Abstract

The statistical ensemble formalism of Kim et al. [J. Opt. Soc. Am. A 4, 433 (1987)] offers a realistic model for characterizing the effect of stochastic nonimage-forming optical media on the state of polarization of transmitted light. With suitable choice of the Jones ensemble, various Mueller transformations—some of which are hitherto unknown—are deduced. It is observed that the ensemble approach is formally identical to the positive-operator-valued measures (POVMs) on the quantum density matrix. This observation, in combination with the recent suggestion by Ahnert and Payne [Phys. Rev. A 71, 012330–1 (2005)] —in the context of generalized quantum measurement on single photon polarization states—that linear optics elements can be employed in setting up all possible POVMs enables us to propose a way of realizing different types of Mueller devices.

© 2008 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |

  1. K. Kim, L. Mandel, and E. Wolf, “Relationship between Jones and Mueller matrices for random media,” J. Opt. Soc. Am. A 4, 433-437 (1987).
    [CrossRef]
  2. S. E. Ahnert and M. C. Payne, “General implementation of all possible positive-operator-value measurements of single photon polarization states,” Phys. Rev. A 71, 012330-1-4 (2005).
    [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. R. Simon, “The connection between Mueller and Jones matrices of polarization optics,” Opt. Commun. 42, 293-297 (1982).
    [CrossRef]
  5. A. B. Kostinski, B. James, and W. M. Boerner, “Optimal reception of partially polarized waves,” J. Opt. Soc. Am. A 5, 58-64 (1988).
    [CrossRef]
  6. A. B. Kostinski, “Depolarization criterion for incoherent scattering,” Appl. Opt. 31, 3506-3508 (1992).
    [CrossRef] [PubMed]
  7. J. J. Gil and E. Bernabeu, “A depolarization criterion in Mueller matrices,” Opt. Acta 32, 259-261 (1985).
    [CrossRef]
  8. R. Simon, “Mueller matrices and depolarization criteria,” J. Mod. Opt. 34, 569-575 (1987).
    [CrossRef]
  9. R. Simon, “Non-depolarizing systems and degree of polarization,” Opt. Commun. 77, 349-354 (1990)
    [CrossRef]
  10. M. Sanjay Kumar and R. Simon, “Characterization of Mueller matrices in polarization optics,” Opt. Commun. 88, 464-470 (1992).
    [CrossRef]
  11. R. Sridhar and R. Simon, “Normal form for Mueller matrices in polarization optics,” J. Mod. Opt. 41, 1903-1915 (1994).
    [CrossRef]
  12. D. G. M. Anderson and R. Barakat, “Necessary and sufficient conditions for a Mueller matrix to be derivable from a Jones matrix,” J. Opt. Soc. Am. A 11, 2305-2319 (1994).
    [CrossRef]
  13. C. V. M. van der Mee and J. W. Hovenier, “Structure of matrices transforming Stokes parameters,” J. Math. Phys. 33, 3574-3584 (1992).
    [CrossRef]
  14. C. R. Givens and A. B. Kostinski, “A simple necessary and sufficient criterion on physically realizable Mueller matrices,” J. Mod. Opt. 40, 471-481 (1993).
    [CrossRef]
  15. C. V. M. van der Mee, “An eigenvalue criterion for matrices transforming Stokes parameters,” J. Math. Phys. 34, 5072-5088 (1993).
    [CrossRef]
  16. S. R. Cloude, “Group theory and polarization algebra,” Optik (Stuttgart) 75, 26-36 (1986).
  17. A. V. Gopala Rao, K. S. Mallesh, and Sudha, “On the algebraic characterization of a Mueller matrix in polarization optics I. Identifying a Mueller matrix from its N matrix,” J. Mod. Opt. 45, 955-987 (1998).
    [CrossRef]
  18. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977).
  19. A real 4-vector is called timelike, spacelike, and null, respectively, if Xt˜GXt>0, Xs˜GXs<0, and Xn˜GXn=0. Here we follow the conventions of K. N. Srinivasa Rao, The Rotation and Lorentz Groups and Their Representations for Physicists (Wiley, 1988).
  20. A linear combination of Mueller matrices Me with nonnegative coefficients pe is also a Mueller matrix. This is because each Mueller matrix {Me} transforms an initial Stokes vector into a final Stokes vector, and a linear combination of Stokes vectors with nonnegative coefficients pe is again a Stokes vector.
  21. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge U. Press, 2002).
  22. L. Mandel and E. Wolf, Quantum Coherence and Quantum Optics (Cambridge U. Press, 1995).
  23. S. R. Cloude, “Conditions for physical realizability of matrix operators in polarimetry,” Proc. SPIE 1166, 177-185 (1989).

