Abstract

We propose a model for the reflectance of a particle medium made of identical, large, spherical, and absorbing particles in a clear binder. A 3D geometrical description of light scattering is developed by relying on the laws of geometrical optics. The amount of light backscattered by a single particle is determined as a function of its absorbance and refractive index. Then, we consider a set of coplanar particles, called a particle sublayer, whose reflectance and transmittance are functions of the particle backscattering ratio and the particle concentration. The reflectance of an infinite particle medium is derived from a description of multiple reflections and transmissions between many superposed particle sublayers. When the binder has a refractive index different from that of air, the medium’s reflectance factor accounts for the multiple reflections occurring beneath the air–binder interface as well as for the measuring geometry. The influences of various parameters, such as the refractive indices and the particle absorption coefficient, are examined.

© 2008 Optical Society of America

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References

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  1. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983).
  2. M. Born and E. Wolf, Principles of Optics, 7th ed. (Pergamon, 1999).
  3. H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981), pp. 200-227.
  4. S. Chandrasekhar, Radiative Transfer (Dover, 1960).
  5. K. Stamnes, S. Chee Tsay, W. Wiscombe, and K. Jayaweera, “Numerically stable algorithm for discrete-ordinate-method radiative transfer in multiple scattering and emitting layer media,” Appl. Opt. 27, 2502-2510 (1988).
    [CrossRef] [PubMed]
  6. L. Simonot, M. Elias, and E. Charron, “Special visual effect of art glazes explained by the radiative transfer equation,” Appl. Opt. 43, 2580-2587 (2004).
    [CrossRef] [PubMed]
  7. P. S. Mudgett and L. W. Richards, “Multiple scattering calculations for technology,” Appl. Opt. 10, 1485-1502 (1971).
    [CrossRef] [PubMed]
  8. W. E. Vargas and G. A. Niklasson, “Applicability conditions of the Kubelka-Munk theory,” Appl. Opt. 36, 5580-5586 (1997).
    [CrossRef] [PubMed]
  9. P. Kubelka and F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. (Leipzig) 12, 593-601 (1931) (in German).
  10. P. Kubelka, “New contributions to the optics of intensely light-scattering material, part I,” J. Opt. Soc. Am. 38, 448-457 (1948).
    [CrossRef] [PubMed]
  11. M. Hébert and R. D. Hersch, “Reflectance and transmittance model for recto-verso halftone prints,” J. Opt. Soc. Am. A 23, 2415-2432 (2006).
    [CrossRef]
  12. G. Stokes, “On the intensity of light reflected from or transmitted through a pile of plates,” Mathematical and Physical Papers of Sir George Stokes, IV (Cambridge U. Press, 1904), pp. 145-156.
  13. P. Kubelka, “New contributions to the optics of intensely light-scattering materials, part II: Non homogeneous layers,” J. Opt. Soc. Am. 44, 330-335 (1954).
    [CrossRef]
  14. G. Kortüm, Reflectance Spectroscopy (Springer-Verlag, 1969).
  15. M. Vöge and K. Simon, “The Kubelka-Munk and Dyck paths,” J. Stat. Mech.: Theory Exp. 2007, P02018 (2007).
    [CrossRef]
  16. K. Simon and B. Trachsler, “A random walk approach for light scattering in material,” Discrete Math. Theor. Comp. Sci. AC, 289-300 (2003).
  17. M. Hébert, R. Hersch, and J.-M. Becker, “Compositional reflectance and transmittance model for multilayer specimens,” J. Opt. Soc. Am. A 24, 2628-2644 (2007).
    [CrossRef]
  18. Z. Bodo, “Some optical properties of luminescent powders,” Acta Phys. Acad. Sci. Hung. 1, 135-150 (1951).
    [CrossRef]
  19. N. T. Melamed, “Optical properties of powders: Part I. Optical absorption coefficients and the absolute value of the diffuse reflectance,” J. Appl. Phys. 34, 560-570 (1963).
    [CrossRef]
  20. A. Mandelis, F. Boroumand, and H. van den Bergh, “Quantitative diffuse reflectance spectroscopy of large powders: The Melamed model revisited,” Appl. Opt. 29, 2853-2860 (1990).
    [CrossRef] [PubMed]
  21. H. Garay, O. Eterradossi, and A. Benhassaine, “Should Melamed's spherical model of size-colour dependence in powders be adapted to non spheric particles?” Powder Technol. 156, 8-18 (2005).
    [CrossRef]
  22. Y. G. Shkuratov, L. Starukhina, H. Hoffmann, and G. Arnold, “A model of spectral albedo of particulate surfaces: Implication to optical properties of the Moon,” Icarus 137, 235-246 (1999).
    [CrossRef]
  23. Y. G. Shkuratov and Y. S. Grynko, “Light scattering by media composed of semitransparent particles of different shapes in ray optics approximation: consequences for spectroscopy, photometry, and polarimetry of planetary regoliths,” Icarus 173, 16-28 (2005).
    [CrossRef]
  24. D. B. Judd, “Fresnel reflection of diffusely incident light,” J. Res. Natl. Bur. Stand. 29, 329-332 (1942).
  25. M. Hébert and R. D. Hersch, “Classical print reflection models: A radiometric approach,” J. Imaging Sci. Technol. 48, 363-374 (2004).
  26. CRC Concise Encyclopedia of Mathematics (CRC Press, 1998), p. 1580.
  27. W. R. McCluney, Introduction to Radiometry and Photometry (Artech House, 1994).
  28. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983), p. 172.
  29. B. Mayer and S. Madronich, “Photolysis frequencies in water droplets: Mie calculations and geometrical optics limits,” Atmos. Chem. Phys. Discuss. 4, 4105-4130 (2004).
    [CrossRef]
  30. W. J. Glantschnig and S. H. Chen, “Light scattering from water droplets in the geometrical optics approximation,” Appl. Opt. 20, 2499-2509 (1981).
    [CrossRef] [PubMed]
  31. L. Simonot, M. Hébert, and R. Hersch, “Extension of the Williams-Clapper model to stacked nondiffusing colored coatings with different refractive indices,” J. Opt. Soc. Am. A 23, 1432-1441 (2006).
    [CrossRef]
  32. J. L. Saunderson, “Calculation of the color pigmented plastics,” J. Opt. Soc. Am. 32, 727-736 (1942).
    [CrossRef]

