Abstract

We show that the method of maximum-likelihood estimation, recently introduced in the context of quantum process tomography, can be applied to the determination of Mueller matrices characterizing the polarization properties of classical optical systems. Contrary to linear reconstruction algorithms, the proposed method yields physically acceptable Mueller matrices even in the presence of uncontrolled experimental errors. We illustrate our method with the case of an unphysical measured Mueller matrix taken from the literature.

© 2006 Optical Society of America

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  1. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).
  2. R. P. Feynamn, F. L. Vernon Jr., and R. W. Hellwarth, J. Appl. Phys. 28, 49 (1957).
    [CrossRef]
  3. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, reprinted 1st ed. (Cambridge U. Press, 2002).
  4. M. Legré, M. Wegmüller, and N. Gisin, Phys. Rev. Lett. 91, 167902 (2003).
    [CrossRef] [PubMed]
  5. N. Brunner, A. Acín, D. Collins, N. Gisin, and V. Scarani, Phys. Rev. Lett. 91, 180402 (2003).
    [CrossRef] [PubMed]
  6. J. B. Altepeter, D. Branning, E. Jeffrey, T. C. Wei, P. G. Kwiat, R. T. Thew, J. L. O'Brien, M. A. Nielsen, and A. G. White, Phys. Rev. Lett. 90, 193601 (2003).
    [CrossRef] [PubMed]
  7. J. Fiurásek and Z. Hradil, Phys. Rev. A 63, 020101(R) (2001).
    [CrossRef]
  8. M. F. Sacchi, Phys. Rev. A 63, 054104 (2001).
    [CrossRef]
  9. R. L. Kosut, I. Walmsley, and H. Rabitz, arXiv.org e-print archive, quant-ph/0411093.
  10. F. Le Roy-Brehonnet and B. Le Jeune, Prog. Quantum Electron. 21, 109 (1997).
    [CrossRef]
  11. In the current literature there is some confusion in the use of terms such as "nominal" Mueller matrix, "physical" Mueller matrix, etc. Here we adopt the terminology of Ref. .
  12. D. G. M. Anderson and R. Barakat, J. Opt. Soc. Am. A 11, 2305 (1994).
    [CrossRef]
  13. A. V. Gopala Rao, K. S. Mallesh, and Sudha, J. Mod. Opt. 45, 955 (1998).
  14. A. V. Gopala Rao, K. S. Mallesh, and Sudha, J. Mod. Opt. 45, 989 (1998).
  15. The literature on ML data analysis and its limitations is vast even in the classical optics context. Many pertinent papers may be found, e.g., by typing "maximum likelihood" on the web search engine of the Journal of the Optical Society of America A.
  16. D. L. Falkoff and J. E. McDonald, J. Opt. Soc. Am. 41, 862 (1951).
    [CrossRef]
  17. U. Fano, Phys. Rev. 93, 121 (1954).
    [CrossRef]
  18. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, 1st ed. (Cambridge U. Press, 1995).
  19. A. Aiello and J. P. Woerdman, arXiv.org e-print archive, math-ph/0412061 (2004).
  20. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in FORTRAN, 2nd ed. (Cambridge U. Press, 1994).
  21. B. J. Howell, Appl. Opt. 18, 1809 (1979).
    [CrossRef]
  22. A. Peres, Quantum Theory, Concepts and Methods (Kluwer Academic, 1998).

2003

M. Legré, M. Wegmüller, and N. Gisin, Phys. Rev. Lett. 91, 167902 (2003).
[CrossRef] [PubMed]

N. Brunner, A. Acín, D. Collins, N. Gisin, and V. Scarani, Phys. Rev. Lett. 91, 180402 (2003).
[CrossRef] [PubMed]

J. B. Altepeter, D. Branning, E. Jeffrey, T. C. Wei, P. G. Kwiat, R. T. Thew, J. L. O'Brien, M. A. Nielsen, and A. G. White, Phys. Rev. Lett. 90, 193601 (2003).
[CrossRef] [PubMed]

2001

J. Fiurásek and Z. Hradil, Phys. Rev. A 63, 020101(R) (2001).
[CrossRef]

M. F. Sacchi, Phys. Rev. A 63, 054104 (2001).
[CrossRef]

1998

A. V. Gopala Rao, K. S. Mallesh, and Sudha, J. Mod. Opt. 45, 955 (1998).

A. V. Gopala Rao, K. S. Mallesh, and Sudha, J. Mod. Opt. 45, 989 (1998).

1997

F. Le Roy-Brehonnet and B. Le Jeune, Prog. Quantum Electron. 21, 109 (1997).
[CrossRef]

1994

1979

B. J. Howell, Appl. Opt. 18, 1809 (1979).
[CrossRef]

1957

R. P. Feynamn, F. L. Vernon Jr., and R. W. Hellwarth, J. Appl. Phys. 28, 49 (1957).
[CrossRef]

1954

U. Fano, Phys. Rev. 93, 121 (1954).
[CrossRef]

1951

Acín, A.

