Abstract

The mathematical inequality which in quantum mechanics gives rise to the uncertainty principle between two non commuting operators is used to develop a spatial step-size selection algorithm for the Split-Step Fourier Method (SSFM) for solving Generalized Non-Linear Schrödinger Equations (G-NLSEs). Numerical experiments are performed to analyze the efficiency of the method in modeling optical-fiber communications systems, showing its advantages relative to other algorithms.

© 2005 Optical Society of America

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References

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  1. G. P. Agrawal, Nonlinear Fiber Optics (London, U.K. Academic, 1995).
  2. Oleg V. Sinkin, Ronald Holzlöhner, John Zweck, and Curtis Menyuk, “Optimization of the Split-Step Fourier Method in Modeling Optical-Fiber Communications Systems” IEEE J. of Lightwave Technol. 21, 61–68 (2003).
    [Crossref]
  3. G.M. Muslu and H.A. Erbay,, “Higher-order split-step Fourier schemes for generalized nonlinear Schrödinger equation” Mathematics and Computers in Simulation (In press).
  4. Malin Premaratne, “Split-Step Spline Method fpr Modeling Optical Fiber Communications Systems” IEEE Photon. Technol. Lett. 16, 1304–1306 (2004).
    [Crossref]
  5. Xueming Liu and Byoungho Lee, “A Fast Method for Nonlinear Schrödinger Equation,” IEEE Photon. Technol. Lett.15, 1549–1551 (2003). See also Xueming Liu and Byoungho Lee, “Effective Algorithms and Their Applications in Fiber Transmission Systems” Japanese Journal of Applied Physics, 43, 2492–2500, (2004).
    [Crossref]
  6. Eq. (6) is a variant of the so-called Baker-Hausdorff formula. See, for example, G.H. Weiss, A.A. Maraudin, and J. Math. Phys, 3, 771–777 (1962).
  7. E. Merzbacher, Quantum Mechanics (Wiley, New York, 1970).
  8. Although N is a non-linear operator, it involves only a multiplication operation and is considered constant in each interval. Eq. (10) follows from applying the Schwartz inequality to the functions [D-<D>]A and [N-<N>]A. Actually, this rigorous derivation determines an even smaller upper bound than that stated by eq. (10): 〈14[D,N]2〉≤ΔDΔN−〈C〉2, where 〈C〉2, the “quantum covariance,” is given by 〈C〉 〈C〉≤12〈DN+ND〉−〈D〉〈N〉.
  9. This is a straightforward consequence of Parseval’s theorem. In quantum mechanics the Fourier transform of an operator is nothing but the same operator expressed in the conjugate representation. Of course, the QM average value of an observable operator can not depend on the representation.
  10. J. Van Roey, J. van der Donk, and P.E. Lagasse, “Beam propagation method: analysis and assessment” J. Opt. Soc. Am.,  71, 808–810 (1981).
    [Crossref]
  11. N.N. Akhmediev, V.I. Korneev, and Yu.V. Kuz’menko, “Excitation of nonlinear surface waves by Gaussian light beams” Sov. Phys. JETP,  61, 62–67 (1985).
  12. B. Fornberg and T.A. Driscoll, “A fast spectral algorithm for nonlinear wave equations with linear dispersion” J. Comp. Phys.,  155, 456–467 (1999).
    [Crossref]
  13. Q. Chang, E. Jia, and W. Suny, “Difference schemes for solving the generalized nonlinear Schrödinger equation” J. Comp. Phys.,  148, 397–415 (1999).
    [Crossref]

2004 (1)

Malin Premaratne, “Split-Step Spline Method fpr Modeling Optical Fiber Communications Systems” IEEE Photon. Technol. Lett. 16, 1304–1306 (2004).
[Crossref]

2003 (1)

Oleg V. Sinkin, Ronald Holzlöhner, John Zweck, and Curtis Menyuk, “Optimization of the Split-Step Fourier Method in Modeling Optical-Fiber Communications Systems” IEEE J. of Lightwave Technol. 21, 61–68 (2003).
[Crossref]

1999 (2)

B. Fornberg and T.A. Driscoll, “A fast spectral algorithm for nonlinear wave equations with linear dispersion” J. Comp. Phys.,  155, 456–467 (1999).
[Crossref]

Q. Chang, E. Jia, and W. Suny, “Difference schemes for solving the generalized nonlinear Schrödinger equation” J. Comp. Phys.,  148, 397–415 (1999).
[Crossref]

1985 (1)

N.N. Akhmediev, V.I. Korneev, and Yu.V. Kuz’menko, “Excitation of nonlinear surface waves by Gaussian light beams” Sov. Phys. JETP,  61, 62–67 (1985).

1981 (1)

J. Van Roey, J. van der Donk, and P.E. Lagasse, “Beam propagation method: analysis and assessment” J. Opt. Soc. Am.,  71, 808–810 (1981).
[Crossref]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (London, U.K. Academic, 1995).

Akhmediev, N.N.

N.N. Akhmediev, V.I. Korneev, and Yu.V. Kuz’menko, “Excitation of nonlinear surface waves by Gaussian light beams” Sov. Phys. JETP,  61, 62–67 (1985).

Chang, Q.

Q. Chang, E. Jia, and W. Suny, “Difference schemes for solving the generalized nonlinear Schrödinger equation” J. Comp. Phys.,  148, 397–415 (1999).
[Crossref]

Driscoll, T.A.

B. Fornberg and T.A. Driscoll, “A fast spectral algorithm for nonlinear wave equations with linear dispersion” J. Comp. Phys.,  155, 456–467 (1999).
[Crossref]

Erbay, H.A.