2007 (2)

2006 (2)

2005 (2)

Y. G. Shkuratov and Y. S. Grynko, “Light scattering by media composed of semitransparent particles of different shapes in ray optics approximation: consequences for spectroscopy, photometry, and polarimetry of planetary regoliths,” Icarus 173, 16-28 (2005).
[CrossRef]

H. Garay, O. Eterradossi, and A. Benhassaine, “Should Melamed's spherical model of size-colour dependence in powders be adapted to non spheric particles?” Powder Technol. 156, 8-18 (2005).
[CrossRef]

2004 (3)

M. Hébert and R. D. Hersch, “Classical print reflection models: A radiometric approach,” J. Imaging Sci. Technol. 48, 363-374 (2004).

B. Mayer and S. Madronich, “Photolysis frequencies in water droplets: Mie calculations and geometrical optics limits,” Atmos. Chem. Phys. Discuss. 4, 4105-4130 (2004).
[CrossRef]

L. Simonot, M. Elias, and E. Charron, “Special visual effect of art glazes explained by the radiative transfer equation,” Appl. Opt. 43, 2580-2587 (2004).
[CrossRef] [PubMed]

2003 (1)

K. Simon and B. Trachsler, “A random walk approach for light scattering in material,” Discrete Math. Theor. Comp. Sci. AC, 289-300 (2003).

