Abstract

In semiconductor and optics fields, some devices are constructed with layered systems including two or three individual layers. Measurement of polarization properties of the individual components of these layered systems is often desired. In this paper, we present methods allowing the simultaneous extraction of the polarization parameters of the individual components by analyzing spectroscopic Mueller matrices (measured at two wavelengths). We have studied both retarder-retarder and retarder-polarizer-retarder systems. The validities of the methods were successfully tested using both simulations and real polarization systems.

© 2016 Optical Society of America

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References

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  1. D. Goldstein, Polarized Light (Marcel Dekker, 2003).
  2. S. Lu and R. A. Chipman, “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A 13(5), 1106–1113 (1996).
    [Crossref]
  3. J. Morio and F. Goudail, “Influence of the order of diattenuator, retarder, and polarizer in polar decomposition of Mueller matrices,” Opt. Lett. 29(19), 2234–2236 (2004).
    [Crossref] [PubMed]
  4. N. Ghosh, M. F. G. Wood, S. H. Li, R. D. Weisel, B. C. Wilson, R. K. Li, and I. A. Vitkin, “Mueller matrix decomposition for polarized light assessment of biological tissues,” J. Biophotonics 2(3), 145–156 (2009).
    [Crossref] [PubMed]
  5. P. Y. Gerligand, M. Smith, and R. Chipman, “Polarimetric images of a cone,” Opt. Express 4(10), 420–430 (1999).
    [Crossref] [PubMed]
  6. S. N. Savenkov, V. V. Marienko, E. A. Oberemok, and O. Sydoruk, “Generalized matrix equivalence theorem for polarization theory,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(5), 056607 (2006).
    [Crossref] [PubMed]
  7. N. Ortega-Quijano and J. L. Arce-Diego, “Mueller matrix differential decomposition,” Opt. Lett. 36(10), 1942–1944 (2011).
    [Crossref] [PubMed]
  8. J. J. Gil and E. Bernabeu, “Obtainment of the polarizing and retardation parameters of a non-depolarizing optical system from the polaridecomposition jof its Mueller matrix,” Optik (Stuttg.) 76, 67–71 (1987).
  9. R. Ossikovski, “Interpretation of nondepolarizing Mueller matrices based on singular-value decomposition,” J. Opt. Soc. Am. A 25(2), 473–482 (2008).
    [Crossref] [PubMed]
  10. L. Jin and E. Kondoh, “Correction of large birefringent effect of windows for in situ ellipsometry measurements,” Opt. Lett. 39(6), 1549–1552 (2014).
    [Crossref] [PubMed]
  11. R. M. A. Azzam, “Photopolarimetric measurement of the Mueller matrix by Fourier analysis of a single detected signal,” Opt. Lett. 2(6), 148–150 (1978).
    [Crossref] [PubMed]
  12. P. S. Hauge, “Mueller matrix ellipsometry with imperfect compensators,” J. Opt. Soc. Am. 68(11), 1519–1528 (1978).
    [Crossref]
  13. D. H. Goldstein and R. A. Chipman, “Errors analysis of a Mueller matrix polarmeter,” J. Opt. Soc. Am. A 7(4), 693–700 (1990).
    [Crossref]
  14. D. H. Goldstein, R. A. Chipman, and D. B. Chenault, “Infrared Spectropolarimetry,” Opt. Eng. 28(2), 282120 (1989).
    [Crossref]
  15. L. Jin, K. Nara, K. Takizawa, and E. Kondoh, “Dispersion measurement of the electro-optic coefficient r22 of the LiNbO3 crystal with Mueller matrix spectropolarimetry,” Jpn. J. Appl. Phys. 54(7), 078003 (2015).
    [Crossref]
  16. C. G. Broyden, “A new double-rank minimization algorithm,” Not. Am. Math. Soc. 16, 670 (1969).
  17. R. Fletcher, “A new approach to variable metric algorithms,” Comput. J. 13(3), 317–322 (1970).
    [Crossref]
  18. D. Goldfarb, “A family of variable metric updates derived by variational means,” Math. Comput. 24(109), 23–26 (1970).
    [Crossref]
  19. D. F. Shanno, “Conditioning of quasi-Newton methods for function minimization,” Math. Comput. 24, 657 (1970).
    [Crossref]
  20. L. Jin, S. Kasuga, E. Kondoh, and B. Gelloz, “Correction of large retardation window effect for ellipsometry measurements using quasi-Newton method,” Appl. Opt. 54(10), 2991–2998 (2015).
    [Crossref] [PubMed]

