Abstract

We propose a compositional model for predicting the reflectance and the transmittance of multilayer specimens composed of layers having possibly distinct refractive indices. The model relies on the laws of geometrical optics and on a description of the multiple reflection–transmission of light between the different layers and interfaces. The highly complex multiple reflection–transmission process occurring between several superposed layers is described by Markov chains. An optical element such as a layer or an interface forms a biface. The multiple reflection–transmission process is developed for a superposition of two bifaces. We obtain general composition formulas for the reflectance and the transmittance of a pair of layers and/or interfaces. Thanks to these compositional expressions, we can calculate the reflectance and the transmittance of three or more superposed bifaces. The model is applicable to regular compositions of bifaces, i.e., multifaces having on each face an angular light distribution that remains constant along successive reflection and transmission events. Kubelka’s layering model, Saunderson’s correction of the Kubelka–Munk model, and the Williams–Clapper model of a color layer superposed on a diffusing substrate are special cases of the proposed compositional model.

© 2007 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. P. Kubelka and F. Munk, "Ein Beitrag zur Optik der Farbanstriche," Z. Tech. Phys. (Leipzig) 12, 593-601 (1931).
  2. P. Kubelka, "New contributions to the optics of intensely light-scattering material. Part I," J. Opt. Soc. Am. 38, 448-457 (1948).
    [CrossRef] [PubMed]
  3. W. E. Vargas and G. A. Niklasson, "Applicability conditions of the Kubelka-Munk theory," Appl. Opt. 36, 5580-5586 (1997).
    [CrossRef] [PubMed]
  4. P. Kubelka, "New contributions to the optics of intensely light-scattering materials. Part II: Nonhomogeneous layers," J. Opt. Soc. Am. 44, 330-335 (1954).
    [CrossRef]
  5. M. Born and E. Wolf, Principle of Optics, 7th ed. (Pergamon, 1999).
  6. J. L. Saunderson, "Calculation of the color pigmented plastics," J. Opt. Soc. Am. 32, 727-736 (1942).
    [CrossRef]
  7. J. W. Ryde, "The scattering of light by turbid media," Proc. R. Soc. London A131, 451-464 (1931).
  8. G. Kortüm, "Phenomenological theories of absorption and scattering of tightly packed particles," in Reflectance Spectroscopy (Springer Verlag, 1969), pp. 103-168.
  9. M. Hébert and R. D. Hersch, "A reflectance and transmittance model for recto-verso halftone prints," J. Opt. Soc. Am. A 22, 1952-1967 (2006).
    [CrossRef]
  10. M. Hébert, "Compositional model for predicting multilayer reflectances and transmittances in color reproduction," Ph.D. Thesis No. 3576 (Ecole Polytechnique Fédérale de Lausanne, 2006).
  11. F. C. Williams and F. R. Clapper, "Multiple internal reflections in photographic color prints," J. Opt. Soc. Am. 29, 595-599 (1953).
    [CrossRef]
  12. J. D. Shore and J. P. Spoonhower, "Reflection density in photographic color prints: generalizations of the Williams-Clapper transform," J. Imaging Sci. Technol. 45, 484-488 (2001).
  13. L. Simonot, M. Hébert, and R. D. 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]
  14. F. R. Clapper and J. A. C. Yule, "The effect of multiple internal reflections on the densities of halftone prints on paper," J. Opt. Soc. Am. 43, 600 (1953).
    [CrossRef]
  15. W. R. McCluney, Introduction to Radiometry and Photometry (Artech House, 1994).
  16. D. B. Judd, "Fresnel reflection of diffusely incident light," J. Res. Natl. Bur. Stand. 29, 329-332 (1942).
  17. H.-H. Perkampus, Encyclopedia of Spectroscopy (VCH, 1995).
  18. W. J. Stewart, Introduction to the Numerical Solution of Markov Chains (Princeton Univ. Press, 1994).
  19. M. Hébert and R. D. Hersch, "Classical print reflection models: a radiometric approach," J. Imaging Sci. Technol. 48, 363-374 (2004).
  20. C. D. Meyer, Matrix Analysis and Applied Linear Algebra (SIAM, 2000), p. 701.
  21. M. Hébert and R. D. Hersch, "Extending the Clapper-Yule model to rough printing supports," J. Opt. Soc. Am. A 22, 1952-1967 (2005).
    [CrossRef]

2006 (2)

2005 (1)

2004 (1)

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

2001 (1)

J. D. Shore and J. P. Spoonhower, "Reflection density in photographic color prints: generalizations of the Williams-Clapper transform," J. Imaging Sci. Technol. 45, 484-488 (2001).