1999 (2)

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

Y. G. Shkuratov, L. Starukhina, H. Hoffmann, and G. Arnold, “A model of spectral albedo of particulate surfaces: Implication to optical properties of the Moon,” Icarus 137, 235-246 (1999).
[CrossRef]

1998 (1)

CRC Concise Encyclopedia of Mathematics (CRC Press, 1998), p. 1580.

1997 (1)

1994 (1)

W. R. McCluney, Introduction to Radiometry and Photometry (Artech House, 1994).

1990 (1)

1988 (1)

1983 (2)

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983), p. 172.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983).

1981 (2)

1971 (1)

1969 (1)

G. Kortüm, Reflectance Spectroscopy (Springer-Verlag, 1969).

1963 (1)

N. T. Melamed, “Optical properties of powders: Part I. Optical absorption coefficients and the absolute value of the diffuse reflectance,” J. Appl. Phys. 34, 560-570 (1963).
[CrossRef]

1960 (1)

S. Chandrasekhar, Radiative Transfer (Dover, 1960).

1954 (1)

1951 (1)

Z. Bodo, “Some optical properties of luminescent powders,” Acta Phys. Acad. Sci. Hung. 1, 135-150 (1951).
[CrossRef]

1948 (1)

1942 (2)

J. L. Saunderson, “Calculation of the color pigmented plastics,” J. Opt. Soc. Am. 32, 727-736 (1942).
[CrossRef]

D. B. Judd, “Fresnel reflection of diffusely incident light,” J. Res. Natl. Bur. Stand. 29, 329-332 (1942).

1931 (1)

P. Kubelka and F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. (Leipzig) 12, 593-601 (1931) (in German).

1904 (1)

G. Stokes, “On the intensity of light reflected from or transmitted through a pile of plates,” Mathematical and Physical Papers of Sir George Stokes, IV (Cambridge U. Press, 1904), pp. 145-156.

Arnold, G.

Y. G. Shkuratov, L. Starukhina, H. Hoffmann, and G. Arnold, “A model of spectral albedo of particulate surfaces: Implication to optical properties of the Moon,” Icarus 137, 235-246 (1999).
[CrossRef]

Becker, J.-M.

Benhassaine, A.

H. Garay, O. Eterradossi, and A. Benhassaine, “Should Melamed's spherical model of size-colour dependence in powders be adapted to non spheric particles?” Powder Technol. 156, 8-18 (2005).
[CrossRef]

Bodo, Z.

Z. Bodo, “Some optical properties of luminescent powders,” Acta Phys. Acad. Sci. Hung. 1, 135-150 (1951).
[CrossRef]

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983), p. 172.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983).

Born, M.

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

Boroumand, F.

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, 1960).

Charron, E.

Chee Tsay, S.

Chen, S. H.

Elias, M.

Eterradossi, O.

H. Garay, O. Eterradossi, and A. Benhassaine, “Should Melamed's spherical model of size-colour dependence in powders be adapted to non spheric particles?” Powder Technol. 156, 8-18 (2005).
[CrossRef]

Garay, H.

H. Garay, O. Eterradossi, and A. Benhassaine, “Should Melamed's spherical model of size-colour dependence in powders be adapted to non spheric particles?” Powder Technol. 156, 8-18 (2005).
[CrossRef]

Glantschnig, W. J.

Grynko, Y. S.

Y. G. Shkuratov and Y. S. Grynko, “Light scattering by media composed of semitransparent particles of different shapes in ray optics approximation: consequences for spectroscopy, photometry, and polarimetry of planetary regoliths,” Icarus 173, 16-28 (2005).
[CrossRef]

Hébert, M.

Hersch, R.

Hersch, R. D.

M. Hébert and R. D. Hersch, “Reflectance and transmittance model for recto-verso halftone prints,” J. Opt. Soc. Am. A 23, 2415-2432 (2006).
[CrossRef]

M. Hébert and R. D. Hersch, “Classical print reflection models: A radiometric approach,” J. Imaging Sci. Technol. 48, 363-374 (2004).

