Abstract

We investigated the performance of mode diversity reception of a polarization-division-multiplexed (PDM) signal with few-mode-fiber (FMF) coupling for high-speed free-space optical communications under atmospheric turbulence. Optical propagation through eigenmodes of a FMF yields coupling between different linearly polarized (LP) modes in orthogonal polarizations, which causes power imbalance and loss of the orthogonality of multiplexed signals within each individual LP mode. Due to this phenomenon, the architecture of mode diversity combining affects the receiver performance. We numerically simulated the power fluctuation coupled to each LP mode after atmospheric propagation and FMF propagation in the condition of an optical downlink from a low-Earth-orbital satellite to the ground. We found that full receiver-side multiple-input multiple-output (Rx-MIMO) architecture in three-mode diversity reception improved the performance by 5 dB compared with selection combining (SC) of signals decoded individually in LP modes, and that it mitigated the required transmitted power by 6 dB compared with reception with single mode fiber (SMF) coupling. We also experimentally confirmed in three-mode diversity reception of a 128 Gb/s PDM-quadrature phase-shift keying with a diffuser plate as a turbulence emulator, that full Rx-MIMO with adaptive filters could work under severe fading and that it outperformed SC.

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

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References

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2018 (3)

S. Miura, T. Watanabe, and Y. Kokubun, “Accurate analysis of crosstalk between LP11 quasi-degenerate modes due to offset connection using true eigenmodes,” IEEE Photonics J. 10(1), 7900811 (2018).
[Crossref]

D. J. Geisler, T. M. Yarnall, G. Lund, C. M. Schieler, M. L. Stevens, N. K. Fontaine, B. S. Robinson, and S. A. Hamilton, “Experimental comparison of 3-mode and single-mode coupling over a 1.6-km free-space link,” Proc. SPIE 10524, 105240H (2018).

M. Arikawa, Y. Ono, and T. Ito, “Mode diversity coherent receiver with few-mode fiber-coupling for high-speed free-space optical communication under atmospheric turbulence,” Proc. SPIE 10524, 1052412 (2018).

2016 (2)

Y. Kokubun, T. Watanabe, S. Miura, and R. Kawata, “What is a mode in few mode fibers?: Proposal of MIMO-free mode division multiplexing using true eigenmodes,” IEICE Electron. Express 13(18), 1–12 (2016).
[Crossref]

H. Takenaka, Y. Koyama, D. Kolev, M. Akioka, N. Iwakiri, H. Kunimori, A. Carrasco-Casado, Y. Munemasa, E. Okamoto, and M. Toyoshima, “In-orbit verification of small optical transponder (SOTA) –Evaluation of satellite-to-ground laser communication links–,” Proc. SPIE 9739, 973903 (2016).
[Crossref]

2015 (2)

A. Biswas, B. V. Oaida, K. S. Andrews, J. M. Kovalik, M. J. Abrahamson, and W. Wright, “Optical payload for lasercomm science (OPALS) link validation during operation from the ISS,” Proc. SPIE 9354, 93540F (2015).
[Crossref]

E. Ip, G. Milione, M.-J. Li, N. Cvijetic, K. Kanonakis, J. Stone, G. Peng, X. Prieto, C. Montero, V. Moreno, and J. Liñares, “SDM transmission of real-time 10GbE traffic using commercial SFP+ transceivers over 0.5km elliptical-core few-mode fiber,” Opt. Express 23(13), 17120–17126 (2015).
[Crossref] [PubMed]

2014 (3)

M. A. Khalighi and M. Uysal, “Survey on free space optical communication: A communication theory perspective,” IEEE Communication Survey & Tutorials 16(4), 2231–2258 (2014).
[Crossref]

D. M. Boroson, B. S. Robinson, D. V. Murphy, D. A. Burianek, F. Khatri, J. M. Kovalik, Z. Sodnik, and D. M. Cornwell, “Overview and results of the lunar laser communication demonstration,” Proc. SPIE 8971, 89710S (2014).
[Crossref]

P. Sillard, M. Bigot-Astruc, and D. Molin, “Few-mode fibers for mode-division-multiplexed systems,” J. Light. Technol. 32(16), 2824–2829 (2014).
[Crossref]

2013 (1)

R. Zhang, J. Wang, G. Zhao, and J. Lv, “Fiber-based free-space optical coherent receiver with vibration compensation mechanism,” Opt. Express 21(15), 18435–18441 (2013).

2012 (2)

R. Maher, D. S. Millar, S. J. Savory, and B. C. Thomsen, “Widely tunable burst mode digital coherent receiver with fast reconfiguration time for 112 Gb/s DP-QPSK WDM networks,” J. Light. Technol. 30(24), 3924–3930 (2012).
[Crossref]

H. Kogelnik and P. J. Winzer, “Modal birefringence in weakly guiding fibers,” J. Light. Technol. 30(14), 2240–2245 (2012).
[Crossref]

2011 (2)

R. A. Fields, D. A. Kozlowski, H. T. Yura, R. L. Wong, J. M. Wicker, C. T. Lunde, M. Gregory, B. K. Wandermoth, F. F. Heine, and J. J. Luna, “5.625 Gbps bidirectional laser communications measurements between the NFIRE satellite and an optical ground station,” Proc. SPIE 8184, 81840D (2011).
[Crossref]

S. Randel, R. Ryf, A. Sierra, P. J. Winzer, A. H. Gnauck, C. A. Bolle, R.-J. Essiambre, D. W. Peckham, A. McCurdy, and R. Lingle, “6×56-Gb/s mode-division multiplexed transmission over 33-km few-mode fiber enabled by 6×6 MIMO equalization,” Opt. Express 19(17), 16697–16707 (2011).
[Crossref] [PubMed]

2010 (2)

S. G. Leon-Saval, A. Argyros, and J. Bland-Hawthorn., “Photonic lanterns: a study of light propagation in multimode to single-mode converters,” Opt. Express 18(8), 8430–8439 (2010).
[Crossref] [PubMed]

M. Toyoshima, H. Takenaka, Y. Shoji, and Y. Takayama, “Frequency characteristics of atmospheric turbulence in space-to-ground laser links,” Proc. SPIE 7685, 76850G (2010).
[Crossref]

2009 (1)

2008 (2)

S. J. Savory, “Digital filters for coherent optical receivers,” Opt. Express 16(2), 804–817 (2008).
[Crossref] [PubMed]

N. Kim and H. Park, “Performance analysis of MIMO system with linear MMSE receiver,” IEEE Transaction on Wirel. Commun. 7(11), 4474–4478 (2008).
[Crossref]

2007 (1)

T. Jono, Y. Takayama, K. Shiratama, I. Mase, B. Demelenne, Z. Sodnik, A. Bird, M. Toyoshima, H. Kunimori, D. Giggenbach, N. Perlot, M. Knapek, and K. Arai, “Overview of the inter-orbit and orbit-to-ground laser communication demonstration by OICETS,” Proc. SPIE 6457, 645702 (2007).
[Crossref]

2006 (1)

V. W. S. Chan, “Free-space optical communications,” J. Light. Technol. 24(12), 4750–4762 (2006).
[Crossref]

2005 (1)

2002 (1)

M. Reyes, S. Chueca, A. Alonso, T. Viera, and Z. Sodnik, “Analysis of the preliminary optical links between ARTEMIS and the optical ground station,” Proc. SPIE 4821, 33–43 (2002).
[Crossref]

1971 (1)

Abrahamson, M. J.

