Abstract

A closed-form expression for the variance of the four-wave mixing (FWM) induced in each subcarrier of a double sideband orthogonal frequency division multiplexing (OFDM) system employing direct detection is proposed and validated. Particularly, using a small signal analysis, equivalent transfer functions that characterize the frequency response of the FWM effect are derived taking into account the walkoff effect between the modulated pump waves and the FWM wave. The accuracy of the variance estimates provided by the closed-form expression is assessed for different sets of system parameters. The closed-form expression provides good variance estimates of the FWM-induced degradation caused by degenerate and nonsymmetric nondegenerate FWM components. For symmetric non-degenerate FWM components, the proposed expression provides reliable but pessimistic variance estimates, not exceeding the actual FWM variance in 1.5 dB for modulation indexes of interest.

© 2014 Optical Society of America

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  8. T. Alves, M. Morant, A. Cartaxo, R. Llorente, “Transmission of OFDM wired-wireless quintuple-play services along WDM LR-PONs using centralized broadband impairment compensation,” Opt. Express 20, 13748–13761 (2012).
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    [CrossRef]
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  15. K. Inoue, “Polarization effect on four-wave mixing efficiency in a single-mode fiber,” J. Quantum Electron. 28, 883–894 (1992).
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  16. R. Tkach, A. Chraplyvy, F. Forghieri, A. Gnauck, R. Derosier, “Four-photon mixing and high-speed WDM systems,” J. Lightwave Technol. 13, 841–849 (1995).
    [CrossRef]
  17. S. Song, C. Allen, K. Demarest, R. Hui, “Intensity-dependent phase-matching effects on four-wave mixing in optical fibers,” J. Lightwave Technol. 17, 2285–2290 (1999).
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  18. H. Song, M. Pearce, “Range of influence and impact of physical impairments in long-haul DWDM systems,” J. Lightwave Technol. 31, 846–854 (2013).
    [CrossRef]
  19. C. Mahon, L. Olofsson, E. Bødtker, G. Jacobsen, “Polarization allocation schemes for minimizing fiber four-wave mixing crosstalk in wavelength division multiplexed optical communication systems,” IEEE Photon. Technol. Lett. 8, 575–577 (1996).
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  23. X. Chen, W. Shieh, “Closed-form expressions for nonlinear transmission performance of densely spaced coherent optical OFDM systems,” Opt. Express 18, 19039–19054 (2010).
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  25. L. Du, M. Morshed, A. Lowery, “Fiber nonlinearity compensation for OFDM super-channels using optical phase conjugation,” Opt. Express 20, 19921–19927 (2012).
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    [CrossRef] [PubMed]
  27. T. Chiang, N. Kagi, M. Marhic, L. Kazovsky, “Cross-phase modulation in fiber links with multiple optical amplifiers and dispersion compensators,” J. Lightwave Technol. 14, 249–260 (1996).
    [CrossRef]
  28. T. Alves, A. Cartaxo, “Semi-analytical approach for performance evaluation of direct-detection OFDM optical communication systems,” Opt. Express 17, 18714–18729 (2009).
    [CrossRef]
  29. R. Llorente, T. Alves, M. Morant, M. Beltran, J. Perez, A. Cartaxo, J. Marti, “Ultra-wideband radio signals distribution in FTTH networks,” IEEE Photon. Technol. Lett. 20, 945–947 (2008).
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    [CrossRef]

2013

2012

2011

2010

2009

2008

R. Llorente, T. Alves, M. Morant, M. Beltran, J. Perez, A. Cartaxo, J. Marti, “Ultra-wideband radio signals distribution in FTTH networks,” IEEE Photon. Technol. Lett. 20, 945–947 (2008).
[CrossRef]

M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, V. Karagodsky, “Phased-array cancellation of nonlinear FWM in coherent OFDM dispersive multi-span links,” Opt. Express 16, 15777–15810 (2008).
[CrossRef] [PubMed]

2007

H. Batshon, I. Djordjevic, B. Vasic, “An improved technique for suppression of intrachannel four-wave mixing in 40-Gb/s optical transmission systems,” IEEE Photon. Technol. Lett. 19, 67–69 (2007).
[CrossRef]

