Abstract

Evolution of next generation wireless networks brings challenges to efficiently transmit a large amount of data from a base station to a remote antenna unit. We investigate a space division multiplexing technique that employs few mode fibers (FMFs) to transmit 3 × 3 MIMO wireless signals, aiming to employ a common digital signal processing (DSP) unit to equalize both the fiber and wireless channel. We optimize system parameters and obtain above 28 dB and 23 dB signal-to-interference and noise ratio (SINR) for 3 meters wireless systems with 500 m and 2 km FMF, which correspond to the transmission capacity of 578 Mb/s and 468 Mb/s using a 20 MHz bandwidth, respectively. Moreover, we analyze that the nonlinear spectrum distortion due to the combined effect of nonlinearity in the directly modulated laser and the differential mode delay in multimode fibers and validate it by simulations.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. J. G. Andrews, S. Buzzi, W. Choi, S. V. Hanly, A. Lozano, A. C. K. Soong, and J. C. Zhang, “What will 5G be?” IEEE J. Sel. Areas Commun. 32(6), 1065–1082 (2014).
    [Crossref]
  2. China Mobile Research Institute, “C-RAN: the road toward green RAN,” White Paper v3.0, Jun. (2014).
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    [Crossref]
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    [Crossref]
  5. J. Kani, J. Terada, K. I. Suzuki, and A. Otaka, “Solutions for future mobile fronthaul and access-network convergence,” J. of Lightw. Tech. 35(3), 527–534 (2017).
  6. P. Chanclou, L. A. Neto, K. Grzybowski, Z. Tayq, F. Saliou, and N. Genay, “Mobile fronthaul architecture and technologies: a RAN equipment assessment,” J. Opt. Commun. Netw. 10(1), A1–A7 (2018).
    [Crossref]
  7. N. J. Gomes, P. Sehier, H. Thomas, P. Chanclou, B. Li, D. Munch, P. Assimakopoulos, S. Dixit, and V. Jungnickel, “Boosting 5G through Ethernet,” IEEE Veh. Technol. Mag. 13, 74–84 (2018).
    [Crossref]
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    [Crossref]
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  12. Z. Wu, J. Li, Y. Tian, D. Ge, J. Zhu, Q. Mo, F. Ren, J. Yu, Z. Li, Z. Chen, and Y. He, “4-mode MDM transmission over MMF with direct detection enabled by cascaded mode-selective couplers,” in Optical Fiber Communication conference (OFC), Th2A.40, (2017).
  13. G. S. D. Gordon, M. J. Crisp, R. V. Penty, T. D. Wilkinson, and I. H. White, “Feasibility demonstration of a mode-division multiplexed MIMO-enabled radio-over-fiber distributed antenna system,” J. Lightwave Technol. 32(20), 3521–3528 (2014).
    [Crossref]
  14. Y. Lei, J. Li, Y. Fan, D. Yu, S. Fu, F. Yin, Y. Dai, and K. Xu, “Space-division-multiplexed transmission of 3x3 multiple-input multiple-output wireless signals over conventional graded-index multimode fiber,” Opt. Express 24(25), 28372–28382 (2016).
    [Crossref] [PubMed]
  15. Q. Mo, J. He, D. Yu, L. Deng, S. Fu, M. Tang, and D. Liu, “2 × 2 MIMO OFDM/OQAM radio signals over an elliptical core few-mode fiber,” Opt. Lett. 41(19), 4546–4549 (2016).
    [Crossref] [PubMed]
  16. T. M. Schmidl and D. C. Cox, “Robust frequency and timing synchronization for OFDM,” IEEE Trans. Commun. 45(12), 1613–1621 (1997).
    [Crossref]
  17. S. Mumtaz, R. J. Essiambre, and G. P. Agrawal, “Nonlinear propagation in multimode and multicore fibers: generalization of the Manakov equations,” J. Lightwave Tech. 31(3), 398–406 (2013).
    [Crossref]
  18. M. B. Shemirani, W. Mao, R. A. Panicker, and J. M. Kahn, “Principal modes in graded-index multimode fiber in presence of spatial- and polarization- mode coupling,” J. Lightwave Technol. 27(10) 1248–1261 (2009).
  19. S. Ö. Arik, J. M. Kahn, and K. P. Ho, “MIMO signal processing for mode-division multiplexing,” IEEE Signal Process. Mag. 25, 34 (2014).

