Abstract

In this paper, a geometrical propagation model is developed that generalizes the classical single-scatter model under the assumption of first-order scattering and non-line-of-sight (NLOS) communication. The generalized model considers the case of a noncoplanar geometry, where it overcomes the restriction that the transmitter and the receiver cone axes lie in the same plane. To verify the model, a Monte Carlo (MC) radiative transfer model based on a photon transport algorithm is constructed. Numerical examples for a wavelength of 266nm are illustrated, which corresponds to a solar-blind NLOS UV communication system. A comparison of the temporal responses of the generalized model and the MC simulation results shows close agreement. Path loss and delay spread are also shown for different pointing directions.

© 2011 Optical Society of America

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References

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    [CrossRef]
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2010 (4)

2009 (2)

2008 (3)

2006 (1)

G. A. Shaw, A. M. Siegel, and J. Model, “Extending the range and performance of non-line-of-sight ultraviolet communication links,” Proc. SPIE 6231, 62310C (2006).
[CrossRef]

2003 (1)

G. A. Shaw, A. M. Siegel, and M. L. Nischan, “Demonstration system and applications for compact wireless ultraviolet communications,” Proc. SPIE 5071, 241–252 (2003).
[CrossRef]

2000 (1)

G. A. Shaw, M. Nischan, M. Iyengar, S. Kaushik, and M. K. Griffin, “NLOS UV communication for distributed sensor systems,” Proc. SPIE 4126, 83–96 (2000).
[CrossRef]

1997 (1)

R. Storn and K. Price, “Differential evolution—a simple and efficient heuristic for global optimization over continuous spaces,” J. Global Optim. 11, 341–359 (1997).
[CrossRef]

1995 (1)

J. M. Kahn, W. J. Krause, and J. B. Carruthers, “Experimental characterization of non-directed indoor infrared channels,” IEEE Trans. Commun. 43, 1613–1623 (1995).
[CrossRef]

1991 (1)

1979 (1)

1978 (1)

Abou-Galala, F.

Abramowitz, M.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables (Dover, 1972).

Bielajew, A. F.

A. F. Bielajew, Fundamentals of the Monte Carlo method for Neutral and Charged Particle Transport (2001), http://www-personal.umich.edu/~bielajew/MCBook/book.pdf.

Boyd, S.

S. Boyd and L. Vandenberghe, Convex Optimization(Cambridge University, 2008).

Briesmeister, J. F.

J. F. Briesmeister, ed., MCNP—A General Monte Carlo N-Particle Transport Code, Version 4C (Los Alamos National Laboratory, 2000).

Carruthers, J. B.

J. M. Kahn, W. J. Krause, and J. B. Carruthers, “Experimental characterization of non-directed indoor infrared channels,” IEEE Trans. Commun. 43, 1613–1623 (1995).
[CrossRef]

Chandrasekhar, S.

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

Chang, S.

Chen, G.

Darbinjan, R.

G. Marchuk, G. Mikhailov, M. Narazaliev, R. Darbinjan, B. Kargin, and B. Elepov, The Monte Carlo Methods in Atmospheric Optics (Springer-Verlag, 1980).

Ding, H.

H. Ding, Z. Xu, and B. M. Sadler, “A path loss model for non-line-of-sight ultraviolet multiple scattering channels,” EURASIP J. Wireless Commun. Network. 2010, 1–12 (2010).
[CrossRef]

Z. Xu, H. Ding, B. M. Sadler, and G. Chen, “Analytical performance study of solar blind non-line-of-sight ultraviolet short-range communication links,” Opt. Lett. 33, 1860–1862 (2008).
[CrossRef]

Elepov, B.

G. Marchuk, G. Mikhailov, M. Narazaliev, R. Darbinjan, B. Kargin, and B. Elepov, The Monte Carlo Methods in Atmospheric Optics (Springer-Verlag, 1980).

Evans, K. F.

K. F. Evans and A. Marshak, 3D Radiative Transfer for Cloudy Atmospheres (Springer-Verlag, 2005).

Fishburne, E. S.

E. S. Fishburne, M. E. Neer, and G. Sandri, “Voice communication via scattered ultraviolet radiation,” Tech. Rep. 274(Aeronautical Research Associates of Princeton, 1976).

Fung, A. K.

F. T. Ulaby, R. K. Moore, and A. K. Fung, Microwave Remote Sensing: Active and Passive (Addison-Wesley, 1981).

Griffin, M. K.

