Abstract

We propose optical injection locking (OIL) to enable compensation of the inter-channel nonlinear phase noise, which is dominated by cross-phase modulation (XPM). In this paper, injection locking is used to create a local oscillator for a homodyne receiver from a residual carrier. The locking is fast enough to follow XPM-phase distortion, but slow enough to reject the signal bands, which are spaced slightly away from the pilot. The homodyne receiver thus partially cancels XPM, as it is common to the signals and the pilot. An experimental 7-channel WDM system gives 1-dB (0.7-dB) improvement in the peak Q of the center channel, for QPSK (16-QAM) modulated OFDM subcarriers, and increased the transmission reach by 320 km. The optimum performance was achieved at an injection ratio of −45 dB, with the injected power as low as −24.5 dBm.

© 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. P. P. Mitra and J. B. Stark, “Nonlinear limits to the information capacity of optical fibre communications,” Nature 411(6841), 1027–1030 (2001).
    [Crossref] [PubMed]
  2. E. Ip and J. M. Khan, “Compensation of dispersion and nonlinear impairments using digital backpropagation,” J. Lightwave Technol. 26(20), 3416–3425 (2008).
    [Crossref]
  3. E. Ip, “Nonlinear compensation using backpropagation for polarization-multiplexed transmission,” J. Lightwave Technol. 28(6), 939–951 (2010).
    [Crossref]
  4. R. Dar and P. J. Winzer, “On the limits of digital back-propagation in fully loaded WDM systems,” IEEE Photonics Technol. Lett. 28(11), 1253–1256 (2016).
    [Crossref]
  5. E. Temprana, E. Myslivets, V. Ataie, B. P.-P. Kuo, N. Alic, V. Vusirikala, V. Dangui, and S. Radic, “Demonstration of coherent transmission reach tripling by frequency-referenced nonlinearity pre-compensation in EDFA-only SMF link,” in 42nd European Conf. on Optical Communication, (ECOC), Dusseldorf, Germany, 2016, pp. 1–3.
  6. E. F. Mateo, F. Yaman, and G. Li, “Efficient compensation of inter-channel nonlinear effects via digital backward propagation in WDM optical transmission,” Opt. Express 18(14), 15144–15154 (2010).
    [Crossref] [PubMed]
  7. L. Zhu, F. Yaman, and G. Li, “Experimental demonstration of XPM compensation for WDM fibre transmission,” Electron. Lett. 46(16), 1140–1141 (2010).
    [Crossref]
  8. R. Dar, M. Feder, A. Mecozzi, and M. Shtaif, “Accumulation of nonlinear interference noise in fiber-optic systems,” Opt. Express 22(12), 14199–14211 (2014).
    [Crossref] [PubMed]
  9. X. Chen and W. Shieh, “Closed-form expressions for nonlinear transmission performance of densely spaced coherent optical OFDM systems,” Opt. Express 18(18), 19039–19054 (2010).
    [Crossref] [PubMed]
  10. T. K. Chiang, N. Kagi, M. E. Marhic, and L. G. Kazovsky, “Cross-phase modulation in fiber links with multiple optical amplifiers and dispersion compensators,” J. Lightwave Technol. 14(3), 249–260 (1996).
    [Crossref]
  11. R. Dar, M. Feder, A. Mecozzi, and M. Shtaif, “Properties of nonlinear noise in long, dispersion-uncompensated fiber links,” Opt. Express 21(22), 25685–25699 (2013).
    [Crossref] [PubMed]
  12. B. Foo, B. Corcoran, and A. Lowery, “Optoelectronic method for inline compensation of XPM in long-haul optical links,” Opt. Express 23(2), 859–872 (2015).
    [Crossref] [PubMed]
  13. Z. Tao, W. Yan, S. Oda, T. Hoshida, and J. C. Rasmussen, “A simplified model for nonlinear cross-phase modulation in hybrid optical coherent system,” Opt. Express 17(16), 13860–13868 (2009).