A. Biswas, B. V. Oaida, K. S. Andrews, J. M. Kovalik, M. J. Abrahamson, and W. Wright, “Optical payload for lasercomm science (OPALS) link validation during operation from the ISS,” Proc. SPIE 9354, 93540F (2015).
[Crossref]

Akioka, M.

H. Takenaka, Y. Koyama, D. Kolev, M. Akioka, N. Iwakiri, H. Kunimori, A. Carrasco-Casado, Y. Munemasa, E. Okamoto, and M. Toyoshima, “In-orbit verification of small optical transponder (SOTA) –Evaluation of satellite-to-ground laser communication links–,” Proc. SPIE 9739, 973903 (2016).
[Crossref]

Alonso, A.

M. Reyes, S. Chueca, A. Alonso, T. Viera, and Z. Sodnik, “Analysis of the preliminary optical links between ARTEMIS and the optical ground station,” Proc. SPIE 4821, 33–43 (2002).
[Crossref]

Andrews, K. S.

A. Biswas, B. V. Oaida, K. S. Andrews, J. M. Kovalik, M. J. Abrahamson, and W. Wright, “Optical payload for lasercomm science (OPALS) link validation during operation from the ISS,” Proc. SPIE 9354, 93540F (2015).
[Crossref]

Andrews, L. C.

L. C. Andrews and R. L. Phillips, Laser beam propagation through random media2nd ed. (SPIE Press, 2005).
[Crossref]

Arai, K.

T. Jono, Y. Takayama, K. Shiratama, I. Mase, B. Demelenne, Z. Sodnik, A. Bird, M. Toyoshima, H. Kunimori, D. Giggenbach, N. Perlot, M. Knapek, and K. Arai, “Overview of the inter-orbit and orbit-to-ground laser communication demonstration by OICETS,” Proc. SPIE 6457, 645702 (2007).
[Crossref]

Argyros, A.

Arikawa, M.

M. Arikawa, Y. Ono, and T. Ito, “Mode diversity coherent receiver with few-mode fiber-coupling for high-speed free-space optical communication under atmospheric turbulence,” Proc. SPIE 10524, 1052412 (2018).

M. Arikawa, T. Ishikawa, K. Hosokawa, S. Takahashi, Y. Ono, and T. Ito, “Demonstration of turbulence-tolerant free-space optical communication receiver using few-mode-fiber coupling and digital combining,” in Proceedings of IEEE Photonics Society Summer Topical Meeting Series (2016), paper TuC3.4.

Barré, N.

J.-F. Morizur, P. Jian, B. Denolle, O. Pinel, N. Barré, and G. Labroille, “Efficient and mode-selective spatial multiplexer based on multi-plane light conversion,” in Optical Fiber Communication Conference (2015), paper W1A.4.
[Crossref]

Bigot-Astruc, M.

P. Sillard, M. Bigot-Astruc, and D. Molin, “Few-mode fibers for mode-division-multiplexed systems,” J. Light. Technol. 32(16), 2824–2829 (2014).
[Crossref]

Bird, A.

T. Jono, Y. Takayama, K. Shiratama, I. Mase, B. Demelenne, Z. Sodnik, A. Bird, M. Toyoshima, H. Kunimori, D. Giggenbach, N. Perlot, M. Knapek, and K. Arai, “Overview of the inter-orbit and orbit-to-ground laser communication demonstration by OICETS,” Proc. SPIE 6457, 645702 (2007).
[Crossref]

Biswas, A.

A. Biswas, B. V. Oaida, K. S. Andrews, J. M. Kovalik, M. J. Abrahamson, and W. Wright, “Optical payload for lasercomm science (OPALS) link validation during operation from the ISS,” Proc. SPIE 9354, 93540F (2015).
[Crossref]

Bland-Hawthorn., J.

Bolle, C. A.

Boroson, D. M.

D. M. Boroson, B. S. Robinson, D. V. Murphy, D. A. Burianek, F. Khatri, J. M. Kovalik, Z. Sodnik, and D. M. Cornwell, “Overview and results of the lunar laser communication demonstration,” Proc. SPIE 8971, 89710S (2014).
[Crossref]

Burianek, D. A.

D. M. Boroson, B. S. Robinson, D. V. Murphy, D. A. Burianek, F. Khatri, J. M. Kovalik, Z. Sodnik, and D. M. Cornwell, “Overview and results of the lunar laser communication demonstration,” Proc. SPIE 8971, 89710S (2014).
[Crossref]

Carrasco-Casado, A.

H. Takenaka, Y. Koyama, D. Kolev, M. Akioka, N. Iwakiri, H. Kunimori, A. Carrasco-Casado, Y. Munemasa, E. Okamoto, and M. Toyoshima, “In-orbit verification of small optical transponder (SOTA) –Evaluation of satellite-to-ground laser communication links–,” Proc. SPIE 9739, 973903 (2016).
[Crossref]

Chan, V. W. S.

V. W. S. Chan, “Free-space optical communications,” J. Light. Technol. 24(12), 4750–4762 (2006).
[Crossref]

Chueca, S.

M. Reyes, S. Chueca, A. Alonso, T. Viera, and Z. Sodnik, “Analysis of the preliminary optical links between ARTEMIS and the optical ground station,” Proc. SPIE 4821, 33–43 (2002).
[Crossref]

Cornwell, D. M.

D. M. Boroson, B. S. Robinson, D. V. Murphy, D. A. Burianek, F. Khatri, J. M. Kovalik, Z. Sodnik, and D. M. Cornwell, “Overview and results of the lunar laser communication demonstration,” Proc. SPIE 8971, 89710S (2014).
[Crossref]

Cvijetic, N.

Davidson, F. M.

Demelenne, B.

