Abstract

The Clapper–Yule model is the only classical spectral reflection model for halftone prints that takes explicitly into account both the multiple internal reflections between the print–air interface and the paper substrate and the lateral propagation of light within the paper bulk. However, the Clapper–Yule model assumes a planar interface and does not take into account the roughness of the print surface. In order to extend the Clapper–Yule model to rough printing supports (e.g., matte coated papers or calendered papers), we model the print surface as a set of randomly oriented microfacets. The influence of the shadowing effect is evaluated and incorporated into the model. By integrating over all incident angles and facet orientations, we are able to express the internal reflectance of the rough interface as a function of the rms facet slope. By considering also the rough interface transmittances both for the incident light and for the emerging light, we obtain a generalization of the Clapper–Yule model for rough interfaces. The comparison between the classical Clapper–Yule model and the model extended to rough surfaces shows that the influence of the surface roughness on the predicted reflectance factor is small. For high-quality papers such as coated and calendered papers, as well as for low-quality papers such as newsprint or copy papers, the influence of surface roughness is negligible, and the classical Clapper–Yule model can be used to predict the halftone-print reflectance factors. The influence of roughness becomes significant only for very rough and thick nondiffusing coatings.

© 2005 Optical Society of America

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References

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  1. J. Yule, W. Nielsen, “The penetration of light into paper and its effect on halftone reproductions,” in Proceedings of the Technical Association of the Graphic Arts (TAGA) Conference, 1951, pp. 65–76. http://www.taga.org.
  2. J. A. S. Viggiano, “Modeling the color of multi-color halftones,” in Proceedings of the Technical Association of the Graphic Arts (TAGA) Conference, 1990, pp. 44–62. http://www.taga.org.
  3. R. Balasubramanian, “Optimization of the spectral Neugebauer model for printer characterization,” J. Electron. Imaging 8, 156–166 (1999).
    [CrossRef]
  4. K. Iino, R. S. Berns, “Building color management modules using linear optimization I. Desktop,” J. Imaging Sci. Technol. 41, 79–94 (1998).
  5. K. Iino, R. S. Berns, “Building color management modules using linear optimization II. Prepress system for offset printing,” J. Imaging Sci. Technol. 42, 99–114 (1998).
  6. F. Ruckdeschel, O. Hauser, “Yule–Nielsen in printing: a physical analysis,” Appl. Opt. 17, 3376–3383 (1978).
    [CrossRef] [PubMed]
  7. J. Arney, “A probability description of the Yule–Nielsen effect I,” J. Imaging Sci. Technol. 41, 633–636 (1997).
  8. G. Rogers, “Optical dot gain: lateral scattering probabilities,” J. Imaging Sci. Technol. 42, 341–345 (1998).
  9. L. Yang, R. Lenz, B. Kruse, “Light scattering and ink penetration effects on tone reproduction,” J. Opt. Soc. Am. A 18, 360–366 (2001).
    [CrossRef]
  10. F. Clapper, J. Yule, “The effect of multiple internal reflections on the densities of halftone prints on paper,” J. Opt. Soc. Am. 43, 600–603 (1953).
    [CrossRef]
  11. G. Rogers, “A generalized Clapper–Yule model of halftone reflectance,” Color Res. Appl. 25, 402–407 (2000).
    [CrossRef]
  12. P. Emmel, R. D. Hersch. “A unified model for color prediction of halftoned prints,” J. Imaging Sci. Technol. 44, 351–359 (2000).
  13. R. D. Hersch, F. Collaud, F. Crété, P. Emmel, “Spectral prediction and dot surface estimation models for halftone prints,” in Color Imaging IX: Processing, Hardcopy, and Applications, R. Eschbach and G. G. Marcu; eds., Proc. SPIE5293, 356–369 (2004).
  14. J. A. Sanchez-Gil, M. Nieto-Vesperinas, “Light scattering from random rough dielectric surfaces,” J. Opt. Soc. Am. A 8, 1270–1286 (1991).
    [CrossRef]
  15. J. Caron, J. Lafait, C. Andraud, “Scalar Kirchhoff’s model for light scattering from dielectric random rough surfaces,” Opt. Commun. 207, 17–28 (2002).
    [CrossRef]
  16. T. Germer, “Polarized light diffusely scattered under smooth and rough interfaces,” Polarization Science and Remote Sensing, J. A. Shaw and J. S. Tyo, eds., Proc. SPIE5158, 193–204 (2003).
  17. S. K. Nayar, K. Ikeuchi, T. Kanade, “Surface reflection: physical and geometrical perspectives,” IEEE Trans. Pattern Anal. Mach. Intell. 13, 611–634 (1991).
    [CrossRef]
  18. M. Born, E. Wolf, Principle of Optics, 7th expanded ed. (Pergamon, 1999), p. 47.
  19. D. B. Judd, “Fresnel reflection of diffusely incident light,” J. Res. Natl. Bur. Stand. 29, 329–332 (1942).
    [CrossRef]
  20. W. R. McCluney, Introduction to Radiometry and Photometry (Artech House, 1994) pp. 7–13.
  21. K. E. Torrance, E. M. Sparrow, “Theory of off-specular reflection from roughened surfaces,” J. Opt. Soc. Am. 57, 1104–1114 (1967).
    [CrossRef]
  22. D. Becker, K. Kasper, “Digital prints: technology, materials, image quality & stability,” http://www.foto.unibas.ch/~rundbrief/les33.htm.
  23. B. G. Smith, “Geometrical shadowing of a random rough surface,” IEEE Trans. Antennas Propag. AP-15, 668–671 (1967).
    [CrossRef]
  24. N. C. Bruce, “On the validity of the inclusion of geometrical shadowing functions in the multiple-scatter Kirchhoff approximation,” Waves Random Media 14, 1–12 (2004).
    [CrossRef]
  25. P. Hansson, “Geometrical modeling of light scattering from paper substrates,” Diploma work (Department of Engineering Sciences, The Ångström Laboratory, Uppsala University, Uppsala, Sweden, 2003). http://www.angstrom.uu.se/solidstatephysics/dissertation_ram/peter_hansson_art/Paper9_v15pdf.
  26. F. C. Williams, F. R. Clapper, “Multiple internal reflections in photographic color prints,” J. Opt. Soc. Am. 43, 595–599 (1953).
    [CrossRef] [PubMed]
  27. M. C. Béland, J. M. Bennett, “Effect of local microroughness on the gloss uniformity of printed paper surfaces,” Appl. Opt. 39, 2719–2726 (2000).
    [CrossRef]