    [Crossref] [PubMed]
  14. M. E. McCarthy, M. A. Z. Al Kahteeb, F. M. Ferreira, and A. D. Ellis, “PMD tolerant nonlinear compensation using in-line phase conjugation,” Opt. Express 24(4), 3385–3392 (2016).
    [Crossref] [PubMed]
  15. S. L. I. Olsson, B. Corcoran, C. Lundström, T. A. Eriksson, M. Karlsson, and P. A. Andrekson, “Phase-sensitive amplified transmission links for improved sensitivity and nonlinearity tolerance,” J. Lightwave Technol. 33(3), 710–721 (2015).
    [Crossref]
  16. L. B. Y. Du and A. J. Lowery, “Pilot-based XPM nonlinearity compensator for CO-OFDM systems,” Opt. Express 19(26), B862–B867 (2011).
    [Crossref] [PubMed]
  17. B. Inan, S. Randel, S. L. Jansen, A. Lobato, S. Adhikari, and N. Hanik, “Pilot-tone-based nonlinearity compensation for optical OFDM systems,” 36th European Conf. and Exhibition on Optical Communication (ECOC), Torino, 2010, paper Tu.4.A.6.
    [Crossref]
  18. A. Lobato, B. Inan, S. Adhikari, and S. L. Jansen, “On the efficiency of RF-Pilot-based nonlinearity compensation for CO-OFDM,” 2011 Optical Fiber Communication Conf. (OFC), Los Angeles, CA, 2011, paper OThF2.
  19. Z. Liu, J. Y. Kim, D. S. Wu, D. J. Richardson, and R. Slavík, “Homodyne OFDM with optical injection locking for carrier recovery,” J. Lightwave Technol. 33(1), 34–41 (2015).
    [Crossref]
  20. O. Lidoyne, P. Gallion, and D. Erasme, “Analysis of a homodyne receiver using an injection locked semiconductor laser,” J. Lightwave Technol. 9(5), 659–665 (1991).
    [Crossref]
  21. J. Jignesh, B. Corcoran, J. Schröder, and A. Lowery, “Polarization independent optical injection locking for carrier recovery in optical communication systems,” Opt. Express 25(18), 21216–21228 (2017).
    [Crossref] [PubMed]
  22. A. Fragkos, A. Bogris, D. Syvridis, and R. Phelan, “Amplitude noise limiting amplifier for phase encoded signals using injection locking in semiconductor lasers,” J. Lightwave Technol. 30(5), 764–771 (2012).
    [Crossref]
  23. O. Lidoyne, P. Gallion, C. Chabran, and G. Debarge, “Locking range, phase noise and power spectrum of an injection-locked semiconductor laser,” IEE Proc., Optoelectron. 137(3), 147–154 (1990).
    [Crossref]
  24. R. A. Shafik, Md. S. Rahman, A. H. M. R. Islam, “On the extended relationships among EVM, BER and SNR as performance metrics,” in Proc. of International Conf. on Electrical and Computer Engineering (2006), Dhaka, Bangladesh, pp. 408–411.
  25. D. Chang, F. Yu, Z. Xiao, N. Stojanovic, F. N. Hauske, Y. Cai, C. Xie, L. Li, X. Xu, and Q. Xiong, “LDPC convolutional codes using layered decoding algorithm for high speed coherent optical transmission,” in Proc. of Optical Fiber Communication Conference (OFC), (2012), paper OW1H.4.
    [Crossref]
  26. R. Rios-Müller, J. Renaudier, P. Brindel, A. Ghazisaeidi, I. Fernandez, P. Tran, C. Simonneau, L. Schmalen, and G. Charlet, “Spectrally-efficient 400-Gb/s single carrier transport over 7200 km,” J. Lightwave Technol. 33(7), 1402–1407 (2015).
    [Crossref]
  27. G. Raybon, A. Adamiecki, P. J. Winzer, C. Xie, A. Konczykowska, F. Jorge, J.-Y. Dupuy, L. L. Buhl, S. Chandrashekhar, A. S. Draving, M. Grove, K. Rush, B. Zhu, and D. W. Peckham, “All-ETDM 107-Gbaud PDM-16QAM (856-Gb/s) transmitter and coherent receiver,” in Proc. of European Conference and Exhibition on Optical Communication (ECOC), (2013), London, paper PD2.D.3.

