Abstract

In large part because of advancements in the design and fabrication of UV LEDs, photodetectors, and filters, significant research interest has recently been focused on non-line-of-sight UV communication systems. This research in, for example, system design and performance prediction, can be greatly aided by accurate channel models that allow for the reproducibility of results, thus facilitating the fair and consistent comparison of different communication approaches. In this paper, we provide a comprehensive derivation of a multiple-scattering Monte Carlo UV channel model, addressing weaknesses in previous treatments. The resulting model can be used to study the contribution of different orders of scattering to the path loss and impulse response functions associated with general UV communication system geometries. Simulation results are provided that demonstrate the benefit of this approach.

© 2011 Optical Society of America

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References

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    [CrossRef]
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  4. D. E. Sunstein, “A scatter communication link at ultraviolet frequencies,” B. S. thesis (Massachusetts Institute of Technology, 1968).
  5. E. S. Fishburne, M. E. Neer, and G. Sandri, “Voice communication via scattered ultraviolet radiation,” Tech. Rep. 274 (Aeronautical Research Associates of Princeton, Inc., Princeton, NJ 1976).
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    [CrossRef]
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    [CrossRef]
  18. V. Adivarahan, W. Fareed, S. Srivastava, T. Katona, M. Gaevski, and A. Khan, “Robust 285 nm deep UV light emitting diodes over metal organic hydride vapor phase epitaxially grown AIN/sapphire templates,” Jpn. J. Appl. Phys. 46, L537 –L579 (2007).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  27. J. H. Mathews, Numerical Methods for Mathematics, Science, and Engineering, 2nd ed. (Prentice Hall, 1992).

2009 (2)

H. Ding, G. Chen, A. K. Majumdar, B. M. Sadler, and Z. Xu, “Modeling of non-line-of-sight ultraviolet scattering channels for communication,” IEEE J. Sel. Areas Commun. 27, 1535–1544(2009).
[CrossRef]

Q. He, B. M. Sadler, and Z. Xu, “Modulation and coding tradeoffs for non-line-of-sight ultraviolet communications,” Proc. SPIE 7464, 74640H (2009).
[CrossRef]

2008 (3)

2007 (2)

V. Adivarahan, W. Fareed, S. Srivastava, T. Katona, M. Gaevski, and A. Khan, “Robust 285 nm deep UV light emitting diodes over metal organic hydride vapor phase epitaxially grown AIN/sapphire templates,” Jpn. J. Appl. Phys. 46, L537 –L579 (2007).
[CrossRef]

S.-C. Shen, Y. Zhang, D. Yoo, J.-B. Limb, J.-H. Ryou, P. D. Yoder, and R. D. Dupuis, “Performance of deep ultraviolet GaN avalanche photodiodes grown by MOCVD,” IEEE Photonics Technol. Lett. 19, 1744–1746 (2007).
[CrossRef]

2006 (2)

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]

D. Kedar and S. Arnon, “Non-line-of-sight optical wireless sensor network operating in multiscattering channel,” Appl. Opt. 45, 8454–8461 (2006).
[CrossRef] [PubMed]

2003 (1)

2002 (1)

M. Shatalov, J. Zhang, A. S. Chitnis, V. Adivarahan, J. Yang, G. Simin, and M. A. Khan, “Deep ultraviolet light-emitting diodes using quaternary AlInGaN multiple quantum wells,” IEEE J. Sel. Top. Quantum Electron. 8, 302–309 (2002).
[CrossRef]

1995 (1)

1994 (1)

1991 (1)

1979 (1)

1978 (1)

Abou-Galaga, F.

Adivarahan, V.

V. Adivarahan, W. Fareed, S. Srivastava, T. Katona, M. Gaevski, and A. Khan, “Robust 285 nm deep UV light emitting diodes over metal organic hydride vapor phase epitaxially grown AIN/sapphire templates,” Jpn. J. Appl. Phys. 46, L537 –L579 (2007).
[CrossRef]

M. Shatalov, J. Zhang, A. S. Chitnis, V. Adivarahan, J. Yang, G. Simin, and M. A. Khan, “Deep ultraviolet light-emitting diodes using quaternary AlInGaN multiple quantum wells,” IEEE J. Sel. Top. Quantum Electron. 8, 302–309 (2002).
[CrossRef]

Arnon, S.

Bucholtz, A.

Chen, G.

Chitnis, A. S.

M. Shatalov, J. Zhang, A. S. Chitnis, V. Adivarahan, J. Yang, G. Simin, and M. A. Khan, “Deep ultraviolet light-emitting diodes using quaternary AlInGaN multiple quantum wells,” IEEE J. Sel. Top. Quantum Electron. 8, 302–309 (2002).
[CrossRef]

Ding, H.

