Abstract

We develop a simple formula for estimating the effect of Four-Wave Mixing (FWM) on received signal quality in coherent optical systems using Orthogonal Frequency Division Multiplexing (OFDM) for dispersion compensation. This shows the nonlinear limit is substantially independent of the number of OFDM subcarriers. Our analysis agrees well with full split-step Fourier method simulations, so allows the nonlinear limit of multi-span systems to be estimated without lengthy simulations.

© 2007 Optical Society of America

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References

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  1. A. R. S. Bahai, B. R. Saltzberg, and M. Ergen, Multi-carrier Digital Communications: Theory and Applications of OFDM, 2nd Edition, (Springer, New York, 2004).
  2. A. J. Lowery and J. Armstrong, “Orthogonal-frequency-division multiplexing for dispersion compensation of long-haul optical systems,” Opt. Express 14, 2079–2084 (2006).
    [CrossRef] [PubMed]
  3. W. Shieh, X. Yi, and Y. Tang, “Experimental Demonstration of Transmission of Coherent Optical OFDM Systems,” in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference, OSA Technical Digest Series (CD) (Optical Society of America, 2007), paper OMP2. http://www.opticsinfobase.org/abstract.cfm?URI=OFC-2007-OMP2
  4. B. J. Schmidt, A. J. Lowery, and J. Armstrong, “Experimental Demonstrations of 20 Gbit/s Direct-Detection Optical OFDM and 12 Gbit/s with a Colorless Transmitter,” in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference, OSA Technical Digest Series (CD) (Optical Society of America, 2007), paper PDP18. http://www.opticsinfobase.org/abstract.cfm?URI=OFC-2007-PDP18
  5. S. L. Jansen, I. Morita, N. Takeda, and H. Tanaka, “20-Gb/s OFDM Transmission over 4,160-km SSMF Enabled by RF-Pilot Tone Phase Noise Compensation,” in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference, OSA Technical Digest Series (CD) (Optical Society of America, 2007), paper PDP15. http://www.opticsinfobase.org/abstract.cfm?URI=OFC-2007-PDP15
  6. A. J. Lowery, L. B. Y. Du, and J. Armstrong, “Performance of optical OFDM in ultralong-haul WDM lightwave systems,” J. Lightwave Technol. 25, 131–138 (2007).
    [CrossRef]
  7. R. W. Tkach, A. R. Chraplyvy, F. Forghieri, A. H. Gnauck, and R. M. Derosier, “Four-photon mixing and high-speed WDM systems,” J. Lightwave Technol. 13, 841–849 (1995).
    [CrossRef]
  8. G. P. Agrawal, Nonlinear Fiber Optics (Optics and Photonics), 3rd Edition, (Academic Press, San Francisco, 2001)
  9. W. Shieh and C. Athaudage, “Coherent optical orthogonal frequency division multiplexing,” Electron. Lett. 42, 587–588 (2006).
    [CrossRef]
  10. K-D. Chang, G-C Yang, and W. C. Kwong, “Determination of FWM products in unequal-spaced-channel WDM lightwave systems,” J. Lightwave Technol. 18, 2113–2122 (2000).
    [CrossRef]

2007 (1)

2006 (2)

2000 (1)

1995 (1)

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

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Optics and Photonics), 3rd Edition, (Academic Press, San Francisco, 2001)

Armstrong, J.

A. J. Lowery, L. B. Y. Du, and J. Armstrong, “Performance of optical OFDM in ultralong-haul WDM lightwave systems,” J. Lightwave Technol. 25, 131–138 (2007).
[CrossRef]

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

B. J. Schmidt, A. J. Lowery, and J. Armstrong, “Experimental Demonstrations of 20 Gbit/s Direct-Detection Optical OFDM and 12 Gbit/s with a Colorless Transmitter,” in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference, OSA Technical Digest Series (CD) (Optical Society of America, 2007), paper PDP18. http://www.opticsinfobase.org/abstract.cfm?URI=OFC-2007-PDP18

Athaudage, C.

W. Shieh and C. Athaudage, “Coherent optical orthogonal frequency division multiplexing,” Electron. Lett. 42, 587–588 (2006).
[CrossRef]

Bahai, A. R. S.