2018 (3)

I. A. Alimi, A. L. Teixeira, and P. P. Monteiro, “Towards an efficient C-RAN optical fronthaul for the future networks: a tutorial on technologies, requirements, challenges, and solutions,” IEEE Commun. Surv. Tutor. 20(1), 708–769 (2018).
[Crossref]

P. Chanclou, L. A. Neto, K. Grzybowski, Z. Tayq, F. Saliou, and N. Genay, “Mobile fronthaul architecture and technologies: a RAN equipment assessment,” J. Opt. Commun. Netw. 10(1), A1–A7 (2018).
[Crossref]

N. J. Gomes, P. Sehier, H. Thomas, P. Chanclou, B. Li, D. Munch, P. Assimakopoulos, S. Dixit, and V. Jungnickel, “Boosting 5G through Ethernet,” IEEE Veh. Technol. Mag. 13, 74–84 (2018).
[Crossref]

2017 (2)

K. N. R. S. V. Prasad, E. Hossain, and V. K. Bhargava, “Energy efficiency in massive MIMO-based 5G networks: opportunities and challenges,” IEEE Wirel. Commun. 24(3), 86–94 (2017).
[Crossref]

J. Kani, J. Terada, K. I. Suzuki, and A. Otaka, “Solutions for future mobile fronthaul and access-network convergence,” J. of Lightw. Tech. 35(3), 527–534 (2017).

2016 (2)

2015 (2)

A. Pizzinat, P. Chanclou, F. Saliou, and T. Diallo, “Things you should know about fronthaul,” J. Lightwave Tech. 33(5), 1077–1083 (2015).
[Crossref]

H. Bogucka, P. Kryszkiewicz, and A. Kliks, “Dynamic spectrum aggregation for future 5G communications,” IEEE Commun. Mag. 53(5), 35–43 (2015).
[Crossref]

2014 (3)

G. S. D. Gordon, M. J. Crisp, R. V. Penty, T. D. Wilkinson, and I. H. White, “Feasibility demonstration of a mode-division multiplexed MIMO-enabled radio-over-fiber distributed antenna system,” J. Lightwave Technol. 32(20), 3521–3528 (2014).
[Crossref]

J. G. Andrews, S. Buzzi, W. Choi, S. V. Hanly, A. Lozano, A. C. K. Soong, and J. C. Zhang, “What will 5G be?” IEEE J. Sel. Areas Commun. 32(6), 1065–1082 (2014).
[Crossref]

S. Ö. Arik, J. M. Kahn, and K. P. Ho, “MIMO signal processing for mode-division multiplexing,” IEEE Signal Process. Mag. 25, 34 (2014).

2013 (1)

S. Mumtaz, R. J. Essiambre, and G. P. Agrawal, “Nonlinear propagation in multimode and multicore fibers: generalization of the Manakov equations,” J. Lightwave Tech. 31(3), 398–406 (2013).
[Crossref]

2009 (1)

1997 (1)

T. M. Schmidl and D. C. Cox, “Robust frequency and timing synchronization for OFDM,” IEEE Trans. Commun. 45(12), 1613–1621 (1997).
[Crossref]

Agrawal, G. P.

S. Mumtaz, R. J. Essiambre, and G. P. Agrawal, “Nonlinear propagation in multimode and multicore fibers: generalization of the Manakov equations,” J. Lightwave Tech. 31(3), 398–406 (2013).
[Crossref]

Alimi, I. A.

I. A. Alimi, A. L. Teixeira, and P. P. Monteiro, “Towards an efficient C-RAN optical fronthaul for the future networks: a tutorial on technologies, requirements, challenges, and solutions,” IEEE Commun. Surv. Tutor. 20(1), 708–769 (2018).
[Crossref]

Andrews, J. G.

J. G. Andrews, S. Buzzi, W. Choi, S. V. Hanly, A. Lozano, A. C. K. Soong, and J. C. Zhang, “What will 5G be?” IEEE J. Sel. Areas Commun. 32(6), 1065–1082 (2014).
[Crossref]

Arik, S. Ö.

S. Ö. Arik, J. M. Kahn, and K. P. Ho, “MIMO signal processing for mode-division multiplexing,” IEEE Signal Process. Mag. 25, 34 (2014).

