Abstract

A product decomposition of a nondepolarizing Mueller matrix consisting of the sequence of three factors — a first linear retarder, a horizontal or vertical “retarding diattenuator,” and a second linear retarder — is proposed. Each matrix factor can be readily identified with one or two basic polarization devices such as partial polarizers and retardation waveplates. The decomposition allows for a straightforward interpretation and parameterization of an experimentally determined Mueller matrix in terms of an arrangement of polarization devices and their characteristic parameters: diattenuations, retardances, and axis azimuths. Its application is illustrated on an experimentally determined Mueller matrix.

© 2008 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |

  1. A. Laskarakis, S. Logothetidis, E. Pavlopoulou, and M. Gioti, "Mueller matrix spectroscopic ellipsometry: formulation and application," Thin Solid Films 455-456, 43-49 (2004).
    [CrossRef]
  2. C. Chen, I. An, G. M. Ferreira, N. J. Podraza, J. A. Zapien, and R. W. Collins, "Multichannel Mueller matrix ellipsometer based on dual rotating compensator principle," Thin Solid Films 455-456, 14-23 (2004).
    [CrossRef]
  3. B. Laude-Boulesteix, A. De Martino, B. Drévillon, and L. Schwartz, "Mueller polarimetric imaging system with liquid crystals," Appl. Opt. 43, 2824-2832 (2004).
    [CrossRef] [PubMed]
  4. N. A. Beaudry, Y. Zhao, and R. A. Chipman, "Dielectric tensor measurement from a single Mueller matrix image," J. Opt. Soc. Am. A 24, 814-824 (2007).
    [CrossRef]
  5. R. A. Synowicki, J. N. Hilfiker, and P. K. Whitman, "Mueller matrix ellipsometry study of uniaxial deuterated potassium dihydrogen phosphate (DKDP)," Thin Solid Films 455-456, 624-627 (2004).
    [CrossRef]
  6. J. Gospodyn and J. C. Sit, "Characterization of dielectric columnar thin films by variable angle Mueller matrix and spectroscopic ellipsometry," Opt. Mater. (Amsterdam, Neth.) 29, 318-325 (2006).
    [CrossRef]
  7. N. J. Podraza, C. Chen, I. An, G. M. Fereira, P. I. Rovira, R. Messier, and R. W. Collins, "Analysis of the optical properties and structure of sculptured thin films from spectroscopic Mueller matrix ellipsometry," Thin Solid Films 455-456, 571-575 (2004).
    [CrossRef]
  8. J. E. Wolfe and R. A. Chipman, "Polarimetric characterization of liquid-crystal-on-silicon panels," Appl. Opt. 45, 1688-1703 (2006).
    [CrossRef] [PubMed]
  9. J. W. Hovenier, "Structure of a general pure Mueller matrix," Appl. Opt. 33, 8318-8324 (1994).
    [CrossRef] [PubMed]
  10. Z.-F. Xing, "On the deterministic and non-deterministic Mueller matrix," J. Mol. Opt. 39, 461-484 (1992).
    [CrossRef]
  11. 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]
  12. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1989), Chap. 2, p. 149; Appendix, pp. 488, 491.
  13. P. Lancaster and M. Tismenetsky, The Theory of Matrices (Academic, 1985), Chap. 5, pp. 190,192.
  14. C. Whitney, "Pauli algebraic operators in polarization optics," J. Opt. Soc. Am. 61, 1207-1213 (1971).
    [CrossRef]
  15. S.-Y. Lu and R. A. Chipman, "Homogeneous and inhomogeneous Jones matrices," J. Opt. Soc. Am. A 11, 766-773 (1994).
    [CrossRef]
  16. J. J. Gil and E. Bernabeu, "Obtainment of the polarizing and retardation parameters of a non-depolarizing optical system from the polar decomposition of its Mueller matrix," Optik (Stuttgart) 76, 67-71 (1987).
  17. S.-Y. Lu and R. A. Chipman, "Interpretation of Mueller matrices based on polar decomposition," J. Opt. Soc. Am. A 13, 1106-1113 (1996).
    [CrossRef]
  18. R. Barakat, "Conditions for the physical realizability of polarization matrices characterizing passive systems," J. Mod. Opt. 34, 1535-1544 (1987).
    [CrossRef]
  19. W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in Pascal (Cambridge U. Press, 1989), Chap. 11, pp. 390-391.
  20. S. R. Cloude, "Conditions for the physical realizability of matrix operators in polarimetry," Proc. SPIE 1166, 177-185 (1989).
  21. R. Ossikovski, A. De Martino, and S. Guyot, "Forward and reverse product decompositions of depolarizing Mueller matrices," Opt. Lett. 32, 689-691 (2007).
    [CrossRef] [PubMed]
  22. R. Ossikovski, E. Garcia-Caurel, and A. De Martino, "Product decompositions of experimentally determined non-depolarizing Mueller matrices," Phys. Status. Solidi A (to be published).

