Abstract

A multichannel free-space optical (FSO) communication system based on orbital angular momentum (OAM)-carrying beams is studied. We numerically analyze the effects of atmospheric turbulence on the system and find that turbulence induces attenuation and crosstalk among channels. Based on a model in which the constituent channels are binary symmetric and crosstalk is a Gaussian noise source, we find optimal sets of OAM states at each turbulence condition studied and determine the aggregate capacity of the multichannel system at those conditions. OAM-multiplexed FSO systems that operate in the weak turbulence regime are found to offer good performance. We verify that the aggregate capacity decreases as the turbulence increases. A per-channel bit-error rate evaluation is presented to show the uneven effects of crosstalk on the constituent channels.

© 2008 Optical Society of America

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    [CrossRef]

2007

J. Wu, H. Li, and Y. Li, “Encoding information as orbital angular momentum states of light for wireless optical communications,” Opt. Eng. 46, 019701 (2007).
[CrossRef]

J. A. Anguita, M. A. Neifeld, and B. V. Vasic, “Spatial correlation and irradiance statistics in a multiple-beam terrestrial free-space optical communication link,” Appl. Opt. 46, 6561-6571 (2007).
[CrossRef] [PubMed]

2006

2005

S. Jafar and A. Goldsmith, “Multiple-antenna capacity in correlated Rayleigh fading with channel covariance information,” IEEE Trans. Wireless Commun. 4, 990-997 (2005).
[CrossRef]

S. S. Muhammad, E. Leitgeb, and O. Koudelkat, “Multilevel modulation and channel codes for terrestrial FSO links,” in Proceedings of the Second International Symposium on Wireless Communication Systems (IEEE, 2005), pp. 795-799.
[CrossRef]

V. P. Aksenov, “Fluctuations of orbital angular momentum of vortex laser-beam in turbulent atmosphere,” Proc. SPIE 5892, 58921Y (2005).
[CrossRef]

C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett. 94, 153901 (2005).
[CrossRef] [PubMed]

2004

2002

M. J. Padgett and L. Allen, “Orbital angular momentum exchange in cylindrical-lens mode converters,” J. Opt. B , 4, S17-S19 (2002).
[CrossRef]

A. Vaziri, G. Weihs, and A. Zeilinger, “Superpositions of the orbital angular momentum for applications in quantum experiments,” J. Opt. B 4, S47-S51 (2002).
[CrossRef]

D. J. Brady, “Multiplex sensors and the constant radiance theorem,” Opt. Lett. 27, 16-18 (2002).
[CrossRef]

2000

1999

L. C. Andrews, R. L. Phillips, C. Y. Hopen, and M. A. Al-Habash, “Theory of optical scintillation,” J. Opt. Soc. Am. A 16, 1417-1429 (1999).
[CrossRef]

J. Courtial and M. J. Padgett, “Performance of a cylindrical lens mode converter for producing Laguerre-Gaussian laser modes,” Opt. Commun. , 159, 13-18 (1999).
[CrossRef]

L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt. 39, 291-372 (1999).
[CrossRef]

1998

Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. , 151, 207-211(1998).
[CrossRef]

1997

D. V. Petrov, F. Canal, and L. Torner, “A simple method to generate optical beams with a screw phase dislocation,” Opt. Commun. , 143, 265-267 (1997).
[CrossRef]

D. Rozas, C. T. Law, and G. A. Swartzlander, Jr., “Propagation dynamics of optical vortices,” J. Opt. Soc. Am. B , 14, 3054-3065 (1997).
[CrossRef]

1995

M. J. Padgett and L. Allen, “The Poynting vector in Laguerre-Gaussian laser modes,” Opt. Commun. 121, 36-40 (1995).
[CrossRef]

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826-829 (1995).
[CrossRef] [PubMed]

1994

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phase plate,” Opt. Commun. , 112, 321-327(1994).
[CrossRef]

1993

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. , 96, 123-132 (1993).
[CrossRef]

1992

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre Gaussian laser modes,” Phys. Rev. A 45, 8185-8189 (1992).
[CrossRef] [PubMed]

V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985-990(1992).
[CrossRef]

L. C. Andrews, “An analytical model for the refractive index power spectrum and its application to optical scintillations in the atmosphere,” J. Mod. Opt. 39, 1849-1853(1992).
[CrossRef]

N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17, 221-223 (1992).
[CrossRef] [PubMed]

Aksenov, V. P.

