Abstract

The recent result obtained by Givens and Kostinski [J. Mod. Opt. 40, 471 (1993)] successfully solves the old and important problem in polarization optics of characterizing a given 4×4 matrix as a Mueller matrix from a mathematical point of view. For practical purposes, however, a further elaboration on this result is needed, namely, the problem of characterizing a matrix whose elements have been empirically obtained after the measurement of a given number of independent quantities that are affected by errors. We solve this problem by first obtaining an alternative form of the Givens–Kostinski theorem that allows us to figure out an algorithm for calculating the error propagation. It turns out that the experimental matrix can be finally regarded as physically meaningful or not, or even undecidable, depending on such errors. As a tool for potential users, a routine (in both fortran and idl languages) that carries out all the numerical calculations is available via ftp at a specified address.

© 1998 Optical Society of America

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References

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  1. C. R. Givens, A. Kostinski, “A simple necessary and sufficient condition on physically realizable Mueller matrices,” J. Mod. Opt. 40, 471–481 (1993).
    [CrossRef]
  2. S. Sridhar, R. Simon, “Normal form for Mueller matrices,” J. Mod. Opt. 41, 1903–1915 (1994).
    [CrossRef]
  3. M. Sanjay Kumar, R. Simon, “Characterization of Mueller matrices in polarization optics,” Opt. Commun. 88, 464–470 (1992).
    [CrossRef]
  4. C. V. M. van der Mee, J. W. Hovenier, “Structure of matrices transforming Stokes parameters,” J. Math. Phys. (N.Y.), 33, 3574–3584 (1992).
    [CrossRef]
  5. A. Kostinski, B. James, W.-M. Boemer, “Optimal reception of partially polarized waves,” J. Opt. Soc. Am. A 5, 58–64 (1988).
    [CrossRef]
  6. B. J. Howell, “Measurement of the polarization effects of an instrument using partially polarized light,” Appl. Opt. 18, 809–812 (1979).
    [CrossRef] [PubMed]
  7. J. Cariou, B. Le Jeune, J. Lotran, Y. Guern, “Polarization effects of seawater and underwater targets,” Appl. Opt. 29, 1689–1695 (1990).
    [CrossRef] [PubMed]
  8. J. van Zyl, C. Papas, C. Elachi, “On the optimum polarizations of incoherently reflected waves,” IEEE Trans. Antennas Propag. AP-35, 818–825 (1987).
    [CrossRef]

1994 (1)

S. Sridhar, R. Simon, “Normal form for Mueller matrices,” J. Mod. Opt. 41, 1903–1915 (1994).
[CrossRef]

1993 (1)

C. R. Givens, A. Kostinski, “A simple necessary and sufficient condition on physically realizable Mueller matrices,” J. Mod. Opt. 40, 471–481 (1993).
[CrossRef]

1992 (2)

M. Sanjay Kumar, R. Simon, “Characterization of Mueller matrices in polarization optics,” Opt. Commun. 88, 464–470 (1992).
[CrossRef]

C. V. M. van der Mee, J. W. Hovenier, “Structure of matrices transforming Stokes parameters,” J. Math. Phys. (N.Y.), 33, 3574–3584 (1992).
[CrossRef]

1990 (1)

1988 (1)

1987 (1)

J. van Zyl, C. Papas, C. Elachi, “On the optimum polarizations of incoherently reflected waves,” IEEE Trans. Antennas Propag. AP-35, 818–825 (1987).
[CrossRef]

1979 (1)

Boemer, W.-M.

Cariou, J.

Elachi, C.

J. van Zyl, C. Papas, C. Elachi, “On the optimum polarizations of incoherently reflected waves,” IEEE Trans. Antennas Propag. AP-35, 818–825 (1987).
[CrossRef]

Givens, C. R.

C. R. Givens, A. Kostinski, “A simple necessary and sufficient condition on physically realizable Mueller matrices,” J. Mod. Opt. 40, 471–481 (1993).
[CrossRef]

Guern, Y.

Hovenier, J. W.

C. V. M. van der Mee, J. W. Hovenier, “Structure of matrices transforming Stokes parameters,” J. Math. Phys. (N.Y.), 33, 3574–3584 (1992).
[CrossRef]

Howell, B. J.

James, B.

Kostinski, A.

C. R. Givens, A. Kostinski, “A simple necessary and sufficient condition on physically realizable Mueller matrices,” J. Mod. Opt. 40, 471–481 (1993).
[CrossRef]

A. Kostinski, B. James, W.-M. Boemer, “Optimal reception of partially polarized waves,” J. Opt. Soc. Am. A 5, 58–64 (1988).
[CrossRef]

Le Jeune, B.

Lotran, J.

Papas, C.

J. van Zyl, C. Papas, C. Elachi, “On the optimum polarizations of incoherently reflected waves,” IEEE Trans. Antennas Propag. AP-35, 818–825 (1987).
[CrossRef]

Sanjay Kumar, M.

M. Sanjay Kumar, R. Simon, “Characterization of Mueller matrices in polarization optics,” Opt. Commun. 88, 464–470 (1992).
[CrossRef]

Simon, R.

