
1

p1ﬃﬃ
2

0

0

1

− p1ﬃﬃ2

0

0

−1

1

0

0

1

−1

0

0

−1

− p1ﬃﬃ2

1

C
0 C
C
C
p1ﬃﬃ C
2 C
0 C
C
C,
0 C
C
C
0 C
C
0 C
A
0

where we have normalized the matrix such that the first columnof the matrix sums to 2, to agree with the overall light level
implicit in the previous full Stokes polarimeters. The full
PG-noise covariance matrix, which is simple enough to present
here in its entirety, is
1
0
s1
s2
s3
s 0  2ϵ v dC
1 B s1
4s 0  16
0
0
ϵ vd
C,
K PG  B
16
A@0
4s 0  ϵ v d
0
s2
2ϵ
8
s3
0
0
2s0  ϵ v dand which in IG noise takes on the particularly simple form01
2 0 0 0BC
0 8 0 0C
vd B
BC:
K IG  2 B
ϵ @0 0 8 0CA
0 0 0 4
From the covariance matrix, we can see that the normalization of the measurement matrix—multiplying W by 1/2 to set
the overall amount of light sampled at 4s0 —indicates that thissystem is less efficient with measuring the input light than the
RRFP instruments are. However, one of the benefits of
Shibata’s approach is that it uses only two retarder positions
to measure, which one can argue boosts its ability to collect
light by a factor of 2, so that the parametervariances are repﬃﬃﬃ
duced by a factor of 2 in IG noise, 2 in PP noise. With thisscaling, the polarization camera-based system’s performance iscomparable to that of the RRFP instruments.
5. DUAL-RETARDER FULL STOKESPOLARIMETERS
Whereas the previous systems used single-retarder elements with
fixed retardance δ but variable orientation θ, another common
setup for polarimetry is to use variable retarders, such as tunableliquid crystal retarders and acousto-optic modulators, which
typically have fixed orientation. Because fast modulators, suchas electro-, magneto-, and acousto-optic elements, typically use amore involved estimation procedure than the slower liquid crystal elements do, we will focus on the latter here.
Whereas single-variable-retarder elements have a limited
ability to sample polarization space, dual-variable-retarderfixed-polarizer (DVRFP) systems can access a larger range ofpolarization states than the RRFP or rotating-retarder polarization-camera (RRPC) systems do [20]. Thus, they have a wider
range of configurations capable of reaching optimal performance. An example given by Tyo [3] is analyzed in the next

Table 4. Instrument Configurations for the Variable
Retarder Systems
Configuration Measurement #

θ1

δ1

θ2

δ2

Tyo

1
2
3
4

22.5° 158°
22.5° 127°
22.5° 47°
22.5° 0.66°

45°
45°
45°
45°

50.6°
−178°
−16.9°
126°

Zallat1
2
3
4−41.4°
41.4°−41.4°
41.4°−20.3°−20.3°
20.3°
20.3°

90°
90°
90°
90°

90°
90°
90°
90°

section, though many other optimized configurations are
possible [21]. In a similar fashion, Zallat et al. demonstrate thatusing two dual-rotating-retarder fixed polarizers (DRRFPs)
increases the flexibility of an RRFP polarimeter, and this system
is analyzed in the following section [16].
A. Tyo’s DVRFP

Tyo’s DVRFP configuration [2] is listed in Table 4 and gives ameasurement matrix
01
0.5 −0.090789 0.408154
0.274167
B
C
B 0.5 −0.089631 −0.410064 0.271687 CC:
W  ϵB
B 0.5 0.477507
0.000899 −0.148277 C@
A
0.5 −0.297178 0.003285 −0.402088
This produces the full PG covariance matrix given in Table 9,
converging to the following in the limit of IG noise:
0
1
1.00
0
0
0
v B 0
3.01
0
0 C
C:
K IG  2d B
@
0
0
2.99
0 A
ϵ
0
0
0
3.01
B. Zallat’s DRRFP

Zallat et al. proposed a setup with a pair of rotating quarterwave plates followed by a fixed polarizer (DRRFP) [16].
The configuration details are listed in Table 4, and the setup
produces a measurement matrix
0 0.5 −0.287573−0.3198
0.255009 1
B 0.5
BW  ϵB@ 0.5