1997 (1)

1954 (1)

1953 (2)

F. C. Williams and F. R. Clapper, "Multiple internal reflections in photographic color prints," J. Opt. Soc. Am. 29, 595-599 (1953).
[CrossRef]

F. R. Clapper and J. A. C. Yule, "The effect of multiple internal reflections on the densities of halftone prints on paper," J. Opt. Soc. Am. 43, 600 (1953).
[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 (2)

J. W. Ryde, "The scattering of light by turbid media," Proc. R. Soc. London A131, 451-464 (1931).

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

Born, M.

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

Clapper, F. R.

F. C. Williams and F. R. Clapper, "Multiple internal reflections in photographic color prints," J. Opt. Soc. Am. 29, 595-599 (1953).
[CrossRef]

F. R. Clapper and J. A. C. Yule, "The effect of multiple internal reflections on the densities of halftone prints on paper," J. Opt. Soc. Am. 43, 600 (1953).
[CrossRef]

Hébert, M.

Hersch, R. D.

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, "Phenomenological theories of absorption and scattering of tightly packed particles," in Reflectance Spectroscopy (Springer Verlag, 1969), pp. 103-168.

Kubelka, P.

McCluney, W. R.

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

Meyer, C. D.

C. D. Meyer, Matrix Analysis and Applied Linear Algebra (SIAM, 2000), p. 701.

Munk, F.

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

Niklasson, G. A.

Perkampus, H.-H.

H.-H. Perkampus, Encyclopedia of Spectroscopy (VCH, 1995).

Ryde, J. W.

J. W. Ryde, "The scattering of light by turbid media," Proc. R. Soc. London A131, 451-464 (1931).

Saunderson, J. L.

Shore, J. D.

J. D. Shore and J. P. Spoonhower, "Reflection density in photographic color prints: generalizations of the Williams-Clapper transform," J. Imaging Sci. Technol. 45, 484-488 (2001).

Simonot, L.

Spoonhower, J. P.

J. D. Shore and J. P. Spoonhower, "Reflection density in photographic color prints: generalizations of the Williams-Clapper transform," J. Imaging Sci. Technol. 45, 484-488 (2001).

Stewart, W. J.

W. J. Stewart, Introduction to the Numerical Solution of Markov Chains (Princeton Univ. Press, 1994).

Vargas, W. E.

Williams, F. C.

F. C. Williams and F. R. Clapper, "Multiple internal reflections in photographic color prints," J. Opt. Soc. Am. 29, 595-599 (1953).
[CrossRef]

Wolf, E.

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

Yule, J. A. C.

Appl. Opt. (1)

J. Imaging Sci. Technol. (2)

J. D. Shore and J. P. Spoonhower, "Reflection density in photographic color prints: generalizations of the Williams-Clapper transform," J. Imaging Sci. Technol. 45, 484-488 (2001).

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. (5)

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).

Proc. R. Soc. London (1)

J. W. Ryde, "The scattering of light by turbid media," Proc. R. Soc. London A131, 451-464 (1931).

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

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

Other (7)

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

M. Hébert, "Compositional model for predicting multilayer reflectances and transmittances in color reproduction," Ph.D. Thesis No. 3576 (Ecole Polytechnique Fédérale de Lausanne, 2006).

H.-H. Perkampus, Encyclopedia of Spectroscopy (VCH, 1995).

