Abstract

A general theory of the parametric gain of amplified spontaneous emission (ASE) noise in periodic dispersion-managed (DM) optical links is presented, based on a linearization of the nonlinear Schrödinger equation around a constant-wave input signal. Closed-form expressions are presented of the in-phase and quadrature ASE power spectral densities (PSDs), valid in the limit of infinitely many spans, for a limited total cumulated nonlinear phase and in-line dispersion, a typical case for nonsoliton systems. PSDs are shown to solely depend on the in-line cumulated dispersion and on the so-called DM kernel. Kernel expressions for both typical terrestrial and submarine DM links are provided. By Taylor expanding the kernel in frequency, we introduce a definition of DM map strength that is more appropriate for limited nonlinear phase DM systems with lossy transmission fibers than the standard definition for soliton systems. Various important special cases of PSDs are discussed in detail. Novel insights, to our knowledge, into the effect of a postdispersion-compensating fiber on such PSDs are included. Finally, examples of application of the PSD formulas to the performance evaluation of both on–off keying and differential phase keying modulated systems are provided.

© 2007 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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  26. A comparison of DM systems with different span lengths at a constant nonlinear phase can make sense, for instance, in a metropolitan network environment where, although the fiber links are short, a big lumped loss, such as a demultiplexer or an add-drop module, is inserted at the end of each span connecting two consecutive nodes.
  27. R. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge U. Press, 1999).
  28. A. Bononi, P. Serena, J.-C. Antona, and S. Bigo, "Implications of nonlinear interactions of signal and noise in low-OSNR transmission systems with FEC," in Optical Fiber Communication Conference, OSA Trends in Optics and Photonics Series (Optical Society of America, 2005), paper OTHW5.
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  34. G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, "Parametric gain in multiwavelength systems: a new approach to noise enhancement analysis," IEEE Photon. Technol. Lett. 9, 1135-1137 (1999).
    [CrossRef]

2006 (1)

2005 (1)

2004 (2)

2003 (2)

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]

K.-P. Ho, "Probability density of nonlinear phase noise," J. Opt. Soc. Am. B 20, 1875-1879 (2003).
[CrossRef]

2002 (5)

2001 (1)

1999 (2)

E. Ciaramella and M. Tamburrini, "Modulation instability in long amplified links with strong dispersion compensation," IEEE Photon. Technol. Lett. 11, 1608-1610 (1999).
[CrossRef]

G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, "Parametric gain in multiwavelength systems: a new approach to noise enhancement analysis," IEEE Photon. Technol. Lett. 9, 1135-1137 (1999).
[CrossRef]

1998 (2)

E. Ciaramella, "Effect of non-uniform chromatic dispersion fibre link in determining system limitations due to four wave mixing," Electron. Lett. 34, 202-204 (1998).
[CrossRef]

M. Midrio, P. Franco, and M. Romagnoli, "Noise statistics in transmission systems with grating dispersion compensation," J. Opt. Soc. Am. B 15, 2748-2756 (1998).
[CrossRef]

1997 (3)

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

C. Lorattanasane and K. Kikuchi, "Parametric instability of optical amplifier noise in long-distance optical transmission systems," IEEE J. Quantum Electron. 33, 1068-1074 (1997).
[CrossRef]

1996 (2)

1995 (1)

1993 (1)

1991 (1)

1990 (1)

Ablowitz, M. J.

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2001).

Antona, J. C.

Antona, J.-C.

A. Bononi, P. Serena, J.-C. Antona, and S. Bigo, "Implications of nonlinear interactions of signal and noise in low-OSNR transmission systems with FEC," in Optical Fiber Communication Conference, OSA Trends in Optics and Photonics Series (Optical Society of America, 2005), paper OTHW5.

Benedetto, S.

G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, "Parametric gain in multiwavelength systems: a new approach to noise enhancement analysis," IEEE Photon. Technol. Lett. 9, 1135-1137 (1999).
[CrossRef]

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

Bigo, S.