N. Brunner, A. Acín, D. Collins, N. Gisin, and V. Scarani, Phys. Rev. Lett. 91, 180402 (2003).
[CrossRef] [PubMed]

Aiello, A.

A. Aiello and J. P. Woerdman, arXiv.org e-print archive, math-ph/0412061 (2004).

Altepeter, J. B.

J. B. Altepeter, D. Branning, E. Jeffrey, T. C. Wei, P. G. Kwiat, R. T. Thew, J. L. O'Brien, M. A. Nielsen, and A. G. White, Phys. Rev. Lett. 90, 193601 (2003).
[CrossRef] [PubMed]

Anderson, D. G. M.

Barakat, R.

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

Branning, D.

J. B. Altepeter, D. Branning, E. Jeffrey, T. C. Wei, P. G. Kwiat, R. T. Thew, J. L. O'Brien, M. A. Nielsen, and A. G. White, Phys. Rev. Lett. 90, 193601 (2003).
[CrossRef] [PubMed]

Brunner, N.

N. Brunner, A. Acín, D. Collins, N. Gisin, and V. Scarani, Phys. Rev. Lett. 91, 180402 (2003).
[CrossRef] [PubMed]

Chuang, I. L.

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, reprinted 1st ed. (Cambridge U. Press, 2002).

Collins, D.

N. Brunner, A. Acín, D. Collins, N. Gisin, and V. Scarani, Phys. Rev. Lett. 91, 180402 (2003).
[CrossRef] [PubMed]

Falkoff, D. L.

Fano, U.

U. Fano, Phys. Rev. 93, 121 (1954).
[CrossRef]

Feynamn, R. P.

R. P. Feynamn, F. L. Vernon Jr., and R. W. Hellwarth, J. Appl. Phys. 28, 49 (1957).
[CrossRef]

Fiurásek, J.

J. Fiurásek and Z. Hradil, Phys. Rev. A 63, 020101(R) (2001).
[CrossRef]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in FORTRAN, 2nd ed. (Cambridge U. Press, 1994).

Gisin, N.

N. Brunner, A. Acín, D. Collins, N. Gisin, and V. Scarani, Phys. Rev. Lett. 91, 180402 (2003).
[CrossRef] [PubMed]

M. Legré, M. Wegmüller, and N. Gisin, Phys. Rev. Lett. 91, 167902 (2003).
[CrossRef] [PubMed]

Gopala Rao, A. V.

A. V. Gopala Rao, K. S. Mallesh, and Sudha, J. Mod. Opt. 45, 955 (1998).

A. V. Gopala Rao, K. S. Mallesh, and Sudha, J. Mod. Opt. 45, 989 (1998).

Hellwarth, R. W.

R. P. Feynamn, F. L. Vernon Jr., and R. W. Hellwarth, J. Appl. Phys. 28, 49 (1957).
[CrossRef]

Howell, B. J.

B. J. Howell, Appl. Opt. 18, 1809 (1979).
[CrossRef]

Hradil, Z.

J. Fiurásek and Z. Hradil, Phys. Rev. A 63, 020101(R) (2001).
[CrossRef]

Jeffrey, E.

J. B. Altepeter, D. Branning, E. Jeffrey, T. C. Wei, P. G. Kwiat, R. T. Thew, J. L. O'Brien, M. A. Nielsen, and A. G. White, Phys. Rev. Lett. 90, 193601 (2003).
[CrossRef] [PubMed]

Kosut, R. L.

R. L. Kosut, I. Walmsley, and H. Rabitz, arXiv.org e-print archive, quant-ph/0411093.

Kwiat, P. G.

