Abstract

We develop an analytic model of Coherent Optical Orthogonal Frequency Division Multiplexing (OFDM) propagation and detection over multi-span long-haul fiber links, comprehensively and rigorously analyzing the impairments due the combined effects of FWM, Dispersion and ASE noise. Consistent with prior work of Innoe and Schadt in the WDM context, our new closed-form expressions for the total FWM received power fluctuations in the wake of dispersive phase mismatch in OFDM transmission, indicate that the FWM contributions of the multitude of spans build-up on a phased-array basis. For particular ultra-long haul link designs, the effectiveness of dispersion in reducing FWM is far greater than previously assumed in OFDM system analysis. The key is having the dominant FWM intermodulation products due to the multiple spans, destructively interfere, mutually cancelling their FWM intermodulation products, analogous to operating at the null of a phased-array antenna system. By applying the new analysis tools, this mode of effectively mitigating the FWM impairment, is shown under specific dispersion and spectral management conditions, to substantially suppress the FWM power fluctuations. Accounting for the phased-array concept and applying the compact OFDM design formulas developed here, we analyzed system performance of a 40 Gbps coherent OFDM system, over standard G.652 fiber, with cyclic prefix based electronic dispersion compensation but no optical compensation along the link. The transmission range for 10-3 target BER is almost tripled from 2560 km to 6960 km, relative to a reference system performing optical dispersion compensation in every span (ideally accounting for FWM and ASE noise and the cyclic prefix overhead, but excluding additional impairments).

© 2008 Optical Society of America

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References

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  1. W. Shieh and C. Athaudage, “Coherent optical orthogonal frequency division multiplexing,” Electron. Lett. 42, 587–589 (2006)
    [Crossref]
  2. J. Lowery, S. Wang, M. Premaratne, and J. Lowery, “Fiber nonlinearity pre- and post-compensation for long-haul optical links using OFDM” Opt. Express15, 12965–12970, (2007), http://www.opticsexpress.org/abstract.cfm?id=125469.
    [Crossref] [PubMed]
  3. W. Shieh, X. Yi, and Y. Tang, “Transmission experiment of multi-gigabit coherent optical OFDM systems over 1000 km SSMF fiber,” Electron. Lett. 43, 183–185 (2007).
    [Crossref]
  4. W. Shieh, X. Yi, Y. Ma, and Y. Tang, “Theoretical and experimental study on PMD-supported transmission using polarization diversity in coherent optical OFDM systems,” Opt. Express 15, 9936–9947 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-16-9936.
    [Crossref] [PubMed]
  5. W. Shieh, R. S. Tucker, W. Chen, X. Yi, and G. Pendock, “Optical performance monitoring in coherent optical OFDM systems,” Opt. Express 15, 350–356 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-2-350.
    [Crossref] [PubMed]
  6. H. Bao and W. Shieh, “Transmission simulation of coherent optical OFDM signals in WDM systems,” Opt. Express 15, 4410–4418 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-8-4410.
    [Crossref] [PubMed]
  7. W. Shieh, H. Bao, and Y. Tang, “Coherent optical OFDM: theory and design,” Opt. Express 16, 841–859 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-2-841.
    [Crossref] [PubMed]
  8. J. Lowery, “Amplified-spontaneous noise limit of optical OFDM lightwave systems,” Opt. Express 16, 860–865, (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-2-860.
    [Crossref] [PubMed]
  9. K. Inoue, “Phase mismatching characteristic of four-wave mixing in fiber lines with multistage optical amplifiers” Opt. Lett 17, 801 (1992).
    [Crossref] [PubMed]
  10. D. G. Schadt, “Effect of amplifier spacing on four wave mixing in multichannel coherent communications,” Electron. Lett 27, 805–7 (1991).
    [Crossref]
  11. T. Schneider, Nonlinear Optics in Telecommunications (Springer2004).
  12. M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, and V. Karagodsky, “The FWM Impairment in Coherent OFDM Compounds on a Phased-Array Basis over Dispersive Multi-Span Links” COTA’08, Boston, July 13–16, (2008).
  13. Please notice that in the published proceedings of [12] an error fell in the stated system performance. The early simulations did not account for the coherent addition of transposed pairs of non-degenerate intermods, as properly carried out in section 6 of this paper.
  14. T.-K. Chiang, N. Kagi, M. E. Marhic, and L. G. Kazovsky, “Cross-phase modulation in fiber links with multiple optical amplifiers and dispersion compensation,” J. Lightwave Technol. 14249–260 (1996).
    [Crossref]
  15. M. E. Marhic, N. Kagi, T.-K. Chiang, and L. G. Kazovsky, “Optimizing the location of dispersion compensators in periodically amplified fiber links in the presence of third-order nonlinear effects” Photon Technol. Lett. 8, 145–147 (1996).
    [Crossref]
  16. M. Eiselt, “Limits on WDM Systems Due to Four-Wave Mixing: A Statistical Approach,” J. Lightwave Technol. 17, 2261–2267 (1999).
    [Crossref]
  17. G.P. Agrawal, Lightwave Technology: Telecommunication Systems (Wiley2005).
    [Crossref]
  18. T. Yamamoto and M. Nakazawa “Highly efficient four-wave mixing in an optical fiber with intensity dependent phase matching,” J. Lightwave Technol. 9, 327–329 (1997).
  19. S. Song, C. T. Allen, K. R. Demarest, and R. Hui, “Intensity-Dependent Phase-Matching Effects on Four-Wave Mixing in Optical Fibers,” J. Lightwave Technol. 17, 2285–2290 (1999).
    [Crossref]
  20. R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics, Vol. I, (Addison-Wesley1965).
  21. W. Zeiler, F. D. Pasquale, P. Bayvl, and J. E. Midwinter, “Modeling of four-wave mixng and gain peaking in amplified WDM optical communication systems and networks,” J. Lightwave Technol. 14, 1933–1942 (1996).
    [Crossref]
  22. Y. Atzmon and M. Nazarathy, “A Gaussian Polar Model for Error Rates of Differential Phase Detection Impaired by Linear, Non-Linear and Laser Phase Noises,” J. Lightwave Technol. (acccepted for publication).
  23. C. Hiew, F. M. Abbou, H. T. Chuah, S. P. Majumder, and A. A. R. Hairul, “BER estimation of optical WDM RZ-DPSK systems through the differential phase Q,” IEEE Photon. Technol. Lett. 16, 2619–2621 (2004).
    [Crossref]

2008 (2)

2007 (4)

2006 (1)

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

2004 (1)

C. Hiew, F. M. Abbou, H. T. Chuah, S. P. Majumder, and A. A. R. Hairul, “BER estimation of optical WDM RZ-DPSK systems through the differential phase Q,” IEEE Photon. Technol. Lett. 16, 2619–2621 (2004).
[Crossref]

1999 (2)

1997 (1)

T. Yamamoto and M. Nakazawa “Highly efficient four-wave mixing in an optical fiber with intensity dependent phase matching,” J. Lightwave Technol. 9, 327–329 (1997).