2007

2006

J. E. Wolfe and R. A. Chipman, "Polarimetric characterization of liquid-crystal-on-silicon panels," Appl. Opt. 45, 1688-1703 (2006).
[CrossRef] [PubMed]

J. Gospodyn and J. C. Sit, "Characterization of dielectric columnar thin films by variable angle Mueller matrix and spectroscopic ellipsometry," Opt. Mater. (Amsterdam, Neth.) 29, 318-325 (2006).
[CrossRef]

2004

N. J. Podraza, C. Chen, I. An, G. M. Fereira, P. I. Rovira, R. Messier, and R. W. Collins, "Analysis of the optical properties and structure of sculptured thin films from spectroscopic Mueller matrix ellipsometry," Thin Solid Films 455-456, 571-575 (2004).
[CrossRef]

A. Laskarakis, S. Logothetidis, E. Pavlopoulou, and M. Gioti, "Mueller matrix spectroscopic ellipsometry: formulation and application," Thin Solid Films 455-456, 43-49 (2004).
[CrossRef]

C. Chen, I. An, G. M. Ferreira, N. J. Podraza, J. A. Zapien, and R. W. Collins, "Multichannel Mueller matrix ellipsometer based on dual rotating compensator principle," Thin Solid Films 455-456, 14-23 (2004).
[CrossRef]

R. A. Synowicki, J. N. Hilfiker, and P. K. Whitman, "Mueller matrix ellipsometry study of uniaxial deuterated potassium dihydrogen phosphate (DKDP)," Thin Solid Films 455-456, 624-627 (2004).
[CrossRef]

B. Laude-Boulesteix, A. De Martino, B. Drévillon, and L. Schwartz, "Mueller polarimetric imaging system with liquid crystals," Appl. Opt. 43, 2824-2832 (2004).
[CrossRef] [PubMed]

1996

1994

1992

Z.-F. Xing, "On the deterministic and non-deterministic Mueller matrix," J. Mol. Opt. 39, 461-484 (1992).
[CrossRef]

1989

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

1987

J. J. Gil and E. Bernabeu, "Obtainment of the polarizing and retardation parameters of a non-depolarizing optical system from the polar decomposition of its Mueller matrix," Optik (Stuttgart) 76, 67-71 (1987).

R. Barakat, "Conditions for the physical realizability of polarization matrices characterizing passive systems," J. Mod. Opt. 34, 1535-1544 (1987).
[CrossRef]

1971

Appl. Opt.

J. Mod. Opt.

R. Barakat, "Conditions for the physical realizability of polarization matrices characterizing passive systems," J. Mod. Opt. 34, 1535-1544 (1987).
[CrossRef]

J. Mol. Opt.

Z.-F. Xing, "On the deterministic and non-deterministic Mueller matrix," J. Mol. Opt. 39, 461-484 (1992).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Lett.