T. Jono, Y. Takayama, K. Shiratama, I. Mase, B. Demelenne, Z. Sodnik, A. Bird, M. Toyoshima, H. Kunimori, D. Giggenbach, N. Perlot, M. Knapek, and K. Arai, “Overview of the inter-orbit and orbit-to-ground laser communication demonstration by OICETS,” Proc. SPIE 6457, 645702 (2007).
[Crossref]

Denolle, B.

J.-F. Morizur, P. Jian, B. Denolle, O. Pinel, N. Barré, and G. Labroille, “Efficient and mode-selective spatial multiplexer based on multi-plane light conversion,” in Optical Fiber Communication Conference (2015), paper W1A.4.
[Crossref]

Dikmelik, Y.

Essiambre, R.-J.

Fields, R. A.

R. A. Fields, D. A. Kozlowski, H. T. Yura, R. L. Wong, J. M. Wicker, C. T. Lunde, M. Gregory, B. K. Wandermoth, F. F. Heine, and J. J. Luna, “5.625 Gbps bidirectional laser communications measurements between the NFIRE satellite and an optical ground station,” Proc. SPIE 8184, 81840D (2011).
[Crossref]

Fontaine, N. K.

D. J. Geisler, T. M. Yarnall, G. Lund, C. M. Schieler, M. L. Stevens, N. K. Fontaine, B. S. Robinson, and S. A. Hamilton, “Experimental comparison of 3-mode and single-mode coupling over a 1.6-km free-space link,” Proc. SPIE 10524, 105240H (2018).

Geisler, D. J.

D. J. Geisler, T. M. Yarnall, G. Lund, C. M. Schieler, M. L. Stevens, N. K. Fontaine, B. S. Robinson, and S. A. Hamilton, “Experimental comparison of 3-mode and single-mode coupling over a 1.6-km free-space link,” Proc. SPIE 10524, 105240H (2018).

T. M. Yarnall, D. J. Geisler, C. M. Schieler, and R. Yip, “Analysis of free-space coupling to photonic lanterns in the presence of tilt errors,” in Proceedings of IEEE Photonics Conference (2017), paper ThH1.5.

Giggenbach, D.

T. Jono, Y. Takayama, K. Shiratama, I. Mase, B. Demelenne, Z. Sodnik, A. Bird, M. Toyoshima, H. Kunimori, D. Giggenbach, N. Perlot, M. Knapek, and K. Arai, “Overview of the inter-orbit and orbit-to-ground laser communication demonstration by OICETS,” Proc. SPIE 6457, 645702 (2007).
[Crossref]

Gloge, D.

Gnauck, A. H.

Gregory, M.

R. A. Fields, D. A. Kozlowski, H. T. Yura, R. L. Wong, J. M. Wicker, C. T. Lunde, M. Gregory, B. K. Wandermoth, F. F. Heine, and J. J. Luna, “5.625 Gbps bidirectional laser communications measurements between the NFIRE satellite and an optical ground station,” Proc. SPIE 8184, 81840D (2011).
[Crossref]

Hamilton, S. A.

D. J. Geisler, T. M. Yarnall, G. Lund, C. M. Schieler, M. L. Stevens, N. K. Fontaine, B. S. Robinson, and S. A. Hamilton, “Experimental comparison of 3-mode and single-mode coupling over a 1.6-km free-space link,” Proc. SPIE 10524, 105240H (2018).

Heine, F. F.

R. A. Fields, D. A. Kozlowski, H. T. Yura, R. L. Wong, J. M. Wicker, C. T. Lunde, M. Gregory, B. K. Wandermoth, F. F. Heine, and J. J. Luna, “5.625 Gbps bidirectional laser communications measurements between the NFIRE satellite and an optical ground station,” Proc. SPIE 8184, 81840D (2011).
[Crossref]

Hoshida, T.

L. Liu, Z. Tao, W. Yan, S. Oda, T. Hoshida, and J. C. Rasmussen, “Initial tap setup of constant modulus algorithm for polarization de-multiplexing in optical coherent receivers,” in Optical Fiber Communication Conference (2009), paper OMT.2.

Hosokawa, K.

M. Arikawa, T. Ishikawa, K. Hosokawa, S. Takahashi, Y. Ono, and T. Ito, “Demonstration of turbulence-tolerant free-space optical communication receiver using few-mode-fiber coupling and digital combining,” in Proceedings of IEEE Photonics Society Summer Topical Meeting Series (2016), paper TuC3.4.

Ip, E.

Ishikawa, T.

M. Arikawa, T. Ishikawa, K. Hosokawa, S. Takahashi, Y. Ono, and T. Ito, “Demonstration of turbulence-tolerant free-space optical communication receiver using few-mode-fiber coupling and digital combining,” in Proceedings of IEEE Photonics Society Summer Topical Meeting Series (2016), paper TuC3.4.

Ito, T.

M. Arikawa, Y. Ono, and T. Ito, “Mode diversity coherent receiver with few-mode fiber-coupling for high-speed free-space optical communication under atmospheric turbulence,” Proc. SPIE 10524, 1052412 (2018).

M. Arikawa, T. Ishikawa, K. Hosokawa, S. Takahashi, Y. Ono, and T. Ito, “Demonstration of turbulence-tolerant free-space optical communication receiver using few-mode-fiber coupling and digital combining,” in Proceedings of IEEE Photonics Society Summer Topical Meeting Series (2016), paper TuC3.4.

Iwakiri, N.

H. Takenaka, Y. Koyama, D. Kolev, M. Akioka, N. Iwakiri, H. Kunimori, A. Carrasco-Casado, Y. Munemasa, E. Okamoto, and M. Toyoshima, “In-orbit verification of small optical transponder (SOTA) –Evaluation of satellite-to-ground laser communication links–,” Proc. SPIE 9739, 973903 (2016).
[Crossref]

Jian, P.

J.-F. Morizur, P. Jian, B. Denolle, O. Pinel, N. Barré, and G. Labroille, “Efficient and mode-selective spatial multiplexer based on multi-plane light conversion,” in Optical Fiber Communication Conference (2015), paper W1A.4.
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T. Jono, Y. Takayama, K. Shiratama, I. Mase, B. Demelenne, Z. Sodnik, A. Bird, M. Toyoshima, H. Kunimori, D. Giggenbach, N. Perlot, M. Knapek, and K. Arai, “Overview of the inter-orbit and orbit-to-ground laser communication demonstration by OICETS,” Proc. SPIE 6457, 645702 (2007).
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Kawata, R.

Y. Kokubun, T. Watanabe, S. Miura, and R. Kawata, “What is a mode in few mode fibers?: Proposal of MIMO-free mode division multiplexing using true eigenmodes,” IEICE Electron. Express 13(18), 1–12 (2016).
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M. A. Khalighi and M. Uysal, “Survey on free space optical communication: A communication theory perspective,” IEEE Communication Survey & Tutorials 16(4), 2231–2258 (2014).
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Khatri, F.