1996 (3)

T.-K. Chiang, N. Kagi, M. E. Marhic, and L. G. Kazovsky, “Cross-phase modulation in fiber links with multiple optical amplifiers and dispersion compensation,” J. Lightwave Technol. 14249–260 (1996).
[Crossref]

M. E. Marhic, N. Kagi, T.-K. Chiang, and L. G. Kazovsky, “Optimizing the location of dispersion compensators in periodically amplified fiber links in the presence of third-order nonlinear effects” Photon Technol. Lett. 8, 145–147 (1996).
[Crossref]

W. Zeiler, F. D. Pasquale, P. Bayvl, and J. E. Midwinter, “Modeling of four-wave mixng and gain peaking in amplified WDM optical communication systems and networks,” J. Lightwave Technol. 14, 1933–1942 (1996).
[Crossref]

1992 (1)

K. Inoue, “Phase mismatching characteristic of four-wave mixing in fiber lines with multistage optical amplifiers” Opt. Lett 17, 801 (1992).
[Crossref] [PubMed]

1991 (1)

D. G. Schadt, “Effect of amplifier spacing on four wave mixing in multichannel coherent communications,” Electron. Lett 27, 805–7 (1991).
[Crossref]

Abbou, F. M.

C. Hiew, F. M. Abbou, H. T. Chuah, S. P. Majumder, and A. A. R. Hairul, “BER estimation of optical WDM RZ-DPSK systems through the differential phase Q,” IEEE Photon. Technol. Lett. 16, 2619–2621 (2004).
[Crossref]

Agrawal, G.P.

G.P. Agrawal, Lightwave Technology: Telecommunication Systems (Wiley2005).
[Crossref]

Allen, C. T.

Athaudage, C.

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

Atzmon, Y.

Y. Atzmon and M. Nazarathy, “A Gaussian Polar Model for Error Rates of Differential Phase Detection Impaired by Linear, Non-Linear and Laser Phase Noises,” J. Lightwave Technol. (acccepted for publication).

Bao, H.

Bayvl, P.

W. Zeiler, F. D. Pasquale, P. Bayvl, and J. E. Midwinter, “Modeling of four-wave mixng and gain peaking in amplified WDM optical communication systems and networks,” J. Lightwave Technol. 14, 1933–1942 (1996).
[Crossref]

Chen, W.

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 compensation,” J. Lightwave Technol. 14249–260 (1996).
[Crossref]

M. E. Marhic, N. Kagi, T.-K. Chiang, and L. G. Kazovsky, “Optimizing the location of dispersion compensators in periodically amplified fiber links in the presence of third-order nonlinear effects” Photon Technol. Lett. 8, 145–147 (1996).
[Crossref]

Cho, P.

M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, and V. Karagodsky, “The FWM Impairment in Coherent OFDM Compounds on a Phased-Array Basis over Dispersive Multi-Span Links” COTA’08, Boston, July 13–16, (2008).

Chuah, H. T.

C. Hiew, F. M. Abbou, H. T. Chuah, S. P. Majumder, and A. A. R. Hairul, “BER estimation of optical WDM RZ-DPSK systems through the differential phase Q,” IEEE Photon. Technol. Lett. 16, 2619–2621 (2004).
[Crossref]

Demarest, K. R.

Eiselt, M.

Feynman, R. P.

R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics, Vol. I, (Addison-Wesley1965).

Hairul, A. A. R.

C. Hiew, F. M. Abbou, H. T. Chuah, S. P. Majumder, and A. A. R. Hairul, “BER estimation of optical WDM RZ-DPSK systems through the differential phase Q,” IEEE Photon. Technol. Lett. 16, 2619–2621 (2004).
[Crossref]

Hiew, C.

C. Hiew, F. M. Abbou, H. T. Chuah, S. P. Majumder, and A. A. R. Hairul, “BER estimation of optical WDM RZ-DPSK systems through the differential phase Q,” IEEE Photon. Technol. Lett. 16, 2619–2621 (2004).
[Crossref]

Hui, R.

Inoue, K.

K. Inoue, “Phase mismatching characteristic of four-wave mixing in fiber lines with multistage optical amplifiers” Opt. Lett 17, 801 (1992).
[Crossref] [PubMed]

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 compensation,” J. Lightwave Technol. 14249–260 (1996).
[Crossref]

M. E. Marhic, N. Kagi, T.-K. Chiang, and L. G. Kazovsky, “Optimizing the location of dispersion compensators in periodically amplified fiber links in the presence of third-order nonlinear effects” Photon Technol. Lett. 8, 145–147 (1996).
[Crossref]

Karagodsky, V.

M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, and V. Karagodsky, “The FWM Impairment in Coherent OFDM Compounds on a Phased-Array Basis over Dispersive Multi-Span Links” COTA’08, Boston, July 13–16, (2008).

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 compensation,” J. Lightwave Technol. 14249–260 (1996).
[Crossref]

M. E. Marhic, N. Kagi, T.-K. Chiang, and L. G. Kazovsky, “Optimizing the location of dispersion compensators in periodically amplified fiber links in the presence of third-order nonlinear effects” Photon Technol. Lett. 8, 145–147 (1996).
[Crossref]

Khurgin, J.

M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, and V. Karagodsky, “The FWM Impairment in Coherent OFDM Compounds on a Phased-Array Basis over Dispersive Multi-Span Links” COTA’08, Boston, July 13–16, (2008).

Leighton, R. B.

R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics, Vol. I, (Addison-Wesley1965).

Lowery, J.

J. Lowery, “Amplified-spontaneous noise limit of optical OFDM lightwave systems,” Opt. Express 16, 860–865, (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-2-860.
[Crossref] [PubMed]

J. Lowery, S. Wang, M. Premaratne, and J. Lowery, “Fiber nonlinearity pre- and post-compensation for long-haul optical links using OFDM” Opt. Express15, 12965–12970, (2007), http://www.opticsexpress.org/abstract.cfm?id=125469.
[Crossref] [PubMed]

J. Lowery, S. Wang, M. Premaratne, and J. Lowery, “Fiber nonlinearity pre- and post-compensation for long-haul optical links using OFDM” Opt. Express15, 12965–12970, (2007), http://www.opticsexpress.org/abstract.cfm?id=125469.
[Crossref] [PubMed]

Ma, Y.

Majumder, S. P.

C. Hiew, F. M. Abbou, H. T. Chuah, S. P. Majumder, and A. A. R. Hairul, “BER estimation of optical WDM RZ-DPSK systems through the differential phase Q,” IEEE Photon. Technol. Lett. 16, 2619–2621 (2004).
[Crossref]

Marhic, M. E.

M. E. Marhic, N. Kagi, T.-K. Chiang, and L. G. Kazovsky, “Optimizing the location of dispersion compensators in periodically amplified fiber links in the presence of third-order nonlinear effects” Photon Technol. Lett. 8, 145–147 (1996).
[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 compensation,” J. Lightwave Technol. 14249–260 (1996).
[Crossref]

Meiman, Y.

M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, and V. Karagodsky, “The FWM Impairment in Coherent OFDM Compounds on a Phased-Array Basis over Dispersive Multi-Span Links” COTA’08, Boston, July 13–16, (2008).

Midwinter, J. E.

W. Zeiler, F. D. Pasquale, P. Bayvl, and J. E. Midwinter, “Modeling of four-wave mixng and gain peaking in amplified WDM optical communication systems and networks,” J. Lightwave Technol. 14, 1933–1942 (1996).
[Crossref]

Nakazawa, M.

T. Yamamoto and M. Nakazawa “Highly efficient four-wave mixing in an optical fiber with intensity dependent phase matching,” J. Lightwave Technol. 9, 327–329 (1997).

Nazarathy, M.

