Abstract

The differential Stokes–Mueller matrix formalism expresses the local evolution of the Mueller matrix or the Stokes parameters for light propagating through a homogeneous optical medium. This paper presents a historical revision of the development of the differential Stokes–Mueller matrix formalism and highlights several important early contributions that have been overlooked. Particularly relevant is that this formalism was pioneered as early as 1929 by Paul Soleillet, almost 50 years earlier than it has been usually assumed. This historical revision demonstrates that several different authors independently formulated the differential Stokes–Mueller formalism during the 20th century and they found equivalent results studying the propagation problem from different approaches.

© 2017 Optical Society of America

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References

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  1. R. Ossikovski, “Differential matrix formalism for depolarizing anisotropic media,” Opt. Lett. 36, 2330–2332 (2011).
    [Crossref]
  2. N. Ortega-Quijano and J. L. Arce-Diego, “Depolarizing differential Mueller matrices,” Opt. Lett. 36, 2429–2431 (2011).
    [Crossref]
  3. R. Ossikovski, “Differential and product Mueller matrix decompositions: a formal comparison,” Opt. Lett. 37, 220–222 (2012).
    [Crossref]
  4. M. Villiger and B. E. Bouma, “Practical decomposition for physically admissible differential Mueller matrices,” Opt. Lett. 39, 1779–1782 (2014).
    [Crossref]
  5. V. Devlaminck, “Depolarizing differential Mueller matrix of homogeneous media under Gaussian fluctuation hypothesis,” J. Opt. Soc. Am. A 32, 1736–1743 (2015).
    [Crossref]
  6. R. Ossikovski and O. Arteaga, “Statistical meaning of the differential Mueller matrix of depolarizing homogeneous media,” Opt. Lett. 39, 4470–4473 (2014).
    [Crossref]
  7. O. Arteaga and B. Kahr, “Characterization of homogenous depolarizing media based on Mueller matrix differential decomposition,” Opt. Lett. 38, 1134–1136 (2013).
    [Crossref]
  8. O. Arteaga, “Number of independent parameters in the Mueller matrix representation of homogeneous depolarizing media,” Opt. Lett. 38, 1131–1133 (2013).
    [Crossref]
  9. N. Agarwal, J. Yoon, E. Garcia-Caurel, T. Novikova, J.-C. Vanel, A. Pierangelo, A. Bykov, A. Popov, I. Meglinski, and R. Ossikovski, “Spatial evolution of depolarization in homogeneous turbid media within the differential Mueller matrix formalism,” Opt. Lett. 40, 5634–5637 (2015).
    [Crossref]
  10. M. Villiger, N. Lippok, and B. Bouma, “Differential Mueller-matrix formalism for polarization sensitive optical coherence tomography,” in Conference on Lasers and Electro-Optics (CLEO) (Optical Society of America, 2015), paper AW1J.4.
  11. H. Arwin, A. Mendoza-Galván, R. Magnusson, A. Andersson, J. Landin, K. Järrendahl, E. Garcia-Caurel, and R. Ossikovski, “Structural circular birefringence and dichroism quantified by differential decomposition of spectroscopic transmission Mueller matrices from Cetonia aurata,” Opt. Lett. 41, 3293–3296 (2016).
    [Crossref]
  12. X. Cui, S. Nichols, O. Arteaga, J. Freudenthal, F. Paula, A. G. Shutukenberg, and B. Kahr, “Dichroism in helicoidal crystals,” J. Am. Chem. Soc. 138, 12211–12218 (2016).
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    [Crossref]
