Abstract

We study and experimentally validate the vector theory of four-wave mixing (FWM) in semiconductor optical amplifiers (SOA). We use the vector theory of FWM to design a polarization insensitive all-optical wavelength converter, suitable for advanced modulation formats, using non-degenerate FWM in SOAs and parallelly polarized pumps. We demonstrate the wavelength conversion of polarization-multiplexed (PM)-QPSK, PM-16QAM and a Nyquist WDM super-channel modulated with PM-QPSK signals at a baud rate of 12.5 GBaud, with total data rates of 50 Gbps, 100 Gbps and 200 Gbps respectively.

© 2016 Optical Society of America

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References

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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  25. B. Filion, An. T. Nguyen, L. A. Rusch, and S. LaRochelle, “Digital post-compensation of nonlinear distortions in wavelength conversion based on four-wave mixing in a semiconductor optical amplifier,” J. Lightwave Technol. 33, 3254–3264 (2015).
    [Crossref]
  26. J. Gong, J. Xu, M. Luo, X. Li, Y. Qiu, Q. Yang, X. Zhang, and S. Yu, “All-optical wavelength conversion for mode division multiplexed superchannels,” Opt. Express 24, 8926–8939 (2016).
    [Crossref] [PubMed]

2016 (2)

2015 (6)

B. Filion, An. T. Nguyen, L. A. Rusch, and S. LaRochelle, “Digital post-compensation of nonlinear distortions in wavelength conversion based on four-wave mixing in a semiconductor optical amplifier,” J. Lightwave Technol. 33, 3254–3264 (2015).
[Crossref]

P. J. Winzer, “Scaling optical fiber networks: challenges and solutions,” Opt. Photon. News 26(3), 28–35 (2015).
[Crossref]

A. P. Anthur, R. T. Watts, S. ODuill, R. Zhou, D. Venkitesh, and L. P. Barry, “Impact of nonlinear phase noise on all-optical wavelength conversion of 10.7-GBaud QPSK data using dual correlated pumps,” IEEE J. Quantum Electron. 51, 9100105 (2015).

S. T. Naimi, S. O’Duill, and L. P. Barry, “All optical wavelength conversion of Nyquist-WDM superchannels using FWM in SOAs,” J. Lightwave Technol. 33, 3959–3967 (2015).
[Crossref]

C. Li, M. Luo, Z. He, H. Li, J. Xu, S. You, Q. Yang, and S. Yu, “Phase noise canceled polarization-insensitive all-optical wavelength conversion of 557-Gb/s PDM-OFDM signal using coherent dual-pump,” J. Lightwave Technol. 33, 2848–2854 (2015).
[Crossref]

S. P. O Duill, P. M. Anandarajah, R. Zhou, and L. P. Barry, “Numerical investigation into the injection-locking phenomena of gain switched lasers for optical frequency comb generation,” Appl. Phys. Lett. 106, 211105 (2015).
[Crossref]

2014 (2)

A. P. Anthur, R. T. Watts, R. Zhou, P. Anandarajah, D. Venkitesh, and L. P. Barry, “Penalty-free wavelength conversion with variable channel separation using gain-switched comb source,” Opt. Commun. 324, 69–72 (2014).
[Crossref]

R. Zhou, T. N. Huynh, V. Vujicic, P. M. Anandarajah, and L. P. Barry, “Phase noise analysis of injected gain switched comb source for coherent communications,” Opt. Express 22, 8120–8125 (2014).
[Crossref] [PubMed]

2013 (3)

S. P. O’Duill, S. T. Naimi, A. P. Anthur, T. N. Huynh, D. Venkitesh, and L. P. Barry, “Simulations of an OSNR-limited all-optical wavelength conversion scheme,” IEEE Photon. Technol. Lett. 25, 2311–2314 (2013).
[Crossref]

H. Zhou, J. Yu, J. Tang, and L. Chen, “Polarization insensitive wavelength conversion for polarization multiplexing non-return-to-zero quadrature phase shift keying signals based on four-wave mixing in a semiconductor optical amplifier using digital coherent detection,” Opt. Eng. 52, 025001 (2013).
[Crossref]