A. Lowery, S. Wang, M. Premaratne, “Calculation of power limit due to fiber nonlinearity in optical OFDM systems,” Opt. Express 15, 13282–13287 (2007).
[CrossRef] [PubMed]

2006

1999

1996

C. Mahon, L. Olofsson, E. Bødtker, G. Jacobsen, “Polarization allocation schemes for minimizing fiber four-wave mixing crosstalk in wavelength division multiplexed optical communication systems,” IEEE Photon. Technol. Lett. 8, 575–577 (1996).
[CrossRef]

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

1995

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

1992

K. Inoue, “Polarization effect on four-wave mixing efficiency in a single-mode fiber,” J. Quantum Electron. 28, 883–894 (1992).
[CrossRef]

1987

N. Shibata, R. Braun, R. Waarts, “Phase-mismatch dependence of efficiency of wave generation through four-wave mixing in a single-mode optical fiber,” J. Quantum Electron. 23, 1205–1210 (1987).
[CrossRef]

1978

K. Hill, D. Johnson, B. Kawasaki, R. MacDonald, “CW three-wave mixing in single-mode optical fibers,” J. Appl. Phys. 49, 5098–5106 (1978).
[CrossRef]

Akhtar, A.

Allen, C.

Alves, T.

Armstrong, J.

Athaudage, C.

W. Shieh, C. Athaudage, “Coherent optical orthogonal frequency division multiplexing,” Electron. Lett. 42, 587–589 (2006).
[CrossRef]

Batshon, H.

H. Batshon, I. Djordjevic, B. Vasic, “An improved technique for suppression of intrachannel four-wave mixing in 40-Gb/s optical transmission systems,” IEEE Photon. Technol. Lett. 19, 67–69 (2007).
[CrossRef]

Beltran, M.

R. Llorente, T. Alves, M. Morant, M. Beltran, J. Perez, A. Cartaxo, J. Marti, “Ultra-wideband radio signals distribution in FTTH networks,” IEEE Photon. Technol. Lett. 20, 945–947 (2008).
[CrossRef]

Bødtker, E.

C. Mahon, L. Olofsson, E. Bødtker, G. Jacobsen, “Polarization allocation schemes for minimizing fiber four-wave mixing crosstalk in wavelength division multiplexed optical communication systems,” IEEE Photon. Technol. Lett. 8, 575–577 (1996).
[CrossRef]

Braun, R.

N. Shibata, R. Braun, R. Waarts, “Phase-mismatch dependence of efficiency of wave generation through four-wave mixing in a single-mode optical fiber,” J. Quantum Electron. 23, 1205–1210 (1987).
[CrossRef]

Carpenter, J.

L. Du, J. Schröder, J. Carpenter, B. Eggleton, A. Lowery, “Flexible all-optical OFDM using WSSs,” in Optical Fiber Communication Conference, OSA Technical Digest Series (CD) (Optical Society of America, 2013), paper PDP5B.9.
[CrossRef]

Cartaxo, A.

Che, D.

Chen, X.

Chiang, T.

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

Cho, P.

Chraplyvy, A.

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

Cvijetic, N.

Demarest, K.

Derosier, R.

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

Djordjevic, I.

H. Batshon, I. Djordjevic, B. Vasic, “An improved technique for suppression of intrachannel four-wave mixing in 40-Gb/s optical transmission systems,” IEEE Photon. Technol. Lett. 19, 67–69 (2007).
[CrossRef]

W. Shieh, I. Djordjevic, OFDM for Optical Communications (Elsevier, San Diego, 2010).

Du, L.

L. Du, M. Morshed, A. Lowery, “Fiber nonlinearity compensation for OFDM super-channels using optical phase conjugation,” Opt. Express 20, 19921–19927 (2012).
[CrossRef] [PubMed]

L. Du, J. Schröder, J. Carpenter, B. Eggleton, A. Lowery, “Flexible all-optical OFDM using WSSs,” in Optical Fiber Communication Conference, OSA Technical Digest Series (CD) (Optical Society of America, 2013), paper PDP5B.9.
[CrossRef]

Eggleton, B.