  15. Paul Soleillet (1902–1992) was born in Marseille. The publication discussed herein was his Ph.D. thesis, which he defended at the École Normale Supérieure in 1929 and became his most relevant research work. Subsequently, he held positions at the Universities of Strasbourg, Poitiers, and la Sorbonne.
  16. F. Perrin, “Polarization of light scattered by isotropic opalescent media,” J. Chem. Phys. 10, 415–427 (1942).
    [Crossref]
  17. H. Mueller, “The foundations of optics,” J. Opt. Soc. Am. 38, 661–662 (1948).
  18. K. Järrendahl and B. Kahr, “Hans Mueller (1900–1965),” in Newsletter (Woollam, 2011), pp. 8–9.
  19. P. Soleillet, “Sur les paramètres caractérisant la polarisation partielle de la lumière dans les phénomènes de fluorescence,” Ann. Phys. 12, 23–59 (1929).
  20. R. Clark Jones, “Frederic Ives Medalist for 1972,” J. Opt. Soc. Am. 63, 519–522 (1973).
    [Crossref]
  21. J.-C. Pecker, “Soleillet,” in Annuaire de l’Association des Anciens Éleves et Amis de l’École Normale Supérieur” (École Normale Supérieur, 1994), pp. 415–417, http://www.mmpolarimetry.com/Soleillet.pdf .
  22. L. M. Barkovskii, “On the tensor of refractive indices in crystal optics,” Sov. Phys. Crystallogr. 21, 245–247 (1976).
  23. G. G. Stokes, “On the composition and resolution of streams of polarized light from different sources,” Trans. Cambridge Philos. Soc. 9, 399–416 (1851).
  24. D. Goldstein, Polarized Light, Revised and Expanded, 2nd ed., Optical Science and Engineering (CRC Press, 2003).
  25. E. Verdet, Leçons d’Optique Physique (Imprimeri Impériale, 1869), Vol. 2.
  26. S. Chandrasekhar, “On the radiative equilibrium of a stellar atmosphere: XVII,” Astrophys. J. 105, 441–460 (1947).
    [Crossref]
  27. H. Poincaré, Théorie Mathématique de la Lumière (Jacques Gabay, 1892).
  28. J. Schellman and H. P. Jensen, “Optical spectroscopy of oriented molecules,” Chem. Rev. 87, 1359–1399 (1987).
    [Crossref]
  29. J. R. Lakowicz, Principles of Fluorescence Spectroscopy, 3rd ed. (Springer, 2006).
  30. O. Arteaga, S. Nichols, and B. Kahr, “Mueller matrices in fluorescence scattering,” Opt. Lett. 37, 2835–2837 (2012).
    [Crossref]
  31. N. Go, “Optical activity of anisotropic solutions: II,” J. Phys. Soc. Jpn. 23, 88–97 (1967).
    [Crossref]
  32. T. C. Troxell and H. A. Scheraga, “Electric dichroism and polymer conformation. I: theory of optical properties of anisotropic media, and method of measurement,” Macromolecules 4, 519–527 (1971).
    [Crossref]
  33. H. P. Jensen, “Differential polarization spectroscopy on single crystals of transition metal complexes,” Appl. Spectrosc. Rev. 18, 305–327 (1982).
    [Crossref]
  34. H. P. Jensen, J. A. Schellman, and T. Troxell, “Modulation techniques in polarization spectroscopy,” Appl. Spectrosc. 32, 192–200 (1978).
    [Crossref]
  35. A. Schönhofer, H. G. Kuball, and C. Puebla, “Optical activity of oriented molecules. IX: phenomenological Mueller matrix description of thick samples and of optical elements,” Chem. Phys. 76, 453–467 (1983).
    [Crossref]
  36. R. Barakat, “Exponential versions of the Jones and Mueller–Jones polarization matrices,” J. Opt. Soc. Am. A 13, 158–163 (1996).
    [Crossref]
  37. O. Arteaga and A. Canillas, “Analytic inversion of the Mueller–Jones polarization matrices for homogeneous media,” Opt. Lett. 35, 559–561 (2010).
    [Crossref]
  38. O. Arteaga, “Mueller matrix polarimetry of anisotropic chiral media,” Ph.D. thesis (University of Barcelona, 2010).
  39. L. D. Barron, Molecular Light Scattering and Optical Activity (Cambridge University, 2004).