J. B. Altepeter, D. Branning, E. Jeffrey, T. C. Wei, P. G. Kwiat, R. T. Thew, J. L. O'Brien, M. A. Nielsen, and A. G. White, Phys. Rev. Lett. 90, 193601 (2003).
[CrossRef] [PubMed]

Le Jeune, B.

F. Le Roy-Brehonnet and B. Le Jeune, Prog. Quantum Electron. 21, 109 (1997).
[CrossRef]

Le Roy-Brehonnet, F.

F. Le Roy-Brehonnet and B. Le Jeune, Prog. Quantum Electron. 21, 109 (1997).
[CrossRef]

Legré, M.

M. Legré, M. Wegmüller, and N. Gisin, Phys. Rev. Lett. 91, 167902 (2003).
[CrossRef] [PubMed]

Mallesh, K. S.

A. V. Gopala Rao, K. S. Mallesh, and Sudha, J. Mod. Opt. 45, 955 (1998).

A. V. Gopala Rao, K. S. Mallesh, and Sudha, J. Mod. Opt. 45, 989 (1998).

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, 1st ed. (Cambridge U. Press, 1995).

McDonald, J. E.

Nielsen, M. A.

J. B. Altepeter, D. Branning, E. Jeffrey, T. C. Wei, P. G. Kwiat, R. T. Thew, J. L. O'Brien, M. A. Nielsen, and A. G. White, Phys. Rev. Lett. 90, 193601 (2003).
[CrossRef] [PubMed]

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, reprinted 1st ed. (Cambridge U. Press, 2002).

O'Brien, J. L.

J. B. Altepeter, D. Branning, E. Jeffrey, T. C. Wei, P. G. Kwiat, R. T. Thew, J. L. O'Brien, M. A. Nielsen, and A. G. White, Phys. Rev. Lett. 90, 193601 (2003).
[CrossRef] [PubMed]

Peres, A.

A. Peres, Quantum Theory, Concepts and Methods (Kluwer Academic, 1998).

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in FORTRAN, 2nd ed. (Cambridge U. Press, 1994).

Rabitz, H.

R. L. Kosut, I. Walmsley, and H. Rabitz, arXiv.org e-print archive, quant-ph/0411093.

Sacchi, M. F.

M. F. Sacchi, Phys. Rev. A 63, 054104 (2001).
[CrossRef]

Scarani, V.

N. Brunner, A. Acín, D. Collins, N. Gisin, and V. Scarani, Phys. Rev. Lett. 91, 180402 (2003).
[CrossRef] [PubMed]

Sudha,

A. V. Gopala Rao, K. S. Mallesh, and Sudha, J. Mod. Opt. 45, 989 (1998).

A. V. Gopala Rao, K. S. Mallesh, and Sudha, J. Mod. Opt. 45, 955 (1998).

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in FORTRAN, 2nd ed. (Cambridge U. Press, 1994).

Thew, R. T.

J. B. Altepeter, D. Branning, E. Jeffrey, T. C. Wei, P. G. Kwiat, R. T. Thew, J. L. O'Brien, M. A. Nielsen, and A. G. White, Phys. Rev. Lett. 90, 193601 (2003).
[CrossRef] [PubMed]

Vernon, F. L.

R. P. Feynamn, F. L. Vernon Jr., and R. W. Hellwarth, J. Appl. Phys. 28, 49 (1957).
[CrossRef]

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in FORTRAN, 2nd ed. (Cambridge U. Press, 1994).

Walmsley, I.

R. L. Kosut, I. Walmsley, and H. Rabitz, arXiv.org e-print archive, quant-ph/0411093.

Wegmüller, M.

M. Legré, M. Wegmüller, and N. Gisin, Phys. Rev. Lett. 91, 167902 (2003).
[CrossRef] [PubMed]

Wei, T. C.

J. B. Altepeter, D. Branning, E. Jeffrey, T. C. Wei, P. G. Kwiat, R. T. Thew, J. L. O'Brien, M. A. Nielsen, and A. G. White, Phys. Rev. Lett. 90, 193601 (2003).
[CrossRef] [PubMed]

White, A. G.

J. B. Altepeter, D. Branning, E. Jeffrey, T. C. Wei, P. G. Kwiat, R. T. Thew, J. L. O'Brien, M. A. Nielsen, and A. G. White, Phys. Rev. Lett. 90, 193601 (2003).
[CrossRef] [PubMed]

Woerdman, J. P.

