Abstract

In a previous paper, four-flux models for solving the scattering transfer equation in terms of Lorenz-Mie parameters were designed for Lorenz-Mie scatter centers embedded in a slab. Formulas for the various transmittances and reflectances were established. The special cases of transparent and nonscattering particles require special treatments which are described in the present work and lead to simpler formulas. The special case of completely opaque atmospheres is also considered. Three- and two-flux models are derived from the general formulas. The connection with classical literature is pointed out.

© 1986 Optical Society of America

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References

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  1. B. Maheu, J. N. Letoulouzan, G. Gouesbet, “Four-Flux Models to Solve the Scattering Transfer Equation in Terms of Lorenz-Mie Parameters,” Appl. Opt. 23, 3353 (1984).
    [CrossRef] [PubMed]
  2. G. Gouesbet, G. Grehan, B. Maheu, “Single Scattering Characteristics of Volume Elements in Coal Clouds,” Appl. Opt. 22, 2038 (1983).
    [CrossRef] [PubMed]
  3. P. Gougeon, J. N. Letoulouzan, G. Gouesbet, C. Thenard, “Optical Diagnosis in Multiple Scattering Media Using a Visible/Infrared Double Extinction Technique,” in preparation.
  4. P. Gougeon, J. N. Letoulouzan, C. Thenard, G. Gouesbet, “Simultaneous Measurements of Sizes and Concentrations of Coal Particles in Multiple Scattering Media by Means of a Double Extinction Technique,” P.S.A. Conference, 16–19 Sept. 1985, Bradford, Yorkshire, England.
  5. G. Gouesbet, M. Ledoux, “Supermicronic and Submicronic Optical Sizing, Including a Discussion of Densely Laden Flows,” Opt. Eng. 23, 631 (1984).
    [CrossRef]
  6. B. Maheu, G. Gouesbet, “Four-Flux Models to Solve the Scattering Transfer Equation in Terms of Lorenz-Mie Parameters,” Internal Report MADO/MG/1/84/I.
  7. G. Kortüm, Reflectance Spectroscopy (Springer, New York, 1969).
    [CrossRef]
  8. J. W. Ryde, “The Scattering of Light by Turbid Media. Part I,” Proc. R. Soc. London Ser. A 131, 451 (1931).
    [CrossRef]
  9. J. W. Ryde, B. S. Cooper, “The Scattering of Light by Turbid Media. Part. II,” Proc. R. Soc. London Ser. A 131, 464 (1931).
    [CrossRef]
  10. P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. 11a, 593 (1931).
  11. P. Kubelka, “New Contributions to the Optics of Intensely Light Scattering Materials. Part I,” J. Opt. Soc. Am. 38, 448 (1948).
    [CrossRef] [PubMed]
  12. C. Sagan, J. B. Pollack, “Anisotropic Nonconservative Scattering and the Clouds of Venus,” J. Geophys. Res. 72, 469 (1967).
    [CrossRef]
  13. A. Schuster, “Radiation Through a Foggy Atmosphere,” Astrophys. J. 21, 1 (1905).
    [CrossRef]

1984 (2)

G. Gouesbet, M. Ledoux, “Supermicronic and Submicronic Optical Sizing, Including a Discussion of Densely Laden Flows,” Opt. Eng. 23, 631 (1984).
[CrossRef]

B. Maheu, J. N. Letoulouzan, G. Gouesbet, “Four-Flux Models to Solve the Scattering Transfer Equation in Terms of Lorenz-Mie Parameters,” Appl. Opt. 23, 3353 (1984).
[CrossRef] [PubMed]

1983 (1)

1967 (1)

C. Sagan, J. B. Pollack, “Anisotropic Nonconservative Scattering and the Clouds of Venus,” J. Geophys. Res. 72, 469 (1967).
[CrossRef]

1948 (1)

1931 (3)

J. W. Ryde, “The Scattering of Light by Turbid Media. Part I,” Proc. R. Soc. London Ser. A 131, 451 (1931).
[CrossRef]

J. W. Ryde, B. S. Cooper, “The Scattering of Light by Turbid Media. Part. II,” Proc. R. Soc. London Ser. A 131, 464 (1931).
[CrossRef]

P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. 11a, 593 (1931).