Hoffmann, H.

Y. G. Shkuratov, L. Starukhina, H. Hoffmann, and G. Arnold, “A model of spectral albedo of particulate surfaces: Implication to optical properties of the Moon,” Icarus 137, 235-246 (1999).
[CrossRef]

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983).

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983), p. 172.

Jayaweera, K.

Judd, D. B.

D. B. Judd, “Fresnel reflection of diffusely incident light,” J. Res. Natl. Bur. Stand. 29, 329-332 (1942).

Kortüm, G.

G. Kortüm, Reflectance Spectroscopy (Springer-Verlag, 1969).

Kubelka, P.

Madronich, S.

B. Mayer and S. Madronich, “Photolysis frequencies in water droplets: Mie calculations and geometrical optics limits,” Atmos. Chem. Phys. Discuss. 4, 4105-4130 (2004).
[CrossRef]

Mandelis, A.

Mayer, B.

B. Mayer and S. Madronich, “Photolysis frequencies in water droplets: Mie calculations and geometrical optics limits,” Atmos. Chem. Phys. Discuss. 4, 4105-4130 (2004).
[CrossRef]

McCluney, W. R.

W. R. McCluney, Introduction to Radiometry and Photometry (Artech House, 1994).

Melamed, N. T.

N. T. Melamed, “Optical properties of powders: Part I. Optical absorption coefficients and the absolute value of the diffuse reflectance,” J. Appl. Phys. 34, 560-570 (1963).
[CrossRef]

Mudgett, P. S.

Munk, F.

P. Kubelka and F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. (Leipzig) 12, 593-601 (1931) (in German).

Niklasson, G. A.

Richards, L. W.

Saunderson, J. L.

Shkuratov, Y. G.

Y. G. Shkuratov and Y. S. Grynko, “Light scattering by media composed of semitransparent particles of different shapes in ray optics approximation: consequences for spectroscopy, photometry, and polarimetry of planetary regoliths,” Icarus 173, 16-28 (2005).
[CrossRef]

Y. G. Shkuratov, L. Starukhina, H. Hoffmann, and G. Arnold, “A model of spectral albedo of particulate surfaces: Implication to optical properties of the Moon,” Icarus 137, 235-246 (1999).
[CrossRef]

Simon, K.

M. Vöge and K. Simon, “The Kubelka-Munk and Dyck paths,” J. Stat. Mech.: Theory Exp. 2007, P02018 (2007).
[CrossRef]

K. Simon and B. Trachsler, “A random walk approach for light scattering in material,” Discrete Math. Theor. Comp. Sci. AC, 289-300 (2003).

Simonot, L.

Stamnes, K.

Starukhina, L.

Y. G. Shkuratov, L. Starukhina, H. Hoffmann, and G. Arnold, “A model of spectral albedo of particulate surfaces: Implication to optical properties of the Moon,” Icarus 137, 235-246 (1999).
[CrossRef]

Stokes, G.

G. Stokes, “On the intensity of light reflected from or transmitted through a pile of plates,” Mathematical and Physical Papers of Sir George Stokes, IV (Cambridge U. Press, 1904), pp. 145-156.

Trachsler, B.

K. Simon and B. Trachsler, “A random walk approach for light scattering in material,” Discrete Math. Theor. Comp. Sci. AC, 289-300 (2003).

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981), pp. 200-227.

van den Bergh, H.

Vargas, W. E.

Vöge, M.

M. Vöge and K. Simon, “The Kubelka-Munk and Dyck paths,” J. Stat. Mech.: Theory Exp. 2007, P02018 (2007).
[CrossRef]

Wiscombe, W.

Wolf, E.

