Abstract

Coherent superposition of light waves has long been used in various fields of science, and recent advances in digital coherent detection and space-division multiplexing have enabled the coherent superposition of information-carrying optical signals to achieve better communication fidelity on amplified-spontaneous-noise limited communication links. However, fiber nonlinearity introduces highly correlated distortions on identical signals and diminishes the benefit of coherent superposition in nonlinear transmission regime. Here we experimentally demonstrate that through coordinated scrambling of signal constellations at the transmitter, together with appropriate unscrambling at the receiver, the full benefit of coherent superposition is retained in the nonlinear transmission regime of a space-diversity fiber link based on an innovatively engineered multi-core fiber. This scrambled coherent superposition may provide the flexibility of trading communication capacity for performance in future optical fiber networks, and may open new possibilities in high-performance and secure optical communications.

© 2012 OSA

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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  19. W. Shieh and X. Chen, “Information spectral efficiency and launch power density limits due to fiber nonlinearity for coherent optical OFDM system,” IEEE Photon. J.3(2), 158–173 (2011).
    [CrossRef]
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    [CrossRef]
  21. A. Mecozzi, C. B. Clausen, M. Shtaif, S.-G. Park, and A. H. Gnauck, “Cancellation of timing and amplitude jitter in symmetric links using highly dispersed pulses,” IEEE Photon. Technol. Lett.13(5), 445–447 (2001).
    [CrossRef]
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  24. D. M. Millar, S. Makovejs, V. Mikhailov, R. I. Killey, P. Bayvel, and S. J. Savory, “Experimental comparison of nonlinear compensation in long-haul PDM-QPSK transmission at 42.7 and 85.4 Gb/s,” in Proceedings of the 2009 European Conference on Optical Communication (Vienna, Austria), paper 9.4.4.
  25. X. Liu, S. Chandrasekhar, A. H. Gnauck, P. J. Winzer, S. Randel, S. Corteselli, B. Zhu, T. Taunay, and M. Fishteyn, “Digital coherent superposition for performance improvement of spatially multiplexed 676-Gb/s OFDM-16QAM superchannels,”in Proceedings of the 2012 European Conference on Optical Communication (Amsterdam, Netherlands), paper Tu.3.C.2 (2012).
  26. A. K. Ekert, “Quantum cryptography based on Bell’s theorem,” Phys. Rev. Lett.67(6), 661–663 (1991).
    [CrossRef] [PubMed]
  27. H.-K. Lo and H. F. Chau, “Unconditional security of quantum key distribution over arbitrarily long distances,” Science283(5410), 2050–2056 (1999).
    [CrossRef] [PubMed]
  28. H. Takesue, S. W. Nam, Q. Zhang, R. H. Hadfield, T. Honjo, K. Tamaki, and Y. Yamamoto, “Quantum key distribution over 40 dB channel loss using superconducting single photon detectors,” Nat. Photonics1(6), 343–348 (2007).
    [CrossRef]

2012 (1)

2011 (4)

2010 (2)

S. J. Savory, “Digital coherent optical receivers: algorithms and subsystems,” IEEE J. Sel. Top. Quantum Electron.16(5), 1164–1179 (2010).
[CrossRef]

D. J. Richardson, “Applied physics. Filling the light pipe,” Science330(6002), 327–328 (2010).
[CrossRef] [PubMed]

2008 (1)

R.-J. Essiambre, G. J. Foschini, G. Kramer, and P. J. Winzer, “Capacity limits of information transport in fiber-optic networks,” Phys. Rev. Lett.101(16), 163901 (2008).
[CrossRef] [PubMed]

2007 (1)

H. Takesue, S. W. Nam, Q. Zhang, R. H. Hadfield, T. Honjo, K. Tamaki, and Y. Yamamoto, “Quantum key distribution over 40 dB channel loss using superconducting single photon detectors,” Nat. Photonics1(6), 343–348 (2007).
[CrossRef]

2006 (1)

P. J. Winzer and R.-J. Essiambre, “Advanced optical modulation formats,” Proc. IEEE94(5), 952–985 (2006).
[CrossRef]

2001 (2)

P. P. Mitra and J. B. Stark, “Nonlinear limits to the information capacity of optical fibre communications,” Nature411(6841), 1027–1030 (2001).
[CrossRef] [PubMed]