1905 (1)

A. Schuster, “Radiation Through a Foggy Atmosphere,” Astrophys. J. 21, 1 (1905).
[CrossRef]

Cooper, B. S.

J. W. Ryde, B. S. Cooper, “The Scattering of Light by Turbid Media. Part. II,” Proc. R. Soc. London Ser. A 131, 464 (1931).
[CrossRef]

Gouesbet, G.

B. Maheu, J. N. Letoulouzan, G. Gouesbet, “Four-Flux Models to Solve the Scattering Transfer Equation in Terms of Lorenz-Mie Parameters,” Appl. Opt. 23, 3353 (1984).
[CrossRef] [PubMed]

G. Gouesbet, M. Ledoux, “Supermicronic and Submicronic Optical Sizing, Including a Discussion of Densely Laden Flows,” Opt. Eng. 23, 631 (1984).
[CrossRef]

G. Gouesbet, G. Grehan, B. Maheu, “Single Scattering Characteristics of Volume Elements in Coal Clouds,” Appl. Opt. 22, 2038 (1983).
[CrossRef] [PubMed]

P. Gougeon, J. N. Letoulouzan, C. Thenard, G. Gouesbet, “Simultaneous Measurements of Sizes and Concentrations of Coal Particles in Multiple Scattering Media by Means of a Double Extinction Technique,” P.S.A. Conference, 16–19 Sept. 1985, Bradford, Yorkshire, England.

B. Maheu, G. Gouesbet, “Four-Flux Models to Solve the Scattering Transfer Equation in Terms of Lorenz-Mie Parameters,” Internal Report MADO/MG/1/84/I.

P. Gougeon, J. N. Letoulouzan, G. Gouesbet, C. Thenard, “Optical Diagnosis in Multiple Scattering Media Using a Visible/Infrared Double Extinction Technique,” in preparation.

Gougeon, P.

P. Gougeon, J. N. Letoulouzan, G. Gouesbet, C. Thenard, “Optical Diagnosis in Multiple Scattering Media Using a Visible/Infrared Double Extinction Technique,” in preparation.

P. Gougeon, J. N. Letoulouzan, C. Thenard, G. Gouesbet, “Simultaneous Measurements of Sizes and Concentrations of Coal Particles in Multiple Scattering Media by Means of a Double Extinction Technique,” P.S.A. Conference, 16–19 Sept. 1985, Bradford, Yorkshire, England.

Grehan, G.

Kortüm, G.

G. Kortüm, Reflectance Spectroscopy (Springer, New York, 1969).
[CrossRef]

Kubelka, P.

P. Kubelka, “New Contributions to the Optics of Intensely Light Scattering Materials. Part I,” J. Opt. Soc. Am. 38, 448 (1948).
[CrossRef] [PubMed]

P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. 11a, 593 (1931).

Ledoux, M.

G. Gouesbet, M. Ledoux, “Supermicronic and Submicronic Optical Sizing, Including a Discussion of Densely Laden Flows,” Opt. Eng. 23, 631 (1984).
[CrossRef]

Letoulouzan, J. N.

B. Maheu, J. N. Letoulouzan, G. Gouesbet, “Four-Flux Models to Solve the Scattering Transfer Equation in Terms of Lorenz-Mie Parameters,” Appl. Opt. 23, 3353 (1984).
[CrossRef] [PubMed]

P. Gougeon, J. N. Letoulouzan, G. Gouesbet, C. Thenard, “Optical Diagnosis in Multiple Scattering Media Using a Visible/Infrared Double Extinction Technique,” in preparation.

P. Gougeon, J. N. Letoulouzan, C. Thenard, G. Gouesbet, “Simultaneous Measurements of Sizes and Concentrations of Coal Particles in Multiple Scattering Media by Means of a Double Extinction Technique,” P.S.A. Conference, 16–19 Sept. 1985, Bradford, Yorkshire, England.