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

Acta Phys. Acad. Sci. Hung. (1)

Z. Bodo, “Some optical properties of luminescent powders,” Acta Phys. Acad. Sci. Hung. 1, 135-150 (1951).
[CrossRef]

Appl. Opt. (6)

Atmos. Chem. Phys. Discuss. (1)

B. Mayer and S. Madronich, “Photolysis frequencies in water droplets: Mie calculations and geometrical optics limits,” Atmos. Chem. Phys. Discuss. 4, 4105-4130 (2004).
[CrossRef]

Discrete Math. Theor. Comp. Sci. (1)

K. Simon and B. Trachsler, “A random walk approach for light scattering in material,” Discrete Math. Theor. Comp. Sci. AC, 289-300 (2003).

Icarus (2)

Y. G. Shkuratov, L. Starukhina, H. Hoffmann, and G. Arnold, “A model of spectral albedo of particulate surfaces: Implication to optical properties of the Moon,” Icarus 137, 235-246 (1999).
[CrossRef]

Y. G. Shkuratov and Y. S. Grynko, “Light scattering by media composed of semitransparent particles of different shapes in ray optics approximation: consequences for spectroscopy, photometry, and polarimetry of planetary regoliths,” Icarus 173, 16-28 (2005).
[CrossRef]

J. Appl. Phys. (1)

N. T. Melamed, “Optical properties of powders: Part I. Optical absorption coefficients and the absolute value of the diffuse reflectance,” J. Appl. Phys. 34, 560-570 (1963).
[CrossRef]

J. Imaging Sci. Technol. (1)

M. Hébert and R. D. Hersch, “Classical print reflection models: A radiometric approach,” J. Imaging Sci. Technol. 48, 363-374 (2004).

J. Opt. Soc. Am. (3)

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

J. Res. Natl. Bur. Stand. (1)

D. B. Judd, “Fresnel reflection of diffusely incident light,” J. Res. Natl. Bur. Stand. 29, 329-332 (1942).

J. Stat. Mech.: Theory Exp. (1)

M. Vöge and K. Simon, “The Kubelka-Munk and Dyck paths,” J. Stat. Mech.: Theory Exp. 2007, P02018 (2007).
[CrossRef]

Powder Technol. (1)

H. Garay, O. Eterradossi, and A. Benhassaine, “Should Melamed's spherical model of size-colour dependence in powders be adapted to non spheric particles?” Powder Technol. 156, 8-18 (2005).
[CrossRef]

Z. Tech. Phys. (Leipzig) (1)

P. Kubelka and F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. (Leipzig) 12, 593-601 (1931) (in German).

Other (9)

G. Kortüm, Reflectance Spectroscopy (Springer-Verlag, 1969).

G. Stokes, “On the intensity of light reflected from or transmitted through a pile of plates,” Mathematical and Physical Papers of Sir George Stokes, IV (Cambridge U. Press, 1904), pp. 145-156.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983).

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

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981), pp. 200-227.

S. Chandrasekhar, Radiative Transfer (Dover, 1960).

CRC Concise Encyclopedia of Mathematics (CRC Press, 1998), p. 1580.

W. R. McCluney, Introduction to Radiometry and Photometry (Artech House, 1994).

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983), p. 172.

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Figures (12)

Fig. 1
Fig. 1

Light ray coming from direction L hitting a surface element d A located according to vector G. The bold half-sphere represents the area illuminated by the L directed light rays.

Fig. 2
Fig. 2

Multiple reflection of light within a spherical transparent particle of refractive index n 2 surrounded by a medium of refractive index n 1 < n 2 . All light rays belong to a same plane through the center of the particle.

Fig. 3
Fig. 3

Evolution of r 1 , r 2 , r 3 , r 4 + , and r S as functions of diametrical absorbance.

Fig. 4
Fig. 4

Backward component r S , diffuse nonabsorbance f S and backscattering ratio x = r S f S as functions of diametrical absorbance.

Fig. 5
Fig. 5

Evolution of the backscattering ratio as functions of diametrical absorbance for various relative refractive indices.

Fig. 6
Fig. 6

Infinitely thick particle medium modeled as an infinite number of particle sublayers.