0.296629−0.248116

0.296629

0.248116−0.316938 CCC:
0.316938 A

0.5−0.287573

0.319800−0.255009

The full PG covariance matrix for this is given in Table 10,
converging to the following in the limit of IG noise:0 1.000
0
0 1
K IG 

vd B 0
@
0
ϵ2
0

2.93
0
0

0
3.05
0

0 C
A:
0
3.02

6. IDEALIZED FULL STOKES POLARIMETERS
Iniesta and Collados introduced an alternative metric for evaluating polarimeter performance that they call the polarimetricResearch Article

Vol. 57, No. 15 / 20 May 2018 / Applied Optics

efficiency, from which they derive two idealized measurement
matrices for maximizing their efficiency metric [22]. The first
ideal matrix is a four-measurement scheme that was later presented by Goudail in the context of a ideal measurement in thepresence of Poisson noise, apparently derived independently ofIniesta’s work [10]. This is the first ideal system presented in thenext section. Because Iniesta and Goudail both refer to themeasurement matrix alone without discussing an instrument
configuration capable of achieving it, we refer to this as an“idealized” polarimeter, though Mu et al. later demonstrated
that this measurement can be achieved using a polarimeter witha fixed polarizer and a pair of rotating quarter-wave plates [23].
In addition to the first system, Iniesta and Collados also presented an idealized six-measurement system in which each of
the Stokes s1 through s3 parameters is analyzed individually by apair of measurements. This, they show, achieves the samepolarimetric efficiency as the four-measurement ideal system,
albeit at the cost of requiring an added two measurements.
Analyzing the performance of these two idealized systems
provides an interesting view into the polarimetric design space.
A. Iniesta’s Four-Measurement System

The first of Iniesta’s idealized systems was identified by
Goudail as having the special characteristic that its performanceis independent of the input SOP. The system measurement
matrix is
1
0 pﬃﬃﬃ
3 1 −1 −1C
B pﬃﬃﬃ
3 1
1
1 C
ϵ B
C,
B
p
ﬃﬃ
ﬃ
W
pﬃﬃﬃ
C
2 3B
@ 3 −1 1 −1 A
pﬃﬃﬃ
3 −1 −1 1for which the full PG covariance matrix K PG is shown in
Table 11. In the limit of IG and PP noise regimes, K PG
becomes
01 0 0 01
vd B 0 3@
ϵ2 0 0
0 0

0
3
0

1

s˜1

s˜3

3
pﬃﬃﬃ
3s˜3
pﬃﬃﬃ
3s˜2

s˜2
pﬃﬃﬃ
3s˜3

K IG 
and

0

B
s˜1
s0 B
K PP  B
B
2ϵ @ s˜2

3
pﬃﬃﬃ
3s˜1

0C
A
0
3
1
s˜3
pﬃﬃﬃ C3s˜2 C:pﬃﬃﬃ C3s˜1 CA
3

Here we can see that this system achieves Goudail’s aim of performance that is independent of the input SOP, but we do findthat the off-diagonal covariances retain some SOP dependence.
B. Iniesta’s Six-Measurement System

Although it was not identified as such by its authors, the secondof Iniesta’s systems also achieves the characteristic that its performance is independent of the input SOP. Though Iniesta andCollados did not identify instrumentation capable of achieving
this measurement matrix, it is straightforward to see that a system such as that of Ref. [24] has an equivalent measurement
matrix. This system employs a six-cell filter wheel in which4291

pairs of cells contain vertical/horizontal polarizers, 45°∕−45°
polarizer, and right-circular/left-circular polarizers. To standardize the measurement to operate on a light input of 4s0 ,the first column of the measurement matrix is scaled to givea sum of 2 (reduced from 4 due to the polarizer rejecting halfof the light).
For this system, the measurement matrix is
0
1
1 1
0
0
B 1 −1 0
0 C
B
C
ϵB
1
0
1
0 C
B
C,
W B
3 B 1 0 −1 0 C
C
@1 0
0
1 A
1 0
0 −1
giving a covariance matrix
0
1
s1
s2
s3
s 0  3ϵ vd
BCC
3s0  9ϵ v d
0
0
1B
B s1
C
K PG  B
C,
9C
2ϵ B s2
0
3s

