Abstract

We address calibration of Mueller polarimeters in the presence of noise. We compare an extension of the eigenvalue calibration method (ECM) and a maximum likelihood (ML) method. The performances of these two calibration methods are investigated with numerical simulations and real experiments on a broadband infrared polarimeter. It is found that the ML method is superior to the extended ECM in terms of calibration precision and can be used at lower signal-to-noise ratio.

© 2013 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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  15. L. M. S. Aas, P. G. Ellingsen, and M. Kildemo, “Near infrared Mueller matrix imaging system and application to retardance imaging of strain,” Thin Solid Films 519, 2737–2741 (2011).
    [CrossRef]
  16. S. Ben Hatit, M. Foldyna, A. De Martino, and B. Drvillon, “Angle-resolved Mueller polarimeter using a microscope objective,” Phys. Status Solidi A 205, 743–747 (2008).
    [CrossRef]
  17. P. E. Gill, W. Murray, and M. H. Wright, Practical Optimization (Academic, 1981).
  18. F. Goudail, “Noise minimization and equalization for Stokes polarimeters in the presence of signal-dependent poisson shot noise,” Opt. Lett. 34, 647–649 (2009).
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  19. G. Anna and F. Goudail, “Optimal Mueller matrix estimation in the presence of Poisson shot noise,” Opt. Express 20, 21331–21340 (2012).
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  20. J. S. Tyo, “Design of optimal polarimeters: maximization of signal-to-noise ratio and minimization of systematic error,” Appl. Opt. 41, 619–630 (2002).
    [CrossRef]

2012 (2)

C. Macias-Romero and P. Torok, “Eigenvalue calibration methods for polarimetry,” J. Eur. Opt. Soc. Rapid Pub. 7, 12004 (2012).
[CrossRef]

G. Anna and F. Goudail, “Optimal Mueller matrix estimation in the presence of Poisson shot noise,” Opt. Express 20, 21331–21340 (2012).
[CrossRef]

2011 (2)

A. Pierangelo, A. Benali, M.-R. Antonelli, T. Novikova, P. Validire, B. Gayet, and A. D. Martino, “Ex-vivo characterization of human colon cancer by Mueller polarimetric imaging,” Opt. Express 19, 1582–1593 (2011).
[CrossRef]

L. M. S. Aas, P. G. Ellingsen, and M. Kildemo, “Near infrared Mueller matrix imaging system and application to retardance imaging of strain,” Thin Solid Films 519, 2737–2741 (2011).
[CrossRef]

2009 (1)

2008 (1)

S. Ben Hatit, M. Foldyna, A. De Martino, and B. Drvillon, “Angle-resolved Mueller polarimeter using a microscope objective,” Phys. Status Solidi A 205, 743–747 (2008).
[CrossRef]

2007 (1)

2006 (1)

2004 (1)

A. D. Martino, E. Garcia-Caurel, B. Laude, and B. Drvillon, “General methods for optimized design and calibration of Mueller polarimeters,” Thin Solid Films 455-456, 112–119 (2004).
[CrossRef]

2003 (1)

2002 (2)

J. S. Tyo, “Design of optimal polarimeters: maximization of signal-to-noise ratio and minimization of systematic error,” Appl. Opt. 41, 619–630 (2002).
[CrossRef]

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

2000 (1)

S. Breugnot and P. Clémenceau, “Modeling and performances of a polarization active imager at λ=806 nm,” Opt. Eng. 39, 2681–2688 (2000).
[CrossRef]

1999 (1)

1996 (1)

1989 (1)

1981 (1)

1980 (1)

1978 (1)

Aas, L. M. S.

L. M. S. Aas, P. G. Ellingsen, and M. Kildemo, “Near infrared Mueller matrix imaging system and application to retardance imaging of strain,” Thin Solid Films 519, 2737–2741 (2011).
[CrossRef]

Anna, G.

Antonelli, M.-R.

Azzam, R. M. A.

Ben Hatit, S.

S. Ben Hatit, M. Foldyna, A. De Martino, and B. Drvillon, “Angle-resolved Mueller polarimeter using a microscope objective,” Phys. Status Solidi A 205, 743–747 (2008).
[CrossRef]

Benali, A.

Bottiger, J. R.

Breugnot, S.

S. Breugnot and P. Clémenceau, “Modeling and performances of a polarization active imager at λ=806 nm,” Opt. Eng. 39, 2681–2688 (2000).
[CrossRef]

Bueno, J. M.

Campbell, M.

Clémenceau, P.

S. Breugnot and P. Clémenceau, “Modeling and performances of a polarization active imager at λ=806 nm,” Opt. Eng. 39, 2681–2688 (2000).
[CrossRef]

Compain, E.

Cookson, C.

Dainty, C.

De Martino, A.

