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) [CrossRef] ] and Kudenov’s stacked photovoltaic polarimeter [Opt. Express 24, 14737 (2016) [CrossRef] ]. 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.
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Each of the IG regime entries () share a factor, whereas each of the PP regime entries () share a factor.
The dependence on in the column is not a typo.
Table 2.
Compiled SNR Results for the Five Linear Stokes Polarimeter Configurationsa
Configuration
IG Regime:
0-45-90-135
1
0-60-120
0.866
0-45-90
0.707
N-0-45
1
N-N-0-45
1.414
PP Regime:
0-45-90-135
1.414
0-60-120
0.866
0-45-90
1
N-0-45
1
N-N-0-45
2
Each of the IG regime entries () share a factor, whereas each of the PP regime entries () share a factor.
Table 3.
Instrument Configurations for Three RP Polarimeters, Where Indicates the Retarder Angle and the Polarizer Angle
Configuration
Measurement #
Ambirajan
1
90°
0°
2
90°
0°
0°
3
90°
30°
0°
4
90°
60°
0°
Sabatke
1
131.8°
−51.69°
0°
2
131.8°
−15.12°
0°
3
131.8°
15.12°
0°
4
131.8°
51.69°
0°
Shibata
1
90°
22.5°
0°
2
90°
22.5°
45°
3
90°
22.5°
90°
4
90°
22.5°
135°
5
90°
67.5°
0°
6
90°
67.5°
45°
7
90°
67.5°
90°
8
90°
67.5°
135°
Table 4.
Instrument Configurations for the Variable Retarder Systems
Configuration
Measurement #
Tyo
1
22.5°
158°
45°
50.6°
2
22.5°
127°
45°
−178°
3
22.5°
47°
45°
−16.9°
4
22.5°
0.66°
45°
126°
Zallat
1
−41.4°
90°
−20.3°
90°
2
41.4°
90°
−20.3°
90°
3
−41.4°
90°
20.3°
90°
4
41.4°
90°
20.3°
90°
Table 5.
Compiled Variance Results for Eight Full Stokes Polarimeter Configurations, under IG Noisea
Configuration
EWV
Ambirajan-RRFP
2.00
7.25
10.67
1.72
21.64
8.33
3.52
5.89
6.58
Sabatke-RRFP
1.00
2.99
2.99
3.02
10.00
6.03
1.73
4.64
4.47
Tyo-DVRFP
1.00
3.01
2.99
3.01
10.01
5.64
1.73
4.19
4.47
Zallat-DRRFP
1.00
2.93
3.05
3.02
10.00
6.19
1.75
4.73
4.47
Shibata-RRPC
2.00
8.00
8.00
4.00
22.00
5.00
2.00
2.00
4.69
Iniesta-4
1.00
3.00
3.00
3.00
10.00
6.20
1.73
4.73
4.47
Iniesta-6
1.50
4.50
4.50
4.50
15.00
4.00
1.73
2.00
4.47
Azzam*
2.26
5.63
16.69
9.93
34.51
26.00
9.59
25.15
16.18
Kudenov-I*
1.34
7.00
34.07
221.41
263.82
81.14
14.91
74.00
48.49
Kudenov-II*
0.36
2.49
0.75
1.23
4.82
8.92
4.15
11.29
6.67
All entries are in units of . Matrix condition numbers are provided for comparison. The starred configurations indicate that the tabulated results use the normalized photometric models.
Table 6.
Compiled SNR Results for Eight Full Stokes Polarimeter Configurations, under IG Noisea
Configuration
IG regime:)
Ambirajan-RRFP
Sabatke-RRFP
Tyo-DVRFP
Zallat-DRRFP
Shibata-RRPC
Iniesta-4
Iniesta-6
Azzam*
Kudenov-I*
Kudenov-II*
“” indicates a table entry.
Table 7.
PG Covariance Matrix for Ambirajan’s RRFP Polarimeter
1.01
0.14
0.24
2.01
−1.31
−0.45
−0.24
−2.63
0.62
0.07
−0.14
0.83
0.13
−1.31
−0.45
−0.24
−2.63
3.62
2.76
0.24
7.25
−0.62
0.24
0.45
−0.83
0.49
0.62
−0.62
5.33
1.33
−4.62
10.67
−0.62
0.07
−0.14
0.83
0.13
0.24
0.45
−0.83
0.49
−0.62
0.86
0.14
0.10
1.72
Table 8.