2017 (1)

2016 (2)

M. E. McCarthy, M. A. Z. Al Kahteeb, F. M. Ferreira, and A. D. Ellis, “PMD tolerant nonlinear compensation using in-line phase conjugation,” Opt. Express 24(4), 3385–3392 (2016).
[Crossref] [PubMed]

R. Dar and P. J. Winzer, “On the limits of digital back-propagation in fully loaded WDM systems,” IEEE Photonics Technol. Lett. 28(11), 1253–1256 (2016).
[Crossref]

2015 (4)

2014 (1)

2013 (1)

2012 (1)

2011 (1)

2010 (4)

2009 (1)

2008 (1)

2001 (1)

P. P. Mitra and J. B. Stark, “Nonlinear limits to the information capacity of optical fibre communications,” Nature 411(6841), 1027–1030 (2001).
[Crossref] [PubMed]

1996 (1)

T. K. Chiang, N. Kagi, M. E. Marhic, and L. G. Kazovsky, “Cross-phase modulation in fiber links with multiple optical amplifiers and dispersion compensators,” J. Lightwave Technol. 14(3), 249–260 (1996).
[Crossref]

1991 (1)

O. Lidoyne, P. Gallion, and D. Erasme, “Analysis of a homodyne receiver using an injection locked semiconductor laser,” J. Lightwave Technol. 9(5), 659–665 (1991).
[Crossref]

1990 (1)

O. Lidoyne, P. Gallion, C. Chabran, and G. Debarge, “Locking range, phase noise and power spectrum of an injection-locked semiconductor laser,” IEE Proc., Optoelectron. 137(3), 147–154 (1990).
[Crossref]

Al Kahteeb, M. A. Z.

Andrekson, P. A.

Bogris, A.

Brindel, P.

Chabran, C.

O. Lidoyne, P. Gallion, C. Chabran, and G. Debarge, “Locking range, phase noise and power spectrum of an injection-locked semiconductor laser,” IEE Proc., Optoelectron. 137(3), 147–154 (1990).
[Crossref]

Charlet, G.

Chen, X.

Chiang, T. K.

T. K. Chiang, N. Kagi, M. E. Marhic, and L. G. Kazovsky, “Cross-phase modulation in fiber links with multiple optical amplifiers and dispersion compensators,” J. Lightwave Technol. 14(3), 249–260 (1996).
[Crossref]

Corcoran, B.

Dar, R.

Debarge, G.

O. Lidoyne, P. Gallion, C. Chabran, and G. Debarge, “Locking range, phase noise and power spectrum of an injection-locked semiconductor laser,” IEE Proc., Optoelectron. 137(3), 147–154 (1990).
[Crossref]

Du, L. B. Y.

Ellis, A. D.

Erasme, D.

O. Lidoyne, P. Gallion, and D. Erasme, “Analysis of a homodyne receiver using an injection locked semiconductor laser,” J. Lightwave Technol. 9(5), 659–665 (1991).
[Crossref]

Eriksson, T. A.

Feder, M.

Fernandez, I.

Ferreira, F. M.

Foo, B.

Fragkos, A.

Gallion, P.

O. Lidoyne, P. Gallion, and D. Erasme, “Analysis of a homodyne receiver using an injection locked semiconductor laser,” J. Lightwave Technol. 9(5), 659–665 (1991).
[Crossref]

O. Lidoyne, P. Gallion, C. Chabran, and G. Debarge, “Locking range, phase noise and power spectrum of an injection-locked semiconductor laser,” IEE Proc., Optoelectron. 137(3), 147–154 (1990).
[Crossref]

Ghazisaeidi, A.

Hoshida, T.

Ip, E.

Jignesh, J.

Kagi, N.

T. K. Chiang, N. Kagi, M. E. Marhic, and L. G. Kazovsky, “Cross-phase modulation in fiber links with multiple optical amplifiers and dispersion compensators,” J. Lightwave Technol. 14(3), 249–260 (1996).
[Crossref]

Karlsson, M.

Kazovsky, L. G.

T. K. Chiang, N. Kagi, M. E. Marhic, and L. G. Kazovsky, “Cross-phase modulation in fiber links with multiple optical amplifiers and dispersion compensators,” J. Lightwave Technol. 14(3), 249–260 (1996).
[Crossref]

Khan, J. M.

Kim, J. Y.

Li, G.

L. Zhu, F. Yaman, and G. Li, “Experimental demonstration of XPM compensation for WDM fibre transmission,” Electron. Lett. 46(16), 1140–1141 (2010).
[Crossref]

E. F. Mateo, F. Yaman, and G. Li, “Efficient compensation of inter-channel nonlinear effects via digital backward propagation in WDM optical transmission,” Opt. Express 18(14), 15144–15154 (2010).
[Crossref] [PubMed]

Lidoyne, O.