L. Du, J. Schröder, J. Carpenter, B. Eggleton, A. Lowery, “Flexible all-optical OFDM using WSSs,” in Optical Fiber Communication Conference, OSA Technical Digest Series (CD) (Optical Society of America, 2013), paper PDP5B.9.
[CrossRef]

Fair, I.

Forghieri, F.

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

Gnauck, A.

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

He, J.

Hill, K.

K. Hill, D. Johnson, B. Kawasaki, R. MacDonald, “CW three-wave mixing in single-mode optical fibers,” J. Appl. Phys. 49, 5098–5106 (1978).
[CrossRef]

Hu, Q.

Hui, R.

Inoue, K.

K. Inoue, “Polarization effect on four-wave mixing efficiency in a single-mode fiber,” J. Quantum Electron. 28, 883–894 (1992).
[CrossRef]

Jacobsen, G.

C. Mahon, L. Olofsson, E. Bødtker, G. Jacobsen, “Polarization allocation schemes for minimizing fiber four-wave mixing crosstalk in wavelength division multiplexed optical communication systems,” IEEE Photon. Technol. Lett. 8, 575–577 (1996).
[CrossRef]

Ji, P.

W. Wei, D. Xu, D. Qian, P. Ji, T. Wang, C. Qiao, “Demonstration of an optical OFDMA metro ring network with dynamic sub-carrier allocation,” in Optical Fiber Communication Conference, OSA Technical Digest Series (CD) (Optical Society of America, 2009), paper NTuA3.
[CrossRef]

Johnson, D.

K. Hill, D. Johnson, B. Kawasaki, R. MacDonald, “CW three-wave mixing in single-mode optical fibers,” J. Appl. Phys. 49, 5098–5106 (1978).
[CrossRef]

Kagi, N.

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

Karagodsky, V.

Kawasaki, B.

K. Hill, D. Johnson, B. Kawasaki, R. MacDonald, “CW three-wave mixing in single-mode optical fibers,” J. Appl. Phys. 49, 5098–5106 (1978).
[CrossRef]

Kazovsky, L.

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

Khurgin, J.

Kumar, S.

Li, A.

Llorente, R.

T. Alves, M. Morant, A. Cartaxo, R. Llorente, “Transmission of OFDM wired-wireless quintuple-play services along WDM LR-PONs using centralized broadband impairment compensation,” Opt. Express 20, 13748–13761 (2012).
[CrossRef] [PubMed]

R. Llorente, T. Alves, M. Morant, M. Beltran, J. Perez, A. Cartaxo, J. Marti, “Ultra-wideband radio signals distribution in FTTH networks,” IEEE Photon. Technol. Lett. 20, 945–947 (2008).
[CrossRef]

Lowery, A.

MacDonald, R.

K. Hill, D. Johnson, B. Kawasaki, R. MacDonald, “CW three-wave mixing in single-mode optical fibers,” J. Appl. Phys. 49, 5098–5106 (1978).
[CrossRef]

Mahon, C.

C. Mahon, L. Olofsson, E. Bødtker, G. Jacobsen, “Polarization allocation schemes for minimizing fiber four-wave mixing crosstalk in wavelength division multiplexed optical communication systems,” IEEE Photon. Technol. Lett. 8, 575–577 (1996).
[CrossRef]

Marhic, M.

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

Marti, J.

R. Llorente, T. Alves, M. Morant, M. Beltran, J. Perez, A. Cartaxo, J. Marti, “Ultra-wideband radio signals distribution in FTTH networks,” IEEE Photon. Technol. Lett. 20, 945–947 (2008).
[CrossRef]

Meiman, Y.

Morant, M.

T. Alves, M. Morant, A. Cartaxo, R. Llorente, “Transmission of OFDM wired-wireless quintuple-play services along WDM LR-PONs using centralized broadband impairment compensation,” Opt. Express 20, 13748–13761 (2012).
[CrossRef] [PubMed]

R. Llorente, T. Alves, M. Morant, M. Beltran, J. Perez, A. Cartaxo, J. Marti, “Ultra-wideband radio signals distribution in FTTH networks,” IEEE Photon. Technol. Lett. 20, 945–947 (2008).
[CrossRef]

Morshed, M.

