Abstract

Using an alternative approach for evaluating the Bit-Error Rate (BER), we present a numerical and experimental investigation of the performance of phase-modulated optical communication systems in the presence of nonlinear phase noise and dispersion. The numerical method is based on the well known Karhunen-Loève expansion combined with a linearization technique of the Nonlinear Schrödinger Equation (NLSE) to account for the nonlinear interaction between signal and noise. Our numerical results show a good agreement with experiments.

© 2009 Optical Society of America

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    [CrossRef] [PubMed]
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  4. R. Holzlohner and C. R. Menyuk, "Use of multicanonical Monte Carlo simulations to obtain accurate bit error rates in optical communications systems," Opt. Lett. 28, 1894-1896 (2003).
    [CrossRef] [PubMed]
  5. X. Wei, X. Liu, and C. Xu, "Numerical simulation of the SPM penalty in a 10-Gb/s RZ-DPSK system," IEEE Photon. Technol. Lett. 15, 1636-1638 (2003).
    [CrossRef]
  6. A. Mecozzi, "Limits to long-haul coherent transmission set by the Kerr nonlinearity and noise of the in-line amplifiers," J. Lightwave Technol. 12, 1993-2000 (1994).
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  9. K.-P. Ho, "Performance of DPSK Signals With Quadratic Phase Noise," IEEE Trans. Commun. 53, 1361-1365 (2005).
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    [CrossRef]
  15. P. Serena, A. Orlandini, and A. Bononi, "Parametric-gain approach to the analysis of single-channel DPSK/DQPSK systems with nonlinear phase noise," J. Lightwave Technol. 24, 2026-2037 (2006).
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    [CrossRef]
  18. A. V. T. Cartaxo, B. Wedding, and W. Idler, "Influence of fiber nonlinearity on the fiber transfer function: theoretical and experimental analysis," J. Lightwave Technol. 17, 1806-1813 (1999).
    [CrossRef]
  19. R. Holzlohner, V. S. Grigoryan, C. R. Menyuk, and W. L. Kath, "Accurate calculation of eye diagrams and bit error rates in optical transmission systems using linearization," J. Lightwave Technol. 20, 389-400 (2002).
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  20. R. Holzlohner, C. R. Menyuk, W. L. Kath, and 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).
    [CrossRef]
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    [CrossRef]
  23. A. G. Green, P. P. Mitra, and L. G. L. Wegener, "Effect of chromatic dispersion on nonlinear phase noise," Opt. Lett. 28, 2455-2457 (2003).
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  27. M. Kac and A. Siegert, "On the Theory of Noise in Radio Receivers with Square Law Detectors," J. Appl. Phys. 18, 383-397 (1947).
    [CrossRef]
  28. E. Forestieri, "Evaluating the error probability in lightwave systems with chromatic dispersion, arbitrary pulse shape and pre- and postdetection filtering," J. Lightwave Technol. 18, 1493-1503 (2000).
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    [CrossRef]
  33. R. O. Moore, G. Biondini, and W. L. Kath, "A Method to Compute Statistics of Large, Noise-Induced Perturbations of Nonlinear Schr¨odinger Solitons," SIAM Review 50, 523-549 (2008).
    [CrossRef]
  34. S. Kumar and L. Liu, "Reduction of nonlinear phase noise using optical phase conjugation in quasi-linear optical transmission systems," Opt. Express 15, 2166-2177 (2007).
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    [CrossRef]

2008 (2)

R. O. Moore, G. Biondini, and W. L. Kath, "A Method to Compute Statistics of Large, Noise-Induced Perturbations of Nonlinear Schr¨odinger Solitons," SIAM Review 50, 523-549 (2008).
[CrossRef]

M. P. Dlubek, A. J. Phillips, and E. C. Larkins, "Nonlinear Evolution of Gaussian ASE Noise in ZMNL Fiber," J. Lightwave Technol. 26, 891-898 (2008).
[CrossRef]

2007 (3)

2006 (2)

2005 (4)

A. H. Gnauck and P. J. Winzer, "Optical phase-shift-keyed transmission," J. Lightwave Technol. 23, 115-130 (2005).
[CrossRef]

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

K.-P. Ho, "Performance of DPSK Signals With Quadratic Phase Noise," IEEE Trans. Commun. 53, 1361-1365 (2005).
[CrossRef]

