Abstract

The probability density of nonlinear phase noise, often called the Gordon–Mollenauer effect, is derived analytically. The nonlinear phase noise can be accurately modeled as the summation of a Gaussian random variable and a noncentral chi-square random variable with two degrees of freedom. Using the received intensity to correct for the phase noise, the residual nonlinear phase noise can be modeled as the summation of a Gaussian random variable and the difference of two noncentral chi-square random variables with two degrees of freedom. The residual nonlinear phase noise can be approximated by Gaussian distribution better than the nonlinear phase noise without correction.

© 2003 Optical Society of America

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References

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  1. J. P. Gordon and L. F. Mollenauer, “Phase noise in photonic communications systems using linear amplifiers,” Opt. Lett. 15, 1351–1353 (1990).
    [CrossRef] [PubMed]
  2. S. Ryu, “Signal linewidth broadening due to nonlinear Kerr effect in long-haul coherent systems using cascaded optical amplifiers,” J. Lightwave Technol. 10, 1450–1457 (1992).
    [CrossRef]
  3. A. H. Gnauck, G. Raybon, S. Chandrasekhar, J. Leuthold, C. Doerr, L. Stulz, A. Agrawal, S. Banerjee, D. Grosz, S. Hunsche, A. Kung, A. Marhelyuk, D. Maymar, M. Movassaghi, X. Liu, C. Xu, X. Wei, and D. M. Gill, “2.5 Tb/s (64× 42.7 Gb/s) transmission over 40×100 km NZDSF using RZ-DPSK format and all-Raman-amplified spans,” in Optical Fiber Communication Conference (Optical Society of America, Washington, D.C., 2002), postdeadline paper FC2.
  4. R. A. Griffin, R. I. Johnstone, R. G. Walker, J. Hall, S. D. Wadsworth, K. Berry, A. C. Carter, M. J. Wale, P. A. Jerram, and N. J. Parsons, “10 Gb/s optical differential quadrature phase shift key (DQPSK) transmission using GaAs/AlGaAs integration,” in Optical Fiber Communication Conference (Optical Society of America, Washington, D.C., 2002), postdeadline paper FD6.
  5. B. Zhu, L. Leng, A. H. Gnauck, M. O. Pedersen, D. Peckham, L. E. Nelson, S. Stulz, S. Kado, L. Gruner-Nielsen, R. L. Lingle, S. Knudsen, J. Leuthold, C. Doerr, S. Chandrasekhar, G. Baynham, P. Gaarde, Y. Emori, and S. Namiki, “Transmission of 3.2 Tb/s (80×42.7 Gb/s) over 5200 km of UltraWave™ fiber with 100-km dispersion-managed spans using RZ-DPSK format,” presented of the 28th European Conference on Optical Communication, Copenhagen, Denmark, September 9–12, 2002, postdeadline paper PD4.2.
  6. X. Liu, X. Wei, R. E. Slusher, and C. J. McKinstrie, “Improving transmission performance in differential phase-shift-keyed systems by use of lumped nonlinear phase-shift compensation,” Opt. Lett. 27, 1616–1618 (2002).
    [CrossRef]
  7. C. Xu and X. Liu, “Postnonlinearity compensation with data-driven phase modulators in phase-shift keying transmission,” Opt. Lett. 27, 1619–1621 (2002).
    [CrossRef]
  8. K.-P. Ho and J. M. Kahn, “Detection technique to mitigate Kerr effect phase noise,” http://arXiv.org/physics/0211097.
  9. G. L. Turin, “The characteristic function of Hermitian quadratic forms in complex normal variables,” Biometrika 47, 199–201 (1960).
    [CrossRef]
  10. J. G. Proakis, Digital Communications, 4th ed. (McGraw-Hill, New York, 2000).
  11. R. M. Gray, “On the asymptotic eigenvalue distribution of Toeplitz matrices,” IEEE Trans. Inf. Theory IT-18, 725–730 (1972).
    [CrossRef]

2002

1992

S. Ryu, “Signal linewidth broadening due to nonlinear Kerr effect in long-haul coherent systems using cascaded optical amplifiers,” J. Lightwave Technol. 10, 1450–1457 (1992).
[CrossRef]

1990

1972

R. M. Gray, “On the asymptotic eigenvalue distribution of Toeplitz matrices,” IEEE Trans. Inf. Theory IT-18, 725–730 (1972).
[CrossRef]

1960

G. L. Turin, “The characteristic function of Hermitian quadratic forms in complex normal variables,” Biometrika 47, 199–201 (1960).
[CrossRef]

Gordon, J. P.

Gray, R. M.

R. M. Gray, “On the asymptotic eigenvalue distribution of Toeplitz matrices,” IEEE Trans. Inf. Theory IT-18, 725–730 (1972).
[CrossRef]

Liu, X.

McKinstrie, C. J.

Mollenauer, L. F.

Ryu, S.

S. Ryu, “Signal linewidth broadening due to nonlinear Kerr effect in long-haul coherent systems using cascaded optical amplifiers,” J. Lightwave Technol. 10, 1450–1457 (1992).
[CrossRef]

Slusher, R. E.

Turin, G. L.

G. L. Turin, “The characteristic function of Hermitian quadratic forms in complex normal variables,” Biometrika 47, 199–201 (1960).
[CrossRef]

Wei, X.

Xu, C.