2005 (1)

S. E. Ahnert and M. C. Payne, “General implementation of all possible positive-operator-value measurements of single photon polarization states,” Phys. Rev. A 71, 012330-1-4 (2005).
[CrossRef]

1998 (1)

A. V. Gopala Rao, K. S. Mallesh, and Sudha, “On the algebraic characterization of a Mueller matrix in polarization optics I. Identifying a Mueller matrix from its N matrix,” J. Mod. Opt. 45, 955-987 (1998).
[CrossRef]

1994 (2)

1993 (2)

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

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

1992 (3)

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

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

A. B. Kostinski, “Depolarization criterion for incoherent scattering,” Appl. Opt. 31, 3506-3508 (1992).
[CrossRef] [PubMed]

1990 (1)

R. Simon, “Non-depolarizing systems and degree of polarization,” Opt. Commun. 77, 349-354 (1990)
[CrossRef]

1989 (1)

S. R. Cloude, “Conditions for physical realizability of matrix operators in polarimetry,” Proc. SPIE 1166, 177-185 (1989).

1988 (1)

1987 (2)

1986 (1)

S. R. Cloude, “Group theory and polarization algebra,” Optik (Stuttgart) 75, 26-36 (1986).

1985 (1)

J. J. Gil and E. Bernabeu, “A depolarization criterion in Mueller matrices,” Opt. Acta 32, 259-261 (1985).
[CrossRef]

1982 (1)

R. Simon, “The connection between Mueller and Jones matrices of polarization optics,” Opt. Commun. 42, 293-297 (1982).
[CrossRef]

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]

Appl. Opt. (1)

J. Math. Phys. (2)

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

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

J. Mod. Opt. (4)

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

A. V. Gopala Rao, K. S. Mallesh, and Sudha, “On the algebraic characterization of a Mueller matrix in polarization optics I. Identifying a Mueller matrix from its N matrix,” J. Mod. Opt. 45, 955-987 (1998).
[CrossRef]

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

R. Simon, “Mueller matrices and depolarization criteria,” J. Mod. Opt. 34, 569-575 (1987).
[CrossRef]

J. Opt. Soc. Am. A (3)

Opt. Acta (1)

J. J. Gil and E. Bernabeu, “A depolarization criterion in Mueller matrices,” Opt. Acta 32, 259-261 (1985).
[CrossRef]

Opt. Commun. (4)

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

R. Simon, “The connection between Mueller and Jones matrices of polarization optics,” Opt. Commun. 42, 293-297 (1982).
[CrossRef]

R. Simon, “Non-depolarizing systems and degree of polarization,” Opt. Commun. 77, 349-354 (1990)
[CrossRef]

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

Optik (Stuttgart) (1)

S. R. Cloude, “Group theory and polarization algebra,” Optik (Stuttgart) 75, 26-36 (1986).

Phys. Rev. A (1)

S. E. Ahnert and M. C. Payne, “General implementation of all possible positive-operator-value measurements of single photon polarization states,” Phys. Rev. A 71, 012330-1-4 (2005).
[CrossRef]

Proc. SPIE (1)

S. R. Cloude, “Conditions for physical realizability of matrix operators in polarimetry,” Proc. SPIE 1166, 177-185 (1989).

Other (5)

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977).

A real 4-vector is called timelike, spacelike, and null, respectively, if Xt˜GXt>0, Xs˜GXs<0, and Xn˜GXn=0. Here we follow the conventions of K. N. Srinivasa Rao, The Rotation and Lorentz Groups and Their Representations for Physicists (Wiley, 1988).

A linear combination of Mueller matrices Me with nonnegative coefficients pe is also a Mueller matrix. This is because each Mueller matrix {Me} transforms an initial Stokes vector into a final Stokes vector, and a linear combination of Stokes vectors with nonnegative coefficients pe is again a Stokes vector.

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge U. Press, 2002).

