Abstract

Deep (>10 dB) long-duration (>1 ms) scintillation fades, caused by propagation through refractive-index turbulence, are the principal impairment that must be overcome to realize Gbps-class laser communication over line-of-sight atmospheric paths in clear-weather conditions. Spatial diversity reception can ameliorate such fades, to a degree, but current systems typically rely on forward error-correction and interleaving to achieve reliable communication over the atmospheric channel. This paper, together with its companion [A. L. Puryear, J. H. Shapiro, and R. R. Parenti, “Reciprocity-enhanced optical communication through atmospheric turbulence—Part II: Communication architectures and performance,” to be submitted to J. Opt. Commun. Netw.], comprise a two-part study that introduces and analyzes an alternative approach, in which atmospheric reciprocity is exploited to eliminate the need for interleaving and minimize the amount of forward error-correction required. The present work (Part I) first describes the problem setting and then presents proofs for reciprocity principles—with and without phase compensation—that apply under rather general conditions. By specializing to the far-field regime, the optimum (power-transfer maximizing) phase compensation is identified. These results underlie the communication architectures and performance analysis that will be reported in the Part II paper.

© 2012 OSA

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2012 (1)

2011 (1)

B. S. Robinson, D. M. Boroson, D. A. Burianek, and D. V. Murphy, “Overview of the lunar laser communications demonstration,” Proc. SPIE, vol. 7923, pp. 792302-1–792302-4, 2011.

2010 (1)

F. G. Walther, S. Michael, R. R. Parenti, and J. A. Taylor, “Air-to-ground optical communication system demonstration design overview and results summary,” Proc. SPIE, vol. 7814, pp. 78140Y-1–78140Y-9, 2010.

2009 (2)

J. A. Greco, “Design of the high-speed framing, FEC, and interleaving hardware used in a 5.4 km free-space optical communication experiment,” Proc. SPIE, vol. 7464, pp. 746409-1–746409-7, 2009.

J. D. Moores, F. G. Walther, J. A. Greco, S. Michael, W. E. Wilcox Jr., A. M. Volpicelli, R. J. Magliocco, and S. R. Henion, “Architecture overview and data summary of a 5.4 km free-space laser communications experiment,” Proc. SPIE, vol. 7464, pp. 746404-1–746404-9, 2009.

2007 (1)

H. Hemmati, A. Biswas, and D. M. Boroson, “Prospects for improvement of interplanetary communication data rates by 30 dB,” Proc. IEEE, vol. 95, no. 10, pp. 2082–2092, Oct.2007.
[CrossRef]

2006 (1)

2005 (1)

M. Toyoshima, S. Yamakawa, T. Yamawaki, K. Arai, M. R. García-Talavera, A. Alonso, Z. Sodnik, and B. Demelenne, “Long-term statistics of laser beam propagation in an optical ground-to-geostationary satellite communications link,” IEEE Trans. Antennas Propag., vol. 53, no. 2, pp. 842–850, Feb.2005.
[CrossRef]

2004 (1)

D. Kedar and S. Arnon, “Urban optical wireless communication networks: The main challenges and possible solutions,” IEEE Commun. Mag., vol. 42, no. 5, pp. S2–S7, May2004.
[CrossRef]

2003 (1)

2000 (1)

V. W. S. Chan, “Optical space communications,” IEEE J. Sel. Top. Quantum Electron., vol. 6, no. 6, pp. 959–975, Nov./Dec.2000.
[CrossRef]

1998 (1)

D. J. T. Healey, D. R. Wisely, I. Neild, and P. Cochrane, “Optical wireless: The story so far,” IEEE Commun. Mag., vol. 36, no. 12, pp. 72–74, 79–82, Dec.1998.
[CrossRef]

1991 (1)

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, and L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide-star,” Nature, vol. 353, pp. 144–146, Sept.1991.
[CrossRef]

1984 (1)

S. Lin, D. J. Costello Jr., and M. J. Miller, “Automatic-repeat request error-control schemes,” IEEE Commun. Mag., vol. 22, no. 12, pp. 5–17, Dec.1984.
[CrossRef]

