Abstract

Using the variational method, we derive simple, closed-form algebraic expressions that approximate the optimal input rms pulse width and the corresponding minimum output rms width for Gaussian pulses subject to both dispersive and nonlinear effects in single-mode fibers. We present results in both numerical and analytical forms and confirm them by the split-step Fourier numerical method. Our results cover both normal and anomalous dispersion in fibers with gain and loss. For the case of normal dispersion we show that both the optimal input and output widths are asymptotically linearly dependent on distance and dependent on the square roots of the dispersion coefficient and the transmitted power.

© 1999 Optical Society of America

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References

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  1. D. Marcuse and C. Lin, “Low dispersion single-mode fiber transmission—the question of practical versus theoretical maximum bandwidth transmission,” IEEE J. Quantum Electron. QE-17, 860–877 (1981).
  2. G. P. Agrawal, Fiber-Optic Communications (Wiley, New York, 1992).
  3. A. Nata and S. Saito, “In-line amplifier transmission distance determined by self-phase modulation and group-velocity dispersion,” J. Lightwave Technol. 12, 280–287 (1994).
    [CrossRef]
  4. M. J. Potasek, G. P. Agrawal, and S. C. Pinault, “Analytic and numerical study of pulse broadening in nonlinear dispersive fibers,” J. Opt. Soc. Am. B 3, 205–211 (1986).
    [CrossRef]
  5. G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, New York, 1995).
  6. X. Ma, “Analysis and simulation of long haul in-line fiber amplifier transmission systems,” Ph.D. dissertation (Virginia Polytechnic Institute and State University, Blacksburg, Virginia, 1994).
  7. D. Marcuse, “RMS width of pulses in nonlinear dispersive fibers,” J. Lightwave Technol. 10, 17–21 (1992).
    [CrossRef]
  8. D. Anderson, “Variational approach to nonlinear pulse propagation in fibers,” Phys. Rev. A 27, 3135–3145 (1983).
    [CrossRef]
  9. V. S. Grigorian, T. Yu, E. A. Goluvchenko, C. R. Menyuk, and A. N. Pilipetski, “Dispersion-managed soliton dynamics,” Opt. Lett. 22, 1609–1611 (1997).
    [CrossRef]
  10. I. Gabitov and S. K. Turitsyn, “Breathing solitons in optical fiber links,” JETP Lett. 63, 862–866 (1996).
    [CrossRef]
  11. B. A. Malomed, “Ideal amplification of an ultrashort soliton in a dispersion-decreasing fiber,” Opt. Lett. 19, 341–343 (1994).
    [CrossRef] [PubMed]
  12. M. Florjanczyk, “RMS width of pulses in nonlinear dispersive fibers: pulses of arbitrary initial form with chirp,” J. Lightwave Technol. 13, 1801–1806 (1995).
    [CrossRef]
  13. D. Anderson, M. Lisak, and T. Reichel, “Asymptotic propagation properties of pulses in a soliton-based optical-fiber communication system,” J. Opt. Soc. Am. B 5, 207–210 (1988).
    [CrossRef]
  14. D. Anderson and M. Lisak, “Propagation characteristics of frequency-chirped super-Gaussian optical pulses,” Opt. Lett. 11, 569–571 (1986).
    [CrossRef] [PubMed]
  15. D. Anderson, M. Lisak, and P. Anderson, “Nonlinearly enhanced chirp pulse compression in single-mode fibers,” Opt. Lett. 10, 134–136 (1985).
    [CrossRef] [PubMed]
  16. I. Vinogradov, ed., Encyclopedia of Mathematics (Kluwer Academic, Dordrecht, The Netherlands, 1988), Vol. 2, p. 19.
  17. J.-H. Lee, “Analysis and characterization of fiber nonlinearities with deterministic and stochastic signal sources,” Ph.D. dissertation proposal (Electrical Engineering Department, Virginia Polytechnic Institute and State University, Blacksburg, Virginia, 1998).
  18. P. Belanger and C. Pare, “Second-order moment analysis of dispersion-managed solitons,” J. Lightwave Technol. 17, 445–451 (1999).
    [CrossRef]
  19. S. Turitsyn, T. Schafer, and V. Mezentsev, “Generalized root-mean-square momentum method to describe chirped return-to-zero signal propagation in dispersion-managed fiber links,” IEEE Photonics Technol. Lett. 11, 203–205 (1999).
    [CrossRef]
  20. T. Yang, W. Kath, and S. Evangelides, “Optimal prechirping for dispersion-managed transmission of return-to-zero pulses,” in Digest of Optical Fiber Communications Conference and International Conference and Integrated Optics and Optical Fiber Communication (Optical Society of America, Washington, D.C., 1999), Feb. 25, 1999, pp. 249–254.
  21. T. Lakoba, J. Yang, D. Kaup, and B. Malomed, “Conditions for stationary pulse propagation in the strong dispersion management regime,” Optics Commun. 149, 366–375 (1998).
    [CrossRef]
  22. V. Afanasjev, B. Malomed, P. Chu, and M. Islam, “Generalized variational approximations for the optical soliton,” Optics Commun. 147, 317–322 (1998).
    [CrossRef]
  23. J. Kutz, P. Holmes, S. Evangelides, and J. Gordon, “Hamiltonian dynamics of dispersion-managed breathers,” J. Opt. Soc. Am. B 15, 87–96 (1998).
    [CrossRef]
  24. I. N. Herstein, Topics in Algebra (Blaisdell, New York, 1964).
  25. S. Wikes, Mathematical Statistics (Wiley, New York, 1962).

