Abstract

We derive analytic formulas for the improvement in effective optical signal-to-noise ratio brought by a digital nonlinear compensator for dispersion uncompensated links. By assuming Gaussian distributed nonlinear noise, we are able to take both nonlinear signal-to-signal and nonlinear signal-to-noise interactions into account. In the limit of weak nonlinear signal-to-noise interactions, we derive an upper boundary of the OSNR improvement. This upper boundary only depends on fiber parameters as well as on the total bandwidth of the considered wavelength-division multiplexing (WDM) signal and the bandwidth available for back propagation. We discuss the dependency of the upper boundary on different fiber types and also the OSNR improvement in practical system conditions. Furthermore, the analytical formulas are validated by numerical simulations.

© 2012 OSA

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References

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  1. T. Tanimura, T. Hoshida, S. Oda, T. Tanaka, C. Ohsima, and J. C. Rasmussen, “Systematic analysis on multi-segment dual-polarisation nonlinear compensation in 112 Gb/s DP-QPSK coherent receiver,” in Tech. Digest of European Conference on Optical Communication,2009, paper 9.4.5.
  2. S. Oda, T. Tanimura, T. Hoshida, C. Ohshima, H. Nakashima, Z. Tao, and J. C. Rasmussen, “112 Gb/s DP-QPSK transmission using a novel nonlinear compensator in digital coherent receiver,” in Tech. Digest of Optical Fiber Communication Conference,2009, paper OThR6.
  3. E. Ip, “Nonlinear compensation using backpropagation for polarization-multiplexed transmission,” J. Lightwave Technol.28(6), 939–951 (2010).
    [CrossRef]
  4. D. S. Millar, S. Makovejs, C. Behrens, S. Hellerbrand, R. I. Killey, P. Bayvel, and S. J. Savory, “Mitigation of fiber nonlinearity using a digital coherent receiver,” IEEE J. Sel. Top. Quantum Electron.16(5), 1217–1226 (2010).
    [CrossRef]
  5. D. Rafique and A. D. Ellis, “Impact of signal-ASE four-wave mixing on the effectiveness of digital back-propagation in 112 Gb/s PM-QPSK systems,” Opt. Express19(4), 3449–3454 (2011).
    [CrossRef] [PubMed]
  6. L. Beygi, E. Agrell, P. Johannisson, M. Karlsson, and H. Wymeersch, “The limits of digital backpropagation in nonlinear coherent fiber-optic links,” in Tech. Digest of European Conference on Optical Communication,2012, P4.14.
  7. G. Gao, X. Chen, and W. Shieh, “Limitation of fiber nonlinearity compensation using digital back propagation in the presence of PMD,” in Tech. Digest of Optical Fiber Communication Conference,2012, paper OM3A.5.
  8. T. Tanimura, S. Oda, T. Hoshida, L. Li, Z. Tao, and J. C. Rasmussen, “Experimental characterization of nonlinearity mitigation by digital back propagation and nonlinear polarization crosstalk canceller under high PMD condition,” in Tech. Digest of Optical Fiber Communication Conference,2011, JWA020.
  9. A. Splett, C. Kurtzke, and K. Petermann, “Ultimate transmission capacity of amplified optical fiber communication systems taking into account fiber nonlinearities,” in Tech. Digest of European Conference on Optical Communication,1993, paper MoC2.4.
  10. X. Chen and W. Shieh, “Closed-form expressions for nonlinear transmission performance of densely spaced coherent optical OFDM systems,” Opt. Express18(18), 19039–19054 (2010).
    [CrossRef] [PubMed]
  11. P. Poggiolini, A. Carena, V. Curri, G. Bosco, and F. Forghieri, “Analytical modeling of non-linear propagation in uncompensated optical transmission links,” IEEE Photon. Technol. Lett.23(11), 742–744 (2011).
    [CrossRef]
  12. G. Bosco, P. Poggiolini, A. Carena, V. Curri, and F. Forghieri, “Analytical results on channel capacity in uncompensated optical links with coherent detection,” Opt. Express19(26), B440–B451 (2011).
    [CrossRef]
  13. A. Carena, V. Curri, G. Bosco, P. Poggiolini, and F. Forghieri, “Modeling of the impact of nonlinear propagation effects in uncompensated optical coherent transmission links,” J. Lightwave Technol.30(10), 1524–1539 (2012).
    [CrossRef]
  14. T. Tanimura, M. Nölle, J. K. Fischer, and C. Schubert, “Analytical results on back propagation nonlinear compensator in coherent transmission systems,” in Tech. Digest of European Conference on Optical Communication,2012, P3.11.
  15. E. Torrengo, R. Cigliutti, G. Bosco, A. Carena, V. Curri, P. Poggiolini, A. Nespola, D. Zeolla, and F. Forghieri, “Experimental validation of an analytical model for nonlinear propagation in uncompensated optical links,” in Tech. Digest of European Conference on Optical Communication,2011, paper We.7.B.2.