A. P. Anthur, R. T. Watts, K. Shi, J. O Carroll, D. Venkitesh, and L. P. Barry, “Dual correlated pumping scheme for phase noise preservation in all-optical wavelength conversion,” Opt. Express 21, 15568–15579 (2013).
[Crossref] [PubMed]

2012 (2)

P. P. Baveja, D. N. Maywar, and G. P. Agrawal, “Interband four-wave mixing in semiconductor optical amplifiers with ASE-enhanced gain recovery,” IEEE J. Sel. Top. Quantum Electron. 8, 899–908 (2012).
[Crossref]

T. N. Huynh, F. Smyth, L. Nguyen, and L. P. Barry, “Effects of phase noise of monolithic tunable laser on coherent communication systems,” Opt. Express 20, B244–B249 (2012).
[Crossref] [PubMed]

2011 (1)

2009 (2)

Y. Mori, C. Zhang, K. Igarashi, K. Katoh, and K. Kikuchi, “Unrepeated 200-km transmission of 40-Gbit/s 16-QAM signals using digital coherent receiver”, Opt. Express 17, 1435–1441 (2009).
[Crossref] [PubMed]

J. Lu, Z. Dong, L. Chen, and J. Yu, “Polarization insensitive wavelength conversion based on four-wave mixing for polarization multiplexing signal in high-nonlinear fiber,” Opt. Commun. 282, 1274–1280 (2009).
[Crossref]

2004 (1)

2000 (1)

J. M. H. Elmirghani and H. T. Mouftah, “All-optical wavelength conversion: technologies and applications in DWDM networks,” IEEE Commun. Mag. 38(3), 86–92 (2000).
[Crossref]

1997 (1)

T. Ito, N. Yoshimoto, K. Magari, K. Kishi, and Y. Kondo, “Extremely low power consumption semiconductor optical amplier gate for WDM applications,” Electron. Lett. 33, 1791–1792 (1997).
[Crossref]

1994 (1)

K. Inoue, “Polarization independent wavelength conversion using fiber four-wave mixing with two orthogonal pump lights of different frequencies,” J. Lightwave Technol. 12, 1916–1920 (1994).
[Crossref]

1988 (1)

Agrawal, G. P.

Anandarajah, P.

A. P. Anthur, R. T. Watts, R. Zhou, P. Anandarajah, D. Venkitesh, and L. P. Barry, “Penalty-free wavelength conversion with variable channel separation using gain-switched comb source,” Opt. Commun. 324, 69–72 (2014).
[Crossref]

Anandarajah, P. M.

S. P. O Duill, P. M. Anandarajah, R. Zhou, and L. P. Barry, “Numerical investigation into the injection-locking phenomena of gain switched lasers for optical frequency comb generation,” Appl. Phys. Lett. 106, 211105 (2015).
[Crossref]

R. Zhou, T. N. Huynh, V. Vujicic, P. M. Anandarajah, and L. P. Barry, “Phase noise analysis of injected gain switched comb source for coherent communications,” Opt. Express 22, 8120–8125 (2014).
[Crossref] [PubMed]

Anthur, A. P.

A. P. Anthur, R. T. Watts, S. ODuill, R. Zhou, D. Venkitesh, and L. P. Barry, “Impact of nonlinear phase noise on all-optical wavelength conversion of 10.7-GBaud QPSK data using dual correlated pumps,” IEEE J. Quantum Electron. 51, 9100105 (2015).

A. P. Anthur, R. T. Watts, R. Zhou, P. Anandarajah, D. Venkitesh, and L. P. Barry, “Penalty-free wavelength conversion with variable channel separation using gain-switched comb source,” Opt. Commun. 324, 69–72 (2014).
[Crossref]

S. P. O’Duill, S. T. Naimi, A. P. Anthur, T. N. Huynh, D. Venkitesh, and L. P. Barry, “Simulations of an OSNR-limited all-optical wavelength conversion scheme,” IEEE Photon. Technol. Lett. 25, 2311–2314 (2013).
[Crossref]

A. P. Anthur, R. T. Watts, K. Shi, J. O Carroll, D. Venkitesh, and L. P. Barry, “Dual correlated pumping scheme for phase noise preservation in all-optical wavelength conversion,” Opt. Express 21, 15568–15579 (2013).
[Crossref] [PubMed]

Barry, L. P.