2016 (2)

2015 (2)

2014 (2)

2013 (2)

2012 (2)

2011 (2)

2010 (1)

1996 (1)

1987 (1)

J. Schellman and H. P. Jensen, “Optical spectroscopy of oriented molecules,” Chem. Rev. 87, 1359–1399 (1987).
[Crossref]

1983 (1)

A. Schönhofer, H. G. Kuball, and C. Puebla, “Optical activity of oriented molecules. IX: phenomenological Mueller matrix description of thick samples and of optical elements,” Chem. Phys. 76, 453–467 (1983).
[Crossref]

1982 (1)

H. P. Jensen, “Differential polarization spectroscopy on single crystals of transition metal complexes,” Appl. Spectrosc. Rev. 18, 305–327 (1982).
[Crossref]

1978 (2)

1976 (1)

L. M. Barkovskii, “On the tensor of refractive indices in crystal optics,” Sov. Phys. Crystallogr. 21, 245–247 (1976).

1973 (1)

1971 (1)

T. C. Troxell and H. A. Scheraga, “Electric dichroism and polymer conformation. I: theory of optical properties of anisotropic media, and method of measurement,” Macromolecules 4, 519–527 (1971).
[Crossref]

1967 (1)

N. Go, “Optical activity of anisotropic solutions: II,” J. Phys. Soc. Jpn. 23, 88–97 (1967).
[Crossref]

1948 (2)

1947 (1)

S. Chandrasekhar, “On the radiative equilibrium of a stellar atmosphere: XVII,” Astrophys. J. 105, 441–460 (1947).
[Crossref]

1942 (1)

F. Perrin, “Polarization of light scattered by isotropic opalescent media,” J. Chem. Phys. 10, 415–427 (1942).
[Crossref]

1929 (1)

P. Soleillet, “Sur les paramètres caractérisant la polarisation partielle de la lumière dans les phénomènes de fluorescence,” Ann. Phys. 12, 23–59 (1929).

1851 (1)

G. G. Stokes, “On the composition and resolution of streams of polarized light from different sources,” Trans. Cambridge Philos. Soc. 9, 399–416 (1851).

Agarwal, N.

Andersson, A.

Arce-Diego, J. L.

Arteaga, O.

Arwin, H.

Azzam, R. M. A.

Barakat, R.

Barkovskii, L. M.

L. M. Barkovskii, “On the tensor of refractive indices in crystal optics,” Sov. Phys. Crystallogr. 21, 245–247 (1976).

Barron, L. D.

L. D. Barron, Molecular Light Scattering and Optical Activity (Cambridge University, 2004).

Bouma, B.

M. Villiger, N. Lippok, and B. Bouma, “Differential Mueller-matrix formalism for polarization sensitive optical coherence tomography,” in Conference on Lasers and Electro-Optics (CLEO) (Optical Society of America, 2015), paper AW1J.4.

Bouma, B. E.

Bykov, A.

Canillas, A.

Chandrasekhar, S.

S. Chandrasekhar, “On the radiative equilibrium of a stellar atmosphere: XVII,” Astrophys. J. 105, 441–460 (1947).
[Crossref]

Clark Jones, R.

Cui, X.

X. Cui, S. Nichols, O. Arteaga, J. Freudenthal, F. Paula, A. G. Shutukenberg, and B. Kahr, “Dichroism in helicoidal crystals,” J. Am. Chem. Soc. 138, 12211–12218 (2016).

Devlaminck, V.

Freudenthal, J.

X. Cui, S. Nichols, O. Arteaga, J. Freudenthal, F. Paula, A. G. Shutukenberg, and B. Kahr, “Dichroism in helicoidal crystals,” J. Am. Chem. Soc. 138, 12211–12218 (2016).

Garcia-Caurel, E.

Go, N.

N. Go, “Optical activity of anisotropic solutions: II,” J. Phys. Soc. Jpn. 23, 88–97 (1967).
[Crossref]

Goldstein, D.

D. Goldstein, Polarized Light, Revised and Expanded, 2nd ed., Optical Science and Engineering (CRC Press, 2003).

Järrendahl, K.

Jensen, H. P.

J. Schellman and H. P. Jensen, “Optical spectroscopy of oriented molecules,” Chem. Rev. 87, 1359–1399 (1987).
[Crossref]

H. P. Jensen, “Differential polarization spectroscopy on single crystals of transition metal complexes,” Appl. Spectrosc. Rev. 18, 305–327 (1982).
[Crossref]

H. P. Jensen, J. A. Schellman, and T. Troxell, “Modulation techniques in polarization spectroscopy,” Appl. Spectrosc. 32, 192–200 (1978).
[Crossref]

Jones, R. C.

Kahr, B.

X. Cui, S. Nichols, O. Arteaga, J. Freudenthal, F. Paula, A. G. Shutukenberg, and B. Kahr, “Dichroism in helicoidal crystals,” J. Am. Chem. Soc. 138, 12211–12218 (2016).

O. Arteaga and B. Kahr, “Characterization of homogenous depolarizing media based on Mueller matrix differential decomposition,” Opt. Lett. 38, 1134–1136 (2013).
[Crossref]

O. Arteaga, S. Nichols, and B. Kahr, “Mueller matrices in fluorescence scattering,” Opt. Lett. 37, 2835–2837 (2012).
[Crossref]

K. Järrendahl and B. Kahr, “Hans Mueller (1900–1965),” in Newsletter (Woollam, 2011), pp. 8–9.

Kuball, H. G.