2012 (1)

2011 (3)

2010 (3)

Bayvel, P.

D. S. Millar, S. Makovejs, C. Behrens, S. Hellerbrand, R. I. Killey, P. Bayvel, and S. J. Savory, “Mitigation of fiber nonlinearity using a digital coherent receiver,” IEEE J. Sel. Top. Quantum Electron.16(5), 1217–1226 (2010).
[CrossRef]

Behrens, C.

D. S. Millar, S. Makovejs, C. Behrens, S. Hellerbrand, R. I. Killey, P. Bayvel, and S. J. Savory, “Mitigation of fiber nonlinearity using a digital coherent receiver,” IEEE J. Sel. Top. Quantum Electron.16(5), 1217–1226 (2010).
[CrossRef]

Bosco, G.

Carena, A.

Chen, X.

Curri, V.

Ellis, A. D.

Forghieri, F.

Hellerbrand, S.

D. S. Millar, S. Makovejs, C. Behrens, S. Hellerbrand, R. I. Killey, P. Bayvel, and S. J. Savory, “Mitigation of fiber nonlinearity using a digital coherent receiver,” IEEE J. Sel. Top. Quantum Electron.16(5), 1217–1226 (2010).
[CrossRef]

Ip, E.

Killey, R. I.

D. S. Millar, S. Makovejs, C. Behrens, S. Hellerbrand, R. I. Killey, P. Bayvel, and S. J. Savory, “Mitigation of fiber nonlinearity using a digital coherent receiver,” IEEE J. Sel. Top. Quantum Electron.16(5), 1217–1226 (2010).
[CrossRef]

Makovejs, S.

D. S. Millar, S. Makovejs, C. Behrens, S. Hellerbrand, R. I. Killey, P. Bayvel, and S. J. Savory, “Mitigation of fiber nonlinearity using a digital coherent receiver,” IEEE J. Sel. Top. Quantum Electron.16(5), 1217–1226 (2010).
[CrossRef]

Millar, D. S.

D. S. Millar, S. Makovejs, C. Behrens, S. Hellerbrand, R. I. Killey, P. Bayvel, and S. J. Savory, “Mitigation of fiber nonlinearity using a digital coherent receiver,” IEEE J. Sel. Top. Quantum Electron.16(5), 1217–1226 (2010).
[CrossRef]

Poggiolini, P.

Rafique, D.

Savory, S. J.

D. S. Millar, S. Makovejs, C. Behrens, S. Hellerbrand, R. I. Killey, P. Bayvel, and S. J. Savory, “Mitigation of fiber nonlinearity using a digital coherent receiver,” IEEE J. Sel. Top. Quantum Electron.16(5), 1217–1226 (2010).
[CrossRef]

Shieh, W.

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

D. S. Millar, S. Makovejs, C. Behrens, S. Hellerbrand, R. I. Killey, P. Bayvel, and S. J. Savory, “Mitigation of fiber nonlinearity using a digital coherent receiver,” IEEE J. Sel. Top. Quantum Electron.16(5), 1217–1226 (2010).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

P. Poggiolini, A. Carena, V. Curri, G. Bosco, and F. Forghieri, “Analytical modeling of non-linear propagation in uncompensated optical transmission links,” IEEE Photon. Technol. Lett.23(11), 742–744 (2011).
[CrossRef]

J. Lightwave Technol. (2)

Opt. Express (3)

Other (8)

T. Tanimura, M. Nölle, J. K. Fischer, and C. Schubert, “Analytical results on back propagation nonlinear compensator in coherent transmission systems,” in Tech. Digest of European Conference on Optical Communication,2012, P3.11.

E. Torrengo, R. Cigliutti, G. Bosco, A. Carena, V. Curri, P. Poggiolini, A. Nespola, D. Zeolla, and F. Forghieri, “Experimental validation of an analytical model for nonlinear propagation in uncompensated optical links,” in Tech. Digest of European Conference on Optical Communication,2011, paper We.7.B.2.

T. Tanimura, T. Hoshida, S. Oda, T. Tanaka, C. Ohsima, and J. C. Rasmussen, “Systematic analysis on multi-segment dual-polarisation nonlinear compensation in 112 Gb/s DP-QPSK coherent receiver,” in Tech. Digest of European Conference on Optical Communication,2009, paper 9.4.5.