S. P. O Duill, P. M. Anandarajah, R. Zhou, and L. P. Barry, “Numerical investigation into the injection-locking phenomena of gain switched lasers for optical frequency comb generation,” Appl. Phys. Lett. 106, 211105 (2015).
[Crossref]

A. P. Anthur, R. T. Watts, S. ODuill, R. Zhou, D. Venkitesh, and L. P. Barry, “Impact of nonlinear phase noise on all-optical wavelength conversion of 10.7-GBaud QPSK data using dual correlated pumps,” IEEE J. Quantum Electron. 51, 9100105 (2015).

S. T. Naimi, S. O’Duill, and L. P. Barry, “All optical wavelength conversion of Nyquist-WDM superchannels using FWM in SOAs,” J. Lightwave Technol. 33, 3959–3967 (2015).
[Crossref]

R. Zhou, T. N. Huynh, V. Vujicic, P. M. Anandarajah, and L. P. Barry, “Phase noise analysis of injected gain switched comb source for coherent communications,” Opt. Express 22, 8120–8125 (2014).
[Crossref] [PubMed]

A. P. Anthur, R. T. Watts, R. Zhou, P. Anandarajah, D. Venkitesh, and L. P. Barry, “Penalty-free wavelength conversion with variable channel separation using gain-switched comb source,” Opt. Commun. 324, 69–72 (2014).
[Crossref]

S. P. O’Duill, S. T. Naimi, A. P. Anthur, T. N. Huynh, D. Venkitesh, and L. P. Barry, “Simulations of an OSNR-limited all-optical wavelength conversion scheme,” IEEE Photon. Technol. Lett. 25, 2311–2314 (2013).
[Crossref]

A. P. Anthur, R. T. Watts, K. Shi, J. O Carroll, D. Venkitesh, and L. P. Barry, “Dual correlated pumping scheme for phase noise preservation in all-optical wavelength conversion,” Opt. Express 21, 15568–15579 (2013).
[Crossref] [PubMed]

T. N. Huynh, F. Smyth, L. Nguyen, and L. P. Barry, “Effects of phase noise of monolithic tunable laser on coherent communication systems,” Opt. Express 20, B244–B249 (2012).
[Crossref] [PubMed]

Baveja, P. P.

P. P. Baveja, D. N. Maywar, and G. P. Agrawal, “Interband four-wave mixing in semiconductor optical amplifiers with ASE-enhanced gain recovery,” IEEE J. Sel. Top. Quantum Electron. 8, 899–908 (2012).
[Crossref]

Bergman, K.

Bhopalwala, M.

Carroll, J. O

Chen, L.

H. Zhou, J. Yu, J. Tang, and L. Chen, “Polarization insensitive wavelength conversion for polarization multiplexing non-return-to-zero quadrature phase shift keying signals based on four-wave mixing in a semiconductor optical amplifier using digital coherent detection,” Opt. Eng. 52, 025001 (2013).
[Crossref]

J. Lu, Z. Dong, L. Chen, and J. Yu, “Polarization insensitive wavelength conversion based on four-wave mixing for polarization multiplexing signal in high-nonlinear fiber,” Opt. Commun. 282, 1274–1280 (2009).
[Crossref]

Chen, Y.

A. Leven, N. Kaneda, and Y. Chen, “A real-time CMA-based 10 Gb/s polarization demultiplexing coherent receiver implemented in an FPGA,” in Proceedings of OFC/NFOEC (2008), paper OTuO2.

Dong, Z.

J. Lu, Z. Dong, L. Chen, and J. Yu, “Polarization insensitive wavelength conversion based on four-wave mixing for polarization multiplexing signal in high-nonlinear fiber,” Opt. Commun. 282, 1274–1280 (2009).
[Crossref]

Duill, S. P. O

S. P. O Duill, P. M. Anandarajah, R. Zhou, and L. P. Barry, “Numerical investigation into the injection-locking phenomena of gain switched lasers for optical frequency comb generation,” Appl. Phys. Lett. 106, 211105 (2015).
[Crossref]

Elmirghani, J. M. H.

J. M. H. Elmirghani and H. T. Mouftah, “All-optical wavelength conversion: technologies and applications in DWDM networks,” IEEE Commun. Mag. 38(3), 86–92 (2000).
[Crossref]

Filion, B.