V. P. Aksenov, “Fluctuations of orbital angular momentum of vortex laser-beam in turbulent atmosphere,” Proc. SPIE 5892, 58921Y (2005).
[CrossRef]

Al-Habash, M. A.

Allen, L.

M. J. Padgett and L. Allen, “Orbital angular momentum exchange in cylindrical-lens mode converters,” J. Opt. B , 4, S17-S19 (2002).
[CrossRef]

L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt. 39, 291-372 (1999).
[CrossRef]

M. J. Padgett and L. Allen, “The Poynting vector in Laguerre-Gaussian laser modes,” Opt. Commun. 121, 36-40 (1995).
[CrossRef]

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. , 96, 123-132 (1993).
[CrossRef]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre Gaussian laser modes,” Phys. Rev. A 45, 8185-8189 (1992).
[CrossRef] [PubMed]

Andrews, L. C.

L. C. Andrews, R. L. Phillips, C. Y. Hopen, and M. A. Al-Habash, “Theory of optical scintillation,” J. Opt. Soc. Am. A 16, 1417-1429 (1999).
[CrossRef]

L. C. Andrews, “An analytical model for the refractive index power spectrum and its application to optical scintillations in the atmosphere,” J. Mod. Opt. 39, 1849-1853(1992).
[CrossRef]

Anguita, J. A.

Babiker, M.

L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt. 39, 291-372 (1999).
[CrossRef]

Bazhenov, V. Y.

V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985-990(1992).
[CrossRef]

Beijersbergen, M. W.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phase plate,” Opt. Commun. , 112, 321-327(1994).
[CrossRef]

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. , 96, 123-132 (1993).
[CrossRef]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre Gaussian laser modes,” Phys. Rev. A 45, 8185-8189 (1992).
[CrossRef] [PubMed]

Belmonte, A.

Bouchal, Z.

Z. Bouchal and R. Celechovsky, “Mixed vortex states of light as information carriers,” New J. Phys. 6, 131-145 (2004).
[CrossRef]

Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. , 151, 207-211(1998).
[CrossRef]

Brady, D. J.

Burge, R. E.

Caetano, D. P.

Canal, F.

D. V. Petrov, F. Canal, and L. Torner, “A simple method to generate optical beams with a screw phase dislocation,” Opt. Commun. , 143, 265-267 (1997).
[CrossRef]

Celechovsky, R.

Z. Bouchal and R. Celechovsky, “Mixed vortex states of light as information carriers,” New J. Phys. 6, 131-145 (2004).
[CrossRef]

Chlup, M.

Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. , 151, 207-211(1998).
[CrossRef]

Coerwinkel, R. P. C.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phase plate,” Opt. Commun. , 112, 321-327(1994).
[CrossRef]

Courtial, J.

J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, “Laser beams: knotted threads of darkness,” Nature 432, 165 (2004),
[CrossRef] [PubMed]

G. Gibson, J. Courtial, and M. J. Padgett, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12, 5448-5456 (2004).
[CrossRef] [PubMed]

J. Courtial and M. J. Padgett, “Performance of a cylindrical lens mode converter for producing Laguerre-Gaussian laser modes,” Opt. Commun. , 159, 13-18 (1999).
[CrossRef]

Dennis, M. R.

J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, “Laser beams: knotted threads of darkness,” Nature 432, 165 (2004),
[CrossRef] [PubMed]

Dholakia, K. C.

Ding, J.

C.-S. Guo, D.-M. Xuea, Y.-J. Hana, and J. Ding, “Optimal phase steps of multi-level spiral phase plates,” Opt. Commun. , 268, 235-239 (2006).
[CrossRef]

Eliel, E. R.

Flatté, S. M.

Friese, M. E. J.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826-829 (1995).
[CrossRef] [PubMed]

Gerber, J. S.

Gibson, G.

Goldsmith, A.