D. M. Boroson, B. S. Robinson, D. V. Murphy, D. A. Burianek, F. Khatri, J. M. Kovalik, Z. Sodnik, and D. M. Cornwell, “Overview and results of the lunar laser communication demonstration,” Proc. SPIE 8971, 89710S (2014).
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T. Jono, Y. Takayama, K. Shiratama, I. Mase, B. Demelenne, Z. Sodnik, A. Bird, M. Toyoshima, H. Kunimori, D. Giggenbach, N. Perlot, M. Knapek, and K. Arai, “Overview of the inter-orbit and orbit-to-ground laser communication demonstration by OICETS,” Proc. SPIE 6457, 645702 (2007).
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H. Kogelnik and P. J. Winzer, “Modal birefringence in weakly guiding fibers,” J. Light. Technol. 30(14), 2240–2245 (2012).
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S. Miura, T. Watanabe, and Y. Kokubun, “Accurate analysis of crosstalk between LP11 quasi-degenerate modes due to offset connection using true eigenmodes,” IEEE Photonics J. 10(1), 7900811 (2018).
[Crossref]

Y. Kokubun, T. Watanabe, S. Miura, and R. Kawata, “What is a mode in few mode fibers?: Proposal of MIMO-free mode division multiplexing using true eigenmodes,” IEICE Electron. Express 13(18), 1–12 (2016).
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Kolev, D.

H. Takenaka, Y. Koyama, D. Kolev, M. Akioka, N. Iwakiri, H. Kunimori, A. Carrasco-Casado, Y. Munemasa, E. Okamoto, and M. Toyoshima, “In-orbit verification of small optical transponder (SOTA) –Evaluation of satellite-to-ground laser communication links–,” Proc. SPIE 9739, 973903 (2016).
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A. Biswas, B. V. Oaida, K. S. Andrews, J. M. Kovalik, M. J. Abrahamson, and W. Wright, “Optical payload for lasercomm science (OPALS) link validation during operation from the ISS,” Proc. SPIE 9354, 93540F (2015).
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D. M. Boroson, B. S. Robinson, D. V. Murphy, D. A. Burianek, F. Khatri, J. M. Kovalik, Z. Sodnik, and D. M. Cornwell, “Overview and results of the lunar laser communication demonstration,” Proc. SPIE 8971, 89710S (2014).
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M. Toyoshima, H. Takenaka, Y. Shoji, Y. Takayama, Y. Koyama, and H. Kunimori, “Polarization measurements through space-to-ground atmospheric propagation paths by using a highly polarized laser source in space,” Opt. Express 17(25), 22333–22340 (2009).
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R. A. Fields, D. A. Kozlowski, H. T. Yura, R. L. Wong, J. M. Wicker, C. T. Lunde, M. Gregory, B. K. Wandermoth, F. F. Heine, and J. J. Luna, “5.625 Gbps bidirectional laser communications measurements between the NFIRE satellite and an optical ground station,” Proc. SPIE 8184, 81840D (2011).
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H. Takenaka, Y. Koyama, D. Kolev, M. Akioka, N. Iwakiri, H. Kunimori, A. Carrasco-Casado, Y. Munemasa, E. Okamoto, and M. Toyoshima, “In-orbit verification of small optical transponder (SOTA) –Evaluation of satellite-to-ground laser communication links–,” Proc. SPIE 9739, 973903 (2016).
[Crossref]

M. Toyoshima, H. Takenaka, Y. Shoji, Y. Takayama, Y. Koyama, and H. Kunimori, “Polarization measurements through space-to-ground atmospheric propagation paths by using a highly polarized laser source in space,” Opt. Express 17(25), 22333–22340 (2009).
[Crossref]

T. Jono, Y. Takayama, K. Shiratama, I. Mase, B. Demelenne, Z. Sodnik, A. Bird, M. Toyoshima, H. Kunimori, D. Giggenbach, N. Perlot, M. Knapek, and K. Arai, “Overview of the inter-orbit and orbit-to-ground laser communication demonstration by OICETS,” Proc. SPIE 6457, 645702 (2007).
[Crossref]

Labroille, G.

J.-F. Morizur, P. Jian, B. Denolle, O. Pinel, N. Barré, and G. Labroille, “Efficient and mode-selective spatial multiplexer based on multi-plane light conversion,” in Optical Fiber Communication Conference (2015), paper W1A.4.
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Li, M.-J.

Liñares, J.

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L. Liu, Z. Tao, W. Yan, S. Oda, T. Hoshida, and J. C. Rasmussen, “Initial tap setup of constant modulus algorithm for polarization de-multiplexing in optical coherent receivers,” in Optical Fiber Communication Conference (2009), paper OMT.2.

Luna, J. J.

R. A. Fields, D. A. Kozlowski, H. T. Yura, R. L. Wong, J. M. Wicker, C. T. Lunde, M. Gregory, B. K. Wandermoth, F. F. Heine, and J. J. Luna, “5.625 Gbps bidirectional laser communications measurements between the NFIRE satellite and an optical ground station,” Proc. SPIE 8184, 81840D (2011).
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Lund, G.

D. J. Geisler, T. M. Yarnall, G. Lund, C. M. Schieler, M. L. Stevens, N. K. Fontaine, B. S. Robinson, and S. A. Hamilton, “Experimental comparison of 3-mode and single-mode coupling over a 1.6-km free-space link,” Proc. SPIE 10524, 105240H (2018).

Lunde, C. T.

R. A. Fields, D. A. Kozlowski, H. T. Yura, R. L. Wong, J. M. Wicker, C. T. Lunde, M. Gregory, B. K. Wandermoth, F. F. Heine, and J. J. Luna, “5.625 Gbps bidirectional laser communications measurements between the NFIRE satellite and an optical ground station,” Proc. SPIE 8184, 81840D (2011).
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Lv, J.

R. Zhang, J. Wang, G. Zhao, and J. Lv, “Fiber-based free-space optical coherent receiver with vibration compensation mechanism,” Opt. Express 21(15), 18435–18441 (2013).

Maher, R.

R. Maher, D. S. Millar, S. J. Savory, and B. C. Thomsen, “Widely tunable burst mode digital coherent receiver with fast reconfiguration time for 112 Gb/s DP-QPSK WDM networks,” J. Light. Technol. 30(24), 3924–3930 (2012).
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Mase, I.

