Abstract

A method for using liquid-crystal variable retarders (LCVRs) with continually varying voltage to measure the complete Mueller matrix of a general sample is presented. The LCVRs are usually employed with fixed retardance values due to the nonlinear voltage–retardance behavior that they show. For the measurement method presented here, the nonlinear voltage–retardance relationship is first measured, and then a linear fit of the known retardance terms to the detected signal is performed. For a gap of air, the measurement error in the Mueller-matrix polarimeter is estimated at 1%–10%, depending on the Mueller-matrix element. Also, we present experimental results for a Glan–Thompson prism polarizer as a test sample, and we use the measured Mueller parameters as functions of the orientation of the optical axes of the polarizer as an indication of the quality of the polarimeter. In addition, results are compared to a typical step-voltage method to measure the Mueller matrix. Both methods give good results.

© 2014 Optical Society of America

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References

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  2. R. M. A. Azzam, “Oscillating-analyzer ellipsometer,” Rev. Sci. Instrum. 47, 624–628 (1976).
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    [CrossRef]
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    [CrossRef]
  8. N. C. Bruce, A. Dominguez-Báez, T. Santana-Sánchez, X. Téllez-Díaz, A. Nogueira-Jiménez, and R. Nava-Sandoval, “Design of a scanning polarimetric scatterometer for rough surface scattering measurements,” J. Phys. Conf. Ser. 274, 012135 (2011).
    [CrossRef]
  9. T. Santana-Sánchez, R. Nava-Sandoval, N. Bruce, D. Domínguez-Báez, and X. B. Téllez-Díaz, “Development of a goniometric scatterometer used for polarized light scattering from rough surfaces,” presented at the 1st International Congress on Instrumentation and Applied Sciences, CCADET-UNAM, Cancún, Mexico, 26–29 October2010.
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  14. A. De Martino, Y.-K. Kim, E. Garcia-Caurel, B. Laude, and B. Drévillon, “Optimized Mueller polarimeter with liquid crystals,” Opt. Lett. 28, 616–618 (2003).
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  15. M. Mujat and A. Dogariu, “Real-time measurement of the polarization transfer function,” Appl. Opt. 40, 34–44 (2001).
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  16. See http://www.meadowlark.com for details on the fabrication and operation of liquid crystal variable retarders.
  17. J. M. López-Téllez and N. C. Bruce, “Stokes polarimetry using analysis of the nonlinear voltage-retardance relationship for liquid-crystal variable retarders,” Rev. Sci. Instrum. 85, 033104 (2014).
    [CrossRef]
  18. W. S. Bickel and W. M. Bailey, “Stokes vectors, Mueller matrices and polarized scattered light,” Am. J. Phys. 53, 468–478 (1985).
    [CrossRef]
  19. P. Terrier, J. M. Charbois, and V. Devlaminck, “Fast-axis orientation dependence on driving voltage for a Stokes polarimeter based on concrete liquid-crystal variable retarders,” Appl. Opt. 49, 4278–4283 (2010).
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2014 (1)

J. M. López-Téllez and N. C. Bruce, “Stokes polarimetry using analysis of the nonlinear voltage-retardance relationship for liquid-crystal variable retarders,” Rev. Sci. Instrum. 85, 033104 (2014).
[CrossRef]

2011 (2)

G. Martínez-Ponce, C. Solano, and C. Pérez-Barrios, “Hybrid complete Mueller polarimeter based on phase modulators,” Opt. Lasers Eng. 49, 723–728 (2011).
[CrossRef]

N. C. Bruce, A. Dominguez-Báez, T. Santana-Sánchez, X. Téllez-Díaz, A. Nogueira-Jiménez, and R. Nava-Sandoval, “Design of a scanning polarimetric scatterometer for rough surface scattering measurements,” J. Phys. Conf. Ser. 274, 012135 (2011).
[CrossRef]

2010 (4)

2007 (1)

2006 (1)

2003 (1)

2002 (1)

S. L. Jacques, J. C. Ramella-Roman, and K. Lee, “Imaging skin pathology with polarized light,” J. Biomed. Opt. 7, 329–340 (2002).
[CrossRef]

2001 (1)

1999 (1)

1997 (1)

1992 (1)

1985 (1)

W. S. Bickel and W. M. Bailey, “Stokes vectors, Mueller matrices and polarized scattered light,” Am. J. Phys. 53, 468–478 (1985).
[CrossRef]

1978 (1)

1976 (1)

R. M. A. Azzam, “Oscillating-analyzer ellipsometer,” Rev. Sci. Instrum. 47, 624–628 (1976).
[CrossRef]

Álvarez-Herrero, A.