O. Lidoyne, P. Gallion, and D. Erasme, “Analysis of a homodyne receiver using an injection locked semiconductor laser,” J. Lightwave Technol. 9(5), 659–665 (1991).
[Crossref]

O. Lidoyne, P. Gallion, C. Chabran, and G. Debarge, “Locking range, phase noise and power spectrum of an injection-locked semiconductor laser,” IEE Proc., Optoelectron. 137(3), 147–154 (1990).
[Crossref]

Liu, Z.

Lowery, A.

Lowery, A. J.

Lundström, C.

Marhic, M. E.

T. K. Chiang, N. Kagi, M. E. Marhic, and L. G. Kazovsky, “Cross-phase modulation in fiber links with multiple optical amplifiers and dispersion compensators,” J. Lightwave Technol. 14(3), 249–260 (1996).
[Crossref]

Mateo, E. F.

McCarthy, M. E.

Mecozzi, A.

Mitra, P. P.

P. P. Mitra and J. B. Stark, “Nonlinear limits to the information capacity of optical fibre communications,” Nature 411(6841), 1027–1030 (2001).
[Crossref] [PubMed]

Oda, S.

Olsson, S. L. I.

Phelan, R.

Rasmussen, J. C.

Renaudier, J.

Richardson, D. J.

Rios-Müller, R.

Schmalen, L.

Schröder, J.

Shieh, W.

Shtaif, M.

Simonneau, C.

Slavík, R.

Stark, J. B.

P. P. Mitra and J. B. Stark, “Nonlinear limits to the information capacity of optical fibre communications,” Nature 411(6841), 1027–1030 (2001).
[Crossref] [PubMed]

Syvridis, D.

Tao, Z.

Tran, P.

Winzer, P. J.

R. Dar and P. J. Winzer, “On the limits of digital back-propagation in fully loaded WDM systems,” IEEE Photonics Technol. Lett. 28(11), 1253–1256 (2016).
[Crossref]

Wu, D. S.

Yaman, F.

L. Zhu, F. Yaman, and G. Li, “Experimental demonstration of XPM compensation for WDM fibre transmission,” Electron. Lett. 46(16), 1140–1141 (2010).
[Crossref]

E. F. Mateo, F. Yaman, and G. Li, “Efficient compensation of inter-channel nonlinear effects via digital backward propagation in WDM optical transmission,” Opt. Express 18(14), 15144–15154 (2010).
[Crossref] [PubMed]

Yan, W.

Zhu, L.

L. Zhu, F. Yaman, and G. Li, “Experimental demonstration of XPM compensation for WDM fibre transmission,” Electron. Lett. 46(16), 1140–1141 (2010).
[Crossref]

Electron. Lett. (1)

L. Zhu, F. Yaman, and G. Li, “Experimental demonstration of XPM compensation for WDM fibre transmission,” Electron. Lett. 46(16), 1140–1141 (2010).
[Crossref]

IEE Proc., Optoelectron. (1)

O. Lidoyne, P. Gallion, C. Chabran, and G. Debarge, “Locking range, phase noise and power spectrum of an injection-locked semiconductor laser,” IEE Proc., Optoelectron. 137(3), 147–154 (1990).
[Crossref]

IEEE Photonics Technol. Lett. (1)

R. Dar and P. J. Winzer, “On the limits of digital back-propagation in fully loaded WDM systems,” IEEE Photonics Technol. Lett. 28(11), 1253–1256 (2016).
[Crossref]

J. Lightwave Technol. (8)

E. Ip and J. M. Khan, “Compensation of dispersion and nonlinear impairments using digital backpropagation,” J. Lightwave Technol. 26(20), 3416–3425 (2008).
[Crossref]

E. Ip, “Nonlinear compensation using backpropagation for polarization-multiplexed transmission,” J. Lightwave Technol. 28(6), 939–951 (2010).
[Crossref]

T. K. Chiang, N. Kagi, M. E. Marhic, and L. G. Kazovsky, “Cross-phase modulation in fiber links with multiple optical amplifiers and dispersion compensators,” J. Lightwave Technol. 14(3), 249–260 (1996).
[Crossref]