S. Oda, T. Tanimura, T. Hoshida, C. Ohshima, H. Nakashima, Z. Tao, and J. C. Rasmussen, “112 Gb/s DP-QPSK transmission using a novel nonlinear compensator in digital coherent receiver,” in Tech. Digest of Optical Fiber Communication Conference,2009, paper OThR6.

L. Beygi, E. Agrell, P. Johannisson, M. Karlsson, and H. Wymeersch, “The limits of digital backpropagation in nonlinear coherent fiber-optic links,” in Tech. Digest of European Conference on Optical Communication,2012, P4.14.

G. Gao, X. Chen, and W. Shieh, “Limitation of fiber nonlinearity compensation using digital back propagation in the presence of PMD,” in Tech. Digest of Optical Fiber Communication Conference,2012, paper OM3A.5.

T. Tanimura, S. Oda, T. Hoshida, L. Li, Z. Tao, and J. C. Rasmussen, “Experimental characterization of nonlinearity mitigation by digital back propagation and nonlinear polarization crosstalk canceller under high PMD condition,” in Tech. Digest of Optical Fiber Communication Conference,2011, JWA020.

A. Splett, C. Kurtzke, and K. Petermann, “Ultimate transmission capacity of amplified optical fiber communication systems taking into account fiber nonlinearities,” in Tech. Digest of European Conference on Optical Communication,1993, paper MoC2.4.

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Figures (5)

Fig. 1
Fig. 1

Block diagram of the optical transmission system under study.

Fig. 2
Fig. 2

(a) Simulation setup, MUX: multiplexer, EDFA: erbium doped fiber amplifiers, SMF: single-mode fiber, BPF: band-pass filter, EQ: equalizer, FOC: frequency offset compensator, CPR: carrier phase recovery, (b) bandwidth used for NLC in a WDM system.

Fig. 3
Fig. 3

Q-factor after transmission as a function of the fiber launched power. NLC using only a single channel (BNLC = 35 GHz).

Fig. 4
Fig. 4

Net Q-gain brought by NLC as a function of the bandwidth used for NLC with BWDM = 11 × 37.5GHz.

Fig. 5
Fig. 5

(a) Upper boundary of OSNR improvement brought by NLC as a function of the bandwidth used for NLC with the systems consisting of SMF (α = 0.2 dB/km, n2 = 2.7 × 10−20 m2/W, Aeff = 80 μm2, D = 17 ps/nm/km), NZ-DSF (α = 0.2 dB/km, n2 = 2.5 × 10−20 m2/W, Aeff = 55 μm2, D = 3.5 ps/nm/km), and ULAF (α = 0.16 dB/km, n2 = 2.2 × 10−20 m2/W, Aeff = 133 μm2, D = 21 ps/nm/km). (b) OSNR improvement brought by NLC vs. bandwidth used for NLC for systems consisting of 80-km SMF per span, Rs = 28 GHz, Δf = 37.5 GHz, NF = 5 dB, BWDM = 4 THz. The results are shown for different transmission length (number of spans).

Equations (9)

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EffectiveOSNR= P Tx,ch P ASE + P NL
P ASE = N s F( A span 1 )hν B n ,
P NL = P NL,sigsig + P NL,sigASE .
P NL,sigsig,w/oNLC = N s γ 2 πα| β 2 | R s Δ f 2 P Tx,ch 3 B n B o /2 B WDM /2 (1/f)df = N s γ 2 πα| β 2 | R s Δ f 2 ln( B WDM B o ) P Tx,ch 3 B n
P NL,sigASE = (1+ N s ) γ 2 2πα| β 2 | R s Δf ln( B WDM ) P Tx,ch 2 P ASE .
P NL,sigsig,w/NLC = N s γ 2 πα| β 2 | R s Δ f 2 ln( B WDM B NLC ) P Tx,ch 3 B n ,
OSN R max,w/oNLC = 2 3 P ASE 2/3 ( 2πα| β 2 |Δ f 2 R s γ 2 N s B n ln( B WDM / B o ) ) 1/3
OSN R max,w/NLC = 2 3 P ASE 2/3 ( 2πα| β 2 |Δ f 2 R s γ 2 N s B n ln( B WDM / B NLC ) ) 1/3 .
OSNRimprovementbyNLC= 10 3 log 10 ( ln( B WDM )ln( B o ) ln( B WDM )ln( B NLC ) ).

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