1999

S. Turitsyn, T. Schafer, and V. Mezentsev, “Generalized root-mean-square momentum method to describe chirped return-to-zero signal propagation in dispersion-managed fiber links,” IEEE Photonics Technol. Lett. 11, 203–205 (1999).
[CrossRef]

P. Belanger and C. Pare, “Second-order moment analysis of dispersion-managed solitons,” J. Lightwave Technol. 17, 445–451 (1999).
[CrossRef]

1998

J. Kutz, P. Holmes, S. Evangelides, and J. Gordon, “Hamiltonian dynamics of dispersion-managed breathers,” J. Opt. Soc. Am. B 15, 87–96 (1998).
[CrossRef]

T. Lakoba, J. Yang, D. Kaup, and B. Malomed, “Conditions for stationary pulse propagation in the strong dispersion management regime,” Optics Commun. 149, 366–375 (1998).
[CrossRef]

V. Afanasjev, B. Malomed, P. Chu, and M. Islam, “Generalized variational approximations for the optical soliton,” Optics Commun. 147, 317–322 (1998).
[CrossRef]

1997

1996

I. Gabitov and S. K. Turitsyn, “Breathing solitons in optical fiber links,” JETP Lett. 63, 862–866 (1996).
[CrossRef]

1995

M. Florjanczyk, “RMS width of pulses in nonlinear dispersive fibers: pulses of arbitrary initial form with chirp,” J. Lightwave Technol. 13, 1801–1806 (1995).
[CrossRef]

1994

A. Nata and S. Saito, “In-line amplifier transmission distance determined by self-phase modulation and group-velocity dispersion,” J. Lightwave Technol. 12, 280–287 (1994).
[CrossRef]

B. A. Malomed, “Ideal amplification of an ultrashort soliton in a dispersion-decreasing fiber,” Opt. Lett. 19, 341–343 (1994).
[CrossRef] [PubMed]

1992

D. Marcuse, “RMS width of pulses in nonlinear dispersive fibers,” J. Lightwave Technol. 10, 17–21 (1992).
[CrossRef]

1988

1986

1985

1983

D. Anderson, “Variational approach to nonlinear pulse propagation in fibers,” Phys. Rev. A 27, 3135–3145 (1983).
[CrossRef]

1981

D. Marcuse and C. Lin, “Low dispersion single-mode fiber transmission—the question of practical versus theoretical maximum bandwidth transmission,” IEEE J. Quantum Electron. QE-17, 860–877 (1981).

Afanasjev, V.

V. Afanasjev, B. Malomed, P. Chu, and M. Islam, “Generalized variational approximations for the optical soliton,” Optics Commun. 147, 317–322 (1998).
[CrossRef]

Agrawal, G. P.

Anderson, D.

Anderson, P.

Belanger, P.

Chu, P.