Gong, J.

He, Z.

C. Li, M. Luo, Z. He, H. Li, J. Xu, S. You, Q. Yang, and S. Yu, “Phase noise canceled polarization-insensitive all-optical wavelength conversion of 557-Gb/s PDM-OFDM signal using coherent dual-pump,” J. Lightwave Technol. 33, 2848–2854 (2015).
[Crossref]

Huynh, T. N.

Igarashi, K.

Inoue, K.

K. Inoue, “Polarization independent wavelength conversion using fiber four-wave mixing with two orthogonal pump lights of different frequencies,” J. Lightwave Technol. 12, 1916–1920 (1994).
[Crossref]

Ito, T.

T. Ito, N. Yoshimoto, K. Magari, K. Kishi, and Y. Kondo, “Extremely low power consumption semiconductor optical amplier gate for WDM applications,” Electron. Lett. 33, 1791–1792 (1997).
[Crossref]

Kaneda, N.

A. Leven, N. Kaneda, and Y. Chen, “A real-time CMA-based 10 Gb/s polarization demultiplexing coherent receiver implemented in an FPGA,” in Proceedings of OFC/NFOEC (2008), paper OTuO2.

Katoh, K.

Kikuchi, K.

Kilper, D. C.

Kishi, K.

T. Ito, N. Yoshimoto, K. Magari, K. Kishi, and Y. Kondo, “Extremely low power consumption semiconductor optical amplier gate for WDM applications,” Electron. Lett. 33, 1791–1792 (1997).
[Crossref]

Kondo, Y.

T. Ito, N. Yoshimoto, K. Magari, K. Kishi, and Y. Kondo, “Extremely low power consumption semiconductor optical amplier gate for WDM applications,” Electron. Lett. 33, 1791–1792 (1997).
[Crossref]

LaRochelle, S.

Leven, A.

A. Leven, N. Kaneda, and Y. Chen, “A real-time CMA-based 10 Gb/s polarization demultiplexing coherent receiver implemented in an FPGA,” in Proceedings of OFC/NFOEC (2008), paper OTuO2.

Li, C.

C. Li, M. Luo, Z. He, H. Li, J. Xu, S. You, Q. Yang, and S. Yu, “Phase noise canceled polarization-insensitive all-optical wavelength conversion of 557-Gb/s PDM-OFDM signal using coherent dual-pump,” J. Lightwave Technol. 33, 2848–2854 (2015).
[Crossref]

Li, H.

C. Li, M. Luo, Z. He, H. Li, J. Xu, S. You, Q. Yang, and S. Yu, “Phase noise canceled polarization-insensitive all-optical wavelength conversion of 557-Gb/s PDM-OFDM signal using coherent dual-pump,” J. Lightwave Technol. 33, 2848–2854 (2015).
[Crossref]

Li, X.

Lin, Q.

Lu, J.

J. Lu, Z. Dong, L. Chen, and J. Yu, “Polarization insensitive wavelength conversion based on four-wave mixing for polarization multiplexing signal in high-nonlinear fiber,” Opt. Commun. 282, 1274–1280 (2009).
[Crossref]

Luo, M.

J. Gong, J. Xu, M. Luo, X. Li, Y. Qiu, Q. Yang, X. Zhang, and S. Yu, “All-optical wavelength conversion for mode division multiplexed superchannels,” Opt. Express 24, 8926–8939 (2016).
[Crossref] [PubMed]

C. Li, M. Luo, Z. He, H. Li, J. Xu, S. You, Q. Yang, and S. Yu, “Phase noise canceled polarization-insensitive all-optical wavelength conversion of 557-Gb/s PDM-OFDM signal using coherent dual-pump,” J. Lightwave Technol. 33, 2848–2854 (2015).
[Crossref]

Magari, K.

T. Ito, N. Yoshimoto, K. Magari, K. Kishi, and Y. Kondo, “Extremely low power consumption semiconductor optical amplier gate for WDM applications,” Electron. Lett. 33, 1791–1792 (1997).
[Crossref]

Maywar, D. N.

