Abstract

In this article, we address the question of significance of the parameters of differential Mueller matrix formalism. We show how the concept of mean value and uncertainty of the optical properties recently introduced to depict this differential matrix can be related to the random fluctuations of these optical properties. From the layered-medium interpretation introduced by Jones [J. Opt. Soc. Am. 38, 671 (1948)] and extended to Mueller–Jones matrix by Azzam [J. Opt. Soc. Am. 68, 1756 (1978)], a generalization to depolarizing Mueller matrices is proposed. Based on the random Mueller–Jones matrix approach, the obtained parameterization perfectly fits the previous results from the literature. Necessary conditions of positivity on specific coefficients imposed in order to have physical Mueller matrix are introduced in a natural way and not inferred a posteriori. Interpretations of the underlying physical processes are also presented. An illustrative experimental example is provided from literature data.

© 2013 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. C. Jones, “A new calculus for the treatment of optical systems. VII. Properties of the N-matrices,” J. Opt. Soc. Am. 38, 671–685 (1948).
    [CrossRef]
  2. R. M. A. Azzam, “Propagation of partially polarized light through anisotropic media with or without depolarization: a differential 4×4 matrix calculus,” J. Opt. Soc. Am. 68, 1756–1767 (1978).
    [CrossRef]
  3. R. Barakat, “Exponential versions of the Jones and Mueller–Jones polarization matrices,” J. Opt. Soc. Am. A 13, 158–163 (1996).
    [CrossRef]
  4. R. Ossikovski, “Differential matrix formalism for depolarizing anisotropic media,” Opt. Lett. 36, 2330–2332 (2011).
    [CrossRef]
  5. N. Ortega-Quijano and J. L. Arce-Diego, “Mueller matrix differential decomposition,” Opt. Lett. 36, 1942–1944 (2011).
    [CrossRef]
  6. N. Ortega-Quijano and J. L. Arce-Diego, “Depolarizing differential Mueller matrices,” Opt. Lett. 36, 2429–2431 (2011).
    [CrossRef]
  7. A. Chipman, Handbook of Optics, 3rd ed. (McGraw-Hill, 2009), Vol. 1.
  8. H. D. Noble and R. A. Chipman, “Mueller matrix roots algorithm and computational considerations,” Opt. Express 20, 17–31 (2012).
    [CrossRef]
  9. R. Ossikovski, “Differential and product Mueller matrix decompositions: a formal comparison,” Opt. Lett. 37, 220–222 (2012).
    [CrossRef]
  10. T. Germer, “Realizable differential matrices for depolarizing media,” Opt. Lett. 37, 921–923 (2012).
    [CrossRef]
  11. B. N. Simon, S. Simon, N. Mukunda, F. Gori, M. Santarsiero, R. Borghi, and R. Simon, “A complete characterization of pre-Mueller and Mueller matrices in polarization optics,” J. Opt. Soc. Am. A 27, 188–199 (2010).
    [CrossRef]
  12. V. Devlaminck, P. Terrier, and J. M. Charbois, “Differential matrix physically admissible for depolarizing media: the case of diagonal matrices,” Opt. Lett. 38, 1497–1499 (2013).
    [CrossRef]
  13. V. Devlaminck, P. Terrier, and J. M. Charbois, “A physically admissible parameterization for differential Mueller matrix of uniform media,” Opt. Lett. 38, 1410–1412 (2013).
    [CrossRef]
  14. N. Ortega-Quijano, B. Haj-Ibrahim, E. García-Caurel, J. Arce-Diego, and R. Ossikovski, “Experimental validation of Mueller matrix differential decomposition,” Opt. Express 20, 1151–1163 (2012).
    [CrossRef]
  15. N. Ghosh, M. F. G. Wood, S. Li, R. D. Weisel, B. C. Wilson, R. Li, and I. A. Vitkin, “Mueller matrix decomposition for polarized light assessment of biological tissues,” J. Biophotonics 2, 145–156 (2009).
    [CrossRef]
  16. 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]
  17. S. Manhas, M. K. Swami, P. Buddhiwant, N. Ghosh, P. K. Gupta, and K. Singh, “Mueller matrix approach for determination of optical rotation in chiral turbid media in backscattering geometry,” Opt. Express 14, 190–202 (2006).
    [CrossRef]
  18. S. Kumar, H. Purwar, R. Ossikovski, I. A. Vitkin, and N. Ghosh, “Comparative study of differential matrix and extended polar decomposition formalisms for polarimetric characterization of complex tissue-like turbid media,” J. Biomed. Opt. 17, 105006 (2012).
    [CrossRef]

2013 (2)

2012 (5)

2011 (3)

2010 (1)

2009 (1)

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

2006 (1)

1996 (2)

1978 (1)

1948 (1)

Arce-Diego, J.