Fig. 7
Fig. 7

Infinite particle medium reflectance as a function of diametrical absorbance for various relative refractive indices with a shadowing ratio a = 0.5 .

Fig. 8
Fig. 8

Evolution of the infinite particle medium reflectance as a function of diametrical absorbance for various shadowing ratios.

Fig. 9
Fig. 9

Spherical transparent particles in a clear binding medium forming a flat interface with a different external medium.

Fig. 10
Fig. 10

Infinite particle medium reflectance factor as a function of the diametrical absorbance for various binder refractive indices n 1 ( n 2 = 1.65 ) and for a diffuse 0 ° measuring geometry in air.

Fig. 11
Fig. 11

Diffuse nonabsorbance f S given by our model (solid curve), Melamed’s model (dashed curve), and the model of Shkuratov et al. (dotted curve) as functions of the diametrical absorbance.

Fig. 12
Fig. 12

Backward component r s and contribution r a of the second and following scattered rays according to our model (solid curve) and to the model of Shkuratov et al. (dashed curve) as functions of the particle absorbance.

Equations (80)

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α ( λ ) = 4 π n ( λ ) κ ( λ ) λ .
t ( λ ) = e α ( λ ) d .
n i sin θ i = n j sin θ j .
R i j p ( θ i ) = tan 2 ( θ i θ j ) tan 2 ( θ i + θ j )
T i j p ( θ i ) = 1 R i j p ( θ i ) .
R i j s ( θ i ) = sin 2 ( θ i θ j ) sin 2 ( θ i + θ j )
T i j s ( θ i ) = 1 R i j s ( θ i ) .
R i j u ( θ i ) = 1 2 ( R i j s ( θ i ) + R i j p ( θ i ) ) .
T j i ( θ j ) = T i j ( θ i )
R j i ( θ j ) = R i j ( θ i ) .
R j i ( θ j ) = 1 ,
T j i ( θ j ) = 0 .
r i j = θ i = 0 π 2 R i j ( θ i ) sin 2 θ i d θ i .
t i j = 1 r i j .
t j i = ( n i n j ) 2 t i j .
1 r j i = ( n i n j ) 2 ( 1 r i j ) .
L = ( sin ψ cos ϕ sin ψ sin ϕ cos ψ )
L = ( sin ψ 0 cos ψ ) .
G = ( sin θ 1 cos ϕ 1 sin θ 1 sin ϕ 1 cos θ 1 ) ,
L = ( 0 0 1 ) .
G = ( cos ψ 0 sin ψ 0 1 0 sin ψ 0 cos ψ ) ( sin θ 1 cos ϕ 1 sin θ 1 sin ϕ 1 cos θ 1 ) = ( cos ψ sin θ 1 cos ϕ 1 + sin ψ cos θ 1 sin θ 1 sin ϕ 1 cos ψ cos θ 1 sin ψ sin θ 1 cos ϕ 1 ) ,