P. P. Baveja, D. N. Maywar, and G. P. Agrawal, “Interband four-wave mixing in semiconductor optical amplifiers with ASE-enhanced gain recovery,” IEEE J. Sel. Top. Quantum Electron. 8, 899–908 (2012).
[Crossref]

Mori, Y.

Mouftah, H. T.

J. M. H. Elmirghani and H. T. Mouftah, “All-optical wavelength conversion: technologies and applications in DWDM networks,” IEEE Commun. Mag. 38(3), 86–92 (2000).
[Crossref]

Naimi, S. T.

S. T. Naimi, S. O’Duill, and L. P. Barry, “All optical wavelength conversion of Nyquist-WDM superchannels using FWM in SOAs,” J. Lightwave Technol. 33, 3959–3967 (2015).
[Crossref]

S. P. O’Duill, S. T. Naimi, A. P. Anthur, T. N. Huynh, D. Venkitesh, and L. P. Barry, “Simulations of an OSNR-limited all-optical wavelength conversion scheme,” IEEE Photon. Technol. Lett. 25, 2311–2314 (2013).
[Crossref]

Nguyen, An. T.

Nguyen, L.

O’Duill, S.

O’Duill, S. P.

S. P. O’Duill, S. T. Naimi, A. P. Anthur, T. N. Huynh, D. Venkitesh, and L. P. Barry, “Simulations of an OSNR-limited all-optical wavelength conversion scheme,” IEEE Photon. Technol. Lett. 25, 2311–2314 (2013).
[Crossref]

ODuill, S.

A. P. Anthur, R. T. Watts, S. ODuill, R. Zhou, D. Venkitesh, and L. P. Barry, “Impact of nonlinear phase noise on all-optical wavelength conversion of 10.7-GBaud QPSK data using dual correlated pumps,” IEEE J. Quantum Electron. 51, 9100105 (2015).

Qiu, Y.

Rastegarfar, H.

Rusch, L. A.

Shi, K.

Smyth, F.

Tang, J.

H. Zhou, J. Yu, J. Tang, and L. Chen, “Polarization insensitive wavelength conversion for polarization multiplexing non-return-to-zero quadrature phase shift keying signals based on four-wave mixing in a semiconductor optical amplifier using digital coherent detection,” Opt. Eng. 52, 025001 (2013).
[Crossref]

Venkitesh, D.

A. P. Anthur, R. T. Watts, S. ODuill, R. Zhou, D. Venkitesh, and L. P. Barry, “Impact of nonlinear phase noise on all-optical wavelength conversion of 10.7-GBaud QPSK data using dual correlated pumps,” IEEE J. Quantum Electron. 51, 9100105 (2015).

A. P. Anthur, R. T. Watts, R. Zhou, P. Anandarajah, D. Venkitesh, and L. P. Barry, “Penalty-free wavelength conversion with variable channel separation using gain-switched comb source,” Opt. Commun. 324, 69–72 (2014).
[Crossref]

S. P. O’Duill, S. T. Naimi, A. P. Anthur, T. N. Huynh, D. Venkitesh, and L. P. Barry, “Simulations of an OSNR-limited all-optical wavelength conversion scheme,” IEEE Photon. Technol. Lett. 25, 2311–2314 (2013).
[Crossref]

A. P. Anthur, R. T. Watts, K. Shi, J. O Carroll, D. Venkitesh, and L. P. Barry, “Dual correlated pumping scheme for phase noise preservation in all-optical wavelength conversion,” Opt. Express 21, 15568–15579 (2013).
[Crossref] [PubMed]

Vujicic, V.

Wang, M.

Watts, R. T.

A. P. Anthur, R. T. Watts, S. ODuill, R. Zhou, D. Venkitesh, and L. P. Barry, “Impact of nonlinear phase noise on all-optical wavelength conversion of 10.7-GBaud QPSK data using dual correlated pumps,” IEEE J. Quantum Electron. 51, 9100105 (2015).

A. P. Anthur, R. T. Watts, R. Zhou, P. Anandarajah, D. Venkitesh, and L. P. Barry, “Penalty-free wavelength conversion with variable channel separation using gain-switched comb source,” Opt. Commun. 324, 69–72 (2014).
[Crossref]

A. P. Anthur, R. T. Watts, K. Shi, J. O Carroll, D. Venkitesh, and L. P. Barry, “Dual correlated pumping scheme for phase noise preservation in all-optical wavelength conversion,” Opt. Express 21, 15568–15579 (2013).
[Crossref] [PubMed]

Winzer, P. J.