W. J. Stewart, Introduction to the Numerical Solution of Markov Chains (Princeton Univ. Press, 1994).

G. Kortüm, "Phenomenological theories of absorption and scattering of tightly packed particles," in Reflectance Spectroscopy (Springer Verlag, 1969), pp. 103-168.

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

C. D. Meyer, Matrix Analysis and Applied Linear Algebra (SIAM, 2000), p. 701.

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.


Figures (9)

Fig. 1
Fig. 1

Multiple reflection–transmission of light within two superposed nonsymmetrical layers.

Fig. 2
Fig. 2

Multiface representing a transparent layer bordered by two flat interfaces. When directional incident light illuminates such a multiface composed of transparent bifaces, every face receives directional light along step-independent orientations given by Snell’s laws: n 0 sin θ 0 = n 1 sin θ 1 = n 2 sin θ 2 .

Fig. 3
Fig. 3

Infinite graph showing the multiple reflection–transmission of light within a quadriface, with light incident at the upper side (lower source off).

Fig. 4
Fig. 4

Markov chain representing the multiple reflection–transmission process within a quadriface.

Fig. 5
Fig. 5

Markov chain representing the multiple reflection–transmission process in a hexaface.

Fig. 6
Fig. 6

Two possible decompositions of a biface.

Fig. 7
Fig. 7

Reflection and transmission of directional incident light at the upper side and the lower side of a colored interface, composed of an ink layer and its interface with air.

Fig. 8
Fig. 8

Markov chain representing the multiple reflection–transmission process of light in the colored interface.

Fig. 9
Fig. 9

Markov chain representing the multiple reflection–transmission of light in a print.

Tables (3)

Tables Icon

Table 1 Correspondence between Multiple Reflection–Transmission Processes and Markov Chains

Tables Icon

Table 2 Particular Transfer Matrix of a Transparent Biface According to the Illumination and Measuring Geometries at Its Upper Side a

Tables Icon

Table 3 Particular Transfer Matrix of a Transparent Biface According to the Illumination and Measuring Geometries at Its Lower Side a

Equations (97)