S. Sridhar, R. Simon, “Normal form for Mueller matrices,” J. Mod. Opt. 41, 1903–1915 (1994).
[CrossRef]

M. Sanjay Kumar, R. Simon, “Characterization of Mueller matrices in polarization optics,” Opt. Commun. 88, 464–470 (1992).
[CrossRef]

Sridhar, S.

S. Sridhar, R. Simon, “Normal form for Mueller matrices,” J. Mod. Opt. 41, 1903–1915 (1994).
[CrossRef]

van der Mee, C. V. M.

C. V. M. van der Mee, J. W. Hovenier, “Structure of matrices transforming Stokes parameters,” J. Math. Phys. (N.Y.), 33, 3574–3584 (1992).
[CrossRef]

van Zyl, J.

J. van Zyl, C. Papas, C. Elachi, “On the optimum polarizations of incoherently reflected waves,” IEEE Trans. Antennas Propag. AP-35, 818–825 (1987).
[CrossRef]

Appl. Opt. (2)

IEEE Trans. Antennas Propag. (1)

J. van Zyl, C. Papas, C. Elachi, “On the optimum polarizations of incoherently reflected waves,” IEEE Trans. Antennas Propag. AP-35, 818–825 (1987).
[CrossRef]

J. Math. Phys. (N.Y.) (1)

C. V. M. van der Mee, J. W. Hovenier, “Structure of matrices transforming Stokes parameters,” J. Math. Phys. (N.Y.), 33, 3574–3584 (1992).
[CrossRef]

J. Mod. Opt. (2)

C. R. Givens, A. Kostinski, “A simple necessary and sufficient condition on physically realizable Mueller matrices,” J. Mod. Opt. 40, 471–481 (1993).
[CrossRef]

S. Sridhar, R. Simon, “Normal form for Mueller matrices,” J. Mod. Opt. 41, 1903–1915 (1994).
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Commun. (1)

M. Sanjay Kumar, R. Simon, “Characterization of Mueller matrices in polarization optics,” Opt. Commun. 88, 464–470 (1992).
[CrossRef]

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Figures (1)

Fig. 1
Fig. 1

Function f(x) corresponding to matrix (24) plotted against x. The function diverges for the values of x equal to the three eigenvalues (asymptotes are marked with vertical dashed lines). Note that in this case the function has only four real solutions (crossings with the horizontal dashed line). x0 is marked with an arrow.

Equations (45)

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

I0,1-p12-p22-p320,
IIQUV=MI=Im00m10m20m03m01m11m21m31m02m12m22m32m03m13m23m331p1p2p3
I0,
P2(p1, p2, p3)(I/I)2-(Q/I)2-(U/I)2-(V/I)20.
δmij=μ=1Nmij(ξμ)ξμ2σμ21/2.
δPmin2=i, j=03Pmin2mij2(δmij)21/2,
P2(p)=a00+2bTp+pTCp,
Aa00bbTC.
aij=qiTGqj,
XCXTD,
P2(π)=a00+2βTπ+πTDπ,
πXp,βXb.
P2(π1, π2, π3)=a00+2i=13βiπi+i=13γiπi2,
βi+γiπi=0,i=1, 2, 3.
γi>0,i=1, 2, 3.
P2-xi=13πi2-1,
πimin=-βi(γi-x),i=1, 2, 3,
γix,i=1, 2, 3.
f(x)i=13 βi2(γi-x)2-1=0,
x1γ1-(β12+β22+β32)1/2,
x2γ1-|β1|.
Pmin2=a00+x0-i=13 βi2(γi-x0),
a00+x0-i=13 βi2(γi-x0)0,
M=0.7599-0.06230.02950.1185-0.05730.4687-0.1811-0.18630.0384-0.17140.53940.02820.1240-0.2168-0.01200.6608.
M=1.0000-0.01180.02790.00010.00450.99560.00130.03500.00120.03410.98380.00830.00920.0178-0.00020.9956.
M=1.00000.07620.13990.02640.07620.78620.3832-0.06150.13990.3832-0.23020.05960.0264-0.06150.05960.4619.
c11=m012-t12,c12=c21=m01m02-t1·t2,
c22=m022-t22,c13=c31=m01m03-t1·t3,
c33=m032-t32, c23=c32=m02m03-t2·t3,
γ3-Tγ2+Sγ-D=0,
c11=μ12,c12=c21=μ1μ2,c22=μ22-τ22,c13=c31=μ1μ3,c33=μ32-τ32,c23=c32=μ2μ3-τ2·τ3.
T=μ12+μ22-τ22+μ32-τ32,
S=-μ12τ22-μ12τ32+(μ22-τ22)(μ32-τ32)-(μ2μ3-τ2·τ3)2,
D=μ12[τ22τ32-(τ2·τ3)2].
c22=μ22-τ220,
c33=μ32-τ320.
c22=μ22-τ220,c33=μ32-τ320orc22=μ22-τ220,c33=μ32-τ320.
c22=μ22-τ220,
c33=μ32-τ320.
(μ2μ3-τ2·τ3)2(μ2μ3-τ2τ3)2,
S-μ12τ22-μ12τ32-(μ3τ2-μ2τ3)20.
S0.
S-μ12(τ22+τ32)+(μ22-τ22)(μ32-τ32).
-μ12<(μ22-τ22)+(μ32-τ32),
S<-τ22(τ22-μ22)-τ32(τ32-μ32)-(τ22τ32-μ22μ32)<0.

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