Assimakopoulos, P.

N. J. Gomes, P. Sehier, H. Thomas, P. Chanclou, B. Li, D. Munch, P. Assimakopoulos, S. Dixit, and V. Jungnickel, “Boosting 5G through Ethernet,” IEEE Veh. Technol. Mag. 13, 74–84 (2018).
[Crossref]

Bhargava, V. K.

K. N. R. S. V. Prasad, E. Hossain, and V. K. Bhargava, “Energy efficiency in massive MIMO-based 5G networks: opportunities and challenges,” IEEE Wirel. Commun. 24(3), 86–94 (2017).
[Crossref]

Bogucka, H.

H. Bogucka, P. Kryszkiewicz, and A. Kliks, “Dynamic spectrum aggregation for future 5G communications,” IEEE Commun. Mag. 53(5), 35–43 (2015).
[Crossref]

Buzzi, S.

J. G. Andrews, S. Buzzi, W. Choi, S. V. Hanly, A. Lozano, A. C. K. Soong, and J. C. Zhang, “What will 5G be?” IEEE J. Sel. Areas Commun. 32(6), 1065–1082 (2014).
[Crossref]

Chanclou, P.

N. J. Gomes, P. Sehier, H. Thomas, P. Chanclou, B. Li, D. Munch, P. Assimakopoulos, S. Dixit, and V. Jungnickel, “Boosting 5G through Ethernet,” IEEE Veh. Technol. Mag. 13, 74–84 (2018).
[Crossref]

P. Chanclou, L. A. Neto, K. Grzybowski, Z. Tayq, F. Saliou, and N. Genay, “Mobile fronthaul architecture and technologies: a RAN equipment assessment,” J. Opt. Commun. Netw. 10(1), A1–A7 (2018).
[Crossref]

A. Pizzinat, P. Chanclou, F. Saliou, and T. Diallo, “Things you should know about fronthaul,” J. Lightwave Tech. 33(5), 1077–1083 (2015).
[Crossref]

Choi, W.

J. G. Andrews, S. Buzzi, W. Choi, S. V. Hanly, A. Lozano, A. C. K. Soong, and J. C. Zhang, “What will 5G be?” IEEE J. Sel. Areas Commun. 32(6), 1065–1082 (2014).
[Crossref]

Cox, D. C.

T. M. Schmidl and D. C. Cox, “Robust frequency and timing synchronization for OFDM,” IEEE Trans. Commun. 45(12), 1613–1621 (1997).
[Crossref]

Crisp, M. J.

Dai, Y.

Deng, L.

Diallo, T.

A. Pizzinat, P. Chanclou, F. Saliou, and T. Diallo, “Things you should know about fronthaul,” J. Lightwave Tech. 33(5), 1077–1083 (2015).
[Crossref]

Dixit, S.

N. J. Gomes, P. Sehier, H. Thomas, P. Chanclou, B. Li, D. Munch, P. Assimakopoulos, S. Dixit, and V. Jungnickel, “Boosting 5G through Ethernet,” IEEE Veh. Technol. Mag. 13, 74–84 (2018).
[Crossref]

Essiambre, R. J.

S. Mumtaz, R. J. Essiambre, and G. P. Agrawal, “Nonlinear propagation in multimode and multicore fibers: generalization of the Manakov equations,” J. Lightwave Tech. 31(3), 398–406 (2013).
[Crossref]

Fan, Y.

Fu, S.

Genay, N.

Gomes, N. J.

N. J. Gomes, P. Sehier, H. Thomas, P. Chanclou, B. Li, D. Munch, P. Assimakopoulos, S. Dixit, and V. Jungnickel, “Boosting 5G through Ethernet,” IEEE Veh. Technol. Mag. 13, 74–84 (2018).
[Crossref]

Gordon, G. S. D.

Grzybowski, K.

Hanly, S. V.

J. G. Andrews, S. Buzzi, W. Choi, S. V. Hanly, A. Lozano, A. C. K. Soong, and J. C. Zhang, “What will 5G be?” IEEE J. Sel. Areas Commun. 32(6), 1065–1082 (2014).
[Crossref]

He, J.

Ho, K. P.

S. Ö. Arik, J. M. Kahn, and K. P. Ho, “MIMO signal processing for mode-division multiplexing,” IEEE Signal Process. Mag. 25, 34 (2014).