P. J. Winzer, “Scaling optical fiber networks: challenges and solutions,” Opt. Photon. News 26(3), 28–35 (2015).
[Crossref]

Xu, J.

J. Gong, J. Xu, M. Luo, X. Li, Y. Qiu, Q. Yang, X. Zhang, and S. Yu, “All-optical wavelength conversion for mode division multiplexed superchannels,” Opt. Express 24, 8926–8939 (2016).
[Crossref] [PubMed]

C. Li, M. Luo, Z. He, H. Li, J. Xu, S. You, Q. Yang, and S. Yu, “Phase noise canceled polarization-insensitive all-optical wavelength conversion of 557-Gb/s PDM-OFDM signal using coherent dual-pump,” J. Lightwave Technol. 33, 2848–2854 (2015).
[Crossref]

Yang, Q.

J. Gong, J. Xu, M. Luo, X. Li, Y. Qiu, Q. Yang, X. Zhang, and S. Yu, “All-optical wavelength conversion for mode division multiplexed superchannels,” Opt. Express 24, 8926–8939 (2016).
[Crossref] [PubMed]

C. Li, M. Luo, Z. He, H. Li, J. Xu, S. You, Q. Yang, and S. Yu, “Phase noise canceled polarization-insensitive all-optical wavelength conversion of 557-Gb/s PDM-OFDM signal using coherent dual-pump,” J. Lightwave Technol. 33, 2848–2854 (2015).
[Crossref]

Yoshimoto, N.

T. Ito, N. Yoshimoto, K. Magari, K. Kishi, and Y. Kondo, “Extremely low power consumption semiconductor optical amplier gate for WDM applications,” Electron. Lett. 33, 1791–1792 (1997).
[Crossref]

You, S.

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A. P. Anthur, R. T. Watts, S. ODuill, R. Zhou, D. Venkitesh, and L. P. Barry, “Impact of nonlinear phase noise on all-optical wavelength conversion of 10.7-GBaud QPSK data using dual correlated pumps,” IEEE J. Quantum Electron. 51, 9100105 (2015).

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S. P. O Duill, P. M. Anandarajah, R. Zhou, and L. P. Barry, “Numerical investigation into the injection-locking phenomena of gain switched lasers for optical frequency comb generation,” Appl. Phys. Lett. 106, 211105 (2015).
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Figures (10)

Fig. 1
Fig. 1 (a) Vector representation of mixing frequencies, (b) Spectral representation of the mixing frequencies and the generated frequency.
Fig. 2
Fig. 2 Schematic of the experimental setup for validating the vector theory of FWM in SOA.
Fig. 3
Fig. 3 (Case-a)Spectrum at the output of SOA when the signal polarization is parallel to that of the pump-1, (Case-b)Spectrum at the output of SOA when the signal polarization is parallel to that of the pump-2.
Fig. 4
Fig. 4 Vector representation of mixing frequencies when the signal is polarization multiplexed and the pumps are (a) parallelly polarized, (b) orthogonally polarized.
Fig. 5
Fig. 5 Schematic of the experimental setup for the demonstration of the wavelength conversion of polarization multiplexed signals.
Fig. 6
Fig. 6 BER as a function of OSNR for the input signal (b2b), signal after SOA (asoa) and the wavelength converted signal (wc) for polarization-multiplexed QPSK signal at 12.5 GBaud.
Fig. 7
Fig. 7 BER as a function of OSNR for the input signal (b2b), signal after SOA (asoa) and the wavelength converted signal (wc) for single polarization 16-QAM signal at 12.5 GBaud.
Fig. 8
Fig. 8 BER as a function of OSNR for the input signal (b2b), signal after SOA (asoa) and the wavelength converted signal (wc) for polarization-multiplexed 16-QAM signal at 12.5 GBaud.
Fig. 9
Fig. 9 Optical spectra at the input and output of the SOA for the wavelength conversion of 12.5 GBaud four-channel Nyquist PM-QPSK superchannel.
Fig. 10
Fig. 10 BER as a function of OSNR for the input signal (b2b), signal after SOA (asoa) and the wavelength converted signal (wc) for four channel Nyquist WDM super-channel modulated with PM-QPSK signals at 12.5 GBaud.