V. Afanasjev, B. Malomed, P. Chu, and M. Islam, “Generalized variational approximations for the optical soliton,” Optics Commun. 147, 317–322 (1998).
[CrossRef]

Evangelides, S.

Florjanczyk, M.

M. Florjanczyk, “RMS width of pulses in nonlinear dispersive fibers: pulses of arbitrary initial form with chirp,” J. Lightwave Technol. 13, 1801–1806 (1995).
[CrossRef]

Gabitov, I.

I. Gabitov and S. K. Turitsyn, “Breathing solitons in optical fiber links,” JETP Lett. 63, 862–866 (1996).
[CrossRef]

Goluvchenko, E. A.

Gordon, J.

Grigorian, V. S.

Holmes, P.

Islam, M.

V. Afanasjev, B. Malomed, P. Chu, and M. Islam, “Generalized variational approximations for the optical soliton,” Optics Commun. 147, 317–322 (1998).
[CrossRef]

Kaup, D.

T. Lakoba, J. Yang, D. Kaup, and B. Malomed, “Conditions for stationary pulse propagation in the strong dispersion management regime,” Optics Commun. 149, 366–375 (1998).
[CrossRef]

Kutz, J.

Lakoba, T.

T. Lakoba, J. Yang, D. Kaup, and B. Malomed, “Conditions for stationary pulse propagation in the strong dispersion management regime,” Optics Commun. 149, 366–375 (1998).
[CrossRef]

Lin, C.

D. Marcuse and C. Lin, “Low dispersion single-mode fiber transmission—the question of practical versus theoretical maximum bandwidth transmission,” IEEE J. Quantum Electron. QE-17, 860–877 (1981).

Lisak, M.

Malomed, B.

V. Afanasjev, B. Malomed, P. Chu, and M. Islam, “Generalized variational approximations for the optical soliton,” Optics Commun. 147, 317–322 (1998).
[CrossRef]

T. Lakoba, J. Yang, D. Kaup, and B. Malomed, “Conditions for stationary pulse propagation in the strong dispersion management regime,” Optics Commun. 149, 366–375 (1998).
[CrossRef]

Malomed, B. A.

Marcuse, D.

D. Marcuse, “RMS width of pulses in nonlinear dispersive fibers,” J. Lightwave Technol. 10, 17–21 (1992).
[CrossRef]

D. Marcuse and C. Lin, “Low dispersion single-mode fiber transmission—the question of practical versus theoretical maximum bandwidth transmission,” IEEE J. Quantum Electron. QE-17, 860–877 (1981).

Menyuk, C. R.

Mezentsev, V.

S. Turitsyn, T. Schafer, and V. Mezentsev, “Generalized root-mean-square momentum method to describe chirped return-to-zero signal propagation in dispersion-managed fiber links,” IEEE Photonics Technol. Lett. 11, 203–205 (1999).
[CrossRef]

Nata, A.

A. Nata and S. Saito, “In-line amplifier transmission distance determined by self-phase modulation and group-velocity dispersion,” J. Lightwave Technol. 12, 280–287 (1994).
[CrossRef]

Pare, C.

Pilipetski, A. N.

Pinault, S. C.

Potasek, M. J.

Reichel, T.

Saito, S.

A. Nata and S. Saito, “In-line amplifier transmission distance determined by self-phase modulation and group-velocity dispersion,” J. Lightwave Technol. 12, 280–287 (1994).
[CrossRef]

Schafer, T.

S. Turitsyn, T. Schafer, and V. Mezentsev, “Generalized root-mean-square momentum method to describe chirped return-to-zero signal propagation in dispersion-managed fiber links,” IEEE Photonics Technol. Lett. 11, 203–205 (1999).
[CrossRef]

Turitsyn, S.

S. Turitsyn, T. Schafer, and V. Mezentsev, “Generalized root-mean-square momentum method to describe chirped return-to-zero signal propagation in dispersion-managed fiber links,” IEEE Photonics Technol. Lett. 11, 203–205 (1999).
[CrossRef]

Turitsyn, S. K.

I. Gabitov and S. K. Turitsyn, “Breathing solitons in optical fiber links,” JETP Lett. 63, 862–866 (1996).
[CrossRef]

Yang, J.