J.-A. Huang, K.-P. Ho, H.-K. Chen, S. K. Liaw, and H.-C. Wang, "Impact of nonlinear phase noise to DPSK signals: experimental verification of a simplified theoretical model," IEEE Photon. Technol. Lett. 17, 2236-2238 (2005).
[CrossRef]

2004 (1)

G. Bosco and 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).
[CrossRef]

2003 (5)

X. Wei, X. Liu, and C. Xu, "Numerical simulation of the SPM penalty in a 10-Gb/s RZ-DPSK system," IEEE Photon. Technol. Lett. 15, 1636-1638 (2003).
[CrossRef]

H. Kim and 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).
[CrossRef]

R. Holzlohner, C. R. Menyuk, W. L. Kath, and 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).
[CrossRef]

R. Holzlohner and C. R. Menyuk, "Use of multicanonical Monte Carlo simulations to obtain accurate bit error rates in optical communications systems," Opt. Lett. 28, 1894-1896 (2003).
[CrossRef] [PubMed]

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

2002 (2)

2000 (1)

1999 (1)

1998 (1)

1997 (1)

R. Hui, M. O’Sullivan, A. Robinson, and M. Taylor, "Modulation instability and its impact in multispan optical amplified IMDD systems: theory and experiments," J. Lightwave Technol. 15, 1071-1082 (1997).
[CrossRef]

1994 (2)

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

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

1990 (2)

J. P. Gordon and L. F. Mollenauer, "Phase noise in photonic communications systems using linear amplifiers," Opt. Lett. 15, 1351-1353 (1990).
[CrossRef] [PubMed]

D. Marcuse, "Derivation of analytical expressions for the bit-error probability in lightwave systems with optical amplifiers," J. Lightwave Technol. 8, 1816-1823 (1990).
[CrossRef]

1947 (1)

M. Kac and A. Siegert, "On the Theory of Noise in Radio Receivers with Square Law Detectors," J. Appl. Phys. 18, 383-397 (1947).
[CrossRef]

Biondini, G.

R. O. Moore, G. Biondini, and W. L. Kath, "A Method to Compute Statistics of Large, Noise-Induced Perturbations of Nonlinear Schr¨odinger Solitons," SIAM Review 50, 523-549 (2008).
[CrossRef]

Bononi, A.

Bosco, G.

G. Bosco and 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).
[CrossRef]

Cartaxo, A. V. T.

Chen, H.-K.

J.-A. Huang, K.-P. Ho, H.-K. Chen, S. K. Liaw, and H.-C. Wang, "Impact of nonlinear phase noise to DPSK signals: experimental verification of a simplified theoretical model," IEEE Photon. Technol. Lett. 17, 2236-2238 (2005).
[CrossRef]

Demir, A.

Dlubek, M. P.

Forestieri, E.

Gnauck, A. H.

A. H. Gnauck and P. J. Winzer, "Optical phase-shift-keyed transmission," J. Lightwave Technol. 23, 115-130 (2005).
[CrossRef]

H. Kim and 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).
[CrossRef]

Gordon, J. P.

Green, A. G.

Grigoryan, V. S.

R. Holzlohner, C. R. Menyuk, W. L. Kath, and 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).
[CrossRef]

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

Hanik, N.

N. Hanik, "Modelling of nonlinear optical wave propagation including linear mode-coupling and birefringence," Opt. Commun. 214, 207-230 (2002).
[CrossRef]

Ho, K.-P.

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

K.-P. Ho, "Performance of DPSK Signals With Quadratic Phase Noise," IEEE Trans. Commun. 53, 1361-1365 (2005).
[CrossRef]

J.-A. Huang, K.-P. Ho, H.-K. Chen, S. K. Liaw, and H.-C. Wang, "Impact of nonlinear phase noise to DPSK signals: experimental verification of a simplified theoretical model," IEEE Photon. Technol. Lett. 17, 2236-2238 (2005).
[CrossRef]

Holzlohner, R.

Huang, J.-A.

J.-A. Huang, K.-P. Ho, H.-K. Chen, S. K. Liaw, and H.-C. Wang, "Impact of nonlinear phase noise to DPSK signals: experimental verification of a simplified theoretical model," IEEE Photon. Technol. Lett. 17, 2236-2238 (2005).
[CrossRef]

Hui, R.