1976 (1)

1975 (2)

R. L. Fante, “Electromagnetic beam propagation in turbulent media,” Proc. IEEE, vol. 63, pp. 1669–1692, Dec.1975.
[CrossRef]

J. H. Shapiro, “Point-ahead limitation on reciprocity tracking,” J. Opt. Soc. Am., vol. 65, pp. 65–68, Jan.1975.
[CrossRef]

1974 (1)

1971 (3)

1965 (1)

D. Slepian, “Analytical solution to two apodization problems,” J. Opt. Soc. Am., vol. 55, pp. 1110–1114, Sept.1965.
[CrossRef]

Alonso, A.

M. Toyoshima, S. Yamakawa, T. Yamawaki, K. Arai, M. R. García-Talavera, A. Alonso, Z. Sodnik, and B. Demelenne, “Long-term statistics of laser beam propagation in an optical ground-to-geostationary satellite communications link,” IEEE Trans. Antennas Propag., vol. 53, no. 2, pp. 842–850, Feb.2005.
[CrossRef]

Ameer, G. A.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, and L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide-star,” Nature, vol. 353, pp. 144–146, Sept.1991.
[CrossRef]

Arai, K.

M. Toyoshima, S. Yamakawa, T. Yamawaki, K. Arai, M. R. García-Talavera, A. Alonso, Z. Sodnik, and B. Demelenne, “Long-term statistics of laser beam propagation in an optical ground-to-geostationary satellite communications link,” IEEE Trans. Antennas Propag., vol. 53, no. 2, pp. 842–850, Feb.2005.
[CrossRef]

Arnon, S.

D. Kedar and S. Arnon, “Urban optical wireless communication networks: The main challenges and possible solutions,” IEEE Commun. Mag., vol. 42, no. 5, pp. S2–S7, May2004.
[CrossRef]

Biswas, A.

H. Hemmati, A. Biswas, and D. M. Boroson, “Prospects for improvement of interplanetary communication data rates by 30 dB,” Proc. IEEE, vol. 95, no. 10, pp. 2082–2092, Oct.2007.
[CrossRef]

Boeke, B. R.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, and L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide-star,” Nature, vol. 353, pp. 144–146, Sept.1991.
[CrossRef]

Boroson, D. M.

B. S. Robinson, D. M. Boroson, D. A. Burianek, and D. V. Murphy, “Overview of the lunar laser communications demonstration,” Proc. SPIE, vol. 7923, pp. 792302-1–792302-4, 2011.

H. Hemmati, A. Biswas, and D. M. Boroson, “Prospects for improvement of interplanetary communication data rates by 30 dB,” Proc. IEEE, vol. 95, no. 10, pp. 2082–2092, Oct.2007.
[CrossRef]

Browne, S. L.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, and L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide-star,” Nature, vol. 353, pp. 144–146, Sept.1991.
[CrossRef]

Burianek, D. A.

B. S. Robinson, D. M. Boroson, D. A. Burianek, and D. V. Murphy, “Overview of the lunar laser communications demonstration,” Proc. SPIE, vol. 7923, pp. 792302-1–792302-4, 2011.

Chan, V. W. S.

Cochrane, P.

D. J. T. Healey, D. R. Wisely, I. Neild, and P. Cochrane, “Optical wireless: The story so far,” IEEE Commun. Mag., vol. 36, no. 12, pp. 72–74, 79–82, Dec.1998.
[CrossRef]

Costello Jr., D. J.

S. Lin, D. J. Costello Jr., and M. J. Miller, “Automatic-repeat request error-control schemes,” IEEE Commun. Mag., vol. 22, no. 12, pp. 5–17, Dec.1984.
[CrossRef]

Demelenne, B.

M. Toyoshima, S. Yamakawa, T. Yamawaki, K. Arai, M. R. García-Talavera, A. Alonso, Z. Sodnik, and B. Demelenne, “Long-term statistics of laser beam propagation in an optical ground-to-geostationary satellite communications link,” IEEE Trans. Antennas Propag., vol. 53, no. 2, pp. 842–850, Feb.2005.
[CrossRef]

Fante, R. L.