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

R = R 1 + T 1 R 2 T 1 + T 1 R 2 R 1 R 2 T 1 + T 1 R 2 ( R 1 R 2 ) 2 T 1 + = R 1 + T 1 T 1 R 2 1 1 R 1 R 2 .
T = T 1 T 2 + T 1 R 2 R 1 T 2 + T 1 ( R 2 R 1 ) 2 T 2 + = T 1 T 2 1 1 R 1 R 2 .
R = k 1 + ( 1 k 1 ) R ( 1 k 2 ) + ( 1 k 1 ) R k 2 R ( 1 k 2 ) + = k 1 + ( 1 k 1 ) ( 1 k 2 ) R 1 k 2 R .
d 2 Φ i ( θ , ϕ ) = d s E i π cos θ sin θ d θ d ϕ .
d E r ( θ , ϕ ) = R ( θ ) E i π cos θ sin θ d θ d ϕ .
E r = ϕ = 0 2 π θ = 0 π 2 R ( θ ) E i π cos θ sin θ d θ d ϕ .
r = θ = 0 π 2 R ( θ ) sin 2 θ d θ .
[ T ( θ ) R ( θ ) R ( θ ) T ( θ ) ] .
r = θ = 0 π 2 R ( θ ) sin 2 θ d θ .
t = θ = 0 π 2 T ( θ ) sin 2 θ d θ
[ T ( ψ ) R ( ψ ) r t ] .
T ( θ ) = t 1 cos θ .
[ t 1 cos θ 0 0 t 1 cos θ ] .
[ T 01 ( θ ) R 01 ( θ ) R 10 ( θ ) T 10 ( θ ) ] .
[ τ ρ ρ τ ] .
G = [ T U R U R V T V ] .
[ p u s u r u x u ] , [ x v r v s v p v ] .
M = [ 0 0 p u 0 0 s u 0 0 0 p v s v 0 0 0 0 r v x v 0 0 0 r u 0 0 x u 0 0 0 0 1 0 0 0 0 0 0 1 ] .
e 0 = [ 1 , 0 , 0 , 0 , 0 , 0 ] .
e 1 = e 0 M = [ 0 , 0 , p u , 0 , 0 , s u ] .
e 2 = e 1 M = e 0 M 2 = [ 0 , 0 , 0 , p u r v , p u x v , s u ] .
e 3 = e 2 M = e 0 M 3 = [ 0 , 0 , p u r u r v , 0 , p u x v , s u + p u r v x u ] .
e = e 0 M = [ 0 , 0 , 0 , 0 , m 15 , m 16 ] ,
M = lim k M k .
e 0 = [ 0 , 1 , 0 , 0 , 0 , 0 ] .
e = e 0 M = [ 0 , 0 , 0 , 0 , m 25 , m 26 ] .
[ T U R U R V T V ] = [ m 15 m 16 m 25 m 26 ] .
[ T U R U R V T V ] = [ p u x v 1 r u r v s u + p u x u r v 1 r u r v s v + p v x v r u 1 r u r v p v x u 1 r u r v ] .
[ p u s u r u x u ] [ x v r v s v p v ] = [ p u x v 1 r u r v s u + p u x u r v 1 r u r v s v + p v x v r u 1 r u r v p v x u 1 r u r v ] .
( F 1 F 2 ) F 3 = F 1 ( F 2 F 3 ) = F 1 F 2 F 3 .
[ p u s u r u x u ] , [ τ ρ ρ τ ] , [ x v r v s v p v ] .
[ p u s u r u x u ] [ τ ρ ρ τ ] = [ p u τ 1 r u ρ s u + p u x u ρ 1 r u ρ ρ + r u τ τ 1 r u ρ x u τ 1 r u ρ ] .
[ p u s u r u x u ] [ τ ρ ρ τ ] [ x v r v s v p v ]
= [ p u x v τ D s u + p u x u ρ r v ( ρ ρ τ τ ) D s v + p v x v ρ r u ( ρ ρ τ τ ) D p v x u τ D ] ,
[ τ ρ ρ τ ] [ x v r v s v p v ] = [ τ x v 1 ρ r v ρ + τ τ r v 1 ρ r v s v + p v x v ρ 1 ρ r v τ p v 1 ρ r v ]
[ a b 0 d ] [ 1 0 c 1 ] = [ 1 b 0 1 ] [ a 0 c d ] = [ a b c d ] .