A. Schönhofer, H. G. Kuball, and C. Puebla, “Optical activity of oriented molecules. IX: phenomenological Mueller matrix description of thick samples and of optical elements,” Chem. Phys. 76, 453–467 (1983).
[Crossref]

Lakowicz, J. R.

J. R. Lakowicz, Principles of Fluorescence Spectroscopy, 3rd ed. (Springer, 2006).

Landin, J.

Lippok, N.

M. Villiger, N. Lippok, and B. Bouma, “Differential Mueller-matrix formalism for polarization sensitive optical coherence tomography,” in Conference on Lasers and Electro-Optics (CLEO) (Optical Society of America, 2015), paper AW1J.4.

Magnusson, R.

Meglinski, I.

Mendoza-Galván, A.

Mueller, H.

H. Mueller, “The foundations of optics,” J. Opt. Soc. Am. 38, 661–662 (1948).

Nichols, S.

X. Cui, S. Nichols, O. Arteaga, J. Freudenthal, F. Paula, A. G. Shutukenberg, and B. Kahr, “Dichroism in helicoidal crystals,” J. Am. Chem. Soc. 138, 12211–12218 (2016).

O. Arteaga, S. Nichols, and B. Kahr, “Mueller matrices in fluorescence scattering,” Opt. Lett. 37, 2835–2837 (2012).
[Crossref]

Novikova, T.

Ortega-Quijano, N.

Ossikovski, R.

Paula, F.

X. Cui, S. Nichols, O. Arteaga, J. Freudenthal, F. Paula, A. G. Shutukenberg, and B. Kahr, “Dichroism in helicoidal crystals,” J. Am. Chem. Soc. 138, 12211–12218 (2016).

Pecker, J.-C.

J.-C. Pecker, “Soleillet,” in Annuaire de l’Association des Anciens Éleves et Amis de l’École Normale Supérieur” (École Normale Supérieur, 1994), pp. 415–417, http://www.mmpolarimetry.com/Soleillet.pdf .

Perrin, F.

F. Perrin, “Polarization of light scattered by isotropic opalescent media,” J. Chem. Phys. 10, 415–427 (1942).
[Crossref]

Pierangelo, A.

Poincaré, H.

H. Poincaré, Théorie Mathématique de la Lumière (Jacques Gabay, 1892).

Popov, A.

Puebla, C.

A. Schönhofer, H. G. Kuball, and C. Puebla, “Optical activity of oriented molecules. IX: phenomenological Mueller matrix description of thick samples and of optical elements,” Chem. Phys. 76, 453–467 (1983).
[Crossref]

Schellman, J.

J. Schellman and H. P. Jensen, “Optical spectroscopy of oriented molecules,” Chem. Rev. 87, 1359–1399 (1987).
[Crossref]

Schellman, J. A.

Scheraga, H. A.

T. C. Troxell and H. A. Scheraga, “Electric dichroism and polymer conformation. I: theory of optical properties of anisotropic media, and method of measurement,” Macromolecules 4, 519–527 (1971).
[Crossref]

Schönhofer, A.

A. Schönhofer, H. G. Kuball, and C. Puebla, “Optical activity of oriented molecules. IX: phenomenological Mueller matrix description of thick samples and of optical elements,” Chem. Phys. 76, 453–467 (1983).
[Crossref]

Shutukenberg, A. G.

X. Cui, S. Nichols, O. Arteaga, J. Freudenthal, F. Paula, A. G. Shutukenberg, and B. Kahr, “Dichroism in helicoidal crystals,” J. Am. Chem. Soc. 138, 12211–12218 (2016).

Soleillet, P.

P. Soleillet, “Sur les paramètres caractérisant la polarisation partielle de la lumière dans les phénomènes de fluorescence,” Ann. Phys. 12, 23–59 (1929).

Stokes, G. G.

G. G. Stokes, “On the composition and resolution of streams of polarized light from different sources,” Trans. Cambridge Philos. Soc. 9, 399–416 (1851).

Troxell, T.

Troxell, T. C.

T. C. Troxell and H. A. Scheraga, “Electric dichroism and polymer conformation. I: theory of optical properties of anisotropic media, and method of measurement,” Macromolecules 4, 519–527 (1971).
[Crossref]

Vanel, J.-C.

Verdet, E.

E. Verdet, Leçons d’Optique Physique (Imprimeri Impériale, 1869), Vol. 2.

Villiger, M.