P. Serena, A. Bononi, J. C. Antona, and S. Bigo, "Parametric gain in the strongly nonlinear regime and its impact on 10-Gb/s NRZ systems with forward-error correction," J. Lightwave Technol. 23, 2352-2363 (2005).
[CrossRef]

A. Bononi, P. Serena, J.-C. Antona, and S. Bigo, "Implications of nonlinear interactions of signal and noise in low-OSNR transmission systems with FEC," in Optical Fiber Communication Conference, OSA Trends in Optics and Photonics Series (Optical Society of America, 2005), paper OTHW5.

Bononi, A.

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]

P. Serena, A. Bononi, J. C. Antona, and S. Bigo, "Parametric gain in the strongly nonlinear regime and its impact on 10-Gb/s NRZ systems with forward-error correction," J. Lightwave Technol. 23, 2352-2363 (2005).
[CrossRef]

A. Orlandini, P. Serena, and A. Bononi, "An alternative analysis of nonlinear phase noise impact on DPSK systems," in Proceedings of the IEEE European Conference on Optical Communication (IEEE, 2006), paper Th3.2.6.
[CrossRef]

A. Bononi, P. Serena, J.-C. Antona, and S. Bigo, "Implications of nonlinear interactions of signal and noise in low-OSNR transmission systems with FEC," in Optical Fiber Communication Conference, OSA Trends in Optics and Photonics Series (Optical Society of America, 2005), paper OTHW5.

Bosco, G.

G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, "Parametric gain in multiwavelength systems: a new approach to noise enhancement analysis," IEEE Photon. Technol. Lett. 9, 1135-1137 (1999).
[CrossRef]

Brandt-Pearce, M.

Bronski, J. C.

Capobianco, A.-D.

F. Consolandi, C. De Angelis, A.-D. Capobianco, and A. Tonello, "Parametric gain in fiber systems with periodic dispersion management," Opt. Commun. 208, 309-320 (2002).
[CrossRef]

Carena, A.

G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, "Parametric gain in multiwavelength systems: a new approach to noise enhancement analysis," IEEE Photon. Technol. Lett. 9, 1135-1137 (1999).
[CrossRef]

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

Chaplyvy, A. R.

C. J. McKinstrie, S. Radic, and A. R. Chaplyvy, "Parametric amplifiers driven by two pump waves," IEEE J. Sel. Top. Quantum Electron. 8, 538-547 (2002).
[CrossRef]

Chávez Boggio, J. M.

Chen, C.-T.

C.-T. Chen, Linear System Theory and Design (Holt-Saunders, 1984), pp. 49-50.

Ciaramella, E.

E. Ciaramella and M. Tamburrini, "Modulation instability in long amplified links with strong dispersion compensation," IEEE Photon. Technol. Lett. 11, 1608-1610 (1999).
[CrossRef]

E. Ciaramella, "Effect of non-uniform chromatic dispersion fibre link in determining system limitations due to four wave mixing," Electron. Lett. 34, 202-204 (1998).
[CrossRef]

Consolandi, F.

F. Consolandi, C. De Angelis, A.-D. Capobianco, and A. Tonello, "Parametric gain in fiber systems with periodic dispersion management," Opt. Commun. 208, 309-320 (2002).
[CrossRef]

Curri, V.

G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, "Parametric gain in multiwavelength systems: a new approach to noise enhancement analysis," IEEE Photon. Technol. Lett. 9, 1135-1137 (1999).
[CrossRef]

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

De Angelis, C.

F. Consolandi, C. De Angelis, A.-D. Capobianco, and A. Tonello, "Parametric gain in fiber systems with periodic dispersion management," Opt. Commun. 208, 309-320 (2002).
[CrossRef]

Doran, N. J.

Evans, L. C.

L. C. Evans, "An introduction to stochastic differential equations," http://math.berkeley/edu/~evans/.

Fragnito, H. L.

Franco, P.

Gaudino, R.

G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, "Parametric gain in multiwavelength systems: a new approach to noise enhancement analysis," IEEE Photon. Technol. Lett. 9, 1135-1137 (1999).
[CrossRef]

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

Gnauck, A. 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]

Gordon, J.-P.

Grigoryan, V. S.

Haus, H. A.

Hirooka, T.

Ho, K.-P.

Holzlöhner, R.

Horn, R. A.

R. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge U. Press, 1999).

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]

Inoue, T.