T. Lakoba, J. Yang, D. Kaup, and B. Malomed, “Conditions for stationary pulse propagation in the strong dispersion management regime,” Optics Commun. 149, 366–375 (1998).
[CrossRef]

Yu, T.

IEEE J. Quantum Electron.

D. Marcuse and C. Lin, “Low dispersion single-mode fiber transmission—the question of practical versus theoretical maximum bandwidth transmission,” IEEE J. Quantum Electron. QE-17, 860–877 (1981).

IEEE Photonics Technol. Lett.

S. Turitsyn, T. Schafer, and V. Mezentsev, “Generalized root-mean-square momentum method to describe chirped return-to-zero signal propagation in dispersion-managed fiber links,” IEEE Photonics Technol. Lett. 11, 203–205 (1999).
[CrossRef]

J. Lightwave Technol.

P. Belanger and C. Pare, “Second-order moment analysis of dispersion-managed solitons,” J. Lightwave Technol. 17, 445–451 (1999).
[CrossRef]

A. Nata and S. Saito, “In-line amplifier transmission distance determined by self-phase modulation and group-velocity dispersion,” J. Lightwave Technol. 12, 280–287 (1994).
[CrossRef]

D. Marcuse, “RMS width of pulses in nonlinear dispersive fibers,” J. Lightwave Technol. 10, 17–21 (1992).
[CrossRef]

M. Florjanczyk, “RMS width of pulses in nonlinear dispersive fibers: pulses of arbitrary initial form with chirp,” J. Lightwave Technol. 13, 1801–1806 (1995).
[CrossRef]

J. Opt. Soc. Am. B

JETP Lett.

I. Gabitov and S. K. Turitsyn, “Breathing solitons in optical fiber links,” JETP Lett. 63, 862–866 (1996).
[CrossRef]

Opt. Lett.

Optics Commun.

T. Lakoba, J. Yang, D. Kaup, and B. Malomed, “Conditions for stationary pulse propagation in the strong dispersion management regime,” Optics Commun. 149, 366–375 (1998).
[CrossRef]

V. Afanasjev, B. Malomed, P. Chu, and M. Islam, “Generalized variational approximations for the optical soliton,” Optics Commun. 147, 317–322 (1998).
[CrossRef]

Phys. Rev. A

D. Anderson, “Variational approach to nonlinear pulse propagation in fibers,” Phys. Rev. A 27, 3135–3145 (1983).
[CrossRef]

Other

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

X. Ma, “Analysis and simulation of long haul in-line fiber amplifier transmission systems,” Ph.D. dissertation (Virginia Polytechnic Institute and State University, Blacksburg, Virginia, 1994).

G. P. Agrawal, Fiber-Optic Communications (Wiley, New York, 1992).

I. Vinogradov, ed., Encyclopedia of Mathematics (Kluwer Academic, Dordrecht, The Netherlands, 1988), Vol. 2, p. 19.

J.-H. Lee, “Analysis and characterization of fiber nonlinearities with deterministic and stochastic signal sources,” Ph.D. dissertation proposal (Electrical Engineering Department, Virginia Polytechnic Institute and State University, Blacksburg, Virginia, 1998).

I. N. Herstein, Topics in Algebra (Blaisdell, New York, 1964).

S. Wikes, Mathematical Statistics (Wiley, New York, 1962).

T. Yang, W. Kath, and S. Evangelides, “Optimal prechirping for dispersion-managed transmission of return-to-zero pulses,” in Digest of Optical Fiber Communications Conference and International Conference and Integrated Optics and Optical Fiber Communication (Optical Society of America, Washington, D.C., 1999), Feb. 25, 1999, pp. 249–254.

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Figures (4)

Fig. 1
Fig. 1

Bottom curves: solid curve, optimal normalized input width s0,opt that is computed by variational method; dotted curve, by split-step Fourier method. Top curves: solid curve, minimum normalized output width that is computed by variational method; dotted curve, by split-step Fourier method.

Fig. 2
Fig. 2

Normalized output widths s(ξ) as functions of ξ in the vicinity of the minimum for ξ=0.2, 10, and 20. Solid curves, computed by variational method; dotted curves, by split-step Fourier method.