Hossain, E.

K. N. R. S. V. Prasad, E. Hossain, and V. K. Bhargava, “Energy efficiency in massive MIMO-based 5G networks: opportunities and challenges,” IEEE Wirel. Commun. 24(3), 86–94 (2017).
[Crossref]

Jungnickel, V.

N. J. Gomes, P. Sehier, H. Thomas, P. Chanclou, B. Li, D. Munch, P. Assimakopoulos, S. Dixit, and V. Jungnickel, “Boosting 5G through Ethernet,” IEEE Veh. Technol. Mag. 13, 74–84 (2018).
[Crossref]

Kahn, J. M.

Kani, J.

J. Kani, J. Terada, K. I. Suzuki, and A. Otaka, “Solutions for future mobile fronthaul and access-network convergence,” J. of Lightw. Tech. 35(3), 527–534 (2017).

Kliks, A.

H. Bogucka, P. Kryszkiewicz, and A. Kliks, “Dynamic spectrum aggregation for future 5G communications,” IEEE Commun. Mag. 53(5), 35–43 (2015).
[Crossref]

Kryszkiewicz, P.

H. Bogucka, P. Kryszkiewicz, and A. Kliks, “Dynamic spectrum aggregation for future 5G communications,” IEEE Commun. Mag. 53(5), 35–43 (2015).
[Crossref]

Lei, Y.

Li, B.

N. J. Gomes, P. Sehier, H. Thomas, P. Chanclou, B. Li, D. Munch, P. Assimakopoulos, S. Dixit, and V. Jungnickel, “Boosting 5G through Ethernet,” IEEE Veh. Technol. Mag. 13, 74–84 (2018).
[Crossref]

Li, J.

Liu, D.

Lozano, A.

J. G. Andrews, S. Buzzi, W. Choi, S. V. Hanly, A. Lozano, A. C. K. Soong, and J. C. Zhang, “What will 5G be?” IEEE J. Sel. Areas Commun. 32(6), 1065–1082 (2014).
[Crossref]

Mao, W.

Mo, Q.

Monteiro, P. P.

I. A. Alimi, A. L. Teixeira, and P. P. Monteiro, “Towards an efficient C-RAN optical fronthaul for the future networks: a tutorial on technologies, requirements, challenges, and solutions,” IEEE Commun. Surv. Tutor. 20(1), 708–769 (2018).
[Crossref]

Mumtaz, S.

S. Mumtaz, R. J. Essiambre, and G. P. Agrawal, “Nonlinear propagation in multimode and multicore fibers: generalization of the Manakov equations,” J. Lightwave Tech. 31(3), 398–406 (2013).
[Crossref]

Munch, D.

N. J. Gomes, P. Sehier, H. Thomas, P. Chanclou, B. Li, D. Munch, P. Assimakopoulos, S. Dixit, and V. Jungnickel, “Boosting 5G through Ethernet,” IEEE Veh. Technol. Mag. 13, 74–84 (2018).
[Crossref]

Neto, L. A.

Otaka, A.

J. Kani, J. Terada, K. I. Suzuki, and A. Otaka, “Solutions for future mobile fronthaul and access-network convergence,” J. of Lightw. Tech. 35(3), 527–534 (2017).

Panicker, R. A.

Penty, R. V.

Pizzinat, A.

A. Pizzinat, P. Chanclou, F. Saliou, and T. Diallo, “Things you should know about fronthaul,” J. Lightwave Tech. 33(5), 1077–1083 (2015).
[Crossref]

Prasad, K. N. R. S. V.

K. N. R. S. V. Prasad, E. Hossain, and V. K. Bhargava, “Energy efficiency in massive MIMO-based 5G networks: opportunities and challenges,” IEEE Wirel. Commun. 24(3), 86–94 (2017).
[Crossref]

Saliou, F.

Schmidl, T. M.

T. M. Schmidl and D. C. Cox, “Robust frequency and timing synchronization for OFDM,” IEEE Trans. Commun. 45(12), 1613–1621 (1997).
[Crossref]

Sehier, P.

N. J. Gomes, P. Sehier, H. Thomas, P. Chanclou, B. Li, D. Munch, P. Assimakopoulos, S. Dixit, and V. Jungnickel, “Boosting 5G through Ethernet,” IEEE Veh. Technol. Mag. 13, 74–84 (2018).
[Crossref]

Shemirani, M. B.