2004 (1)

N. C. Bruce, “On the validity of the inclusion of geometrical shadowing functions in the multiple-scatter Kirchhoff approximation,” Waves Random Media 14, 1–12 (2004).
[CrossRef]

2002 (1)

J. Caron, J. Lafait, C. Andraud, “Scalar Kirchhoff’s model for light scattering from dielectric random rough surfaces,” Opt. Commun. 207, 17–28 (2002).
[CrossRef]

2001 (1)

2000 (3)

M. C. Béland, J. M. Bennett, “Effect of local microroughness on the gloss uniformity of printed paper surfaces,” Appl. Opt. 39, 2719–2726 (2000).
[CrossRef]

G. Rogers, “A generalized Clapper–Yule model of halftone reflectance,” Color Res. Appl. 25, 402–407 (2000).
[CrossRef]

P. Emmel, R. D. Hersch. “A unified model for color prediction of halftoned prints,” J. Imaging Sci. Technol. 44, 351–359 (2000).

1999 (1)

R. Balasubramanian, “Optimization of the spectral Neugebauer model for printer characterization,” J. Electron. Imaging 8, 156–166 (1999).
[CrossRef]

1998 (3)

K. Iino, R. S. Berns, “Building color management modules using linear optimization I. Desktop,” J. Imaging Sci. Technol. 41, 79–94 (1998).

K. Iino, R. S. Berns, “Building color management modules using linear optimization II. Prepress system for offset printing,” J. Imaging Sci. Technol. 42, 99–114 (1998).

