Analysis of Phase Error of Retarders in Spectrally Encoded Mueller Matrix Measurement

Shuqiang Huang; Xiumei Jiang; Yuanhua Feng

doi:10.3788/LOP202158.1926001

Journals >Laser & Optoelectronics Progress >Volume 58 >Issue 19 >Page 1926001 > Article

Laser & Optoelectronics Progress
Vol. 58, Issue 19, 1926001 (2021)

Analysis of Phase Error of Retarders in Spectrally Encoded Mueller Matrix Measurement

Shuqiang Huang¹, Xiumei Jiang¹, and Yuanhua Feng^2、3、*

Author Affiliations

¹Department of Optoelectronic Engineering, College of Science and Engineering, Jinan University, Guangzhou , Guangdong 510632, China

²Department of Electronic Engineering, College of Information Science and Technology, Jinan University, Guangzhou , Guangdong 510632, China

³State Key Laboratory of Integrated Optoelectronics, Beijing 100083, China

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DOI: 10.3788/LOP202158.1926001 Cite this Article Set citation alerts

Shuqiang Huang, Xiumei Jiang, Yuanhua Feng. Analysis of Phase Error of Retarders in Spectrally Encoded Mueller Matrix Measurement[J]. Laser & Optoelectronics Progress, 2021, 58(19): 1926001 Copy Citation Text

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Fig. 1. Schematic of experimental setup of spectrally encoded Mueller matrix measurement system

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Fig. 2. Measurement spectrum with 0° polarizer as sample

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Fig. 3. Influence of phase

ϕ_{w}

on the Mueller matrix for different samples

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Fig. 4. The value of each element of the Muller matrix with the compensation of polarizer from 0° to 90°

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$ϕ_{2} / r a d$		$ϕ_{3} / r a d$	$ϕ_{4} / r a d$	$E_{R M S}$ of vacuum		$E_{R M S}$ of polarizer（ $θ = 45 °$ ）		$E_{R M S}$ of quarter waveplate（ $θ = 45 °$ ）
$ϕ_{2} / r a d$		$ϕ_{3} / r a d$	$ϕ_{4} / r a d$	Without compensation	With compensation	Without compensation	With compensation	Without compensation	With compensation
0.01	0		0	0.0030	0.0013	0.0040	0.0024	0.0051	0.0023
0.1	0		0	0.0330	0.0013	0.0351	0.0024	0.0366	0.0023
0	0.01		0	0.0030	0.0013	0.0040	0.0024	0.0045	0.0023
0	0.1		0	0.0330	0.0013	0.0351	0.0024	0.0358	0.0023
0	0		0.01	0.0030	0.0013	0.0040	0.0024	0.0051	0.0023
0	0		0.1	0.0330	0.0013	0.0351	0.0024	0.0436	0.0023
0.01	0.01		0.01	0.0106	0.0013	0.0079	0.0024	0.0093	0.0023
0.1	0.1		0.1	0.1083	0.0013	0.0790	0.0024	0.0836	0.0023
0.1	0.1		-0.1	0.0432	0.0013	0.0355	0.0024	0.0444	0.0023