PG Covariance Matrix for Sabatke’s RRFP Polarimeter
0.50
1.00
0.50
0.50
0.50
0.50
1.50
3.00
0.50
−0.71
−0.71
−0.50
0.50
0.50
−0.71
1.50
0.50
3.00
−0.71
0.50
−0.71
−0.50
−0.71
1.50
−0.50
3.00
Table 9.
PG Covariance Matrix for Tyo’s DVRFP Polarimeter
0.50
1.00
0.50
0.50
—
0.50
0.01
0.50
1.50
0.73
−0.59
3.01
−0.27
0.01
−0.58
−0.46
0.01
0.50
−0.27
0.01
1.49
−0.27
−0.01
0.82
2.99
0.82
0.01
0.01
0.50
0.01
−0.58
−0.46
0.01
0.82
0.01
0.01
1.50
−0.46
0.01
−0.24
3.01
Table 10.
PG Covariance Matrix for Zallat’s DRRFP Polarimeter
0.50
1.00
−0.01
0.50
−0.03
0.50
−0.01
−0.01
0.50
−0.01
0.50
−0.03
1.47
0.01
2.93
−0.20
0.84
0.83
0.20
0.50
−0.01
−0.20
0.84
1.53
−0.18
3.05
0.03
0.86
0.05
−0.01
0.50
0.83
0.20
0.03
0.86
0.05
1.51
0.23
3.02
Table 11.
PG Covariance Matrix for Iniesta’s Four-Measurement Idealized Polarimeter
0.50
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
Table 12.
PG Covariance Matrix for Azzam’s Four-Detector Polarimeter
1.07
0.19
−0.09
2.26
1.53
0.33
−0.15
−0.03
3.37
−1.02
−0.55
0.25
−0.06
−2.64
0.27
0.05
−0.03
0.16
0.26
1.53
0.33
−0.15
−0.03
3.37
3.05
−0.01
−0.26
−0.15
5.63
−0.77
−1.51
0.11
−0.60
−3.12
−0.27
0.34
−0.21
0.30
−0.50
−1.02
−0.55
0.25
−0.06
−2.64
−0.77
−1.51
0.11
−0.60
−3.12
11.24
−4.11
0.68
−2.05
16.69
−2.38
0.30
−0.37
2.42
−8.41
0.27
0.05
−0.03
0.16
0.26
−0.27
0.34
−0.21
0.30
−0.50
−2.38
0.30
−0.37
2.42
−8.41
4.98
−1.84
0.97
−2.09
9.93
Table 13.
PG Covariance Matrix for the Kudenov-I Stacked Photovoltaic Polarimeter
0.36
0.11
−0.03
1.34
0.41
0.43
−0.08
2.89
−0.23
−0.15
0.31
0.08
−1.42
−1.59
−1.19
0.06
0.74
−13.87
0.41
0.43
−0.08
2.89
2.01
0.49
−0.18
7.00
−0.79
−0.21
0.71
0.19
−3.35
−2.47
−3.21
0.13
1.68
−31.10
−0.23
−0.15
0.31
0.08
−1.42
−0.79
−0.21
0.71
0.19
−3.35
12.90
4.84
3.02
0.86
34.07
5.12
4.53
2.49
−1.68
42.67
−1.59
−1.19
0.06
0.74
−13.87
−2.47
−3.21
0.13
1.68
−31.10
5.12
4.53
2.49
−1.68
42.67
41.42
23.11
−0.86
−7.29
221.41
Table 14.
PG Covariance Matrix for the Kudenov-II Stacked Photovoltaic Polarimeter
0.34
0.01
0.14
0.36
−0.32
0.14
−0.17
−0.04
−0.31
0.26
0.04
0.38
0.03
0.29
0.03
0.06
0.15
0.06
−0.32
0.14
−0.17
−0.04
−0.31
2.43
0.16
−0.76
0.18
2.49
−0.87
−0.30
−0.46
−0.11
−0.98
−0.19
0.06
−0.16
0.80
−0.06
0.26
0.04
0.38
0.03
0.29
−0.87
−0.30
−0.46
−0.11
−0.98
0.70
0.10
0.36
0.07
0.75
0.09
−0.01
0.15
−0.27
0.05
0.03
0.06
0.15
0.06
−0.19
0.06
−0.16
0.80
−0.06
0.09
−0.01
0.15
−0.27
0.05
1.17
0.40
−0.35
−0.09
1.23
Tables (14)
Table 1.