G. Rogers, “Optical dot gain: lateral scattering probabilities,” J. Imaging Sci. Technol. 42, 341–345 (1998).

1997 (1)

J. Arney, “A probability description of the Yule–Nielsen effect I,” J. Imaging Sci. Technol. 41, 633–636 (1997).

1991 (2)

S. K. Nayar, K. Ikeuchi, T. Kanade, “Surface reflection: physical and geometrical perspectives,” IEEE Trans. Pattern Anal. Mach. Intell. 13, 611–634 (1991).
[CrossRef]

J. A. Sanchez-Gil, M. Nieto-Vesperinas, “Light scattering from random rough dielectric surfaces,” J. Opt. Soc. Am. A 8, 1270–1286 (1991).
[CrossRef]

1978 (1)

1967 (2)

K. E. Torrance, E. M. Sparrow, “Theory of off-specular reflection from roughened surfaces,” J. Opt. Soc. Am. 57, 1104–1114 (1967).
[CrossRef]

B. G. Smith, “Geometrical shadowing of a random rough surface,” IEEE Trans. Antennas Propag. AP-15, 668–671 (1967).
[CrossRef]

1953 (2)

1942 (1)

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

Andraud, C.

J. Caron, J. Lafait, C. Andraud, “Scalar Kirchhoff’s model for light scattering from dielectric random rough surfaces,” Opt. Commun. 207, 17–28 (2002).
[CrossRef]

Arney, J.

J. Arney, “A probability description of the Yule–Nielsen effect I,” J. Imaging Sci. Technol. 41, 633–636 (1997).

Balasubramanian, R.

R. Balasubramanian, “Optimization of the spectral Neugebauer model for printer characterization,” J. Electron. Imaging 8, 156–166 (1999).
[CrossRef]

Becker, D.

D. Becker, K. Kasper, “Digital prints: technology, materials, image quality & stability,” http://www.foto.unibas.ch/~rundbrief/les33.htm.

Béland, M. C.

Bennett, J. M.

Berns, R. S.

K. Iino, R. S. Berns, “Building color management modules using linear optimization I. Desktop,” J. Imaging Sci. Technol. 41, 79–94 (1998).

K. Iino, R. S. Berns, “Building color management modules using linear optimization II. Prepress system for offset printing,” J. Imaging Sci. Technol. 42, 99–114 (1998).

Born, M.

M. Born, E. Wolf, Principle of Optics, 7th expanded ed. (Pergamon, 1999), p. 47.

Bruce, N. C.

N. C. Bruce, “On the validity of the inclusion of geometrical shadowing functions in the multiple-scatter Kirchhoff approximation,” Waves Random Media 14, 1–12 (2004).
[CrossRef]

Caron, J.

J. Caron, J. Lafait, C. Andraud, “Scalar Kirchhoff’s model for light scattering from dielectric random rough surfaces,” Opt. Commun. 207, 17–28 (2002).
[CrossRef]

Clapper, F.

Clapper, F. R.

Collaud, F.

R. D. Hersch, F. Collaud, F. Crété, P. Emmel, “Spectral prediction and dot surface estimation models for halftone prints,” in Color Imaging IX: Processing, Hardcopy, and Applications, R. Eschbach and G. G. Marcu; eds., Proc. SPIE5293, 356–369 (2004).

Crété, F.

R. D. Hersch, F. Collaud, F. Crété, P. Emmel, “Spectral prediction and dot surface estimation models for halftone prints,” in Color Imaging IX: Processing, Hardcopy, and Applications, R. Eschbach and G. G. Marcu; eds., Proc. SPIE5293, 356–369 (2004).

Emmel, P.

P. Emmel, R. D. Hersch. “A unified model for color prediction of halftoned prints,” J. Imaging Sci. Technol. 44, 351–359 (2000).

R. D. Hersch, F. Collaud, F. Crété, P. Emmel, “Spectral prediction and dot surface estimation models for halftone prints,” in Color Imaging IX: Processing, Hardcopy, and Applications, R. Eschbach and G. G. Marcu; eds., Proc. SPIE5293, 356–369 (2004).

Germer, T.

T. Germer, “Polarized light diffusely scattered under smooth and rough interfaces,” Polarization Science and Remote Sensing, J. A. Shaw and J. S. Tyo, eds., Proc. SPIE5158, 193–204 (2003).