[ a b 0 1 ] [ 1 0 c d ] = [ 1 b 0 d ] [ a 0 c 1 ] = [ a b c d ] .
[ T 01 ( θ 0 ) R 01 ( θ 0 ) R 10 ( θ 1 ) T 10 ( θ 1 ) ] ,
[ t 1 cos θ 1 0 0 t 1 cos θ 1 ] .
[ T 01 ( θ 0 ) R 01 ( θ 0 ) R 10 ( θ 1 ) T 10 ( θ 1 ) ] [ t 1 cos θ 1 0 0 t 1 cos θ 1 ] = [ T 01 ( θ 0 ) t 1 cos θ 1 R 01 ( θ 0 ) R 10 ( θ 1 ) t 2 cos θ 1 T 10 ( θ 1 ) t 1 cos θ 1 ] .
[ T 01 ( θ 0 ) t 1 1 ( sin θ 0 n ) 2 R 01 ( θ 0 ) R 10 ( θ 1 ) t 2 cos θ 1 T 10 ( θ 1 ) t 1 cos θ 1 ] .
[ T i n R S r 1 T e x ] ,
[ τ ρ ρ τ ] ,
[ T 10 ( θ ) R 10 ( θ ) R 01 ( θ ) T 01 ( θ ) ] ,
[ T e x r 2 R S T i n ] ,
[ T i n R s r 1 T e x ] [ τ ρ ρ τ ] [ T e x r 2 R s T i n ] .
[ T i n T e x τ D R s + T i n T e x ρ r 2 ( ρ ρ τ 2 ) D R s + T i n T e x ρ r 1 ( ρ ρ τ 2 ) D T i n T e x τ D ] ,
[ T i n R s r 1 T e x ] = [ T i n R s 0 T e x ] [ 1 0 r 1 1 ] ,
[ T e x r 2 R s T i n ] = [ 1 r 2 0 1 ] [ T e x 0 R s T i n ] .
[ T i n R s 0 T e x ] [ 1 0 r 1 1 ] [ τ ρ ρ τ ] [ 1 r 2 0 1 ] [ T e x 0 R s T i n ] .
[ τ D ρ r 2 ( ρ ρ τ 2 ) D ρ r 1 ( ρ ρ τ 2 ) D τ D ] ,
[ T i n R s 0 T e x ] [ τ D ρ r 2 ( ρ ρ τ 2 ) D ρ r 1 ( ρ ρ τ 2 ) D τ D ] [ T e x 0 R s T i n ] .
[ T i n R S r 1 T e x ] = d i r ( 45 ° ) [ T 01 ( θ ) t 1 1 ( sin θ n ) 2 R 01 ( θ ) R 10 ( θ ) t 2 cos θ T 10 ( θ ) t 1 cos θ ] L r a d ( 0 ° ) = [ T 01 ( 45 ° ) t n n 2 1 2 0 r ( t ) T 10 ( 0 ) t π n 2 ] ,
r ( t ) = θ = 0 π 2 R 10 ( θ ) t 2 cos θ sin 2 θ d θ .
[ T e x r 2 R S T i n ] = L r a d ( ψ = 0 ° ) [ T 10 ( θ 1 ) R 10 ( θ 1 ) R 01 ( θ 0 ) T 01 ( θ 0 ) ] L = [ T 10 ( 0 ° ) π n 2 r 10 R 01 ( 0 ° ) π t 01 ] .
r 10 = θ = 0 π 2 R 10 ( θ ) sin 2 θ d θ ,
t 01 = θ = 0 π 2 T 01 ( θ ) sin 2 θ d θ ,
[ T 01 ( 45 ) T 10 ( 0 ) π n 2 t n n 2 1 2 τ D T 01 ( 45 ) T 10 ( 0 ) π n 2 t 1 + n n 2 1 2 ρ r 10 [ ρ ρ τ 2 ] D R 01 ( 0 ) π + t 01 T 10 ( 0 ) π n 2 ρ r 10 ( t ) [ ρ ρ τ 2 ] D t 01 T 10 ( 0 ) π n 2 t τ D ] ,
[ τ 1 ρ 1 ρ 1 τ 1 ] , [ τ 3 ρ 3 ρ 3 τ 3 ] .
[ t 2 1 cos θ 0 0 t 2 1 cos θ ] L L = [ t 2 ¯ 0 0 t 2 ¯ ] ,
t 2 ¯ = θ = 0 π 2 t 2 1 cos θ sin 2 θ d θ .
[ τ 1 ρ 1 ρ 1 τ 1 ] [ t 2 ¯ 0 0 t 2 ¯ ] [ τ 3 ρ 3 ρ 3 τ 3 ] = [ τ 1 τ 3 t 2 ¯ 1 ρ 1 ρ 3 t 2 ¯ 2 ρ 1 + τ 1 2 ρ 3 t 2 ¯ 2 1 ρ 1 ρ 3 t 2 ¯ 2 ρ 3 + τ 3 2 ρ 1 t 2 ¯ 2 1 ρ 1 ρ 3 t 2 ¯ 2 τ 1 τ 3 t 2 ¯ 1 ρ 1 ρ 3 t 2 ¯ 2 ] .