H. Ding, G. Chen, A. K. Majumdar, B. M. Sadler, and Z. Xu, “Modeling of non-line-of-sight ultraviolet scattering channels for communication,” IEEE J. Sel. Areas Commun. 27, 1535–1544(2009).
[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] [PubMed]

Dupuis, R. D.

S.-C. Shen, Y. Zhang, D. Yoo, J.-B. Limb, J.-H. Ryou, P. D. Yoder, and R. D. Dupuis, “Performance of deep ultraviolet GaN avalanche photodiodes grown by MOCVD,” IEEE Photonics Technol. Lett. 19, 1744–1746 (2007).
[CrossRef]

Elterman, L.

L. Elterman, “UV, visible, and IR attenuation for altitudes to 50 km, 1968,” Tech. Rep. AFCRL-68-0153 (Air Force Cambridge Research Laboratories, Bedford, MA 1968).

Fareed, W.

V. Adivarahan, W. Fareed, S. Srivastava, T. Katona, M. Gaevski, and A. Khan, “Robust 285 nm deep UV light emitting diodes over metal organic hydride vapor phase epitaxially grown AIN/sapphire templates,” Jpn. J. Appl. Phys. 46, L537 –L579 (2007).
[CrossRef]

Feller, W.

W. Feller, “Densities in higher dimensions. Normal densities and processes,” in An Introduction to Probability Theory and its Applications (Wiley, 1966), Vol. II, pp. 65–100.

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, Inc., Princeton, NJ 1976).

Gaevski, M.

V. Adivarahan, W. Fareed, S. Srivastava, T. Katona, M. Gaevski, and A. Khan, “Robust 285 nm deep UV light emitting diodes over metal organic hydride vapor phase epitaxially grown AIN/sapphire templates,” Jpn. J. Appl. Phys. 46, L537 –L579 (2007).
[CrossRef]

Harvey, G. L.

G. L. Harvey, “A survey of ultraviolet communication systems,” Tech. Rep. NRL-6037 (U. S. Naval Research Laboratory, Washington, DC 1964).

He, Q.

Q. He, B. M. Sadler, and Z. Xu, “Modulation and coding tradeoffs for non-line-of-sight ultraviolet communications,” Proc. SPIE 7464, 74640H (2009).
[CrossRef]

Hombs, B.

D. Moriarty and B. Hombs, “System design of tactical communications with solar blind ultraviolet non line-of-sight systems,” in IEEE Military Communications Conference (IEEE, 2009), 1–7.

Junge, D. M.

D. M. Junge, “Non-line-of-sight electro-optic laser communications in the middle ultraviolet,” M. S. thesis (Naval Postgraduate School, Monterey, CA 1977).

Katona, T.

V. Adivarahan, W. Fareed, S. Srivastava, T. Katona, M. Gaevski, and A. Khan, “Robust 285 nm deep UV light emitting diodes over metal organic hydride vapor phase epitaxially grown AIN/sapphire templates,” Jpn. J. Appl. Phys. 46, L537 –L579 (2007).
[CrossRef]

Kedar, D.

Kennedy, R. S.

W. S. Ross and R. S. Kennedy, “An investigation of atmospheric optically scattered non-line-of-sight communication links,” Tech. Rep. ARO-15365.2-A.EL (U. S. Army Research Office, Research Triangle Park, NJ 1980).

Khan, A.

V. Adivarahan, W. Fareed, S. Srivastava, T. Katona, M. Gaevski, and A. Khan, “Robust 285 nm deep UV light emitting diodes over metal organic hydride vapor phase epitaxially grown AIN/sapphire templates,” Jpn. J. Appl. Phys. 46, L537 –L579 (2007).
[CrossRef]

Khan, M. A.

M. Shatalov, J. Zhang, A. S. Chitnis, V. Adivarahan, J. Yang, G. Simin, and M. A. Khan, “Deep ultraviolet light-emitting diodes using quaternary AlInGaN multiple quantum wells,” IEEE J. Sel. Top. Quantum Electron. 8, 302–309 (2002).
[CrossRef]

Kopeika, N. S.

Limb, J.-B.

S.-C. Shen, Y. Zhang, D. Yoo, J.-B. Limb, J.-H. Ryou, P. D. Yoder, and R. D. Dupuis, “Performance of deep ultraviolet GaN avalanche photodiodes grown by MOCVD,” IEEE Photonics Technol. Lett. 19, 1744–1746 (2007).
[CrossRef]

Liu, J. S.