R. Hui, M. O’Sullivan, A. Robinson, and M. Taylor, "Modulation instability and its impact in multispan optical amplified IMDD systems: theory and experiments," J. Lightwave Technol. 15, 1071-1082 (1997).
[CrossRef]

Idler, W.

Kac, M.

M. Kac and A. Siegert, "On the Theory of Noise in Radio Receivers with Square Law Detectors," J. Appl. Phys. 18, 383-397 (1947).
[CrossRef]

Kath, W. L.

R. O. Moore, G. Biondini, and W. L. Kath, "A Method to Compute Statistics of Large, Noise-Induced Perturbations of Nonlinear Schr¨odinger Solitons," SIAM Review 50, 523-549 (2008).
[CrossRef]

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

R. Holzlohner, C. R. Menyuk, W. L. Kath, and 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).
[CrossRef]

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

Kim, H.

H. Kim and 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).
[CrossRef]

Kumar, S.

Larkins, E. C.

Lee, J.-S.

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

Liaw, S. K.

J.-A. Huang, K.-P. Ho, H.-K. Chen, S. K. Liaw, and H.-C. Wang, "Impact of nonlinear phase noise to DPSK signals: experimental verification of a simplified theoretical model," IEEE Photon. Technol. Lett. 17, 2236-2238 (2005).
[CrossRef]

Liu, L.

Liu, X.

X. Wei, X. Liu, and C. Xu, "Numerical simulation of the SPM penalty in a 10-Gb/s RZ-DPSK system," IEEE Photon. Technol. Lett. 15, 1636-1638 (2003).
[CrossRef]

Marcuse, D.

D. Marcuse, "Derivation of analytical expressions for the bit-error probability in lightwave systems with optical amplifiers," J. Lightwave Technol. 8, 1816-1823 (1990).
[CrossRef]

McKinstrie, C. J.

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

Mecozzi, A.

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

Menyuk, C. R.

Mitra, P. P.

Mollenauer, L. F.

Moore, R. O.

R. O. Moore, G. Biondini, and W. L. Kath, "A Method to Compute Statistics of Large, Noise-Induced Perturbations of Nonlinear Schr¨odinger Solitons," SIAM Review 50, 523-549 (2008).
[CrossRef]

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

O’Sullivan, M.

R. Hui, M. O’Sullivan, A. Robinson, and M. Taylor, "Modulation instability and its impact in multispan optical amplified IMDD systems: theory and experiments," J. Lightwave Technol. 15, 1071-1082 (1997).
[CrossRef]

Orlandini, A.

Phillips, A. J.

Poggiolini, P.

G. Bosco and 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).
[CrossRef]

Robinson, A.

R. Hui, M. O’Sullivan, A. Robinson, and M. Taylor, "Modulation instability and its impact in multispan optical amplified IMDD systems: theory and experiments," J. Lightwave Technol. 15, 1071-1082 (1997).
[CrossRef]

Serena, P.

Shim, C.-S.

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

Siegert, A.

M. Kac and A. Siegert, "On the Theory of Noise in Radio Receivers with Square Law Detectors," J. Appl. Phys. 18, 383-397 (1947).
[CrossRef]

Spiller, E. T.

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

Taylor, M.

R. Hui, M. O’Sullivan, A. Robinson, and M. Taylor, "Modulation instability and its impact in multispan optical amplified IMDD systems: theory and experiments," J. Lightwave Technol. 15, 1071-1082 (1997).
[CrossRef]

Wang, H.-C.

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

J.-A. Huang, K.-P. Ho, H.-K. Chen, S. K. Liaw, and H.-C. Wang, "Impact of nonlinear phase noise to DPSK signals: experimental verification of a simplified theoretical model," IEEE Photon. Technol. Lett. 17, 2236-2238 (2005).
[CrossRef]

Wedding, B.

Wegener, L. G. L.

Wei, X.

X. Wei, X. Liu, and C. Xu, "Numerical simulation of the SPM penalty in a 10-Gb/s RZ-DPSK system," IEEE Photon. Technol. Lett. 15, 1636-1638 (2003).
[CrossRef]

Winzer, P. J.

Xu, C.