A. Mecozzi, C. B. Clausen, M. Shtaif, S.-G. Park, and A. H. Gnauck, “Cancellation of timing and amplitude jitter in symmetric links using highly dispersed pulses,” IEEE Photon. Technol. Lett.13(5), 445–447 (2001).
[CrossRef]

2000 (1)

H. R. Stuart, “Dispersive multiplexing in multimode optical fiber,” Science289(5477), 281–283 (2000).
[CrossRef] [PubMed]

1999 (1)

H.-K. Lo and H. F. Chau, “Unconditional security of quantum key distribution over arbitrarily long distances,” Science283(5410), 2050–2056 (1999).
[CrossRef] [PubMed]

1996 (1)

G. J. Foschini, “Layered space-time architecture for wireless communication in a fading environment when using multi-element antennas,” Bell Labs Tech. J.1(2), 41–59 (1996).
[CrossRef]

1995 (1)

R. W. Tkach, A. R. Chraplyvy, F. Forghieri, A. H. Gnauck, and R. M. Derosier, “Four-photon mixing and high speed WDM systems,” J. Lightwave Technol.13(5), 841–849 (1995).
[CrossRef]

1991 (1)

A. K. Ekert, “Quantum cryptography based on Bell’s theorem,” Phys. Rev. Lett.67(6), 661–663 (1991).
[CrossRef] [PubMed]

1948 (1)

C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J.27, 379–423 (1948).

1804 (1)

T. Young, “Experimental demonstration of the general law of the interference of light,” Philos. Trans. R. Soc. Lond.94, 1-16.(1804).

Burrows, E. C.

Chandrasekhar, S.

Chau, H. F.

H.-K. Lo and H. F. Chau, “Unconditional security of quantum key distribution over arbitrarily long distances,” Science283(5410), 2050–2056 (1999).
[CrossRef] [PubMed]

Chen, X.

Chraplyvy, A. R.

R. W. Tkach, A. R. Chraplyvy, F. Forghieri, A. H. Gnauck, and R. M. Derosier, “Four-photon mixing and high speed WDM systems,” J. Lightwave Technol.13(5), 841–849 (1995).
[CrossRef]

Clausen, C. B.

A. Mecozzi, C. B. Clausen, M. Shtaif, S.-G. Park, and A. H. Gnauck, “Cancellation of timing and amplitude jitter in symmetric links using highly dispersed pulses,” IEEE Photon. Technol. Lett.13(5), 445–447 (2001).
[CrossRef]

Derosier, R. M.

R. W. Tkach, A. R. Chraplyvy, F. Forghieri, A. H. Gnauck, and R. M. Derosier, “Four-photon mixing and high speed WDM systems,” J. Lightwave Technol.13(5), 841–849 (1995).
[CrossRef]

Dimarcello, F. V.

Ekert, A. K.

A. K. Ekert, “Quantum cryptography based on Bell’s theorem,” Phys. Rev. Lett.67(6), 661–663 (1991).
[CrossRef] [PubMed]

Essiambre, R.-J.

R.-J. Essiambre, G. J. Foschini, G. Kramer, and P. J. Winzer, “Capacity limits of information transport in fiber-optic networks,” Phys. Rev. Lett.101(16), 163901 (2008).
[CrossRef] [PubMed]

P. J. Winzer and R.-J. Essiambre, “Advanced optical modulation formats,” Proc. IEEE94(5), 952–985 (2006).
[CrossRef]

Fini, J. M.

Fishteyn, M.

Forghieri, F.

R. W. Tkach, A. R. Chraplyvy, F. Forghieri, A. H. Gnauck, and R. M. Derosier, “Four-photon mixing and high speed WDM systems,” J. Lightwave Technol.13(5), 841–849 (1995).
[CrossRef]

Foschini, G. J.

R.-J. Essiambre, G. J. Foschini, G. Kramer, and P. J. Winzer, “Capacity limits of information transport in fiber-optic networks,” Phys. Rev. Lett.101(16), 163901 (2008).
[CrossRef] [PubMed]

G. J. Foschini, “Layered space-time architecture for wireless communication in a fading environment when using multi-element antennas,” Bell Labs Tech. J.1(2), 41–59 (1996).
[CrossRef]

Gnauck, A. H.