Y. Atzmon and M. Nazarathy, “A Gaussian Polar Model for Error Rates of Differential Phase Detection Impaired by Linear, Non-Linear and Laser Phase Noises,” J. Lightwave Technol. (acccepted for publication).

M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, and V. Karagodsky, “The FWM Impairment in Coherent OFDM Compounds on a Phased-Array Basis over Dispersive Multi-Span Links” COTA’08, Boston, July 13–16, (2008).

Noe, R.

M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, and V. Karagodsky, “The FWM Impairment in Coherent OFDM Compounds on a Phased-Array Basis over Dispersive Multi-Span Links” COTA’08, Boston, July 13–16, (2008).

Pasquale, F. D.

W. Zeiler, F. D. Pasquale, P. Bayvl, and J. E. Midwinter, “Modeling of four-wave mixng and gain peaking in amplified WDM optical communication systems and networks,” J. Lightwave Technol. 14, 1933–1942 (1996).
[Crossref]

Pendock, G.

Premaratne, M.

J. Lowery, S. Wang, M. Premaratne, and J. Lowery, “Fiber nonlinearity pre- and post-compensation for long-haul optical links using OFDM” Opt. Express15, 12965–12970, (2007), http://www.opticsexpress.org/abstract.cfm?id=125469.
[Crossref] [PubMed]

Sands, M.

R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics, Vol. I, (Addison-Wesley1965).

Schadt, D. G.

D. G. Schadt, “Effect of amplifier spacing on four wave mixing in multichannel coherent communications,” Electron. Lett 27, 805–7 (1991).
[Crossref]

Schneider, T.

T. Schneider, Nonlinear Optics in Telecommunications (Springer2004).

Shieh, W.

Shpantzer, I.

M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, and V. Karagodsky, “The FWM Impairment in Coherent OFDM Compounds on a Phased-Array Basis over Dispersive Multi-Span Links” COTA’08, Boston, July 13–16, (2008).

Song, S.

Tang, Y.

Tucker, R. S.

Wang, S.

J. Lowery, S. Wang, M. Premaratne, and J. Lowery, “Fiber nonlinearity pre- and post-compensation for long-haul optical links using OFDM” Opt. Express15, 12965–12970, (2007), http://www.opticsexpress.org/abstract.cfm?id=125469.
[Crossref] [PubMed]

Weidenfeld, R.

M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, and V. Karagodsky, “The FWM Impairment in Coherent OFDM Compounds on a Phased-Array Basis over Dispersive Multi-Span Links” COTA’08, Boston, July 13–16, (2008).

Yamamoto, T.

T. Yamamoto and M. Nakazawa “Highly efficient four-wave mixing in an optical fiber with intensity dependent phase matching,” J. Lightwave Technol. 9, 327–329 (1997).

Yi, X.

Zeiler, W.

W. Zeiler, F. D. Pasquale, P. Bayvl, and J. E. Midwinter, “Modeling of four-wave mixng and gain peaking in amplified WDM optical communication systems and networks,” J. Lightwave Technol. 14, 1933–1942 (1996).
[Crossref]

Electron. Lett (1)

D. G. Schadt, “Effect of amplifier spacing on four wave mixing in multichannel coherent communications,” Electron. Lett 27, 805–7 (1991).
[Crossref]

Electron. Lett. (2)

W. Shieh, X. Yi, and Y. Tang, “Transmission experiment of multi-gigabit coherent optical OFDM systems over 1000 km SSMF fiber,” Electron. Lett. 43, 183–185 (2007).
[Crossref]

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

IEEE Photon. Technol. Lett. (1)

C. Hiew, F. M. Abbou, H. T. Chuah, S. P. Majumder, and A. A. R. Hairul, “BER estimation of optical WDM RZ-DPSK systems through the differential phase Q,” IEEE Photon. Technol. Lett. 16, 2619–2621 (2004).
[Crossref]

J. Lightwave Technol. (6)

W. Zeiler, F. D. Pasquale, P. Bayvl, and J. E. Midwinter, “Modeling of four-wave mixng and gain peaking in amplified WDM optical communication systems and networks,” J. Lightwave Technol. 14, 1933–1942 (1996).
[Crossref]

Y. Atzmon and M. Nazarathy, “A Gaussian Polar Model for Error Rates of Differential Phase Detection Impaired by Linear, Non-Linear and Laser Phase Noises,” J. Lightwave Technol. (acccepted for publication).

T.-K. Chiang, N. Kagi, M. E. Marhic, and L. G. Kazovsky, “Cross-phase modulation in fiber links with multiple optical amplifiers and dispersion compensation,” J. Lightwave Technol. 14249–260 (1996).
[Crossref]

M. Eiselt, “Limits on WDM Systems Due to Four-Wave Mixing: A Statistical Approach,” J. Lightwave Technol. 17, 2261–2267 (1999).
[Crossref]

T. Yamamoto and M. Nakazawa “Highly efficient four-wave mixing in an optical fiber with intensity dependent phase matching,” J. Lightwave Technol. 9, 327–329 (1997).

S. Song, C. T. Allen, K. R. Demarest, and R. Hui, “Intensity-Dependent Phase-Matching Effects on Four-Wave Mixing in Optical Fibers,” J. Lightwave Technol. 17, 2285–2290 (1999).
[Crossref]

Opt. Express (5)

Opt. Lett (1)

K. Inoue, “Phase mismatching characteristic of four-wave mixing in fiber lines with multistage optical amplifiers” Opt. Lett 17, 801 (1992).
[Crossref] [PubMed]

Photon Technol. Lett. (1)

M. E. Marhic, N. Kagi, T.-K. Chiang, and L. G. Kazovsky, “Optimizing the location of dispersion compensators in periodically amplified fiber links in the presence of third-order nonlinear effects” Photon Technol. Lett. 8, 145–147 (1996).
[Crossref]

Other (6)

J. Lowery, S. Wang, M. Premaratne, and J. Lowery, “Fiber nonlinearity pre- and post-compensation for long-haul optical links using OFDM” Opt. Express15, 12965–12970, (2007), http://www.opticsexpress.org/abstract.cfm?id=125469.
[Crossref] [PubMed]

R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics, Vol. I, (Addison-Wesley1965).

G.P. Agrawal, Lightwave Technology: Telecommunication Systems (Wiley2005).
[Crossref]

T. Schneider, Nonlinear Optics in Telecommunications (Springer2004).

M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, and V. Karagodsky, “The FWM Impairment in Coherent OFDM Compounds on a Phased-Array Basis over Dispersive Multi-Span Links” COTA’08, Boston, July 13–16, (2008).

Please notice that in the published proceedings of [12] an error fell in the stated system performance. The early simulations did not account for the coherent addition of transposed pairs of non-degenerate intermods, as properly carried out in section 6 of this paper.

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Fig. 1.
Fig. 1.