G. A. Shaw, M. Nischan, M. Iyengar, S. Kaushik, and M. K. Griffin, “NLOS UV communication for distributed sensor systems,” Proc. SPIE 4126, 83–96 (2000).
[CrossRef]

Harris, T. E.

H. Kahn and T. E. Harris, “Estimation of particle transmission by random sampling,” in Monte Carlo Method, National Bureau of Standards, Applied Mathematics Series (U.S. Government Printing Office, 1951), Vol. 12, pp. 27–30.

He, Q.

Huffman, R. E.

R. E. Huffman, Atmospheric Ultraviolet Remote Sensing(Academic, 1992).

Iyengar, M.

G. A. Shaw, M. Nischan, M. Iyengar, S. Kaushik, and M. K. Griffin, “NLOS UV communication for distributed sensor systems,” Proc. SPIE 4126, 83–96 (2000).
[CrossRef]

Jacques, S. L.

S. A. Prahl, M. Keijzer, S. L. Jacques, and A. J. Welch, “A Monte Carlo model of light propagation in tissue,” in Dosimetry of Laser Radiation in Medicine and BiologySPIE Institutes for Advanced Optical Technologies (SPIE, 1989), Vol. IS 5, pp. 102–111.

Jia, H.

Kahn, H.

H. Kahn and T. E. Harris, “Estimation of particle transmission by random sampling,” in Monte Carlo Method, National Bureau of Standards, Applied Mathematics Series (U.S. Government Printing Office, 1951), Vol. 12, pp. 27–30.

Kahn, J. M.

J. M. Kahn, W. J. Krause, and J. B. Carruthers, “Experimental characterization of non-directed indoor infrared channels,” IEEE Trans. Commun. 43, 1613–1623 (1995).
[CrossRef]

Kargin, B.

G. Marchuk, G. Mikhailov, M. Narazaliev, R. Darbinjan, B. Kargin, and B. Elepov, The Monte Carlo Methods in Atmospheric Optics (Springer-Verlag, 1980).

Kaushik, S.

G. A. Shaw, M. Nischan, M. Iyengar, S. Kaushik, and M. K. Griffin, “NLOS UV communication for distributed sensor systems,” Proc. SPIE 4126, 83–96 (2000).
[CrossRef]

Keijzer, M.

S. A. Prahl, M. Keijzer, S. L. Jacques, and A. J. Welch, “A Monte Carlo model of light propagation in tissue,” in Dosimetry of Laser Radiation in Medicine and BiologySPIE Institutes for Advanced Optical Technologies (SPIE, 1989), Vol. IS 5, pp. 102–111.

Krause, W. J.

J. M. Kahn, W. J. Krause, and J. B. Carruthers, “Experimental characterization of non-directed indoor infrared channels,” IEEE Trans. Commun. 43, 1613–1623 (1995).
[CrossRef]

Lewis, E. E.

E. E. Lewis and W. F. Miller, Computational Methods of Neutron Transport (Wiley, 1984).

Luettgen, M. R.

Marchuk, G.

G. Marchuk, G. Mikhailov, M. Narazaliev, R. Darbinjan, B. Kargin, and B. Elepov, The Monte Carlo Methods in Atmospheric Optics (Springer-Verlag, 1980).

Marshak, A.

K. F. Evans and A. Marshak, 3D Radiative Transfer for Cloudy Atmospheres (Springer-Verlag, 2005).

Mikhailov, G.

G. Marchuk, G. Mikhailov, M. Narazaliev, R. Darbinjan, B. Kargin, and B. Elepov, The Monte Carlo Methods in Atmospheric Optics (Springer-Verlag, 1980).

Miller, W. F.

E. E. Lewis and W. F. Miller, Computational Methods of Neutron Transport (Wiley, 1984).

Model, J.

G. A. Shaw, A. M. Siegel, and J. Model, “Extending the range and performance of non-line-of-sight ultraviolet communication links,” Proc. SPIE 6231, 62310C (2006).
[CrossRef]

Moore, R. K.

F. T. Ulaby, R. K. Moore, and A. K. Fung, Microwave Remote Sensing: Active and Passive (Addison-Wesley, 1981).

Narazaliev, M.

G. Marchuk, G. Mikhailov, M. Narazaliev, R. Darbinjan, B. Kargin, and B. Elepov, The Monte Carlo Methods in Atmospheric Optics (Springer-Verlag, 1980).