S. Chandrasekhar, A. H. Gnauck, X. Liu, P. J. Winzer, Y. Pan, E. C. Burrows, T. F. Taunay, B. Zhu, M. Fishteyn, M. F. Yan, J. M. Fini, E. M. Monberg, and F. V. Dimarcello, “WDM/SDM transmission of 10 x 128-Gb/s PDM-QPSK over 2688-km 7-core fiber with a per-fiber net aggregate spectral-efficiency distance product of 40,320 km·b/s/Hz,” Opt. Express20(2), 706–711 (2012).
[CrossRef] [PubMed]

A. Mecozzi, C. B. Clausen, M. Shtaif, S.-G. Park, and A. H. Gnauck, “Cancellation of timing and amplitude jitter in symmetric links using highly dispersed pulses,” IEEE Photon. Technol. Lett.13(5), 445–447 (2001).
[CrossRef]

R. W. Tkach, A. R. Chraplyvy, F. Forghieri, A. H. Gnauck, and R. M. Derosier, “Four-photon mixing and high speed WDM systems,” J. Lightwave Technol.13(5), 841–849 (1995).
[CrossRef]

Hadfield, R. H.

H. Takesue, S. W. Nam, Q. Zhang, R. H. Hadfield, T. Honjo, K. Tamaki, and Y. Yamamoto, “Quantum key distribution over 40 dB channel loss using superconducting single photon detectors,” Nat. Photonics1(6), 343–348 (2007).
[CrossRef]

Honjo, T.

H. Takesue, S. W. Nam, Q. Zhang, R. H. Hadfield, T. Honjo, K. Tamaki, and Y. Yamamoto, “Quantum key distribution over 40 dB channel loss using superconducting single photon detectors,” Nat. Photonics1(6), 343–348 (2007).
[CrossRef]

Kramer, G.

R.-J. Essiambre, G. J. Foschini, G. Kramer, and P. J. Winzer, “Capacity limits of information transport in fiber-optic networks,” Phys. Rev. Lett.101(16), 163901 (2008).
[CrossRef] [PubMed]

Li, G.

Liu, X.

Lo, H.-K.

H.-K. Lo and H. F. Chau, “Unconditional security of quantum key distribution over arbitrarily long distances,” Science283(5410), 2050–2056 (1999).
[CrossRef] [PubMed]

Mecozzi, A.

A. Mecozzi, C. B. Clausen, M. Shtaif, S.-G. Park, and A. H. Gnauck, “Cancellation of timing and amplitude jitter in symmetric links using highly dispersed pulses,” IEEE Photon. Technol. Lett.13(5), 445–447 (2001).
[CrossRef]

Mitra, P. P.

P. P. Mitra and J. B. Stark, “Nonlinear limits to the information capacity of optical fibre communications,” Nature411(6841), 1027–1030 (2001).
[CrossRef] [PubMed]

Monberg, E. M.

Nam, S. W.

H. Takesue, S. W. Nam, Q. Zhang, R. H. Hadfield, T. Honjo, K. Tamaki, and Y. Yamamoto, “Quantum key distribution over 40 dB channel loss using superconducting single photon detectors,” Nat. Photonics1(6), 343–348 (2007).
[CrossRef]

Pan, Y.

Park, S.-G.

A. Mecozzi, C. B. Clausen, M. Shtaif, S.-G. Park, and A. H. Gnauck, “Cancellation of timing and amplitude jitter in symmetric links using highly dispersed pulses,” IEEE Photon. Technol. Lett.13(5), 445–447 (2001).
[CrossRef]

Richardson, D. J.

D. J. Richardson, “Applied physics. Filling the light pipe,” Science330(6002), 327–328 (2010).
[CrossRef] [PubMed]

Savory, S. J.

S. J. Savory, “Digital coherent optical receivers: algorithms and subsystems,” IEEE J. Sel. Top. Quantum Electron.16(5), 1164–1179 (2010).
[CrossRef]

Shannon, C. E.

C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J.27, 379–423 (1948).

Shieh, W.

W. Shieh and X. Chen, “Information spectral efficiency and launch power density limits due to fiber nonlinearity for coherent optical OFDM system,” IEEE Photon. J.3(2), 158–173 (2011).
[CrossRef]

Shtaif, M.