Johnson, C. R.

R. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge U. Press, 1999).

Karlsson, M.

Kath, W. L.

Kikuchi, K.

C. Lorattanasane and K. Kikuchi, "Parametric instability of optical amplifier noise in long-distance optical transmission systems," IEEE J. Quantum Electron. 33, 1068-1074 (1997).
[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]

Kutz, J. N.

Lorattanasane, C.

C. Lorattanasane and K. Kikuchi, "Parametric instability of optical amplifier noise in long-distance optical transmission systems," IEEE J. Quantum Electron. 33, 1068-1074 (1997).
[CrossRef]

Matera, F.

McKinstrie, C. J.

C. J. McKinstrie, S. Radic, and A. R. Chaplyvy, "Parametric amplifiers driven by two pump waves," IEEE J. Sel. Top. Quantum Electron. 8, 538-547 (2002).
[CrossRef]

Mecozzi, A.

Menyuk, C. R.

Midrio, M.

Moeser, J.

M. J. Ablowitz and J. Moeser, "Dispersion management for randomly varying optical fibers," Opt. Lett. 8, 821-823 (2004).
[CrossRef]

Mollenauer, L. F.

Orlandini, A.

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]

A. Orlandini, P. Serena, and A. Bononi, "An alternative analysis of nonlinear phase noise impact on DPSK systems," in Proceedings of the IEEE European Conference on Optical Communication (IEEE, 2006), paper Th3.2.6.
[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]

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd ed. (Mc-Graw-Hill, 1991).

Poggiolini, P.

G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, "Parametric gain in multiwavelength systems: a new approach to noise enhancement analysis," IEEE Photon. Technol. Lett. 9, 1135-1137 (1999).
[CrossRef]

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

Radic, S.

C. J. McKinstrie, S. Radic, and A. R. Chaplyvy, "Parametric amplifiers driven by two pump waves," IEEE J. Sel. Top. Quantum Electron. 8, 538-547 (2002).
[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]

Romagnoli, M.

Serena, P.

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]

P. Serena, A. Bononi, J. C. Antona, and S. Bigo, "Parametric gain in the strongly nonlinear regime and its impact on 10-Gb/s NRZ systems with forward-error correction," J. Lightwave Technol. 23, 2352-2363 (2005).
[CrossRef]

A. Orlandini, P. Serena, and A. Bononi, "An alternative analysis of nonlinear phase noise impact on DPSK systems," in Proceedings of the IEEE European Conference on Optical Communication (IEEE, 2006), paper Th3.2.6.
[CrossRef]

A. Bononi, P. Serena, J.-C. Antona, and S. Bigo, "Implications of nonlinear interactions of signal and noise in low-OSNR transmission systems with FEC," in Optical Fiber Communication Conference, OSA Trends in Optics and Photonics Series (Optical Society of America, 2005), paper OTHW5.

Settembre, M.

Smith, N. J.

Tamburrini, M.

E. Ciaramella and M. Tamburrini, "Modulation instability in long amplified links with strong dispersion compensation," IEEE Photon. Technol. Lett. 11, 1608-1610 (1999).
[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]

Tenenbaum, S.

Tonello, A.

F. Consolandi, C. De Angelis, A.-D. Capobianco, and A. Tonello, "Parametric gain in fiber systems with periodic dispersion management," Opt. Commun. 208, 309-320 (2002).
[CrossRef]

Xu, B.

Zwillinger, D.

D. Zwillinger, Handbook of Differential Equations (Academic, 1998).

Electron. Lett. (1)

E. Ciaramella, "Effect of non-uniform chromatic dispersion fibre link in determining system limitations due to four wave mixing," Electron. Lett. 34, 202-204 (1998).
[CrossRef]

IEEE J. Quantum Electron. (1)

C. Lorattanasane and K. Kikuchi, "Parametric instability of optical amplifier noise in long-distance optical transmission systems," IEEE J. Quantum Electron. 33, 1068-1074 (1997).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

C. J. McKinstrie, S. Radic, and A. R. Chaplyvy, "Parametric amplifiers driven by two pump waves," IEEE J. Sel. Top. Quantum Electron. 8, 538-547 (2002).
[CrossRef]

