Abstract

Fiber nonlinearities can degrade the performance of a wavelength-division multiplexing optical network. For high input power, a low chromatic dispersion coefficient, or low channel spacing, the most severe penalties are due to four-wave mixing (FWM). To compute the bit-error rate that is due to FWM noise, one must evaluate accurately the probability-density functions (pdf) of both the space and the mark states. An accurate evaluation of the pdf of the FWM noise in the space state is given, for the first time to the authors’ knowledge, by use of Monte Carlo simulations. Additionally, it is shown that the pdf in the mark state is not symmetric as had been assumed in previous studies. Diagrams are presented that permit estimation of the pdf, given the number of channels in the system. The accuracy of the previous models is also investigated, and finally the results of this study are used to estimate the power limits of a wavelength-division multiplexing system.

© 2004 Optical Society of America

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References

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  1. H. J. Thiele, R. I. Killey, P. Bayvel, “Investigation of XPM distortion in transmission over installed fiber,” IEEE Photon. Technol. Lett. 12, 669–671 (2000).
    [CrossRef]
  2. M. Shtaif, M. Eiselt, “Analysis of intensity interference caused by cross-phase-modulation in dispersive optical fibers,” IEEE Photon. Technol. Lett. 10, 979–981 (1998).
    [CrossRef]
  3. K. Inoue, K. Nakanishi, K. Oda, H. Toba, “Crosstalk and power penalty due to fiber four-wave mixing in multichannel transmissions,” J. Lightwave Technol. 12, 1423–1439 (1994).
    [CrossRef]
  4. M. Eiselt, “Limits on WDM systems due to four-wave mixing: a statistical approach,” J. Lightwave Technol. 17, 2261–2267 (1999).
    [CrossRef]
  5. J. G. Proakis, Digital Communications, 4th ed. (McGraw-Hill, New York, 2000).
  6. K. Inoue, “Polarization effect on four-wave mixing efficiency in a single-mode fiber,” IEEE J. Quantum Electron. 28, 883–894 (1992).
    [CrossRef]
  7. N. Shibata, K. Nosu, K. Iwashita, Y. Azuma, “Transmission limitations due to fiber nonlinearities in optical FDM systems,” IEEE J. Sel. Area Commun. 8, 1068–1077 (1990).
    [CrossRef]
  8. G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, New York, 1995).
  9. S. Song, C. T. Allen, K. R. Demarest, R. Hui, “Intensity-dependent phase-matching effects on four-wave mixing in optical fibers,” J. Lightwave Technol. 17, 2285–2290 (1999).
    [CrossRef]
  10. K. O. Hill, D. C. Johnson, B. S. Kawasaki, R. I. MacDonald, “cw three-wave mixing in single-mode optical fibers,” J. Appl. Phys. 49, 5098–5106 (1978).
    [CrossRef]
  11. G. H. Einarsson, Principles of Lightwave Communications (Wiley, Chichester, UK, 1996).
  12. J. Tang, C. K. Siew, L. Zhang, “Optical nonlinear effects on the performance of IP traffic over GMPLS-based DWDM networks,” Computer Commun. 26, 1330–1340 (2003).
    [CrossRef]
  13. P. E. Green, Fiber Optic Networks (Prentice-Hall, Englewood Cliffs, N.J., 1993).
  14. K. Inoue, “Phase-mismatching characteristic of four-wave mixing in fiber lines with multistage optical amplifiers,” Opt. Lett. 17, 801–802 (1992).
    [CrossRef] [PubMed]
  15. T. Kamalakis, T. Sphicopoulos, “Asymptotic behavior of in-band crosstalk noise in WDM networks,” IEEE Photon. Technol. Lett. 15, 476–478 (2003).
    [CrossRef]

2003 (2)

J. Tang, C. K. Siew, L. Zhang, “Optical nonlinear effects on the performance of IP traffic over GMPLS-based DWDM networks,” Computer Commun. 26, 1330–1340 (2003).
[CrossRef]

T. Kamalakis, T. Sphicopoulos, “Asymptotic behavior of in-band crosstalk noise in WDM networks,” IEEE Photon. Technol. Lett. 15, 476–478 (2003).
[CrossRef]

2000 (1)

H. J. Thiele, R. I. Killey, P. Bayvel, “Investigation of XPM distortion in transmission over installed fiber,” IEEE Photon. Technol. Lett. 12, 669–671 (2000).
[CrossRef]

1999 (2)

1998 (1)

