Soe-Mie F. Nee, Chan Yoo, Teresa Cole, and Dennis Burge, "Characterization for imperfect polarizers under imperfect conditions," Appl. Opt. 37, 54-64 (1998)
The principles for measuring the extinction ratio and
transmittance of a polarizer are formulated by use of the principal
Mueller matrix, which includes both polarization and
depolarization. The extinction ratio is about half of the
depolarization, and the contrast is the inverse of the extinction
ratio. Errors in the extinction ratio caused by partially polarized
incident light and the misalignment of polarizers can be corrected by
the devised zone average method and the null method. Used with a
laser source, the null method can measure contrasts for very good
polarizers. Correct algorithms are established to deduce the
depolarization for three comparable polarizers calibrated
mutually. These methods are tested with wire-grid polarizers used
in the 3–5-μm wavelength region with a laser source and also a lamp
source. The contrasts obtained from both methods agree.
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Measured Transmittance T, Depolarization ,
Copolarized and Cross-Polarized NSS u and v for
Two Wire-grid Polarizersa
Polarizer P (deg)
Analyzer A (deg)
Intensity (mV)
Deviation (mV)
Deduced Parameters
10073
52
Tp = 0.342
90
6891
33
ℰp = 9.06 × 10-5
0
0.625
0.003
Cp = 11026
90
5418
26
p = 1.81 × 10-4
0
0.522
0.003
Ta = 0.269
90
90
3697
20
ℰa = 9.63 × 10-5
0
0
0.0059
0.0003
Ca = 10379
0
90
0.331
0.002
vp + δ = 3 × 10-6
90
0
0.36
0.002
up = 1.79 × 10-4
Incident light was a He–Ne laser at
3.39-μm wavelength passing through a high-contrast polarizer oriented
at 90°.
Table 2
Measured Intensities for Light Passing through a Wire-Grid
Polarizer at λ = 3.39 μma
Polarizer
I(0°) (μV)
I(90°) (μV)
α (%)
P
1728.78 ± 0.85
1470.11 ± 0.57
8.09 ± 0.05
A
1531.33 ± 0.58
1291.78 ± 0.63
8.49 ± 0.05
R
2052.92 ± 0.93
1744.03 ± 1.89
8.14 ± 0.08
Incident light was generated from a lamp
source and a monochromator; α was calculated from Eqs. (12).
Table 3
Intensities for Light Passing through a Pair of Wire-Grid
Polarizers Measured at λ = 3.39 μm for the Zone Average Method
with a Lamp Source
Pair
I(0°, 0)° (μV)
I(90°, 90°) (μV)
I(0, 90) (μV)
I(90, 0) (μV)
PA
1077.72 ± 0.45
913.28 ± 0.45
0.210 ± 0.020
0.219 ± 0.020
RA
1304.28 ± 0.45
1107.00 ± 0.58
0.437 ± 0.010
0.400 ± 0.009
PR
1475.06 ± 0.71
1256.22 ± 0.79
0.489 ± 0.008
0.491 ± 0.010
Table 4
T, α, ℰ, and C for Wire-Grid
Polarizers at λ = 3.39 μm Determined from Data of Table 3 by the
Zone Average Method
Polarizer
T (%)
α (%)
ℰ
C
% Error in ℰ & C
A
31.45 ± .02
8.18 ± .03
1.016 × 10-4
9813
7.03
P
35.62 ± .02
8.26 ± .03
1.137 × 10-4
8802
7.84
R
42.71 ± .02
8.01 ± .04
2.455 × 10-4
4076
3.26
Table 5
Intensities for Light Passing through Three Wire-Grid
Polarizers at Different Orientations, Measured by the Null Method with
a He–Ne Laser Source at λ 3.39 μm
±45°
0°
90°
I(A0 - 2°) (μV)
I(A0) (μV)
I(A0 + 2°) (μV)
A+
R
P
25.600 ± 0.032
4.420 ± 0.044
24.845 ± 0.047
A-
R
P
49.488 ± 0.049
8.717 ± 0.047
49.006 ± 0.037
P+
R
A
23.867 ± 0.045
4.133 ± 0.037
24.283 ± 0.051
P-
R
A
47.808 ± 0.045
7.875 ± 0.046
45.929 ± 0.030
R+
P
A
20.681 ± 0.021
2.363 ± 0.047
19.850 ± 0.059
R-
P
A
43.863 ± 0.022
5.745 ± 0.034
40.612 ± 0.019
Table 6
Intensities for Light Passing Through a Pair of Wire-Grid
Polarizers Measured Using the Zone Average
Methoda
Pair
I(0°, 0)° (μV)
I(90°, 90°) (μV)
I(0, 90) (μV)
I(90, 0) (μV)
PA
26083 ± 37
27883 ± 37
5.298 ± 0.054
4.903 ± 0.054
RA
25316 ± 37
24783 ± 37
6.734 ± 0.039
6.871 ± 0.048
RP
27433 ± 47
26917 ± 37
7.618 ± 0.043
7.468 ± 0.047
Incident light was generated from a
He–Ne laser at λ = 3.39 μm and passing through the third polarizer
oriented at -45°.