S. Jafar and A. Goldsmith, “Multiple-antenna capacity in correlated Rayleigh fading with channel covariance information,” IEEE Trans. Wireless Commun. 4, 990-997 (2005).
[CrossRef]

Guo, C.-S.

C.-S. Guo, D.-M. Xuea, Y.-J. Hana, and J. Ding, “Optimal phase steps of multi-level spiral phase plates,” Opt. Commun. , 268, 235-239 (2006).
[CrossRef]

Gutiérrez-Vega, J. C.

Hana, Y.-J.

C.-S. Guo, D.-M. Xuea, Y.-J. Hana, and J. Ding, “Optimal phase steps of multi-level spiral phase plates,” Opt. Commun. , 268, 235-239 (2006).
[CrossRef]

He, H.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826-829 (1995).
[CrossRef] [PubMed]

Heckenberg, N. R.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826-829 (1995).
[CrossRef] [PubMed]

N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17, 221-223 (1992).
[CrossRef] [PubMed]

Hershcovitz, O.

Hickmann, J. M.

Hopen, C. Y.

Jafar, S.

S. Jafar and A. Goldsmith, “Multiple-antenna capacity in correlated Rayleigh fading with channel covariance information,” IEEE Trans. Wireless Commun. 4, 990-997 (2005).
[CrossRef]

Khonina, S. N.

Kloosterboer, J. G.

Kotlyar, V. V.

Koudelkat, O.

S. S. Muhammad, E. Leitgeb, and O. Koudelkat, “Multilevel modulation and channel codes for terrestrial FSO links,” in Proceedings of the Second International Symposium on Wireless Communication Systems (IEEE, 2005), pp. 795-799.
[CrossRef]

Kovalev, A. A.

Kristensen, M.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phase plate,” Opt. Commun. , 112, 321-327(1994).
[CrossRef]

Law, C. T.

Leach, J.

J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, “Laser beams: knotted threads of darkness,” Nature 432, 165 (2004),
[CrossRef] [PubMed]

Leitgeb, E.

S. S. Muhammad, E. Leitgeb, and O. Koudelkat, “Multilevel modulation and channel codes for terrestrial FSO links,” in Proceedings of the Second International Symposium on Wireless Communication Systems (IEEE, 2005), pp. 795-799.
[CrossRef]

Li, H.

J. Wu, H. Li, and Y. Li, “Encoding information as orbital angular momentum states of light for wireless optical communications,” Opt. Eng. 46, 019701 (2007).
[CrossRef]

Li, Y.

J. Wu, H. Li, and Y. Li, “Encoding information as orbital angular momentum states of light for wireless optical communications,” Opt. Eng. 46, 019701 (2007).
[CrossRef]

Lin, J.

Lipson, S. G.

López-Mariscal, C.

McDuff, R.

Milne, G. C.

Miyaji, G.

Miyanaga, N.

Moed, S.

Muhammad, S. S.

S. S. Muhammad, E. Leitgeb, and O. Koudelkat, “Multilevel modulation and channel codes for terrestrial FSO links,” in Proceedings of the Second International Symposium on Wireless Communication Systems (IEEE, 2005), pp. 795-799.
[CrossRef]

Nakatsuka, M.

Neifeld, M. A.

Oemrawsingh, S. S. R.

Padgett, M. J.

G. Gibson, J. Courtial, and M. J. Padgett, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12, 5448-5456 (2004).
[CrossRef] [PubMed]

J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, “Laser beams: knotted threads of darkness,” Nature 432, 165 (2004),
[CrossRef] [PubMed]

M. J. Padgett and L. Allen, “Orbital angular momentum exchange in cylindrical-lens mode converters,” J. Opt. B , 4, S17-S19 (2002).
[CrossRef]

L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt. 39, 291-372 (1999).
[CrossRef]

J. Courtial and M. J. Padgett, “Performance of a cylindrical lens mode converter for producing Laguerre-Gaussian laser modes,” Opt. Commun. , 159, 13-18 (1999).
[CrossRef]

M. J. Padgett and L. Allen, “The Poynting vector in Laguerre-Gaussian laser modes,” Opt. Commun. 121, 36-40 (1995).
[CrossRef]

Paterson, C.