Arce-Diego, J. L.

Azzam, R. M. A.

Barakat, R.

Borghi, R.

Buddhiwant, P.

Charbois, J. M.

Chipman, A.

A. Chipman, Handbook of Optics, 3rd ed. (McGraw-Hill, 2009), Vol. 1.

Chipman, R. A.

Devlaminck, V.

García-Caurel, E.

Germer, T.

Ghosh, N.

S. Kumar, H. Purwar, R. Ossikovski, I. A. Vitkin, and N. Ghosh, “Comparative study of differential matrix and extended polar decomposition formalisms for polarimetric characterization of complex tissue-like turbid media,” J. Biomed. Opt. 17, 105006 (2012).
[CrossRef]

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

S. Manhas, M. K. Swami, P. Buddhiwant, N. Ghosh, P. K. Gupta, and K. Singh, “Mueller matrix approach for determination of optical rotation in chiral turbid media in backscattering geometry,” Opt. Express 14, 190–202 (2006).
[CrossRef]

Gori, F.

Gupta, P. K.

Haj-Ibrahim, B.

Jones, R. C.

Kumar, S.

S. Kumar, H. Purwar, R. Ossikovski, I. A. Vitkin, and N. Ghosh, “Comparative study of differential matrix and extended polar decomposition formalisms for polarimetric characterization of complex tissue-like turbid media,” J. Biomed. Opt. 17, 105006 (2012).
[CrossRef]

Li, R.

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

Li, S.

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

Lu, S. Y.

Manhas, S.

Mukunda, N.

Noble, H. D.

Ortega-Quijano, N.

Ossikovski, R.

Purwar, H.

S. Kumar, H. Purwar, R. Ossikovski, I. A. Vitkin, and N. Ghosh, “Comparative study of differential matrix and extended polar decomposition formalisms for polarimetric characterization of complex tissue-like turbid media,” J. Biomed. Opt. 17, 105006 (2012).
[CrossRef]

Santarsiero, M.

Simon, B. N.

Simon, R.

Simon, S.

Singh, K.

Swami, M. K.

Terrier, P.

Vitkin, I. A.

S. Kumar, H. Purwar, R. Ossikovski, I. A. Vitkin, and N. Ghosh, “Comparative study of differential matrix and extended polar decomposition formalisms for polarimetric characterization of complex tissue-like turbid media,” J. Biomed. Opt. 17, 105006 (2012).
[CrossRef]

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

Weisel, R. D.

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

Wilson, B. C.

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

Wood, M. F. G.

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

J. Biomed. Opt. (1)

S. Kumar, H. Purwar, R. Ossikovski, I. A. Vitkin, and N. Ghosh, “Comparative study of differential matrix and extended polar decomposition formalisms for polarimetric characterization of complex tissue-like turbid media,” J. Biomed. Opt. 17, 105006 (2012).
[CrossRef]

J. Biophotonics (1)

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

J. Opt. Soc. Am. (2)

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

Opt. Express (3)

Opt. Lett. (7)

Other (1)

A. Chipman, Handbook of Optics, 3rd ed. (McGraw-Hill, 2009), Vol. 1.

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.


Tables (1)

Tables Icon

Table 1. Mueller Matrices and Corresponding Lm and Lu Matrices, along with the ki Coefficients Calculated from Lu Matrices

Equations (36)

