Abstract

The classification of polarization elements, the polarization affecting optical devices that have a Jones-matrix representation according to the type of eigenvectors they possess, is given a new visit through the group-theoretical connection of polarization elements. The diattenuators and retarders are recognized as the elements corresponding to boosts and rotations, respectively. The structure of homogeneous elements other than diattenuators and retarders are identified by giving the quaternion corresponding to these elements. The set of degenerate polarization elements is identified with the so-called null elements of the Lorentz group. Singular polarization elements are examined in their more illustrative Mueller-matrix representation, and, finally the eigenstructure of a special class of singular Mueller matrices is studied.

© 2001 Optical Society of America

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References

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  1. W. A. Shurcliff, Polarized Light, Production and Use (Harvard U. Press, Cambridge, Mass., 1962).
  2. S.-Y. Lu, R. A. Chipman, “Homogeneous and inhomogeneous Jones matrices,” J. Opt. Soc. Am. A 11, 766–773 (1994).
    [CrossRef]
  3. R. Barakat, “Bilinear constraints between elements of the 4×4 Mueller–Jones transfer matrix of polarization theory,” Opt. Commun. 38, 159–161 (1981).
    [CrossRef]
  4. S. R. Cloude, “Group theory and polarisation algebra,” Optik 75, 26–36 (1986).
  5. C. V. M. van der Mee, “An eigenvalue criterion for the matrices transforming Jones matrices,” J. Math. Phys. 34, 5072–5088 (1993).
    [CrossRef]
  6. R. Sridhar, R. Simon, “Normal form for Mueller matrices in polarization optics,” J. Mod. Opt. 41, 1903–1915 (1994).
    [CrossRef]
  7. C. S. Brown, A. E. Bak, “Unified formalism for polarization optics with application to polarimetry on a twisted optical fiber,” Opt. Eng. 35, 1624–1635 (1995).
  8. D. Han, Y. S. Kim, M. E. Noz, “Polarization optics and bilinear representation of the Lorentz group,” Phys. Lett. A 219, 26–32 (1996).
    [CrossRef]
  9. K. N. Srinivasa Rao, Linear Algebra and Group Theory for Physicists (Wiley, New York, 1996).
  10. P. Lancaster, Tismenetsky, The Theory of Matrices, 2nd ed. (Academic, San Diego, Calif., 1985).
  11. Sudha, Some Algebraic Aspects of Relativity and Polarization Optics, Ph.D. thesis (available from the author at the address on the title page).
  12. S.-Y. Lu, R. A. Chipman, “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A 13, 1106–1113 (1996).
    [CrossRef]
  13. S.-Y. Lu, R. A. Chipman, “Mueller matrices and the degree of polarization,” Opt. Commun. 146, 11–14 (1998).
    [CrossRef]

1998 (1)

S.-Y. Lu, R. A. Chipman, “Mueller matrices and the degree of polarization,” Opt. Commun. 146, 11–14 (1998).
[CrossRef]

1996 (2)

D. Han, Y. S. Kim, M. E. Noz, “Polarization optics and bilinear representation of the Lorentz group,” Phys. Lett. A 219, 26–32 (1996).
[CrossRef]

S.-Y. Lu, R. A. Chipman, “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A 13, 1106–1113 (1996).
[CrossRef]

1995 (1)

C. S. Brown, A. E. Bak, “Unified formalism for polarization optics with application to polarimetry on a twisted optical fiber,” Opt. Eng. 35, 1624–1635 (1995).

1994 (2)

S.-Y. Lu, R. A. Chipman, “Homogeneous and inhomogeneous Jones matrices,” J. Opt. Soc. Am. A 11, 766–773 (1994).
[CrossRef]

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

1993 (1)

C. V. M. van der Mee, “An eigenvalue criterion for the matrices transforming Jones matrices,” J. Math. Phys. 34, 5072–5088 (1993).
[CrossRef]

1986 (1)

S. R. Cloude, “Group theory and polarisation algebra,” Optik 75, 26–36 (1986).

1981 (1)

R. Barakat, “Bilinear constraints between elements of the 4×4 Mueller–Jones transfer matrix of polarization theory,” Opt. Commun. 38, 159–161 (1981).
[CrossRef]

Bak, A. E.

C. S. Brown, A. E. Bak, “Unified formalism for polarization optics with application to polarimetry on a twisted optical fiber,” Opt. Eng. 35, 1624–1635 (1995).

Barakat, R.

R. Barakat, “Bilinear constraints between elements of the 4×4 Mueller–Jones transfer matrix of polarization theory,” Opt. Commun. 38, 159–161 (1981).
[CrossRef]

Brown, C. S.

C. S. Brown, A. E. Bak, “Unified formalism for polarization optics with application to polarimetry on a twisted optical fiber,” Opt. Eng. 35, 1624–1635 (1995).

Chipman, R. A.

Cloude, S. R.

S. R. Cloude, “Group theory and polarisation algebra,” Optik 75, 26–36 (1986).

Han, D.