Table 1. Influence of

Δ ϕ_{2}

Δ ϕ_{3}

,and

Δ ϕ_{4}

on the Mueller matrix for different samples and the compensation effect

Sample	Theoretical Muellermatrix	Experimental Mueller matrix		Mueller matrix element error
Sample	Theoretical Muellermatrix	Without compensation	With compensation	$E_{R M S}$	E_max
Vacuum	$[\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}]$	$[\begin{matrix} 1.0000 & 0.0431 & - 0.0518 & 0.1440 \\ 0.0439 & 0.9774 & 0.0142 & 0.0095 \\ 0.0265 & 0.0261 & 0.8505 & - 0.3377 \\ 0.0175 & 0.0351 & 0.3343 & 0.9301 \end{matrix}]$	$[\begin{matrix} 1.0000 & 0.0487 & - 0.0494 & - 0.0019 \\ 0.0473 & 0.9899 & - 0.0153 & - 0.0143 \\ 0.0074 & - 0.0071 & 0.9669 & - 0.0106 \\ 0.0304 & 0.0431 & - 0.0605 & 0.9314 \end{matrix}]$	0.0354	0.0686
Polarizer 0°	$[\begin{matrix} 1 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}]$	$[\begin{matrix} 1.0000 & 0.9922 & 0.0077 & 0.0129 \\ 0.9823 & 0.9820 & 0.0067 & - 0.0048 \\ - 0.0314 & 0.0405 & 0.0431 & 0.0286 \\ 0.0136 & - 0.0157 & - 0.0148 & 0.0035 \end{matrix}]$	$[\begin{matrix} 1.0000 & 0.9943 & 0.0227 & 0.0094 \\ 0.9859 & 0.9838 & - 0.0067 & - 0.0058 \\ 0.0156 & 0.0003 & - 0.0671 & 0.0537 \\ - 0.0212 & 0.0070 & - 0.0330 & - 0.0597 \end{matrix}]$	0.0199	0.0471
Polarizer 45°	$[\begin{matrix} 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}]$	$[\begin{matrix} 1.0000 & 0.0362 & 1.2229 & - 0.5937 \\ 0.0174 & - 0.0043 & 0.0415 & - 0.0556 \\ - 0.1386 & - 0.0399 & - 0.1625 & - 0.0154 \\ 1.3922 & 0.0492 & 1.4881 & - 0.7976 \end{matrix}]$	$[\begin{matrix} 1.0000 & 0.0411 & 0.9998 & 0.0080 \\ 0.0113 & - 0.0061 & 0.0042 & 0.0035 \\ 1.0584 & 0.0056 & 1.0248 & - 0.0449 \\ - 0.0044 & 0.0332 & - 0.0409 & 0.0324 \end{matrix}]$	0.0271	0.0584
Quarter-wave plate 0°	$[\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & - 1 \\ 0 & 0 & 1 & 0 \end{matrix}]$	$[\begin{matrix} 1.0000 & - 2024 & 0.0630 & - 0.0780 \\ 0.0266 & - 0.0853 & - 0.8329 & - 0.3081 \\ 0.0013 & 0.4204 & 0.3130 & - 0.6268 \\ 0.0958 & 0.9229 & - 0.1780 & 0.7461 \end{matrix}]$	$[\begin{matrix} 1.0000 & 0.0616 & - 0.0022 & - 0.0603 \\ 0.0584 & 0.9860 & 0.0181 & - 0.0035 \\ 0.0014 & 0.0430 & 0.0041 & 1.0197 \\ 0.0236 & 0.0036 & - 0.9539 & - 0.0676 \end{matrix}]$	0.0389	0.0676
Quarter-wave plate 90°	$[\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & - 1 \\ 0 & 0 & 1 & 0 \end{matrix}]$	$[\begin{matrix} 1.0000 & 0.0564 & - 0.1682 & 0.0825 \\ 0.0480 & 0.9769 & 0.0150 & 0.0102 \\ 0.0097 & 0.0028 & - 0.4047 & - 0.7883 \\ 0.0066 & 0.0400 & 0.8655 & - 0.3518 \end{matrix}]$	$[\begin{matrix} 1.0000 & 0.0651 & - 0.0252 & 0.0612 \\ 0.0541 & 0.9911 & - 0.0144 & 0.0126 \\ 0.0007 & - 0.0243 & 0.0193 & - 0.9452 \\ 0.0066 & 0.0319 & 0.9688 & 0.0630 \end{matrix}]$	0.0375	0.0630
Half-wave plate 0°	$[\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & - 1 & 0 \\ 0 & 0 & 0 & - 1 \end{matrix}]$	$[\begin{matrix} 1.0000 & 0.0867 & 0.0537 & 0.1629 \\ 0.0676 & 0.9870 & 0.0020 & - 0.0112 \\ - 0.0323 & 0.0003 & - 0.8732 & 0.3927 \\ - 0.0260 & 0.0124 & - 0.2852 & - 0.9353 \end{matrix}]$	$[\begin{matrix} 1.0000 & 0.0694 & 0.0394 & 0.0190 \\ 0.0625 & 0.9972 & 0.0020 & - 0.0002 \\ - 0.0054 & - 0.0087 & - 1.0308 & - 0.0211 \\ - 0.0408 & 0.0089 & 0.0340 & - 0.9569 \end{matrix}]$	0.0325	0.0694

Table 2. Comparison with and without compensation of different samples using polarizer method

Sample	Theoretical Mueller matrix	Experimental Mueller matrix（with compensation）	Mueller Matrix element error
Sample	Theoretical Mueller matrix	Experimental Mueller matrix（with compensation）	$E_{R M S}$	E_max
Vacuum	$[\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}]$	$[\begin{matrix} 1.0000 & 0.0454 & 0.0343 & - 0.0381 \\ 0.0470 & 0.9905 & - 0.0136 & - 0.0137 \\ - 0.0340 & - 0.0076 & 0.9772 & 0.0086 \\ 0.0484 & 0.0406 & - 0.0783 & 0.9281 \end{matrix}]$	0.0391	0.0783
Polarizer 0°	$[\begin{matrix} 1 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}]$	$[\begin{matrix} 1.0000 & 1.0015 & - 0.0331 & 0.0076 \\ 0.9887 & 0.9859 & 0.0108 & - 0.0039 \\ 0.0070 & 0.0156 & - 0.0451 & 0.0698 \\ 0.0296 & 0.0231 & - 0.0771 & - 0.0365 \end{matrix}]$	0.0331	0.0698
Half-wave plate 0°	$[\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & - 1 & 0 \\ 0 & 0 & 0 & - 1 \end{matrix}]$	$[\begin{matrix} 1.0000 & 0.0590 & - 0.0490 & 0.0516 \\ 0.0707 & 0.9943 & 0.0066 & - 0.0011 \\ 0.0383 & - 0.0090 & - 1.0275 & - 0.0469 \\ - 0.0534 & 0.0135 & 0.0423 & - 0.9543 \end{matrix}]$	0.0396	0.0707
Quarter-wave plate 0°	$[\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & - 1 \\ 0 & 0 & 1 & 0 \end{matrix}]$	$[\begin{matrix} 1.0000 & 0.0658 & 0.0345 & 0.0294 \\ 0.0569 & 0.9870 & 0.0193 & - 0.0024 \\ 0.0119 & 0.0418 & - 0.0176 & 1.0281 \\ 0.0628 & 0.0074 & - 0.9530 & - 0.0635 \end{matrix}]$	0.0382	0.0658

Table 3. Comparison of compensation results for different samples with 45° quarter-wave plate

Shuqiang Huang, Xiumei Jiang, Yuanhua Feng. Analysis of Phase Error of Retarders in Spectrally Encoded Mueller Matrix Measurement[J]. Laser & Optoelectronics Progress, 2021, 58(19): 1926001

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