Maheu, B.

Munk, F.

P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. 11a, 593 (1931).

Pollack, J. B.

C. Sagan, J. B. Pollack, “Anisotropic Nonconservative Scattering and the Clouds of Venus,” J. Geophys. Res. 72, 469 (1967).
[CrossRef]

Ryde, J. W.

J. W. Ryde, B. S. Cooper, “The Scattering of Light by Turbid Media. Part. II,” Proc. R. Soc. London Ser. A 131, 464 (1931).
[CrossRef]

J. W. Ryde, “The Scattering of Light by Turbid Media. Part I,” Proc. R. Soc. London Ser. A 131, 451 (1931).
[CrossRef]

Sagan, C.

C. Sagan, J. B. Pollack, “Anisotropic Nonconservative Scattering and the Clouds of Venus,” J. Geophys. Res. 72, 469 (1967).
[CrossRef]

Schuster, A.

A. Schuster, “Radiation Through a Foggy Atmosphere,” Astrophys. J. 21, 1 (1905).
[CrossRef]

Thenard, C.

P. Gougeon, J. N. Letoulouzan, G. Gouesbet, C. Thenard, “Optical Diagnosis in Multiple Scattering Media Using a Visible/Infrared Double Extinction Technique,” in preparation.

P. Gougeon, J. N. Letoulouzan, C. Thenard, G. Gouesbet, “Simultaneous Measurements of Sizes and Concentrations of Coal Particles in Multiple Scattering Media by Means of a Double Extinction Technique,” P.S.A. Conference, 16–19 Sept. 1985, Bradford, Yorkshire, England.

Appl. Opt. (2)

Astrophys. J. (1)

A. Schuster, “Radiation Through a Foggy Atmosphere,” Astrophys. J. 21, 1 (1905).
[CrossRef]

J. Geophys. Res. (1)

C. Sagan, J. B. Pollack, “Anisotropic Nonconservative Scattering and the Clouds of Venus,” J. Geophys. Res. 72, 469 (1967).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Eng. (1)

G. Gouesbet, M. Ledoux, “Supermicronic and Submicronic Optical Sizing, Including a Discussion of Densely Laden Flows,” Opt. Eng. 23, 631 (1984).
[CrossRef]

Proc. R. Soc. London Ser. A (2)

J. W. Ryde, “The Scattering of Light by Turbid Media. Part I,” Proc. R. Soc. London Ser. A 131, 451 (1931).
[CrossRef]

J. W. Ryde, B. S. Cooper, “The Scattering of Light by Turbid Media. Part. II,” Proc. R. Soc. London Ser. A 131, 464 (1931).
[CrossRef]

Z. Tech. Phys. (1)

P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. 11a, 593 (1931).

Other (4)

B. Maheu, G. Gouesbet, “Four-Flux Models to Solve the Scattering Transfer Equation in Terms of Lorenz-Mie Parameters,” Internal Report MADO/MG/1/84/I.

G. Kortüm, Reflectance Spectroscopy (Springer, New York, 1969).
[CrossRef]

P. Gougeon, J. N. Letoulouzan, G. Gouesbet, C. Thenard, “Optical Diagnosis in Multiple Scattering Media Using a Visible/Infrared Double Extinction Technique,” in preparation.

P. Gougeon, J. N. Letoulouzan, C. Thenard, G. Gouesbet, “Simultaneous Measurements of Sizes and Concentrations of Coal Particles in Multiple Scattering Media by Means of a Double Extinction Technique,” P.S.A. Conference, 16–19 Sept. 1985, Bradford, Yorkshire, England.

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Figures (1)

Fig. 1
Fig. 1

Scattering slab is limited by planes I (in) and O (out) located at z = Z and z = 0, respectively. The background surface S is parallel to the slab and located at a negative z. Incident radiation is constituted of (1) a collimated beam of intensity IcZ with infinite lateral extension hitting the slab perpendicularly and (2) a semi-isotropic diffuse radiation of intensity IdZ.