(a): Graphical (j,k)-plane representation of the “FWM set” S[i] of 12033 intermods (FWM mixing products) falling onto sub-channel i=64, for an OFDM system with M=128 sub-channels on a regular frequency grid. (b),(c): Array Factor Dinc function parameterized by N span =83 (section 5). (b) is a zoomed-in version of (c). (a): The points corresponding to the intermods in S[64], reside on a dense 2D grid (the individual “beat” points may be resolved in the e-version by sufficiently zooming in), forming an elongated tilted hexagon with two right angles, in turn partitioned into two subsets: the star-shaped “mainlobe” region consisting of the black points, and its complement, the “sidelobes” region, containing the orange points (appearing gray in colorless print). The excluded SPM/XPM intermods are the missing points in the white crosshairs at j=64 or k=64. A point at [j,k] then represents the triplet of sub-channels indexed [j,k,j+k-64], generating a FWM beat at sub-channel 64. The total FWM power at sub-channel 64 is then the sum of 12033 contributions from each of the intermods. The relative strength of the FWM contribution of each beat (the amount of FWM attenuation relative to the dispersionless case) is indirectly represented by the contour map blue overlay. The rest of this caption describing (b),(c) and the blue overlay, refers to sections 5,8. It is shown there that the points of the mainlobe/sidelobe regions (j,k)-plane in the respectively map into the first mainlobe (|u|<1)/sidelobes (|u|>1) of the dinc function representing the phased-array FWM cancellation factor. Accordingly, the orange (gray) intermods in the “sidelobe” set experience high FWM attenuation, when accounting for the phased-array effect, making very weak contributions to the overall buildup of FWM at sub-channel i=64 (for an 83-spans system, the peaks of the dinc sidelobes range from -13.3 dB to -38.4 dB). Most of the black points in the star-shaped “mainlobe” make strong FWM contributions, however there are just 320 such points (or 3.2% of the 12033 intermods), hence, fortunately, the overall (rms-average) FWM suppression is dominated by the very weak contributions (high FWM attenuation) of the vastly larger sidelobes region occupying 96.8%. The overall FWM suppression attained with this configuration turns out to be 18.4 dB. The blue contour lines are hyperbolas, with their labels, u, representing the normalized hyperbolic distances between each point and the [64, 64] center point (normalized by the critical HD, d h crit for 200 MHz sub-channel separation and standard G.652 fiber). Six such contours are actually plotted with labels u=1,5,35,82,84,125, however one should visualize the infinite family of contour lines passing through every point. Consider the contour line passing through a given beat point. As indicated by the elbowed arrows connecting between the graphs, the contour line labels u represent the argument values whereat the dinc function (b) or (c) is sampled in order to obtain the corresponding array factor (FWM attenuation) for the given beat. E.g., any beat point on the displayed integer-labeled contour lines generates no FWM (as the dinc is sampled at its zero crossings). The thicker contours lines labeled 1, 82, 84 have special significance. The boundary between the main mainlobe (black) and sidelobes (orange) region is the thick blue contour line with label u=1. As seen in (b) the abscissa u=1 corresponds to the first zero-crossing separating the mainlobe of the dinc function from its first sidelobe. The narrow stripe between the two thick contour lines labeled {82,84}=83±1 represents the second mainlobe, |u-83|<1, for a 83-spans system treated here. Most intermods within this secondary mainlobe, yields high FWM, but there are very few of these intermods. Indeed, this region, consisting of a very thin, curved stripe, captures just a few grid points (in a zoom-in not reproduced here, just 23 beat points were counted within either of the two thin curved stripes in the second and fourth quadrants, with many of these point landing quite close to the stripe boundaries (the contour lines labeled 82 and 84), hence potentially experiencing higher attenuation even though these are mainlobe points. This effect will be confirmed in Fig. 3, plotting the amount of FWM attenuation experienced by all intermods.

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Fig. 2.
Fig. 2.

Array Factor (dinc function) as resultant (red arrow) adding up N span phasors (black arrows), each of length 1/N span ,, regularly dephased by an angle θ=β ijk L span =2π/N coh =2πu ijk N span between successive phasors. (a): N span =12, θ=10°, 14°, 18°, 26°, 30°. In the last case the polygon closes upon itself (12·30°=360°) corresponding to the first zero crossing of the dinc. The other five points sample the dinc in its mainlobe. (b): θ=18°, N span =1,4,8,16,32,64. In last two cases the polygon retraces itself, making several revolutions. In fact N span =20 accomplishes one full revolution. 64 modulo 12=4, hence the resultants for N span =4,64 are parallel. The condition for making one complete revolution (which yields zero resultant) is N span θ=2π or N span =N coh . The condition for zero resultant (possibly making multiple complete revolutions) is that N coh divide N span . When N span <N coh (N span >N coh ) the dinc is sampled in its mainlobe (sidelobes). In the dinc sidelobes, the polygon curls up upon itself, completing at least one full revolution, while becoming quite small.

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Fig. 3.
Fig. 3.

(a): Single-span FWM attenuation L̂ FWM ijk (contour plot labels are in dB units) for the [j,k] intermods falling onto sub-channel i=64, over the elongated tilted hexagonal-shaped S[64] domain. The relevant link parameters are stated in the text. A key parameter is the 200 MHz OFDM subcarrier separation. Even at the dispersive phase phase mismatch corresponding to this large frequency separation, most of the mixing products experience less than 0.5 dB FWM attenuation. The average single-span FWM suppression over all intermods is just 1 dB (relative to a dispersion-free system). Figs. (b)–(e) address an 83-span system with the same parameters, but with the phased-array effect at work, plotting the D̂ FWM ijk =F ijk L̂ FWM ijk FWM efficiency (attenuation) over all intermods. In particular, Figs. (b),(c) provide respective 3D and 2D top views (with contour lines at 2 dB intervals) over the [-18.4,0] dB vertical range, whereas (d),(e) provide a deeper cross-section of the FWM attenuation function over the [-30.4,-18.4] dB range. (b) should be visualized laid on top of (d). The cross-sectional level of -18.4 dB coincides with the average FWM suppression over all intermods. Consistent with this average value, the efficiencies of most intermods are seen to fall under the -18.4 dB floor in (b) or (c) (most of the area is white in (c) and most of the area of the bottom facet of the box is visible in (b); conversely the cross-shaped mainlobe occupies a relatively small area; Notice that (b),(c) also feature the first sidelobe barely sticking up through the bottom, in the four quadrants). Further inspecting (d), (e) it is apparent that many of the intermods have FWM efficiencies falling even lower than -30.4 dB. All this is indicative of the very favorable averaging of FWM attenuation, making it plausible that the large overall 18.4 dB suppression be attained. Figs. (b)–(e) also display “topographic features” consisting of “ridges/spikes/thin sheets” sticking out in the second and fourth quadrant. The origin of this effect is the sparse second-mainlobe region, |u-83|<1, as detailed in the caption of Fig. 1.

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Fig. 4.
Fig. 4.