Hansson, P.

P. Hansson, “Geometrical modeling of light scattering from paper substrates,” Diploma work (Department of Engineering Sciences, The Ångström Laboratory, Uppsala University, Uppsala, Sweden, 2003). http://www.angstrom.uu.se/solidstatephysics/dissertation_ram/peter_hansson_art/Paper9_v15pdf.

Hauser, O.

Hersch, R. D.

P. Emmel, R. D. Hersch. “A unified model for color prediction of halftoned prints,” J. Imaging Sci. Technol. 44, 351–359 (2000).

R. D. Hersch, F. Collaud, F. Crété, P. Emmel, “Spectral prediction and dot surface estimation models for halftone prints,” in Color Imaging IX: Processing, Hardcopy, and Applications, R. Eschbach and G. G. Marcu; eds., Proc. SPIE5293, 356–369 (2004).

Iino, K.

K. Iino, R. S. Berns, “Building color management modules using linear optimization II. Prepress system for offset printing,” J. Imaging Sci. Technol. 42, 99–114 (1998).

K. Iino, R. S. Berns, “Building color management modules using linear optimization I. Desktop,” J. Imaging Sci. Technol. 41, 79–94 (1998).

Ikeuchi, K.

S. K. Nayar, K. Ikeuchi, T. Kanade, “Surface reflection: physical and geometrical perspectives,” IEEE Trans. Pattern Anal. Mach. Intell. 13, 611–634 (1991).
[CrossRef]

Judd, D. B.

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

Kanade, T.

S. K. Nayar, K. Ikeuchi, T. Kanade, “Surface reflection: physical and geometrical perspectives,” IEEE Trans. Pattern Anal. Mach. Intell. 13, 611–634 (1991).
[CrossRef]

Kasper, K.

D. Becker, K. Kasper, “Digital prints: technology, materials, image quality & stability,” http://www.foto.unibas.ch/~rundbrief/les33.htm.

Kruse, B.

Lafait, J.

J. Caron, J. Lafait, C. Andraud, “Scalar Kirchhoff’s model for light scattering from dielectric random rough surfaces,” Opt. Commun. 207, 17–28 (2002).
[CrossRef]

Lenz, R.

McCluney, W. R.

W. R. McCluney, Introduction to Radiometry and Photometry (Artech House, 1994) pp. 7–13.

Nayar, S. K.

S. K. Nayar, K. Ikeuchi, T. Kanade, “Surface reflection: physical and geometrical perspectives,” IEEE Trans. Pattern Anal. Mach. Intell. 13, 611–634 (1991).
[CrossRef]

Nielsen, W.

J. Yule, W. Nielsen, “The penetration of light into paper and its effect on halftone reproductions,” in Proceedings of the Technical Association of the Graphic Arts (TAGA) Conference, 1951, pp. 65–76. http://www.taga.org.

Nieto-Vesperinas, M.

Rogers, G.

G. Rogers, “A generalized Clapper–Yule model of halftone reflectance,” Color Res. Appl. 25, 402–407 (2000).
[CrossRef]

G. Rogers, “Optical dot gain: lateral scattering probabilities,” J. Imaging Sci. Technol. 42, 341–345 (1998).

Ruckdeschel, F.

Sanchez-Gil, J. A.

Smith, B. G.

B. G. Smith, “Geometrical shadowing of a random rough surface,” IEEE Trans. Antennas Propag. AP-15, 668–671 (1967).
[CrossRef]

Sparrow, E. M.

K. E. Torrance, E. M. Sparrow, “Theory of off-specular reflection from roughened surfaces,” J. Opt. Soc. Am. 57, 1104–1114 (1967).
[CrossRef]

Torrance, K. E.

K. E. Torrance, E. M. Sparrow, “Theory of off-specular reflection from roughened surfaces,” J. Opt. Soc. Am. 57, 1104–1114 (1967).
[CrossRef]

Viggiano, J. A. S.

J. A. S. Viggiano, “Modeling the color of multi-color halftones,” in Proceedings of the Technical Association of the Graphic Arts (TAGA) Conference, 1990, pp. 44–62. http://www.taga.org.

Williams, F. C.

Wolf, E.

