Abstract

Gordon and Mollenauer, in their famous paper published in 1990, laid out how the interplay between the nonlinear Kerr effect in optical fibers and the amplified spontaneous-emission (ASE) noise from the optical-amplifiers results in enhanced levels of noise and degrades the performance of modulation schemes that encode information in, particularly, the phase of the optical carrier. This phenomenon has been termed as nonlinear phase noise in the literature. In this paper, we first present a comparative and critical review of previous techniques that have been proposed for the analysis of nonlinear phase noise by forming a classification framework that reveals some key underlying features. We then present a unifying theory and a comprehensive methodology and computational techniques for the analysis and characterization of nonlinear phase noise and its impact on system performance by building on and extending previous work that we identify as most favorable and systematic. In our treatment, we consider a multichannel multispan optically amplified dense wavelength-division multiplexed system and develop general techniques for the analysis of the intricate interplay among Kerr nonlinearity, chromatic dispersion, and ASE noise, and for computing the bit-error-ratio performance of differential phase-shift-keying (DPSK) systems. By means of the extensive results we present, we demonstrate and argue that correlated noise behavior plays a most significant role in understanding nonlinear phase noise and its impact on DPSK system performance.

© 2007 IEEE

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2007 (1)

J. A. C. Weideman, L. N. Trefethen, "Parabolic and hyperbolic contours for computing the Bromwich integral," Math. Comput. (2007).

2006 (6)

J. A. C. Weideman, "Optimizing Talbot's contours for the inversion of the Laplace transform," BIT Numer. Math. 46, 653-670 (2006).

L. N. Trefethen, J. A. C. Weideman, T. Schmelzer, "Talbot quadratures and rational approximations," SIAM J. Numer. Anal. 44, 2342-2362 (2006).

A. Demir, "Computing timing jitter from phase noise spectra for oscillators and phase-locked loops with white and $1/f$ noise," IEEE Trans. Circuits Syst. I, Reg. Papers 53, 1869-1884 (2006).

K.-P. Ho, H.-C. Wang, "Effect of dispersion on nonlinear phase noise," Opt. Lett. 31, 2109-2111 (2006).

F. Zhang, C.-A. Bunge, K. Petermann, "Analysis of nonlinear phase noise in single-channel return-to-zero differential phase-shift keying transmission systems," Opt. Lett. 31, 1038-1040 (2006).

P. Serena, A. Orlandini, A. Bononi, "Parametric-gain approach to the analysis of single-channel DPSK/DQPSK systems with nonlinear phase noise," J. Lightw. Technol. 24, 2026-2037 (2006).

2005 (6)

W. Pellegrini, J. Zweck, C. R. Menyuk, R. Holzlöhner, "Computation of bit error ratios for a dense WDM system using the noise covariance matrix and multicanonical Monte Carlo methods," IEEE Photon. Technol. Lett. 17, 1644-1646 (2005).

E. T. Spiller, W. L. Kath, R. O. Moore, C. J. McKinstrie, "Computing large signal distortions and bit-error ratios in DPSK transmission systems ," IEEE Photon. Technol. Lett. 17, 1022-1024 (2005).

K.-P. Ho, "Comparison of nonlinear phase noise and intrachannel four-wave mixing for RZ-DPSK signals in dispersive transmission systems," IEEE Photon. Technol. Lett. 17, 1426-1428 (2005).

K.-P. Ho, "Performance of DPSK signals with quadratic phase noise," IEEE Trans. Commun. 53, 1361-1365 (2005).

X. Huang, L. Zhang, M. Zhang, P. Ye, "Impact of nonlinear phase noise on direct-detection DQPSK WDM systems," IEEE Photon. Technol. Lett. 17, 1423-1425 (2005).

A. K. Kassam, L. N. Trefethen, "Fourth-order time-stepping for stiff PDEs," SIAM J. Sci. Comput. 26, 1214-1233 (2005).

2004 (11)

M. Kang, M.-S. Alouini, "Quadratic forms in complex Gaussian matrices and performance analysis of MIMO systems with cochannel interference," IEEE Trans. Wireless Commun. 3, 418-431 (2004).

G. Bosco, P. Poggiolini, "On the Q factor inaccuracy in the performance analysis of optical direct-detection DPSK systems ," IEEE Photon. Technol. Lett. 16, 665-667 (2004).