BER performance of two OFDM links both links attaining BER=10-3, a BER level still amenable to Forward Error Correction, compatible with OFDM: (a): 33-span link carrying 40 Gb/s, with dispersion compensated at the end of every span, hence not using a cyclic prefix - does not display the phased-array effect, as its fully compensated spans add up coherently. (b): 87-span link with no in-line dispersion compensation — dispersion is electronically compensated in the receiver, based on the cyclic prefix inserted at the transmitter. The solid curves in (a) and (b) are the BERs related to FWM, ASE-induced phase noise, and both effects combined (TOTAL). The dotted curves in (a) are the FWM-induced BER, and total (FWM+ASE) BER for a 40 Gb/s system either in the absence of dispersion or for weak dispersion, e.g. when M=512 sub-channels are used in the system (a) to carry 40 Gb/s over 11.11 GHz such that the subcarrier separation is 11.11 GHz/512=21.7 MHz, in which case the effect of dispersion is unnoticeable with the densely packed subcarriers. The cyclic-prefix-free design (a) is superior in its spectral efficiency, relative to the ultra-long-haul system (b), which has its spectral efficiency reduced by a factor of 0.35, relative to that of system (a), as explained in the text. Therefore the total bandwidth required by system (a) is 1/0.35=2.83 times smaller. To make a fair comparison between the two systems, the spectral efficiency advantage of the cyclic-prefix-free system (a) is utilized to carry 2.83 times more data over this system, 40 Gb/s×2.83=113 Gb/s, while bringing the bandwidths of the two uneven-spectralefficiency systems to be equal (both 31.4 GHz), by increasing the spectral separation between the subcarriers of system (a) by a factor of 2.83 to 21.7 MHz×2.83=61.33 MHz (to coincide with that of system (b)) while retaining the 512-point FFT. Having increased spectral separation between subcarriers — see solid FWM curve in (a) — actually helps system (a) providing some dispersive phase mismatch even over a single span, ameliorating the FWM impairment by almost 1 dB in this span-by-span configuration —with no phased-array effect. The resulting TOTAL solid curve there is slightly better than the dispersion-free or weak dispersion case (the dotted lines), accounting for an increase in reach by one span (from 32 to 33 spans). Once all in-line dispersion compensation is removed (b), the phased-array effect is seen to provide dramatically enhanced reach (87 spans), almost tripling transmission range relative to (a), by virtue of the large FWM suppression generated via the phased-array effect. We conclude that a conventional design (a), with span-by-span dispersion compensation, is very inefficient in reach, and lacks flexibility in provisioning, however such cyclic-prefix-free design is superior in its spectral efficiency, relative to the ultra-long-haul system (b), which has its spectral efficiency reduced by a factor of 0.35, relative to that of system (a), as explained in the text.

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Equations (142)