Soong, A. C. K.

J. G. Andrews, S. Buzzi, W. Choi, S. V. Hanly, A. Lozano, A. C. K. Soong, and J. C. Zhang, “What will 5G be?” IEEE J. Sel. Areas Commun. 32(6), 1065–1082 (2014).
[Crossref]

Suzuki, K. I.

J. Kani, J. Terada, K. I. Suzuki, and A. Otaka, “Solutions for future mobile fronthaul and access-network convergence,” J. of Lightw. Tech. 35(3), 527–534 (2017).

Tang, M.

Tayq, Z.

Teixeira, A. L.

I. A. Alimi, A. L. Teixeira, and P. P. Monteiro, “Towards an efficient C-RAN optical fronthaul for the future networks: a tutorial on technologies, requirements, challenges, and solutions,” IEEE Commun. Surv. Tutor. 20(1), 708–769 (2018).
[Crossref]

Terada, J.

J. Kani, J. Terada, K. I. Suzuki, and A. Otaka, “Solutions for future mobile fronthaul and access-network convergence,” J. of Lightw. Tech. 35(3), 527–534 (2017).

Thomas, H.

N. J. Gomes, P. Sehier, H. Thomas, P. Chanclou, B. Li, D. Munch, P. Assimakopoulos, S. Dixit, and V. Jungnickel, “Boosting 5G through Ethernet,” IEEE Veh. Technol. Mag. 13, 74–84 (2018).
[Crossref]

White, I. H.

Wilkinson, T. D.

Xu, K.

Yin, F.

Yu, D.

Zhang, J. C.

J. G. Andrews, S. Buzzi, W. Choi, S. V. Hanly, A. Lozano, A. C. K. Soong, and J. C. Zhang, “What will 5G be?” IEEE J. Sel. Areas Commun. 32(6), 1065–1082 (2014).
[Crossref]

IEEE Commun. Mag. (1)

H. Bogucka, P. Kryszkiewicz, and A. Kliks, “Dynamic spectrum aggregation for future 5G communications,” IEEE Commun. Mag. 53(5), 35–43 (2015).
[Crossref]

IEEE Commun. Surv. Tutor. (1)

I. A. Alimi, A. L. Teixeira, and P. P. Monteiro, “Towards an efficient C-RAN optical fronthaul for the future networks: a tutorial on technologies, requirements, challenges, and solutions,” IEEE Commun. Surv. Tutor. 20(1), 708–769 (2018).
[Crossref]

IEEE J. Sel. Areas Commun. (1)

J. G. Andrews, S. Buzzi, W. Choi, S. V. Hanly, A. Lozano, A. C. K. Soong, and J. C. Zhang, “What will 5G be?” IEEE J. Sel. Areas Commun. 32(6), 1065–1082 (2014).
[Crossref]

IEEE Signal Process. Mag. (1)

S. Ö. Arik, J. M. Kahn, and K. P. Ho, “MIMO signal processing for mode-division multiplexing,” IEEE Signal Process. Mag. 25, 34 (2014).

IEEE Trans. Commun. (1)

T. M. Schmidl and D. C. Cox, “Robust frequency and timing synchronization for OFDM,” IEEE Trans. Commun. 45(12), 1613–1621 (1997).
[Crossref]

IEEE Veh. Technol. Mag. (1)

N. J. Gomes, P. Sehier, H. Thomas, P. Chanclou, B. Li, D. Munch, P. Assimakopoulos, S. Dixit, and V. Jungnickel, “Boosting 5G through Ethernet,” IEEE Veh. Technol. Mag. 13, 74–84 (2018).
[Crossref]

IEEE Wirel. Commun. (1)

K. N. R. S. V. Prasad, E. Hossain, and V. K. Bhargava, “Energy efficiency in massive MIMO-based 5G networks: opportunities and challenges,” IEEE Wirel. Commun. 24(3), 86–94 (2017).
[Crossref]

J. Lightwave Tech. (2)

S. Mumtaz, R. J. Essiambre, and G. P. Agrawal, “Nonlinear propagation in multimode and multicore fibers: generalization of the Manakov equations,” J. Lightwave Tech. 31(3), 398–406 (2013).
[Crossref]

A. Pizzinat, P. Chanclou, F. Saliou, and T. Diallo, “Things you should know about fronthaul,” J. Lightwave Tech. 33(5), 1077–1083 (2015).
[Crossref]

J. Lightwave Technol. (2)

J. of Lightw. Tech. (1)

J. Kani, J. Terada, K. I. Suzuki, and A. Otaka, “Solutions for future mobile fronthaul and access-network convergence,” J. of Lightw. Tech. 35(3), 527–534 (2017).