v
0
0
d
ϵ
@
A
s3
0
0
3s 0  9ϵ v dwhich in the IG limit becomes
03
2

B
0
vd B
K IG  2 B
B
ϵ @0
0

0 0
9
2

0

0

9
2

0 0

0

1

C
0C
C:
0C
A
9
2

7. DIATTENUATING FULL STOKESPOLARIMETERS
Developing a photometric model for polarimetric measurement
allows us to characterize the statistical behavior of systems having different detection efficiencies and, in particular, systems
that incorporate diattenuating elements instead of ideal (ornear-ideal) polarizer elements. The first system is Azzam’s
four-detector polarimeter [Fig. 2(e)], in which the incoming
light is reflected off three detector elements oriented at specificorientation angles for each to act as diattenuators and partial
detectors [17]. The light remaining after the third detector
is absorbed at the fourth, placed at normal incidence to thebeam. The following calculations are apparently the first tocompare the performance of Azzam’s system to other polarimetric techniques.
The second instrument we analyze is the stacked photovoltaic polarimeter developed by Kudenov et al. [Fig. 2(f )][18,25]. This instrument consists of a stack of partiallydetecting organic photovoltaic elements sandwiched with wave
plates. Here too we encounter an instrument employing diattenuator elements, so that quantitative comparison with othertechniques requires use of a photometric model.
In both cases, researchers provide experimentally obtained
measurement matrices rather than theoretical models, so thatthe detection efficiency ϵ is already incorporated into W. We
can also note that whereas most of the other polarimeter types
scan in time to collect the four measurements needed to reconstruct the Stokes vector, Azzam’s and Kudenov’s instruments

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split the input light among their four detectors. Thus, the diattenuating systems’ measurement model naturally assumes theamount of light available for measurement is s0 rather than 4s0 .
This artificially penalizes the noise in the reconstructed Stokes
parameters. Thus, to compare performance under similar measurement assumptions, we augment the measurement matrices
to assume 4s0 as input. This produces entries that are greaterthan 1 in the first column, which appears unphysical unless wekeep in mind this scaling by a factor of 4.
A. Azzam’s Four-Detector Polarimeter

The measurement matrix for the four-detector polarimeter isprovided by Azzam in Ref. [26] as
0 0.746 0.157 −0.071 −0.001 1
B 2.433
W@
0.694
2.597

−1.180−0.244−1.9340.516
0.137−0.2960.447 CA:
−0.405−0.429However, we can see that this matrix is not provided in photometric form—the sum of the entries in the first column is 6.47,
which is unphysical for a photometric model. If we assume thatthe overall detection efficiency of this system approaches 1, andthat the input light level is 4s 0 , then we can normalize thematrix to
0 0.461 0.097 −0.044 −0.001 1
B 1.504 −0.730 0.319
W  @
0.429 −0.151 0.085
1.606 −1.196 −0.183

0.276 CA:
−0.250−0.265

The full PG covariance matrix for W  is shown in Table 12. In
the limit of IG noise, the covariance matrix becomes
0 2.26 3.37 −2.64 0.26 1
B 3.37 5.63 −3.12 −0.50 CK IG  vd @A:
−2.64 −3.12 16.69 −8.41
0.26 −0.50 −8.41 9.93

Whereas the normalization of the measurement matrix has
no effect on the condition numbers, it increases the PP-regime
variances by 1.6 and the IG-regime variances by 2.6.
B. Kudenov’s Stacked Photovoltaic Polarimeter

The measurement matrix for this system is provided by
Yang et al. in Ref. [18] as
0 0.0753 −0.0331
100
B 0.0280W@
0.01050.00450.00570.00430.0030−0.0046 0.0038 CA:
0.00420.0008−0.0001 −0.0012

However, as with Azzam’s four-detector polarimeter, thereported matrix is also not given in photometric form, as its
elements are given in units of amperes. If we normalize thematrix so that the first column sums to 4, we obtain
0 2.546 −1.119 0.000 0.000 1
B 0.947W  @
0.355
0.152

0.193
0.145
0.101

−0.1560.142−0.0030.128 CA:
0.027
−0.041This gives the covariance matrix shown in Table 13. In thelimit of IG noise, the covariance matrix becomes