J. Wang, J. M. Kahn, "Impact of chromatic and polarization-mode dispersion on DPSK systems using interferometric detection," J. Lightw. Technol. 22, 362-371 (2004).

K.-P. Ho, "Error probability of DPSK signals with cross-phase modulation induced nonlinear phase noise ," IEEE J. Sel. Topics Quantum Electron. 10, 421-427 (2004).

K.-P. Ho, J. M. Kahn, "Electronic compensation technique to mitigate nonlinear phase noise," J. Lightw. Technol. 22, 779-783 (2004).

K.-P. Ho, "Impact of nonlinear phase noise to DPSK signals: A comparison of different models," IEEE Photon. Technol. Lett. 16, 1403-1405 (2004).

Y. Yadin, M. Shtaif, M. Orenstein, "Nonlinear phase noise in phase-modulated WDM fiber-optic communications," IEEE Photon. Technol. Lett. 16, 1307-1309 (2004).

A. Demir, "Non-Monte Carlo formulations and computational techniques for the stochastic nonlinear Schrodinger equation," J. Comput. Phys. 201, 148-171 (2004).

J. Wang, J. M. Kahn, "Accurate bit-error-ratio computation in nonlinear CRZ-OOK and CRZ-DPSK systems," IEEE Photon. Technol. Lett. 16, 2165-2167 (2004).

C. Xu, X. Liu, X. Wei, "Differential phase-shift keying for high spectral efficiency optical transmissions," IEEE J. Sel. Topics Quantum Electron. 10, 281-293 (2004).

J. M. Kahn, K.-P. Ho, "Spectral efficiency limits and modulation/detection techniques for DWDM systems," IEEE J. Sel. Topics Quantum Electron. 10, 259-272 (2004).

2003 (10)

R. Holzlöhner, C. R. Menyuk, W. L. Kath, V. S. Grigoryan, "A covariance matrix method to compute bit error rates in a highly nonlinear dispersion-managed soliton system," IEEE Photon. Technol. Lett. 15, 688-690 (2003).

A. G. Green, P. P. Mitra, L. G. L. Wegener, "Effect of chromatic dispersion on nonlinear phase noise," Opt. Lett. 28, 2455-2457 (2003).

B. Xu, M. Brandt-Pearce, "Comparison of FWM- and XPM-induced crosstalk using Volterra series transfer function method ," J. Lightw. Technol. 21, 40-53 (2003).

H. Kim, A. H. Gnauck, "Experimental investigation of the performance limitation of DPSK systems due to nonlinear phase noise," IEEE Photon. Technol. Lett. 15, 320-322 (2003).

H. Kim, "Cross-phase-modulation-induced nonlinear phase noise in WDM direct-detection DPSK systems ," J. Lightw. Technol. 21, 1770-1774 (2003).

K.-P. Ho, "Performance degradation of phase-modulated systems due to nonlinear phase noise," IEEE Photon. Technol. Lett. 15, 1213-1215 (2003).

K.-P. Ho, "Compensation improvement of DPSK signal with nonlinear phase noise," IEEE Photon. Technol. Lett. 15, 1216-1218 (2003).

T. Mizuochi, K. Ishida, T. Kobayashi, J. Abe, K. Kinjo, K. Motoshima, K. Kasahara, "A comparative study of DPSK and OOK WDM transmission over transoceanic distances and their performance degradations due to nonlinear phase noise," J. Lightw. Technol. 21, 1933-1943 (2003).

E. Forestieri, "Correction to ‘Evaluating the error probability in lightwave systems with chromatic dispersion, arbitrary pulse shape and pre- and postdetection filtering’," J. Lightw. Technol. 21, 1592 (2003).

P. Winzer, H. Kim, "Degradations in balanced DPSK receivers," IEEE Photon. Technol. Lett. 15, 1282-1284 (2003).

2002 (4)

Y. Ma, T. J. Lim, S. Pasupath, "Error probability for coherent and differential PSK over arbitrary Rician fading channels with multiple cochannel interferers," IEEE Trans. Commun. 50, 429-441 (2002).

C. J. McKinstrie, C. Xie, "Phase jitter in single-channel solution systems with constant dispersion," IEEE J. Sel. Topics Quantum Electron. 8, 616-625 (2002).

R. Holzlöhner, V. S. Grigoryan, C. R. Menyuk, W. L. Kath, "Accurate calculation of eye diagrams and bit error rates in optical transmission systems using linearization," J. Lightw. Technol. 20, 389-400 (2002).