IEEE Photon. Technol. Lett. (4)

G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, "Parametric gain in multiwavelength systems: a new approach to noise enhancement analysis," IEEE Photon. Technol. Lett. 9, 1135-1137 (1999).
[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]

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

E. Ciaramella and M. Tamburrini, "Modulation instability in long amplified links with strong dispersion compensation," IEEE Photon. Technol. Lett. 11, 1608-1610 (1999).
[CrossRef]

J. Lightwave Technol. (4)

J. Opt. Soc. Am. B (8)

Opt. Commun. (1)

F. Consolandi, C. De Angelis, A.-D. Capobianco, and A. Tonello, "Parametric gain in fiber systems with periodic dispersion management," Opt. Commun. 208, 309-320 (2002).
[CrossRef]

Opt. Lett. (5)

Other (9)

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2001).

A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd ed. (Mc-Graw-Hill, 1991).

D. Zwillinger, Handbook of Differential Equations (Academic, 1998).

A. Orlandini, P. Serena, and A. Bononi, "An alternative analysis of nonlinear phase noise impact on DPSK systems," in Proceedings of the IEEE European Conference on Optical Communication (IEEE, 2006), paper Th3.2.6.
[CrossRef]

L. C. Evans, "An introduction to stochastic differential equations," http://math.berkeley/edu/~evans/.

C.-T. Chen, Linear System Theory and Design (Holt-Saunders, 1984), pp. 49-50.

A comparison of DM systems with different span lengths at a constant nonlinear phase can make sense, for instance, in a metropolitan network environment where, although the fiber links are short, a big lumped loss, such as a demultiplexer or an add-drop module, is inserted at the end of each span connecting two consecutive nodes.

R. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge U. Press, 1999).

A. Bononi, P. Serena, J.-C. Antona, and S. Bigo, "Implications of nonlinear interactions of signal and noise in low-OSNR transmission systems with FEC," in Optical Fiber Communication Conference, OSA Trends in Optics and Photonics Series (Optical Society of America, 2005), paper OTHW5.

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Figures (15)

Fig. 1
Fig. 1

Power gain profile f ( ζ ) within a single period of a general periodic DM link. The overall link is the repetition of N s periods.

Fig. 2
Fig. 2

Plot of r (solid curves) and 1 ( ω ω c ) 2 (dashed curve) versus normalized frequency f n = ω 2 π , at Φ NL = 0.5 π and ξ in = 0.1 . Critical frequency f c ω c 2 π .

Fig. 3
Fig. 3

Normalized ASE PSDs (top: S ̂ pp and S ̂ qq ; bottom: S ̂ pq ) versus normalized frequency f n in an uncompensated system ( S = 0 ) with Φ NL = 0.5 π , and ξ in = 0.1275 (left), ξ in = 0.1275 (right). Solid curves: SSFM simulations for N s = 5 , 10 , 15 , 50 . Dashed curves: theory [Eqs. (29)].

Fig. 4
Fig. 4

Normalized ASE PSDs versus normalized frequency f n in a DM system with full in-line compensation, with Φ NL = 0.5 π , S = 0.1275 (left), and S = 0.1275 (right). Solid curves: SSFM simulations for N s = 5 , 10 , 15 , 50 . Dashed curves: theory [Eqs. (29)].

Fig. 5
Fig. 5

Normalized ASE PSD S ̂ pp and S ̂ qq versus frequency f n , in a DM system with ξ in = 0.1275 ( 1000 ps nm at 10 Gb s ) and with Φ NL = 0.72 π rad for the (left) SMF system ( S = 0.0485 ) ; (center) NZDSF + system ( S = 0.0095 ) ; (right) NZDSF system ( S = 0.0058 ) . Solid curves: SSFM simulations for N s = 5 , 10 , 15 , 50 . Dashed curves: theory [Eqs. (29)].

Fig. 6
Fig. 6

(solid curves) ASE PSDs versus frequency for a long-span terrestrial link with Φ NL = 0.5 π , ξ in = 0.1 , and both S = 0.1 and S = 0.001 . (dashed curves) S = 0 unmapped case.