0 1.34B 2.89K IG  v d @−1.42−13.87

2.89
7.00
−3.35
−31.10

−1.42
−3.35
34.07
42.67

−13.87 1−31.10 CA:
42.67
221.41

Yang et al. are aware of the low performance that this systemdelivers, and they show that optimizing the photovoltaic
element characteristics should provide a substantial improvement. They model an optimized measurement matrix as
011.187 −0.770 −1.636 0.000
B 1.043 0.314 −0.254 −0.672 CCW II  B
@ 0.923 0.111 0.785 0.067 A,
0.847 0.346 −0.367 0.604
where we have normalized the matrix given in Ref. [18] into
idealized photometric form. The corresponding covariance matrix is given in Table 14, which in the limit of IG noise becomes
01
0.36 −0.31 0.29
0.06
B −0.31 2.49 −0.98 −0.06 CC:
K II IG  v d B@ 0.29 −0.98 0.75
0.05 A
0.06 −0.06 0.05
1.22
In contrast to the Azzam instrument, the photometric normalization of the measurement matrix here has improved theresults significantly. Without knowing the measurement modelfor this system in more detail, however, we cannot determine
the correct scaling factor to use here. The low quantum efficiency and significant reflectivity of the photovoltaic detectors
themselves means that, unlike the Azzam four-detector system,
an assumption of ideal external quantum efficiency seems
unrealistic [27].
8. COMPARISONS OF FULL STOKESPOLARIMETERS
Table 5 shows the variances for all eight of the full Stokes polarimeter configurations, assuming that the total amount of lightavailable for measurement is 4s 0 . To compensate for the factthat the Azzam and Kudenov system variances are for actual
measurement matrices rather than for ideal measurement
matrices, we used the efficiency-normalized (starred) versionsof the measurement model here.
From the results shown in Table 5, we can clearly see theimprovement of Sabatke’s RRFP with δ  132° in place ofAmbirajan’s use of the much more common quarter-wave plate(δ  90°). Ambirajan’s setup appears to be particularly good atmeasuring circular states (s 3 ), at which it surpasses all of the
optimized systems.
The results also show that all of the instruments in the
middle of the table—Sabatke’s RRFP, Tyo’s DVRFP,
Zallat’s DRRFP, and Iniesta’s two idealized systems—enjoy optimal performance, in which the parameter variances give a1∶3∶3∶3 ratio, not only just in IG noise but across all noise
regimes (PG regime). Goudail proposed the system labeled here
as “Iniesta-4” as having the unique properties of having homogeneous noise properties independent of the input SOP, but
we can see that this is in fact a property shared by all of the
optimized designs [10]. Thus, due to the generality of the derivation, it appears that any polarimeter optimized under IG
noise is simultaneously optimized in any noise regime that

Research Article
Table 5.

Vol. 57, No. 15 / 20 May 2018 / Applied Optics

4293

Compiled Variance Results for Eight Full Stokes Polarimeter Configurations, under IG Noisea

Configurationvarŝ 0 

varŝ 1 

varŝ 2 

varŝ 3 

EWV

κ1

κ2

κ∞

κF

Ambirajan-RRFP
Sabatke-RRFP
Tyo-DVRFP
Zallat-DRRFP
Shibata-RRPC
Iniesta-4
Iniesta-6
Azzam*
Kudenov-I*
Kudenov-II*

2.00
1.00
1.00
1.00
2.00
1.00
1.50
2.26
1.34
0.36

7.25
2.99
3.01
2.93
8.00
3.00
4.50
5.63
7.00
2.49

10.67
2.99
2.99
3.05
8.00
3.00
4.50
16.69
34.07
0.75

1.72
3.02
3.01
3.02
4.00
3.00
4.50
9.93
221.41
1.23

21.64
10.00
10.01
10.00
22.00
10.00
15.00
34.51
263.82
4.82

8.33
6.03
5.64
6.19
5.00
6.20
4.00
26.00
81.14
8.92

3.52
1.73
1.73
1.75
2.00
1.73
1.73
9.59
14.91
4.15

5.89
4.64
4.19
4.73
2.00
4.73
2.00
25.15
74.00
11.29

6.58
4.47
4.47
4.47
4.69
4.47
4.47
16.18
48.49
6.67

a
All entries are in units of v d ∕ϵ2 . Matrix condition numbers are provided for comparison. The starred configurations indicate that the tabulated results use thenormalized photometric models.