Neer, M. E.

E. S. Fishburne, M. E. Neer, and G. Sandri, “Voice communication via scattered ultraviolet radiation,” Tech. Rep. 274(Aeronautical Research Associates of Princeton, 1976).

Nischan, M.

G. A. Shaw, M. Nischan, M. Iyengar, S. Kaushik, and M. K. Griffin, “NLOS UV communication for distributed sensor systems,” Proc. SPIE 4126, 83–96 (2000).
[CrossRef]

Nischan, M. L.

G. A. Shaw, A. M. Siegel, and M. L. Nischan, “Demonstration system and applications for compact wireless ultraviolet communications,” Proc. SPIE 5071, 241–252 (2003).
[CrossRef]

Prahl, S. A.

S. A. Prahl, M. Keijzer, S. L. Jacques, and A. J. Welch, “A Monte Carlo model of light propagation in tissue,” in Dosimetry of Laser Radiation in Medicine and BiologySPIE Institutes for Advanced Optical Technologies (SPIE, 1989), Vol. IS 5, pp. 102–111.

Price, K.

R. Storn and K. Price, “Differential evolution—a simple and efficient heuristic for global optimization over continuous spaces,” J. Global Optim. 11, 341–359 (1997).
[CrossRef]

Reilly, D. M.

Sadler, B. M.

Sandri, G.

E. S. Fishburne, M. E. Neer, and G. Sandri, “Voice communication via scattered ultraviolet radiation,” Tech. Rep. 274(Aeronautical Research Associates of Princeton, 1976).

Shapiro, J. H.

Shaw, G. A.

G. A. Shaw, A. M. Siegel, and J. Model, “Extending the range and performance of non-line-of-sight ultraviolet communication links,” Proc. SPIE 6231, 62310C (2006).
[CrossRef]

G. A. Shaw, A. M. Siegel, and M. L. Nischan, “Demonstration system and applications for compact wireless ultraviolet communications,” Proc. SPIE 5071, 241–252 (2003).
[CrossRef]

G. A. Shaw, M. Nischan, M. Iyengar, S. Kaushik, and M. K. Griffin, “NLOS UV communication for distributed sensor systems,” Proc. SPIE 4126, 83–96 (2000).
[CrossRef]

Siegel, A. M.

G. A. Shaw, A. M. Siegel, and J. Model, “Extending the range and performance of non-line-of-sight ultraviolet communication links,” Proc. SPIE 6231, 62310C (2006).
[CrossRef]

G. A. Shaw, A. M. Siegel, and M. L. Nischan, “Demonstration system and applications for compact wireless ultraviolet communications,” Proc. SPIE 5071, 241–252 (2003).
[CrossRef]

Sobol, I. M.

I. M. Sobol, A Primer for the Monte Carlo Method (CRC, 1994).

Stegun, I. A.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables (Dover, 1972).

Storn, R.

R. Storn and K. Price, “Differential evolution—a simple and efficient heuristic for global optimization over continuous spaces,” J. Global Optim. 11, 341–359 (1997).
[CrossRef]

Ulaby, F. T.

F. T. Ulaby, R. K. Moore, and A. K. Fung, Microwave Remote Sensing: Active and Passive (Addison-Wesley, 1981).

Vandenberghe, L.

S. Boyd and L. Vandenberghe, Convex Optimization(Cambridge University, 2008).

Wang, L.

Warde, C.

Welch, A. J.

S. A. Prahl, M. Keijzer, S. L. Jacques, and A. J. Welch, “A Monte Carlo model of light propagation in tissue,” in Dosimetry of Laser Radiation in Medicine and BiologySPIE Institutes for Advanced Optical Technologies (SPIE, 1989), Vol. IS 5, pp. 102–111.

Xu, H. D. Z.

Xu, Z.

Yang, J.

Yin, H.

Zachor, A. S.