M. Born, E. Wolf, Principle of Optics, 7th expanded ed. (Pergamon, 1999), p. 47.

Yang, L.

Yule, J.

F. Clapper, J. Yule, “The effect of multiple internal reflections on the densities of halftone prints on paper,” J. Opt. Soc. Am. 43, 600–603 (1953).
[CrossRef]

J. Yule, W. Nielsen, “The penetration of light into paper and its effect on halftone reproductions,” in Proceedings of the Technical Association of the Graphic Arts (TAGA) Conference, 1951, pp. 65–76. http://www.taga.org.

Appl. Opt. (2)

Color Res. Appl. (1)

G. Rogers, “A generalized Clapper–Yule model of halftone reflectance,” Color Res. Appl. 25, 402–407 (2000).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

B. G. Smith, “Geometrical shadowing of a random rough surface,” IEEE Trans. Antennas Propag. AP-15, 668–671 (1967).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell. (1)

S. K. Nayar, K. Ikeuchi, T. Kanade, “Surface reflection: physical and geometrical perspectives,” IEEE Trans. Pattern Anal. Mach. Intell. 13, 611–634 (1991).
[CrossRef]

J. Electron. Imaging (1)

R. Balasubramanian, “Optimization of the spectral Neugebauer model for printer characterization,” J. Electron. Imaging 8, 156–166 (1999).
[CrossRef]

J. Imaging Sci. Technol. (5)

K. Iino, R. S. Berns, “Building color management modules using linear optimization I. Desktop,” J. Imaging Sci. Technol. 41, 79–94 (1998).

K. Iino, R. S. Berns, “Building color management modules using linear optimization II. Prepress system for offset printing,” J. Imaging Sci. Technol. 42, 99–114 (1998).

P. Emmel, R. D. Hersch. “A unified model for color prediction of halftoned prints,” J. Imaging Sci. Technol. 44, 351–359 (2000).

J. Arney, “A probability description of the Yule–Nielsen effect I,” J. Imaging Sci. Technol. 41, 633–636 (1997).

G. Rogers, “Optical dot gain: lateral scattering probabilities,” J. Imaging Sci. Technol. 42, 341–345 (1998).

J. Opt. Soc. Am. (3)

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

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

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

Opt. Commun. (1)

J. Caron, J. Lafait, C. Andraud, “Scalar Kirchhoff’s model for light scattering from dielectric random rough surfaces,” Opt. Commun. 207, 17–28 (2002).
[CrossRef]

Waves Random Media (1)

N. C. Bruce, “On the validity of the inclusion of geometrical shadowing functions in the multiple-scatter Kirchhoff approximation,” Waves Random Media 14, 1–12 (2004).
[CrossRef]

Other (8)

P. Hansson, “Geometrical modeling of light scattering from paper substrates,” Diploma work (Department of Engineering Sciences, The Ångström Laboratory, Uppsala University, Uppsala, Sweden, 2003). http://www.angstrom.uu.se/solidstatephysics/dissertation_ram/peter_hansson_art/Paper9_v15pdf.

D. Becker, K. Kasper, “Digital prints: technology, materials, image quality & stability,” http://www.foto.unibas.ch/~rundbrief/les33.htm.

W. R. McCluney, Introduction to Radiometry and Photometry (Artech House, 1994) pp. 7–13.

M. Born, E. Wolf, Principle of Optics, 7th expanded ed. (Pergamon, 1999), p. 47.

T. Germer, “Polarized light diffusely scattered under smooth and rough interfaces,” Polarization Science and Remote Sensing, J. A. Shaw and J. S. Tyo, eds., Proc. SPIE5158, 193–204 (2003).

R. D. Hersch, F. Collaud, F. Crété, P. Emmel, “Spectral prediction and dot surface estimation models for halftone prints,” in Color Imaging IX: Processing, Hardcopy, and Applications, R. Eschbach and G. G. Marcu; eds., Proc. SPIE5293, 356–369 (2004).

J. Yule, W. Nielsen, “The penetration of light into paper and its effect on halftone reproductions,” in Proceedings of the Technical Association of the Graphic Arts (TAGA) Conference, 1951, pp. 65–76. http://www.taga.org.

