Abstract

In this paper, we propose a criterion for bounding the relative errors associated with the determination of the Stokes vector that describes the state of polarization of a light beam. No assumptions about the magnitude, origin, or statistical behavior of the errors are made. It is shown that figures of merit such as the condition number and the equally weighted variance naturally arise as optimization parameters. Moreover, a third optimization parameter emerges, which takes into account errors associated with the matrix that represents the selected configuration of analyzers. Finally, a new and more general figure of merit is derived from this analysis and is applied in an optimization process of a very well known polarimeter.

© 2013 Optical Society of America

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References

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  1. N. Ghosh and I. A. Vitkin, “Tissue polarimetry: concepts, challenges, applications and outlook,” J. Biomed. Opt. 16, 110801 (2011).
    [CrossRef]
  2. P. Shukla and A. Pradhan, “Mueller decomposition images for cervical tissue: potential for discriminating normal and dysplastic states,” Opt. Express 17, 1600–1609 (2009).
    [CrossRef]
  3. M. F. G. Wood, N. Ghosh, X. Guo, and I. A. Vitkin, “Towards noninvasive glucose sensing using polarization analysis of multiply scattered light,” in Handbook of Optical Sensing of Glucose in Biological Fluids and Tissues, V. V. Tuchin, ed., Vol. 12 of Series in Medical Physics and Biomedical Engineering (Taylor & Francis, 2008), Chap. 17.
  4. D. Côté and I. A. Vitkin, “Robust concentration determination of optically active molecules in turbid media with validated three-dimensional polarization sensitive Monte Carlo calculations,” Opt. Express 13, 148–163 (2005).
    [CrossRef]
  5. S. Tan and R. M. Narayanan, “Design and performance of a multiwavelength airborne polarimetric lidar for vegetation remote sensing,” Appl. Opt. 43, 2360–2368 (2004).
    [CrossRef]
  6. G. Horvath, A. Barta, J. Gal, B. Suhai, and O. Haiman, “Ground-based full-sky imaging polarimetry of rapidly changing skies and its use for polarimetric cloud detection,” Appl. Opt. 41, 543–559 (2002).
    [CrossRef]
  7. D. S. Sabatke, M. R. Descour, E. L. Dereniak, W. C. Sweatt, S. A. Kemme, and G. S. Phipps, “Optimization of retardance for a complete Stokes polarimeter,” Opt. Lett. 25, 802–804 (2000).
    [CrossRef]
  8. 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]
  9. A. Peinado, A. Lizana, J. Vidal, C. Iemmi, and J. Campos, “Optimization and performance criteria of a Stokes polarimeter based on two variable retarders,” Opt. Express 18, 9815–9830 (2010).
    [CrossRef]
  10. A. Ben-Israel and T. N. E. Greville, Generalized Inverses: Theory and Applications (Springer, 2003).
  11. A. Peinado, A. Lizana, J. Vidal, C. Iemmi, and J. Campos, “Optimized Stokes polarimeters based on a single twisted nematic liquid-crystal device for the minimization of noise propagation,” Appl. Opt. 50, 5437–5445 (2011).
    [CrossRef]
  12. J. M. Daniels, “Optimum design of a polarimeter for visible light,” Rev. Sci. Instrum. 57, 1570–1573 (1986).
    [CrossRef]
  13. K. M. Twietmeyer and R. A. Chipman, “Optimization of Mueller matrix polarimeters in the presence of error sources,” Opt. Express 16, 11589–11603 (2008).
    [CrossRef]
  14. I. J. Vaughn and B. G. Hoover, “Noise reduction in a laser polarimeter based on discrete waveplate rotations,” Opt. Express 16, 2091–2108 (2008).
    [CrossRef]
  15. D. Layden, M. F. G. Wood, and I. A. Vitkin, “Optimum selection of input polarization states in determining the sample Mueller matrix: a dual photoelastic polarimeter approach,” Opt. Express 20, 20466–20481 (2012).
    [CrossRef]
  16. T. M. Apostol, Calculus, Volume II: Multi-Variable Calculus and Linear Algebra, with Applications to Differential Equations and Probability (Wiley, 1967).
  17. G. H. Golub and C. F. Van Loan, Matrix Computations (Johns Hopkins University, 1996).
  18. J. T. Schwartz, Introduction to Matrices and Vectors (Dover, 2001).

2012 (1)

2011 (2)

2010 (1)

2009 (1)

2008 (2)

2005 (1)

2004 (1)

2002 (2)

2000 (1)

1986 (1)

J. M. Daniels, “Optimum design of a polarimeter for visible light,” Rev. Sci. Instrum. 57, 1570–1573 (1986).
[CrossRef]

Apostol, T. M.

T. M. Apostol, Calculus, Volume II: Multi-Variable Calculus and Linear Algebra, with Applications to Differential Equations and Probability (Wiley, 1967).

Barta, A.

Ben-Israel, A.

A. Ben-Israel and T. N. E. Greville, Generalized Inverses: Theory and Applications (Springer, 2003).

Campos, J.

Chipman, R. A.