satisfies independence of measurements and has a finite
variance.
We can also see that both Azzam’s four-detector polarimeterand the stacked photovoltaic polarimeter have higher variances,with particular difficulty detecting s2 and s3 . However, if the
optimized version of Kudenov’s stacked-detector system canbe implemented, we can see that its performance surpasses
all of the other systems by a wide margin due to its efficientuse of the input light.
From the condition numbers listed in Table 5, we can notethat the 2-norm and Frobenius-norm condition numbers givethe same result for every optimized system, whereas the 1-norm
and infinity-norm condition numbers are more erratic, showing preference for one system over another even though thecovariance metric shows that the performance is unchanged.
Thus, if any condition number is to be used as a metric, thenthe former pair is to be preferred over the latter.
In addition, it is useful to note that despite the effort we took
to efficiency normalize the Azzam and Kudenov systems, thecondition numbers for these are unchanged because any definition for κW is invariant to a scalar factor in W. Looking at theform of the covariance matrices, we can go even further and saythat the appearance of ϵ as a scalar factor applied to K in the IG
and PP noise regimes more or less justifies the past use of thecondition number as a proxy for system performance. This
works well in the limits of IG and PP noise but holds only approximately in the intermediate mixed PG noise regime.
Table 6. Compiled SNR Results for Eight Full Stokes
Polarimeter Configurations, under IG Noisea
Configuration

SNRŝ 0 

SNRŝ 1 

SNRŝ 2 

SNRŝ 3 

IG regime: □ ϵ∕σd )Ambirajan-RRFP
Sabatke-RRFP
Tyo-DVRFP
Zallat-DRRFP
Shibata-RRPC
Iniesta-4
Iniesta-6
Azzam*
Kudenov-I*
Kudenov-II*a0.71s01.00s01.00s01.00s00.71s01.00s00.82s00.67s00.86s01.67s0“□” indicates a table entry.

0.37s 1
0.58s 1
0.58s 1
0.58s 10.35s 1
0.58s 1
0.47s 1
0.42s 10.38s 1
0.63s 10.31s20.58s20.58s20.57s20.35s20.58s20.47s20.24s20.17s2
1.15s20.76s30.58s30.58s30.57s30.50s30.58s30.47s30.32s30.07s30.91s3

From the tabulated variances, Table 6 shows the corresponding SNRs of each system analyzed in the IG regime.
As we expect from the variances listed in Table 5, we see thatthe idealized stacked detector has the highest performance. Itsfour SNR values (1.67, 0.63, 1.115, and 0.91) look somewhat
close to (1.5, 1, 1, and 1), suggesting a possible limit for the“perfect polarimeter” when the input measurement light isgiven as 4s 0 .
9. CONCLUSIONS
Although recent research has shown how the variances of the
Stokes parameters depend on the input SOP in PP noise, andthat the variance behavior is generally different in the IG and
PP noise regimes, several important features have gone unnoticed. In both the linear and the full Stokes polarimeters, we seethat optimized designs exhibit the same covariance matrix
structure, either 1∶2∶2 (for linear Stokes) or 1∶3∶3∶3 (for full
Stokes) ratios of variances, in both IG and PP noise regimes.
This justifies the use of the simpler IG noise regime for calculating polarimeter optimization and to an extent also exonerates
the use of matrix condition numbers for a proxy of system
performance. Any system optimized under IG noise will be“equivariant” under PP noise. We can also note that no polarimeter design is able to diagonalize the covariance matrix under
PP noise, so that even the idealized systems exhibit correlated
noise characteristics.
Goudail has shown that, for PP noise, the maximumlikelihood estimator (MLE) can provide a significant boost
in performance over the least squares estimators that we haveused throughout the preceding analysis [28]. However, only
those users who operate at very low light levels will be able
to take advantage of this because the MLE provides better performance only when the number of measured photoelectrons
ϵI is small (e.g., less than 10)—a level below which theGaussian distribution can no longer be used as a good approximation to the Poisson or the PG mixed distribution.
An unexpected result in the exploration of linear Stokes
polarimeter designs was the unique behavior of the optimized
0-60-120 configuration, in which the parameter variances depend on the amplitude of s˜1 but not of s˜2 . All other optimized
linear configurations—the 0-45-90-135 and any other setupsthat evenly sample the half-circle of angles—exhibit no dependence on the input SOP.