Compiled Variance Results for the Five Linear Stokes Polarimeter Configurationsa
Each of the IG regime entries () share a factor, whereas each of the PP regime entries () share a factor.
The dependence on in the column is not a typo.
Table 2.
Compiled SNR Results for the Five Linear Stokes Polarimeter Configurationsa
Configuration
IG Regime:
0-45-90-135
1
0-60-120
0.866
0-45-90
0.707
N-0-45
1
N-N-0-45
1.414
PP Regime:
0-45-90-135
1.414
0-60-120
0.866
0-45-90
1
N-0-45
1
N-N-0-45
2
Each of the IG regime entries () share a factor, whereas each of the PP regime entries () share a factor.
Table 3.
Instrument Configurations for Three RP Polarimeters, Where Indicates the Retarder Angle and the Polarizer Angle
Configuration
Measurement #
Ambirajan
1
90°
0°
2
90°
0°
0°
3
90°
30°
0°
4
90°
60°
0°
Sabatke
1
131.8°
−51.69°
0°
2
131.8°
−15.12°
0°
3
131.8°
15.12°
0°
4
131.8°
51.69°
0°
Shibata
1
90°
22.5°
0°
2
90°
22.5°
45°
3
90°
22.5°
90°
4
90°
22.5°
135°
5
90°
67.5°
0°
6
90°
67.5°
45°
7
90°
67.5°
90°
8
90°
67.5°
135°
Table 4.
Instrument Configurations for the Variable Retarder Systems
Configuration
Measurement #
Tyo
1
22.5°
158°
45°
50.6°
2
22.5°
127°
45°
−178°
3
22.5°
47°
45°
−16.9°
4
22.5°
0.66°
45°
126°
Zallat
1
−41.4°
90°
−20.3°
90°
2
41.4°
90°
−20.3°
90°
3
−41.4°
90°
20.3°
90°
4
41.4°
90°
20.3°
90°
Table 5.
Compiled Variance Results for Eight Full Stokes Polarimeter Configurations, under IG Noisea
Configuration
EWV
Ambirajan-RRFP
2.00
7.25
10.67
1.72
21.64
8.33
3.52
5.89
6.58
Sabatke-RRFP
1.00
2.99
2.99
3.02
10.00
6.03
1.73
4.64
4.47
Tyo-DVRFP
1.00
3.01
2.99
3.01
10.01
5.64
1.73
4.19
4.47
Zallat-DRRFP
1.00
2.93
3.05
3.02
10.00
6.19
1.75
4.73
4.47
Shibata-RRPC
2.00
8.00
8.00
4.00
22.00
5.00
2.00
2.00
4.69
Iniesta-4
1.00
3.00
3.00
3.00
10.00
6.20
1.73
4.73
4.47
Iniesta-6
1.50
4.50
4.50
4.50
15.00
4.00
1.73
2.00
4.47
Azzam*
2.26
5.63
16.69
9.93
34.51
26.00
9.59
25.15
16.18
Kudenov-I*
1.34
7.00
34.07
221.41
263.82
81.14
14.91
74.00
48.49
Kudenov-II*
0.36
2.49
0.75
1.23
4.82
8.92
4.15
11.29
6.67
All entries are in units of . Matrix condition numbers are provided for comparison. The starred configurations indicate that the tabulated results use the normalized photometric models.
Table 6.
Compiled SNR Results for Eight Full Stokes Polarimeter Configurations, under IG Noisea
Configuration
IG regime:)
Ambirajan-RRFP
Sabatke-RRFP
Tyo-DVRFP
Zallat-DRRFP
Shibata-RRPC
Iniesta-4
Iniesta-6
Azzam*
Kudenov-I*
Kudenov-II*
“” indicates a table entry.