Appl. Opt. (1)

EURASIP J. Wireless Commun. Network. (1)

H. Ding, Z. Xu, and B. M. Sadler, “A path loss model for non-line-of-sight ultraviolet multiple scattering channels,” EURASIP J. Wireless Commun. Network. 2010, 1–12 (2010).
[CrossRef]

IEEE Commun. Mag. (1)

Z. Xu and B. M. Sadler, “Ultraviolet communications: potential and state-of-the-art,” IEEE Commun. Mag. 46, 67–73 (2008).
[CrossRef]

IEEE Trans. Commun. (1)

J. M. Kahn, W. J. Krause, and J. B. Carruthers, “Experimental characterization of non-directed indoor infrared channels,” IEEE Trans. Commun. 43, 1613–1623 (1995).
[CrossRef]

J. Global Optim. (1)

R. Storn and K. Price, “Differential evolution—a simple and efficient heuristic for global optimization over continuous spaces,” J. Global Optim. 11, 341–359 (1997).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Express (4)

Opt. Lett. (2)

Proc. SPIE (3)

G. A. Shaw, A. M. Siegel, and M. L. Nischan, “Demonstration system and applications for compact wireless ultraviolet communications,” Proc. SPIE 5071, 241–252 (2003).
[CrossRef]

G. A. Shaw, M. Nischan, M. Iyengar, S. Kaushik, and M. K. Griffin, “NLOS UV communication for distributed sensor systems,” Proc. SPIE 4126, 83–96 (2000).
[CrossRef]

G. A. Shaw, A. M. Siegel, and J. Model, “Extending the range and performance of non-line-of-sight ultraviolet communication links,” Proc. SPIE 6231, 62310C (2006).
[CrossRef]

Other (17)

R. E. Huffman, Atmospheric Ultraviolet Remote Sensing(Academic, 1992).

D. M. Reilly, “Atmospheric optical communications in the middle ultraviolet,” M.S. thesis (Massachusetts Institute of Technology, 1976).

F. T. Ulaby, R. K. Moore, and A. K. Fung, Microwave Remote Sensing: Active and Passive (Addison-Wesley, 1981).

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables (Dover, 1972).

S. Boyd and L. Vandenberghe, Convex Optimization(Cambridge University, 2008).

Wolfram Research, Inc., “Wolfram research: mathematica, technical and scientific software,” http://www.wolfram.com.

I. M. Sobol, A Primer for the Monte Carlo Method (CRC, 1994).

E. E. Lewis and W. F. Miller, Computational Methods of Neutron Transport (Wiley, 1984).

A. F. Bielajew, Fundamentals of the Monte Carlo method for Neutral and Charged Particle Transport (2001), http://www-personal.umich.edu/~bielajew/MCBook/book.pdf.

H. Kahn and T. E. Harris, “Estimation of particle transmission by random sampling,” in Monte Carlo Method, National Bureau of Standards, Applied Mathematics Series (U.S. Government Printing Office, 1951), Vol. 12, pp. 27–30.

S. A. Prahl, M. Keijzer, S. L. Jacques, and A. J. Welch, “A Monte Carlo model of light propagation in tissue,” in Dosimetry of Laser Radiation in Medicine and BiologySPIE Institutes for Advanced Optical Technologies (SPIE, 1989), Vol. IS 5, pp. 102–111.

J. F. Briesmeister, ed., MCNP—A General Monte Carlo N-Particle Transport Code, Version 4C (Los Alamos National Laboratory, 2000).

G. Marchuk, G. Mikhailov, M. Narazaliev, R. Darbinjan, B. Kargin, and B. Elepov, The Monte Carlo Methods in Atmospheric Optics (Springer-Verlag, 1980).

K. F. Evans and A. Marshak, 3D Radiative Transfer for Cloudy Atmospheres (Springer-Verlag, 2005).

The MathWorks, Inc., “Mathworks—Matlab and Simulink for technical computing,” http://www.mathworks.com.

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

E. S. Fishburne, M. E. Neer, and G. Sandri, “Voice communication via scattered ultraviolet radiation,” Tech. Rep. 274(Aeronautical Research Associates of Princeton, 1976).

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Figures (8)

Fig. 1
Fig. 1

Example of the geometry of an NLOS FSO link.

Fig. 2
Fig. 2

Example of coordinate transformation for an Rx cone.

Fig. 3
Fig. 3

Example for the coplanar and noncoplanar geometries with a cutting plane Z o normal to the z-axis at an arbitrary point z o . (a) three-dimensional view, (b) cross section in Z o .

Fig. 4
Fig. 4

Example of ϕ ( ξ , η ) limits for (a) coplanar and (b) noncoplanar cases where z = r 2 ξ η .

Fig. 5
Fig. 5

Possible geometrical cases for ϕ ( ξ , η ) limits.