C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett. 94, 153901 (2005).
[CrossRef] [PubMed]

Petrov, D. V.

D. V. Petrov, F. Canal, and L. Torner, “A simple method to generate optical beams with a screw phase dislocation,” Opt. Commun. , 143, 265-267 (1997).
[CrossRef]

Phillips, R. L.

Rotschild, C.

Rozas, D.

Rubinsztein-Dunlop, H.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826-829 (1995).
[CrossRef] [PubMed]

Smith, C. P.

Soares, W. C.

Soifer, V. A.

Soskin, M. S.

V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985-990(1992).
[CrossRef]

Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre Gaussian laser modes,” Phys. Rev. A 45, 8185-8189 (1992).
[CrossRef] [PubMed]

Sueda, K.

Swartzlander, G. A.

't Hooft, G. W.

Tao, S. H.

Torner, L.

D. V. Petrov, F. Canal, and L. Torner, “A simple method to generate optical beams with a screw phase dislocation,” Opt. Commun. , 143, 265-267 (1997).
[CrossRef]

van der Veen, H. E. L. O.

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. , 96, 123-132 (1993).
[CrossRef]

van Houwelingen, J. A. W.

Vasic, B. V.

Vasnetsov, M. V.

V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985-990(1992).
[CrossRef]

Vaziri, A.

A. Vaziri, G. Weihs, and A. Zeilinger, “Superpositions of the orbital angular momentum for applications in quantum experiments,” J. Opt. B 4, S47-S51 (2002).
[CrossRef]

Verstegen, E. J. K.

Wagner, J.

Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. , 151, 207-211(1998).
[CrossRef]

Weihs, G.

A. Vaziri, G. Weihs, and A. Zeilinger, “Superpositions of the orbital angular momentum for applications in quantum experiments,” J. Opt. B 4, S47-S51 (2002).
[CrossRef]

White, A. G.

Woerdman, J. P.

S. S. R. Oemrawsingh, J. A. W. van Houwelingen, E. R. Eliel, J. P. Woerdman, E. J. K. Verstegen, J. G. Kloosterboer, and G. W. 't Hooft, “Production and characterization of spiral phase plates for optical wavelengths,” Appl. Opt. , 43, 688-694(2004).
[CrossRef] [PubMed]

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phase plate,” Opt. Commun. , 112, 321-327(1994).
[CrossRef]

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Figures (13)

Fig. 1
Fig. 1

(a) Intensity distribution of a LG beam with OAM state m = 5 and radius w 0 = 3 cm . The frame is 20 cm on a side. (b) Phase distribution of the beam at z = 0 . (c) Phase distribution of the beam at z = 1 km .

Fig. 2
Fig. 2

Diagram of a free-space optical communication link using multiplexed OAM channels.

Fig. 3
Fig. 3

(a) Intensity distribution of a distorted LG beam with initial OAM state m = 5 and radius w 0 = 3 cm . The frame is 20 cm on a side. (b) The distorted phase distribution after z = 1 km still preserves some of its helical structure. In this example C n 2 = 10 15 m 2 / 3 .

Fig. 4
Fig. 4

Transmission efficiency η mm as a function of the OAM state for several values of turbulence strength C n 2 .

Fig. 5
Fig. 5

Average crosstalk η mn observed on OAM channels n S induced by transmit channel m { + 1 , + 5 , + 10 , + 15 } : (a)  C n 2 = 10 16 m 2 / 3 , (b)  C n 2 = 10 15 m 2 / 3 , (c)  C n 2 = 10 14 m 2 / 3 . The insets show a comparison between our simulations and the results of [15].

Fig. 6
Fig. 6

Average crosstalk η mn observed on OAM channels n S induced by transmit channel m { + 1 , + 5 , + 10 , + 20 } : (a)  C n 2 = 3 × 10 14 m 2 / 3 and (b)  C n 2 = 10 13 m 2 / 3 .

Fig. 7
Fig. 7

(a) One-dimensional intensity profile ( ϕ = 0 ) of some LG modes. (b) Overlap integral of intensity distribution of OAM state + 5 and those of other OAM states. Note the similarity with the crosstalk for m = + 5 in Fig. 6b.