S. L. I. Olsson, B. Corcoran, C. Lundström, T. A. Eriksson, M. Karlsson, and P. A. Andrekson, “Phase-sensitive amplified transmission links for improved sensitivity and nonlinearity tolerance,” J. Lightwave Technol. 33(3), 710–721 (2015).
[Crossref]

Z. Liu, J. Y. Kim, D. S. Wu, D. J. Richardson, and R. Slavík, “Homodyne OFDM with optical injection locking for carrier recovery,” J. Lightwave Technol. 33(1), 34–41 (2015).
[Crossref]

O. Lidoyne, P. Gallion, and D. Erasme, “Analysis of a homodyne receiver using an injection locked semiconductor laser,” J. Lightwave Technol. 9(5), 659–665 (1991).
[Crossref]

R. Rios-Müller, J. Renaudier, P. Brindel, A. Ghazisaeidi, I. Fernandez, P. Tran, C. Simonneau, L. Schmalen, and G. Charlet, “Spectrally-efficient 400-Gb/s single carrier transport over 7200 km,” J. Lightwave Technol. 33(7), 1402–1407 (2015).
[Crossref]

A. Fragkos, A. Bogris, D. Syvridis, and R. Phelan, “Amplitude noise limiting amplifier for phase encoded signals using injection locking in semiconductor lasers,” J. Lightwave Technol. 30(5), 764–771 (2012).
[Crossref]

Nature (1)

P. P. Mitra and J. B. Stark, “Nonlinear limits to the information capacity of optical fibre communications,” Nature 411(6841), 1027–1030 (2001).
[Crossref] [PubMed]

Opt. Express (9)

E. F. Mateo, F. Yaman, and G. Li, “Efficient compensation of inter-channel nonlinear effects via digital backward propagation in WDM optical transmission,” Opt. Express 18(14), 15144–15154 (2010).
[Crossref] [PubMed]

R. Dar, M. Feder, A. Mecozzi, and M. Shtaif, “Accumulation of nonlinear interference noise in fiber-optic systems,” Opt. Express 22(12), 14199–14211 (2014).
[Crossref] [PubMed]

X. Chen and W. Shieh, “Closed-form expressions for nonlinear transmission performance of densely spaced coherent optical OFDM systems,” Opt. Express 18(18), 19039–19054 (2010).
[Crossref] [PubMed]

L. B. Y. Du and A. J. Lowery, “Pilot-based XPM nonlinearity compensator for CO-OFDM systems,” Opt. Express 19(26), B862–B867 (2011).
[Crossref] [PubMed]

R. Dar, M. Feder, A. Mecozzi, and M. Shtaif, “Properties of nonlinear noise in long, dispersion-uncompensated fiber links,” Opt. Express 21(22), 25685–25699 (2013).
[Crossref] [PubMed]

B. Foo, B. Corcoran, and A. Lowery, “Optoelectronic method for inline compensation of XPM in long-haul optical links,” Opt. Express 23(2), 859–872 (2015).
[Crossref] [PubMed]

Z. Tao, W. Yan, S. Oda, T. Hoshida, and J. C. Rasmussen, “A simplified model for nonlinear cross-phase modulation in hybrid optical coherent system,” Opt. Express 17(16), 13860–13868 (2009).
[Crossref] [PubMed]

M. E. McCarthy, M. A. Z. Al Kahteeb, F. M. Ferreira, and A. D. Ellis, “PMD tolerant nonlinear compensation using in-line phase conjugation,” Opt. Express 24(4), 3385–3392 (2016).
[Crossref] [PubMed]

J. Jignesh, B. Corcoran, J. Schröder, and A. Lowery, “Polarization independent optical injection locking for carrier recovery in optical communication systems,” Opt. Express 25(18), 21216–21228 (2017).
[Crossref] [PubMed]

Other (6)

R. A. Shafik, Md. S. Rahman, A. H. M. R. Islam, “On the extended relationships among EVM, BER and SNR as performance metrics,” in Proc. of International Conf. on Electrical and Computer Engineering (2006), Dhaka, Bangladesh, pp. 408–411.