Fig. 6
Fig. 6

Simulation results for temporal response h ( t ) at r = 100 m , β T = β R = 45 ° , θ T = θ R = 15 ° , and α = { 0 ° , 10 ° , 20 ° , 30 ° , 40 ° } with detector of radius a = 1.5 cm . Results are generated with MC simulation (indicated by discrete points •) and the model from Eq. (6).

Fig. 7
Fig. 7

Simulation results for the path loss P L and the value of the common volume | V | at r = 100 m , θ T = θ R = 15 ° , β T = β R = 45 ° , and α in the range [ 0 ° , 42 ° ] with detector of radius a = 1.5 cm .

Fig. 8
Fig. 8

Simulation results for the rms delay spread D at r = 100 m , θ T = θ R = 15 ° , β T = β R = 45 ° , and α in the range [ 0 ° , 40 ° ] with detector of radius a = 1.5 cm .

Tables (2)

Tables Icon

Table 1 ϕ ( ξ , η ) Limits in Fig. 5

Tables Icon

Table 2 t min and t max Corresponding to the Geometric Configuration r = 100 m , β T = β R = 45 ° , θ T = θ R = 15 ° , and α = 0 ° , 10 ° , 20 ° , 30 ° , 40 °

Equations (55)

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

k s P ( Ω T Ω R ) cos ζ A d 4 π Ω T r T 2 r R 2 exp [ k e ( r T + r R ) ] δ V .
cos ζ = sin β R sin ψ R ( cos α R cos ϕ + sin α R sin ϕ ) + cos β R cos ψ R
P ( Ω T Ω R ) = P ( cos Θ s ) ,
cos Θ s = Ω T · Ω R = cos ( ψ T + ψ R ) .
k s P ( cos Θ s ) cos ζ A d 2 π Ω T r ( ξ 2 η 2 ) exp ( k e r ξ ) δ ϕ δ η δ ξ .
g ( ξ ) δ ξ = k s A d exp ( k e r ξ ) 2 π Ω T r η 1 ( ξ ) η 2 ( ξ ) P ( cos Θ s ) ξ 2 η 2 ϕ 1 ( ξ , η ) ϕ 2 ( ξ , η ) cos ζ d ϕ d η δ ξ .
h ( t ) = c r g ( c t r ) .
H o = h ( t ) d t .
P L = 10 log 10 ( H o ) .
D = [ ( t μ ) 2 h 2 ( t ) d t h 2 ( t ) d t ] 1 / 2 ,
μ = t h 2 ( t ) d t h 2 ( t ) d t .
x 2 + y 2 = tan 2 θ R z 2 .
[ x y z ] = R y ( β R ) R z ( α R ) [ x y z + r / 2 ] ,
R y ( β R ) R z ( α R ) = [ cos β R cos α R cos β R sin α R sin β R sin α R cos α R 0 sin β R cos α R sin β R sin α R cos β R ] .
A x 2 + B x y + C y 2 + D x ( z + r / 2 ) + E y ( z + r / 2 ) + F ( z + r / 2 ) 2 = 0 ,
A = sin 2 α R + cos 2 α R ( cos 2 β R tan 2 θ R sin 2 β R ) ,
B = ( 1 + tan 2 θ R ) sin 2 α R sin 2 β R ,
C = cos 2 α R + sin 2 α R ( cos 2 β R tan 2 θ R sin 2 β R ) ,
D = ( 1 + tan 2 θ R ) cos α R sin 2 β R ,
E = ( 1 + tan 2 θ R ) sin α R sin 2 β R ,
F = sin 2 β R tan 2 θ R cos 2 β R .
ξ min c = { 1 , ( β T θ T ) 0 or ( β R θ R ) 0 a + ( a 2 1 ) 1 / 2 , 0 < ( β T θ T ) + ( β R θ R ) π , π ( β T θ T ) + ( β R θ R ) ,