Nazarathy, M.

Noe, R.

Olofsson, L.

C. Mahon, L. Olofsson, E. Bødtker, G. Jacobsen, “Polarization allocation schemes for minimizing fiber four-wave mixing crosstalk in wavelength division multiplexed optical communication systems,” IEEE Photon. Technol. Lett. 8, 575–577 (1996).
[CrossRef]

Pavel, L.

Pearce, M.

Pechenkin, V.

Perez, J.

R. Llorente, T. Alves, M. Morant, M. Beltran, J. Perez, A. Cartaxo, J. Marti, “Ultra-wideband radio signals distribution in FTTH networks,” IEEE Photon. Technol. Lett. 20, 945–947 (2008).
[CrossRef]

Premaratne, M.

Qian, D.

W. Wei, D. Xu, D. Qian, P. Ji, T. Wang, C. Qiao, “Demonstration of an optical OFDMA metro ring network with dynamic sub-carrier allocation,” in Optical Fiber Communication Conference, OSA Technical Digest Series (CD) (Optical Society of America, 2009), paper NTuA3.
[CrossRef]

Qiao, C.

W. Wei, D. Xu, D. Qian, P. Ji, T. Wang, C. Qiao, “Demonstration of an optical OFDMA metro ring network with dynamic sub-carrier allocation,” in Optical Fiber Communication Conference, OSA Technical Digest Series (CD) (Optical Society of America, 2009), paper NTuA3.
[CrossRef]

Schröder, J.

L. Du, J. Schröder, J. Carpenter, B. Eggleton, A. Lowery, “Flexible all-optical OFDM using WSSs,” in Optical Fiber Communication Conference, OSA Technical Digest Series (CD) (Optical Society of America, 2013), paper PDP5B.9.
[CrossRef]

Shibata, N.

N. Shibata, R. Braun, R. Waarts, “Phase-mismatch dependence of efficiency of wave generation through four-wave mixing in a single-mode optical fiber,” J. Quantum Electron. 23, 1205–1210 (1987).
[CrossRef]

Shieh, W.

Shpantzer, I.

Song, H.

Song, S.

Tkach, R.

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

Vasic, B.

H. Batshon, I. Djordjevic, B. Vasic, “An improved technique for suppression of intrachannel four-wave mixing in 40-Gb/s optical transmission systems,” IEEE Photon. Technol. Lett. 19, 67–69 (2007).
[CrossRef]

Waarts, R.

N. Shibata, R. Braun, R. Waarts, “Phase-mismatch dependence of efficiency of wave generation through four-wave mixing in a single-mode optical fiber,” J. Quantum Electron. 23, 1205–1210 (1987).
[CrossRef]

Wang, S.

Wang, T.

W. Wei, D. Xu, D. Qian, P. Ji, T. Wang, C. Qiao, “Demonstration of an optical OFDMA metro ring network with dynamic sub-carrier allocation,” in Optical Fiber Communication Conference, OSA Technical Digest Series (CD) (Optical Society of America, 2009), paper NTuA3.
[CrossRef]

Wang, Y.

Wei, W.

W. Wei, D. Xu, D. Qian, P. Ji, T. Wang, C. Qiao, “Demonstration of an optical OFDMA metro ring network with dynamic sub-carrier allocation,” in Optical Fiber Communication Conference, OSA Technical Digest Series (CD) (Optical Society of America, 2009), paper NTuA3.
[CrossRef]

Weidenfeld, R.

Xu, D.

W. Wei, D. Xu, D. Qian, P. Ji, T. Wang, C. Qiao, “Demonstration of an optical OFDMA metro ring network with dynamic sub-carrier allocation,” in Optical Fiber Communication Conference, OSA Technical Digest Series (CD) (Optical Society of America, 2009), paper NTuA3.
[CrossRef]

Electron. Lett.

W. Shieh, C. Athaudage, “Coherent optical orthogonal frequency division multiplexing,” Electron. Lett. 42, 587–589 (2006).
[CrossRef]

IEEE Photon. Technol. Lett.