R. L. Fante, “Electromagnetic beam propagation in turbulent media,” Proc. IEEE, vol. 63, pp. 1669–1692, Dec.1975.
[CrossRef]

Frazier, B. W.

R. K. Tyson and B. W. Frazier, Field Guide to Adaptive Optics. SPIE, Bellingham, 2012.

Fried, D. L.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, and L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide-star,” Nature, vol. 353, pp. 144–146, Sept.1991.
[CrossRef]

D. P. Greenwood and D. L. Fried, “Power spectra requirements for wave-front-compensation systems,” J. Opt. Soc. Am., vol. 66, no. 3, pp. 193–206, 1976.
[CrossRef]

Fugate, R. Q.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, and L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide-star,” Nature, vol. 353, pp. 144–146, Sept.1991.
[CrossRef]

García-Talavera, M. R.

M. Toyoshima, S. Yamakawa, T. Yamawaki, K. Arai, M. R. García-Talavera, A. Alonso, Z. Sodnik, and B. Demelenne, “Long-term statistics of laser beam propagation in an optical ground-to-geostationary satellite communications link,” IEEE Trans. Antennas Propag., vol. 53, no. 2, pp. 842–850, Feb.2005.
[CrossRef]

Gloge, D.

D. Marcuse, D. Gloge, and E. A. J. Marcatili, “Guiding properties of fibers,” in Optical Fiber Telecommunications. S. E. Miller and A. G. Chuynoweth, Eds., Academic, New York, 1979.

Greco, J. A.

Greco, J. A.

J. D. Moores, F. G. Walther, J. A. Greco, S. Michael, W. E. Wilcox Jr., A. M. Volpicelli, R. J. Magliocco, and S. R. Henion, “Architecture overview and data summary of a 5.4 km free-space laser communications experiment,” Proc. SPIE, vol. 7464, pp. 746404-1–746404-9, 2009.

J. A. Greco, “Design of the high-speed framing, FEC, and interleaving hardware used in a 5.4 km free-space optical communication experiment,” Proc. SPIE, vol. 7464, pp. 746409-1–746409-7, 2009.

Greenwood, D. P.

Healey, D. J. T.

D. J. T. Healey, D. R. Wisely, I. Neild, and P. Cochrane, “Optical wireless: The story so far,” IEEE Commun. Mag., vol. 36, no. 12, pp. 72–74, 79–82, Dec.1998.
[CrossRef]

Hemmati, H.

H. Hemmati, A. Biswas, and D. M. Boroson, “Prospects for improvement of interplanetary communication data rates by 30 dB,” Proc. IEEE, vol. 95, no. 10, pp. 2082–2092, Oct.2007.
[CrossRef]

Henion, S. R.

J. D. Moores, F. G. Walther, J. A. Greco, S. Michael, W. E. Wilcox Jr., A. M. Volpicelli, R. J. Magliocco, and S. R. Henion, “Architecture overview and data summary of a 5.4 km free-space laser communications experiment,” Proc. SPIE, vol. 7464, pp. 746404-1–746404-9, 2009.

Kedar, D.

D. Kedar and S. Arnon, “Urban optical wireless communication networks: The main challenges and possible solutions,” IEEE Commun. Mag., vol. 42, no. 5, pp. S2–S7, May2004.
[CrossRef]

Levitt, B. K.

B. K. Levitt, “Variable-rate optical communication through the turbulent atmosphere,” Tech. Rep. 483, MIT Cambridge Research Lab of Electronics, Aug.20, 1971.

Lin, S.

S. Lin, D. J. Costello Jr., and M. J. Miller, “Automatic-repeat request error-control schemes,” IEEE Commun. Mag., vol. 22, no. 12, pp. 5–17, Dec.1984.
[CrossRef]

Lutomirski, R. F.

Magliocco, R. J.

J. D. Moores, F. G. Walther, J. A. Greco, S. Michael, W. E. Wilcox Jr., A. M. Volpicelli, R. J. Magliocco, and S. R. Henion, “Architecture overview and data summary of a 5.4 km free-space laser communications experiment,” Proc. SPIE, vol. 7464, pp. 746404-1–746404-9, 2009.