Azzam, R. M. A.

Bailey, W. M.

W. S. Bickel and W. M. Bailey, “Stokes vectors, Mueller matrices and polarized scattered light,” Am. J. Phys. 53, 468–478 (1985).
[CrossRef]

Belenguer, T.

Bickel, W. S.

W. S. Bickel and W. M. Bailey, “Stokes vectors, Mueller matrices and polarized scattered light,” Am. J. Phys. 53, 468–478 (1985).
[CrossRef]

Bruce, N.

T. Santana-Sánchez, R. Nava-Sandoval, N. Bruce, D. Domínguez-Báez, and X. B. Téllez-Díaz, “Development of a goniometric scatterometer used for polarized light scattering from rough surfaces,” presented at the 1st International Congress on Instrumentation and Applied Sciences, CCADET-UNAM, Cancún, Mexico, 26–29 October2010.

Bruce, N. C.

J. M. López-Téllez and N. C. Bruce, “Stokes polarimetry using analysis of the nonlinear voltage-retardance relationship for liquid-crystal variable retarders,” Rev. Sci. Instrum. 85, 033104 (2014).
[CrossRef]

N. C. Bruce, A. Dominguez-Báez, T. Santana-Sánchez, X. Téllez-Díaz, A. Nogueira-Jiménez, and R. Nava-Sandoval, “Design of a scanning polarimetric scatterometer for rough surface scattering measurements,” J. Phys. Conf. Ser. 274, 012135 (2011).
[CrossRef]

Charbois, J. M.

Compain, E.

Dainty, C.

De Martino, A.

Delplancke, F.

Devlaminck, V.

Dogariu, A.

Dominguez-Báez, A.

N. C. Bruce, A. Dominguez-Báez, T. Santana-Sánchez, X. Téllez-Díaz, A. Nogueira-Jiménez, and R. Nava-Sandoval, “Design of a scanning polarimetric scatterometer for rough surface scattering measurements,” J. Phys. Conf. Ser. 274, 012135 (2011).
[CrossRef]

Domínguez-Báez, D.

T. Santana-Sánchez, R. Nava-Sandoval, N. Bruce, D. Domínguez-Báez, and X. B. Téllez-Díaz, “Development of a goniometric scatterometer used for polarized light scattering from rough surfaces,” presented at the 1st International Congress on Instrumentation and Applied Sciences, CCADET-UNAM, Cancún, Mexico, 26–29 October2010.

Drevillon, B.

Drévillon, B.

Foldyna, M.

I. S. Nerbo, S. LeRoy, M. Foldyna, M. Kildemo, and E. Sondergard, “Characterization of inclined GaSb nanopillars by Mueller matrix ellipsometry,” J. Appl. Phys. 108, 014307 (2010).
[CrossRef]

Garcia-Caurel, E.

Goldstein, D.

D. Goldstein, Polarized Light, 2nd ed. (Dekker, 2003).

Goldstein, D. H.

Heredero, R. L.

Hoover, B. G.

Jacques, S. L.

S. L. Jacques, J. C. Ramella-Roman, and K. Lee, “Imaging skin pathology with polarized light,” J. Biomed. Opt. 7, 329–340 (2002).
[CrossRef]

Johnson, S. J.

Kildemo, M.

I. S. Nerbo, S. LeRoy, M. Foldyna, M. Kildemo, and E. Sondergard, “Characterization of inclined GaSb nanopillars by Mueller matrix ellipsometry,” J. Appl. Phys. 108, 014307 (2010).
[CrossRef]

Kim, Y.-K.

Lara, D.

Laude, B.

Lee, K.

S. L. Jacques, J. C. Ramella-Roman, and K. Lee, “Imaging skin pathology with polarized light,” J. Biomed. Opt. 7, 329–340 (2002).
[CrossRef]

LeRoy, S.