2015 (2)

L. Jin, K. Nara, K. Takizawa, and E. Kondoh, “Dispersion measurement of the electro-optic coefficient r22 of the LiNbO3 crystal with Mueller matrix spectropolarimetry,” Jpn. J. Appl. Phys. 54(7), 078003 (2015).
[Crossref]

L. Jin, S. Kasuga, E. Kondoh, and B. Gelloz, “Correction of large retardation window effect for ellipsometry measurements using quasi-Newton method,” Appl. Opt. 54(10), 2991–2998 (2015).
[Crossref] [PubMed]

2014 (1)

2011 (1)

2009 (1)

N. Ghosh, M. F. G. Wood, S. H. Li, R. D. Weisel, B. C. Wilson, R. K. Li, and I. A. Vitkin, “Mueller matrix decomposition for polarized light assessment of biological tissues,” J. Biophotonics 2(3), 145–156 (2009).
[Crossref] [PubMed]

2008 (1)

2006 (1)

S. N. Savenkov, V. V. Marienko, E. A. Oberemok, and O. Sydoruk, “Generalized matrix equivalence theorem for polarization theory,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(5), 056607 (2006).
[Crossref] [PubMed]

2004 (1)

1999 (1)

1996 (1)

1990 (1)

1989 (1)

D. H. Goldstein, R. A. Chipman, and D. B. Chenault, “Infrared Spectropolarimetry,” Opt. Eng. 28(2), 282120 (1989).
[Crossref]

1987 (1)

J. J. Gil and E. Bernabeu, “Obtainment of the polarizing and retardation parameters of a non-depolarizing optical system from the polaridecomposition jof its Mueller matrix,” Optik (Stuttg.) 76, 67–71 (1987).

1978 (2)

1970 (3)

R. Fletcher, “A new approach to variable metric algorithms,” Comput. J. 13(3), 317–322 (1970).
[Crossref]

D. Goldfarb, “A family of variable metric updates derived by variational means,” Math. Comput. 24(109), 23–26 (1970).
[Crossref]

D. F. Shanno, “Conditioning of quasi-Newton methods for function minimization,” Math. Comput. 24, 657 (1970).
[Crossref]

1969 (1)

C. G. Broyden, “A new double-rank minimization algorithm,” Not. Am. Math. Soc. 16, 670 (1969).

Arce-Diego, J. L.

Azzam, R. M. A.

Bernabeu, E.

J. J. Gil and E. Bernabeu, “Obtainment of the polarizing and retardation parameters of a non-depolarizing optical system from the polaridecomposition jof its Mueller matrix,” Optik (Stuttg.) 76, 67–71 (1987).

Broyden, C. G.

C. G. Broyden, “A new double-rank minimization algorithm,” Not. Am. Math. Soc. 16, 670 (1969).

Chenault, D. B.

D. H. Goldstein, R. A. Chipman, and D. B. Chenault, “Infrared Spectropolarimetry,” Opt. Eng. 28(2), 282120 (1989).
[Crossref]

Chipman, R.

Chipman, R. A.

Fletcher, R.

R. Fletcher, “A new approach to variable metric algorithms,” Comput. J. 13(3), 317–322 (1970).
[Crossref]

Gelloz, B.

Gerligand, P. Y.

Ghosh, N.

N. Ghosh, M. F. G. Wood, S. H. Li, R. D. Weisel, B. C. Wilson, R. K. Li, and I. A. Vitkin, “Mueller matrix decomposition for polarized light assessment of biological tissues,” J. Biophotonics 2(3), 145–156 (2009).
[Crossref] [PubMed]

Gil, J. J.

J. J. Gil and E. Bernabeu, “Obtainment of the polarizing and retardation parameters of a non-depolarizing optical system from the polaridecomposition jof its Mueller matrix,” Optik (Stuttg.) 76, 67–71 (1987).

Goldfarb, D.

D. Goldfarb, “A family of variable metric updates derived by variational means,” Math. Comput. 24(109), 23–26 (1970).
[Crossref]

Goldstein, D. H.