A. R. S. Bahai, B. R. Saltzberg, and M. Ergen, Multi-carrier Digital Communications: Theory and Applications of OFDM, 2nd Edition, (Springer, New York, 2004).

Chang, K-D.

Chraplyvy, A. R.

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

Derosier, R. M.

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

Du, L. B. Y.

Ergen, M.

A. R. S. Bahai, B. R. Saltzberg, and M. Ergen, Multi-carrier Digital Communications: Theory and Applications of OFDM, 2nd Edition, (Springer, New York, 2004).

Forghieri, F.

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

Gnauck, A. H.

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

Jansen, S. L.

S. L. Jansen, I. Morita, N. Takeda, and H. Tanaka, “20-Gb/s OFDM Transmission over 4,160-km SSMF Enabled by RF-Pilot Tone Phase Noise Compensation,” in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference, OSA Technical Digest Series (CD) (Optical Society of America, 2007), paper PDP15. http://www.opticsinfobase.org/abstract.cfm?URI=OFC-2007-PDP15

Kwong, W. C.

Lowery, A. J.

A. J. Lowery, L. B. Y. Du, and J. Armstrong, “Performance of optical OFDM in ultralong-haul WDM lightwave systems,” J. Lightwave Technol. 25, 131–138 (2007).
[CrossRef]

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

B. J. Schmidt, A. J. Lowery, and J. Armstrong, “Experimental Demonstrations of 20 Gbit/s Direct-Detection Optical OFDM and 12 Gbit/s with a Colorless Transmitter,” in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference, OSA Technical Digest Series (CD) (Optical Society of America, 2007), paper PDP18. http://www.opticsinfobase.org/abstract.cfm?URI=OFC-2007-PDP18

Morita, I.

S. L. Jansen, I. Morita, N. Takeda, and H. Tanaka, “20-Gb/s OFDM Transmission over 4,160-km SSMF Enabled by RF-Pilot Tone Phase Noise Compensation,” in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference, OSA Technical Digest Series (CD) (Optical Society of America, 2007), paper PDP15. http://www.opticsinfobase.org/abstract.cfm?URI=OFC-2007-PDP15

Saltzberg, B. R.

A. R. S. Bahai, B. R. Saltzberg, and M. Ergen, Multi-carrier Digital Communications: Theory and Applications of OFDM, 2nd Edition, (Springer, New York, 2004).

Schmidt, B. J.

B. J. Schmidt, A. J. Lowery, and J. Armstrong, “Experimental Demonstrations of 20 Gbit/s Direct-Detection Optical OFDM and 12 Gbit/s with a Colorless Transmitter,” in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference, OSA Technical Digest Series (CD) (Optical Society of America, 2007), paper PDP18. http://www.opticsinfobase.org/abstract.cfm?URI=OFC-2007-PDP18

Shieh, W.

W. Shieh and C. Athaudage, “Coherent optical orthogonal frequency division multiplexing,” Electron. Lett. 42, 587–588 (2006).
[CrossRef]

W. Shieh, X. Yi, and Y. Tang, “Experimental Demonstration of Transmission of Coherent Optical OFDM Systems,” in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference, OSA Technical Digest Series (CD) (Optical Society of America, 2007), paper OMP2. http://www.opticsinfobase.org/abstract.cfm?URI=OFC-2007-OMP2

Takeda, N.

S. L. Jansen, I. Morita, N. Takeda, and H. Tanaka, “20-Gb/s OFDM Transmission over 4,160-km SSMF Enabled by RF-Pilot Tone Phase Noise Compensation,” in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference, OSA Technical Digest Series (CD) (Optical Society of America, 2007), paper PDP15. http://www.opticsinfobase.org/abstract.cfm?URI=OFC-2007-PDP15

Tanaka, H.

S. L. Jansen, I. Morita, N. Takeda, and H. Tanaka, “20-Gb/s OFDM Transmission over 4,160-km SSMF Enabled by RF-Pilot Tone Phase Noise Compensation,” in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference, OSA Technical Digest Series (CD) (Optical Society of America, 2007), paper PDP15. http://www.opticsinfobase.org/abstract.cfm?URI=OFC-2007-PDP15

Tang, Y.