J. A. S. Viggiano, “Modeling the color of multi-color halftones,” in Proceedings of the Technical Association of the Graphic Arts (TAGA) Conference, 1990, pp. 44–62. http://www.taga.org.

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Figures (5)

Fig. 1
Fig. 1

Diagram of the interaction of light with the print according to the Clapper–Yule model.

Fig. 2
Fig. 2

Element d s of a planar interface, of normal vector N, receives a radiance from direction V , within a solid angle d V N = d V N = sin θ d θ d ϕ . The radiance transmitted into the air, in direction R within a solid angle d R N = sin θ r d θ r d ϕ r , is equal to the radiance received by the surface d s d of the detector within its solid angle d R R . Vectors V, N, and R are related according to Snell’s refraction law.

Fig. 3
Fig. 3

(a) Cross section of an ink-jet print on uncoated paper (paper thickness of 106 μ m ), (b) cross section of an ink-jet print on coated paper (paper thickness of 118 μ m ), (c) cross section of an ink-jet print on resin-coated paper (paper thickness of 157 μ m ); courtesy of Becker and Kasper.[22]

Fig. 4
Fig. 4

Smith’s illumination probability function for interface’s rms slopes of m = 0.05 (solid curve) and m = 0.2 (dashed curve). The illumination probability of facets inclined with an angle of 45° is plotted as a function of the light incidence angle θ. The illumination probability is zero for incident angles θ < 45 ° .

Fig. 5
Fig. 5

Inclined facet elements d s of a rough interface, of normal vector H, receives a radiance from direction V , within a solid angle d V H = d V H = sin θ d θ d ϕ . The radiance transmitted into the air, in direction R within a solid angle d R H = sin θ r d θ r d ϕ r , is equal to the radiance received by the surface d s d of the detector within its solid angle d R R . Vectors V, H, and R are related according to Snell’s refraction law.

Tables (6)

Tables Icon

Table 1 Main Expressions Resulting from the Classical and Extended Clapper-Yule Models a

Tables Icon

Table 2 Rms Slopes of Various Paper Types a

Tables Icon

Table 3 Evaluation of Roughness-Dependent Terms for Various Roughnesses (rms Slope m) a

Tables Icon

Table 4 Deviations of Reflectance Factor R Ω w a

Tables Icon

Table 5 Deviations of Reflectance Factor R δ w a

Tables Icon

Table 6 Deviations of Reflectance Factor R p = R Ω p = R δ p a

Equations (84)