C. Mahon, L. Olofsson, E. Bødtker, G. Jacobsen, “Polarization allocation schemes for minimizing fiber four-wave mixing crosstalk in wavelength division multiplexed optical communication systems,” IEEE Photon. Technol. Lett. 8, 575–577 (1996).
[CrossRef]

H. Batshon, I. Djordjevic, B. Vasic, “An improved technique for suppression of intrachannel four-wave mixing in 40-Gb/s optical transmission systems,” IEEE Photon. Technol. Lett. 19, 67–69 (2007).
[CrossRef]

R. Llorente, T. Alves, M. Morant, M. Beltran, J. Perez, A. Cartaxo, J. Marti, “Ultra-wideband radio signals distribution in FTTH networks,” IEEE Photon. Technol. Lett. 20, 945–947 (2008).
[CrossRef]

J. Appl. Phys.

K. Hill, D. Johnson, B. Kawasaki, R. MacDonald, “CW three-wave mixing in single-mode optical fibers,” J. Appl. Phys. 49, 5098–5106 (1978).
[CrossRef]

J. Lightwave Technol.

A. Akhtar, L. Pavel, S. Kumar, “Modeling and analysis of the contribution of channel walk-off to nondegenerate and degenerate four-wave-mixing noise in RZ-OOK optical transmission systems,” J. Lightwave Technol. 24, 4269–4285 (2006).
[CrossRef]

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

S. Song, C. Allen, K. Demarest, R. Hui, “Intensity-dependent phase-matching effects on four-wave mixing in optical fibers,” J. Lightwave Technol. 17, 2285–2290 (1999).
[CrossRef]

H. Song, M. Pearce, “Range of influence and impact of physical impairments in long-haul DWDM systems,” J. Lightwave Technol. 31, 846–854 (2013).
[CrossRef]

J. Armstrong, “OFDM for optical communications,” J. Lightwave Technol. 27, 189–204 (2009).
[CrossRef]

N. Cvijetic, “OFDM for next generation optical access networks,” J. Lightwave Technol. 30, 384–398 (2012).
[CrossRef]

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

T. Alves, A. Cartaxo, “Transmission of multiband OFDM-UWB signals along LR-PONs employing a Mach-Zehnder modulator biased at the quasi-minimum power transmission point,” J. Lightwave Technol. 30, 1587–1594 (2012).
[CrossRef]

T. Alves, A. Cartaxo, “Distribution of double-sideband OFDM-UWB radio signals in dispersion compensated long-reach PONs,” J. Lightwave Technol. 29, 2467–2474 (2011).
[CrossRef]

J. Opt. Commun. Netw.

J. Quantum Electron.

N. Shibata, R. Braun, R. Waarts, “Phase-mismatch dependence of efficiency of wave generation through four-wave mixing in a single-mode optical fiber,” J. Quantum Electron. 23, 1205–1210 (1987).
[CrossRef]

K. Inoue, “Polarization effect on four-wave mixing efficiency in a single-mode fiber,” J. Quantum Electron. 28, 883–894 (1992).
[CrossRef]

Opt. Express

X. Chen, A. Li, D. Che, Q. Hu, Y. Wang, J. He, W. Shieh, “Blockwise phase switching for double-sideband direct-detected optical OFDM signals,” Opt. Express 21, 13436–13441 (2013).
[CrossRef] [PubMed]

A. Lowery, J. Armstrong, “Orthogonal-frequency-division multiplexing for dispersion compensation of long-haul optical systems,” Opt. Express 14, 2079–2084 (2006).
[CrossRef] [PubMed]

T. Alves, M. Morant, A. Cartaxo, R. Llorente, “Transmission of OFDM wired-wireless quintuple-play services along WDM LR-PONs using centralized broadband impairment compensation,” Opt. Express 20, 13748–13761 (2012).
[CrossRef] [PubMed]

L. Du, M. Morshed, A. Lowery, “Fiber nonlinearity compensation for OFDM super-channels using optical phase conjugation,” Opt. Express 20, 19921–19927 (2012).
[CrossRef] [PubMed]

M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, V. Karagodsky, “Phased-array cancellation of nonlinear FWM in coherent OFDM dispersive multi-span links,” Opt. Express 16, 15777–15810 (2008).
[CrossRef] [PubMed]

A. Lowery, S. Wang, M. Premaratne, “Calculation of power limit due to fiber nonlinearity in optical OFDM systems,” Opt. Express 15, 13282–13287 (2007).
[CrossRef] [PubMed]

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

T. Alves, A. Cartaxo, “Semi-analytical approach for performance evaluation of direct-detection OFDM optical communication systems,” Opt. Express 17, 18714–18729 (2009).
[CrossRef]

Opt. Lett.