M. Shtaif, M. Eiselt, “Analysis of intensity interference caused by cross-phase-modulation in dispersive optical fibers,” IEEE Photon. Technol. Lett. 10, 979–981 (1998).
[CrossRef]

1994 (1)

K. Inoue, K. Nakanishi, K. Oda, H. Toba, “Crosstalk and power penalty due to fiber four-wave mixing in multichannel transmissions,” J. Lightwave Technol. 12, 1423–1439 (1994).
[CrossRef]

1992 (2)

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

K. Inoue, “Phase-mismatching characteristic of four-wave mixing in fiber lines with multistage optical amplifiers,” Opt. Lett. 17, 801–802 (1992).
[CrossRef] [PubMed]

1990 (1)

N. Shibata, K. Nosu, K. Iwashita, Y. Azuma, “Transmission limitations due to fiber nonlinearities in optical FDM systems,” IEEE J. Sel. Area Commun. 8, 1068–1077 (1990).
[CrossRef]

1978 (1)

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

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, New York, 1995).

Allen, C. T.

Azuma, Y.

N. Shibata, K. Nosu, K. Iwashita, Y. Azuma, “Transmission limitations due to fiber nonlinearities in optical FDM systems,” IEEE J. Sel. Area Commun. 8, 1068–1077 (1990).
[CrossRef]

Bayvel, P.

H. J. Thiele, R. I. Killey, P. Bayvel, “Investigation of XPM distortion in transmission over installed fiber,” IEEE Photon. Technol. Lett. 12, 669–671 (2000).
[CrossRef]

Demarest, K. R.

Einarsson, G. H.

G. H. Einarsson, Principles of Lightwave Communications (Wiley, Chichester, UK, 1996).

Eiselt, M.

M. Eiselt, “Limits on WDM systems due to four-wave mixing: a statistical approach,” J. Lightwave Technol. 17, 2261–2267 (1999).
[CrossRef]

M. Shtaif, M. Eiselt, “Analysis of intensity interference caused by cross-phase-modulation in dispersive optical fibers,” IEEE Photon. Technol. Lett. 10, 979–981 (1998).
[CrossRef]

Green, P. E.

P. E. Green, Fiber Optic Networks (Prentice-Hall, Englewood Cliffs, N.J., 1993).

Hill, K. O.

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

Hui, R.

Inoue, K.

K. Inoue, K. Nakanishi, K. Oda, H. Toba, “Crosstalk and power penalty due to fiber four-wave mixing in multichannel transmissions,” J. Lightwave Technol. 12, 1423–1439 (1994).
[CrossRef]

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

K. Inoue, “Phase-mismatching characteristic of four-wave mixing in fiber lines with multistage optical amplifiers,” Opt. Lett. 17, 801–802 (1992).
[CrossRef] [PubMed]

Iwashita, K.

N. Shibata, K. Nosu, K. Iwashita, Y. Azuma, “Transmission limitations due to fiber nonlinearities in optical FDM systems,” IEEE J. Sel. Area Commun. 8, 1068–1077 (1990).
[CrossRef]

Johnson, D. C.

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

Kamalakis, T.

T. Kamalakis, T. Sphicopoulos, “Asymptotic behavior of in-band crosstalk noise in WDM networks,” IEEE Photon. Technol. Lett. 15, 476–478 (2003).
[CrossRef]

Kawasaki, B. S.

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

Killey, R. I.

H. J. Thiele, R. I. Killey, P. Bayvel, “Investigation of XPM distortion in transmission over installed fiber,” IEEE Photon. Technol. Lett. 12, 669–671 (2000).
[CrossRef]

MacDonald, R. I.

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

Nakanishi, K.

K. Inoue, K. Nakanishi, K. Oda, H. Toba, “Crosstalk and power penalty due to fiber four-wave mixing in multichannel transmissions,” J. Lightwave Technol. 12, 1423–1439 (1994).
[CrossRef]

Nosu, K.

N. Shibata, K. Nosu, K. Iwashita, Y. Azuma, “Transmission limitations due to fiber nonlinearities in optical FDM systems,” IEEE J. Sel. Area Commun. 8, 1068–1077 (1990).
[CrossRef]

Oda, K.

K. Inoue, K. Nakanishi, K. Oda, H. Toba, “Crosstalk and power penalty due to fiber four-wave mixing in multichannel transmissions,” J. Lightwave Technol. 12, 1423–1439 (1994).
[CrossRef]

Proakis, J. G.