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Vol. 57, No. 15 / 20 May 2018 / Applied Optics

The incorporation of a photometric measurement model provides a foundation for building a mixed PG noise model andalso enables the quantitative comparison of unusual systems,
such as Azzam’s four-detector polarimeter and Kudenov’s
stacked photovoltaic polarimeter, with more common instruments, such as the RRFP. The variance results indicate thatthe diattenuating designs so far have a lower performance,
but they also have a lot of room for improvement, and so thisis not an indication of any fundamental gap between the techniques. See the Code 1, Ref. [29] (written in ScientificPython)used for the covariance matrix calculations in the paper.
APPENDIX A: COVARIANCE MATRICES UNDERMIXED PG NOISE
Tables 7–14 give the full PG noise model covariance matrices
for the full Stokes polarimeter systems compared. Each covariance matrix has been formatted into a table for readability, with

Table 9. PG Covariance Matrix for Tyo’s DVRFP
Polarimeter
K ij 

□s 0 ∕ϵ

K 00 
K 01 
K 02 
K 03 
K 10 
K 11 
K 12 
K 13 
K 20 
K 21 
K 22 
K 23 
K 30 
K 31 
K 32 
K 33 

0.50

□s 1 ∕ϵ□s 2 ∕ϵ□s 3 ∕ϵ

1.00
0.50
0.50
0.50

1.50

0.50
0.73
−0.58

1.49

−0.27

−0.27
0.50
−0.27
−0.01
0.82

−0.58
1.50

□vd ∕ϵ2

−0.46

0.82
0.01

−0.59

—
0.01

−0.46

3.01
0.01
0.01

0.82
0.01
0.50
−0.46
0.01
−0.24

0.01
2.99
0.01
0.01
0.01
0.01
3.01

Table 7. PG Covariance Matrix for Ambirajan’s RRFP
Polarimeter

Table 10. PG Covariance Matrix for Zallat’s DRRFP
Polarimeter

K ij 

□s 0 ∕ϵ□s 1 ∕ϵ

K 00 
K 01 
K 02 
K 03 
K 10 
K 11 
K 12 
K 13 
K 20 
K 21 
K 22 
K 23 
K 30 
K 31 
K 32 
K 33 

1.01
−1.31

0.14
−0.45

0.07
−1.31
3.62

−0.14
−0.45
2.76

0.24

0.45

□s 2 ∕ϵ

1.33

0.07
0.24

−0.14
0.45

0.86

0.14

□vd ∕ϵ2

K ij 

□s 0 ∕ϵ□s 1 ∕ϵ

0.24
−0.24

2.01
−2.63

0.50
−0.01

0.50

0.83
−0.24
0.24

0.13
−2.63
7.25

−0.83

0.49

−4.62

10.67

0.83
−0.83

0.13
0.49

0.10

1.72

K 00 
K 01 
K 02 
K 03 
K 10 
K 11 
K 12 
K 13 
K 20 
K 21 
K 22 
K 23 
K 30 
K 31 
K 32 
K 33 

0.62

−0.62
0.62
−0.62

5.33

□s 3 ∕ϵ

−0.62
−0.62

−0.01
1.47

1.53
0.03
0.03
1.51

0.50
0.01

−0.18
0.86

□s 2 ∕ϵ□s 3 ∕ϵ

0.50
−0.01

−0.01
0.50

−0.20
0.83
0.50
−0.20

0.84
0.20
−0.01
0.84

−0.01
0.83

□vd ∕ϵ2
1.00
−0.03
−0.03
2.93

3.05
0.05
0.50
0.20

0.86
0.23

0.05
3.02

Table 8. PG Covariance Matrix for Sabatke’s RRFP
Polarimeter

Table 11. PG Covariance Matrix for Iniesta’s FourMeasurement Idealized Polarimeter