T. Jono, Y. Takayama, K. Shiratama, I. Mase, B. Demelenne, Z. Sodnik, A. Bird, M. Toyoshima, H. Kunimori, D. Giggenbach, N. Perlot, M. Knapek, and K. Arai, “Overview of the inter-orbit and orbit-to-ground laser communication demonstration by OICETS,” Proc. SPIE 6457, 645702 (2007).
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Matsumoto, W.

Y. Miyata, K. Sugihara, W. Matsumoto, K. Onohara, T. Sugihara, K. Kubo, H. Yoshida, and T. Mizuoch, “A triple-concatenated FEC using soft-decision decoding for 100 Gb/s optical transmission,” in Optical Fiber Communication Conference (2010), paper OThL.3.
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McCurdy, A.

Milione, G.

Millar, D. S.

R. Maher, D. S. Millar, S. J. Savory, and B. C. Thomsen, “Widely tunable burst mode digital coherent receiver with fast reconfiguration time for 112 Gb/s DP-QPSK WDM networks,” J. Light. Technol. 30(24), 3924–3930 (2012).
[Crossref]

Miura, S.

S. Miura, T. Watanabe, and Y. Kokubun, “Accurate analysis of crosstalk between LP11 quasi-degenerate modes due to offset connection using true eigenmodes,” IEEE Photonics J. 10(1), 7900811 (2018).
[Crossref]

Y. Kokubun, T. Watanabe, S. Miura, and R. Kawata, “What is a mode in few mode fibers?: Proposal of MIMO-free mode division multiplexing using true eigenmodes,” IEICE Electron. Express 13(18), 1–12 (2016).
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Miyata, Y.

Y. Miyata, K. Sugihara, W. Matsumoto, K. Onohara, T. Sugihara, K. Kubo, H. Yoshida, and T. Mizuoch, “A triple-concatenated FEC using soft-decision decoding for 100 Gb/s optical transmission,” in Optical Fiber Communication Conference (2010), paper OThL.3.
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Mizuoch, T.

Y. Miyata, K. Sugihara, W. Matsumoto, K. Onohara, T. Sugihara, K. Kubo, H. Yoshida, and T. Mizuoch, “A triple-concatenated FEC using soft-decision decoding for 100 Gb/s optical transmission,” in Optical Fiber Communication Conference (2010), paper OThL.3.
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Moreno, V.

Morizur, J.-F.

J.-F. Morizur, P. Jian, B. Denolle, O. Pinel, N. Barré, and G. Labroille, “Efficient and mode-selective spatial multiplexer based on multi-plane light conversion,” in Optical Fiber Communication Conference (2015), paper W1A.4.
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H. Takenaka, Y. Koyama, D. Kolev, M. Akioka, N. Iwakiri, H. Kunimori, A. Carrasco-Casado, Y. Munemasa, E. Okamoto, and M. Toyoshima, “In-orbit verification of small optical transponder (SOTA) –Evaluation of satellite-to-ground laser communication links–,” Proc. SPIE 9739, 973903 (2016).
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Murphy, D. V.

D. M. Boroson, B. S. Robinson, D. V. Murphy, D. A. Burianek, F. Khatri, J. M. Kovalik, Z. Sodnik, and D. M. Cornwell, “Overview and results of the lunar laser communication demonstration,” Proc. SPIE 8971, 89710S (2014).
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Oaida, B. V.

A. Biswas, B. V. Oaida, K. S. Andrews, J. M. Kovalik, M. J. Abrahamson, and W. Wright, “Optical payload for lasercomm science (OPALS) link validation during operation from the ISS,” Proc. SPIE 9354, 93540F (2015).
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Oda, S.

L. Liu, Z. Tao, W. Yan, S. Oda, T. Hoshida, and J. C. Rasmussen, “Initial tap setup of constant modulus algorithm for polarization de-multiplexing in optical coherent receivers,” in Optical Fiber Communication Conference (2009), paper OMT.2.

Okamoto, E.

H. Takenaka, Y. Koyama, D. Kolev, M. Akioka, N. Iwakiri, H. Kunimori, A. Carrasco-Casado, Y. Munemasa, E. Okamoto, and M. Toyoshima, “In-orbit verification of small optical transponder (SOTA) –Evaluation of satellite-to-ground laser communication links–,” Proc. SPIE 9739, 973903 (2016).
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M. Arikawa, T. Ishikawa, K. Hosokawa, S. Takahashi, Y. Ono, and T. Ito, “Demonstration of turbulence-tolerant free-space optical communication receiver using few-mode-fiber coupling and digital combining,” in Proceedings of IEEE Photonics Society Summer Topical Meeting Series (2016), paper TuC3.4.

Onohara, K.

Y. Miyata, K. Sugihara, W. Matsumoto, K. Onohara, T. Sugihara, K. Kubo, H. Yoshida, and T. Mizuoch, “A triple-concatenated FEC using soft-decision decoding for 100 Gb/s optical transmission,” in Optical Fiber Communication Conference (2010), paper OThL.3.
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N. Kim and H. Park, “Performance analysis of MIMO system with linear MMSE receiver,” IEEE Transaction on Wirel. Commun. 7(11), 4474–4478 (2008).
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Peng, G.

Perlot, N.

T. Jono, Y. Takayama, K. Shiratama, I. Mase, B. Demelenne, Z. Sodnik, A. Bird, M. Toyoshima, H. Kunimori, D. Giggenbach, N. Perlot, M. Knapek, and K. Arai, “Overview of the inter-orbit and orbit-to-ground laser communication demonstration by OICETS,” Proc. SPIE 6457, 645702 (2007).
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Randel, S.

Rasmussen, J. C.

L. Liu, Z. Tao, W. Yan, S. Oda, T. Hoshida, and J. C. Rasmussen, “Initial tap setup of constant modulus algorithm for polarization de-multiplexing in optical coherent receivers,” in Optical Fiber Communication Conference (2009), paper OMT.2.

Reyes, M.

M. Reyes, S. Chueca, A. Alonso, T. Viera, and Z. Sodnik, “Analysis of the preliminary optical links between ARTEMIS and the optical ground station,” Proc. SPIE 4821, 33–43 (2002).
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Robinson, B. S.

D. J. Geisler, T. M. Yarnall, G. Lund, C. M. Schieler, M. L. Stevens, N. K. Fontaine, B. S. Robinson, and S. A. Hamilton, “Experimental comparison of 3-mode and single-mode coupling over a 1.6-km free-space link,” Proc. SPIE 10524, 105240H (2018).

D. M. Boroson, B. S. Robinson, D. V. Murphy, D. A. Burianek, F. Khatri, J. M. Kovalik, Z. Sodnik, and D. M. Cornwell, “Overview and results of the lunar laser communication demonstration,” Proc. SPIE 8971, 89710S (2014).
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Ryf, R.

Savory, S. J.