X. Wei, X. Liu, and C. Xu, "Numerical simulation of the SPM penalty in a 10-Gb/s RZ-DPSK system," IEEE Photon. Technol. Lett. 15, 1636-1638 (2003).
[CrossRef]

IEEE Photon. Technol. Lett. (6)

H. Kim and 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).
[CrossRef]

X. Wei, X. Liu, and C. Xu, "Numerical simulation of the SPM penalty in a 10-Gb/s RZ-DPSK system," IEEE Photon. Technol. Lett. 15, 1636-1638 (2003).
[CrossRef]

J.-A. Huang, K.-P. Ho, H.-K. Chen, S. K. Liaw, and H.-C. Wang, "Impact of nonlinear phase noise to DPSK signals: experimental verification of a simplified theoretical model," IEEE Photon. Technol. Lett. 17, 2236-2238 (2005).
[CrossRef]

R. Holzlohner, C. R. Menyuk, W. L. Kath, and 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).
[CrossRef]

G. Bosco and 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).
[CrossRef]

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

IEEE Trans. Commun. (1)

K.-P. Ho, "Performance of DPSK Signals With Quadratic Phase Noise," IEEE Trans. Commun. 53, 1361-1365 (2005).
[CrossRef]

J. Appl. Phys. (1)

M. Kac and A. Siegert, "On the Theory of Noise in Radio Receivers with Square Law Detectors," J. Appl. Phys. 18, 383-397 (1947).
[CrossRef]

J. Lightwave Technol. (12)

D. Marcuse, "Derivation of analytical expressions for the bit-error probability in lightwave systems with optical amplifiers," J. Lightwave Technol. 8, 1816-1823 (1990).
[CrossRef]

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

R. Hui, M. O’Sullivan, A. Robinson, and M. Taylor, "Modulation instability and its impact in multispan optical amplified IMDD systems: theory and experiments," J. Lightwave Technol. 15, 1071-1082 (1997).
[CrossRef]

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

A. Demir, "Nonlinear Phase Noise in Optical-Fiber-Communication Systems," J. Lightwave Technol. 25, 2002-2032 (2007).
[CrossRef]

M. P. Dlubek, A. J. Phillips, and E. C. Larkins, "Nonlinear Evolution of Gaussian ASE Noise in ZMNL Fiber," J. Lightwave Technol. 26, 891-898 (2008).
[CrossRef]

A. H. Gnauck and P. J. Winzer, "Optical phase-shift-keyed transmission," J. Lightwave Technol. 23, 115-130 (2005).
[CrossRef]

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

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

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

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

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

J. Opt. Soc. Am. B (1)

Opt. Commun. (1)

N. Hanik, "Modelling of nonlinear optical wave propagation including linear mode-coupling and birefringence," Opt. Commun. 214, 207-230 (2002).
[CrossRef]

Opt. Express (1)

Opt. Lett. (4)

SIAM Review (1)

R. O. Moore, G. Biondini, and W. L. Kath, "A Method to Compute Statistics of Large, Noise-Induced Perturbations of Nonlinear Schr¨odinger Solitons," SIAM Review 50, 523-549 (2008).
[CrossRef]

Other (9)

M. Ohm, R. J. Essiambre, and P. J. Winzer, "Nonlinear phase noise and distortion in 42.7-Gbit/s RZ-DPSK systems," in 31st European Conference on Optical Communication, ECOC 2005 (Glasgow, Scotland, 2005).
[CrossRef]

P. Serena, A. Orlandini, and A. Bononi, "A parametric gain approach to DPSK performance evaluation in presence of nonlinear phase noise," in 30th European Conference on Optical Communication, ECOC 2004 (Stockholm, Sweden, 2004).

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G. Strang, Linear Algebra and its Applications, 3rd ed. (Saunders, 1988).

A. Orlandini, P. Serena, and A. Bononi, "An Alternative Analysis of Nonlinear Phase Noise Impact on DPSK Systems," in 32nd European Conference on Optical Communication, ECOC 2006 (Cannes, France, 2006).
[CrossRef]

L. D. Coelho, L. Molle, D. Gross, N. Hanik, R. Freund, C. Caspar, and E.-D. Schmidt, "Numerical and Experimental Investigation of the Effect of Dispersion on Nonlinear Phase Noise in RZ-DPSK Systems," in 33rd European Conference on Optical Communication, ECOC 2007 (Berlin, Germany, 2007).
[CrossRef]

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Figures (7)

Fig. 1.
Fig. 1.