K ij 

□s 0 ∕ϵ□vd ∕ϵ2

K ij 

□s 0 ∕ϵ

K 00 
K 01 
K 02 
K 03 
K 10 
K 11 
K 12 
K 13 
K 20 
K 21 
K 22 
K 23 
K 30 
K 31 
K 32 
K 33 

0.50

1.00

K 00 
K 01 
K 02 
K 03 
K 10 
K 11 
K 12 
K 13 
K 20 
K 21 
K 22 
K 23 
K 30 
K 31 
K 32 
K 33 

0.50

□s 1 ∕ϵ□s 2 ∕ϵ□s 3 ∕ϵ

0.50
0.50
0.50
0.50
1.50
0.50
−0.71
0.50
0.50
1.50

1.50

−0.71
−0.50
−0.71

0.50
−0.71
−0.71
−0.50

3.00

3.00
−0.71

0.50
−0.50
3.00

□s 1 ∕ϵ□s 2 ∕ϵ□s 3 ∕ϵ□vd ∕ϵ2
1.00

0.50
0.50
0.50
0.50
1.50

3.00
0.87
0.87
0.50
0.87

1.50

3.00
0.87
0.50
0.87
0.87

1.50

3.00

Research Article

Vol. 57, No. 15 / 20 May 2018 / Applied Optics

Table 12. PG Covariance Matrix K for Azzam’s FourDetector Polarimeter
K ij 

□s 0

□s 1

□s 2

□s 3

□vd

K 00 
K 01 
K 02 
K 03 
K 10 
K 11 
K 12 
K 13 
K 20 
K 21 
K 22 
K 23 
K 30 
K 31 
K 32 
K 33 

1.07
1.53
−1.02
0.27
1.53
3.05
−0.77
−0.27
−1.02
−0.77
11.24
−2.38
0.27
−0.27
−2.38
4.98

0.19
0.33
−0.55
0.05
0.33
−0.01
−1.51
0.34
−0.55
−1.51
−4.11
0.30
0.05
0.34
0.30
−1.84

−0.09
−0.15
0.25
−0.03
−0.15
−0.26
0.11
−0.21
0.25
0.11
0.68
−0.37
−0.03
−0.21
−0.37
0.97

−0.03
−0.06
0.16
−0.03
−0.15
−0.60
0.30
−0.06
−0.60
−2.05
2.42
0.16
0.30
2.42
−2.09

2.26
3.37
−2.64
0.26
3.37
5.63
−3.12
−0.50
−2.64
−3.12
16.69
−8.41
0.26
−0.50
−8.41
9.93

Table 13. PG Covariance Matrix K for the Kudenov-I
Stacked Photovoltaic Polarimeter
K ij 

□s 0

□s 1

K 00 
K 01 
K 02 
K 03 
K 10 
K 11 
K 12 
K 13 
K 20 
K 21 
K 22 
K 23 
K 30 
K 31 
K 32 
K 33 

0.36
0.41
−0.23
−1.59
0.41
2.01
−0.79
−2.47
−0.23
−0.79
12.90
5.12
−1.59
−2.47
5.12
41.42

0.11
0.43
−0.15
−1.19
0.43
0.49
−0.21
−3.21
−0.15
−0.21
4.84
4.53
−1.19
−3.21
4.53
23.11

□s 2

0.31
0.06
0.71
0.13
0.31
0.71
3.02
2.49
0.06
0.13
2.49
−0.86

□s 3

□vd

−0.03
−0.08
0.08
0.74
−0.08
−0.18
0.19
1.68
0.08
0.19
0.86
−1.68
0.74
1.68
−1.68
−7.29

1.34
2.89
−1.42
−13.87
2.89
7.00
−3.35
−31.10
−1.42
−3.35
34.07
42.67
−13.87−31.1042.67
221.41

Table 14. PG Covariance Matrix KII  for the Kudenov-II
Stacked Photovoltaic Polarimeter
K ij 

□s 0

□s 1

□s 2

□s 3

□vd

K 00 
K 01 
K 02 
K 03 
K 10 
K 11 
K 12 
K 13 
K 20 
K 21 
K 22 
K 23 
K 30 
K 31 
K 32 
K 33 

0.34
−0.32
0.26
0.03
−0.32
2.43
−0.87
−0.19
0.26
−0.87
0.70
0.09
0.03
−0.19
0.09
1.17

0.01
0.14
0.04

0.14
−0.17
0.38
0.06
−0.17
−0.76
−0.46
−0.16
0.38
−0.46
0.36
0.15
0.06
−0.16
0.15
−0.35

−0.04
0.03
0.15
−0.04
0.18
−0.11
0.80
0.03
−0.11
0.07
−0.27
0.15
0.80
−0.27
−0.09