J. G. Proakis, Digital Communications, 4th ed. (McGraw-Hill, New York, 2000).

Shibata, N.

N. Shibata, K. Nosu, K. Iwashita, Y. Azuma, “Transmission limitations due to fiber nonlinearities in optical FDM systems,” IEEE J. Sel. Area Commun. 8, 1068–1077 (1990).
[CrossRef]

Shtaif, M.

M. Shtaif, M. Eiselt, “Analysis of intensity interference caused by cross-phase-modulation in dispersive optical fibers,” IEEE Photon. Technol. Lett. 10, 979–981 (1998).
[CrossRef]

Siew, C. K.

J. Tang, C. K. Siew, L. Zhang, “Optical nonlinear effects on the performance of IP traffic over GMPLS-based DWDM networks,” Computer Commun. 26, 1330–1340 (2003).
[CrossRef]

Song, S.

Sphicopoulos, T.

T. Kamalakis, T. Sphicopoulos, “Asymptotic behavior of in-band crosstalk noise in WDM networks,” IEEE Photon. Technol. Lett. 15, 476–478 (2003).
[CrossRef]

Tang, J.

J. Tang, C. K. Siew, L. Zhang, “Optical nonlinear effects on the performance of IP traffic over GMPLS-based DWDM networks,” Computer Commun. 26, 1330–1340 (2003).
[CrossRef]

Thiele, H. J.

H. J. Thiele, R. I. Killey, P. Bayvel, “Investigation of XPM distortion in transmission over installed fiber,” IEEE Photon. Technol. Lett. 12, 669–671 (2000).
[CrossRef]

Toba, H.

K. Inoue, K. Nakanishi, K. Oda, H. Toba, “Crosstalk and power penalty due to fiber four-wave mixing in multichannel transmissions,” J. Lightwave Technol. 12, 1423–1439 (1994).
[CrossRef]

Zhang, L.

J. Tang, C. K. Siew, L. Zhang, “Optical nonlinear effects on the performance of IP traffic over GMPLS-based DWDM networks,” Computer Commun. 26, 1330–1340 (2003).
[CrossRef]

Computer Commun. (1)

J. Tang, C. K. Siew, L. Zhang, “Optical nonlinear effects on the performance of IP traffic over GMPLS-based DWDM networks,” Computer Commun. 26, 1330–1340 (2003).
[CrossRef]

IEEE J. Quantum Electron. (1)

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

IEEE J. Sel. Area Commun. (1)

N. Shibata, K. Nosu, K. Iwashita, Y. Azuma, “Transmission limitations due to fiber nonlinearities in optical FDM systems,” IEEE J. Sel. Area Commun. 8, 1068–1077 (1990).
[CrossRef]

IEEE Photon. Technol. Lett. (3)

H. J. Thiele, R. I. Killey, P. Bayvel, “Investigation of XPM distortion in transmission over installed fiber,” IEEE Photon. Technol. Lett. 12, 669–671 (2000).
[CrossRef]

M. Shtaif, M. Eiselt, “Analysis of intensity interference caused by cross-phase-modulation in dispersive optical fibers,” IEEE Photon. Technol. Lett. 10, 979–981 (1998).
[CrossRef]

T. Kamalakis, T. Sphicopoulos, “Asymptotic behavior of in-band crosstalk noise in WDM networks,” IEEE Photon. Technol. Lett. 15, 476–478 (2003).
[CrossRef]

J. Appl. Phys. (1)

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

J. Lightwave Technol. (3)

Opt. Lett. (1)

Other (4)

P. E. Green, Fiber Optic Networks (Prentice-Hall, Englewood Cliffs, N.J., 1993).

J. G. Proakis, Digital Communications, 4th ed. (McGraw-Hill, New York, 2000).

G. H. Einarsson, Principles of Lightwave Communications (Wiley, Chichester, UK, 1996).

G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, New York, 1995).

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Figures (6)

Fig. 1
Fig. 1

Pdfs of (a) the space and (b) the mark states. The exponential fittings for the pdfs are shown by solid curves.

Fig. 2
Fig. 2

(a), (b) Parameters A and b in the space state for the central channel with respect to the number of channels N. (c), (d) Parameters A and b in the mark state for the central channel with respect to the number of channels N.