J. S. Liu, Monte Carlo Strategies in Scientific Computing (Springer, 2001).

Luettgen, M. R.

Majumdar, A. K.

H. Ding, G. Chen, A. K. Majumdar, B. M. Sadler, and Z. Xu, “Modeling of non-line-of-sight ultraviolet scattering channels for communication,” IEEE J. Sel. Areas Commun. 27, 1535–1544(2009).
[CrossRef]

Mathews, J. H.

J. H. Mathews, Numerical Methods for Mathematics, Science, and Engineering, 2nd ed. (Prentice Hall, 1992).

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]

Moriarty, D.

D. Moriarty and B. Hombs, “System design of tactical communications with solar blind ultraviolet non line-of-sight systems,” in IEEE Military Communications Conference (IEEE, 2009), 1–7.

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, Inc., Princeton, NJ 1976).

Reilly, D. M.

D. M. Reilly, “Temporal characteristics of single-scatter radiation,” J. Opt. Soc. Am. 69, 464–470 (1979).
[CrossRef]

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

Ross, W. S.

W. S. Ross and R. S. Kennedy, “An investigation of atmospheric optically scattered non-line-of-sight communication links,” Tech. Rep. ARO-15365.2-A.EL (U. S. Army Research Office, Research Triangle Park, NJ 1980).

Ryou, J.-H.

S.-C. Shen, Y. Zhang, D. Yoo, J.-B. Limb, J.-H. Ryou, P. D. Yoder, and R. D. Dupuis, “Performance of deep ultraviolet GaN avalanche photodiodes grown by MOCVD,” IEEE Photonics Technol. Lett. 19, 1744–1746 (2007).
[CrossRef]

Sadler, B. M.

Q. He, B. M. Sadler, and Z. Xu, “Modulation and coding tradeoffs for non-line-of-sight ultraviolet communications,” Proc. SPIE 7464, 74640H (2009).
[CrossRef]

H. Ding, G. Chen, A. K. Majumdar, B. M. Sadler, and Z. Xu, “Modeling of non-line-of-sight ultraviolet scattering channels for communication,” IEEE J. Sel. Areas Commun. 27, 1535–1544(2009).
[CrossRef]

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

G. Chen, F. Abou-Galaga, Z. Xu, and B. M. Sadler, “Experimental evaluation of LED-based solar blind NLOS communication links,” Opt. Express 16, 15059–15068 (2008).
[CrossRef] [PubMed]

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, Inc., Princeton, NJ 1976).

Shapiro, J. H.

Shatalov, M.

M. Shatalov, J. Zhang, A. S. Chitnis, V. Adivarahan, J. Yang, G. Simin, and M. A. Khan, “Deep ultraviolet light-emitting diodes using quaternary AlInGaN multiple quantum wells,” IEEE J. Sel. Top. Quantum Electron. 8, 302–309 (2002).
[CrossRef]

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]

Shen, S.-C.

S.-C. Shen, Y. Zhang, D. Yoo, J.-B. Limb, J.-H. Ryou, P. D. Yoder, and R. D. Dupuis, “Performance of deep ultraviolet GaN avalanche photodiodes grown by MOCVD,” IEEE Photonics Technol. Lett. 19, 1744–1746 (2007).
[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]

Simin, G.

M. Shatalov, J. Zhang, A. S. Chitnis, V. Adivarahan, J. Yang, G. Simin, and M. A. Khan, “Deep ultraviolet light-emitting diodes using quaternary AlInGaN multiple quantum wells,” IEEE J. Sel. Top. Quantum Electron. 8, 302–309 (2002).
[CrossRef]

Srivastava, S.

V. Adivarahan, W. Fareed, S. Srivastava, T. Katona, M. Gaevski, and A. Khan, “Robust 285 nm deep UV light emitting diodes over metal organic hydride vapor phase epitaxially grown AIN/sapphire templates,” Jpn. J. Appl. Phys. 46, L537 –L579 (2007).
[CrossRef]

Sunstein, D. E.

D. E. Sunstein, “A scatter communication link at ultraviolet frequencies,” B. S. thesis (Massachusetts Institute of Technology, 1968).

Xu, Z.

H. Ding, G. Chen, A. K. Majumdar, B. M. Sadler, and Z. Xu, “Modeling of non-line-of-sight ultraviolet scattering channels for communication,” IEEE J. Sel. Areas Commun. 27, 1535–1544(2009).
[CrossRef]

Q. He, B. M. Sadler, and Z. Xu, “Modulation and coding tradeoffs for non-line-of-sight ultraviolet communications,” Proc. SPIE 7464, 74640H (2009).
[CrossRef]

G. Chen, F. Abou-Galaga, Z. Xu, and B. M. Sadler, “Experimental evaluation of LED-based solar blind NLOS communication links,” Opt. Express 16, 15059–15068 (2008).
[CrossRef] [PubMed]

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] [PubMed]

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

Yang, J.