Fig. 7
Fig. 7

Normalized ASE PSD S ̂ pp and S ̂ qq versus frequency f n at Φ NL = 0.72 π rad and cumulated in-line dispersion ξ in = 0.025 ( 200 ps nm at R = 10 Gb s ). Left: theory [Eqs. (29)] (dashed curves) and SSFM simulations (symbols) for 20-span terrestrial map with D T = 8 ps ( nm km ) , D C = 100 ps ( nm km ) , and various lengths L T = 10 , 50 , 100 km . Right: PSDs for L T = 10 km case as in left plot (dashed curves) and equivalent long-span system with D eq = 1.9 ps ( nm km ) (solid curves).

Fig. 8
Fig. 8

Normalized ASE PSD S ̂ pp and S ̂ qq versus frequency f n , in a system with Φ NL = 0.72 π rad , ξ in = 0.025 , and terrestrial map with L T = 100 km . Left: two-section span with DCF within dual-stage amplifier, with D T = 8 ps ( nm km ) , D C = 100 ps ( nm km ) , and various ratios P DCF P = 20 , 3 , 0 dB . Right: PSDs for the P DCF P = 0 dB case as in left figure (solid curves) and equivalent long-span system (dashed curves) with D eq = 12.8 ps ( nm km ) .

Fig. 9
Fig. 9

S 1 S T versus transmission fiber length L T for varying η c . Same attenuation and dispersion values as in Fig. 7.

Fig. 10
Fig. 10

S ̂ pp (concave) and S ̂ qq (convex) versus frequency f n for a three-section span with 15 km at dispersion + D , 30 km at D , and 15 km at + D , and D is 8 ps ( nm km ) . Φ NL = 0.72 π rad . PSDs of equivalent long-span fit with D eq = 1.6 ps ( nm km ) also shown.

Fig. 11
Fig. 11

S ̂ pp (concave) and S ̂ qq (convex) versus frequency f n for a fully compensated two-section span with 50 km at dispersion + D and slope + S and 50 km at D and slope S , with D = 0.1 ps ( nm km ) and S = 0.058 ps ( nm 2 km ) . PSDs ignoring dispersion slope are shown, as dashed curves. Φ NL = 0.72 π rad .

Fig. 12
Fig. 12

ASE PSD for different postcompensations. (a) ξ post = 0.025 ; (b) ξ post = 0 ; (c) ξ post = 0.025 . Index 1: in-phase component; index 2: quadrature component. Dashed curves: eigenvalues λ m , λ M . S = 0.01 , ξ in = 0 , Φ NL = 0.6 π .

Fig. 13
Fig. 13

Normalized in-phase ASE variance versus ξ post and ξ in . Terrestrial long-span DM system with S = 0.022 and Φ NL = 0.6 π . Circles: ξ post opt in Eq. (42).

Fig. 14
Fig. 14

Q factor versus Φ NL for a NRZ-DPSK signal after propagation into a 20 × 100 km fully compensated NZDSF + system at R = 10 Gb s . Dashed curves: evaluation using formulas (29). Solid curves: evaluation using Monte Carlo PSDs. Dashed–dotted line with crosses: Q factor in the absence of PG.

Fig. 15
Fig. 15

Q factor versus Φ NL for a NRZ-OOK signal after propagation into a 20 × 100 km fully compensated NZDSF + system at R = 10 Gb s . Dashed curves: evaluation using formulas (29). Solid curves: evaluation using simulated PSDs. Dashed–dotted lines with crosses: Q factor in the absence of PG.

Equations (63)