d 2 Φ i ( L , G ) = E i π d A cos θ 1 d ω ,
d 2 Φ i ( L , G ) = E i π r 2 cos θ 1 sin θ 1 d θ 1 d ϕ 1 sin ψ d ψ d ϕ .
Φ i = r 2 E i π ϕ = 0 2 π ψ = 0 π 2 [ ϕ 1 = 0 2 π θ 1 = 0 π 2 cos θ 1 sin θ 1 d θ 1 d ϕ 1 ] sin ψ d ψ d ϕ = 2 π r 2 E i .
Φ r = r 2 E i π ϕ = 0 2 π ψ = 0 π 2 [ ϕ 1 = 0 2 π θ 1 = 0 π 2 R 12 ( θ 1 ) cos θ 1 sin θ 1 d θ 1 d ϕ 1 ] sin ψ d ψ d ϕ .
Φ r = 2 π r 2 E i θ 1 = 0 π 2 R 12 ( θ 1 ) sin 2 θ 1 d θ 1 .
r 12 = θ 1 = 0 π 2 R 12 ( θ 1 ) sin 2 θ 1 d θ 1 .
t ( θ 1 ) = e α d cos θ 2 = e α d 1 ( n 1 sin θ 1 n 2 ) 2 .
F S ( θ 1 ) = R 12 ( θ 1 ) + T 12 ( θ 1 ) T 21 ( θ 2 ) t ( θ 1 ) k = 0 [ R 21 ( θ 2 ) t ( θ 1 ) ] k .
F S ( θ 1 ) = R 12 ( θ 1 ) + ( 1 R 12 ( θ 1 ) ) 2 t ( θ 1 ) 1 R 12 ( θ 1 ) t ( θ 1 ) .
d 2 Φ S ( L , G ) = F S ( θ 1 ) d 2 Φ i ( L , G ) .
Φ S = r 2 E i π ϕ = 0 2 π ψ = 0 π 2 ϕ 1 = 0 2 π θ 1 = 0 π 2 F S ( θ 1 ) cos θ 1 sin θ 1 sin ψ d θ 1 d ϕ 1 d ψ d ϕ = 2 π r 2 E i θ 1 = 0 π 2 F S ( θ 1 ) sin 2 θ 1 d θ 1 .
f S = θ 1 = 0 π 2 F S ( θ 1 ) sin 2 θ 1 d θ 1 ,
f S = r 12 + θ 1 = 0 π 2 ( 1 R 12 ( θ 1 ) ) 2 t ( θ 1 ) 1 R 12 ( θ 1 ) t ( θ 1 ) sin 2 θ 1 d θ 1 .
r S = x f S ,
t S = f S r S = ( 1 x ) f S .
θ ( 1 ) = π 2 θ 1 ,
θ ( N ) θ ( N 1 ) = 2 θ 2 π , N 1 ,
θ ( N ) = 2 ( N 1 ) θ 2 2 θ 1 ( N 2 ) π mod ( 2 π ) .
I = G × L G × L = ( cos ψ sin ϕ 1 cos ϕ 1 sin ψ sin ϕ 1 ) .
L N = cos θ ( N ) ( L ) + sin θ ( N ) I × ( L ) , N 1 ,
L N = ( cos θ ( N ) sin ψ + sin θ ( N ) cos ψ cos ϕ 1 sin θ ( N ) sin ϕ 1 cos θ ( N ) cos ψ sin θ ( N ) sin ψ cos ϕ 1 ) , N 1 .
cos ψ ( N ) = ( cos θ ( N ) cos ψ + sin θ ( N ) sin ψ cos ϕ 1 ) , N 1 .
H ( cos ψ ( N ) ) = { 1 if cos ψ ( N ) > 0 0 otherwise .
F 1 ( θ 1 ) = R 12 ( θ 1 ) .
F N ( θ 1 ) = T 12 2 ( θ 1 ) R 12 N 2 ( θ 1 ) t N 1 ( θ 1 ) .
Φ N = r 2 E i ψ = 0 π 2 ϕ 1 = 0 2 π θ 1 = 0 π 2 H ( cos ψ ( N ) ) F N ( θ 1 ) sin 2 θ 1 sin ψ d θ 1 d ϕ 1 d ψ .
r N = Φ N Φ i ,
r S = N = 1 r N .
r S = r 1 + r 2 + r 3 + r 4 + .