W. Shieh, X. Yi, and Y. Tang, “Experimental Demonstration of Transmission of Coherent Optical OFDM Systems,” in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference, OSA Technical Digest Series (CD) (Optical Society of America, 2007), paper OMP2. http://www.opticsinfobase.org/abstract.cfm?URI=OFC-2007-OMP2

Tkach, R. W.

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

Yang, G-C

Yi, X.

W. Shieh, X. Yi, and Y. Tang, “Experimental Demonstration of Transmission of Coherent Optical OFDM Systems,” in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference, OSA Technical Digest Series (CD) (Optical Society of America, 2007), paper OMP2. http://www.opticsinfobase.org/abstract.cfm?URI=OFC-2007-OMP2

Electron. Lett. (1)

W. Shieh and C. Athaudage, “Coherent optical orthogonal frequency division multiplexing,” Electron. Lett. 42, 587–588 (2006).
[CrossRef]

J. Lightwave Technol. (3)

Opt. Express (1)

Other (5)

W. Shieh, X. Yi, and Y. Tang, “Experimental Demonstration of Transmission of Coherent Optical OFDM Systems,” in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference, OSA Technical Digest Series (CD) (Optical Society of America, 2007), paper OMP2. http://www.opticsinfobase.org/abstract.cfm?URI=OFC-2007-OMP2

B. J. Schmidt, A. J. Lowery, and J. Armstrong, “Experimental Demonstrations of 20 Gbit/s Direct-Detection Optical OFDM and 12 Gbit/s with a Colorless Transmitter,” in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference, OSA Technical Digest Series (CD) (Optical Society of America, 2007), paper PDP18. http://www.opticsinfobase.org/abstract.cfm?URI=OFC-2007-PDP18

S. L. Jansen, I. Morita, N. Takeda, and H. Tanaka, “20-Gb/s OFDM Transmission over 4,160-km SSMF Enabled by RF-Pilot Tone Phase Noise Compensation,” in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference, OSA Technical Digest Series (CD) (Optical Society of America, 2007), paper PDP15. http://www.opticsinfobase.org/abstract.cfm?URI=OFC-2007-PDP15

G. P. Agrawal, Nonlinear Fiber Optics (Optics and Photonics), 3rd Edition, (Academic Press, San Francisco, 2001)

A. R. S. Bahai, B. R. Saltzberg, and M. Ergen, Multi-carrier Digital Communications: Theory and Applications of OFDM, 2nd Edition, (Springer, New York, 2004).

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Figures (4)

Fig. 1.
Fig. 1.

Coherent optical OFDM system block diagram showing typical optical spectra at the input and output of the fiber link.

Fig. 2.
Fig. 2.

number of degenerate and non-degenerate FWM products falling on and outside the OFDM subcarrier band for 512 subcarriers in the OFDM subcarrier band.

Fig. 3.
Fig. 3.

Simulated (dots) and theoretical (lines) estimates of Q for each OFDM subcarrier. Left: 64 subcarriers; Right: 512 subcarriers.

Fig. 4.
Fig. 4.

Multi-span system simulation results (circles) compared with theory (lines) for a number of span input powers. Short spans at lower input powers have been omitted for clarity.

Equations (13)

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P ijk = D 2 ijk 9 ( γL eff ) 2 exp ( αL ) P i P j P k 1 + ( 2 πc λ 2 α ( λ i λ k ) ( λ j λ k ) D ) 2
γ = 2 πn 2 λA eff
L eff = ( l e αL ) α
P ijk D ijk 2 9 ( γL eff ) 2 P SC 3
M = ( N 3 N 2 ) 2
P FWM / SC = ( M N ) P ijk
( q elec ) 2 = P SC P FWM / SC = P SC ( M N ) P ijk
q elec 2 1 2 ( γ L e P total ) 2
M DG = ( N 2 ) 1 ; M NDG ( i ) = ( 3 N 2 10 N 4 i ( i 1 ) + 8 ) 8
M NDG i = 0 = ( 3 N 2 + 10 N + 8 ) 8 0.375 N 2
M NDG i = N 2 = ( 2 N 2 8 N + 8 ) 8 0.250 N 2
M NDGrms = M 2 NDG ( 7.2 N 2 ) 8 0.335 N 2
L ES = 1 e αL last α + ( S 1 ) L eff

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