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A z = j β 2 ( z ) 2 2 A t 2 + β 3 ( z ) 6 3 A t 3 j γ ( z ) A 2 A + g ( z ) 2 A + W A ( z , t ) ,
U z = j 1 2 L d 2 U τ 2 + 1 6 L d 3 U τ 3 j f ( z ) L NL U 2 U + W U ( z , τ ) ,
U ( z , τ ) = [ 1 + u ( z , τ ) ] exp [ j Φ NL ( z ) ] ,
u z = j 1 2 L d 2 u τ 2 + 1 6 L d 3 u τ 3 j f ( z ) L NL ( u + u * ) + W U ,
u ̃ z = j ω 2 2 L d ( z ) u ̃ ( z , ω ) j ω 3 6 L d ( z ) u ̃ ( z , ω ) j f ( z ) L NL ( z ) [ u ̃ ( z , ω ) + u ̃ * ( z , ω ) ] + W ̃ U ( z , ω ) .
u ̃ ( z , ω ) = a ̃ ( z , ω ) exp [ j Θ Δ ( z , ω ) 2 ] ,
Θ Δ = ω 2 0 z [ 1 L Δ ( x ) + ω 3 L Δ ( x ) ] d x .
a ̃ z = j ω 2 2 L D a ̃ ( z , ω ) j ω 3 6 L D a ̃ ( z , ω ) j f ( z ) L NL ( z ) [ a ̃ ( z , ω ) + a ̃ * ( z , ω ) exp ( j Θ Δ ) ] + W ̃ U ( z , ω ) ,
R ( ω ) f ( z ) e j Θ Δ ( z , ω ) L NL ( z ) = 1 L 0 L f ( x ) e j Θ Δ ( x , ω ) L NL ( x ) d x ,
a ̃ z = j [ L ( ω ) a ̃ ( z , ω ) + R ( ω ) a ̃ * ( z , ω ) ] + W ̃ ( z , ω ) ,
E { W ̃ ( z 1 , ω ) W ̃ * ( z 2 , ω ) } = 2 σ 2 δ ( z 1 z 2 ) ,
L p z = Φ NL ω 2 2 ξ in ,
L q z = j ω 3 6 ξ in ,
a ( z , ω ) z = M ( ω ) a ( z , ω ) + W ( z , ω ) ,
A = [ R q L p R p L p R p R q ] .
a ( z , ω ) = e M ( ω ) z a ( 0 , ω ) + 0 z e M ( ω ) x W ( z x , ω ) d x .
k = R p 2 + R q 2 L p 2 .
e M z = e L q z [ cosh ( k z ) I + sinh ( k z ) k z A z ] ,
S ( z , ω ) = lim T 0 E { a ( z , ω ) a ( z , ω ) } T 0 [ S pp S pq S qp S qq ] ,
S ̂ S σ 2 z = 1 z 0 z e M x e M t x d x = 1 z 0 z e A x e A x d x ,
S ̂ = f 1 I + f 2 A A + f 3 * A z + f 3 A z ,
f 1 = 1 z 0 z cosh ( k x ) 2 d x = 1 2 [ sinh ( 2 k r z ) 2 k r z + sin ( 2 k i z ) 2 k i z ] ,
f 2 = 1 z 3 0 z sinh ( k x ) k 2 d x = 1 2 k z 2 [ sinh ( 2 k r z ) 2 k r z sin ( 2 k i z ) 2 k i z ] ,
f 3 = 1 z 2 0 z cosh ( k x ) sinh ( k * x ) k * d x = 1 2 k * z [ cosh ( 2 k r z ) 1 2 k r z + j cos ( 2 k i z ) 1 2 k i z ] ,
S ̂ pp = f 1 + f 2 ( R q 2 + L p R p 2 ) z 2 + 2 R { f 3 R q * } z ,
S ̂ qq = f 1 + f 2 ( R q 2 + L p + R p 2 ) z 2 2 R { f 3 R q * } z ,
S ̂ pq = 2 [ R { f 3 R p * } z + f 2 R { R q } L p z 2 + j ( I { f 3 } L p z + f 2 I { R q R p * } z 2 ) ] ,
S ̂ = [ 1 Φ NL Φ NL 1 + 4 3 Φ NL 2 ] ,
f ( ζ ) = f ( ζ k ) exp [ ( ζ ζ k ) L A k ] , ζ k ζ < ζ k + 1 ,