Tables (1)

Tables Icon

Table 1 Frequencies of the idlers generated by different FWM processes. The frequencies generated in our experiments for the cases, (a) ϕ = 0° where the signal is co-polarised with the pump1, and (b) ϕ = 90°, where the signal is co-polarised with pump-2, are indicated as ticks. The FWM frequencies that are beating to generate a grating are given in red and the frequency that is diffracted by this beat frequency is given in black in column two.

Equations (15)

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E p 1 = A p 1 exp ( j ( ω p 1 t ϕ p 1 ) ) a ^ x ,
E p 2 = A p 2 exp ( j ( ω p 2 t ϕ p 2 ) ) a ^ y ,
E s = A s exp ( j ( ω s t ϕ s ) ) [ cos ( ϕ ) a ^ x + sin ( ϕ ) a ^ y ] ,
E i = η 1 ( E p 1 . E p 2 * ) E s + η 2 ( E s . E p 2 * ) E p 1 ,
E i = η 2 A s A p 1 A p 2 * exp ( j ( ω p 1 ω p 2 + ω s ) t ( ϕ p 1 ϕ p 2 + ϕ s ) ) sin ( ϕ ) a ^ x .
E p 1 = A p 1 exp ( j ( ω p 1 t ϕ p 1 ) ) a ^ x ,
E p 2 = A p 2 exp ( j ( ω p 2 t ϕ p 2 ) ) a ^ x ,
E s = exp ( j ( ω s t ϕ s ) ) [ ( A s 1 cos ( ϕ ) + A s 2 cos ( 90 + ϕ ) ) ] a ^ x + exp ( j ( ω s t ϕ s ) ) [ ( A s 1 sin ( ϕ ) + A s 2 cos ( ϕ ) ) ] a ^ y ,
E i = A p 1 A p 2 * A s exp ( j ( ω p 1 ω p 2 + ω s ) t ( ϕ p 1 ϕ p 2 + ϕ s ) ) ( η 1 + η 2 ) [ A s 1 cos ( ϕ ) + A s 2 cos ( 90 + ϕ ) ] a ^ x + η 1 A p 1 A p 2 * A s exp ( j ( ω p 1 ω p 2 + ω s ) t ( ϕ p 1 ϕ p 2 + ϕ s ) ) [ A s 1 sin ( ϕ ) + A s 2 s i n ( 90 + ϕ ) ] a ^ y .
[ A i 1 A i 2 ] = η 1 A p 1 A p 2 * A s exp ( j ( ω p 1 ω p 2 + ω s ) t ( ϕ p 1 ϕ p 2 + ϕ s ) ) [ cos ( ϕ ) sin ( ϕ ) sin ( ϕ ) cos ( ϕ ) ] [ A s 1 A s 2 ] .
E i = η 2 ( E s . E p 2 * ) E p 1 ,
E i = η 2 A s A p 2 * A p 1 exp ( j ( ω p 1 ω p 2 + ω s ) t ( ϕ p 1 ϕ p 2 + ϕ s ) ) × [ A s 1 sin ( ϕ ) + A s 2 cos ( ϕ ) ] a ^ x .
E i 2 = η 1 ( E p 1 . E s * ) E p 2 + η 2 ( E p 2 . E s * ) E p 1 ,
E i 2 = A p 1 A p 2 A s * exp ( j ( ω p 1 ω p 2 + ω s ) t ( ϕ p 1 ϕ p 2 + ϕ s ) ) × [ η 1 ( A s 1 cos ( ϕ ) + A s 2 cos ( ϕ ) ) a ^ x + η 1 ( A s 1 sin ( ϕ ) + A s 2 cos ( ϕ ) ) a ^ y ] .
[ A i 1 A i 2 ] = η 1 A p 1 A p 2 A s * exp ( j ( ω p 1 ω p 2 + ω s ) t ( ϕ p 1 ϕ p 2 + ϕ s ) ) [ η 1 cos ( ϕ ) η 1 sin ( ϕ ) η 2 sin ( ϕ ) η 2 cos ( ϕ ) ] [ A s 1 A s 2 ] ,

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