Marcatili, E. A. J.

D. Marcuse, D. Gloge, and E. A. J. Marcatili, “Guiding properties of fibers,” in Optical Fiber Telecommunications. S. E. Miller and A. G. Chuynoweth, Eds., Academic, New York, 1979.

Marcuse, D.

D. Marcuse, D. Gloge, and E. A. J. Marcatili, “Guiding properties of fibers,” in Optical Fiber Telecommunications. S. E. Miller and A. G. Chuynoweth, Eds., Academic, New York, 1979.

Michael, S.

F. G. Walther, S. Michael, R. R. Parenti, and J. A. Taylor, “Air-to-ground optical communication system demonstration design overview and results summary,” Proc. SPIE, vol. 7814, pp. 78140Y-1–78140Y-9, 2010.

J. D. Moores, F. G. Walther, J. A. Greco, S. Michael, W. E. Wilcox Jr., A. M. Volpicelli, R. J. Magliocco, and S. R. Henion, “Architecture overview and data summary of a 5.4 km free-space laser communications experiment,” Proc. SPIE, vol. 7464, pp. 746404-1–746404-9, 2009.

R. R. Parenti, S. Michael, J. M. Roth, and T. M. Yarnall, “Observations of power-in-fiber statistics in two recent free-space communication link experiments,” in Applications of Lasers for Sensing and Free Space Communications, 2010, LSMB3.

Miller, M. J.

S. Lin, D. J. Costello Jr., and M. J. Miller, “Automatic-repeat request error-control schemes,” IEEE Commun. Mag., vol. 22, no. 12, pp. 5–17, Dec.1984.
[CrossRef]

Moores, J. D.

J. D. Moores, F. G. Walther, J. A. Greco, S. Michael, W. E. Wilcox Jr., A. M. Volpicelli, R. J. Magliocco, and S. R. Henion, “Architecture overview and data summary of a 5.4 km free-space laser communications experiment,” Proc. SPIE, vol. 7464, pp. 746404-1–746404-9, 2009.

Murphy, D. V.

B. S. Robinson, D. M. Boroson, D. A. Burianek, and D. V. Murphy, “Overview of the lunar laser communications demonstration,” Proc. SPIE, vol. 7923, pp. 792302-1–792302-4, 2011.

Neild, I.

D. J. T. Healey, D. R. Wisely, I. Neild, and P. Cochrane, “Optical wireless: The story so far,” IEEE Commun. Mag., vol. 36, no. 12, pp. 72–74, 79–82, Dec.1998.
[CrossRef]

Parenti, R. R.

R. R. Parenti, J. M. Roth, J. H. Shapiro, F. G. Walther, and J. A. Greco, “Experimental observations of channel reciprocity in single-mode free-space optical links,” Opt. Express, vol. 20, no. 19, pp. 21635–21644, Sept.2012.
[CrossRef] [PubMed]

F. G. Walther, S. Michael, R. R. Parenti, and J. A. Taylor, “Air-to-ground optical communication system demonstration design overview and results summary,” Proc. SPIE, vol. 7814, pp. 78140Y-1–78140Y-9, 2010.

R. R. Parenti, J. M. Roth, J. H. Shapiro, and F. G. Walther, “Observations of channel reciprocity in optical free-space communications experiments,” in Applications of Lasers for Sensing and Free Space Communications, 2011, LTuD3.

R. R. Parenti, S. Michael, J. M. Roth, and T. M. Yarnall, “Observations of power-in-fiber statistics in two recent free-space communication link experiments,” in Applications of Lasers for Sensing and Free Space Communications, 2010, LSMB3.

A. L. Puryear, J. H. Shapiro, and R. R. Parenti, “Reciprocity- enhanced optical communication through atmospheric turbulence—Part II: Communication architectures and perfor mance,” to be submitted to J. Opt. Commun. Netw.

Puryear, A. L.