Table 7
Extinction Ratios and Contrasts of Wire-Grid Polarizers
for the Null and the Zone Average Methods with a Laser Source at λ =
3.39 μm
Polarizer
ℰnull
Cnull
Error %
ℰzone-ave
Czone-ave
Error %
A
8.254 × 10-5
12115
1.61
9.152 × 10-5
10926
1.24
P
9.371 × 10-5
10671
1.42
9.754 × 10-5
10252
1.16
R
1.667 × 10-4
6000
0.80
1.801 × 10-4
5552
0.63
Tables (7)
Table 1
Measured Transmittance T, Depolarization ,
Copolarized and Cross-Polarized NSS u and v for
Two Wire-grid Polarizersa
Polarizer P (deg)
Analyzer A (deg)
Intensity (mV)
Deviation (mV)
Deduced Parameters
10073
52
Tp = 0.342
90
6891
33
ℰp = 9.06 × 10-5
0
0.625
0.003
Cp = 11026
90
5418
26
p = 1.81 × 10-4
0
0.522
0.003
Ta = 0.269
90
90
3697
20
ℰa = 9.63 × 10-5
0
0
0.0059
0.0003
Ca = 10379
0
90
0.331
0.002
vp + δ = 3 × 10-6
90
0
0.36
0.002
up = 1.79 × 10-4
Incident light was a He–Ne laser at
3.39-μm wavelength passing through a high-contrast polarizer oriented
at 90°.
Table 2
Measured Intensities for Light Passing through a Wire-Grid
Polarizer at λ = 3.39 μma
Polarizer
I(0°) (μV)
I(90°) (μV)
α (%)
P
1728.78 ± 0.85
1470.11 ± 0.57
8.09 ± 0.05
A
1531.33 ± 0.58
1291.78 ± 0.63
8.49 ± 0.05
R
2052.92 ± 0.93
1744.03 ± 1.89
8.14 ± 0.08
Incident light was generated from a lamp
source and a monochromator; α was calculated from Eqs. (12).
Table 3
Intensities for Light Passing through a Pair of Wire-Grid
Polarizers Measured at λ = 3.39 μm for the Zone Average Method
with a Lamp Source
Pair
I(0°, 0)° (μV)
I(90°, 90°) (μV)
I(0, 90) (μV)
I(90, 0) (μV)
PA
1077.72 ± 0.45
913.28 ± 0.45
0.210 ± 0.020
0.219 ± 0.020
RA
1304.28 ± 0.45
1107.00 ± 0.58
0.437 ± 0.010
0.400 ± 0.009
PR
1475.06 ± 0.71
1256.22 ± 0.79
0.489 ± 0.008
0.491 ± 0.010
Table 4
T, α, ℰ, and C for Wire-Grid
Polarizers at λ = 3.39 μm Determined from Data of Table 3 by the
Zone Average Method
Polarizer
T (%)
α (%)
ℰ
C
% Error in ℰ & C
A
31.45 ± .02
8.18 ± .03
1.016 × 10-4
9813
7.03
P
35.62 ± .02
8.26 ± .03
1.137 × 10-4
8802
7.84
R
42.71 ± .02
8.01 ± .04
2.455 × 10-4
4076
3.26
Table 5
Intensities for Light Passing through Three Wire-Grid
Polarizers at Different Orientations, Measured by the Null Method with
a He–Ne Laser Source at λ 3.39 μm
±45°
0°
90°
I(A0 - 2°) (μV)
I(A0) (μV)
I(A0 + 2°) (μV)
A+
R
P
25.600 ± 0.032
4.420 ± 0.044
24.845 ± 0.047
A-
R
P
49.488 ± 0.049
8.717 ± 0.047
49.006 ± 0.037
P+
R
A
23.867 ± 0.045
4.133 ± 0.037
24.283 ± 0.051
P-
R
A
47.808 ± 0.045
7.875 ± 0.046
45.929 ± 0.030
R+
P
A
20.681 ± 0.021
2.363 ± 0.047
19.850 ± 0.059
R-
P
A
43.863 ± 0.022
5.745 ± 0.034
40.612 ± 0.019
Table 6
Intensities for Light Passing Through a Pair of Wire-Grid
Polarizers Measured Using the Zone Average
Methoda
Pair
I(0°, 0)° (μV)
I(90°, 90°) (μV)
I(0, 90) (μV)
I(90, 0) (μV)
PA
26083 ± 37
27883 ± 37
5.298 ± 0.054
4.903 ± 0.054
RA
25316 ± 37
24783 ± 37
6.734 ± 0.039
6.871 ± 0.048
RP
27433 ± 47
26917 ± 37
7.618 ± 0.043
7.468 ± 0.047
Incident light was generated from a
He–Ne laser at λ = 3.39 μm and passing through the third polarizer
oriented at -45°.
Table 7
Extinction Ratios and Contrasts of Wire-Grid Polarizers
for the Null and the Zone Average Methods with a Laser Source at λ =
3.39 μm