A. Mecozzi, C. B. Clausen, M. Shtaif, S.-G. Park, and A. H. Gnauck, “Cancellation of timing and amplitude jitter in symmetric links using highly dispersed pulses,” IEEE Photon. Technol. Lett.13(5), 445–447 (2001).
[CrossRef]

Stark, J. B.

P. P. Mitra and J. B. Stark, “Nonlinear limits to the information capacity of optical fibre communications,” Nature411(6841), 1027–1030 (2001).
[CrossRef] [PubMed]

Stuart, H. R.

H. R. Stuart, “Dispersive multiplexing in multimode optical fiber,” Science289(5477), 281–283 (2000).
[CrossRef] [PubMed]

Takesue, H.

H. Takesue, S. W. Nam, Q. Zhang, R. H. Hadfield, T. Honjo, K. Tamaki, and Y. Yamamoto, “Quantum key distribution over 40 dB channel loss using superconducting single photon detectors,” Nat. Photonics1(6), 343–348 (2007).
[CrossRef]

Tamaki, K.

H. Takesue, S. W. Nam, Q. Zhang, R. H. Hadfield, T. Honjo, K. Tamaki, and Y. Yamamoto, “Quantum key distribution over 40 dB channel loss using superconducting single photon detectors,” Nat. Photonics1(6), 343–348 (2007).
[CrossRef]

Taunay, T. F.

Tkach, R. W.

R. W. Tkach, A. R. Chraplyvy, F. Forghieri, A. H. Gnauck, and R. M. Derosier, “Four-photon mixing and high speed WDM systems,” J. Lightwave Technol.13(5), 841–849 (1995).
[CrossRef]

Winzer, P. J.

Yamamoto, Y.

H. Takesue, S. W. Nam, Q. Zhang, R. H. Hadfield, T. Honjo, K. Tamaki, and Y. Yamamoto, “Quantum key distribution over 40 dB channel loss using superconducting single photon detectors,” Nat. Photonics1(6), 343–348 (2007).
[CrossRef]

Yan, M. F.

Young, T.

T. Young, “Experimental demonstration of the general law of the interference of light,” Philos. Trans. R. Soc. Lond.94, 1-16.(1804).

Zhang, Q.

H. Takesue, S. W. Nam, Q. Zhang, R. H. Hadfield, T. Honjo, K. Tamaki, and Y. Yamamoto, “Quantum key distribution over 40 dB channel loss using superconducting single photon detectors,” Nat. Photonics1(6), 343–348 (2007).
[CrossRef]

Zhu, B.

Bell Labs Tech. J. (1)

G. J. Foschini, “Layered space-time architecture for wireless communication in a fading environment when using multi-element antennas,” Bell Labs Tech. J.1(2), 41–59 (1996).
[CrossRef]

Bell Syst. Tech. J. (1)

C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J.27, 379–423 (1948).

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

S. J. Savory, “Digital coherent optical receivers: algorithms and subsystems,” IEEE J. Sel. Top. Quantum Electron.16(5), 1164–1179 (2010).
[CrossRef]

IEEE Photon. J. (1)

W. Shieh and X. Chen, “Information spectral efficiency and launch power density limits due to fiber nonlinearity for coherent optical OFDM system,” IEEE Photon. J.3(2), 158–173 (2011).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

A. Mecozzi, C. B. Clausen, M. Shtaif, S.-G. Park, and A. H. Gnauck, “Cancellation of timing and amplitude jitter in symmetric links using highly dispersed pulses,” IEEE Photon. Technol. Lett.13(5), 445–447 (2001).
[CrossRef]

J. Lightwave Technol. (1)

R. W. Tkach, A. R. Chraplyvy, F. Forghieri, A. H. Gnauck, and R. M. Derosier, “Four-photon mixing and high speed WDM systems,” J. Lightwave Technol.13(5), 841–849 (1995).
[CrossRef]

Nat. Photonics (1)

H. Takesue, S. W. Nam, Q. Zhang, R. H. Hadfield, T. Honjo, K. Tamaki, and Y. Yamamoto, “Quantum key distribution over 40 dB channel loss using superconducting single photon detectors,” Nat. Photonics1(6), 343–348 (2007).
[CrossRef]

Nature (1)

P. P. Mitra and J. B. Stark, “Nonlinear limits to the information capacity of optical fibre communications,” Nature411(6841), 1027–1030 (2001).
[CrossRef] [PubMed]

Opt. Express (4)

Philos. Trans. R. Soc. Lond. (1)

T. Young, “Experimental demonstration of the general law of the interference of light,” Philos. Trans. R. Soc. Lond.94, 1-16.(1804).