Opt. Mater. (Amsterdam, Neth.)

J. Gospodyn and J. C. Sit, "Characterization of dielectric columnar thin films by variable angle Mueller matrix and spectroscopic ellipsometry," Opt. Mater. (Amsterdam, Neth.) 29, 318-325 (2006).
[CrossRef]

Optik (Stuttgart)

J. J. Gil and E. Bernabeu, "Obtainment of the polarizing and retardation parameters of a non-depolarizing optical system from the polar decomposition of its Mueller matrix," Optik (Stuttgart) 76, 67-71 (1987).

Proc. SPIE

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

Thin Solid Films

N. J. Podraza, C. Chen, I. An, G. M. Fereira, P. I. Rovira, R. Messier, and R. W. Collins, "Analysis of the optical properties and structure of sculptured thin films from spectroscopic Mueller matrix ellipsometry," Thin Solid Films 455-456, 571-575 (2004).
[CrossRef]

A. Laskarakis, S. Logothetidis, E. Pavlopoulou, and M. Gioti, "Mueller matrix spectroscopic ellipsometry: formulation and application," Thin Solid Films 455-456, 43-49 (2004).
[CrossRef]

C. Chen, I. An, G. M. Ferreira, N. J. Podraza, J. A. Zapien, and R. W. Collins, "Multichannel Mueller matrix ellipsometer based on dual rotating compensator principle," Thin Solid Films 455-456, 14-23 (2004).
[CrossRef]

R. A. Synowicki, J. N. Hilfiker, and P. K. Whitman, "Mueller matrix ellipsometry study of uniaxial deuterated potassium dihydrogen phosphate (DKDP)," Thin Solid Films 455-456, 624-627 (2004).
[CrossRef]

Other

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1989), Chap. 2, p. 149; Appendix, pp. 488, 491.

P. Lancaster and M. Tismenetsky, The Theory of Matrices (Academic, 1985), Chap. 5, pp. 190,192.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in Pascal (Cambridge U. Press, 1989), Chap. 11, pp. 390-391.

R. Ossikovski, E. Garcia-Caurel, and A. De Martino, "Product decompositions of experimentally determined non-depolarizing Mueller matrices," Phys. Status. Solidi A (to be published).

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.


Equations (66)