D. Han, Y. S. Kim, M. E. Noz, “Polarization optics and bilinear representation of the Lorentz group,” Phys. Lett. A 219, 26–32 (1996).
[CrossRef]

Kim, Y. S.

D. Han, Y. S. Kim, M. E. Noz, “Polarization optics and bilinear representation of the Lorentz group,” Phys. Lett. A 219, 26–32 (1996).
[CrossRef]

Lancaster, P.

P. Lancaster, Tismenetsky, The Theory of Matrices, 2nd ed. (Academic, San Diego, Calif., 1985).

Lu, S.-Y.

Noz, M. E.

D. Han, Y. S. Kim, M. E. Noz, “Polarization optics and bilinear representation of the Lorentz group,” Phys. Lett. A 219, 26–32 (1996).
[CrossRef]

Shurcliff, W. A.

W. A. Shurcliff, Polarized Light, Production and Use (Harvard U. Press, Cambridge, Mass., 1962).

Simon, R.

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

Sridhar, R.

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

Srinivasa Rao, K. N.

K. N. Srinivasa Rao, Linear Algebra and Group Theory for Physicists (Wiley, New York, 1996).

Sudha,

Sudha, Some Algebraic Aspects of Relativity and Polarization Optics, Ph.D. thesis (available from the author at the address on the title page).

Tismenetsky,

P. Lancaster, Tismenetsky, The Theory of Matrices, 2nd ed. (Academic, San Diego, Calif., 1985).

van der Mee, C. V. M.

C. V. M. van der Mee, “An eigenvalue criterion for the matrices transforming Jones matrices,” J. Math. Phys. 34, 5072–5088 (1993).
[CrossRef]

J. Math. Phys. (1)

C. V. M. van der Mee, “An eigenvalue criterion for the matrices transforming Jones matrices,” J. Math. Phys. 34, 5072–5088 (1993).
[CrossRef]

J. Mod. Opt. (1)

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

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

Opt. Commun. (2)

S.-Y. Lu, R. A. Chipman, “Mueller matrices and the degree of polarization,” Opt. Commun. 146, 11–14 (1998).
[CrossRef]

R. Barakat, “Bilinear constraints between elements of the 4×4 Mueller–Jones transfer matrix of polarization theory,” Opt. Commun. 38, 159–161 (1981).
[CrossRef]

Opt. Eng. (1)

C. S. Brown, A. E. Bak, “Unified formalism for polarization optics with application to polarimetry on a twisted optical fiber,” Opt. Eng. 35, 1624–1635 (1995).

Optik (1)

S. R. Cloude, “Group theory and polarisation algebra,” Optik 75, 26–36 (1986).

Phys. Lett. A (1)

D. Han, Y. S. Kim, M. E. Noz, “Polarization optics and bilinear representation of the Lorentz group,” Phys. Lett. A 219, 26–32 (1996).
[CrossRef]

Other (4)

K. N. Srinivasa Rao, Linear Algebra and Group Theory for Physicists (Wiley, New York, 1996).

P. Lancaster, Tismenetsky, The Theory of Matrices, 2nd ed. (Academic, San Diego, Calif., 1985).

Sudha, Some Algebraic Aspects of Relativity and Polarization Optics, Ph.D. thesis (available from the author at the address on the title page).

W. A. Shurcliff, Polarized Light, Production and Use (Harvard U. Press, Cambridge, Mass., 1962).

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Tables (2)

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Table 1 Nonsingular Polarization Elements

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Table 2 Singular Polarization Elements

Equations (26)

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Tr=q0-iq3-iq1-q2-iq1+q2q0+iq3,
X1=1a1+ia2a3-1;X2=1a1+ia2a3+1,
Tb=q0-iq3-iq1-q2-iq1+q2q0+iq3,
X1=1n1+in2n3+1;X2=1n1+in2n3-1,
Mb=A(TbTb*)A-1;A=1001100-101100i-i0,
S1=A(X1X1*)={1,n3,n1,n2},
S2=A(X2X2*)={1,-n3,-n1,-n2}.
S1={1,-a3,-a1,-a2},S2={1,a3,a1,a2},
T=TrTb=TbTr.
q=qrqb;q=(q0, q);
q0=cosθr2coshθb2-i sinθr2sinhθb2,
q=cosθr2sinhθb2+sinθr2coshθb2aˆ.
SYX˜-XY˜;X˜Y=0,
qn=(q0, q);q12+q22+q32+q02=1,
q12+q22+q32=0.
X0=1-iq3iq1+q2,
M=m00XY˜G;X={1,x,y,z};
X˜GX=0,Y={1,-p,-q,-r};
Y˜GY=0,G=diag(1,-1,-1,-1).
λ1=m00(1+px+qy+rz),λ2=0,0,0.
M1=m00XY˜G;X˜GX=0,
Y˜GY>0,G=diag(1,-1,-1,-1).
M2=mooXY˜G;X˜GX>0,
Y˜GY=0,G=diag(1,-1,-1,-1).
M3=m00XY˜G;X˜GX>0,
Y˜GY>0,G=diag(1,-1,-1,-1).

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