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W i = T n ν ( L , N ) V i ,
W 0 = ρ B ( 1 a + a t ) W i .
W 0 r = ( 1 a + a t ) W 0 .
W 1 = r i ρ B ( 1 a + a t 2 ) W 0 .
W k = ( r i ρ B ( 1 a + a t 2 ) ) k W 0 , k = 1 , 2 , 3 .
W k r = ( 1 a + a t ) W k , k = 1 , 2 , 3 ,
W r = k = 0 W k r = ( 1 a + a t ) [ k = 0 ( r i ρ B ( 1 a + a t 2 ) ) k ] W 0 ,
W r = ρ B ( 1 a + a t ) 2 1 r i ρ B ( 1 a + a t 2 ) W i .
d 2 Φ i = E i π d s V N d V N ,
d 2 Φ r = F 1 n ν ( V , N ) E i π d s V N d V N .
E r = d Φ r d s = E i π Ω N F 1 n ν ( V , N ) V N d V N .
r i = 1 π Ω N F 1 n ν ( V , N ) V N d V N .
r i = 1 π ϕ = 0 2 π θ = 0 π 2 F 1 n ν ( θ ) cos θ sin θ d θ d ϕ .
r i = θ = 0 π 2 F 1 n ν ( θ ) sin 2 θ d θ .
t i = 1 r i .
V r = ( 1 r i ) W r .
V r = T n ν ( L , N ) ρ B ( 1 a + a t ) 2 1 r i ρ B ( 1 a + a t 2 ) ( 1 r i ) V i .
L r = d 2 Φ d s d d R R = d 2 Φ d s R N d R N ,
d 2 Φ = T n ν ( R , N ) d 2 Φ i .
ϕ r = ϕ ,
sin θ r = n ν sin θ .
W r π = d 2 Φ i d s V N d V N .
d ϕ r = d ϕ ,
cos θ r d θ r = n ν cos θ d θ .
V N d V N R N d R N = cos θ sin θ d θ d ϕ cos θ r sin θ r d θ r d ϕ r = 1 n ν 2 .
L r = T n ν ( R , N ) 1 n ν 2 W r π ,
L r = T n ν ( L , N ) ρ B ( 1 a + a t ) 2 1 r i ρ B ( 1 a + a t 2 ) T n ν ( R , N ) n ν 2 V i π .
R Ω w = T n ν ( L , N ) ( 1 r i ) ρ B ( 1 a + a t λ ) 2 1 r i ρ B ( 1 a + a t λ 2 ) .
R Ω p = V r V i V r ( a = 0 ) V i = ( 1 r i ρ B ) ( 1 a + a t ) 2 1 r i ρ B ( 1 a + a t 2 ) .
R δ w = T n ν ( L , N ) T n ν ( R , N ) n ν 2 ρ B ( 1 a + a t ) 2 1 r i ρ B ( 1 a + a t 2 ) .
R δ p = L r L r ( a = 0 ) = ( 1 r i ρ B ) ( 1 a + a t ) 2 1 r i ρ B ( 1 a + a t 2 ) .
( sin θ h cos ϕ h ) x + ( sin θ h sin ϕ h ) y + ( cos θ h ) z = 0 .
z = ( tan θ h cos ϕ h ) x ( tan θ h sin ϕ h ) y .
s x = z x = tan θ h cos ϕ h ,
s y = z y = tan θ h sin ϕ h .
s = s x 2 + s y 2 = tan θ h .
f ( s x ) = exp ( s x 2 2 m 2 ) 2 π m , f ( s y ) = exp ( s y 2 2 m 2 ) 2 π m .
P ( s ) = P ( s x ) P ( s y ) = exp ( s 2 2 m 2 ) 2 π m 2 d s x d s y .
d s x d s y d H N = d s x d s y sin θ h d θ h d ϕ h = 1 sin θ h s x θ h s x ϕ h s y θ h s y ϕ h = 1 cos 3 θ h .
P ( H ) = D ( H ) d H N ,
D ( H ) = exp ( tan 2 θ h 2 m 2 ) 2 π m 2 cos 3 θ h .
Ω N D ( H ) d H N = 1 .
G m ( V , H ) = h ( V H ) Λ m ( θ ) + 1 = { 1 Λ m ( θ ) + 1 ( if V Ω H ) 0 ( if V Ω H ) } ,
Λ m ( θ ) = 1 2 [ 1 π 2 m cot θ exp ( cot 2 θ 2 m 2 ) erfc ( cot θ 2 m ) ] .
θ shad = π 2 arctan ( 2 m ) .
W i = V i Ω N T n ν ( L , H ) D ( H ) d H N .
τ n ν ( L ) = Ω N T n ν ( L , H ) D ( H ) d H N .
W r = ρ B ( 1 a + a t ) 2 1 r ¯ i ρ B ( 1 a + a t 2 ) W i .
d 2 Φ i ( V ) = L i d s V N d V N .