ξ max c = { b + ( b 2 1 ) 1 / 2 , 0 < ( β T + θ T ) + ( β R + θ R ) < π , π ( β T + θ T ) + ( β R + θ R ) ,
a = 1 + cos ( β T θ T ) cos ( β R θ R ) cos ( β T θ T ) + cos ( β R θ R ) ,
b = 1 + cos ( β T + θ T ) cos ( β R + θ R ) cos ( β T + θ T ) + cos ( β R + θ R ) .
maximize ξ ( x , y , z ) subject to h T ( x , y , z ) = 0 , h R ( x , y , z ) = 0 , z T ( x , y , z ) 0 , z R ( x , y , z ) 0 ,
ξ ( x , y , z ) = [ x 2 + y 2 + ( z + r / 2 ) 2 ] 1 / 2 + [ x 2 + y 2 + ( z r / 2 ) 2 ] 1 / 2 r .
η 1 T ( ξ ) = { 1 , β T θ T 0 1 ξ cos ( β T θ T ) ξ cos ( β T θ T ) , β T θ T 0 ,
η 2 T ( ξ ) = { 1 , β T + θ T π 1 ξ cos ( β T + θ T ) ξ cos ( β T + θ T ) , β T + θ T π ,
η 1 R ( ξ ) = { 1 , β R + θ R π ξ cos ( β R + θ R ) 1 ξ cos ( β R + θ R ) , β R + θ R π ,
η 2 R ( ξ ) = { 1 , β R θ R 0 ξ cos ( β R θ R ) 1 ξ cos ( β R θ R ) , β R θ R 0 .
η 1 ( ξ ) = max [ η 1 T ( ξ ) , η 1 R ( ξ ) ] ,
η 2 ( ξ ) = min [ η 2 T ( ξ ) , η 2 R ( ξ ) ] .
R ( ξ , η ) = [ r 2 4 ( ξ 2 1 ) ( 1 η 2 ) ] 1 / 2 .
x 1 , 2 c ( ξ , η ) = r ( 1 + ξ η ) D 4 ( A C ) ± { r 2 ( 1 + ξ η ) 2 D 2 4 ( A C ) [ 4 C R 2 ( ξ , η ) + r 2 ( 1 + ξ η ) 2 F ] } 1 / 2 4 ( A C ) ,
y 1 , 2 c ( ξ , η ) = ± [ R 2 ( ξ , η ) x 1 2 ( ξ , η ) ] 1 / 2 , y 3 , 4 c ( ξ , η ) = ± [ R 2 ( ξ , η ) x 2 2 ( ξ , η ) ] 1 / 2 ,
x * c = { x 1 c , ( | x 1 c | R < | x 2 c | ) or [ ( | x 1 c | R ) and ( | x 2 c | R ) and ( x 2 c < x 1 c ) ] x 2 c , ( | x 2 c | R < | x 1 c | ) or [ ( | x 1 c | R ) and ( | x 2 c | R ) and ( x 1 c x 2 c ) ] ,
y * 1 , 2 c = ± ( R 2 x * c 2 ) 1 / 2 .
ϕ 2 R , 1 R c ( ξ , η ) = { tan 1 ( y * 1 , 2 c x * c ) , [ C ( ξ , η ) S R c ( z ) ] and [ β R 0 ] ± π , otherwise .
ϕ 1 R ( ξ , η ) = ϕ 1 R c ( ξ , η ) + α ,
ϕ 2 R ( ξ , η ) = ϕ 2 R c ( ξ , η ) + α .
ω T = ( sin Θ T cos Φ T , sin Θ T sin Φ T , cos Θ T ) ,
cos Θ T = [ 1 G 1 ( 1 cos θ T ) ] ,
Φ T = 2 π G 2 ,
Ω T = [ R y ( π β T ) R z ( α T ) ] T ω T ,
r T = 1 k e ln ( 1 G 3 ) ,
p q = p T + r T Ω T .
W = ϖ o W 0 .
W r n = W n P ( cos Θ s n ) 4 π cos ζ n A d r R n 2 exp ( k e r R n ) I ( ζ n ) ,
I ( ζ n ) = { 1 , ζ n θ R 0 , ζ n θ R ,
r R n = [ x q n 2 + y q n 2 + ( z q n + r / 2 ) 2 ] 1 / 2 .
h ( t i ) Δ t i W r n N Δ t i .
P R ( cos Θ s ) = 3 [ 1 + 3 γ + ( 1 γ ) cos 2 Θ s ] 4 ( 1 + 2 γ ) ,
P M ( cos Θ s ) = ( 1 g 2 ) [ 1 ( 1 + g 2 2 g cos Θ s ) 3 / 2 + f 0.5 ( 3 cos 2 Θ s 1 ) ( 1 + g 2 ) 3 / 2 ] ,
P ( cos Θ s ) = k s r k s P R ( cos Θ s ) + k s m k s P M ( cos Θ s ) ,

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