M. Villiger and B. E. Bouma, “Practical decomposition for physically admissible differential Mueller matrices,” Opt. Lett. 39, 1779–1782 (2014).
[Crossref]

M. Villiger, N. Lippok, and B. Bouma, “Differential Mueller-matrix formalism for polarization sensitive optical coherence tomography,” in Conference on Lasers and Electro-Optics (CLEO) (Optical Society of America, 2015), paper AW1J.4.

Yoon, J.

Ann. Phys. (1)

P. Soleillet, “Sur les paramètres caractérisant la polarisation partielle de la lumière dans les phénomènes de fluorescence,” Ann. Phys. 12, 23–59 (1929).

Appl. Spectrosc. (1)

Appl. Spectrosc. Rev. (1)

H. P. Jensen, “Differential polarization spectroscopy on single crystals of transition metal complexes,” Appl. Spectrosc. Rev. 18, 305–327 (1982).
[Crossref]

Astrophys. J. (1)

S. Chandrasekhar, “On the radiative equilibrium of a stellar atmosphere: XVII,” Astrophys. J. 105, 441–460 (1947).
[Crossref]

Chem. Phys. (1)

A. Schönhofer, H. G. Kuball, and C. Puebla, “Optical activity of oriented molecules. IX: phenomenological Mueller matrix description of thick samples and of optical elements,” Chem. Phys. 76, 453–467 (1983).
[Crossref]

Chem. Rev. (1)

J. Schellman and H. P. Jensen, “Optical spectroscopy of oriented molecules,” Chem. Rev. 87, 1359–1399 (1987).
[Crossref]

J. Am. Chem. Soc. (1)

X. Cui, S. Nichols, O. Arteaga, J. Freudenthal, F. Paula, A. G. Shutukenberg, and B. Kahr, “Dichroism in helicoidal crystals,” J. Am. Chem. Soc. 138, 12211–12218 (2016).

J. Chem. Phys. (1)

F. Perrin, “Polarization of light scattered by isotropic opalescent media,” J. Chem. Phys. 10, 415–427 (1942).
[Crossref]

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (2)

J. Phys. Soc. Jpn. (1)

N. Go, “Optical activity of anisotropic solutions: II,” J. Phys. Soc. Jpn. 23, 88–97 (1967).
[Crossref]

Macromolecules (1)

T. C. Troxell and H. A. Scheraga, “Electric dichroism and polymer conformation. I: theory of optical properties of anisotropic media, and method of measurement,” Macromolecules 4, 519–527 (1971).
[Crossref]

Opt. Lett. (11)

H. Arwin, A. Mendoza-Galván, R. Magnusson, A. Andersson, J. Landin, K. Järrendahl, E. Garcia-Caurel, and R. Ossikovski, “Structural circular birefringence and dichroism quantified by differential decomposition of spectroscopic transmission Mueller matrices from Cetonia aurata,” Opt. Lett. 41, 3293–3296 (2016).
[Crossref]

O. Arteaga, S. Nichols, and B. Kahr, “Mueller matrices in fluorescence scattering,” Opt. Lett. 37, 2835–2837 (2012).
[Crossref]

O. Arteaga and A. Canillas, “Analytic inversion of the Mueller–Jones polarization matrices for homogeneous media,” Opt. Lett. 35, 559–561 (2010).
[Crossref]

R. Ossikovski and O. Arteaga, “Statistical meaning of the differential Mueller matrix of depolarizing homogeneous media,” Opt. Lett. 39, 4470–4473 (2014).
[Crossref]

O. Arteaga and B. Kahr, “Characterization of homogenous depolarizing media based on Mueller matrix differential decomposition,” Opt. Lett. 38, 1134–1136 (2013).
[Crossref]

O. Arteaga, “Number of independent parameters in the Mueller matrix representation of homogeneous depolarizing media,” Opt. Lett. 38, 1131–1133 (2013).
[Crossref]

N. Agarwal, J. Yoon, E. Garcia-Caurel, T. Novikova, J.-C. Vanel, A. Pierangelo, A. Bykov, A. Popov, I. Meglinski, and R. Ossikovski, “Spatial evolution of depolarization in homogeneous turbid media within the differential Mueller matrix formalism,” Opt. Lett. 40, 5634–5637 (2015).
[Crossref]

R. Ossikovski, “Differential matrix formalism for depolarizing anisotropic media,” Opt. Lett. 36, 2330–2332 (2011).
[Crossref]