J. Opt. Commun. Netw. (1)

Opt. Express (1)

Opt. Lett. (1)

Other (4)

H. Liu, H. Wen, J. C. A. Zacarias, J. E. Antonio-Lopez, N. Wang, P. Sillard, A. A. Correa, R. Amezcua-Correa, and G. Li “3×10 Gb/s mode group-multiplexed transmission over a 20 km few-mode fiber using photonic lanterns,” in Optical Fiber Communication conference (OFC), M2D.5, (2017).

Z. Wu, J. Li, Y. Tian, D. Ge, J. Zhu, Q. Mo, F. Ren, J. Yu, Z. Li, Z. Chen, and Y. He, “4-mode MDM transmission over MMF with direct detection enabled by cascaded mode-selective couplers,” in Optical Fiber Communication conference (OFC), Th2A.40, (2017).

Common Public Radio Interface (CPRI), CPRI specification V7.0, Oct. 2015.

China Mobile Research Institute, “C-RAN: the road toward green RAN,” White Paper v3.0, Jun. (2014).

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Figures (9)

Fig. 1
Fig. 1 Experimental setup (AWG: arbitrary waveform generator, BPF: band pass filter, Att: attenuator, Tx: transmitter, PC: polarization controller, PL-Mux: photonic lantern multiplexer, PL-Demux: photonic lantern demultiplexer, FMF: quasi-single mode fiber, Rx: receiver, PA: power amplifier, LNA: low noise amplifier, DSP: digital signal processing).
Fig. 2
Fig. 2 Optical intensity distributions at the output port of PLs and FMF (Beam sizes are given in Table 1).
Fig. 3
Fig. 3 Signal spectrums for systems with different lengths of FMF (a ~f) and single mode fiber (g and h) (OFDM (a, b, c, and g) or two tone (d, e, f, and h) signals are used as inputs).
Fig. 4
Fig. 4 Impact of RF attenuation on the signal spectrum. ((a~c) use OFDM signal as input, (d~f) use two tone signal as input, FMF length = 2 km).
Fig. 5
Fig. 5 SIR and SINR vs. input power to the optical transmitters (FMF length = 2 km, no air link).
Fig. 6
Fig. 6 Signal spectrum obtained by simulations.
Fig. 7
Fig. 7 Simulation and experimental results of SIR vs. FMF length.
Fig. 8
Fig. 8 3 × 3 MIMO transmission performance of different system configurations.
Fig. 9
Fig. 9 (a) SINR vs. FMF length, (b) optimal RF input signal power and capacity vs. FMF length (air link = 3 m).

Tables (1)

Tables Icon

Table 1 Beam size

Equations (12)

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

P s (t)= P 0 [ 1+m×u(t) ],
s(t)=A 1+m×u(t) ,
r(t)=R | s(t) | 2 =K[ 1+m×u(t) ],
U out = P ^ tot U in ,
P ^ tot = P ^ K P ^ K-1 P ^ 2 P ^ 1 ,
P ^ k = P ^ pr P ^ ln ,
P ^ pr =[ cosθ I MxM sinθ e jφ I MxM -sinθ e -jφ I MxM cosθ I MxM ],
P ^ ln =[ e ( - Γ X +jC )Δz 0 0 e ( - Γ Y +jC )Δz ],
Γ Xm =αΔz/2+j β 1,Xm ωΔz+j β 2,Xm Δz ω 2 /2+jωΔ τ Xm ,
Γ Ym =αΔz/2+j β 1,Ym ωΔz+j β 2,Ym Δz ω 2 /2+jωΔ τ Ym ,
C mn ={ 0,m=n k 0 2 2 β 0 I m I n ( n r 2 n r0 2 ) F m F n dxdy,mn .
r perturb (ϕ)= r 0 + a 1 cos( ϕ ϕ 1 )+ a 2 cos[ 2( ϕ ϕ 2 ) ].

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