D. H. Goldstein and R. A. Chipman, “Errors analysis of a Mueller matrix polarmeter,” J. Opt. Soc. Am. A 7(4), 693–700 (1990).
[Crossref]

D. H. Goldstein, R. A. Chipman, and D. B. Chenault, “Infrared Spectropolarimetry,” Opt. Eng. 28(2), 282120 (1989).
[Crossref]

Goudail, F.

Hauge, P. S.

Jin, L.

Kasuga, S.

Kondoh, E.

Li, R. K.

N. Ghosh, M. F. G. Wood, S. H. Li, R. D. Weisel, B. C. Wilson, R. K. Li, and I. A. Vitkin, “Mueller matrix decomposition for polarized light assessment of biological tissues,” J. Biophotonics 2(3), 145–156 (2009).
[Crossref] [PubMed]

Li, S. H.

N. Ghosh, M. F. G. Wood, S. H. Li, R. D. Weisel, B. C. Wilson, R. K. Li, and I. A. Vitkin, “Mueller matrix decomposition for polarized light assessment of biological tissues,” J. Biophotonics 2(3), 145–156 (2009).
[Crossref] [PubMed]

Lu, S.

Marienko, V. V.

S. N. Savenkov, V. V. Marienko, E. A. Oberemok, and O. Sydoruk, “Generalized matrix equivalence theorem for polarization theory,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(5), 056607 (2006).
[Crossref] [PubMed]

Morio, J.

Nara, K.

L. Jin, K. Nara, K. Takizawa, and E. Kondoh, “Dispersion measurement of the electro-optic coefficient r22 of the LiNbO3 crystal with Mueller matrix spectropolarimetry,” Jpn. J. Appl. Phys. 54(7), 078003 (2015).
[Crossref]

Oberemok, E. A.

S. N. Savenkov, V. V. Marienko, E. A. Oberemok, and O. Sydoruk, “Generalized matrix equivalence theorem for polarization theory,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(5), 056607 (2006).
[Crossref] [PubMed]

Ortega-Quijano, N.

Ossikovski, R.

Savenkov, S. N.

S. N. Savenkov, V. V. Marienko, E. A. Oberemok, and O. Sydoruk, “Generalized matrix equivalence theorem for polarization theory,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(5), 056607 (2006).
[Crossref] [PubMed]

Shanno, D. F.

D. F. Shanno, “Conditioning of quasi-Newton methods for function minimization,” Math. Comput. 24, 657 (1970).
[Crossref]

Smith, M.

Sydoruk, O.

S. N. Savenkov, V. V. Marienko, E. A. Oberemok, and O. Sydoruk, “Generalized matrix equivalence theorem for polarization theory,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(5), 056607 (2006).
[Crossref] [PubMed]

Takizawa, K.

L. Jin, K. Nara, K. Takizawa, and E. Kondoh, “Dispersion measurement of the electro-optic coefficient r22 of the LiNbO3 crystal with Mueller matrix spectropolarimetry,” Jpn. J. Appl. Phys. 54(7), 078003 (2015).
[Crossref]

Vitkin, I. A.

N. Ghosh, M. F. G. Wood, S. H. Li, R. D. Weisel, B. C. Wilson, R. K. Li, and I. A. Vitkin, “Mueller matrix decomposition for polarized light assessment of biological tissues,” J. Biophotonics 2(3), 145–156 (2009).
[Crossref] [PubMed]

Weisel, R. D.

N. Ghosh, M. F. G. Wood, S. H. Li, R. D. Weisel, B. C. Wilson, R. K. Li, and I. A. Vitkin, “Mueller matrix decomposition for polarized light assessment of biological tissues,” J. Biophotonics 2(3), 145–156 (2009).
[Crossref] [PubMed]

Wilson, B. C.

N. Ghosh, M. F. G. Wood, S. H. Li, R. D. Weisel, B. C. Wilson, R. K. Li, and I. A. Vitkin, “Mueller matrix decomposition for polarized light assessment of biological tissues,” J. Biophotonics 2(3), 145–156 (2009).
[Crossref] [PubMed]

Wood, M. F. G.