I. S. Nerbo, S. LeRoy, M. Foldyna, M. Kildemo, and E. Sondergard, “Characterization of inclined GaSb nanopillars by Mueller matrix ellipsometry,” J. Appl. Phys. 108, 014307 (2010).
[CrossRef]

López-Téllez, J. M.

J. M. López-Téllez and N. C. Bruce, “Stokes polarimetry using analysis of the nonlinear voltage-retardance relationship for liquid-crystal variable retarders,” Rev. Sci. Instrum. 85, 033104 (2014).
[CrossRef]

Martínez Pillet, V.

Martínez-Ponce, G.

G. Martínez-Ponce, C. Solano, and C. Pérez-Barrios, “Hybrid complete Mueller polarimeter based on phase modulators,” Opt. Lasers Eng. 49, 723–728 (2011).
[CrossRef]

Mujat, M.

Nava-Sandoval, R.

N. C. Bruce, A. Dominguez-Báez, T. Santana-Sánchez, X. Téllez-Díaz, A. Nogueira-Jiménez, and R. Nava-Sandoval, “Design of a scanning polarimetric scatterometer for rough surface scattering measurements,” J. Phys. Conf. Ser. 274, 012135 (2011).
[CrossRef]

T. Santana-Sánchez, R. Nava-Sandoval, N. Bruce, D. Domínguez-Báez, and X. B. Téllez-Díaz, “Development of a goniometric scatterometer used for polarized light scattering from rough surfaces,” presented at the 1st International Congress on Instrumentation and Applied Sciences, CCADET-UNAM, Cancún, Mexico, 26–29 October2010.

Nerbo, I. S.

I. S. Nerbo, S. LeRoy, M. Foldyna, M. Kildemo, and E. Sondergard, “Characterization of inclined GaSb nanopillars by Mueller matrix ellipsometry,” J. Appl. Phys. 108, 014307 (2010).
[CrossRef]

Nogueira-Jiménez, A.

N. C. Bruce, A. Dominguez-Báez, T. Santana-Sánchez, X. Téllez-Díaz, A. Nogueira-Jiménez, and R. Nava-Sandoval, “Design of a scanning polarimetric scatterometer for rough surface scattering measurements,” J. Phys. Conf. Ser. 274, 012135 (2011).
[CrossRef]

Pérez-Barrios, C.

G. Martínez-Ponce, C. Solano, and C. Pérez-Barrios, “Hybrid complete Mueller polarimeter based on phase modulators,” Opt. Lasers Eng. 49, 723–728 (2011).
[CrossRef]

Poirier, S.

Ramella-Roman, J. C.

S. L. Jacques, J. C. Ramella-Roman, and K. Lee, “Imaging skin pathology with polarized light,” J. Biomed. Opt. 7, 329–340 (2002).
[CrossRef]

Ramos, G.

Reina, M.

Rodríguez-Herrera, O. G.

Sánchez, A.

Santana-Sánchez, T.

N. C. Bruce, A. Dominguez-Báez, T. Santana-Sánchez, X. Téllez-Díaz, A. Nogueira-Jiménez, and R. Nava-Sandoval, “Design of a scanning polarimetric scatterometer for rough surface scattering measurements,” J. Phys. Conf. Ser. 274, 012135 (2011).
[CrossRef]

T. Santana-Sánchez, R. Nava-Sandoval, N. Bruce, D. Domínguez-Báez, and X. B. Téllez-Díaz, “Development of a goniometric scatterometer used for polarized light scattering from rough surfaces,” presented at the 1st International Congress on Instrumentation and Applied Sciences, CCADET-UNAM, Cancún, Mexico, 26–29 October2010.

Solano, C.

G. Martínez-Ponce, C. Solano, and C. Pérez-Barrios, “Hybrid complete Mueller polarimeter based on phase modulators,” Opt. Lasers Eng. 49, 723–728 (2011).
[CrossRef]

Sondergard, E.

I. S. Nerbo, S. LeRoy, M. Foldyna, M. Kildemo, and E. Sondergard, “Characterization of inclined GaSb nanopillars by Mueller matrix ellipsometry,” J. Appl. Phys. 108, 014307 (2010).
[CrossRef]

Téllez-Díaz, X.