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

M = M R M D = M D M R ,
M D = T u [ 1 d T d D ] ,
D = 1 d 2 I + ( 1 1 d 2 ) d ̂ d ̂ T ,
M R = [ 1 0 T 0 R ] ,
δ = cos 1 [ 1 2 tr ( M R ) 1 ] ,
a i = 1 2 sin δ j , k = 1 3 ε i j k R j k , i = 1 , 2 , 3 ,
M = [ M 11 b T a N ] ,
M U = [ 1 0 T 0 U ] , M V = [ 1 0 T 0 V ] ,
M U T MM V = [ 1 0 T 0 U T ] [ M 11 b T a N ] [ 1 0 T 0 V ] = [ M 11 b T V U T a U T NV ] = M .
M = [ M 11 ε a 0 0 ε a M 11 0 0 0 0 c f 0 0 f c ] ,
M = M R D [ 1 2 ( T + t ) 1 2 ε ( T t ) 0 0 1 2 ε ( T t ) 1 2 ( T + t ) 0 0 0 0 T t cos δ T t sin δ 0 0 T t sin δ T t cos δ ] ,
T = M 11 + a , t = M 11 a ;
cos δ = f M 11 2 a 2 , sin δ = f M 11 2 a 2
M R D = M D M R = [ 1 2 ( T + t ) 1 2 ε ( T t ) 0 0 1 2 ε ( T t ) 1 2 ( T + t ) 0 0 0 0 T t 0 0 0 0 T t ] [ 1 0 0 0 0 1 0 0 0 0 cos δ sin δ 0 0 sin δ cos δ ] .
M H 1 = [ 1 0 T 0 H 1 ] , M H 2 = [ 1 0 T 0 H 2 ] ,
H i = I 2 h i h i T h i 2 ,
h 1 = a ε a e 1 , h 2 = b ε b e 1 ; e 1 = [ 1 0 0 ] T
M U = M H 1 D , M V = M H 2 D ,
D = [ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ] .
M = M U M R D M V T .
M R D = M ψ Δ 1 2 ( T + t ) [ 1 cos 2 ψ 0 0 cos 2 ψ 1 0 0 0 0 sin 2 ψ cos Δ sin 2 ψ sin Δ 0 0 sin 2 ψ sin Δ sin 2 ψ cos Δ ] ,
cos 2 ψ = ε t T T + t , sin 2 ψ = 2 T t T + t , Δ = δ .
M 2 = M U M 1 M V T ,
M U M V ,
M R = MM D 1 = M D 1 M ,
M = λ 1 M 1 + λ 2 M 2 + λ 3 M 3 + λ 4 M 4 ,
C = j , k = 1 4 M j k ( σ j × σ k * )
M i ( j k ) = 1 4 tr [ C i ( σ j × σ k * ) ] ,
M = [ 0.9997 0.0620 0.0247 0.7699 0.2394 0.3734 0.4407 0.3558 0.3258 0.5146 0.3041 0.3868 0.6550 0.0182 0.3479 0.8410 ] .
M U = [ 1 0 0 0 0 0.3110 0.4233 0.8509 0 0.4233 0.7400 0.5228 0 0.8509 0.5228 0.0510 ] ,
M V T = [ 1 0 0 0 0 0.0802 0.0320 0.9963 0 0.0320 0.9991 0.0295 0 0.9963 0.0295 0.0812 ]
M R D = [ 0.9997 0.7728 0.0000 0.0000 0.7697 0.9942 0.0017 0.0064 0.0000 0.0032 0.6002 0.2141 0.0000 0.0030 0.2165 0.5952 ]
sin 2 ψ = 1 2 ( M 33 2 + M 34 2 + M 43 2 + M 44 2 ) M 11 ,
cos Δ = 1 2 ( M 33 + M 44 ) M 11 sin 2 ψ ,
sin Δ = 1 2 ( M 34 M 43 ) M 11 sin 2 ψ ,
λ 1 M 1 = [ 0.9981 0.0643 0.0240 0.7673 0.2401 0.3751 0.4381 0.3574 0.3246 0.5155 0.3023 0.3885 0.6561 0.0200 0.3464 0.8443 ] .
M U = [ 1 0 0 0 0 0.3116 0.4213 0.8517 0 0.4213 0.7421 0.5213 0 0.8517 0.5213 0.0538 ] ,
M V T = [ 1 0 0 0 0 0.0835 0.0312 0.9960 0 0.0312 0.9991 0.0287 0 0.9960 0.0287 0.0843 ]