Equations (78)

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A 1 = 2 k [ k + 2 ( 1 ζ ) s ] ,
A 2 = s [ k ζ + s ( 1 ζ ) + ζ ( k + s ) ] ,
A 3 = s ( 1 ζ ) ( k + s ) ( 1 ) ,
A 4 = [ k + ( 1 ζ ) s ] ,
A 5 = ( 1 ζ ) s .
A 6 = r c ( 1 2 r c b ) + r c b 1 r c r c b
A 7 = r d i + r d b ( 1 r d e r d i ) 1 r d b r d e
τ cc = τ c ( 1 r c b ) ( 1 r c ) 2 exp [ ( k + s ) Z ] ( 1 r c r c b ) r c ( r c + r c b 2 r c r c b ) exp [ 2 ( k + s ) Z ] · I c Z I c Z + I d Z ,
τ cd = ( 1 r d b ) ( 1 r d i ) ( 1 r c ) τ d exp [ ( k + s ) Z ] ( 1 r d e r d b ) [ A 1 ( k + s ) 2 ] { 1 r c A 6 exp [ 2 ( k + s ) Z ] } I c Z I c Z + I d Z N D ,
N = A 1 [ r d i A 3 A 2 + A 6 ( r d i A 2 A 3 ) ] ch [ A 1 Z ] + [ ( A 5 r d i A 4 ) ( A 3 + A 2 A 6 ) ( A 4 r d i A 5 ) ( A 2 + A 3 A 6 ) sh [ A 1 Z ] + A 1 { ( A 2 r d i A 3 ) exp [ ( k + s ) Z ] + A 6 ( A 3 r d i A 2 ) exp [ ( k + s ) Z ] } ,
D = A 1 ( r d i A 7 1 ) ch [ A 1 Z ] + [ A 7 ( A 5 r d i A 4 ) + r d i A 5 A 4 ) ] sh [ A 1 Z ] ,
τ dd = ( 1 r d b ) ( 1 r d i ) ( 1 r d e ) τ d 1 r d e r d b · A 1 { A 1 ( 1 r d i A 7 ) ch [ A 1 Z ] + A 4 ( 1 + r d i A 7 ) A 5 ( A 7 + r d i ) sh [ A 1 Z ] } · I d Z I c Z + I d Z .
cc = { r c + ( 1 r c ) 2 ( r c + r c b 2 r c r c b ) exp [ 2 ( k + s ) Z ] ( 1 r c r c b ) r c ( r c + r c b 2 r c r c b ) exp [ 2 ( k + s ) Z ] } I c Z I c Z + I d Z ,
cd = ( 1 r d i ) ( 1 r c ) exp [ ( k + s ) Z ] [ A 1 ( k + s ) 2 ] { 1 r c A 6 exp [ 2 ( k + s ) Z ] } · I c Z I c Z + I d Z 1 A 1 ( r d i A 7 1 ) ch [ A 1 Z ] + [ A 5 ( A 7 + r d i ) A 4 ( 1 + r d i A 7 ) sh [ A 1 Z ] ( A 1 [ A 3 + A 2 A 6 A 7 ( A 2 + A 3 A 6 ) ] + { A 1 ( A 2 A 7 A 3 ) ch [ A 1 · Z ] + [ A 2 ( A 5 A 4 A 7 ) + A 3 ( A 5 A 7 A 4 ) ] sh [ A 1 Z ] } exp [ ( k + s ) Z ] + A 6 [ A 1 ( A 3 A 7 A 2 ) ch [ A 1 Z ] + [ A 3 ( A 5 A 4 A 7 ) + A 2 ( A 5 A 7 A 4 ) ] sh [ A 1 Z ] ] exp [ ( k + s ) Z ] ) ,
