• Laser & Optoelectronics Progress
  • Vol. 58, Issue 19, 1926001 (2021)
Shuqiang Huang1, Xiumei Jiang1, and Yuanhua Feng2、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 show less

    Abstract

    Spectrally encoded Mueller matrix measurement has the advantages of high measurement speed, compact structure, low loss, and no moving parts. Furthermore, all elements in the Mueller matrix can be obtained with only a single measurement. The main principle of this method is to use a set of phase retarders with a specific thickness ratio to modulate the Mueller matrix elements to the frequency channel of the spectrum; then, the Mueller matrix is demodulated through the Fourier transform of the spectrum. However, the thickness or phase error of the retarders causes a large error in the demodulated Mueller matrix elements. In this work, we theoretically obtained the general expression of light intensity with phase error and then calculated the phase error using a single sample. This method can avoid the influence of the different initial phases of different samples and improve the calculation accuracy of the phase error. We calculated the influence of phase errors of the retarders via simulation and experimentally verified the feasibility of the error calculation and calibration methods.
    Sout=Pπ2·R5ϕ+Δϕ4,π4·R5ϕ+Δϕ3,0·M·Rϕ+Δϕ2,0·Rϕ,π4P0·Sin
    Iλ=16m00+8m02cosΔϕ2-8m03sinΔϕ2-8m20cosΔϕ3-Δϕ4-8m30sinΔϕ3-Δϕ4-4m22cosΔϕ2cosΔϕ3-Δϕ4+4m33sinΔϕ2sinΔϕ3-Δϕ4+4m23sinΔϕ2cosΔϕ3-Δϕ4-4m32cosΔϕ2sinΔϕ3-Δϕ4+cosf0λ+ϕw8m01-8m21cosΔϕ3-Δϕ4-4m31sinΔϕ3-Δϕ4+cos2f0λ+2ϕw+Δϕ2-4m02+2m22cosΔϕ3-Δϕ4+2m32sinΔϕ3-Δϕ4-sin2f0λ+2ϕw+Δϕ2×4m03-2m33sinΔϕ3-Δϕ4-2m23cosΔϕ3-Δϕ4+cos3f0λ+3ϕw-Δϕ2+Δϕ42m12-sin3f0λ+3ϕw-Δϕ2+Δϕ42m13-cos4f0λ+4ϕw+Δϕ44m11-cos5f0λ+5ϕw+Δϕ4×-8m10-4m12cosΔϕ2+4m13sinΔϕ2-cos6f0λ+6ϕw+Δϕ44m11+cos7f0λ+7ϕw+Δϕ2+Δϕ4×2m12+sin7f0λ+7ϕw+Δϕ2+Δϕ42m13-cos8f0λ+8ϕw-Δϕ2+Δϕ3+Δϕ4m22-m33+sin8f0λ+8ϕw-Δϕ2+Δϕ3+Δϕ4m23-m32+cos9f0λ+9ϕw+Δϕ3+Δϕ42m21-sin9f0λ+9ϕw+Δϕ3+Δϕ42m31+cos10f0λ+10ϕw+Δϕ3+Δϕ44m20+2m22cosΔϕ2-2m23sinΔϕ2-sin10f0λ+10ϕw+Δϕ3+Δϕ44m30+2m33sinΔϕ2+2m32cosΔϕ2+cos11f0λ+11ϕw+Δϕ3+Δϕ4×2m21-sin11f0λ+11ϕw+Δϕ3+Δϕ42m31-cos12f0λ+12ϕw+Δϕ2+Δϕ3+Δϕ4m22+m33-sin12f0λ+12ϕw+Δϕ2+Δϕ3+Δϕ4m23-m32,
    I0λ=16+8cosf0λ+ϕw-4cos4f0λ+4ϕw+Δϕ4+8cos5f0λ+5ϕw+Δϕ4-4cos6f0λ+6ϕw+Δϕ4
    ϕw=Pphasef04ϕw+Δϕ4=Pphase4f05ϕw+Δϕ4=Pphase5f06ϕw+Δϕ4=Pphase6f0
    I45λ=16+8cosΔϕ2-8cosΔϕ3-Δϕ4-4cosΔϕ2cosΔϕ3-Δϕ4-4-2cosΔϕ3-Δϕ4×cos2f0λ+2ϕw+Δϕ2-cos8f0λ+8ϕw-Δϕ2+Δϕ3+Δϕ4+4+2cosΔϕ2×cos10f0λ+10ϕw+Δϕ3+Δϕ4-cos12f0λ+12ϕw+Δϕ2+Δϕ3+Δϕ4
    2ϕw+Δϕ2=Pphase2f08ϕw-Δϕ2+Δϕ3+Δϕ4=Pphase8f010ϕw+Δϕ3+Δϕ4=Pphase10f012ϕw+Δϕ2+Δϕ3+Δϕ4=Pphase12f0
    Iλ4,45λ=16-4cosΔϕ2cosΔϕ3-Δϕ4+2cosΔϕ3-Δϕ4cos2f0λ+2ϕw+Δϕ2+4sinΔϕ2cos5f0λ+5ϕw+Δϕ4-cos8f0λ+8ϕw-Δϕ2+Δϕ3+Δϕ4+2cosΔϕ2cos10f0λ+10ϕw+Δϕ3+Δϕ4-cos12f0λ+12ϕw+Δϕ2+Δϕ3+Δϕ4+2sin3f0λ+3ϕw-Δϕ2+Δϕ4-2sin7f0λ+7ϕw+Δϕ2+Δϕ4-2sin9f0λ+9ϕw+Δϕ3+Δϕ4-2sin11f0λ+11ϕw+Δϕ3+Δϕ4
    3ϕw-Δϕ2+Δϕ4=Pphase3f07ϕw+Δϕ2+Δϕ4=Pphase7f08ϕw-Δϕ2+Δϕ3+Δϕ4=Pphase8f09ϕw+Δϕ3+Δϕ4=Pphase9f0
    φ0=0φ1=ϕwφ2=2ϕw+Δϕ2φ3=3ϕw-Δϕ2+Δϕ4φ4=4ϕw+Δϕ4φ5=5ϕw+Δϕ4φ6=6ϕw+Δϕ4φ7=7ϕw+Δϕ2+Δϕ4φ8=8ϕw-Δϕ2+Δϕ3+Δϕ4φ9=9ϕw+Δϕ3+Δϕ4φ10=10ϕw+Δϕ3+Δϕ4φ11=11ϕw+Δϕ3+Δϕ4φ12=12ϕw+Δϕ2+Δϕ3+Δϕ4
    X=AT·A-1·AT·VIkf0λ×exp-iφn
    ERMS=i,j=03mij-mij'2/16
    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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