Other

L. Du, J. Schröder, J. Carpenter, B. Eggleton, A. Lowery, “Flexible all-optical OFDM using WSSs,” in Optical Fiber Communication Conference, OSA Technical Digest Series (CD) (Optical Society of America, 2013), paper PDP5B.9.
[CrossRef]

W. Wei, D. Xu, D. Qian, P. Ji, T. Wang, C. Qiao, “Demonstration of an optical OFDMA metro ring network with dynamic sub-carrier allocation,” in Optical Fiber Communication Conference, OSA Technical Digest Series (CD) (Optical Society of America, 2009), paper NTuA3.
[CrossRef]

Towards 2020 - Photonics driving economic growth in Europe, Brussels, Belgium: European Technology Platform Photonics21, http://www.photonics21.org/downloads , (2013).

W. Shieh, I. Djordjevic, OFDM for Optical Communications (Elsevier, San Diego, 2010).

High Rate UltraWideband PHY and MAC Standard, 2ECMA Int. (2007).

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Figures (8)

Fig. 1
Fig. 1

Illustration of the spectrum (optical carrier and modulated DSB-OFDM signal) of 3 optical channels of a WDM signal at the fiber input, and corresponding carrier-carrier and carrier-signal FWM components at the fiber output.

Fig. 2
Fig. 2

Illustration of the three in-band FWM waves generated by a 3-channel WDM signal. It is assumed that the spacing between channels a and b is the same as the spacing between channels b and c.

Fig. 3
Fig. 3

Amplitude response of the equivalent FWM transfer functions of the FWM-wave generated at the frequency of channel (a) a (b) b and (c) c. SSMF lengths of 75 km and 100 km are considered. The power of the optical carrier per channel at the fiber input is 12 dBm. ν0 is the central frequency of each optical channel and ν is the optical frequency.

Fig. 4
Fig. 4

(a) Power of Āeq,FWM normalized to the power of the optical carrier at the fiber output as a function of the fiber length. (b) and (c) Amplitude response of Heq,2 (L, f) and Heq,4 (L, f), respectively, normalized by the power loss introduced by the SSMF, as a function of the fiber length. The FWM component is being generated at the frequency of channel a (see Fig. 2) and the power of the optical carrier per channel at the fiber input is 12 dBm. In (b) and (c): results obtained for a frequency of 2 GHz (dashed-dotted line), 3.7 GHz (continuous line), 4.4 GHz (dashed line) and 12.5 GHz (dotted line).

Fig. 5
Fig. 5

EVM of each optical channel as a function of the modulation index for (a) and (c) UWB 2, and (b) and (d) UWB 3. The average optical power per channel at fiber input is 12 dBm and the channel spacing is 25 GHz. In (a) and (b): 3-channel WDM signal, SSMF length of 75 km. In (c) and (d): 4-channel WDM signal, SSMF length of 90 km.

Fig. 6
Fig. 6

EVM of each optical channel as a function of the fiber length for (a) and (c) UWB 2, and (b) and (d) UWB 3. A 3-channel WDM signal and a modulation index of 8% are considered. (a) and (b) Channel spacing of 25 GHz and average optical power per channel at fiber input of 12 dBm; (c) and (d) Channel spacing of 50 GHz and average optical power per channel at fiber input of 17 dBm.

Fig. 7
Fig. 7

EVM of each optical channel as a function of the average optical power per channel at fiber input for (a) UWB 2 and (b) UWB 3. A WDM signal comprising 4 channels, a SSMF length of 100 km and a modulation index of 8% are considered.

Fig. 8
Fig. 8

EVM of UWB 3 as a function of each channel of a WDM signal comprising 5, 7 and 9 optical channels. A modulation index of 8%, a fiber length of 60 km, an average optical power at fiber input of 12 dBm and a channel spacing of 25 GHz are considered.