Biometrika

G. L. Turin, “The characteristic function of Hermitian quadratic forms in complex normal variables,” Biometrika 47, 199–201 (1960).
[CrossRef]

IEEE Trans. Inf. Theory

R. M. Gray, “On the asymptotic eigenvalue distribution of Toeplitz matrices,” IEEE Trans. Inf. Theory IT-18, 725–730 (1972).
[CrossRef]

J. Lightwave Technol.

S. Ryu, “Signal linewidth broadening due to nonlinear Kerr effect in long-haul coherent systems using cascaded optical amplifiers,” J. Lightwave Technol. 10, 1450–1457 (1992).
[CrossRef]

Opt. Lett.

Other

K.-P. Ho and J. M. Kahn, “Detection technique to mitigate Kerr effect phase noise,” http://arXiv.org/physics/0211097.

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

A. H. Gnauck, G. Raybon, S. Chandrasekhar, J. Leuthold, C. Doerr, L. Stulz, A. Agrawal, S. Banerjee, D. Grosz, S. Hunsche, A. Kung, A. Marhelyuk, D. Maymar, M. Movassaghi, X. Liu, C. Xu, X. Wei, and D. M. Gill, “2.5 Tb/s (64× 42.7 Gb/s) transmission over 40×100 km NZDSF using RZ-DPSK format and all-Raman-amplified spans,” in Optical Fiber Communication Conference (Optical Society of America, Washington, D.C., 2002), postdeadline paper FC2.

R. A. Griffin, R. I. Johnstone, R. G. Walker, J. Hall, S. D. Wadsworth, K. Berry, A. C. Carter, M. J. Wale, P. A. Jerram, and N. J. Parsons, “10 Gb/s optical differential quadrature phase shift key (DQPSK) transmission using GaAs/AlGaAs integration,” in Optical Fiber Communication Conference (Optical Society of America, Washington, D.C., 2002), postdeadline paper FD6.

B. Zhu, L. Leng, A. H. Gnauck, M. O. Pedersen, D. Peckham, L. E. Nelson, S. Stulz, S. Kado, L. Gruner-Nielsen, R. L. Lingle, S. Knudsen, J. Leuthold, C. Doerr, S. Chandrasekhar, G. Baynham, P. Gaarde, Y. Emori, and S. Namiki, “Transmission of 3.2 Tb/s (80×42.7 Gb/s) over 5200 km of UltraWave™ fiber with 100-km dispersion-managed spans using RZ-DPSK format,” presented of the 28th European Conference on Optical Communication, Copenhagen, Denmark, September 9–12, 2002, postdeadline paper PD4.2.

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Figures (5)

Fig. 1
Fig. 1

pdf of both ϕNL and ϕRES.

Fig. 2
Fig. 2

pdf of ϕNL is the convolution of a Gaussian pdf and a noncentral χ2 pdf with two degrees of freedom.

Fig. 3
Fig. 3

Cumulative tail probability of ϕNL as compared with the model of Fig. 2 and a Gaussian approximation.

Fig. 4
Fig. 4

pdf of ϕRES is the convolution of a Gaussian pdf and two noncentral χ2 pdf’s with two degrees of freedom.

Fig. 5
Fig. 5

Cumulative tail probability of ϕRES as compared with the model of Fig. 4 and a Gaussian approximation.

Equations (27)

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ϕNL=|A+n1|2+|A+n1+n2|2++|A+n1++nN|2,
φ1=|A+x1|2+|A+x1+x2|2++|A+x1++xN|2.
φ2=y12+|y1+y2|2++|y1++yN|2
φ1=NA2+2AwTx+xTCx,
M=1000110011101111.
Ψφ1(ν)=exp(jνNA2)(2πσ2)N/2exp(2jνAwTx-xTΓx)dx,
xTΓx-2jνAwTx=(x-jνAΓ-1w)TΓ(x-jνAΓ-1w)+ν2A2wTΓ-1w,
Ψφ1(ν)=exp(jνNA2-ν2A2wTΓ-1w)(2σ2)N/2det(Γ)1/2,
Ψφ1(ν)=exp[jνNA2-2σ2ν2A2wT(I-2jνσ2C)-1w]det(I-2jνσ2C)1/2.
ΨϕNL(ν)
=exp[jνNA2-2σ2ν2A2wT(I-2jνσ2C)-1w]det(I-2jνσ2C).
ΨϕNL(ν)=expjνNA2-2σ2ν2A2k=1N(vkTw)21-2jνσ2λkk=1N(1-2jνσ2λk)
ΨϕNL(ν)=k=1N11-2jνσ2λkexpjνA2(vkTw)2/λk1-2jνσ2λk.
C-1=1-1000-12-1000-1200000-12
1λk21-cos(2k+1)π2N=4 sin2(2k-1)π4N,k=1,, N.
ϕRES=|A+n1|2+|A+n1+n2|2++|A+n1++nN-1|2-(αopt-1)|A+n1++nN|2.
(N-αopt)A2+2AwrTx+xTCrx,
Cr=(M-L)T(M-L)-(αopt-1)LTL,
L=000000001111.
ΨϕRES(ν)=exp[-jν(N-αopt)A2-2σ2ν2A2wrT(I-2jνσ2Cr)-1wr]det(I-2jνσ2Cr).
1λk4 sin2(k-1.25)π2(N-1),k=2,, N,
λ1-k=2Nλk.
αopt=N+12A2+(2N+1)σ2/3A2+Nσ2N+12.
σϕRES2=(N-1)N(N+1)σ2×A4+2Nσ2A2+(2N2+1)σ4/33(A2+Nσ2)
σϕNL2=43 N(N+1)σ2N+12A2+(N2+N+1)σ2.
ϕNL¯=N[A2+(N+1)σ2].
ϕRES¯=ϕNL¯-αopt(A2+2Nσ2).

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