Φ 4 + = r 2 E i π ϕ = 0 2 π ψ = 0 π 2 [ ϕ 1 = 0 2 π θ 1 = 0 π 2 ( T 12 ( θ 1 ) T 21 ( θ 2 ) t ( θ 1 ) k = 2 [ R 21 ( θ 2 ) t ( θ 1 ) ] k ) cos θ 1 sin θ 1 d θ 1 d ϕ 1 ] sin ψ d ψ d ϕ .
Φ 4 + = 2 π r 2 E i θ 1 = 0 π 2 T 12 2 ( θ 1 ) R 12 2 ( θ 1 ) t 3 ( θ 1 ) 1 R 12 ( θ 1 ) t ( θ 1 ) sin 2 θ 1 d θ 1 .
r 4 + = 1 2 Φ 4 + Φ i = 1 2 θ 1 = 0 π 2 T 12 2 ( θ 1 ) R 12 2 ( θ 1 ) t 3 ( θ 1 ) 1 R 12 ( θ 1 ) t ( θ 1 ) sin 2 θ 1 d θ 1 .
r 1 = 1 2 π ψ = 0 π 2 ϕ 1 = 0 2 π θ 1 = 0 π 2 H ( cos ψ ( 1 ) ) R 12 ( θ 1 ) sin 2 θ 1 sin ψ d θ 1 d ϕ 1 d ψ
r L = a r S = a x f S ,
t L = 1 a + a ( 1 x ) f S .
t M = θ = 0 π 2 e α M d cos θ sin 2 θ d θ .
t L = ( 1 a ) t M + a ( 1 x ) f S .
r L = k = 1 N a k x k f S k
t L = 1 k = 1 N a k + k = 1 N a k ( 1 x k ) f S k .
r 1 + = r L + t L 2 r 1 r L r .
r 2 1 + r L 2 t L 2 r L r + 1 = 0
r = 1 + r L 2 t L 2 2 r L ( 1 + r L 2 t L 2 2 r L ) 2 1 .
r = A A 2 1
A = ( 1 f S ) 2 x f S + ( 1 a ( 1 f S ) ) ( 1 + ( 1 f S ) 2 x f S ) .
ρ ( θ 0 , θ 0 ) = ( n 0 n 1 ) 2 T 01 ( θ 0 ) T 01 ( θ 0 ) r 1 r r 10 .
ρ ( d , 0 ) = ( n 0 n 1 ) 2 t 01 T 01 ( 0 ) r 1 r r 10 .
ρ ( d , 0 ) = ( 1 r 10 ) T 01 ( 0 ) r 1 r r 10 .
M = e α d ¯ ,
d ¯ = θ 2 = 0 π 2 d cos θ 2 sin 2 θ 2 d θ 2 = 2 d 3 .
f S = r 12 + t 12 t 21 M k = 0 [ r 21 M ] k = r 12 + t 12 t 21 M 1 r 21 M .
M = θ 2 = 0 π 2 e α d cos θ 2 sin 2 θ 2 d θ 2 = 2 ( α d ) 2 ( 1 ( α d + 1 ) e α d ) .
r 1 = θ 1 = 0 π 4 R 12 ( θ 1 ) sin 2 θ 1 d θ 1 .
r ̃ 21 = θ 2 = 0 arcsin ( n 0 n 1 ) R 21 ( θ 2 ) sin 2 θ 2 d θ 2 θ 2 = 0 arcsin ( n 1 n 2 ) sin 2 θ 2 d θ 2 = ( n 1 n 2 ) 2 r 12 ( n 1 n 2 ) 2 = r 12
f S = r 12 + ( 1 r 12 ) 2 M 1 r 12 M .
d ¯ = θ = 0 arcsin ( n 1 n 2 ) d cos θ sin 2 θ d θ θ = 0 arcsin ( n 1 n 2 ) sin 2 θ d θ ,
M = exp ( α d ¯ ) = exp [ ( 2 3 ) α d ( n 2 n 1 ) 2 ( 1 μ 3 ) ]
μ = 1 ( n 2 n 1 ) 2 .
M = θ = 0 arcsin ( n 1 n 2 ) e α d cos θ sin 2 θ d θ θ = 0 arcsin ( n 1 n 2 ) sin 2 θ d θ ,
M = 2 ( n 2 n 1 ) 2 ( α d ) 2 [ e α d μ ( 1 + α d μ ) e α d ( 1 + α d ) ] .

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