N. Ortega-Quijano and J. L. Arce-Diego, “Depolarizing differential Mueller matrices,” Opt. Lett. 36, 2429–2431 (2011).
[Crossref]

R. Ossikovski, “Differential and product Mueller matrix decompositions: a formal comparison,” Opt. Lett. 37, 220–222 (2012).
[Crossref]

M. Villiger and B. E. Bouma, “Practical decomposition for physically admissible differential Mueller matrices,” Opt. Lett. 39, 1779–1782 (2014).
[Crossref]

Sov. Phys. Crystallogr. (1)

L. M. Barkovskii, “On the tensor of refractive indices in crystal optics,” Sov. Phys. Crystallogr. 21, 245–247 (1976).

Trans. Cambridge Philos. Soc. (1)

G. G. Stokes, “On the composition and resolution of streams of polarized light from different sources,” Trans. Cambridge Philos. Soc. 9, 399–416 (1851).

Other (10)

D. Goldstein, Polarized Light, Revised and Expanded, 2nd ed., Optical Science and Engineering (CRC Press, 2003).

E. Verdet, Leçons d’Optique Physique (Imprimeri Impériale, 1869), Vol. 2.

J. R. Lakowicz, Principles of Fluorescence Spectroscopy, 3rd ed. (Springer, 2006).

H. Poincaré, Théorie Mathématique de la Lumière (Jacques Gabay, 1892).

O. Arteaga, “Mueller matrix polarimetry of anisotropic chiral media,” Ph.D. thesis (University of Barcelona, 2010).

L. D. Barron, Molecular Light Scattering and Optical Activity (Cambridge University, 2004).

M. Villiger, N. Lippok, and B. Bouma, “Differential Mueller-matrix formalism for polarization sensitive optical coherence tomography,” in Conference on Lasers and Electro-Optics (CLEO) (Optical Society of America, 2015), paper AW1J.4.

Paul Soleillet (1902–1992) was born in Marseille. The publication discussed herein was his Ph.D. thesis, which he defended at the École Normale Supérieure in 1929 and became his most relevant research work. Subsequently, he held positions at the Universities of Strasbourg, Poitiers, and la Sorbonne.

J.-C. Pecker, “Soleillet,” in Annuaire de l’Association des Anciens Éleves et Amis de l’École Normale Supérieur” (École Normale Supérieur, 1994), pp. 415–417, http://www.mmpolarimetry.com/Soleillet.pdf .

K. Järrendahl and B. Kahr, “Hans Mueller (1900–1965),” in Newsletter (Woollam, 2011), pp. 8–9.

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Figures (1)

Fig. 1.
Fig. 1. Photos of early pioneers of the optical polarization calculus during the first half of the 20th century. From left to right: Francis Perrin (1901–1992) (AIP Emilio Segrè Visual Archives), Hans Mueller (1900–1965), and R. Clark Jones (1916–2004) [20]. Despite many attempts, I have been unable to find a photo of Paul Soleillet (1902–1992). Biographical information about him is also extraordinarily scarce, with the exception of a note that appeared in 1994 in an alumni association publication [21].

Equations (11)

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

d M d z = mM ,
d S d z = mS ,
d I = h ( i I + m M + c C + s S ) d r , d M = h ( m I + i M ) d r , d C = h ( c I + i C ) d r , d S = h ( s I + i S ) d r ,
d I = 0 , d M = ( q S r C ) θ d r , d C = ( r M p S ) θ d r , d S = ( p C q M ) θ d r ,
m = [ h i h m h c h s h m h i r θ q θ h c r θ h i p θ h s q θ p θ h i ] ,
i = K + K h , m = K K h α , c = K K h β , s = K K h γ ,
d d l F ( l ) = lim Δ l 0 F ( Δ l ) I Δ l F ( l ) = 2 η H F ( l ) ,
F ( l ) = exp ( 2 η l H ) .
a α β = [ a x x a x y a x z a y x a y y a y z a z x a z y a z z ] .
a α β = α α β + ψ α β γ n γ + ,
m = k [ I ( a x x + a y y ) I ( a x x a y y ) I ( a x y + a y x ) R ( a y x a x y ) I ( a x x a y y ) I ( a x x + a y y ) I ( a y x a x y ) R ( a x y + a y x ) I ( a x y + a y x ) I ( a y x a x y ) I ( a x x + a y y ) R ( a x x a y y ) R ( a y x a x y ) R ( a x y + a y x ) R ( a x x a y y ) I ( a x x + a y y ) ] ,

Metrics