Fig. 3
Fig. 3

Comparison of normalized RMS output widths as a function of normalized distance z/LD that are computed by variational method (solid curve), split-step Fourier method (dotted curve), one-step method [Eq. (14)] (dashed–dotted curve), and two-step method [Eq. (15)] (dashed curve).

Fig. 4
Fig. 4

Top curves: optimal normalized input widths s0,opt that are computed by variational method (solid curve) and split-step Fourier method (dotted curve) for the anomalous dispersion case. Bottom curves: minimum normalized output widths that are computed by variational method (solid curve) and split-step Fourier method (dotted curve) in the anomalous dispersion case.

Tables (1)

Tables Icon

Table 1 Comparison of Output rms Widths at 9600 km Computed by Eqs. (23) and (24)

Equations (42)

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j Uz=(β2/2) 2UT2-γP0|U(z, T)|2U(z, T),
L=(j/2)U U*z-U* Uz-(β2/2)UT2-(γP0/2)|U|4
U(z, T)=A(z)expT2-12a2(z)+jb(z),
az=-2β2a(z)b(z),a(z)|A(z)|2=a0,
bz=2β2b2(z)-β22a4(z)-γP0a022a3(z),
2az2=β22a3(z)+a0β2γP02a2(z),
az2=β221a02-1a2(z)+2γP0β21-a0a(z),
2(a2)z2=2β22a02+22β2γP0-2a0β2γP0a(z),
s(ξ)=σ(ξ)(β2LNL/2)1/2,LNL=1γP0,ξ=zLNL,
s0=a0/(β2LNL)1/2=(LD/LNL)1/2=N,
2(s)ξ2=1s3(ξ)+s02s2(ξ),
2(s2)ξ2=2s02+22-2 s0s(ξ),
σ2σ02=1+2 z2LDLNL+1+433z2LNL2 z2LD2.
s2(ξ)=c0+c1ξ+c2ξ2+c3ξ3+c4ξ4+ .
s2(ξ)s02+1s02+12ξ2+224s04+124s02ξ4.
y3-ξ21+ξ224y-ξ4212=0,y=s02.
α3-1+24ξ2α-83ξ2=0.
s2(ξ)(2ξ2/24)+(ξ2/2)=ξ2(2+12)/24
α(ξ)=231+24ξ21/2 cos13arccos36ξ(ξ2+24)3/2,
(2/3)cos{(1/3)arccos[36ξ/(ξ2+24)3/2]}
σ0,opt0.32zβ2/LNL1+24LNL2z21/2,
σ0,opt2=β2z2224LNLα(ξ),
2(s2)(s0)2=2+6ξ2s041+ξ224+5ξ426s06,
2(s2)(s0)2=2s 2s(s0)2+2ss02,
s2(ξ)=s02+1s02+2ξ2+433ξ4s02.
s2(ξ)=s02+12[1+(ξ2/4s04)]1/2ξ2+1+ξ233[1+(ξ2/4s04)]2 ξ2s02.
2Wx2=C1W3+C2W2,
-(1/30)[(32/8)s0-8+(17/24)s0-6+(2/6)s0-4]ξ6.
j Ux=sgn(β2)22Uτ2-N2|U(x, τ)|2U(x, τ),
x=z/LD,τ=T/T0,
2(s2)ξ2=2s02±222 s0s(ξ),
s2(ξ)s02+1s02±12ξ2+±224s04+124s02ξ4,
α3-1+24ξ2α83ξ2=0.
f±(α)=α3-1+24ξ2α83ξ2,
α(ξ)=-231+24ξ21/2×cos13arccos36ξ(ξ2+24)3/2-π3,
j Uz=sgn(β2)2LD2Uτ2-exp(-z)LNL|U(z, τ)|2U(z, τ),
Pavg=(1/za)0zaP0 exp(-w)dw,
j Uz=sgn(β2)2LD2Uτ2-1L¯NL|U(z, τ)|2U(z, τ).
Rbz=z/(4σmin)0.25z/(2.34σ0,opt)=0.11z/(0.32zβ2/LNL)=0.33LNL/β2.
s2(ξ)=ξ224α+1α+23-Cξ2α224.
s0,optξ=α(24)1/4=1(72)1/4=0.34,
smin2=ξ22433=ξ29/8,smin=1.03ξ.

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