A. L. Puryear, J. H. Shapiro, and R. R. Parenti, “Reciprocity- enhanced optical communication through atmospheric turbulence—Part II: Communication architectures and perfor mance,” to be submitted to J. Opt. Commun. Netw.

Roberts, P. H.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, and L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide-star,” Nature, vol. 353, pp. 144–146, Sept.1991.
[CrossRef]

Robinson, B. S.

B. S. Robinson, D. M. Boroson, D. A. Burianek, and D. V. Murphy, “Overview of the lunar laser communications demonstration,” Proc. SPIE, vol. 7923, pp. 792302-1–792302-4, 2011.

Roth, J. M.

R. R. Parenti, J. M. Roth, J. H. Shapiro, F. G. Walther, and J. A. Greco, “Experimental observations of channel reciprocity in single-mode free-space optical links,” Opt. Express, vol. 20, no. 19, pp. 21635–21644, Sept.2012.
[CrossRef] [PubMed]

R. R. Parenti, J. M. Roth, J. H. Shapiro, and F. G. Walther, “Observations of channel reciprocity in optical free-space communications experiments,” in Applications of Lasers for Sensing and Free Space Communications, 2011, LTuD3.

R. R. Parenti, S. Michael, J. M. Roth, and T. M. Yarnall, “Observations of power-in-fiber statistics in two recent free-space communication link experiments,” in Applications of Lasers for Sensing and Free Space Communications, 2010, LSMB3.

Ruane, R. E.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, and L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide-star,” Nature, vol. 353, pp. 144–146, Sept.1991.
[CrossRef]

Shapiro, J. H.

R. R. Parenti, J. M. Roth, J. H. Shapiro, F. G. Walther, and J. A. Greco, “Experimental observations of channel reciprocity in single-mode free-space optical links,” Opt. Express, vol. 20, no. 19, pp. 21635–21644, Sept.2012.
[CrossRef] [PubMed]

J. H. Shapiro, “Point-ahead limitation on reciprocity tracking,” J. Opt. Soc. Am., vol. 65, pp. 65–68, Jan.1975.
[CrossRef]

J. H. Shapiro, “Normal-mode approach to wave propagation in the turbulent atmosphere,” Appl. Opt., vol. 13, pp. 2614–2619, Nov.1974.
[CrossRef] [PubMed]

J. H. Shapiro, “Reciprocity of the turbulent atmosphere,” J. Opt. Soc. Am., vol. 61, no. 4, pp. 492–495, Apr.1971.
[CrossRef]

J. H. Shapiro, “Optimal power transfer through atmospheric turbulence using state knowledge,” IEEE Trans. Commun. Technol., vol. 19, pp. 410–414, Aug.1971.
[CrossRef]

A. L. Puryear, J. H. Shapiro, and R. R. Parenti, “Reciprocity- enhanced optical communication through atmospheric turbulence—Part II: Communication architectures and perfor mance,” to be submitted to J. Opt. Commun. Netw.

J. H. Shapiro, “Imaging and optical communication through atmospheric turbulence,” in Laser Beam Propagation in the Atmosphere. J. W. Strohbehn, Ed., Springer-Verlag, Berlin, 1978.

R. R. Parenti, J. M. Roth, J. H. Shapiro, and F. G. Walther, “Observations of channel reciprocity in optical free-space communications experiments,” in Applications of Lasers for Sensing and Free Space Communications, 2011, LTuD3.

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M. Toyoshima, S. Yamakawa, T. Yamawaki, K. Arai, M. R. García-Talavera, A. Alonso, Z. Sodnik, and B. Demelenne, “Long-term statistics of laser beam propagation in an optical ground-to-geostationary satellite communications link,” IEEE Trans. Antennas Propag., vol. 53, no. 2, pp. 842–850, Feb.2005.
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J. D. Moores, F. G. Walther, J. A. Greco, S. Michael, W. E. Wilcox Jr., A. M. Volpicelli, R. J. Magliocco, and S. R. Henion, “Architecture overview and data summary of a 5.4 km free-space laser communications experiment,” Proc. SPIE, vol. 7464, pp. 746404-1–746404-9, 2009.