Fig. 3
Fig. 3

(a) pdf of the photocurrent for N = 16, D = 10 (ps/nm)/km, P in = 5 dBm, and Δf = 25 GHz. (b) Error probability P e1 for the mark state with respect to the receiver threshold for the same parameters. (c) P e1 as a function of receiver threshold for N = 8, D = 5 (ps/nm)/km, P in = 18 dBm, and Δf = 100 GHz. (d) Standard deviation of the sum I m for the central channel with respect to the number of channels N for the symmetrical distribution and the numerically computed distribution.

Fig. 4
Fig. 4

(a) Error probability in the mark state with respect to the receiver threshold for N = 16, D = 10 (ps/nm)/km, P in = 5 dBm, and Δf = 25 GHz for the Gaussian and numerical models. (b) BER dependence with respect to the input power for the Gaussian and the numerical models.

Fig. 5
Fig. 5

Pdfs of the photocurrent in the mark and the space states for N = 16, D = 0 (ps/nm)/km, dD/dλ = 0.07 (ps/nm2)/km, P in = 0 dBm, and Δf = 100 GHz.

Fig. 6
Fig. 6

BER as a function of input peak power P in in the mark state for (a) N = 8, (b) N = 16, and (c) N = 32. The values of the chromatic dispersion used are D 1 = 2 (ps/nm)/km, D 2 = 5 (ps/nm)/km, and D 3 = 10 (ps/nm)/km.

Equations (27)

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WNL=xˆ½  Wi expjkiz-ωit+c.c.=xˆ½  Wi expjθi+c.c.,
Wn= 3εo4 χxxxx3|En|2En+ 3εo4 χxxxx3pqr2|Ep|2+|Eq|2+|Er|2En+ 3εo2 χxxxx3pqr EpEqEr* expjθp+θq-θr-θn+,
Ppqr= γ29 dpqr2PpPqPr exp-aLLeff2η,
η= a2a2+Δβ21+ 4 exp-aLsin2ΔβL/21-exp-aL2.
Δβ= 2πλ2cfp-frfq-frDλo+ dDλodλλ22c× fp-fo+fq-fo= 2πλ2c Δf2p-rq-rDλo+ dDλodλλ22cΔfp-o+q-o,
Δβ 2πλ2Dc Δf2p-rq-r.
Em= Pn exp-aLexpjθn+ PFIMmexpjθFIMmmark,
Es= PFIMsexpjθFIMsspace,
PFIMmexpjθFIMm=pqrn BpBqBrPpqr×expjθpqr+pqr=n BpBqPpqn×expjθpqn+p=qr BpBrPppr×expjθppr,
PFIMsexpjθFIMs=pqrn BpBqBrPpqr×expjθpqr+p=qr BpBrPppr×expjθpqr,
Sm=k|Em|2kPn exp-aL+2kδ Pn exp-aLIm,
Ss=k|Es|2kδ2Is,
δ= γc2πλ2DΔf2 Pin3/2 exp-aL/2,
Im= 13pqr BpBqBrdpqr|p-nq-n|cosθpqr-θn,
Is=13pqrrn BpBqBrdpqr|p-nq-n|cos θpqr2+13pqrrn BpBqBrdpqr|p-nq-n|sin θpqr2.
fSmSm= 12kδPn exp-aL fIm×Sm-kPn exp-aL2kδPn exp-aL,
fSsSs= 1kδ2 fIsSskδ2,
Pe1=-Q fSmξdξ,
fexpIm= 12σexp-2|Im|σ,
Pe= 12πQg exp- t22dt= 12erfcQg2,
Qg= Sm-Ssσm+σs kPn exp-aL-k pqrn Ppqr+¼ p=qr Pppr2k2Pn exp-aL{ pqrn Ppqr+¼ pqr=n Ppqn+¼ p=qr Pppr1/2.
Pe=½ Q fSsξdξ+½ -Q fSmξdξ,
MZs=EMZ|Xs=11-NosμMXGs1-Nos.
MXs Ab+sexpb+sIs,max-1,
Im=13pqr epqr=13pqrdpqrBpBqBr|p-nq-n| ×cosθpqr-θn= 13pqrdpqr|p-nq-n|BpBqBr× cosθpqr-θn=0
Im3=13pqrdpqrBpBqBr|p-nq-n|cosθpqr-θn3=13pqr epqr3= 127pqr epqr3+ 19pqrpqrepqr2epqr+ 29pqrpqrpqrepqrepqrepqr,
θ=θpqr-n+θpqr-n-θpqr-n=θp+θq-θr-θn+θp+θq-θr-θn-θp-θq+θr+θn.

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