S. Ben Hatit, M. Foldyna, A. De Martino, and B. Drvillon, “Angle-resolved Mueller polarimeter using a microscope objective,” Phys. Status Solidi A 205, 743–747 (2008).
[CrossRef]

Drevillon, B.

Drvillon, B.

S. Ben Hatit, M. Foldyna, A. De Martino, and B. Drvillon, “Angle-resolved Mueller polarimeter using a microscope objective,” Phys. Status Solidi A 205, 743–747 (2008).
[CrossRef]

A. D. Martino, E. Garcia-Caurel, B. Laude, and B. Drvillon, “General methods for optimized design and calibration of Mueller polarimeters,” Thin Solid Films 455-456, 112–119 (2004).
[CrossRef]

Ellingsen, P. G.

L. M. S. Aas, P. G. Ellingsen, and M. Kildemo, “Near infrared Mueller matrix imaging system and application to retardance imaging of strain,” Thin Solid Films 519, 2737–2741 (2011).
[CrossRef]

Engheta, N.

Foldyna, M.

S. Ben Hatit, M. Foldyna, A. De Martino, and B. Drvillon, “Angle-resolved Mueller polarimeter using a microscope objective,” Phys. Status Solidi A 205, 743–747 (2008).
[CrossRef]

Fry, E. S.

Garcia-Caurel, E.

A. D. Martino, E. Garcia-Caurel, B. Laude, and B. Drvillon, “General methods for optimized design and calibration of Mueller polarimeters,” Thin Solid Films 455-456, 112–119 (2004).
[CrossRef]

Gayet, B.

Gill, P. E.

P. E. Gill, W. Murray, and M. H. Wright, Practical Optimization (Academic, 1981).

Goudail, F.

Hauge, P. S.

Hunter, J.

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]

Kildemo, M.

L. M. S. Aas, P. G. Ellingsen, and M. Kildemo, “Near infrared Mueller matrix imaging system and application to retardance imaging of strain,” Thin Solid Films 519, 2737–2741 (2011).
[CrossRef]

Kisilak, M.

Lara, D.

Laude, B.

A. D. Martino, E. Garcia-Caurel, B. Laude, and B. Drvillon, “General methods for optimized design and calibration of Mueller polarimeters,” Thin Solid Films 455-456, 112–119 (2004).
[CrossRef]

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]

Lopez, A. G.

Macias-Romero, C.

C. Macias-Romero and P. Torok, “Eigenvalue calibration methods for polarimetry,” J. Eur. Opt. Soc. Rapid Pub. 7, 12004 (2012).
[CrossRef]

Martino, A. D.

A. Pierangelo, A. Benali, M.-R. Antonelli, T. Novikova, P. Validire, B. Gayet, and A. D. Martino, “Ex-vivo characterization of human colon cancer by Mueller polarimetric imaging,” Opt. Express 19, 1582–1593 (2011).
[CrossRef]

A. D. Martino, E. Garcia-Caurel, B. Laude, and B. Drvillon, “General methods for optimized design and calibration of Mueller polarimeters,” Thin Solid Films 455-456, 112–119 (2004).
[CrossRef]

Murray, W.

P. E. Gill, W. Murray, and M. H. Wright, Practical Optimization (Academic, 1981).

Narasimhan, S. G.

Nayar, S. K.

Novikova, T.

Pierangelo, A.

Poirier, S.

Pugh, E. N.

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]

Rowe, M. P.

Schechner, Y. Y.

Solomon, J. E.

Thompson, R. C.

Torok, P.

C. Macias-Romero and P. Torok, “Eigenvalue calibration methods for polarimetry,” J. Eur. Opt. Soc. Rapid Pub. 7, 12004 (2012).
[CrossRef]

Tyo, J. S.

Validire, P.

Wright, M. H.

P. E. Gill, W. Murray, and M. H. Wright, Practical Optimization (Academic, 1981).

Appl. Opt. (7)

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. Eur. Opt. Soc. Rapid Pub. (1)

C. Macias-Romero and P. Torok, “Eigenvalue calibration methods for polarimetry,” J. Eur. Opt. Soc. Rapid Pub. 7, 12004 (2012).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Opt. Eng. (1)

S. Breugnot and P. Clémenceau, “Modeling and performances of a polarization active imager at λ=806 nm,” Opt. Eng. 39, 2681–2688 (2000).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Phys. Status Solidi A (1)

S. Ben Hatit, M. Foldyna, A. De Martino, and B. Drvillon, “Angle-resolved Mueller polarimeter using a microscope objective,” Phys. Status Solidi A 205, 743–747 (2008).
[CrossRef]

Thin Solid Films (2)

L. M. S. Aas, P. G. Ellingsen, and M. Kildemo, “Near infrared Mueller matrix imaging system and application to retardance imaging of strain,” Thin Solid Films 519, 2737–2741 (2011).
[CrossRef]

A. D. Martino, E. Garcia-Caurel, B. Laude, and B. Drvillon, “General methods for optimized design and calibration of Mueller polarimeters,” Thin Solid Films 455-456, 112–119 (2004).
[CrossRef]

Other (1)

P. E. Gill, W. Murray, and M. H. Wright, Practical Optimization (Academic, 1981).

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Figures (5)

Fig. 1.
Fig. 1.