[ T i n R s r 1 T e x ] [ τ 1 ρ 1 ρ 1 τ 1 ] [ t 2 ¯ 0 0 t 2 ¯ ] [ τ 3 ρ 3 ρ 3 τ 3 ] [ T e x r 2 R s T i n ] .
[ τ 2 ρ 2 ρ 2 τ 2 ] .
[ p u s u r u x u ] , [ x v r v s v p v ] .
M = ( 0 0 p u 0 0 s u 0 0 0 p v s v 0 0 0 0 r v x v 0 0 0 r u 0 0 x u 0 0 0 0 1 0 0 0 0 0 0 1 ) .
M = ( A B 0 24 I ) .
M 2 = ( A B 0 I ) ( A B 0 I ) = ( A 2 AB + B 0 I ) .
M 3 = ( A 2 AB + B 0 I ) ( A B 0 I ) = ( A 3 A 2 B + AB + B 0 I ) .
M k = ( A k j = 0 k 1 A j B 0 I ) ,
M = ( lim k A k j = 0 A j B 0 I ) .
M = ( 0 ( I 4 A ) 1 B 0 I ) .
( G ) = ( I 4 A ) 1 B ,
( I 4 A ) = ( I P 0 I R ) .
( I P 0 I R ) ( I P ( I R ) 1 0 ( I R ) 1 ) = ( I 0 0 I ) .
( I 4 A ) 1 = ( I P ( I R ) 1 0 ( I R ) 1 ) .
( I 4 A ) 1 B = ( I P ( I R ) 1 0 ( I R ) 1 ) ( S X ) = ( S + P ( I R ) 1 X ( I R ) 1 X ) .
G = S + P ( I R ) 1 X .
I R = ( 1 0 0 1 ) ( 0 r v r u 0 ) = ( 1 r v r u 1 ) .
( I R ) 1 = 1 Δ ( 1 r v r u 1 ) .
G = ( 0 s u s v 0 ) + 1 Δ ( p u 0 0 p v ) ( 1 r v r u 1 ) ( x v 0 0 x u ) = ( p u x v 1 r u r v s u + p u x u r v 1 r u r v s v + p v x v r u 1 r u r v p v x u 1 r u r v ) .
M = ( 0 0 p u 0 0 0 0 s u 0 0 0 p v 0 0 s v 0 0 0 0 0 τ ρ 0 0 0 0 0 0 ρ τ 0 0 0 0 0 r v 0 0 x v 0 0 0 r u 0 0 0 0 x u 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 ) .
B = ( S 0 X ) = ( 0 s u s v 0 0 0 0 0 x v 0 0 x u )
M = ( 0 ( I 6 A ) 1 B 0 I ) .
I 6 A = ( I P 0 0 I C 0 R I ) .
( I 6 A ) 1 = ( I P ( I CR ) 1 PC ( I RC ) 1 0 ( I CR ) 1 C ( I RC ) 1 0 R ( I CR ) 1 ( I RC ) 1 ) .
( I 6 A ) 1 B = ( S + PC ( I RC ) 1 X C ( I RC ) 1 X ( I RC ) 1 X ) .
G = S + PC ( I RC ) 1 X .
I RC = ( 1 0 0 1 ) ( 0 r v r u 0 ) ( τ ρ ρ τ ) = ( 1 r v ρ r v τ r u τ 1 r u ρ ) .
D = ( 1 r u ρ ) ( 1 r v ρ ) r u r v τ τ ,
( I RC ) 1 = 1 D ( 1 r u ρ r v τ r u τ 1 r v ρ ) .
G = ( 0 s u s v 0 ) + 1 D ( p u 0 0 p v ) ( τ ρ ρ τ ) ( 1 r u ρ r v τ r u τ 1 r v ρ ) ( x v 0 0 x u ) ,
G = ( p u x v τ D s u + p u x u ρ r v ( ρ ρ τ τ ) D s v + p v x v ρ r u ( ρ ρ τ τ ) D p v x u τ D ) .
[ T ( θ ) R ( θ ) R ( θ ) T ( θ ) ] ,
[ T ( θ ) R ( θ ) R ( θ ) T ( θ ) ] L L = [ θ = 0 π 2 T ( θ ) sin 2 θ d θ θ = 0 π 2 R ( θ ) sin 2 θ d θ θ = 0 π 2 R ( θ ) sin 2 θ d θ θ = 0 π 2 T ( θ ) sin 2 θ d θ ] ,
[ P ( θ ) S ( θ ) R ( θ ) X ( θ ) ] .
n 1 sin ψ 1 = n 0 sin ψ .

Metrics