Phys. Rev. Lett. (2)

R.-J. Essiambre, G. J. Foschini, G. Kramer, and P. J. Winzer, “Capacity limits of information transport in fiber-optic networks,” Phys. Rev. Lett.101(16), 163901 (2008).
[CrossRef] [PubMed]

A. K. Ekert, “Quantum cryptography based on Bell’s theorem,” Phys. Rev. Lett.67(6), 661–663 (1991).
[CrossRef] [PubMed]

Proc. IEEE (1)

P. J. Winzer and R.-J. Essiambre, “Advanced optical modulation formats,” Proc. IEEE94(5), 952–985 (2006).
[CrossRef]

Science (3)

H.-K. Lo and H. F. Chau, “Unconditional security of quantum key distribution over arbitrarily long distances,” Science283(5410), 2050–2056 (1999).
[CrossRef] [PubMed]

H. R. Stuart, “Dispersive multiplexing in multimode optical fiber,” Science289(5477), 281–283 (2000).
[CrossRef] [PubMed]

D. J. Richardson, “Applied physics. Filling the light pipe,” Science330(6002), 327–328 (2010).
[CrossRef] [PubMed]

Other (9)

M. Nakazawa, “Giant leaps in optical communication technologies towards 2030 and beyond,” in Proceedings of the2010European Conference on Optical Communication (Turin, Italy), Plenary Talk.

A. R. Chraplyvy, “The coming capacity crunch,” in Proceedings of the2009European Conference on Optical Communication (Vienna, Austria), Plenary Talk.

X. Liu, S. Chandrasekhar, A. H. Gnauck, P. J. Winzer, A. R. Chraplyvy, B. Zhu, T. Taunay, and M. Fishteyn, “Performance improvement of space-division multiplexed 128-Gb/s PDM-QPSK signals by constructive superposition in a single-input-multiple-output configuration,” in Proceedings of the 2012 Optical Fiber Communication Conference (Optical Society of America, Washington, DC, 2012), OTu1D3.

S. Naderi Shahi and S. Kumar, “Reduction of nonlinear impairments in fiber transmission system using fiber diversity,” in Proceedings of the 2011 OSA Summer Topical Meeting on Signal Processing in Photonic Communications (Toronto, Canada), SPWA3.

J. Sakaguchi, B. J. Puttnam, W. Klaus, Y. Awaji, N. Wada, A. Kanno, T. Kawanishi, K. Imamura, H. Inaba, K. Mukasa, R. Sugizaki, T. Kobayashi, and M. Watanabe, “19-core fiber transmission of 19x100x172-Gb/s SDM-WDM-PDM-QPSK signals at 305Tb/s,” in Proceedings of the 2012 Optical Fiber Communication Conference (Optical Society of America, Washington, DC, 2012), PDP5C.1.

G. P. Agrawal, Nonlinear Fiber Optics (Academic Press 2007).

A. Carena, G. Bosco, G. V. Curri, P. Poggiolini, M. Tapia Taiba, and F. Forghieri, “Statistical characterization of PM-QPSK signals after propagation in uncompensated fiber links,” in Proceedings of the 2010 European Conference on Optical Communication (Turin, Italy), P4.07.

D. M. Millar, S. Makovejs, V. Mikhailov, R. I. Killey, P. Bayvel, and S. J. Savory, “Experimental comparison of nonlinear compensation in long-haul PDM-QPSK transmission at 42.7 and 85.4 Gb/s,” in Proceedings of the 2009 European Conference on Optical Communication (Vienna, Austria), paper 9.4.4.

X. Liu, S. Chandrasekhar, A. H. Gnauck, P. J. Winzer, S. Randel, S. Corteselli, B. Zhu, T. Taunay, and M. Fishteyn, “Digital coherent superposition for performance improvement of spatially multiplexed 676-Gb/s OFDM-16QAM superchannels,”in Proceedings of the 2012 European Conference on Optical Communication (Amsterdam, Netherlands), paper Tu.3.C.2 (2012).