Equations (37)

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A 1 ( z , t ) z = 1 2 α A 1 ( z , t ) β 1 , 1 A 1 ( z , t ) t j 1 3 D γ A 2 ( z , t ) A 3 ( z , t ) A 4 * ( z , t ) exp ( j Δ β z )
A 2 ( z , t ) z = 1 2 α A 2 ( z , t ) β 1 , 2 A 2 ( z , t ) t
A 3 ( z , t ) z = 1 2 α A 3 ( z , t ) β 1 , 3 A 3 ( z , t ) t
A 4 ( z , t ) z = 1 2 α A 4 ( z , t ) β 1 , 4 A 4 ( z , t ) t
A i ( z , t ) = [ A i ¯ + a i ( 0 , t β 1 , i z ) ] exp ( 1 2 α z ) exp ( j ϕ i )
A 2 ( z , t ) A 3 ( z , t ) A 4 * ( z , t ) = [ A 2 ¯ A 3 ¯ A 4 ¯ + A 2 ¯ A 3 ¯ a 4 ( 0 , t β 1 , 4 z ) + A 2 ¯ A 4 ¯ a 3 ( 0 , t β 1 , 3 z ) + + A 3 ¯ A 4 ¯ a 2 ( 0 , t β 1 , 2 z ) ] exp ( 3 2 α z ) exp ( j ϕ g )
A 1 ( L , t ' ) = A ¯ F W M + i = 2 4 a i ( 0 , t ' ) * h i ( L , t ' )
A ¯ F W M = j 3 D γ exp ( j ϕ g ) exp ( 1 2 α L ) A 2 ¯ A 3 ¯ A 4 ¯ j Δ β α [ exp ( j Δ β L α L ) 1 ]
H i ( L , ω ) = j 3 D γ exp ( j ϕ g ) exp ( 1 2 α L ) Π m = 2 m i 4 A m ¯ j ( Δ β + ω d 1 , i ) α [ exp ( j ( Δ β + ω d 1 , i ) L α L ) 1 ]
A s ( L , t ) = [ A s ¯ + a s ( 0 , t ) ] exp ( 1 2 α L ) exp ( j ϕ s )
i t ( t ) = R λ [ | A s ( L , t ) | 2 + A s ( L , t ) A 1 * ( L , t ) + A 1 ( L , t ) A s * ( L , t ) + | A 1 ( L , t ) | 2 ]
i F W M ( t ) R λ [ A s ( L , t ) A 1 * ( L , t ) + A 1 ( L , t ) A s * ( L , t ) ] = = R λ [ A s ¯ + a s ( 0 , t ) ] exp ( 1 2 α L ) [ A ¯ e q , F W M + i = 2 4 a i ( 0 , t ) * h e q , i ( L , t ) ]
σ F W M , p 2 = E [ i F W M 2 ( t ) ] = R λ 2 exp ( α L ) E { [ A s ¯ 2 + 2 A s ¯ a s ( 0 , t ) + a s ( 0 , t ) a s ( 0 , t ) ] × × | A ¯ e q , F W M + i = 2 4 a i ( 0 , t ) * h e q , i ( L , t ) | 2 }
σ F W M , p 2 = σ c , c 2 + σ c , 2 2 + σ s , s 2
σ F W M , p 2 = R λ 2 { exp ( α L ) A s ¯ 2 + A ¯ e q , F W M | 2 + + [ n = 1 5 S F W M , n ( L , f ) ] d t }
S F W M , 1 ( L , f ) = exp ( α L ) | A ¯ e q , F W M | 2 S s ( f )
S F W M , 2 ( L , f ) = exp ( α L ) A s ¯ 2 a d , 2 S 2 ( f ) | H e q , 2 ( L , f ) | 2
S F W M , 3 ( L , f ) = exp ( α L ) A s ¯ 2 a d , 3 S 3 ( f ) | H e q , 3 ( L , f ) | 2
S F W M , 4 ( L , f ) = exp ( α L ) A s ¯ 2 a d , 4 S 4 ( f ) | H e q , 4 ( L , f ) | 2
S F W M , 5 ( L , f ) = exp ( α L ) a n d , s A s ¯ S s ( f ) [ A ¯ e q , F W M H e q , 4 * ( L , f ) + A ¯ e q , F W M * H e q , 4 ( L , f ) ]