Table 7.
PG Covariance Matrix for Ambirajan’s RRFP Polarimeter
1.01
0.14
0.24
2.01
−1.31
−0.45
−0.24
−2.63
0.62
0.07
−0.14
0.83
0.13
−1.31
−0.45
−0.24
−2.63
3.62
2.76
0.24
7.25
−0.62
0.24
0.45
−0.83
0.49
0.62
−0.62
5.33
1.33
−4.62
10.67
−0.62
0.07
−0.14
0.83
0.13
0.24
0.45
−0.83
0.49
−0.62
0.86
0.14
0.10
1.72
Table 8.
PG Covariance Matrix for Sabatke’s RRFP Polarimeter
0.50
1.00
0.50
0.50
0.50
0.50
1.50
3.00
0.50
−0.71
−0.71
−0.50
0.50
0.50
−0.71
1.50
0.50
3.00
−0.71
0.50
−0.71
−0.50
−0.71
1.50
−0.50
3.00
Table 9.
PG Covariance Matrix for Tyo’s DVRFP Polarimeter
0.50
1.00
0.50
0.50
—
0.50
0.01
0.50
1.50
0.73
−0.59
3.01
−0.27
0.01
−0.58
−0.46
0.01
0.50
−0.27
0.01
1.49
−0.27
−0.01
0.82
2.99
0.82
0.01
0.01
0.50
0.01
−0.58
−0.46
0.01
0.82
0.01
0.01
1.50
−0.46
0.01
−0.24
3.01
Table 10.
PG Covariance Matrix for Zallat’s DRRFP Polarimeter
0.50
1.00
−0.01
0.50
−0.03
0.50
−0.01
−0.01
0.50
−0.01
0.50
−0.03
1.47
0.01
2.93
−0.20
0.84
0.83
0.20
0.50
−0.01
−0.20
0.84
1.53
−0.18
3.05
0.03
0.86
0.05
−0.01
0.50
0.83
0.20
0.03
0.86
0.05
1.51
0.23
3.02
Table 11.
PG Covariance Matrix for Iniesta’s Four-Measurement Idealized Polarimeter
0.50
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
Table 12.
PG Covariance Matrix for Azzam’s Four-Detector Polarimeter
1.07
0.19
−0.09
2.26
1.53
0.33
−0.15
−0.03
3.37
−1.02
−0.55
0.25
−0.06
−2.64
0.27
0.05
−0.03
0.16
0.26
1.53
0.33
−0.15
−0.03
3.37
3.05
−0.01
−0.26
−0.15
5.63
−0.77
−1.51
0.11
−0.60
−3.12
−0.27
0.34
−0.21
0.30
−0.50
−1.02
−0.55
0.25
−0.06
−2.64
−0.77
−1.51
0.11
−0.60
−3.12
11.24
−4.11
0.68
−2.05
16.69
−2.38
0.30
−0.37
2.42
−8.41
0.27
0.05
−0.03
0.16
0.26
−0.27
0.34
−0.21
0.30
−0.50
−2.38
0.30
−0.37
2.42
−8.41
4.98
−1.84
0.97
−2.09
9.93
Table 13.
PG Covariance Matrix for the Kudenov-I Stacked Photovoltaic Polarimeter
0.36
0.11
−0.03
1.34
0.41
0.43
−0.08
2.89
−0.23
−0.15
0.31
0.08
−1.42
−1.59
−1.19
0.06
0.74
−13.87
0.41
0.43
−0.08
2.89
2.01
0.49
−0.18
7.00
−0.79
−0.21
0.71
0.19
−3.35
−2.47
−3.21
0.13
1.68
−31.10
−0.23
−0.15
0.31
0.08
−1.42
−0.79
−0.21
0.71
0.19
−3.35
12.90
4.84
3.02
0.86
34.07
5.12
4.53
2.49
−1.68
42.67
−1.59
−1.19
0.06
0.74
−13.87
−2.47
−3.21
0.13
1.68
−31.10
5.12
4.53
2.49
−1.68
42.67
41.42
23.11
−0.86
−7.29
221.41
Table 14.
PG Covariance Matrix for the Kudenov-II Stacked Photovoltaic Polarimeter