Côté, D.

Daniels, J. M.

J. M. Daniels, “Optimum design of a polarimeter for visible light,” Rev. Sci. Instrum. 57, 1570–1573 (1986).
[CrossRef]

Dereniak, E. L.

Descour, M. R.

Gal, J.

Ghosh, N.

N. Ghosh and I. A. Vitkin, “Tissue polarimetry: concepts, challenges, applications and outlook,” J. Biomed. Opt. 16, 110801 (2011).
[CrossRef]

M. F. G. Wood, N. Ghosh, X. Guo, and I. A. Vitkin, “Towards noninvasive glucose sensing using polarization analysis of multiply scattered light,” in Handbook of Optical Sensing of Glucose in Biological Fluids and Tissues, V. V. Tuchin, ed., Vol. 12 of Series in Medical Physics and Biomedical Engineering (Taylor & Francis, 2008), Chap. 17.

Golub, G. H.

G. H. Golub and C. F. Van Loan, Matrix Computations (Johns Hopkins University, 1996).

Greville, T. N. E.

A. Ben-Israel and T. N. E. Greville, Generalized Inverses: Theory and Applications (Springer, 2003).

Guo, X.

M. F. G. Wood, N. Ghosh, X. Guo, and I. A. Vitkin, “Towards noninvasive glucose sensing using polarization analysis of multiply scattered light,” in Handbook of Optical Sensing of Glucose in Biological Fluids and Tissues, V. V. Tuchin, ed., Vol. 12 of Series in Medical Physics and Biomedical Engineering (Taylor & Francis, 2008), Chap. 17.

Haiman, O.

Hoover, B. G.

Horvath, G.

Iemmi, C.

Kemme, S. A.

Layden, D.

Lizana, A.

Narayanan, R. M.

Peinado, A.

Phipps, G. S.

Pradhan, A.

Sabatke, D. S.

Schwartz, J. T.

J. T. Schwartz, Introduction to Matrices and Vectors (Dover, 2001).

Shukla, P.

Suhai, B.

Sweatt, W. C.

Tan, S.

Twietmeyer, K. M.

Tyo, J. S.

Van Loan, C. F.

G. H. Golub and C. F. Van Loan, Matrix Computations (Johns Hopkins University, 1996).

Vaughn, I. J.

Vidal, J.

Vitkin, I. A.

D. Layden, M. F. G. Wood, and I. A. Vitkin, “Optimum selection of input polarization states in determining the sample Mueller matrix: a dual photoelastic polarimeter approach,” Opt. Express 20, 20466–20481 (2012).
[CrossRef]

N. Ghosh and I. A. Vitkin, “Tissue polarimetry: concepts, challenges, applications and outlook,” J. Biomed. Opt. 16, 110801 (2011).
[CrossRef]

D. Côté and I. A. Vitkin, “Robust concentration determination of optically active molecules in turbid media with validated three-dimensional polarization sensitive Monte Carlo calculations,” Opt. Express 13, 148–163 (2005).
[CrossRef]

M. F. G. Wood, N. Ghosh, X. Guo, and I. A. Vitkin, “Towards noninvasive glucose sensing using polarization analysis of multiply scattered light,” in Handbook of Optical Sensing of Glucose in Biological Fluids and Tissues, V. V. Tuchin, ed., Vol. 12 of Series in Medical Physics and Biomedical Engineering (Taylor & Francis, 2008), Chap. 17.

Wood, M. F. G.

D. Layden, M. F. G. Wood, and I. A. Vitkin, “Optimum selection of input polarization states in determining the sample Mueller matrix: a dual photoelastic polarimeter approach,” Opt. Express 20, 20466–20481 (2012).
[CrossRef]

M. F. G. Wood, N. Ghosh, X. Guo, and I. A. Vitkin, “Towards noninvasive glucose sensing using polarization analysis of multiply scattered light,” in Handbook of Optical Sensing of Glucose in Biological Fluids and Tissues, V. V. Tuchin, ed., Vol. 12 of Series in Medical Physics and Biomedical Engineering (Taylor & Francis, 2008), Chap. 17.

Appl. Opt. (4)

J. Biomed. Opt. (1)

N. Ghosh and I. A. Vitkin, “Tissue polarimetry: concepts, challenges, applications and outlook,” J. Biomed. Opt. 16, 110801 (2011).
[CrossRef]

Opt. Express (6)

Opt. Lett. (1)

Rev. Sci. Instrum. (1)

J. M. Daniels, “Optimum design of a polarimeter for visible light,” Rev. Sci. Instrum. 57, 1570–1573 (1986).
[CrossRef]

Other (5)

T. M. Apostol, Calculus, Volume II: Multi-Variable Calculus and Linear Algebra, with Applications to Differential Equations and Probability (Wiley, 1967).