R. Maher, D. S. Millar, S. J. Savory, and B. C. Thomsen, “Widely tunable burst mode digital coherent receiver with fast reconfiguration time for 112 Gb/s DP-QPSK WDM networks,” J. Light. Technol. 30(24), 3924–3930 (2012).
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S. J. Savory, “Digital filters for coherent optical receivers,” Opt. Express 16(2), 804–817 (2008).
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D. J. Geisler, T. M. Yarnall, G. Lund, C. M. Schieler, M. L. Stevens, N. K. Fontaine, B. S. Robinson, and S. A. Hamilton, “Experimental comparison of 3-mode and single-mode coupling over a 1.6-km free-space link,” Proc. SPIE 10524, 105240H (2018).

T. M. Yarnall, D. J. Geisler, C. M. Schieler, and R. Yip, “Analysis of free-space coupling to photonic lanterns in the presence of tilt errors,” in Proceedings of IEEE Photonics Conference (2017), paper ThH1.5.

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T. Jono, Y. Takayama, K. Shiratama, I. Mase, B. Demelenne, Z. Sodnik, A. Bird, M. Toyoshima, H. Kunimori, D. Giggenbach, N. Perlot, M. Knapek, and K. Arai, “Overview of the inter-orbit and orbit-to-ground laser communication demonstration by OICETS,” Proc. SPIE 6457, 645702 (2007).
[Crossref]

Shoji, Y.

Sierra, A.

Sillard, P.

P. Sillard, M. Bigot-Astruc, and D. Molin, “Few-mode fibers for mode-division-multiplexed systems,” J. Light. Technol. 32(16), 2824–2829 (2014).
[Crossref]

Sodnik, Z.

D. M. Boroson, B. S. Robinson, D. V. Murphy, D. A. Burianek, F. Khatri, J. M. Kovalik, Z. Sodnik, and D. M. Cornwell, “Overview and results of the lunar laser communication demonstration,” Proc. SPIE 8971, 89710S (2014).
[Crossref]

T. Jono, Y. Takayama, K. Shiratama, I. Mase, B. Demelenne, Z. Sodnik, A. Bird, M. Toyoshima, H. Kunimori, D. Giggenbach, N. Perlot, M. Knapek, and K. Arai, “Overview of the inter-orbit and orbit-to-ground laser communication demonstration by OICETS,” Proc. SPIE 6457, 645702 (2007).
[Crossref]

M. Reyes, S. Chueca, A. Alonso, T. Viera, and Z. Sodnik, “Analysis of the preliminary optical links between ARTEMIS and the optical ground station,” Proc. SPIE 4821, 33–43 (2002).
[Crossref]

Stevens, M. L.

D. J. Geisler, T. M. Yarnall, G. Lund, C. M. Schieler, M. L. Stevens, N. K. Fontaine, B. S. Robinson, and S. A. Hamilton, “Experimental comparison of 3-mode and single-mode coupling over a 1.6-km free-space link,” Proc. SPIE 10524, 105240H (2018).

Stone, J.

Sugihara, K.

Y. Miyata, K. Sugihara, W. Matsumoto, K. Onohara, T. Sugihara, K. Kubo, H. Yoshida, and T. Mizuoch, “A triple-concatenated FEC using soft-decision decoding for 100 Gb/s optical transmission,” in Optical Fiber Communication Conference (2010), paper OThL.3.
[Crossref]

Sugihara, T.

Y. Miyata, K. Sugihara, W. Matsumoto, K. Onohara, T. Sugihara, K. Kubo, H. Yoshida, and T. Mizuoch, “A triple-concatenated FEC using soft-decision decoding for 100 Gb/s optical transmission,” in Optical Fiber Communication Conference (2010), paper OThL.3.
[Crossref]

Takahashi, S.

M. Arikawa, T. Ishikawa, K. Hosokawa, S. Takahashi, Y. Ono, and T. Ito, “Demonstration of turbulence-tolerant free-space optical communication receiver using few-mode-fiber coupling and digital combining,” in Proceedings of IEEE Photonics Society Summer Topical Meeting Series (2016), paper TuC3.4.

Takayama, Y.

M. Toyoshima, H. Takenaka, Y. Shoji, and Y. Takayama, “Frequency characteristics of atmospheric turbulence in space-to-ground laser links,” Proc. SPIE 7685, 76850G (2010).
[Crossref]

M. Toyoshima, H. Takenaka, Y. Shoji, Y. Takayama, Y. Koyama, and H. Kunimori, “Polarization measurements through space-to-ground atmospheric propagation paths by using a highly polarized laser source in space,” Opt. Express 17(25), 22333–22340 (2009).
[Crossref]

T. Jono, Y. Takayama, K. Shiratama, I. Mase, B. Demelenne, Z. Sodnik, A. Bird, M. Toyoshima, H. Kunimori, D. Giggenbach, N. Perlot, M. Knapek, and K. Arai, “Overview of the inter-orbit and orbit-to-ground laser communication demonstration by OICETS,” Proc. SPIE 6457, 645702 (2007).
[Crossref]

Takenaka, H.

H. Takenaka, Y. Koyama, D. Kolev, M. Akioka, N. Iwakiri, H. Kunimori, A. Carrasco-Casado, Y. Munemasa, E. Okamoto, and M. Toyoshima, “In-orbit verification of small optical transponder (SOTA) –Evaluation of satellite-to-ground laser communication links–,” Proc. SPIE 9739, 973903 (2016).
[Crossref]

M. Toyoshima, H. Takenaka, Y. Shoji, and Y. Takayama, “Frequency characteristics of atmospheric turbulence in space-to-ground laser links,” Proc. SPIE 7685, 76850G (2010).
[Crossref]

M. Toyoshima, H. Takenaka, Y. Shoji, Y. Takayama, Y. Koyama, and H. Kunimori, “Polarization measurements through space-to-ground atmospheric propagation paths by using a highly polarized laser source in space,” Opt. Express 17(25), 22333–22340 (2009).
[Crossref]

Tao, Z.

L. Liu, Z. Tao, W. Yan, S. Oda, T. Hoshida, and J. C. Rasmussen, “Initial tap setup of constant modulus algorithm for polarization de-multiplexing in optical coherent receivers,” in Optical Fiber Communication Conference (2009), paper OMT.2.

Thomsen, B. C.

R. Maher, D. S. Millar, S. J. Savory, and B. C. Thomsen, “Widely tunable burst mode digital coherent receiver with fast reconfiguration time for 112 Gb/s DP-QPSK WDM networks,” J. Light. Technol. 30(24), 3924–3930 (2012).
[Crossref]

Toyoshima, M.