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

dEdz=NE,
dSdz=mNDS,
mND=Λ(NI+IN*)Λ,
Λ=12[1100001i001i1100].
m=[d0d1+d7d2+d8d3+d9d1d7d0d13d12+d6d11d5d2d8d12d6d0d14d10+d4d3d9d11+d5d10d4d0d15].
M(z)=exp(zm).
L=ln(M)=Lm+Lu.
Lm=12(LGLTG)Lu=12(L+GLTG),
[k1+k2+k30000k1k2k30000k2k1k30000k3k1k2],
Lu=Am(i)A1,
m(1)=diag(k1+k2+k3,k1k2k3,k2k1k3,k3k1k2),orm(2)=[0.5(k4+k1)0.5k4000.5k40.5(k4k1)00000.5k100000.5k1].
Mie=exp(τiGi)=I+τiGi+12τi2Gi2+O(τi3),
τi=μi+σi,τi=μiσi=0,
Mie=exp(τiGi)=I+(μi+σi)Gi+(σiμi+12σi2)Gi2+O(τi3).
Me=i{0,6}Mie,
Me=I+i=06μiGi+(i,j){0,6}ijσiσjGiGj+12i{0,6}σi2Gi2+O(τi3).
Me=I+i=06μiGi+12i{0,6}σi2Gi2+O(τi3).
log(Me)=log(i{0,6}Mie)=log(I+τG+O(τ2))=τG+O(τ2G2),
G=1τ(i=06μiGi+(i,j){0,6}ijσiσjGiGj+12i{0,6}σi2Gi2).
Me=exp[τG+O(τ2G2)].
Meq=exp[qτG+O(qτ2G2)]=exp[zG+O(zτG2)].
M(z)=limq+Meq=limτ0Mezτ=exp[zG]
G=i=06μiGi+(i,j){0,6}ijσiσjGiGj+12i{0,6}σi2Gi2,
GiGj=12{Gi,Gj}+12[Gi,Gj],
Gm=i=06(μi+σ0σi)Gi+12[(i,j){1,6}ijσiσj[Gi,Gj]],Gu=12[(i,j){1,6}ijσiσj{Gi,Gj}]+i=16σi2Gi2.
Gu=[σ42+σ52+σ6212(σ2σ6σ3σ5)12(σ3σ4σ1σ6)12(σ1σ5σ2σ4)+12(σ2σ6σ3σ5)σ42σ22σ32+12(σ1σ2+σ4σ5)+12(σ1σ3+σ4σ6)+12(σ3σ4σ1σ6)+12(σ1σ2+σ4σ5)σ52σ12σ32+12(σ2σ3+σ5σ6)+12(σ1σ5σ2σ4)+12(σ1σ3+σ4σ6)+12(σ2σ3+σ5σ6)σ62σ12σ22].
Gm=[μ0+σ02μ4+σ0σ412(σ2σ6σ3σ5)μ5+σ0σ512(σ3σ4σ1σ6)μ6+σ0σ612(σ1σ5σ2σ4)μ4+σ0σ412(σ2σ6σ3σ5)μ0+σ02μ3+σ0σ3+12(σ4σ5σ1σ2)μ2σ0σ2+12(σ4σ6σ1σ3)μ5+σ0σ512(σ3σ4σ1σ6)μ3σ0σ312(σ4σ5σ1σ2)μ0+σ02μ1+σ0σ1+12(σ5σ6σ2σ3)μ6+σ0σ612(σ1σ5σ2σ4)μ2+σ0σ212(σ4σ6σ1σ3)μ1σ0σ112(σ5σ6σ2σ3)μ0+σ02].
Gu=[σ42+σ52+σ620000σ42σ22σ320000σ52σ12σ320000σ62σ12σ22].
GuD1=diag(k1+k2+k3,k1k2k3,k2k1k3,k3k1k2),orGuD2=diag[0.5(k1+k4),0.5(k1k4),0.5k1,0.5k1].
k1=σ12+σ42,k2=σ22+σ52,k3=σ32+σ62.
k1=σ12+σ42,k4=σ22+σ32+σ52+σ62,and0.5(k1+k4)=σ12+σ22+σ32=σ42+σ52+σ62.
Mphantom=[10.01850.00290.00420.01720.75690.04050.04620.00340.05240.54500.54660.00240.00700.62440.5967]=MΔMRMD.
MΔ=[10000.00310.75930.00500.00190.00310.00500.77370.00840.00180.00190.00840.8638]MR=[100000.99720.04700.057800.07420.69640.713800.00670.71610.6980]MD=[10.01850.00290.00420.01851000.002900.999800.0042000.9998].
J1=iG1=[00000000000i00i0]J2=iG2=[0000000i00000i00]J3=iG3=[000000i00i000000]K1=iG4=[0i00i00000000000]K2=iG5=[00i00000i0000000]K3=iG6=[000i00000000i000].
[Jm,Jn]=iαmnkJk,[Km,Kn]=iαmnkKk,[Jm,Kn]=iαmnkJk,
{J1,J1}=[0000000000200002]{K1,K1}=[2000020000000000]{J2,J2}=[0000020000000002]{K2,K2}=[2000000000200000]{J3,J3}=[0000020000000002]{K3,K3}=[2000000000000002]{J1,J2}={K1,K2}=[0000001001000000]{K1,J2}={K2,J1}=[0001000000001000]{J1,J3}={K1,K3}=[0000000100000100]{K1,J3}={K3,J1}=[0010000010000000]{J2,J3}={K2,K3}=[0000000000010010]{K2,J3}={K3,J2}=[0100100000000000].

Metrics