dd = { r d e + ( 1 r d i ) ( 1 r d e ) [ A 7 A 1 ch [ A 1 Z ] + ( A 5 A 4 A 7 ) sh [ A 1 Z ] ] A 1 ( 1 r d i A 7 ) ch [ A 1 Z ] + [ A 4 ( 1 + r d i A 7 ) A 5 ( A 7 + r d i ) ] sh [ A 1 Z ] } I d Z I c Z + I d Z .
A 1 2 2 k ( 1 ζ ) s 0 ,
A 1 A 2 = s 2 [ ( 1 ζ ) + ζ ] ,
A 2 A 3 = s 2 ( 1 ζ ) ( 1 ) ,
A 2 A 5 .
ch ( A 1 Z ) 1 ,
sh ( A 1 Z ) A 1 Z ,
τ cc = ( 1 r c ) 2 ( 1 r c b ) τ c exp ( s Z ) ( 1 r c r c b ) [ 1 r c A 6 exp ( 2 s Z ) ] · I c Z I c Z + I d Z ,
τ cd = ( 1 r d b ) ( 1 r d i ) ( 1 r c ) τ d exp ( s Z ) ( 1 r d e r d b ) s 2 [ 1 r c A 6 exp ( 2 s Z ) ] · 1 [ ( 1 r d i ) ( 1 A 7 ) Z A 5 + ( 1 r d i A 7 ) ] · I c Z I c Z + I d Z · { ( 1 r d i ) Z A 5 ( A 3 A 2 ) ( 1 A 6 ) + ( A 2 r d i A 3 ) [ exp ( s Z ) 1 ] + ( A 3 r d i A 2 ) A 6 [ exp ( s Z ) 1 ] }
τ dd = ( 1 r d b ) ( 1 r d i ) ( 1 r d e ) τ d ( 1 r d e r d b ) [ ( 1 r d i ) ( 1 A 7 ) Z A 5 + ( 1 r d i A 7 ) ] · I d Z I c Z + I d Z .
dd = [ r c + ( 1 r c ) 2 A 6 exp ( 2 s Z ) 1 r c A 6 exp ( 2 s Z ) ] I c Z I c Z + I d Z .
c d = ( 1 r d i ) ( 1 r c ) exp ( s Z ) s 2 [ 1 r c A 6 exp ( 2 s Z ) · 1 ( 1 r d i ) ( 1 A 7 ) Z A 5 + ( 1 r d i A 7 ) I c Z I c Z + I d Z ( [ A 3 + A 2 A 6 A 7 ( A 2 + A 3 A 6 ) ] + { [ ( 1 A 7 ) Z A 5 + A 7 ] A 2 A 3 [ ( 1 A 7 ) Z A 5 + 1 ] } exp ( + s Z ) + A 6 { [ ( 1 A 7 ) Z A 5 + A 7 ] A 3 A 2 [ ( 1 A 7 ) Z A 5 + 1 ] } exp ( s Z ) ) ,
dd = [ 1 ( 1 r d e ) ( 1 A 7 ) ( 1 r d i ) ( 1 A 7 ) Z A 5 + ( 1 r d i A 7 ) ] I d Z I c Z + I d Z .
R cc + R cd + R dd + τ cc + τ cd + τ dd = 1 .
τ cc = τ cd = τ dd = 0 ;
cc = r c I c Z / ( I c Z + I d Z ) ;
cd = ( 1 r c ) I c Z / ( I c Z + I d Z ) ;
dd = I d Z / ( I c Z + I d Z ) .
A 1 = A 4 = k ,
A 2 = A 3 = A 5 = 0 .
τ cc = τ c ( 1 r c b ) ( 1 r c ) 2 exp ( k Z ) ( 1 r c r c b ) r c ( r c + r c b 2 r c r c b ) exp ( 2 k Z ) · I c Z I c Z + I d Z ,
τ c d = 0 ( physically obvious ) ,
τ dd = ( 1 r d b ) ( 1 r d i ) ( 1 r d e ) τ d exp ( k Z ) ( 1 r d e r d b ) [ 1 r d i A 7 exp ( 2 k Z ) ] · I d Z I c Z + I d Z ,