Fig. 8
Fig. 8

Average OAM crosstalk induced by the simultaneous effects of turbulence and misalignment for m = + 5 and C n 2 = 10 15 m 2 / 3 .

Fig. 9
Fig. 9

Optimal OAM channel sets versus P Tx / N 0 for (a)  M = 7 and C n 2 = 10 14 m 2 / 3 , (b)  M = 9 and C n 2 = 10 14 m 2 / 3 , (c)  M = 7 and C n 2 = 3 × 10 14 m 2 / 3 .

Fig. 10
Fig. 10

Aggregate capacity (in bits/M-channels) versus P Tx / N 0 for (a)  C n 2 = 10 15 m 2 / 3 , (b)  C n 2 = 10 14 m 2 / 3 , (c)  C n 2 = 3 × 10 14 m 2 / 3 , (d)  C n 2 = 10 13 m 2 / 3 .

Fig. 11
Fig. 11

BER versus P Tx / N 0 in decibels with M = 7 at (a)  C n 2 = 10 15 m 2 / 3 and (b)  C n 2 = 3 × 10 14 m 2 / 3 .

Fig. 12
Fig. 12

Aggregate capacity (in bits/M-channels) versus P Tx / N 0 (transmit power fixed) for (a)  C n 2 = 10 15 m 2 / 3 , (b)  C n 2 = 10 14 m 2 / 3 , (c)  C n 2 = 3 × 10 14 m 2 / 3 , (d)  C n 2 = 10 13 m 2 / 3 .

Fig. 13
Fig. 13

Aggregate capacity (in bits/M-channels use) versus number of OAM channels M for (a)  C n 2 = 10 15 m 2 / 3 , (b)  C n 2 = 10 14 m 2 / 3 , (c)  C n 2 = 3 × 10 14 m 2 / 3 , (d)  C n 2 = 10 13 m 2 / 3 .

Tables (6)

Tables Icon

Table 2 Coefficients { η mn } for m = [ 0 , 20 ] and n = [ 9 , 20 ] , where m (Along Rows) is the Transmit OAM State and n (along columns) is the Receive OAM State

Tables Icon

Table 1 Crosstalk Induced by Beam Misalignment for m = + 5

Tables Icon

Table 2 Optimal Channel Sets for M = 9 and P Tx / N 0 = 20 dB

Tables Icon

Table 3 Optimal Channel Sets for M = 9 and P Tx / N 0 = 30 dB

Tables Icon

Table 4 Optimal Channel Sets for M = 9 and P Tx / N 0 = 40 dB

Tables Icon

Table 5 Optimum Channel Sets for M = 8 at Various Power Levels

Equations (7)

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

u ( r , ϕ , z ) = 2 p ! π ( p + | m | ) ! 1 w ( z ) [ r 2 w ( z ) ] | m | L p m [ 2 r 2 w 2 ( z ) ] × exp [ r 2 w 2 ( z ) ] exp [ i k r 2 z 2 ( z 2 + z R 2 ) ] × exp [ i ( 2 p + | m | + 1 ) tan 1 z z R ] exp ( i m ϕ ) ,
u n ( r , ϕ , z ) · u m * ( r , ϕ , z ) r d r d ϕ u n ( r , ϕ , z ) u m * ( r , ϕ , z ) = { 0     n m r d r d ϕ | u m ( r , ϕ , z ) | 2 n = m .
Φ ( κ ) = 0.033 C n 2 ( κ 2 + 1 / L 0 2 ) 11 / 6 f ( κ , κ l )       with         f ( κ , κ l ) = exp ( κ 2 / κ l 2 ) [ 1 + 1.802 ( κ / κ l ) 0.254 ( κ / κ l ) 7 / 6 ] ,
γ η mm 2 n O n m η nm 2 + N 0 / P Tx .
O ^ = arg max O S m O C ( p m ) ,
C ( p m ) = 1 + p m log 2 p m + ( 1 p m ) log 2 ( 1 p m )
p m = 1 2 erfc ( γ / 2 ) ,

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