Schematic of reflection configuration for calibration.

Fig. 2.
Fig. 2.

RMSEs of Q^ for the extended ECM and ML method as a function of SNR. 500 realizations at each SNR.

Fig. 3.
Fig. 3.

Functional schema of the Mueller ellipsometer consisting of an input arm, the reflection sample, and an exit arm. The input arm includes a FTIR interferometer, a PSG, and a movable platform for the calibration sample of the linear polarizer. The exit arm consists of another retractable platform for the calibration sample of the linear polarizer, the PSA, and the detector. The PSG includes a linear polarizer and a V-shaped retarder. The same elements, i.e., polarizer and retarder, can be found in the PSA. The FTIR source, the detector, and the motors that control the orientation of the V-shaped retarders are controlled with a computer.

Fig. 4.
Fig. 4.

Typical spectrum recorded during normal operation of the broadband Mueller polarimeter. The spectrum has been normalized to the maximum value at 2100cm1.

Fig. 5.
Fig. 5.

RMSE of Q^ for extended ECM and ML method at different wave numbers at (a) 16 integrations and (b) four integrations with 100% intensity, and at (c) 16 integrations with 10% intensity. The sampling interval is 77cm1.

Equations (32)

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P(τp,θ)=τp2J(θ)[1100110000000000]J(θ),
J(θ)=[10000cos(2θ)sin(2θ)00sin(2θ)cos(2θ)00001].
R(τr,Δ,ψ)=τr×[1cos2ψ00cos2ψ10000sin2ψcosΔsin2ψsinΔ00sin2ψsinΔsin2ψcosΔ],
τp2=trace[(AR1W)1(AR1P2W)].
τr12=0.5(λr1+λr2),
Δ12=0.5arg(λc1/λc2),
ψ12=arctanλr1/λr2,
P2(θ2)XX(AR1W)1(AR1P2W)=0,
R11R2XX(AR1W)1(AR2W)=0.
H1(θ2)vec(W)=0,
H2vec(W)=0,
H1(θ2)=EP2(θ2)[(AR1W)1(AR1P2W)]TE,
H2=E(R11R2)[(AR1W)1(AR2W)]TE,
K(θ2)vec(X)=0,
K(θ2)=H1(θ2)TH1(θ2)+H2TH2.
M(θ,τ,a,b,c)=J(θ)P0J(θ)=τ2J(θ)[1a00a10000bc00cb]J(θ),
[λ1λ2λ3λ4]=τ2[b+icbic1+a1a].
{τ=(λ3+λ4)a=(λ3λ4)/τb=(λ1+λ2)/τc=imag(λ1λ2)/τ.
Ik=βkAMkW,
W(ξW)=12[1I2I3I4p1cos(2ε1)cos(2α1)I2p2cos(2ε2)cos(2α2)I3p3cos(2ε3)cos(2α3)I4p4cos(2ε4)cos(2α4)p1cos(2ε1)sin(2α1)I2p2cos(2ε2)sin(2α2)I3p3cos(2ε3)sin(2α3)I4p4cos(2ε4)sin(2α4)p1sin(2ε1)I2p2sin(2ε2)I3p3sin(2ε3)I4p4sin(2ε4)],
AT(ξA)=12[1I6I7I8p5cos(2ε5)cos(2α5)I6p6cos(2ε6)cos(2α6)I7p7cos(2ε7)cos(2α7)I8p8cos(2ε8)cos(2α8)p5cos(2ε5)sin(2α5)I6p6cos(2ε6)sin(2α6)I7p7cos(2ε7)sin(2α7)I8p8cos(2ε8)sin(2α8)p5sin(2ε5)I6p6sin(2ε6)I7p7sin(2ε7)I8p8sin(2ε8)],
{ξ^m,ξ^W,ξ^A}=argmaxξm,ξW,ξA{L(ξm,ξW,ξA)},
L(β;ξm,ξW,ξA)=12σ2kIkβkAMkW2,
L(β;ξm,ξW,ξA)=12σ2kikβkQmk2,
Lβk=0,β^k=[ik]TQmkQmk2.
L(ξm,ξW,ξA)=k([ik]TQmk)2Qmk2.
W=AT=12[11111/31/31/31/31/31/31/31/31/31/31/31/3].
P0=τ2[10.9000.9100000.050.1000.10.05],
SNR=10log10[β0216σ2],
RMSE=i,j=015(Q^ijQij0)2,
P=τP[1100110000000000];C(θ,δ)=τRJ(θ)[1000010000cosδsinδ00sinδcosδ]J(θ),
Sout(θ,δ)=C(θ,δ)P[1000].

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