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u ( t ; z ) = Re { u ∼ ( t ; z ) e j ω 0 t } = Re { u ˘ ( t ; z ) e − j ( β 0 z − ω 0 t ) }
u ∼ i ( t , z ) = u ˘ i ( t , z ) e − j β 0 z
u ˘ ( t ; z ) = ∑ i = 1 M u ˘ i ( t ; z ) e j Ω i t
S ∼ i ( t ) = u ˘ i ( t , 0 ) e j Ω i t = A c e j Ω i t e j μ 0 ( i ) 1 [ − T 2 , T 2 ] ( t )
G OA = e α L span .
y ( t ) = 1 T ∫ − T 2 T 2 x ( t ) dt .
∂ z u ˘ − j ( β ″ 2 ) ∂ t 2 u ˘ + ( α 2 ) u ˘ = − j γ ∣ u ˘ ∣ 2 u ˘ .
∂ z u ˘ i + β ″ Ω i ∂ t u ˘ i − j β ″ 2 ∂ t 2 u ˘ i = − j β i T u ˘ i − j γ ∑ [ j , k , l ] ∈ S [ i ] u ˘ j u ˘ k u ˘ l *
S [ i ] ≡ { [ j , k , − l ] : ω j + ω k − ω l = ω i , j ≠ i ≠ k } , i = 1 , 2 , … , M ,
β i T ≡ β ″ 2 Ω i 2 − j α 2 + γ ( ∣ u ˘ i ∣ 2 + 2 ∑ k = 1 k ≠ i M ∣ u ˘ k ∣ 2 ) = β i DL − β 0 + 2 γ P T − γ p i
β i DL ≡ β 0 + ( β ″ 2 ) Ω i 2 − j α ( z ) 2 = β i − j α ( z ) 2
β i ≡ β 0 + ( β ″ 2 ) Ω i 2
β i T = β i − β 0 − j α 2 + 2 γ P T − γ p i
u ˘ i = u ˘ i ( 1 ) + u ˘ i ( 3 ) ,
∂ z u ˘ i ( 1 ) + β ″ Ω i ∂ t u ˘ i ( 1 ) − j β ″ 2 ∂ t 2 u ˘ i ( 1 ) + j β i T u ˘ i ( 1 ) = 0
∂ z u ˘ i ( 3 ) + β ″ Ω i ∂ t u ˘ i ( 3 ) − j β ″ 2 ∂ t 2 u ˘ i ( 3 ) + j β i T u ˘ i ( 3 ) = − j γ ∑ [ j , k , l ] ∈ S [ i ] u ˘ j ( 1 ) u ˘ k ( 1 ) u ˘ l ( 1 ) *
β i T ( t , z ) ≡ β ″ 2 Ω i 2 − j α 2 + γ ( ∣ u ˘ i ( 1 ) ∣ 2 + 2 ∑ k = 1 k ≠ i M ∣ u ˘ k ( 1 ) ∣ 2 ) = β i DL − β 0 + 2 γ P T ( t , z ) − γ p i ( 1 ) ( t , z )
∂ z u ˘ i ( 1 ) = − j β i T u ˘ i ( 1 )
∂ z u ˘ i ( 3 ) = − j β i T u ˘ i ( 3 ) − j γ ∑ [ j , k , l ] ∈ S [ i ] u ˘ j ( 1 ) u ˘ k ( 1 ) u ˘ l ( 1 ) *
u ˘ i ( 1 ) ( t , z ) = u ˘ i ( 1 ) ( t , 0 ) e − j ∫ 0 z β i T ( t , z ′ ) dz ′ = A c e j μ 0 ( i ) 1 [ − T 2 , T 2 ] ( t ) e − ( α 0 2 ) z e − j ( β ″ 2 ) Ω i 2 z e − j γ ( 2 M − 1 ) P 0 Z eff
Z eff = ∫ 0 z e − α 0 z ′ dz ′ = ( 1 − e − α 0 z ) α 0
ν ˘ i ( t ; z ) = u ˘ i ( t ; z ) e j ∫ 0 z β i T ( t , z ′ ) dz′
u ˘ i ( t ; z ) = ν ˘ i ( t ; z ) e − j ∫ 0 z β i T ( t , z ′ ) dz′
∂ z ν ˘ i ( 1 ) = 0
∂ z ν ˘ i ( 3 ) = − j γ ∑ [ j , k , l ] ∈ S [ i ] ν ˘ j ( 1 ) ν ˘ k ( 1 ) ν ˘ l ( 1 ) * e − j Δ Φ T ( t , z )
Δ Φ T ( t , z ) = ∫ 0 z Δ β ijkl T ( t , z ′ ) dz ′ = ∫ 0 z Δ β ijkl DL ( t , z ′ ) dz ′ − γ ∫ 0 z Δ p ijkl ( t , z ′ ) dz′
Δ β ijkl T ≡ β j T + β k T − ( β l T ) * − β i T = Δ β i DL − γ Δ p ijkl
Δ β ijkl DL ≡ β j DL + β k DL − ( β l DL ) * − β i DL
= β j + β k − β l − β i − j α = β ″ 2 [ ( Ω j 2 + Ω k 2 − Ω l 2 ) − Ω i 2 ] − j α ( z )
Δ β ijkl ≡ β ″ 2 [ ( Ω j 2 + Ω k 2 − Ω l 2 ) − Ω i 2 ]
Δ p ijkl ( t , z ) ≡ p j ( t , z ) + p k ( t , z ) − p l ( t , z ) − p i ( t , z ) .
α ( z ) = α 0 − α 0 L span Σ s = 1 N span − 1 δ ( z − s L span )
∀ z : ν ˘ i ( 1 ) ( t , z ) = u ˘ i ( 1 ) ( t , 0 ) = p 0 ( t ) e j ϕ i ( t ) ,
∀ z : p i ( t , z ) ≡ ∣ ν ˘ i ( 1 ) ( t , z ) ∣ 2 = p 0 ( t )
∂ z ν ˘ i ( 3 ) = − j γ ∑ [ j , k ] ∈ S [ i ] ν ˘ j ( 1 ) ν ˘ k ( 1 ) ν ˘ j + k − i ( 1 ) * e − j Δ Φ T ( t , z )
S [ i ] = { [ j , k ] ∣ 1 ≤ j , k ≤ M & 0 < j + k − i ≤ M & j ≠ i ≠ k } ,
Δ β ijk ≡ β j + β k − β j + k − i − β i = β ″ ( Δ ω ) 2 2 [ ( j 2 + k 2 − ( j + k − i ) 2 ) − i 2 ]
Δ β ijk = − β ″ ( Δ ω ) 2 ( j − i ) ( k − i ) = − β ″ ( Δ ω ) 2 j ′ k ′
Δ Φ T ( t , z ) = ∫ 0 z Δ β ijk T ( t , z ′ ) dz ′ = ∫ 0 z Δ β ijk DL ( t , z ′ ) dz ′ − γ ∫ 0 z Δ p ijk ( t , z ′ ) dz′
Δ β ijk T ≡ β j T + β k T − β j + k − i T − β i T = Δ β ijk DL − γ Δ p ijk , with
Δ β ijk DL = Δ β ijk − j α ( z ) ,
Δ p ijk ( t , z ) ≡ p j ( t , z ) + p k ( t , z ) − p j + k − i ( t , z ) − p i ( t , z ) .
p j ( 1 ) ( t , z ) = p k ( 1 ) ( t , z ) = p j + k − i ( 1 ) ( t , z ) = p i ( 1 ) ( t , z ) ≡ p ( 1 ) ( t , z ) = p 0 ( t ) ∫ 0 z e − α ( z ′ ) dz ′ .
∂ z ν ˘ i ( 3 ) = − j γ ∑ [ j , k ] ∈ S [ i ] ν ˘ j ( 1 ) ν ˘ k ( 1 ) ν j + k − i ( 1 ) * e − j Δ Φ DL ( t , z )
Δ Φ DL ( t , z ) ≡ ∫ 0 z Δ β ijk DL ( t , z ′ ) dz ′ = ∫ 0 z Δ β ijk ( t , z ′ ) dz ′ − j ∫ 0 z α ( z ′ ) dz′
∂ z ν ˘ i ( 3 ) = − j γ ∑ [ j , k ] ∈ S [ i ] ν ˘ j ( 1 ) ν ˘ k ( 1 ) ν ˘ j + k − i ( 1 ) * e − j Δ β ijk DL z single-span equation
ν ˘ i ( 3 ) ( L ) = − j γ ∑ [ j , k ] ∈ S [ i ] ν ˘ j ( 1 ) ν ˘ k ( 1 ) ν ˘ j + k − i ( 1 ) * L ijk FWM single-span solution
L ijk FWM ≡ ∫ 0 L e − j Δ β ijk DL z dz = e − j Δ β ijk DL L − 1 − j Δ β ijk DL = 1 − e − α L e − j Δ β ijk L j Δ β ijk DL = 1 − e − α L e − j Δ β ijk L j Δ β ijk + α
L ijk FWM ∣ Δ β ijk = 0 = ( 1 − e − α L ) α ≡ L eff .
∣ u ˘ i ( 3 ) ( L ) ∣ 2 = ∣ ν ˘ i ( 3 ) ( L ) ∣ 2 e − α L = γ 2 e − α L p 0 3 ∣ ∑ [ j , k ] ∈ S [ i ] ∣ L ijk FWM ∣ e j ( ϕ j + ϕ k − ϕ j + k − i + ∠ L ijk FWM ( L ) ) ∣ 2