R. Holzlöhner, V. S. Grigoryan, C. R. Menyuk, W. L. Kath, "Efficient and accurate computation of eye diagrams and bit-error rates in a single-channel CRZ system," IEEE Photon. Technol. Lett. 14, 1079-1081 (2002).

2001 (1)

G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, S. Benedetto, "A novel analytical method to the evaluation of the impact of fiber parametric gain on the bit error rate," IEEE Trans. Commun. 49, 2154-2163 (2001).

2000 (3)

E. Forestieri, "Evaluating the error probability in lightwave systems with chromatic dispersion, arbitrary pulse shape and pre- and postdetection filtering," J. Lightw. Technol. 18, 1493-1503 (2000).

G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, S. Benedetto, "A novel analytical method for the BER evaluation in optical systems affected by parametric gain ," IEEE Photon. Technol. Lett. 12, 152-154 (2000).

J.-K. Rhee, D. Chowdhury, K. S. Cheng, U. Gliese, "DPSK $32 \times 10\ \hbox{Gb/s}$ transmission modeling on $5 \times 90\ \hbox{km}$ terrestrial system," IEEE Photon. Technol. Lett. 12, 1627-1629 (2000).

1999 (3)

G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, S. Benedetto, "Parametric gain in multiwavelength systems: A new approach to noise enhancement analysis ," IEEE Photon. Technol. Lett. 11, 1135-1137 (1999).

R. Hui, K. R. Demarest, C. T. Allen, "Cross-phase modulation in multispan WDM optical fiber systems," J. Lightw. Technol. 17, 1018-1026 (1999).

A. V. T. Cartaxo, B. Wedding, W. Idler, "Influence of fiber nonlinearity on the fiber transfer function: Theoretical and experimental analysis," J. Lightw. Technol. 17, 1806-1813 (1999).

1998 (3)

A. V. T. Cartaxo, B. Wedding, W. Idler, "Influence of fiber nonlinearity on the phase noise to intensity noise conversion in fiber transmission: Theoretical and experimental analysis," J. Lightw. Technol. 16, 1187-1194 (1998).

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).

J. Leibrich, C. Wree, W. Rosenkranz, "CF-RZ-DPSK for suppression of XPM on dispersion-managed long-haul optical WDM transmission on standard single-mode fiber," IEEE Photon. Technol. Lett. 14, 155-157 (1998).

1997 (2)

C. W. Helstrom, "Computing the distribution of sums of random sine waves and of Rayleigh-ditributed random variables by saddle-point integration," IEEE Trans. Commun. 45, 1487-1494 (1997).

A. Carena, V. Curri, R. Gaudino, P. Poggiolini, S. Benedetto, "New analytical results on fiber parametric gain and its effects on ASE noise," IEEE Photon. Technol. Lett. 9, 535-537 (1997).

1996 (3)

A. Demir, E. Liu, A. Sangiovanni-Vincentelli, "Time-domain non-Monte Carlo noise simulation for nonlinear dynamic circuits with arbitrary excitations," IEEE Trans. Comput.-Aided Design Integr. Circuits Syst. 15, 493-505 (1996).

T.-K. Chiang, N. Kagi, M. E. Marhic, L. G. Kazovsky, "Cross-phase modulation in fiber links with multiple optical amplifiers and dispersion compensators ," J. Lightw. Technol. 14, 249-260 (1996).

D. Raphaeli, "Distribution of noncentral indefinite quadratic forms in complex normal variables," IEEE Trans. Inf. Theory 42, 1002-1997 (1996).

1994 (3)

J.-S. Lee, C.-S. Shim, "Bit-error-rate analysis of optically preamplified receivers using an eigenfunction expansion method in optical frequency domain," J. Lightw. Technol. 12, 1224-1229 (1994).

A. Mecozzi, "Limits to long-haul coherent transmission set by the Kerr nonlinearity and noise of the in-line amplifiers," J. Lightw. Technol. 12, 1993-2000 (1994).

T.-K. Chiang, N. Kagi, T. K. Fong, M. E. Marhic, L. G. Kazovsky, "Cross-phase modulation in dispersive fibers: Theoretical and experimental investigation of the impact of modulation frequency," IEEE Photon. Technol. Lett. 6, 733-736 (1994).