L. Mandel and E. Wolf, Quantum Coherence and Quantum Optics (Cambridge U. Press, 1995).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (1)

Fig. 1
Fig. 1

Linear optics scheme of Ahnert and Payne [2] corresponding to a two-element Jones assembly. The light beam, characterized by the Stokes vector S, enters at the bottom left corner and exits at 1 and 2. The arrangement corresponds to a Mueller matrix M affecting the polarization state of the incident light beam according to the relation S = MS , where S specifies the polarization state of the output beam and the Mueller matrix M is the result of a two-element Jones ensemble corresponding to a two element POVM { V i , i = 1 , 2 . } of Eq. (13). Here U, U , U are Jones devices corresponding to arbitrary 2 × 2 unitary matrices; P, P are phase-shifter devices with angles γ and ξ, respectively; R, R , R , r, r , r denote polarization rotators, where r, R correspond to variable angles of rotation θ, ϕ, respectively; and the remaining polarization rotators R , R , r , r have π 2 , π, π 2 , and π as angles of rotation. B and M, respectively, denote polarizing beam splitters and mirrors.

Tables (1)

Tables Icon

Table 1 Mueller Matrices Resulting from 2-element Jones Ensemble

Equations (31)

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

C = E E .
σ 1 = [ 1 0 0 1 ] , σ 2 = [ 0 1 1 0 ] , σ 3 = [ 0 i i 0 ]
σ 0 = [ 1 0 0 1 ] ,
C = 1 2 i = 0 3 s i σ i = 1 2 [ s 0 + s 1 s 2 i s 3 s 2 + i s 3 s 0 s 1 ] .
s 0 = Tr ( C σ 0 ) = intensity of the beam ,
s i = Tr ( C σ i ) = components of polarization vector s of the beam.
s 0 > 0 , s 0 2 s 2 0 .
C = JCJ .
S = MS
s i = Tr ( C σ i ) = Tr ( JCJ σ i ) = 1 2 j = 0 3 Tr ( J σ i J σ j ) s j ,
M i j = 1 2 Tr ( J σ i J σ j )
M = [ M 00 M 01 M 02 M 03 M 10 M 11 M 12 M 13 M 20 M 21 M 22 M 23 M 30 M 31 M 32 M 33 ] ,
N = M ̃ GM ,
C a v = e p e ( J e CJ e ) = e p e C e
C a v = 1 n e = 1 n J e CJ e .
ρ = i = 1 n V i ρ V i = i p i ρ i , ρ i = V i ρ V i p i ; p i = Tr [ V i ρ V i ] ,
ρ = ρ H H H H + ρ H V H V + ρ H V * V H + ρ V V V V ,
C = ( a ̂ H a ̂ H a ̂ H a ̂ V a ̂ V a ̂ H a ̂ V a ̂ V ) ,
s 0 = S ̂ 0 = ( a ̂ H a ̂ H + a ̂ V a ̂ V ) = ρ H H + ρ V V = Tr ( ρ ) ,
s 1 = S ̂ 1 = ( a ̂ H a ̂ H a ̂ V a ̂ V ) = ρ H H ρ V V = Tr ( ρ σ 1 ) ,
s 2 = S ̂ 2 = a ̂ H a ̂ V + a ̂ V a ̂ H ) = ρ H V + ρ H V * = Tr ( ρ σ 2 ) ,
s 3 = S ̂ 3 = i ( a ̂ H a ̂ V a ̂ V a ̂ H ) = i ( ρ H V ρ H V * ) = Tr ( ρ σ 3 ) .
V 1 = U D 1 U , V 2 = U D 2 U .
D 1 = [ e i γ cos θ 0 0 cos ϕ ] , D 2 = [ e i ξ sin θ 0 0 sin ϕ ] ,
i = 1 , 2 V i V i = U D 1 D 1 U + U D 2 D 2 U = I ,
M = 1 2 [ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 ] ,
U = I = [ 1 0 0 1 ] , U = [ 0 i i 0 ] ,
U = [ i 0 0 i ]
M = 1 2 [ 1 0 0 0 1 2 0 0 1 2 0 0 1 2 0 0 1 2 0 0 ] ,
U = 1 2 [ 1 i i 1 ] , U = [ i 0 0 i ] ,
U = [ 0 1 1 0 ]

Metrics