H. Takenaka, Y. Koyama, D. Kolev, M. Akioka, N. Iwakiri, H. Kunimori, A. Carrasco-Casado, Y. Munemasa, E. Okamoto, and M. Toyoshima, “In-orbit verification of small optical transponder (SOTA) –Evaluation of satellite-to-ground laser communication links–,” Proc. SPIE 9739, 973903 (2016).
[Crossref]

M. Toyoshima, H. Takenaka, Y. Shoji, and Y. Takayama, “Frequency characteristics of atmospheric turbulence in space-to-ground laser links,” Proc. SPIE 7685, 76850G (2010).
[Crossref]

M. Toyoshima, H. Takenaka, Y. Shoji, Y. Takayama, Y. Koyama, and H. Kunimori, “Polarization measurements through space-to-ground atmospheric propagation paths by using a highly polarized laser source in space,” Opt. Express 17(25), 22333–22340 (2009).
[Crossref]

T. Jono, Y. Takayama, K. Shiratama, I. Mase, B. Demelenne, Z. Sodnik, A. Bird, M. Toyoshima, H. Kunimori, D. Giggenbach, N. Perlot, M. Knapek, and K. Arai, “Overview of the inter-orbit and orbit-to-ground laser communication demonstration by OICETS,” Proc. SPIE 6457, 645702 (2007).
[Crossref]

Uysal, M.

M. A. Khalighi and M. Uysal, “Survey on free space optical communication: A communication theory perspective,” IEEE Communication Survey & Tutorials 16(4), 2231–2258 (2014).
[Crossref]

Viera, T.

M. Reyes, S. Chueca, A. Alonso, T. Viera, and Z. Sodnik, “Analysis of the preliminary optical links between ARTEMIS and the optical ground station,” Proc. SPIE 4821, 33–43 (2002).
[Crossref]

Wandermoth, B. K.

R. A. Fields, D. A. Kozlowski, H. T. Yura, R. L. Wong, J. M. Wicker, C. T. Lunde, M. Gregory, B. K. Wandermoth, F. F. Heine, and J. J. Luna, “5.625 Gbps bidirectional laser communications measurements between the NFIRE satellite and an optical ground station,” Proc. SPIE 8184, 81840D (2011).
[Crossref]

Wang, J.

R. Zhang, J. Wang, G. Zhao, and J. Lv, “Fiber-based free-space optical coherent receiver with vibration compensation mechanism,” Opt. Express 21(15), 18435–18441 (2013).

Watanabe, T.

S. Miura, T. Watanabe, and Y. Kokubun, “Accurate analysis of crosstalk between LP11 quasi-degenerate modes due to offset connection using true eigenmodes,” IEEE Photonics J. 10(1), 7900811 (2018).
[Crossref]

Y. Kokubun, T. Watanabe, S. Miura, and R. Kawata, “What is a mode in few mode fibers?: Proposal of MIMO-free mode division multiplexing using true eigenmodes,” IEICE Electron. Express 13(18), 1–12 (2016).
[Crossref]

Wicker, J. M.

R. A. Fields, D. A. Kozlowski, H. T. Yura, R. L. Wong, J. M. Wicker, C. T. Lunde, M. Gregory, B. K. Wandermoth, F. F. Heine, and J. J. Luna, “5.625 Gbps bidirectional laser communications measurements between the NFIRE satellite and an optical ground station,” Proc. SPIE 8184, 81840D (2011).
[Crossref]

Winzer, P. J.

Wong, R. L.

R. A. Fields, D. A. Kozlowski, H. T. Yura, R. L. Wong, J. M. Wicker, C. T. Lunde, M. Gregory, B. K. Wandermoth, F. F. Heine, and J. J. Luna, “5.625 Gbps bidirectional laser communications measurements between the NFIRE satellite and an optical ground station,” Proc. SPIE 8184, 81840D (2011).
[Crossref]

Wright, W.

A. Biswas, B. V. Oaida, K. S. Andrews, J. M. Kovalik, M. J. Abrahamson, and W. Wright, “Optical payload for lasercomm science (OPALS) link validation during operation from the ISS,” Proc. SPIE 9354, 93540F (2015).
[Crossref]

Yan, W.

L. Liu, Z. Tao, W. Yan, S. Oda, T. Hoshida, and J. C. Rasmussen, “Initial tap setup of constant modulus algorithm for polarization de-multiplexing in optical coherent receivers,” in Optical Fiber Communication Conference (2009), paper OMT.2.

Yarnall, T. M.

D. J. Geisler, T. M. Yarnall, G. Lund, C. M. Schieler, M. L. Stevens, N. K. Fontaine, B. S. Robinson, and S. A. Hamilton, “Experimental comparison of 3-mode and single-mode coupling over a 1.6-km free-space link,” Proc. SPIE 10524, 105240H (2018).

T. M. Yarnall, D. J. Geisler, C. M. Schieler, and R. Yip, “Analysis of free-space coupling to photonic lanterns in the presence of tilt errors,” in Proceedings of IEEE Photonics Conference (2017), paper ThH1.5.

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T. M. Yarnall, D. J. Geisler, C. M. Schieler, and R. Yip, “Analysis of free-space coupling to photonic lanterns in the presence of tilt errors,” in Proceedings of IEEE Photonics Conference (2017), paper ThH1.5.

Yoshida, H.

Y. Miyata, K. Sugihara, W. Matsumoto, K. Onohara, T. Sugihara, K. Kubo, H. Yoshida, and T. Mizuoch, “A triple-concatenated FEC using soft-decision decoding for 100 Gb/s optical transmission,” in Optical Fiber Communication Conference (2010), paper OThL.3.
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R. Zhang, J. Wang, G. Zhao, and J. Lv, “Fiber-based free-space optical coherent receiver with vibration compensation mechanism,” Opt. Express 21(15), 18435–18441 (2013).

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R. Zhang, J. Wang, G. Zhao, and J. Lv, “Fiber-based free-space optical coherent receiver with vibration compensation mechanism,” Opt. Express 21(15), 18435–18441 (2013).

Appl. Opt. (2)

IEEE Communication Survey & Tutorials (1)

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IEEE Photonics J. (1)

S. Miura, T. Watanabe, and Y. Kokubun, “Accurate analysis of crosstalk between LP11 quasi-degenerate modes due to offset connection using true eigenmodes,” IEEE Photonics J. 10(1), 7900811 (2018).
[Crossref]

IEEE Transaction on Wirel. Commun. (1)

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[Crossref]

IEICE Electron. Express (1)

Y. Kokubun, T. Watanabe, S. Miura, and R. Kawata, “What is a mode in few mode fibers?: Proposal of MIMO-free mode division multiplexing using true eigenmodes,” IEICE Electron. Express 13(18), 1–12 (2016).
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[Crossref]

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M. Arikawa, Y. Ono, and T. Ito, “Mode diversity coherent receiver with few-mode fiber-coupling for high-speed free-space optical communication under atmospheric turbulence,” Proc. SPIE 10524, 1052412 (2018).