M. Shatalov, J. Zhang, A. S. Chitnis, V. Adivarahan, J. Yang, G. Simin, and M. A. Khan, “Deep ultraviolet light-emitting diodes using quaternary AlInGaN multiple quantum wells,” IEEE J. Sel. Top. Quantum Electron. 8, 302–309 (2002).
[CrossRef]

Yoder, P. D.

S.-C. Shen, Y. Zhang, D. Yoo, J.-B. Limb, J.-H. Ryou, P. D. Yoder, and R. D. Dupuis, “Performance of deep ultraviolet GaN avalanche photodiodes grown by MOCVD,” IEEE Photonics Technol. Lett. 19, 1744–1746 (2007).
[CrossRef]

Yoo, D.

S.-C. Shen, Y. Zhang, D. Yoo, J.-B. Limb, J.-H. Ryou, P. D. Yoder, and R. D. Dupuis, “Performance of deep ultraviolet GaN avalanche photodiodes grown by MOCVD,” IEEE Photonics Technol. Lett. 19, 1744–1746 (2007).
[CrossRef]

Zachor, A. S.

Zhang, J.

M. Shatalov, J. Zhang, A. S. Chitnis, V. Adivarahan, J. Yang, G. Simin, and M. A. Khan, “Deep ultraviolet light-emitting diodes using quaternary AlInGaN multiple quantum wells,” IEEE J. Sel. Top. Quantum Electron. 8, 302–309 (2002).
[CrossRef]

Zhang, Y.

S.-C. Shen, Y. Zhang, D. Yoo, J.-B. Limb, J.-H. Ryou, P. D. Yoder, and R. D. Dupuis, “Performance of deep ultraviolet GaN avalanche photodiodes grown by MOCVD,” IEEE Photonics Technol. Lett. 19, 1744–1746 (2007).
[CrossRef]

Appl. Opt. (5)

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 J. Sel. Areas Commun. (1)

H. Ding, G. Chen, A. K. Majumdar, B. M. Sadler, and Z. Xu, “Modeling of non-line-of-sight ultraviolet scattering channels for communication,” IEEE J. Sel. Areas Commun. 27, 1535–1544(2009).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

M. Shatalov, J. Zhang, A. S. Chitnis, V. Adivarahan, J. Yang, G. Simin, and M. A. Khan, “Deep ultraviolet light-emitting diodes using quaternary AlInGaN multiple quantum wells,” IEEE J. Sel. Top. Quantum Electron. 8, 302–309 (2002).
[CrossRef]

IEEE Photonics Technol. Lett. (1)

S.-C. Shen, Y. Zhang, D. Yoo, J.-B. Limb, J.-H. Ryou, P. D. Yoder, and R. D. Dupuis, “Performance of deep ultraviolet GaN avalanche photodiodes grown by MOCVD,” IEEE Photonics Technol. Lett. 19, 1744–1746 (2007).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Jpn. J. Appl. Phys. (1)

V. Adivarahan, W. Fareed, S. Srivastava, T. Katona, M. Gaevski, and A. Khan, “Robust 285 nm deep UV light emitting diodes over metal organic hydride vapor phase epitaxially grown AIN/sapphire templates,” Jpn. J. Appl. Phys. 46, L537 –L579 (2007).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Proc. SPIE (2)

Q. He, B. M. Sadler, and Z. Xu, “Modulation and coding tradeoffs for non-line-of-sight ultraviolet communications,” Proc. SPIE 7464, 74640H (2009).
[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 (11)

D. Moriarty and B. Hombs, “System design of tactical communications with solar blind ultraviolet non line-of-sight systems,” in IEEE Military Communications Conference (IEEE, 2009), 1–7.

J. H. Mathews, Numerical Methods for Mathematics, Science, and Engineering, 2nd ed. (Prentice Hall, 1992).

W. Feller, “Densities in higher dimensions. Normal densities and processes,” in An Introduction to Probability Theory and its Applications (Wiley, 1966), Vol. II, pp. 65–100.

W. S. Ross and R. S. Kennedy, “An investigation of atmospheric optically scattered non-line-of-sight communication links,” Tech. Rep. ARO-15365.2-A.EL (U. S. Army Research Office, Research Triangle Park, NJ 1980).