cc = [ r c + ( 1 r c ) 2 ( r c + r c b 2 r c r c b ) exp ( 2 k Z ) ( 1 r c r c b ) r c ( r c + r c b 2 r c r c b ) exp ( 2 k Z ) ] · I c Z I c Z + I d Z ,
cd = 0 ( physically obvious ) ,
dd = [ r d e + ( 1 r d i ) ( 1 r d e ) A 7 exp ( 2 k Z ) 1 r d i A 7 exp ( 2 k Z ) ] · I d Z I c Z + I d Z .
τ cc = τ cd = τ dd = 0 ;
R cc = r c I c Z / ( I c Z + I d Z ) .
cd = ( 1 r d i ) ( 1 r c ) [ A 1 ( k + s ) 2 ] · A 1 ( A 2 A 7 A 3 ) + A 2 ( A 5 A 4 A 7 ) + A 3 ( A 5 A 7 A 4 ) A 1 ( r d i A 7 1 ) + [ A 5 ( A 7 + r d i ) A 4 ( 1 + r d i A 7 ) ] · I c Z I c Z + I d Z ,
dd = [ r d e + ( 1 r d i ) ( 1 r d e ) ( A 7 A 1 + A 5 A 4 A 7 ) A 1 ( 1 r d i A 7 ) + A 4 ( 1 + r d i A 7 ) A 5 ( A 7 + r d i ) ] · I d Z I c Z + I d Z .
cd = ( 1 r d i ) ( 1 r c ) [ A 1 ( k + s ) 2 ] · A 3 ( A 4 + A 1 ) A 2 A 5 ( 1 r d i ) A 5 + ( A 4 + A 1 A 5 ) · I c Z I c Z + I d Z ;
dd = ( 1 r d i ) A 5 + r d e ( A 4 + A 1 A 5 ) ( 1 r d i ) A 5 + ( A 4 + A 1 A 5 ) · I d Z I c Z + I d Z .
τ cc = exp [ ( k + s ) Z ] · I c Z I c Z + I d Z ( Beer - Lambert law ) ,
τ cd = exp [ ( k + s ) Z ] [ A 1 ( k + s ) 2 ] · I c Z I c Z + I d Z · A 1 A 2 { ch A 1 Z ] exp [ ( k + s ) Z ] } + ( A 2 A 4 A 3 A 5 ) sh [ A 1 Z ] A 1 ch [ A 1 Z ] + A 4 sh ] A 1 Z ] ,
τ dd = 1 ch [ A 1 Z ] + A 4 A 1 sh [ A 1 Z ] · I d Z I c Z + I d Z ,
cc = 0 ( no return collimated flux ) ,
cd = exp [ ( k + s ) Z ] [ A 1 ( k + s ) 2 ] · I c Z I c Z + I d Z · { A 3 exp [ ( k + s ) Z ] A 1 A 3 + A 2 A 5 sh [ A 1 Z ] · exp [ ( k + s ) Z ] A 1 ch [ A 1 Z ] + A 4 sh [ A 1 Z ] } ,
dd = A 5 sh [ A 1 Z ] A 1 ch [ A 1 Z ] + A 4 sh [ A 1 Z ] · I d Z I c Z + I d Z .
τ dd = R cc = R dd = 0 .
= cd = exp [ ( k + s ) Z ] [ A 1 ( k + s ) 2 ] · A 1 A 3 + exp [ ( k + s ) Z ] · [ A 1 A 3 ch ] A 1 Z ] + ( A 3 A 4 A 2 A 5 ) sh [ A 1 Z ] ] A 1 ch [ A 1 Z ] + A 4 sh [ A 1 Z ] ,
τ = exp [ ( k + s ) Z ] ( 1 + A 1 A 2 { ch [ A 1 Z ] exp [ ( k + s ) Z ] } + ( A 2 A 4 A 3 A 5 ) sh [ A 1 Z ] [ A 1 ( k + s ) 2 ] [ A 1 ch [ A 1 Z ] + A 4 sh [ A 1 Z ] ] ) .