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J. D. Moores, F. G. Walther, J. A. Greco, S. Michael, W. E. Wilcox Jr., A. M. Volpicelli, R. J. Magliocco, and S. R. Henion, “Architecture overview and data summary of a 5.4 km free-space laser communications experiment,” Proc. SPIE, vol. 7464, pp. 746404-1–746404-9, 2009.

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R. R. Parenti, S. Michael, J. M. Roth, and T. M. Yarnall, “Observations of power-in-fiber statistics in two recent free-space communication link experiments,” in Applications of Lasers for Sensing and Free Space Communications, 2010, LSMB3.

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Figures (2)

Fig. 1
Fig. 1

(Color online) Propagation geometry for bidirectional optical communication through atmospheric turbulence without adaptive optics, using fiber-coupled transceivers that employ diplexers to share a common entrance/exit pupil at each terminal. Transmitted powers at the fiber inputs are P 0 T and P L T , leading to received powers at the fiber outputs equal to P L R and P 0 R , respectively.

Fig. 2
Fig. 2

(Color online) Propagation geometry for bidirectional optical communication through atmospheric turbulence with adaptive optics, using fiber-coupled transceivers that employ diplexers to share a common entrance/exit pupil at each terminal. Transmitted powers at the fiber inputs are P 0 T and P L T , leading to received powers at the fiber outputs equal to P L R and P 0 R , respectively. Each terminal has a single pupil-plane adaptive-optics (AO) element to phase compensate both its transmitter and its receiver.

Equations (39)