M R D = [ 0.9981 0.7704 0.0000 0.0000 0.7704 0.9981 0.0000 0.0000 0.0000 0.0000 0.5963 0.2172 0.0000 0.0000 0.2172 0.5963 ]
M = [ 1 2 ( E 1 + E 2 + E 3 + E 4 ) 1 2 ( E 1 E 2 E 3 + E 4 ) F 13 + F 42 G 13 G 42 1 2 ( E 1 E 2 + E 3 E 4 ) 1 2 ( E 1 + E 2 E 3 E 4 ) F 13 F 42 G 13 + G 42 F 14 + F 32 F 14 F 32 F 12 + F 34 G 12 + G 34 G 14 + G 32 G 14 G 32 G 12 + G 34 F 12 F 34 ] ,
E i = T i 2 , F i j = F j i = Re ( T i * T j ) , G i j = G j i = Im ( T i * T j )
F 14 + F 32 = G 14 + G 32 = 0 , F 13 + F 42 = G 13 + G 42 = 0
ε a = 1 2 ( E 1 E 2 + E 3 E 4 ) , ε b = 1 2 ( E 1 E 2 E 3 + E 4 ) .
T 1 * T 4 = T 3 * T 2 , and T 3 * T 1 = T 2 * T 4 .
( T 1 2 T 2 2 ) T 3 * T 4 = 0 .
M = [ 1 2 ( E 1 + E 2 ) 1 2 ( E 1 E 2 ) 0 0 1 2 ( E 1 E 2 ) 1 2 ( E 1 + E 2 ) 0 0 0 0 F 12 G 12 0 0 G 12 F 12 ] ,
T 1 * T 4 = T 3 * T 2 , T 3 * T 1 = T 2 * T 4 .
M = [ E 1 + E 3 0 0 0 0 E 1 E 3 F 13 F 42 G 13 + G 42 0 F 14 F 32 F 12 + F 34 G 12 + G 34 0 G 14 G 32 G 12 + G 34 F 12 F 34 ] ,
M = ( E 1 + E 3 ) [ 1 0 T 0 N ] ,
r i r j = 0 , i j
r 1 = r 2 = r 3 = 1 ,
( E 1 + E 3 ) 2 r 2 r 3 ( F 14 F 32 ) ( G 14 G 32 ) + ( F 12 + F 34 ) ( G 12 + G 34 ) + ( F 12 F 34 ) ( G 12 + G 34 ) = 2 ( G 32 F 14 + F 32 G 14 ) + 2 ( G 12 F 34 + F 12 G 34 ) = 2 Im ( T 3 * T 2 T 1 * T 4 ) + 2 Im ( T 1 * T 2 T 3 * T 4 ) = 0 ,
( E 1 + E 3 ) 2 r 1 2 ( E 1 E 3 ) 2 + ( F 13 F 42 ) 2 + ( G 13 + G 42 ) 2 = ( E 1 E 3 ) 2 + F 13 2 + G 13 2 + F 42 2 + G 42 2 2 ( F 13 F 42 + G 12 G 42 ) = ( E 1 E 3 ) 2 + E 1 E 3 + E 2 E 4 2 Re ( T 1 * T 3 T 2 * T 4 ) = ( E 1 E 3 ) 2 + 2 E 1 E 3 + 2 Re ( T 1 * T 3 T 3 * T 1 ) = ( E 1 + E 3 ) 2 ,
M U = M V
M R D = ( E 1 + E 3 ) M L H = ( E 1 + E 3 ) [ 1 0 0 0 0 1 0 0 0 0 cos δ sin δ 0 0 sin δ cos δ ] .
R 12 = R 21
H i ( 12 ) = 2 h i ( 1 ) h i ( 2 ) h i 2 = H i ( 21 ) , i = 1 , 2 ,
sin 2 α = a 2 , cos 2 α = a 1 ,
J = J R 1 J D J R 2 + = [ u 1 u 2 * u 2 u 1 * ] [ d 1 0 0 d 2 ] [ v 1 * v 2 * v 2 v 1 ] ,
J L R = [ a + i c i b i b a i c ] ,
J φ = [ e i φ 0 0 e i φ ] ,
J χ = [ e i χ 0 0 e i χ ] ,
J = J R 1 J D J R 2 + = J R 1 J φ J φ + J D J χ J χ + J R 2 + = J U J R D J V + ,
J U J R 1 J φ = [ a 1 + i c 1 i b 1 i b 1 a 1 i c 1 ] ,
J V J R 2 J χ = [ a 2 + i c 2 i b 2 i b 2 a 2 i c 2 ] ,
J R D J φ + J D J χ = [ d 1 e i δ 0 0 d 2 e i δ ] = [ d 1 0 0 d 2 ] [ e i δ 0 0 e i δ ] = J D J δ = J δ J D ,

Metrics