N. Ghosh, M. F. G. Wood, S. H. Li, R. D. Weisel, B. C. Wilson, R. K. Li, and I. A. Vitkin, “Mueller matrix decomposition for polarized light assessment of biological tissues,” J. Biophotonics 2(3), 145–156 (2009).
[Crossref] [PubMed]

Appl. Opt. (1)

Comput. J. (1)

R. Fletcher, “A new approach to variable metric algorithms,” Comput. J. 13(3), 317–322 (1970).
[Crossref]

J. Biophotonics (1)

N. Ghosh, M. F. G. Wood, S. H. Li, R. D. Weisel, B. C. Wilson, R. K. Li, and I. A. Vitkin, “Mueller matrix decomposition for polarized light assessment of biological tissues,” J. Biophotonics 2(3), 145–156 (2009).
[Crossref] [PubMed]

J. Opt. Soc. Am. (1)

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

Jpn. J. Appl. Phys. (1)

L. Jin, K. Nara, K. Takizawa, and E. Kondoh, “Dispersion measurement of the electro-optic coefficient r22 of the LiNbO3 crystal with Mueller matrix spectropolarimetry,” Jpn. J. Appl. Phys. 54(7), 078003 (2015).
[Crossref]

Math. Comput. (2)

D. Goldfarb, “A family of variable metric updates derived by variational means,” Math. Comput. 24(109), 23–26 (1970).
[Crossref]

D. F. Shanno, “Conditioning of quasi-Newton methods for function minimization,” Math. Comput. 24, 657 (1970).
[Crossref]

Not. Am. Math. Soc. (1)

C. G. Broyden, “A new double-rank minimization algorithm,” Not. Am. Math. Soc. 16, 670 (1969).

Opt. Eng. (1)

D. H. Goldstein, R. A. Chipman, and D. B. Chenault, “Infrared Spectropolarimetry,” Opt. Eng. 28(2), 282120 (1989).
[Crossref]

Opt. Express (1)

Opt. Lett. (4)

Optik (Stuttg.) (1)

J. J. Gil and E. Bernabeu, “Obtainment of the polarizing and retardation parameters of a non-depolarizing optical system from the polaridecomposition jof its Mueller matrix,” Optik (Stuttg.) 76, 67–71 (1987).

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

S. N. Savenkov, V. V. Marienko, E. A. Oberemok, and O. Sydoruk, “Generalized matrix equivalence theorem for polarization theory,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(5), 056607 (2006).
[Crossref] [PubMed]

Other (1)

D. Goldstein, Polarized Light (Marcel Dekker, 2003).

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Figures (6)

Fig. 1
Fig. 1 Two birefringent layer structure
Fig. 2
Fig. 2 Multi-layered system of retarder –polarizer-retarder
Fig. 3
Fig. 3 (Upper) Extracted results of individual retardation and azimuth angle of two retarders train, and measurement results of singular retarders. (Lower) Differences of Δδ and Δθ between the extracted and measured results.
Fig. 4
Fig. 4 (Upper) Extracted results of linear birefringence parameters and transmission angle from the train of retarder-polarizer-retarder, and measurement results of single optical elements. (Lower) Differences (Δδ, Δθ, Δϕ) between the extracted and measured results.
Fig. 5
Fig. 5 Muller matrix elements m12-m44 measured for the train of two retarders
Fig. 6
Fig. 6 Muller matrix elements m12-m44 measured for the train of retarder-polarizer-retarder

Tables (2)

Tables Icon

Table 1 Polarization parameters used for simulation and extracted parameters by using proposed methods.

Tables Icon

Table 2 Polarization parameters used for simulation and extracted parameters for the systems including an achromatic or nearly achromatic retarder and a chromatic retarder.