R = R cd = A 3 ( A 4 + A 1 ) A 2 A 5 [ A 1 ( k + s ) 2 ] ( A 4 + A 1 ) = R * ,
τ = ( 1 r d i ) ( 1 r d e ) A 1 { A 1 ( 1 r d i 2 ) ch [ A 1 Z ] + [ A 4 ( 1 + r d i 2 ) 2 r d i A 5 ] sh [ A 1 Z ] } ,
= { r d e + ( 1 r d i ) ( 1 r d e ) [ r d i A 1 ch [ A 1 Z ] + ( A 5 r d i A 4 ) sh [ A 1 Z ] ] A 1 ( 1 r d i 2 ) ch [ A 1 Z ] + [ A 4 ( 1 + r d i 2 ) 2 r d i A 5 ] sh [ A 1 Z ] } .
τ = ( 1 r d b ) τ d A 1 A 1 ch [ A 1 Z ] + ( A 4 r d b A 5 ) sh [ A 1 Z ] ,
= r d b A 1 ch [ A 1 Z ] + ( A 5 r d b A 4 ) sh [ A 1 Z ] A 1 ch [ A 1 Z ] + ( A 4 r d b A 5 ) sh [ A 1 Z ] .
K = 2 k ,
S = 2 ( 1 ζ ) s .
χ = 1 + K S = 1 + a ζ a ( 1 ζ ) ,
τ = ( 1 r d b ) τ d χ 2 1 χ 2 1 ch ( S χ 2 1 Z ) + ( χ r d b ) sh ( S χ 2 1 Z ) ,
= r d b χ 2 1 ch ( S χ 2 1 Z ) + ( 1 r d b χ ) sh ( S χ 2 1 Z ) χ 2 1 ch ( S χ 2 1 Z ) + ( χ r d b ) sh ( S χ 2 1 Z ) .
τ 0 = χ 2 1 χ 2 1 ch ( S χ 2 1 Z ) + χ sh ( S χ 2 1 Z ) ,
R 0 = 1 χ + χ 2 1 coth ( S χ 2 1 Z ) = exp ( 2 S Z χ 2 1 ) 1 ( χ + χ 2 1 ) exp ( 2 S Z χ 2 1 ) ( χ χ 2 1 ) ,
τ 0 = A 1 A 1 ch [ A 1 Z ] + A 4 sh [ A 1 Z ] ,
0 = A 5 A 4 + A 1 coth [ A 1 Z ] .
R = R = r d b ( χ χ 2 1 ) 1 r d b χ χ 2 1 .
R = χ χ 2 1 = 1 χ + χ 2 1 ,
τ SP = χ 2 1 χ 2 1 ch ( 3 2 S χ 2 1 Z ) + χ sh ( 3 2 S χ 2 1 Z ) ,
R SP = exp ( 3 S χ 2 1 Z ) 1 ( χ + χ 2 1 ) exp ( 3 S χ 2 1 Z ) ( χ χ 2 1 ) ,
τ = 2 ( 1 r d b ) τ d 1 a 2 1 a ch [ 2 ( k + s ) 1 a Z ] + 2 [ a ( 1 + r d b ) sh [ 2 ( k + s ) 1 a Z ] ,
= 2 r d b 1 a ch [ 2 ( k + s ) 1 a Z ] + [ a r d b ( 2 a ) sh [ 2 ( k + s ) 1 a Z ] 2 1 a ch [ 2 ( k + s ) 1 a Z ] + [ 2 a ( 1 + r d b ) sh [ 2 ( k + s ) 2 1 a Z ] ,
τ 0 = 2 1 a 2 1 a ch [ 2 ( k + s ) 1 a Z ] + ( 2 a ) sh [ 2 ( k + s ) 1 a Z ] ,
R 0 = a { exp [ 4 ( k + s ) 1 a Z ] 1 } ( 1 + 1 a ) 2 exp [ 4 ( k + s ) 1 a Z ] ( 1 1 a ) 2 = a ( 2 a ) + 2 1 a coth [ 2 ( k + s ) 1 a Z ] .
R = 1 a ( 1 1 a ) 2 = a ( 1 + 1 a ) 2 = 1 1 a 1 + 1 a ,

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