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Figures (4)

Fig. 1
Fig. 1

Schematic of an optical fiber transmission experiment on scrambled coherent superposition. a, Illustration of a space-division-multiplexed (SDM) and wavelength-division-multiplexed (WDM) fiber transmission link. b,Illustration of a polarization-division-multiplexed (PDM) Cartesian modulator used in an optical transmitter (TX), capable of modulating the real and imaginary parts of the complex E-field of each of two orthogonal polarization components of an optical carrier. c, Signal constellations of a PDM-QPSK signal carrying 4 bits per symbol. d, Drawing of seven-core-fiber with fan-out couplers to standard single-mode fibers used for SDM transmission. e, Image of the cross section of a low-crosstalk seven-core fiber used in the experiment. f, Schematic of the digital coherent detection based coherent superposition of multiple SDM signals. The optical local oscillator (LO) beats with the received signal in a 2-by-8 polarization-diversity optical hybrid to obtain the in-phase and quadrature components of two orthogonal polarization components of the signal (usually in a different basis compared to the transmitter). Balanced photo-detectors (BDs) and high-speed analog-to-digital converters (ADCs) are used to reconstruct the received signal in the digital domain. A digital signal processor (offline) is then used to compensate the channel response and perform coherent superposition of multiple SDM signals. g, Mathematical representations of SCS and direct digital coherent superposition (DCS) at the receiver (RX).

Fig. 2
Fig. 2

Correlation and de-correlation of nonlinear distortions. a, Measured moderate correlation between the distortion on the received signal E-field from core 1 of a 76.8-km seven-core fiber and that from core 2 after moderately nonlinear transmission (P = 7 dBm). b, Measured high correlation between the distortions on the two SDM signals after highly nonlinear transmission (P = 10 dBm). c, Measured near-zero correlation between the two SDM signals after highly nonlinear transmission (P = 10 dBm) with constellation scrambling through a delay (τ) of 100 symbols between the two signals. d, Illustration of the signal distortions due to the linear and nonlinear transmission effects in the case of DCS with 3 superimposed signals. e, Illustration of the signal distortions due to the linear and nonlinear effects in the case of SCS with 3 superimposed signals. The final signal variance after SCS is half of that after DCS, showing the benefit of the constellation scrambling.

Fig. 3
Fig. 3

Theoretical and experimental results. a, Theoretical signal quality factor (Q2) dependence on signal launch power. The absolute values of the signal quality factors depend on transmission link conditions and are normalized here for ease of comparison with the experiment. b, Experimentally measured signal quality factor dependence on signal launch power after 2688-km SDM transmission over the 7-core fiber, showing reasonably good agreement with the theoretical results. c, Signal constellation obtained by SCS of 3 SDM signals after 2688-km transmission at P = 4 dBm. d, Signal constellation obtained by DCS of 3 SDM signals after 2688-km transmission at P = 4 dBm, showing more distortions than that obtained by SCS.

Fig. 4
Fig. 4

Performance improvement obtained by SCS. a, Measured signal quality factor improvement by SCS and DCS as a function of the number of superimposed signals after 2688-km SDM transmission. The signal launch power is fixed at 0 dBm per core, which is approximately the optimum power ( P 0 ) for each signal without coherent superposition. The theoretical results are also shown. Reasonably good agreement between theory and experiment is achieved. b-d, Signal constellations obtained by SCS of 2, 3, and 5 SDM signals after 2688-km transmission at P = P 0 , respectively. e-g, Signal constellations obtained by DCS of 2, 3, and 5 SDM signals after 2688-km transmission at P = P 0 , respectively. The additional performance gain of SCS over DCS can be seen from the clearer constellations obtained by SCS than by DCS for the same value of m.

Equations (7)

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ρ= (x x ¯ )(y y ¯ ) ¯ /( σ x σ y )
Q 2 =2 [erf c 1 (2BER)] 2
Q 2 = κ σ L 2 + σ NL 2 = 1 a P 1 +b P 2
Q 0 2 = a 2/3 b 1/3 /( 2 1/3 + 2 2/3 )
P 0 = ( a 2b ) 1/3
Q max,SCS 2 =m Q 0 2
Q max,DCS 2 = m 2/3 Q 0 2

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