S DG [ i ] ≡ { [ j , k ] ∈ S [ i ] | j = k } = { [ j , j ] | j ≤ M & 0 < 2 j − i ≤ M & j ≠ i }
S NDG [ i ] ≡ { [ j , k ] ∈ S [ i ] ∣ j ≠ k } = S > NDG [ i ] ∪ S < NDG [ i ]
S > NDG [ i ] ≡ { [ j , k ] ∈ S [ i ] | j > k } ; S < NDG [ i ] ≡ { [ j , k ] ∈ S [ i ] | j < k }
Σ [ i ] ≡ ∑ [ j , k ] ∈ S [ i ] ∣ L ijk FWM ∣ e j ( ϕ j + ϕ k − ϕ j + k − i + ∠ L ijk FWM ( L ) ) = Σ DG [ i ] + Σ NDG [ i ]
Σ DG [ i ] = ∑ [ j , i ] ∈ S DG [ i ] ∣ L ijj FWM ∣ e j ( ϕ j + ϕ j − ϕ 2 j − i + ∠ L ijj FWM ( L ) )
Σ NDG [ i ] ≡ ∑ [ j , k ] ∈ S > NDG [ i ] ∣ L ijk FWM ∣ { e j ( ϕ j + ϕ k − ϕ j + k − i + ∠ L ijk FWM ( L ) ) + e j ( ϕ k + ϕ j − ϕ k + j − i + ∠ L ikj FWM ( L ) ) }
= ∑ [ j , k ] ∈ S > NDG [ i ] ∣ L ijk FWM ∣ · 2 e j ( ϕ j + ϕ k + ϕ j + k − i ∠ L ijk FWM ( L ) )
〈 e j ( ϕ j + ϕ k − ϕ j + k − i + ∠ L ijk FWM ) e − j ( ϕ j ′ + ϕ k ′ − ϕ j ′ + k ′ − i + ∠ ij′ k ′ FWM ) 〉 = δ jj ′ δ kk ′
σ FWM 2 ≡ σ u ˘ i ( L ) 2 = 〈 ∣ u ˘ i ( 3 ) ( L ) ∣ 2 〉 = γ 2 e − α L p 0 3 { 4 ∑ [ j , k ] ∈ S > NDG [ i ] ∣ L ijk FWM ∣ 2 + ∑ [ j , j ] ∈ S DG [ i ] ∣ L ijj FWM ∣ 2 }
= γ 2 e − α L p 0 3 · 2 { ∑ [ j , k ] ∈ S NDG [ i ] ∣ L ijk FWM ∣ 2 + 1 2 ∑ [ j , j ] ∈ S DG [ i ] ∣ L ijj FWM ∣ 2 }
= ( γ L eff ) 2 e − α L p 0 3 · 2 { ∑ [ j , k ] ∈ S [ i ] ∣ L ˘ ijk FWM ∣ 2 − 1 2 ∑ [ j , j ] ∈ S DG [ i ] ∣ L ˘ ijj FWM ∣ 2 }
L ̂ ijk FWM ≡ L ijk FWM L eff = 1 − e − α L e − j Δ β ijk L j Δ β ijk + α 1 − e − α L α = ( 1 − e − α L e − j Δ β ijk L ) 1 − e − α L j Δ β ijk α + 1
L ̂ rms FWM [ i , M ] ≡ 〈 L ijk FWM 〉 rms = N beats − 1 [ i , M ] ∑ [ j , k ] ∈ S [ i ] ∣ L ̂ ijk FWM ∣ 2
N beats [ i , M ] ≡ ∣ S [ i ] ∣ = ( M 2 − 5 M + 2 ) 2 + ( M + 1 ) i − i 2
L ̂ rms DG − FWM [ i , M ] ≡ 〈 L ijj FWM 〉 rms = 1 N beats DG [ i , M ] ∑ [ j , j ] ∈ S DG [ i ] ∣ L ̂ ijj FWM ∣ 2
L ̂ eff FWM [ i , M ] ≡ 1 N beats [ i , M ] { ∑ [ j , k ] ∈ S [ i ] ∣ L ̂ ijk FWM ∣ 2 − 1 2 ∑ [ j , j ] ∈ S DG [ i ] ∣ L ̂ ijj FWM ∣ 2 }
= ( L ̂ rms FWM [ i , M ] ) 2 − N beats DG [ i , M ] 2 N beats [ i , M ] ( L ̂ rms DG − FWM [ i , M ] ) 2
σ u ˘ i ( L span ) 2 = 2 ( γ L eff L ̂ eff FWM [ i , M ] ) 2 N beats [ i , M ] p 0 3 e − α 0 L span single-span FWM power
r ∼ i ( 1 ) = e ( α 0 2 ) L span u ˘ i ( 1 ) ( L span ) .
σ FWM 2 [ i , M ] ≡ σ r i ∼ 2 = e α 0 L span σ u ˘ i ( L ) 2 = 2 ( γ L eff L ̂ eff FWM [ i , M ] ) 2 N beats [ i , M ] p 0 3
Δ Φ DL ( t , z ) = ∫ 0 z Δ β ijkl DL ( t , z ) dz = Δ β ijkl z − j ∫ 0 z α ( z ′ ) dz′ = Δ β ijkl z + j G log ( z )
G log ( z ) ≡ − ∫ 0 z α ( z ′ ) dz ′ = − α 0 [ z ] L span , 0 ≤ z ≤ L
[ z ] L span ≡ z mod L span .
1 [ 0 , L ] ( z ) e − j Δ Φ DL ( t , z ) = 1 [ 0 , L ] ( z ) e G log ( z ) e − j Δ β ijkl z ≡ G ( z ) e − j Δ β ijkl z
G ( z ) ≡ e G log ( z ) = e − α 0 [ z ] L span 1 [ 0 , L ] ( z ) = Σ S = 0 N span − 1 e − α 0 ( z − s L span ) 1 [ 0 , L span ] ( z − s L span ) .
G ( z ) = Σ S = 0 N span − 1 δ ( z − s L span ) ⊗ e − α 0 z 1 [ 0 , L span ] ( z )
∂ z ν ˘ i ( 3 ) = − j γ ∑ [ j , k ] ∈ S [ i ] ν ˘ j ( 1 ) ν ˘ k ( 1 ) ν ˘ j + k − i ( 1 ) * G ( z ) e − j Δ β ijk z
ν ˘ i ( 3 ) ( L ) = − j γ ∑ [ j , k ] ∈ S [ i ] ν ˘ j ( 1 ) ν ˘ k ( 1 ) ν ˘ j + k − i ( 1 ) * D ijk FWM = − j γ L eff N span ∑ [ j , k ] ∈ S [ i ] ν ˘ j ( 1 ) ν ˘ k ( 1 ) ν ˘ j + k − i ( 1 ) * D ̂ ijk FWM
D ijk FWM ≡ ∫ 0 L G ( z ) e − j Δ β ijkl z dz = ∫ − ∞ ∞ G ( z ) e − j Δ β ijkl z dz = F { G ( z ) } ∣ k = Δ β ijk
F { G ( z ) } ≡ ∫ − ∞ ∞ G ( z ) e − jkz dz
D ̂ ijk FWM ≡ D ijk FWM ( L eff N span ) .
∫ 0 L β i T ( t , z ′ ) dz ′ = ( β i − β 0 ) L − j 1 2 ∫ 0 L α ( z ) dz ′ = ( β i − β 0 ) L − j 1 2 ∫ L − L span L α 0 dz ′
= ( β i − β 0 ) L − ( α 0 2 ) L span
u ˘ i ( 3 ) ( L ) = v ˘ i ( 3 ) ( L ) e − j ( β i − β 0 ) L e ( − α 0 2 ) L span
r ∼ i = u ˘ i ( 3 ) ( L ) e ( α 0 2 ) L span = v ˘ i ( 3 ) ( L ) e − j ( β i − β 0 ) L
r ∼ i = − j γ e − j ( β i − β 0 ) L ∑ [ j , k ] ∈ S [ i ] v ˘ j ( 1 ) v ˘ k ( 1 ) v ˘ j + k − i ( 1 ) * D ijk FWM
F { ∑ s = 0 N span − 1 δ ( z − s L span ) } = ∑ s = 0 N span − 1 e − j Δ β ijkl s L span ≡ N span F ijk FWM
F { e − α 0 z 1 [ 0 , L span ] ( z ) } = ∫ 0 L span e − α 0 z e − j Δ β ijk z dz = ∫ 0 L span e − j Δ β ijk DL z dz ≡ L ijk FWM
D ijk FWM = F ijk N span · L ijk FWM
F ijk ≡ 1 N span ∑ s = 0 N span − 1 e − j Δ β ijkl s L span = e − j Δ β ijk ( L − L span ) 2 sin ⁡ ( Δ β ijk L 2 ) N span sin ⁡ ( Δ β ijk L span 2 )
dinc N [ u ] ≡ sin ⁡ ( π u ) N sin ⁡ ( π u N ) = N − 1 ( 1 + e − j θ + e − j 2 θ + … + e − j ( D − 1 ) θ ∣ θ → 2 π u N )
F ijk [ N span ] = e − j Δ β ijk ( L − L span ) 2 dinc N span [ L Δ β ijk 2 π ]
L coh ijk ≡ 2 π ∣ Δ β ijk ∣ = 2 π β ″ ( Δ ω ) 2 ∣ j − i ∣ · ∣ k − i ∣ ,