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T. M. Yarnall, D. J. Geisler, C. M. Schieler, and R. Yip, “Analysis of free-space coupling to photonic lanterns in the presence of tilt errors,” in Proceedings of IEEE Photonics Conference (2017), paper ThH1.5.

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[Crossref]

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[Crossref]

J.-F. Morizur, P. Jian, B. Denolle, O. Pinel, N. Barré, and G. Labroille, “Efficient and mode-selective spatial multiplexer based on multi-plane light conversion,” in Optical Fiber Communication Conference (2015), paper W1A.4.
[Crossref]

L. Liu, Z. Tao, W. Yan, S. Oda, T. Hoshida, and J. C. Rasmussen, “Initial tap setup of constant modulus algorithm for polarization de-multiplexing in optical coherent receivers,” in Optical Fiber Communication Conference (2009), paper OMT.2.

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Figures (14)

Fig. 1
Fig. 1 Schematic diagram of simulation model for LEO-to-ground optical downlink with FMF coupling. RPSs: random phase screens, FMF: few-mode fiber, SMF: single mode fiber.
Fig. 2
Fig. 2 Dependence of coupling efficiency to LP modes after FMF propagation with fixed random phase patterns of the atmospheric turbulence on the fiber length: (a) LP01, (b) LP11e, and (c) LP11o.
Fig. 3
Fig. 3 Temporal fluctuation of coupling efficiency to LP modes after FMF propagation: (a) LP01, (b) LP11e, and (c) LP11o. Colored regions represent the range of fluctuation with the change of the polarization state, and the solid lines correspond to the average.
Fig. 4
Fig. 4 Probability density function (PDF) of temporal fluctuation of coupling efficiency to LP modes after FMF propagation: (a) LP01, (b) LP11e, and (c) LP11o. Dashed lines are regression curves by the Gamma-Gamma distribution.
Fig. 5
Fig. 5 Two types of architecture for mode diversity combining: (a) selection combining after signals demodulated and decoded individually, and (b) full Rx-MIMO.
Fig. 6
Fig. 6 Simulation results of estimated FEC frame error rate, which is the ratio where BER exceed the FEC threshold against the transmitted power of a 128 Gb/s PDM-QPSK signal.
Fig. 7
Fig. 7 Experimental setup for three-mode diversity reception of 128 Gb/s PDM-QPSK signal with turbulence emulator. EDFA: erbium-doped fiber amplifier, LO: local oscillator, OSC: oscilloscope, DSP: digital signal processing.
Fig. 8
Fig. 8 Offline DSP for mode diversity combining with full Rx-MIMO. DDLMS: decision-directed least mean square, CPE: carrier phase estimation.
Fig. 9
Fig. 9 Evaluation results for back-to-back sensitivity.
Fig. 10
Fig. 10 Variation of coupling efficiency to outputs of mode demultiplexer with polarization scrambling.
Fig. 11
Fig. 11 (a) Temporal fluctuation of coupling efficiency to outputs of mode demultiplexer with rotating the diffuser plate, and (b) its probability density function. Dashed lines are regression curves by the Gamma-Gamma distribution.
Fig. 12
Fig. 12 Cumulated distribution function of temporal fluctuation of coupling efficiency.
Fig. 13
Fig. 13 Evaluation results for (a) BER fluctuation with rotating diffuser plate and (b) constellations at 5 ms. Transmitted power was −20 dBm.
Fig. 14
Fig. 14 Experimental results for estimated FEC frame error rate against transmitted power and its estimated value from results for coupling efficiency shown in Fig. 12.

Equations (8)

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( E HE 11 e , E HE 11 o , E TM 01 , E HE 21 e , E HE 21 o , E TE 01 ) = ( E LP 01 x , E LP 01 y , E LP 11 ex , E LP 11 oy , E LP 11 ox , E LP 11 ey ) M , M = ( I 2 0 0 M 1 ) , M 1 = 1 2 ( 1 1 0 0 1 1 0 0 0 0 1 1 0 0 1 1 ) ,
( E HE 11 e , E HE 11 o , E TM 01 , E HE 21 e , E HE 21 o , E TE 01 ) = ( E HE 11 e , E HE 11 o , E TM 01 , E HE 21 e , E HE 21 o , E TE 01 ) P , P = ( P 0 0 0 P 1 ) , P 0 = ( e i β 0 z 0 0 e i β 0 z ) , P 1 = ( e i β M z 0 0 0 0 1 0 0 0 0 1 0 0 0 0 e i β E z ) .
β E = 2 n 2 k 0 Δ 2 ( U 0 W 0 V 2 ) 2 κ 0 + ( W 0 ) , β M = 2 n 2 k 0 Δ 2 ( U 0 W 0 V 2 ) 2 ( κ 0 + ( W 0 ) 2 κ 2 ( W 0 ) ) .
( E LP 01 x , E LP 01 y , E LP 11 ex , E LP 11 oy , E LP 11 ox , E LP 11 ey ) = ( E LP 01 x , E LP 01 y , E LP 11 ex , E LP 11 oy , E LP 11 ox , E LP 11 ey ) M P M 1 , M P M 1 = ( P 0 0 0 P ˜ 1 ) , P ˜ 1 = 1 2 ( e i β M z + 1 e i β M z 1 0 0 e i β M z 1 e i β M z + 1 0 0 0 0 e i β E z + 1 e i β E z + 1 0 0 e i β E z + 1 e i β E z + 1 ) .
ρ = ( E r E LP 01 x * d S E r E LP 01 y * d S E r E LP 11 ex * d S E r E LP 11 oy * d S E r E LP 11 ox * d S E r E LP 11 ey * d S ) = ( E x U r U LP 01 * d S E y U r U LP 01 * d S E x U r U LP 11 e * d S E y U r U LP 11 o * d S E x U r U LP 11 o * d S E y U r U LP 11 e * d S ) .
ρ ˜ = M P M 1 ρ .
H = ( ρ ˜ | ( E x , E y ) t = ( cos ψ , e i δ sin ψ ) t , ρ ˜ | ( E x , E y ) t = ( sin ψ , e i δ cos ψ ) t ) .
H LP 01 = ( h 11 h 12 h 21 h 22 ) , H LP 11 e = ( h 31 h 32 h 61 h 62 ) , H LP 11 o = ( h 51 h 52 h 41 h 42 ) .

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