J. S. Liu, Monte Carlo Strategies in Scientific Computing (Springer, 2001).

D. M. Junge, “Non-line-of-sight electro-optic laser communications in the middle ultraviolet,” M. S. thesis (Naval Postgraduate School, Monterey, CA 1977).

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

G. L. Harvey, “A survey of ultraviolet communication systems,” Tech. Rep. NRL-6037 (U. S. Naval Research Laboratory, Washington, DC 1964).

L. Elterman, “UV, visible, and IR attenuation for altitudes to 50 km, 1968,” Tech. Rep. AFCRL-68-0153 (Air Force Cambridge Research Laboratories, Bedford, MA 1968).

D. E. Sunstein, “A scatter communication link at ultraviolet frequencies,” B. S. thesis (Massachusetts Institute of Technology, 1968).

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

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Figures (8)

Fig. 1
Fig. 1

UV communication system geometry.

Fig. 2
Fig. 2

Illustration of a planar photon propagation path.

Fig. 3
Fig. 3

Pseudocode describing the overall Monte Carlo UV communication channel model.

Fig. 4
Fig. 4

Pseudocode for function scat_angle.

Fig. 5
Fig. 5

Path loss as a function of (a) range and (b) azimuth offset angle.

Fig. 6
Fig. 6

Comparison of the rate of convergence for two Monte Carlo implementations.

Fig. 8
Fig. 8

Double-peaked IRF.

Equations (18)

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

f T ( α T , ψ T ) = { sin α T 2 π [ 1 cos ( β T / 2 ) ] , if     0 α T β T / 2 0 , otherwise,
f e ( d ) = k e exp { k e d } ,
p ray ( α , ψ ) = 3 [ 1 + 3 γ + ( 1 γ ) cos 2 α ] 16 π ( 1 + 2 γ ) ,
p mie ( α , ψ ) = 1 g 2 4 π [ 1 ( 1 + g 2 2 g cos α ) 3 / 2 + f 3 cos 2 α 1 2 ( 1 + g 2 ) 3 / 2 ] ,
p total ( α , ψ ) = k r k s p ray ( α , ψ ) + k m k s p mie ( α , ψ ) ,
f α , ψ ( α , ψ ) = p total ( α , ψ ) sin α .
P R ( α R , ψ R ) = { 1 , if     α R β R / 2 0 , otherwise,
f D ( P ) = f T ( α T , ψ T ) [ i = 1 n k s k e f e ( d i ) f α , ψ ( α i , ψ i ) ] P e ¯ ( d n + 1 ) P R ( α R , ψ R ) .
Ω f T ( α T , ψ T ) d α T d ψ T = 1 2 π [ 1 cos ( β T / 2 ) ] Ω sin α T d α T d ψ T 1 r ( r 2 + S R / π ) 1 / 2 1 cos ( β T / 2 ) ,
P D , 0 = { P e ¯ ( r ) 1 r ( r 2 + S R / π ) 1 / 2 1 cos ( β T / 2 ) , if     μ 0 , y cos ( β T / 2 ) and μ R , y cos ( β R / 2 ) 0 , otherwise , ,
C 1 = k r k s · 3 ( 1 + 3 γ ) 8 ( 1 + 2 γ ) + k m k s · f ( 1 g 2 ) 4 ( 1 + g 2 ) 3 / 2 ,
C 2 = k r k s · 1 γ 8 ( 1 + 2 γ ) k m k s · f ( 1 g 2 ) 4 ( 1 + g 2 ) 3 / 2 ,
F α ( α ) = C 1 cos α + C 2 cos 3 α + C 3 ( C 4 C 5 cos α ) 1 / 2 + C 6 .
μ x , i = sin α i 1 ( 1 μ z , i 1 2 ) 1 / 2 ( μ x , i 1 μ z , i 1 cos ψ i 1 μ y , i 1 sin ψ i 1 ) + μ x , i 1 cos α i 1 ,
μ y , i = sin α i 1 ( 1 μ z , i 1 2 ) 1 / 2 ( μ y , i 1 μ z , i 1 cos ψ i 1 + μ x , i 1 sin ψ i 1 ) + μ y , i 1 cos α i 1 ,
μ z , i = sin α i 1 cos ψ i 1 ( 1 μ z , i 1 2 ) 1 / 2 + μ z , i 1 cos α i 1 ,
P i R = P e ¯ ( r i ) · 2 π [ 1 r i ( r i 2 + S R / π ) 1 / 2 ] · p total ( α i R , 0 ) ,
P D , i = { p i 1 ( k s k e ) P i R , if     r i · μ R r i cos ( β R / 2 ) 0 , otherwise .

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