0.36
−0.31
0.29
0.06
−0.31
2.49
−0.98
−0.06
0.29
−0.98
0.75
0.05
0.06
−0.06
0.05
1.23

0.14
0.16
−0.30
0.06
0.04
−0.30
0.10
−0.01
0.06
−0.01
0.40

4295

each row listing a matrix entry. In addition, factors shared byevery element in a column are listed at the top of the column,
and entries whose value is zero are left blank. Thus, a table
header and an entry row listed as
K ij  □s0 ∕ϵ □s1 ∕ϵ
K 00  1.01
0.14

□s 2 ∕ϵ□s3 ∕ϵ □vd ∕ϵ2
−0.24
2.01

means that the K 00 entry is written out in full equation form as
s
s
s
s
2.01
K 00  1.01 0  0.14 1  0 2 − 0.24 3  2 v d :
ϵ
ϵ
ϵ
ϵ
ϵ
Funding. Japan Society for the Promotion of Science(JSPS) (15H03941).
Acknowledgment. The authors thank Ruonan Yang andMichael Kudenov for helpful comments on the paper.

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6998

Erratum

Vol. 57, No. 24 / 20 August 2018 / Applied Optics

Stokes polarimeter performance: general noisemodel and analysis: erratum
NATHAN HAGEN,1,*

TINGKUI MU,2 AND YUKITOSHI OTANI1

1

Utsunomiya University, Department of Optical Engineering, Center for Optical Research and Engineering (CORE), 7-1-2 Yoto,
Utsunomiya, Tochigi 321-0954, Japan
2
Institute of Space Optics (ISO), Department of Opto-Electronics Information Science and Engineering, School of Science, Xi’an Jiaotong
University, Xi’an Shaanxi 710049, China*Corresponding author: nh@hagenlab.org
Received 13 July 2018; posted 21 July 2018 (Doc. ID 338738); published 17 August 2018

We correct two errors in Appl. Opt. 57, 4283 (2018).

© 2018 Optical Society of America

OCIS codes: (260.5430) Polarization; (280.0280) Remote sensing and sensors; (230.5440) Polarization-selective devices;(030.4280) Noise in imaging systems.
https://doi.org/10.1364/AO.57.006998

On page 4285 of the original paper [1], the unnumbered equation appearing immediately before the numbered Eq. (5) was
given incorrectly. The covariance matrix should be diagonal, i.e.,01varŝ0 
covŝ 0 , ŝ1  covŝ0 , ŝ2 BCvarŝ1 
covŝ1 , ŝ2  A
K IG  @ covŝ 1 , ŝ0 covŝ 2 , ŝ0  covŝ 2 , ŝ1 
0
1
1 0 0
v B
C
 2d @ 0 2 0 A:ϵ
0 0 2

varŝ2 

In addition, for Sabatke’s RRFP, the covariance matrix in
the PP noise limit given in Section 4.B (page 4289) of theoriginal paper is not correct. Two terms are missing fromthe diagonal, so that the result should be
1
0
s˜p
1
s˜1
2 ﬃﬃﬃ
pﬃﬃﬃs˜3s B s˜1
3pﬃﬃﬃ
s˜2 − 2s˜3 − p
2s˜2ﬃﬃﬃ − s˜3 C
C:K PP  0 B@
3p
ﬃﬃﬃs˜1− 2s˜1 A
2ϵ s˜2 s˜2 −pﬃﬃﬃ 2s˜3
s˜3
− 2s˜2− 2s˜13 − s˜1

1559-128X/18/246998-01 Journal © 2018 Optical Society of America

This change affects one of the assertions given in theConclusion, where it is stated that “Any system optimized
under IG noise will be “equivariant” under PP noise.” Thecorrected form of the matrix is obviously not equivariant.
Likewise, the covariance matrices for Tyo’s DVRFP and Zallat’s
DRRFP polarimeter (Tables 9 and 10 on page 4294) demonstrate that these systems will not be equivariant as well. Thus,
this assertion is incorrect.
The remainder of the paper is unaffected, and these changes
do not affect the code provided with the paper for calculating
the covariance matrices.

REFERENCE
1. N. Hagen and Y. Otani, “Stokes polarimeter performance:
general noise model and analysis,” Appl. Opt. 57, 4283–4296(2018).