Θ Δ ( ζ k , ω ) = n = 1 k 1 ( ω 2 L Δ n + ω 3 3 L Δ n ) L n ,
R ( ω ) z = k = 1 N e j Θ Δ ( ζ k , ω ) H k ( L k , ω ) ,
H k ( x , ω ) = Φ NL k L A k L k eff 1 exp ( x L A k + j ω 2 x L Δ x + j ω 3 3 x L Δ k ) 1 j ω 2 L A k L Δ k j ω 3 3 L A k L Δ k ,
k z = Φ NL r ( ω ) 2 [ 1 sgn ( ξ in ) ( ω ω c ) 2 ] 2 ,
R z = 0 L f ( x ) e j Θ Δ ( x , ω ) L NL ( x ) d x 0 L f ( x ) L NL ( x ) d x = Φ NL ,
r ( ω ) 1 ( ω ω c ) 2 .
r ( ω ) = n = 0 ( j ) n S n n ! ω 2 n = 1 j S 1 ω 2 1 2 S 2 ω 4 + j 1 6 S 3 ω 6 ,
S n f ( z ) L NL ( z ) [ 0 z d x L Δ ( x ) ] n f ( z ) L NL ( z ) .
H k ( L k , ω ) Φ NL k [ 1 + j ω 2 L A k L Δ k ( 1 L k L k eff e L k L A k ) ] ,
S 1 = k = 1 N η k [ n = 1 k 1 L n L Δ n + L A k L Δ k ( 1 L k L k eff e L k L A k ) ] ,
η k Φ NL k Φ NL = L k eff L NL k f ( ζ k ) n = 1 N L n eff L NL n f ( ζ n ) ,
n = 1 N L n L Δ n = 0 .
S ̂ pp ( ω ) = c 0 ( ω ) + [ c 2 ( ω ) r r ( ω ) c 1 ( ω ) r i ( ω ) ] ,
S qq ( ω ) = c 0 ( ω ) + [ c 2 ( ω ) r r ( ω ) c 1 ( ω ) r i ( ω ) ] ,
S pq ( ω ) = [ c 1 ( ω ) r r ( ω ) + c 2 ( ω ) r i ( ω ) ] ,
c 0 = f 1 + f 2 [ Φ NL 2 r 2 + ( L p z ) 2 ] = 1 + Φ NL 2 r 2 [ sinh ( 2 k z ) 2 k z 1 ] ( k z ) 2 ,
c 1 = 2 Φ NL f 3 = 2 Φ NL [ cosh ( 2 k z ) 1 ] ( 2 k z ) 2 ,
c 2 = 2 Φ NL f 2 L p z = Φ NL 2 [ 1 sgn ( ξ in ) ( ω ω c ) 2 ] [ sinh ( 2 k z ) 2 k z 1 ] ( k z ) 2 ,
R z H T ( L T , ω ) = Φ NL 1 + j S ω 2 ,
S = χ α ( D T D n N s L ) ,
S ̂ pp = 1 1 cos ν 2 Φ NL S ω 2 ,
S ̂ qq = 1 + 1 cos ν 2 Φ NL S ω 2 + 2 S 2 ω A ( 1 sin ν ν ) ,
S ̂ pq = 1 cos ν 2 Φ NL S 2 ω 4 + 1 S ω 2 ( 1 sin ν ν ) ,
ω PG ( Φ NL 4 + 25 Φ NL 2 5 ) ω Δ ,
ξ in 4 Φ NL S .
R z = H T ( L T , ω ) + e j ω 2 L T L Δ T H C ( L C , ω ) ,
η c = γ C L C eff P DCF P ( γ T L T eff + γ C L C eff P DCF P ) ,
S 1 = S T { ( 1 η c ) ( 1 L T L T eff e L T L A T ) + η c L T L A T [ 1 L A C L C ( 1 L C L C eff e L C L A C ) ] } .
R z = Φ NL 1 + S ω 2 csch ( G ) sin ( S G ω 2 ) j S ω 2 [ 1 sech ( G ) cos ( S G ω 2 ) ] 1 + S 2 ω 4 ,
S 1 = S [ 1 sech ( G ) ] ,
S 2 = 2 S 2 [ 1 G csch ( G ) ] .
S ̃ = U S ̂ U .
θ ̂ ( ω ) = 1 2 arctan ( 2 S ̂ pq S ̂ pp S ̂ qq ) .
ξ post opt = 1 π 2 arctan [ c 1 ( π ) r r ( π ) + c 2 ( π ) r i ( π ) c 2 ( π ) r r ( π ) c 1 ( π ) r i ( π ) ] .

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