a d , i = { 1 for nondegenerate terms ( ν 2 ν 3 ) 4 for degenerate terms ( ν 2 = ν 3 ) and  i = 2 0 for degenerate terms ( ν 2 = ν 3 ) and i = 3 1 for degenerate terms ( ν 2 = ν 3 ) and  i = 4
a n d , s = { 2 for symmetric nondegenerate terms ( ν 1 = v 4 ) 0 for nonsymmetric nondegenerate ( ν 1 ν 4 ) and degenerate terms ( ν 2 = ν 3 )
σ F W M 2 [ k ] = R λ 2 + n = 1 5 S F W M , n ( f ) | H e q , r ( f , k ) | 2 d f
[ H e q , 1 , ( I , Q ) ( f , k ) H e q , 2 , ( I , Q ) ( f , k ) ] = H r ( f ) 2 exp [ j 2 π f t 0 ] P ( I , Q ) T A e q [ k ]
σ F W M , t 2 [ k ] = i = 1 N F W M σ F M W , i 2 [ k ]
E V ^ M [ k ] σ F W M , n 2 [ k ] = σ F W M , t 2 [ k ] p s [ k ]
σ c , c 2 = R λ 2 exp ( α L ) A s ¯ 2 { | A ¯ e q , F W M | 2 + E [ i = 2 4 a i ( 0 , t ) * h e q , i ( L , t ) l = 2 4 a l ( 0 , t ) * h e q , l * ( L , t ) ] }
σ c , c 2 = R λ 2 exp ( α L ) A s ¯ 2 { | A ¯ e q , F W M | 2 + i = 2 4 + S i ( f ) | H e q , i ( L , f ) | 2 d f }
| H i ( L , f ) | 2 = η i ( f ) ( D γ 3 ) 2 exp ( α L ) P ¯ 2 L e f f 2
| A ¯ F W M | 2 = η i ( 0 ) ( D γ 3 ) 2 exp ( α L ) P ¯ 3 L e f f 2
η i ( f ) = α 2 α 2 + ( Δ β + 2 π f d 1 , i ) 2 { 1 + 4 exp ( α L ) sin 2 [ 1 2 ( Δ β + 2 π f d 1 , i ) L ] ( 1 exp ( α L ) 2 ) }
σ c , c 2 = R λ 2 exp ( α L ) A s ¯ 2 { | A ¯ e q , F W M | 2 + E { [ a 4 ( 0 , t ) * h e q , 4 ( L , t ) ] [ a 4 ( 0 , t ) * h e q , 4 * ( L , t ) ] + + 4 [ a 2 ( 0 , t ) * h e q , 2 ( L , t ) ] [ a 2 ( 0 , t ) * h e q , 2 * ( L , t ) ] } }
σ c , c 2 = R λ 2 exp ( α L ) A s ¯ 2 { | A ¯ e q , F W M | 2 + + S 4 ( f ) | H e q , 4 ( f ) | 2 d f + + 4 + S 2 ( f ) | H e q , 2 ( f ) | 2 d f }
σ c , s 2 = 2 R λ 2 exp ( α L ) A s ¯ E { a s ( 0 , t ) i = 2 4 [ A ¯ e q , F W M h e q , i * ( L , t ) + A ¯ e q , F W M * h e q , i ( L , t ) ] a i ( 0 , t ) }
σ c , s 2 = 2 R λ 2 exp ( α L ) A s ¯ + S s ( f ) [ A ¯ e q , F W M H e q , 4 * ( L , f ) + A ¯ e q , F W M * H e q , 4 ( L , t ) ] d f
A ¯ e q , F W M H e q , 4 * ( L , f ) + A ¯ e q , F W M * H e q , 4 ( L , f ) = 2 [ A ¯ F W M H 4 * ( L , f ) + A ¯ F W M * H 4 ( L , f ) ]
σ s , s 2 = R λ 2 exp ( α L ) | A ¯ e q , F W M | 2 P O F D M

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