1993 (2)

M. Azizoglu, P. A. Humblet, J. S. Young, "Phase noise induced performance limits for DPSK modulation with and without frequency feedback ," J. Lightw. Technol. 11, 290-302 (1993).

K. Kikuchi, "Enhancement of optical-amplifier noise by nonlinear refractive index and group-velocity dispersion of optical fibers," IEEE Photon. Technol. Lett. 5, 221-223 (1993).

1992 (2)

J. Wang, K. Petermann, "Small-signal analysis for dispersive optical fiber communication systems," J. Lightw. Technol. 10, 96-100 (1992).

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1990 (3)

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1986 (3)

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C. W. Helstrom, "Calculating error probabilities for intersymbol and cochannel interference," IEEE Trans. Commun. COM-34, 430-435 (1986).

1984 (1)

A. R. Chraplyvy, D. Marcuse, P. S. Henry, "Carrier-induced phase noise in angle-modulated optical-fiber systems," J. Lightw. Technol. LT-2, 6-10 (1984).

1982 (1)

R. F. Pawula, S. O. Rice, J. H. Roberts, "Distribution of the phase angle between two vectors perturbed by Gaussian noise," IEEE Trans. Commun. COM-30, 1828-1841 (1982).

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K. Kikuchi, "Enhancement of optical-amplifier noise by nonlinear refractive index and group-velocity dispersion of optical fibers," IEEE Photon. Technol. Lett. 5, 221-223 (1993).

G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, S. Benedetto, "A novel analytical method for the BER evaluation in optical systems affected by parametric gain ," IEEE Photon. Technol. Lett. 12, 152-154 (2000).

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A. Demir, "Computing timing jitter from phase noise spectra for oscillators and phase-locked loops with white and $1/f$ noise," IEEE Trans. Circuits Syst. I, Reg. Papers 53, 1869-1884 (2006).

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G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, S. Benedetto, "A novel analytical method to the evaluation of the impact of fiber parametric gain on the bit error rate," IEEE Trans. Commun. 49, 2154-2163 (2001).

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C. W. Helstrom, "Computing the distribution of sums of random sine waves and of Rayleigh-ditributed random variables by saddle-point integration," IEEE Trans. Commun. 45, 1487-1494 (1997).

R. F. Pawula, S. O. Rice, J. H. Roberts, "Distribution of the phase angle between two vectors perturbed by Gaussian noise," IEEE Trans. Commun. COM-30, 1828-1841 (1982).

C. W. Helstrom, "Calculating error probabilities for intersymbol and cochannel interference," IEEE Trans. Commun. COM-34, 430-435 (1986).

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IEEE Trans. Inf. Theory (2)

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S. Ryu, "Signal linewidth broadening due to nonlinear Kerr effect in long-haul coherent systems using cascaded optical amplifiers," J. Lightw. Technol. 10, 1450-1457 (1992).

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T.-K. Chiang, N. Kagi, M. E. Marhic, L. G. Kazovsky, "Cross-phase modulation in fiber links with multiple optical amplifiers and dispersion compensators ," J. Lightw. Technol. 14, 249-260 (1996).

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J. Wang, J. M. Kahn, "Impact of chromatic and polarization-mode dispersion on DPSK systems using interferometric detection," J. Lightw. Technol. 22, 362-371 (2004).

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K.-P. Ho, J. M. Kahn, "Electronic compensation technique to mitigate nonlinear phase noise," J. Lightw. Technol. 22, 779-783 (2004).

R. Hui, K. R. Demarest, C. T. Allen, "Cross-phase modulation in multispan WDM optical fiber systems," J. Lightw. Technol. 17, 1018-1026 (1999).

R. Holzlöhner, V. S. Grigoryan, C. R. Menyuk, W. L. Kath, "Accurate calculation of eye diagrams and bit error rates in optical transmission systems using linearization," J. Lightw. Technol. 20, 389-400 (2002).

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Opt. Lett. (5)

Phys. Rev. A, Gen. Phys. (1)

D. J. Kaup, "Perturbation theory for solitons in optical fibers," Phys. Rev. A, Gen. Phys. 42, 5689-5694 (1990).

SIAM J. Numer. Anal. (1)

L. N. Trefethen, J. A. C. Weideman, T. Schmelzer, "Talbot quadratures and rational approximations," SIAM J. Numer. Anal. 44, 2342-2362 (2006).

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