u ≡ L Δ β ijk 2 π = L L coh ijk = N span L span L coh ijk = N span N coh ijk
N coh ijk ≡ L coh ijk L span
∣ F ijk [ N span ] ∣ = ∣ dinc N span [ L L coh ijk ] ∣ = ∣ dinc N span [ N span N coh ijk ] ∣
D ̂ ijk FWM = F ijk L ̂ ijk FWM
G ( z ) = ∑ s = 0 N span − 1 δ ( z − z s ) ⊗ e − α s z 1 [ z s − z s − 1 ] ( z )
D ijk FWM = ∑ s = 0 N span − 1 e − j Δ β ijkl z s L ijk FWM [ s ] = ∑ s = 0 N span − 1 e − j ( Δ β ijkl z s + ∠ L ijk FWM [ s ] ) ∣ L ijk FWM [ s ] ∣
N coh ijk < N span or L coh ijk < L Large FWM attenuation condition
d h crit < ∣ j − i ∣ · ∣ k − i ∣ ≡ d h { [ j , k ] , [ i , i ] } High-FWM Mattenuation condition
d h { [ x 1 , x 2 ] , [ x 2 , x 2 ] } ≡ ∣ x 1 − x 2 ∣ · ∣ y 1 − y 2 ∣ .
d h crit ≡ 2 π L ∣ β ″ ∣ ( Δ ω ) 2 = [ 2 π L β ″ ( Δ ν ) 2 ] − 1 = M 2 [ 2 π β ″ L span N span W 2 ] − 1
u ijk ≡ N span N coh ijk = d h { [ j , k ] , [ i , i ] } d h crit = 2 π β ″ LW 2 M 2 d h { [ j , k ] , [ i , i ] }
d h { [ j , j ] , [ i , i ] } = ( j − i ) 2 = l 0 2 .
1 = 2 π β ″ ( LW 2 M 2 ) d h { [ j , j ] , [ i , i ] } = 2 π ( LW 2 β ″ M 2 ) l 0 2
σ FWM 2 [ i , M ] = 〈 ∣ r ∼ i ∣ 2 〉 = ( γ L eff N span ) 2 p 0 3 · 2 { ∑ [ j , k ] ∈ S [ i ] ∣ D ̂ ijk FWM ∣ 2 − 1 2 ∑ [ j , j ] ∈ S DG [ i ] ∣ D ̂ ijj FWM ∣ 2 }
D ̂ eff FWM [ i , M , N span ] ≡ ( D ̂ rms FWM [ i , M , N span ] ) 2 − N beats DG [ i , M ] 2 N beats [ i , M ] ( D ̂ rms DG − FWM [ i , M , N span ] ) 2
D ̂ rms FWM [ i , M , N span ] ≡ 1 N beats [ i , M ] ∑ [ j , k ] ∈ S [ i ] ∣ D ̂ ijk FWM ∣ 2
D ̂ rms DG − FWM [ i , M , N span ] ≡ 1 N beats DG [ i , M ] ∑ [ j , j ] ∈ S DG [ i ] ∣ D ̂ ijj FWM ∣ 2
σ FWM 2 = 2 ( γ L eff N span D ̂ eff FWM [ i , M , N span ] ) 2 N beats [ i , M ] p 0 3
σ ∠ FWM 2 [ i , M ] ≡ var { φ i } ≈ var { r i im A } = var { r ∼ i } ( 2 A 2 ) = σ FWM 2 [ i , M ] 2 p 0
p 0 = A 2 = P T M
σ ∠ FWM 2 [ i , M , N span ] = ( γ L eff N span D ̂ eff FWM [ i , M , N span ] ) 2 N beats [ i , M ] p 0 2
σ ∠ FWM 2 [ i , M , N span ] = ( γ L eff N span D ̂ eff FWM [ i , M , N span ] ) 2 N ̂ beats [ i , M ] · P T 2
σ ∠ FWM [ i , M , N span ] = γ L eff N span D ̂ eff FWM [ i , M , N span ] N ̂ beats [ i , M ] · P T dispersive
N ̂ beats [ i , M ] ≡ N beats [ i , M ] M 2 = 0.5 + ( i − 2.5 ) M + ( 1 + i − i 2 ) M 2 .
N ̂ beats [ M 2 , M ] = 3 4 − 2 M + 1 M 2 ; N ̂ beats [ 64 , 128 ] = 0.734 ≈ N ̂ beats [ M 2 , M ]
σ ∠ FWM 2 [ M 2 , M ] ≈ 0.734 ( γ L eff N span P T ) 2 dispersion-free
σ ∠ FWM [ M 2 , M , N span ] ≈ 0.857 γ L eff · N span D ̂ eff FWM [ i , M , N span ] P T dispersive
BER FWM [ i , M ] ≅ 2 Q [ q FWM [ i , M , N span ] ] ,
q FWM [ i , M , N span ] ≡ π m σ ∠ FWM [ i , M , N span ]
q ∠ FWM [ i , M , N span ] ≡ π m κ m σ ∠ FWM = π m γ L eff κ m N ̂ beats [ i , M ] D ̂ eff FWM [ i , M , N span ] N span P T
q ∠ FWM QPSK [ i , M , N span ] ≈ 0.0332 ( D ̂ eff FWM [ i , M , N span ] N span P T ) − 1
q ∠ FWM QPSK [ M 2 , M ] dBE ≈ − 29.6 − N span dBE − 2 [ P T dBm − 30 ] − ∣ D ̂ eff FWM [ i , M , N span ] ∣ dBE
( u ˘ j ( 1 ) + u ˘ j ( 3 ) ) ( u ˘ k ( 1 ) + u ˘ k ( 3 ) ) ( u ˘ l ( 1 ) * + u ˘ l ( 3 ) * ) ≈ u ˘ j ( 1 ) u ˘ k ( 1 ) u ˘ l ( 1 ) * +
u ˘ j ( 3 ) u ˘ k ( 1 ) u ˘ l ( 1 ) * + u ˘ j ( 1 ) u ˘ k ( 3 ) u ˘ l ( 1 ) * + u ˘ j ( 1 ) u ˘ k ( 1 ) u ˘ l ( 3 ) * + higher order terms
1 q ∠ T 2 = 1 q ∠ FWM 2 + 1 q ∠ LN 2 .
A e j ϕ i e j Ω i t + N ∼ 1 ( t ) + N ∼ 2 ( t ) + … + N ∼ N span + 1 ( t ) ,
ρ ( t ) = A + N ∼ 1 ( t ) + N ∼ 2 ( t ) + … + N ∼ N span + 1 ( t )
r ∼ = A + n ∼ 1 + n ∼ 2 + … + n ∼ N span + 1 , n ∼ s ≡ 1 T ∫ − T 2 T 2 N ∼ s ( t ) dt , s = 1 , 2 , … , N span + 1
ϕ LN ≡ ∠ r ∼ = ∠ { A + n ∼ 1 + n ∼ 2 + … + n ∼ N span + 1 } ≈ ∑ s = 1 N span + 1 n ∼ s im A
σ ∠ LN 2 ≡ Var ϕ LN = ( N span + 1 ) 1 2 N 0 / T ∣ A ∣ 2 = ( N span + 1 ) 1 2 N 0 / T P T / M = 1 2 N 0 ( N span + 1 ) W P T
OSNR OFDM ≡ P T 2 ( N span + 1 ) N 0 W ref = P T ( N span + 1 ) F N ( G OA − 1 ) h ν 0 W ref
σ ∠ LN 2 = OSNR OFDM − 1 · B ( 4 W ref ) = 1 4 ( N span + 1 ) F N ( G OA − 1 ) h ν 0 W · P T − 1
q ∠ LN ≡ π m κ m σ ∠ NL = π m OSNR OFDM κ m W ( 4 W ref )
q ∠ LN 2 ≡ ( π m ) 2 κ m σ ∠ NL − 2 = ( 2 π m ) 2 κ m [ F N ( G OA − 1 ) h ν 0 ( N span + 1 ) ] − 1 P T W
q ∠ T = π ( m κ m ( σ ∠ FWM 2 + σ ∠ NL 2 ) )
R b = 40 Gb s = 461 Ch · 21.7 Msym / s . Ch · Pol · 2 b sym · 2 pol
q ∠ FWM [ i , M , N span ] DC after every span = π m γ L eff κ m N ̂ beats [ i , M ] L ̂ eff FWM [ i , M ] N span P T
R b ( M ρ η ) = ( 1 Δ ν + T CP ) − 1
T CP = β ″ L ( 2 π W ) = 2 π β ″ N span L span M Δ ν

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