Multi-Span System

Fig. 2.
Fig. 2.

Balanced Receiver

Fig. 3.
Fig. 3.

Block diagram for signal transmission including nonlinear phase noise

Fig. 4.
Fig. 4.

Experimental set-up in a loop configuration: (a) non- and (b) high-dispersive span.

Fig. 5.
Fig. 5.

Fitting of the 20Gbit/s 50% RZ-DPSK Signal Power Spectrum

Fig. 6.
Fig. 6.

BER vs RX-OSNR for ASE noise added to the signal at the receiver

Fig. 7.
Fig. 7.

BER vs RX-OSNR for ASE noise added to the signal at the transmitter

Tables (1)

Tables Icon

Table 1. Fiber Parameters

Equations (52)

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A ( z , t ) z j 2 β 2 2 A ( z , t ) t 2 1 6 β 3 3 A ( z , t ) t 3 = A ( z , y ) 2 A ( z , t ) · e αz ,
A ( z ) z = P 0 A ( z ) · e αz
A ( z ) = P 0 · e ( z ) ,
A ( z , t ) = ( P 0 + a ( z , t ) ) · e ( z ) .
a ˜ ( z , ω ) z + j 2 β 2 ω 2 a ˜ ( z , ω ) + j 6 β 3 ω 3 a ˜ ( z , ω ) = P 0 e αz ( a ˜ ( z , ω ) + a ˜ * ( z , ω ) ) .
a ˜ p ( z , ω ) z + j a ˜ q ( z , ω ) z + j 2 β 2 ω 2 a ˜ p ( z , ω ) 1 2 β 2 ω 2 a ˜ q ( z , ω ) + j 6 β 3 ω 3 a ˜ p ( z , ω )
1 6 β 3 ω 3 a ˜ q ( z , ω ) = 2 P 0 e αz a ˜ p ( z , ω ) .
a ˜ p ( z , ω ) z = ρ · a ˜ q ( z , ω )
a ˜ q ( z , ω ) z = ( ρ + 2 γ P 0 e αz ) a ˜ p ( z , ω )
M = ( cos ( δL ) ρ δ sin ( δL ) δ ρ sin ( δL ) cos ( δL ) ) ,
M = lim dz 0 i = 1 N sec ( cos ( δ i L ) ρ δ i sin ( δ i L ) δ i ρ sin ( δ i L ) cos ( δ i L ) ) = ( M 11 ( ω ) M 12 ( ω ) M 21 ( ω ) M 22 ( ω ) ) ,
δ i = ρ 2 + 2 ρ γ dz P 0 e α ( i 1 ) dz ,
γ dz = 1 exp ( αdz ) αdz · γ .
G 1 = ( G pp G pq G qp G qq ) lim τ 1 τ E { a ˜ out · a ˜ out H } = lim τ 1 τ ( E { a ˜ p 2 } E { a ˜ p · a ˜ q * } E { a ˜ q · a ˜ p * } E { a ˜ q 2 } ) ,
Φ ( L , ω ) = Φ ASE · exp ( αL ) 2 · ( M 11 ( ω ) 2 + M 12 ( ω ) 2 + M 21 ( ω ) 2 + M 22 ( ω ) 2 )
H i = ( ℜ𝔢 { H ( ω ) } 𝔍𝔪 { H ( ω ) } 𝔍𝔪 { H ( ω ) } ℜ𝔢 { H ( ω ) } )
a ˜ out = i = 1 N M i · a ˜ i + H N + 1 · a ˜ N + 1 ,
G N + 1 ( z , ω ) = Φ ASE N + 1 ( i = 1 N M i ( M i ) H Φ ASE i Φ ASE N + 1 + H N + 1 H N + 1 H ) ,
W ˜ = 1 Φ ASE N + 1 ( G pp 0 ( G pq * G pp ) G qq G pq 2 G pp )
a ˜ out = W ˜ · a ˜ N + 1 = ( W ˜ 11 ( ω ) W ˜ 12 ( ω ) W ˜ 21 ( ω ) W ˜ 22 ( ω ) ) · a ˜ N + 1 .