N. C. Bruce, A. Dominguez-Báez, T. Santana-Sánchez, X. Téllez-Díaz, A. Nogueira-Jiménez, and R. Nava-Sandoval, “Design of a scanning polarimetric scatterometer for rough surface scattering measurements,” J. Phys. Conf. Ser. 274, 012135 (2011).
[CrossRef]

Téllez-Díaz, X. B.

T. Santana-Sánchez, R. Nava-Sandoval, N. Bruce, D. Domínguez-Báez, and X. B. Téllez-Díaz, “Development of a goniometric scatterometer used for polarized light scattering from rough surfaces,” presented at the 1st International Congress on Instrumentation and Applied Sciences, CCADET-UNAM, Cancún, Mexico, 26–29 October2010.

Terrier, P.

Tyo, J. S.

Uribe-Patarroyo, N.

Wang, Z.

Am. J. Phys. (1)

W. S. Bickel and W. M. Bailey, “Stokes vectors, Mueller matrices and polarized scattered light,” Am. J. Phys. 53, 468–478 (1985).
[CrossRef]

Appl. Opt. (8)

J. Appl. Phys. (1)

I. S. Nerbo, S. LeRoy, M. Foldyna, M. Kildemo, and E. Sondergard, “Characterization of inclined GaSb nanopillars by Mueller matrix ellipsometry,” J. Appl. Phys. 108, 014307 (2010).
[CrossRef]

J. Biomed. Opt. (1)

S. L. Jacques, J. C. Ramella-Roman, and K. Lee, “Imaging skin pathology with polarized light,” J. Biomed. Opt. 7, 329–340 (2002).
[CrossRef]

J. Phys. Conf. Ser. (1)

N. C. Bruce, A. Dominguez-Báez, T. Santana-Sánchez, X. Téllez-Díaz, A. Nogueira-Jiménez, and R. Nava-Sandoval, “Design of a scanning polarimetric scatterometer for rough surface scattering measurements,” J. Phys. Conf. Ser. 274, 012135 (2011).
[CrossRef]

Opt. Express (1)

Opt. Lasers Eng. (1)

G. Martínez-Ponce, C. Solano, and C. Pérez-Barrios, “Hybrid complete Mueller polarimeter based on phase modulators,” Opt. Lasers Eng. 49, 723–728 (2011).
[CrossRef]

Opt. Lett. (2)

Rev. Sci. Instrum. (2)

J. M. López-Téllez and N. C. Bruce, “Stokes polarimetry using analysis of the nonlinear voltage-retardance relationship for liquid-crystal variable retarders,” Rev. Sci. Instrum. 85, 033104 (2014).
[CrossRef]

R. M. A. Azzam, “Oscillating-analyzer ellipsometer,” Rev. Sci. Instrum. 47, 624–628 (1976).
[CrossRef]

Other (3)

D. Goldstein, Polarized Light, 2nd ed. (Dekker, 2003).

T. Santana-Sánchez, R. Nava-Sandoval, N. Bruce, D. Domínguez-Báez, and X. B. Téllez-Díaz, “Development of a goniometric scatterometer used for polarized light scattering from rough surfaces,” presented at the 1st International Congress on Instrumentation and Applied Sciences, CCADET-UNAM, Cancún, Mexico, 26–29 October2010.

See http://www.meadowlark.com for details on the fabrication and operation of liquid crystal variable retarders.

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Figures (4)

Fig. 1.
Fig. 1.

Setup for a Mueller-matrix polarimeter. The angles associated with each component refer to the relative angle of the optical axis of that component. ω1, ω2, ω3, and ω4 are the frequencies of the variations of the retardances.

Fig. 2.
Fig. 2.

Retardance versus voltage for a typical liquid-crystal retarder for a wavelength of 633 nm [17].

Fig. 3.
Fig. 3.

Sixteen Mueller-matrix elements (classified as they appear in the matrix) of a Glan–Thompson prism polarizer drawn as a function of its optical angle (in degrees), obtained using a continually varying voltage method of measurement. The dots are the experimental results including device and sample imperfections; the solid curves are the fitted theoretical curves of an ideal polarizer. The Mueller elements are all normalized by the first entry, M11.

Fig. 4.
Fig. 4.

Sixteen Mueller-matrix elements (classified as they appear in the matrix) of a Glan–Thompson prism polarizer drawn as a function of its optical angle (in degrees), obtained using a step-voltage method of measurement. The dots are the experimental results including device and sample imperfections; the solid curves are the fitted theoretical curves of an ideal polarizer. The Mueller elements are all normalized by the first entry, M11.