D. Chang, F. Yu, Z. Xiao, N. Stojanovic, F. N. Hauske, Y. Cai, C. Xie, L. Li, X. Xu, and Q. Xiong, “LDPC convolutional codes using layered decoding algorithm for high speed coherent optical transmission,” in Proc. of Optical Fiber Communication Conference (OFC), (2012), paper OW1H.4.
[Crossref]

G. Raybon, A. Adamiecki, P. J. Winzer, C. Xie, A. Konczykowska, F. Jorge, J.-Y. Dupuy, L. L. Buhl, S. Chandrashekhar, A. S. Draving, M. Grove, K. Rush, B. Zhu, and D. W. Peckham, “All-ETDM 107-Gbaud PDM-16QAM (856-Gb/s) transmitter and coherent receiver,” in Proc. of European Conference and Exhibition on Optical Communication (ECOC), (2013), London, paper PD2.D.3.

B. Inan, S. Randel, S. L. Jansen, A. Lobato, S. Adhikari, and N. Hanik, “Pilot-tone-based nonlinearity compensation for optical OFDM systems,” 36th European Conf. and Exhibition on Optical Communication (ECOC), Torino, 2010, paper Tu.4.A.6.
[Crossref]

A. Lobato, B. Inan, S. Adhikari, and S. L. Jansen, “On the efficiency of RF-Pilot-based nonlinearity compensation for CO-OFDM,” 2011 Optical Fiber Communication Conf. (OFC), Los Angeles, CA, 2011, paper OThF2.

E. Temprana, E. Myslivets, V. Ataie, B. P.-P. Kuo, N. Alic, V. Vusirikala, V. Dangui, and S. Radic, “Demonstration of coherent transmission reach tripling by frequency-referenced nonlinearity pre-compensation in EDFA-only SMF link,” in 42nd European Conf. on Optical Communication, (ECOC), Dusseldorf, Germany, 2016, pp. 1–3.

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Figures (9)

Fig. 1
Fig. 1 a) Scaled XPM efficiency (dB) vs. frequency (GHz), b) OIL-based nonlinear phase noise compensation.
Fig. 2
Fig. 2 Optical injection locking setup with polarization locking module used in this experiment. PBS- polarization beam splitter; PC- polarization controller, SG: signal generator, VOA: variable optical attenuator, ADC: analog-digital convertor, DSP: digital signal processing.
Fig. 3
Fig. 3 Spectral effects of OIL’s phase-transfer bandwidth on received signals when a) BOIL< BXPM and b) BOIL> Bguard.
Fig. 4
Fig. 4 Phase transfer function of the optical injection locking setup; Pinj: Injected power (dBm). Inset: 10-Gbaud modulated signal spectrum. Pinj of −28, −24.5 and −20 dBm correspond to injection ratios of −48, −45 and −40 dB, respectively.
Fig. 5
Fig. 5 Experimental setup: AWG- arbitrary waveform generator; DCM- dispersion compensation module; EDFA- erbium doped fiber amplifier; AOM- acousto-optic modulator; WSS- wavelength selective switch (WaveShaper).
Fig. 6
Fig. 6 Receiver side DSP flow implemented for experiments. FFT- fast Fourier transform; Ix, Qx, Iy, Qy- electrical waveforms from coherent receiver.
Fig. 7
Fig. 7 Q vs. injected power for various launch-powers and link lengths: a) 320 km, b) 1920 km, c) 2560 km, d) 2880 km.
Fig. 8
Fig. 8 Q vs. launch power with QPSK modulation. The injection ratio for the OIL was fixed at −45 dB, i.e. the optimum value shown in Fig. 7. The link distance was also varied to observe the maximum reach that can be achieved while the peak Q is greater than 7% hard FEC limit (Q = 8.5 dB, BER = 3.8 × 10−3). Figure 8 shows the performance plots for QPSK modulation. Similar to our observations in Fig. 5, the performance reaches the maximum at PL = 3 dBm and then drops when increased further in to the nonlinear region (PL > 3 dBm).
Fig. 9
Fig. 9 Q vs. launch-power with 16-QAM modulation.

Tables (1)

Tables Icon

Table 1 Peak-Q-performance gains for QPSK and 16-QAM modulated signals.

Equations (2)

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

η XPM (ω)= α 2 (ωΔβ) 2 + α 2 [ 1+ 4 sin 2 ( ωΔβL 2 ) e αL (1 e αL ) 2 ]| sin( NωΔβL 2 ) sin( ωΔβL 2 ) |.
Q BER (dB)=20 log 10 ( 2(M1) 3 × erfc 1 ( BER× log 2 M ( 1 1 M ) ) ).

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