n ˜ re ( z , ω ) = ℜ𝔢 { a ˜ ( z , ω ) } = a ˜ ( z , ω ) + a ˜ * ( z , ω ) 2
n ˜ im ( z , ω ) = 𝔍𝔪 { a ˜ ( z , ω ) } = a ˜ ( z , ω ) a ˜ * ( z , ω ) 2 j
n ˜ out ( f ) = B 1 ( a ˜ out ( f ) a ˜ out * ( f ) ) = B 1 ( W ˜ 11 ( f ) W ˜ 12 ( f ) W ˜ 21 ( f ) W ˜ 22 ( f ) W ˜ 11 * ( f ) D 0 W ˜ 12 * ( f ) D 0 W ˜ 21 * ( f ) D 0 W ˜ 22 * ( f ) D 0 ) a ˜ N + 1 ( f ) ,
n ˜ out ( f ) = W ( f ) · n ˜ N + 1 ( f ) ,
W ( f ) = B 1 ( W ˜ 11 ( f ) W ˜ 12 ( f ) W ˜ 21 ( f ) W ˜ 22 ( f ) W ˜ 11 * ( f ) D 0 W ˜ 12 * ( f ) D 0 W ˜ 21 * ( f ) D 0 W ˜ 22 * ( f ) D 0 ) B 2 ,
B 1 = 1 2 ( I j I I j I j I I j I I )
B 2 = 1 2 ( I + D 0 j ( I + D 0 ) j ( I D 0 ) I + D 0 )
I ( t k ) = E ˜ * ( f 2 ) K ( f 1 , f 2 ) E ˜ ( f 1 ) e j 2 π ( f 1 f 2 ) t k d f 1 d f 2 ,
K ( f 1 , f 2 ) = H e ( f 1 f 2 ) [ H o * ( f 2 ) H 1 * ( f 2 ) H o ( f 1 ) H 1 ( f 1 ) H o * ( f 2 ) H 2 * ( f 2 ) H o ( f 1 ) H 2 ( f 1 ) ] ,
φ ( f ) = 1 λ K ( f , f ) φ ( f ) d f .
φ m ( f ) φ l * ( f ) df = δ ml ,
E ˜ ( f ) e j 2 πft = i c i ( t ) · φ i ( f ) ,
E ˜ ( f ) e j 2 πft = i ( s i ( t ) + n i ( t ) ) φ i ( f ) .
I ( t k ) = i λ i s i ( t k ) + n i ( t k ) 2 .
I ( t k ) = m = 1 2 M l = 1 2 M e m * K ml e l ,
e m = E ˜ ( f m ) · e j 2 π f m t k Δ f ,
e s , m = s ˜ out ( f m ) · e j 2 π f m t k Δ f ,
e n , m = n ˜ out ( f m ) · e j 2 π f m t k Δ f ,
K ml = K ( f l , f m ) Δ f ,
I ( t k ) = e H Ke ,
λ i · q i = K · q i ,
Ψ I ( t k ) ( s ) = i = 1 2 M e α i s 1 β i s ( 1 β i s ) ξ ,
E [ I ( t k ) ] = i = 1 2 M λ i ( ξ · 2 σ 2 + s i ( t k ) 2 )
Var [ I ( t k ) ] = i = 1 2 M ξ · 4 σ 2 λ i 2 ( σ 2 + s i ( t k ) 2 ) ,
I ( t k ) = e W T K W e W ,
λ i , W · q i , W = K W · q i , W ,
e W = W 1 ( f ) ( ℜ𝔢 { e s } 𝔍𝔴 { e s } ) e W , s + ( ℜ𝔢 { e n } 𝔍𝔴 { e n } ) e W , n
K W = W T ( f ) ( ℜ𝔢 { K } 𝔍𝔴 { K } 𝔍𝔴 { K } ℜ𝔢 { K } ) W ( f ) .
ϕ NL ( a ) = N ( γ DSF 1 L eff DSF 1 P DSF 1 + γ DSF 2 L eff DSF 2 P DSF 2 )
ϕ NL ( b ) = N ( γ SMF L eff SMF P SMF + γ DCF L eff DCF P DCF ) ,
P eff = 1 + H w ( π ) 2 · P peak ,
H w ( ω ) = ( 1 + ( L eff 4 L D ω 2 ) 2 ) 1 ,

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