Tables (1)

Tables Icon

Table 1. Sixteen Measurements Required to Obtain the Complete Mueller Matrix of a General Samplea

Equations (16)

Equations on this page are rendered with MathJax. Learn more.

Sout=MSSin,
MPSG=MR(δ2,0°)MR(δ1,45°)MP(0°),
MPSA=MP(0°)MR(δ4,45°)MR(δ3,0°),
MPSG=12(1100cos(δ1)cos(δ1)00sin(δ1)sin(δ2)sin(δ1)sin(δ2)00sin(δ1)cos(δ2)sin(δ1)cos(δ2)00),
MPSA=12(1cos(δ4)sin(δ3)sin(δ4)cos(δ3)sin(δ4)1cos(δ4)sin(δ3)sin(δ4)cos(δ3)sin(δ4)00000000).
MS=(M11M12M13M14M21M22M23M24M31M32M33M34M41M42M43M44),
MT=MPSAMSMPSG.
Sin=(1000).
Sout=MTSin,
I=S0out=14{M11M12cos(δ1)M13sin(δ1)sin(δ2)M14sin(δ1)cos(δ2)M21cos(δ4)+M22cos(δ1)cos(δ4)+M23sin(δ1)sin(δ2)cos(δ4)+M24sin(δ1)cos(δ2)cos(δ4)+M31sin(δ3)sin(δ4)M32cos(δ1)sin(δ3)sin(δ4)M33sin(δ1)sin(δ2)sin(δ3)sin(δ4)M34sin(δ1)cos(δ2)sin(δ3)sin(δ4)+M41cos(δ3)sin(δ4)M42cos(δ1)cos(δ3)sin(δ4)M43sin(δ1)sin(δ2)cos(δ3)sin(δ4)M44sin(δ1)cos(δ2)cos(δ3)sin(δ4)}.
I=A+Bcos(δ1)+Csin(δ1)sin(δ2)+Dsin(δ1)cos(δ2)+Ecos(δ4)+Fcos(δ1)cos(δ4)+Gsin(δ1)sin(δ2)cos(δ4)+Hsin(δ1)cos(δ2)cos(δ4)+Jsin(δ3)sin(δ4)+Kcos(δ1)sin(δ3)sin(δ4)+Lsin(δ1)sin(δ2)sin(δ3)sin(δ4)+Msin(δ1)cos(δ2)sin(δ3)sin(δ4)+Ncos(δ3)sin(δ4)+Pcos(δ1)cos(δ3)sin(δ4)+Qsin(δ1)sin(δ2)cos(δ3)sin(δ4)+Rsin(δ1)cos(δ2)cos(δ3)sin(δ4),
A=14M11,B=14M12,C=14M13,D=14M14,E=14M21,F=14M22,G=14M23,H=14M24,J=14M31,K=14M32,L=14M33,M=14M34,N=14M41,P=14M42,Q=14M43,R=14M44.
V=Vmin+(VmaxVmin)mod(tti),
MAir=(10.0540.0520.0210.0290.9860.0310.0830.0140.0370.9550.0460.0390.0010.0510.897).
M11=I(H,H)+I(H,V)+I(V,V)+I(V,H),M12=I(H,H)+I(H,V)I(V,V)I(V,H),M13=2{0.5M11I(,H)I(,V)},M14=2{0.5M11I(L,H)I(L,V)},M21=I(H,H)I(H,V)+I(V,V)I(V,H),M22=I(H,H)I(H,V)I(V,V)+I(V,H),M23=2{0.5M21I(,H)+I(,V)},M24=2{0.5M21I(L,H)+I(L,V)},M31=2{I(H,+)+I(V,+)0.5M11},M32=2{I(H,+)I(V,+)0.5M12},M33=M11M13+M314I(,+),M34=M11M14+M314I(L,+),M41=2{0.5M11I(H,L)I(V,L)},M42=2{0.5M12I(H,L)+I(V,L)},M43=4I(,L)M11+M13+M41,M44=4I(L,L)M11+M14+M41.
MAir=(10.0030.0170.0130.0130.9750.0210.0110.0210.0110.9710.0110.0190.0120.0231.026).

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