Equations (12)

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L=( 1 0 0 0 0 1(1cos δ λ ) sin 2 2θ (1cos δ λ )sin2θcos2θ sin δ λ sin2θ 0 (1cos δ λ )sin2θcos2θ 1(1cos δ λ ) cos 2 2θ sin δ λ cos2θ 0 sin δ λ sin2θ sin δ λ cos2θ cos δ λ ),
M= L 2 L 1 =( 1 m 12 m 13 m 14 m 21 m 22 m 23 m 24 m 31 m 32 m 33 m 34 m 41 m 42 m 43 m 44 )=( 1 0 0 0 0 0 M L 3×3 0 ),
m 22,λ =(1C d 2,λ S t 2 2 )(1C d 1,λ S t 1 2 )+C d 2,λ S t 2 C t 2 C d 1,λ S t 1 C t 1 S d 2,λ S t 2 S d 1,λ S t 1 m 23,λ =(1C d 2,λ S t 2 2 )C d 1,λ S t 1 C t 1 +C d 2,λ S t 2 C t 2 (1C d 1,λ C t 1 2 )+S d 2,λ S t 2 S d 1,λ C t 1 m 24,λ =(1C d 2,λ S t 2 2 )S d 1,λ S t 1 +C d 2,λ S t 2 C t 2 S d 1,λ C t 1 S d 2,λ S t 2 (1C d 1,λ ) m 32,λ =C d 2,λ S t 2 C t 2 (1C d 1,λ S t 1 2 )+(1C d 2,λ C t 2 2 )C d 1,λ S t 1 C t 1 +S d 2,λ C t 2 S d 1,λ S t 1 m 33,λ =C d 2,λ S t 2 C t 2 C d 1,λ S t 1 C t 1 +(1C d 2,λ C t 2 2 )(1C d 1,λ C t 1 2 )S d 2,λ C t 2 S d 1,λ C t 1 m 34,λ =C d 2,λ S t 2 C t 2 S d 1,λ S t 1 +(1C d 2,λ C t 2 2 )S d 1,λ C t 1 +S d 2,λ C t 2 (1C d 1,λ ) m 42,λ =S d 2,λ S t 2 (1C d 1,λ S t 1 2 )S d 2,λ C t 2 C d 1,λ S t 1 C t 1 +(1C d 2,λ )S d 1,λ S t 1 m 43,λ =S d 2,λ S t 2 C d 1,λ S t 1 C t 1 S d 2,λ C t 2 (1C d 1,λ C t 1 2 )(1C d 2,λ )S d 1,λ C t 1 m 44,λ =S d 2,λ S t 2 S d 1,λ S t 1 S d 2,λ C t 2 S d 1,λ C t 1 +(1C d 2,λ )(1C d 1,λ ).
e= k=1 2 i=2 4 j=2 4 ( m ij,λk,measured m ij,λk ) 2 ,
δ λ1 =arctan S d λ1 1C d λ1 ,( 180 δ λ1 180 ), δ λ2 =arctan S d λ2 1C d λ2 ,( 180 δ λ2 180 ), θ= 1 2 arctan St Ct ,( 90 θ 90 ).
P= 1 2 ( 1 cos2φ sin2φ 0 cos2φ cos 2 2φ cos2φsin2φ 0 sin2φ cos2φsin2φ sin 2 2φ 0 0 0 0 0 ),
M L2PL1 = L 2 P L 1 = 1 2 ( 1 m 12 m 13 m 14 m 21 m 31 MLP L 3×3 m 41 ),
m 12,λ =cos2φ(1C d 1,λ S t 1 2 )+sin2φ(C d 1 ,λ S t 1 C t 1 ) m 13,λ =cos2φ(C d 1,λ S t 1 C t 1 )+sin2φ(1C d 1,λ C t 1 2 ) m 14,λ =cos2φ(S d 1,λ S t 1 )+sin2φ(S d 1,λ C t 1 ),
m 21,λ =cos2φ(1C d 2,λ S t 3 2 )+sin2φ(C d 2 ,λ S t 2 C t 2 ) m 31,λ =cos2φ(C d 2,λ S t 2 C t 2 )+sin2φ(1C d 2,λ C t 2 2 ) m 41,λ =cos2φ(S d 2,λ S t 2 )sin2φ(S d 2,λ C t 2 ),
e= k=1 2 i=2 4 ( m 1i,λk,measured m 1i,λk ) 2 ,
e= k=1 2 i=2 4 ( m i1,λk,measured m i1,λk ) 2 ,
δ i,λ1 =arccos(1C d i,λ1 ),( 0 δ λ1 180 ), δ i,λ2 =arccos(1C d i,λ2 ),( 0 δ λ2 180 ), θ i = 1 2 arctan S t i C t i ,( 90 θ 90 ), φ= 1 2 arctan sin2φ cos2φ ,( 90 θ 90 ).

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