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P L R / P 0 T = P 0 R / P L T ,
E F 0 xmt ( ρ f , t ) = Re [ E F 0 xmt ( ρ f ) e i ω 0 t i ] ,
P 0 T = F 0 d ρ f | E F 0 xmt ( ρ f ) | 2 .
E F 0 xmt ( ρ f ) = P 0 T ξ ( ρ f ) ,
F 0 d ρ f | ξ ( ρ f ) | 2 = 1 .
E A 0 xmt ( ρ ) = F 0 d ρ f E F 0 xmt ( ρ f ) g A 0 F 0 ( ρ , ρ f ) ,
E A 0 xmt ( ρ ) = F 0 d ρ f E F 0 xmt ( ρ f ) g A 0 F 0 ( ρ , ρ f ) e i θ 0 ( ρ ) .
E A L rcv ( ρ ) = A 0 d ρ E A 0 xmt ( ρ ) h ( ρ , ρ )
E F L rcv ( ρ f ) = A L d ρ E A L rcv ( ρ ) g F L A L ( ρ f , ρ )
E F L rcv ( ρ f ) = A L d ρ E A L rcv ( ρ ) e i θ L ( ρ ) g F L A L ( ρ f , ρ ) ,
P L R = | F L d ρ f E F L rcv ( ρ f ) ξ * ( ρ f ) | 2 .
E F L xmt ( ρ f , t ) = Re [ E F L xmt ( ρ f ) e i ω 0 t i ] ,
P L T = F L d ρ f | E F L xmt ( ρ f ) | 2 .
E A L xmt ( ρ ) = F L d ρ f E F L xmt ( ρ f ) g A L F L ( ρ , ρ f ) ,
E A L xmt ( ρ ) = F L d ρ f E F L xmt ( ρ f ) g A L F L ( ρ , ρ f ) e i θ L ( ρ ) .
E A 0 rcv ( ρ ) = A L d ρ E A L xmt ( ρ ) h ( ρ , ρ )
E F 0 rcv ( ρ f ) = A 0 d ρ E A 0 rcv ( ρ ) g F 0 A 0 ( ρ f , ρ )
E F 0 rcv ( ρ f ) = A 0 d ρ E A 0 rcv ( ρ ) e i θ 0 ( ρ ) g F 0 A 0 ( ρ f , ρ )
P 0 R = | F 0 d ρ f E F 0 rcv ( ρ f ) ξ * ( ρ f ) | 2 .
P L R P 0 T = | F L d ρ f A L d ρ A 0 d ρ F 0 d ρ f ξ ( ρ f ) g A 0 F 0 ( ρ , ρ f ) × e i θ 0 ( ρ ) h ( ρ , ρ ) e i θ L ( ρ ) g F L A L ( ρ f , ρ ) ξ * ( ρ f ) | 2 ,
P 0 R P L T = | F 0 d ρ f A 0 d ρ A L d ρ F L d ρ f ξ ( ρ f ) g A L F L ( ρ , ρ f ) × e i θ L ( ρ ) h ( ρ , ρ ) e i θ 0 ( ρ ) g F 0 A 0 ( ρ f , ρ ) ξ * ( ρ f ) | 2 ,
g 0 ( ρ , ρ f ) = g A 0 F 0 ( ρ , ρ f ) = g F 0 A 0 ( ρ f , ρ ) ,
g L ( ρ , ρ f ) = g A L F L ( ρ , ρ f ) = g F L A L ( ρ f , ρ ) .
P L R P 0 T = | F L d ρ f A L d ρ A 0 d ρ F 0 d ρ f ξ ( ρ f ) g 0 ( ρ , ρ f ) × e i θ 0 ( ρ ) h ( ρ , ρ ) e i θ L ( ρ ) g L ( ρ , ρ f ) ξ ( ρ f ) | 2 ,
P 0 R P L T = | F 0 d ρ f A 0 d ρ A L d ρ F L d ρ f ξ ( ρ f ) g L ( ρ , ρ f ) × e i θ L ( ρ ) h ( ρ , ρ ) e i θ 0 ( ρ ) g 0 ( ρ , ρ f ) ξ ( ρ f ) | 2 ,
ζ 0 ( ρ ) = 4 / π D 0 2 , for  ρ A 0 .
ζ L ( ρ ) = 4 π / D L 2 , for  ρ A L
D f A L d ρ A 0 d ρ | h ( ρ , ρ ) | 2 .
h ( ρ , ρ ) = n = 1 η n ϕ n ( ρ ) Φ n * ( ρ ) , for  ρ A 0 ρ A L .
h ( ρ , ρ ) η 1 ϕ 1 ( ρ ) Φ 1 * ( ρ ) , for  ρ A 0 ρ A L ,
h NT ( ρ ρ ) D f o ζ L ( ρ ) ζ 0 * ( ρ ) , for  ρ A 0 ρ A L
F 0 d ρ f g 0 ( ρ , ρ f ) ξ ( ρ f ) = μ 0 ζ 0 ( ρ ) , for  ρ A 0 ,
F L d ρ f g L ( ρ , ρ f ) ξ ( ρ f ) = μ L ζ L ( ρ ) , for  ρ A L ,
P L R P 0 T = P 0 R P L T = η 1 μ 0 μ L × | A 0 d ρ 4 π D 0 2 Φ 1 * ( ρ ) | 2 | A L d ρ 4 π D L 2 ϕ 1 ( ρ ) | 2
P L R P 0 T = P 0 R P L T = η 1 μ 0 μ L | A 0 d ρ 4 π D 0 2 e i θ 0 ( ρ ) Φ 1 * ( ρ ) | 2 × | A L d ρ 4 π D L 2 e i θ L ( ρ ) ϕ 1 ( ρ ) | 2 .
Φ 1 ( ρ ) = | Φ 1 ( ρ ) | e i Φ 1 ( ρ ) ,
ϕ 1 ( ρ ) = | ϕ 1 ( ρ ) | e i ϕ 1 ( ρ ) ,
P L R P 0 T = P 0 R P L T η 1 μ 0 μ L ( A 0 d ρ 4 π D 0 2 | Φ 1 ( ρ ) | ) 2 × ( A L d ρ 4 π D L 2 | ϕ 1 ( ρ ) | ) 2 ,
P L R P 0 T = P 0 R P L T = η 1 μ 0 μ L | A 0 d ρ 4 π D 0 2 e i ϵ 0 ( ρ ) | Φ 1 ( ρ ) | | 2 × | A L d ρ 4 π D L 2 e i ϵ L ( ρ ) | ϕ 1 ( ρ ) | | 2 ,