d 2 Φ i ( V , H ) = G m ( V , H ) P ( H ) d 2 Φ i ( V ) ,
d 2 Φ i ( V , H ) = L i d s G m ( V , H ) D ( H ) V N d V N d H N .
d 2 Φ r ( V , H ) = L i d s F 1 n ν ( V , H ) G m ( V , H ) D ( H ) V N d V N d H N .
r i = E r E i = H Ω N V Ω N F 1 n ν ( V , H ) G m ( V , H ) D ( H ) V N d V N d H N H Ω N V Ω N G m ( V , H ) D ( H ) V N d V N d H N .
t ¯ i = 1 r ¯ i ,
V r = ( 1 r ¯ i ) W r .
W r = τ n ν ( L ) ρ B ( 1 a + a t ) 2 1 r ¯ i ρ B ( 1 a + a t 2 ) V i .
V r = τ n ν ( L ) ρ B ( 1 a + a t ) 2 1 r ¯ i ρ B ( 1 a + a t 2 ) ( 1 r ¯ i ) V i .
L r = d 2 Φ d s d d R R .
L r = d 2 Φ d s d d R R = Ω N d 2 Φ ( R , H ) D ( H ) d H N d s d d R R .
d 2 Φ ( R , H ) d s d d R R = d 2 Φ ( R , H ) d s R H d R H .
d 2 Φ ( R , H ) = T n ν ( R , H ) d 2 Φ i ( V , H ) ,
d 2 Φ i ( V , H ) = W r π d s V H d V H .
V H d V H R H d R H = 1 n ν 2 .
L r = W r π 1 n ν 2 Ω N T n ν ( R , H ) D ( H ) d H H .
τ n ν ( R ) = Ω N T n ν ( R , H ) D ( H ) d H H .
L r = τ n ν ( L ) ρ B ( 1 a + a t ) 2 1 r ¯ i ρ B ( 1 a + a t 2 ) τ n ν ( R ) n ν 2 V i π .
R Ω w = τ n ν ( L ) ( 1 r ¯ i ) ρ B ( 1 a + a t ) 2 1 r ¯ i ρ B ( 1 a + a t 2 ) .
R Ω p = ( 1 r ¯ i ρ B ) ( 1 a + a t ) 2 1 r ¯ i ρ B ( 1 a + a t 2 ) .
R δ w = τ n ν ( L ) τ n ν ( R ) n ν 2 ρ B ( 1 a + a t ) 2 1 r ¯ i ρ B ( 1 a + a t 2 ) .
R δ p = ( 1 r ¯ i ρ B ) ( 1 a + a t ) 2 1 r ¯ i ρ B ( 1 a + a t 2 ) .
E r = V Ω N H Ω N F 1 n ν ( V , H ) G m ( V , H ) D ( H ) V N d V N d H N
E i = V Ω N H Ω N G m ( V , H ) D ( H ) V N d V N d H N .
V N = cos θ ,
d V N = sin θ d θ d ϕ ,
d H N = sin θ h d θ h d ϕ h .
G m ( V , H ) = h ( V H ) Λ m ( θ ) + 1 ,
Λ m ( θ ) = 1 2 [ 1 π 2 m cot θ exp ( cot 2 θ 2 m 2 ) erfc ( cot θ 2 m ) ] .
V H = cos θ cos θ h + sin θ sin θ h cos ( ϕ ϕ h ) .
D ( H ) = exp ( tan 2 θ h 2 m 2 ) 2 π m 2 cos 3 θ h .
E r = 1 2 π m 2 ϕ h = 0 2 π θ h = 0 π 2 ϕ = 0 2 π θ = 0 π 2 F 1 n ν ( θ v h ) U ( cos θ v h ) Λ m ( θ ) + 1 exp ( tan 2 θ h 2 m 2 ) cos 3 θ h cos θ sin θ sin θ h d θ d ϕ d θ h d ϕ h .
cos θ v h = cos θ cos θ h + sin θ sin θ h cos ϕ ,
E r = 1 m 2 θ h = 0 π 2 ϕ = 0 2 π θ = 0 π 2 F 1 n ν ( θ v h ) U ( cos θ v h ) Λ m ( θ ) + 1 exp ( tan 2 θ h 2 m 2 ) cos 3 θ h cos θ sin θ sin θ h d θ d ϕ d θ h .
E r = 2 m 2 θ h = 0 π 2 ϕ = 0 π θ = 0 π 2 F 1 n ν ( θ v h ) U ( cos θ v h ) Λ m ( θ ) + 1 exp ( tan 2 θ h 2 m 2 ) cos 3 θ h cos θ sin θ sin θ h d θ d ϕ d θ h .
E r = 2 m 2 θ h = 0 π 2 ϕ = 0 π θ = 0 π 2 F 1 n ν ( θ v h ) U ( cos θ v h ) Λ m ( θ ) + 1 exp ( tan 2 θ h 2 m 2 ) cos 3 θ h cos θ sin θ sin θ h Δ θ Δ ϕ Δ θ h .

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