A. Aiello and J. P. Woerdman, arXiv.org e-print archive, math-ph/0412061 (2004).

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, 1st ed. (Cambridge U. Press, 1995).

Appl. Opt.

B. J. Howell, Appl. Opt. 18, 1809 (1979).
[CrossRef]

J. Appl. Phys.

R. P. Feynamn, F. L. Vernon Jr., and R. W. Hellwarth, J. Appl. Phys. 28, 49 (1957).
[CrossRef]

J. Mod. Opt.

A. V. Gopala Rao, K. S. Mallesh, and Sudha, J. Mod. Opt. 45, 955 (1998).

A. V. Gopala Rao, K. S. Mallesh, and Sudha, J. Mod. Opt. 45, 989 (1998).

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Phys. Rev.

U. Fano, Phys. Rev. 93, 121 (1954).
[CrossRef]

Phys. Rev. A

J. Fiurásek and Z. Hradil, Phys. Rev. A 63, 020101(R) (2001).
[CrossRef]

M. F. Sacchi, Phys. Rev. A 63, 054104 (2001).
[CrossRef]

Phys. Rev. Lett.

M. Legré, M. Wegmüller, and N. Gisin, Phys. Rev. Lett. 91, 167902 (2003).
[CrossRef] [PubMed]

N. Brunner, A. Acín, D. Collins, N. Gisin, and V. Scarani, Phys. Rev. Lett. 91, 180402 (2003).
[CrossRef] [PubMed]

J. B. Altepeter, D. Branning, E. Jeffrey, T. C. Wei, P. G. Kwiat, R. T. Thew, J. L. O'Brien, M. A. Nielsen, and A. G. White, Phys. Rev. Lett. 90, 193601 (2003).
[CrossRef] [PubMed]

Prog. Quantum Electron.

F. Le Roy-Brehonnet and B. Le Jeune, Prog. Quantum Electron. 21, 109 (1997).
[CrossRef]

Other

In the current literature there is some confusion in the use of terms such as "nominal" Mueller matrix, "physical" Mueller matrix, etc. Here we adopt the terminology of Ref. .

R. L. Kosut, I. Walmsley, and H. Rabitz, arXiv.org e-print archive, quant-ph/0411093.

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, reprinted 1st ed. (Cambridge U. Press, 2002).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, 1st ed. (Cambridge U. Press, 1995).

A. Aiello and J. P. Woerdman, arXiv.org e-print archive, math-ph/0412061 (2004).

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in FORTRAN, 2nd ed. (Cambridge U. Press, 1994).

The literature on ML data analysis and its limitations is vast even in the classical optics context. Many pertinent papers may be found, e.g., by typing "maximum likelihood" on the web search engine of the Journal of the Optical Society of America A.

A. Peres, Quantum Theory, Concepts and Methods (Kluwer Academic, 1998).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

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Figures (1)

Fig. 1
Fig. 1

Classical polarization tomography (CPT) scheme.

Equations (11)

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

ρ J Tr J .
J i j out = k , l M i j , k l J k l in , i , j , k , l { 0 , 1 } .
p a b ( M ) = Tr [ Π ( a ) J ( b ) out ] Tr J ( b ) in ,
L [ M ] = a , b f a b ln [ p a b ( M ) ] = a , b I a b ln Tr [ Π ( a ) M Π ( b ) ] ,
M = μ , ν 0 , 3 H μ ν [ ϵ ( μ ) ϵ ( ν ) ] ,
p a b ( M ) = μ , ν 0 , 3 H μ ν Tr [ Π ( a ) ϵ ( μ ) Π ( b ) ϵ ( ν ) T ] ,
H = 2 C C Tr ( C C ) ,
C = [ h 1 0 0 0 h 5 + i h 6 h 2 0 0 h 11 + i h 12 h 7 + i h 8 h 3 0 h 15 + i h 16 h 13 + i h 14 h 9 + i h 10 h 4 ] ,
M μ ν = Tr { H opt [ σ ( μ ) σ ( ν ) * ] } ,
M = [ 0.7599 0.0295 0.1185 0.0623 0.0384 0.5394 0.0282 0.1714 0.1240 0.012 0.6608 0.2168 0.0573 0.1811 01863 0.4687 ] .
M ML = [ 0.7599 0.0257 0.1206 0.0576 0.0372 0.5285 0.0001 0.0496 0.1208 0.0001 0.6184 0.1920 0.0554 0.0572 0.1794 0.4822 ] .

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