G. H. Golub and C. F. Van Loan, Matrix Computations (Johns Hopkins University, 1996).

J. T. Schwartz, Introduction to Matrices and Vectors (Dover, 2001).

M. F. G. Wood, N. Ghosh, X. Guo, and I. A. Vitkin, “Towards noninvasive glucose sensing using polarization analysis of multiply scattered light,” in Handbook of Optical Sensing of Glucose in Biological Fluids and Tissues, V. V. Tuchin, ed., Vol. 12 of Series in Medical Physics and Biomedical Engineering (Taylor & Francis, 2008), Chap. 17.

A. Ben-Israel and T. N. E. Greville, Generalized Inverses: Theory and Applications (Springer, 2003).

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Figures (2)

Fig. 1.
Fig. 1.

Basic scheme of the polarimeter used as an example.

Fig. 2.
Fig. 2.

Optimum analyzers obtained in optimization processes nos. 1, 2, and 3.

Tables (1)

Tables Icon

Table 1. Numerical Values of the Four Figures of Merit Calculated for Each of the Optimization Processes

Equations (55)

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I=QS,
Q˜(QTQ)1QT,
S=Q˜I.
S(q˜,i)=(Q˜+q˜)(I+i).
δS(q˜,i)=S(q˜,i)S¯=(Q˜+q˜)(I+i)Q˜I.
δS(q˜,i)=Q˜i+q˜I+q˜i.
δSj(q˜,i)=kQ˜jkik+kq˜jkIk+kq˜jkik,
limδI,δQ˜δS=.
q˜+δQ˜,i+δI,
δS(q,i)Q˜i++q+I+q+i+Q˜i++q+I+q+i+Q˜δI+δQ˜I+δQ˜δI,
QSI.
δSS(δII+δQ˜Q˜+δIIδQ˜Q˜)Q˜Q.
κ(Q)Q˜Q,
ε(S)(ε(I)+ε(Q˜)+ε(I)ε(Q˜))κ(Q).
δQ˜(q)=[(QT+qT)(Q+q)]1(QT+qT)Q˜,
δQ˜(q)=[(QT+qT)(Q+q)]1(QT+qT)Q˜.
δQ˜=[QTQ(In+Γ+)]1(QT+q+T)Q˜,
δQ˜=(In+Γ+)1(QTQ)1(QT+q+T)Q˜=(In+Γ+)1[Q˜+(QTQ)1q+T]Q˜.
δQ˜=(In+Γ+)1[(QTQ)1q+TΓ+Q˜](In+Γ+)1[(QTQ)1q+T+Γ+Q˜](In+Γ+)1[(QTQ)1q+T+Γ+Q˜].
δQ˜(In+Γ+)1(Q˜2q+T+Γ+Q˜),ε(Q˜)(In+Γ+)1(Q˜q+T+Γ+).
Γ+=(QTQ)1[(QT+q+T)q++q+TQ](QTQ)1[(QT+q+T)q++q+TQ](QTQ)1[QT+q+Tq++q+TQ]Q˜2q+(2Q+q+)Q˜2QδQ[2+ε(Q)]κ(Q)2ε(Q)[2+ε(Q)],
Q˜q+TQ˜δQ=κ(Q)ε(Q)κ(Q)2ε(Q).
ε(Q˜)κ(Q)2ε(Q)[3+ε(Q)](In+Γ+)1.
(In+Γ)1i=0Γi
ε(Q˜)κ(Q)2ε(Q)[3+ε(Q)]i=0Γ+i.
i=0Γ+ii=0{κ(Q)2ε(Q)[2+ε(Q)]}i,
ε(Q˜)i=0{[κ(Q)2ε(Q)]i+1[3+ε(Q)][2+ε(Q)]i}.
[3+ε(Q)][2+ε(Q)]i[3+ε(Q)]i+1,
ε(Q˜)i=1{κ(Q)2ε(Q)[3+ε(Q)]}i,
ξ(Q)κ(Q)2ε(Q)[3+ε(Q)]
Ξ(Q)i=1[ξ(Q)]i.
ε(S){ε(I)+[1+ε(I)]Ξ(Q)}κ(Q),
limε(Q),ε(I)0ε(S)=0,
ξ(Q)Q˜δQ[3κ(Q)+Q˜δQ],
ε(S){ε(I)+[1+ε(I)]Q˜δQΞ(Q)}κ(Q),
Ξ(Q)κ(Q)[3+ε(Q)]i=0[ξ(Q)]i.
Λ(x1,x2,,xN)Q˜δQκ(Q),
Q+Q(Y),
QQ(Z),
Qij(Y)Qij(X),Qij(Z)Qij(X),
δQQ+Q,
A0,A=0A=0,
αA=|α|A,
A+BA+B,
ABAB,
A=AT.
AxVAMxV.
Q=UΣVTQ˜=VΣ˜UT,
Q˜=VΣ˜UT=Σ˜.
(QTQ)1=(VΣUTUΣVT)1=(VΣ2VT)1=(VTΣ2V)=Σ2Σ˜2=Q˜2.
(QTQ)1Q˜2.
EWV(Q)j=1r1σj2,
Q˜=Σ˜.
Σ˜[tr(Σ˜TΣ˜)]1/2=(j=1r1σj2)1/2,
Q˜F2=Σ˜2=j=1r1σj2=EWV.

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