G.M. Muslu and H.A. Erbay,, “Higher-order split-step Fourier schemes for generalized nonlinear Schrödinger equation” Mathematics and Computers in Simulation (In press).

Fornberg, B.

B. Fornberg and T.A. Driscoll, “A fast spectral algorithm for nonlinear wave equations with linear dispersion” J. Comp. Phys.,  155, 456–467 (1999).
[Crossref]

Holzlöhner, Ronald

Oleg V. Sinkin, Ronald Holzlöhner, John Zweck, and Curtis Menyuk, “Optimization of the Split-Step Fourier Method in Modeling Optical-Fiber Communications Systems” IEEE J. of Lightwave Technol. 21, 61–68 (2003).
[Crossref]

Jia, E.

Q. Chang, E. Jia, and W. Suny, “Difference schemes for solving the generalized nonlinear Schrödinger equation” J. Comp. Phys.,  148, 397–415 (1999).
[Crossref]

Korneev, V.I.

N.N. Akhmediev, V.I. Korneev, and Yu.V. Kuz’menko, “Excitation of nonlinear surface waves by Gaussian light beams” Sov. Phys. JETP,  61, 62–67 (1985).

Kuz’menko, Yu.V.

N.N. Akhmediev, V.I. Korneev, and Yu.V. Kuz’menko, “Excitation of nonlinear surface waves by Gaussian light beams” Sov. Phys. JETP,  61, 62–67 (1985).

Lagasse, P.E.

J. Van Roey, J. van der Donk, and P.E. Lagasse, “Beam propagation method: analysis and assessment” J. Opt. Soc. Am.,  71, 808–810 (1981).
[Crossref]

Lee, Byoungho

Xueming Liu and Byoungho Lee, “A Fast Method for Nonlinear Schrödinger Equation,” IEEE Photon. Technol. Lett.15, 1549–1551 (2003). See also Xueming Liu and Byoungho Lee, “Effective Algorithms and Their Applications in Fiber Transmission Systems” Japanese Journal of Applied Physics, 43, 2492–2500, (2004).
[Crossref]

Xueming Liu and Byoungho Lee, “A Fast Method for Nonlinear Schrödinger Equation,” IEEE Photon. Technol. Lett.15, 1549–1551 (2003). See also Xueming Liu and Byoungho Lee, “Effective Algorithms and Their Applications in Fiber Transmission Systems” Japanese Journal of Applied Physics, 43, 2492–2500, (2004).
[Crossref]

Liu, Xueming

Xueming Liu and Byoungho Lee, “A Fast Method for Nonlinear Schrödinger Equation,” IEEE Photon. Technol. Lett.15, 1549–1551 (2003). See also Xueming Liu and Byoungho Lee, “Effective Algorithms and Their Applications in Fiber Transmission Systems” Japanese Journal of Applied Physics, 43, 2492–2500, (2004).
[Crossref]

Xueming Liu and Byoungho Lee, “A Fast Method for Nonlinear Schrödinger Equation,” IEEE Photon. Technol. Lett.15, 1549–1551 (2003). See also Xueming Liu and Byoungho Lee, “Effective Algorithms and Their Applications in Fiber Transmission Systems” Japanese Journal of Applied Physics, 43, 2492–2500, (2004).
[Crossref]

Maraudin, A.A.

Eq. (6) is a variant of the so-called Baker-Hausdorff formula. See, for example, G.H. Weiss, A.A. Maraudin, and J. Math. Phys, 3, 771–777 (1962).

Math, J.

Eq. (6) is a variant of the so-called Baker-Hausdorff formula. See, for example, G.H. Weiss, A.A. Maraudin, and J. Math. Phys, 3, 771–777 (1962).

Menyuk, Curtis

Oleg V. Sinkin, Ronald Holzlöhner, John Zweck, and Curtis Menyuk, “Optimization of the Split-Step Fourier Method in Modeling Optical-Fiber Communications Systems” IEEE J. of Lightwave Technol. 21, 61–68 (2003).
[Crossref]

Merzbacher, E.

E. Merzbacher, Quantum Mechanics (Wiley, New York, 1970).

Muslu, G.M.

G.M. Muslu and H.A. Erbay,, “Higher-order split-step Fourier schemes for generalized nonlinear Schrödinger equation” Mathematics and Computers in Simulation (In press).

Premaratne, Malin

Malin Premaratne, “Split-Step Spline Method fpr Modeling Optical Fiber Communications Systems” IEEE Photon. Technol. Lett. 16, 1304–1306 (2004).
[Crossref]

Sinkin, Oleg V.

Oleg V. Sinkin, Ronald Holzlöhner, John Zweck, and Curtis Menyuk, “Optimization of the Split-Step Fourier Method in Modeling Optical-Fiber Communications Systems” IEEE J. of Lightwave Technol. 21, 61–68 (2003).
[Crossref]

Suny, W.

Q. Chang, E. Jia, and W. Suny, “Difference schemes for solving the generalized nonlinear Schrödinger equation” J. Comp. Phys.,  148, 397–415 (1999).
[Crossref]

van der Donk, J.

J. Van Roey, J. van der Donk, and P.E. Lagasse, “Beam propagation method: analysis and assessment” J. Opt. Soc. Am.,  71, 808–810 (1981).
[Crossref]

Van Roey, J.

J. Van Roey, J. van der Donk, and P.E. Lagasse, “Beam propagation method: analysis and assessment” J. Opt. Soc. Am.,  71, 808–810 (1981).
[Crossref]

Weiss, G.H.

Eq. (6) is a variant of the so-called Baker-Hausdorff formula. See, for example, G.H. Weiss, A.A. Maraudin, and J. Math. Phys, 3, 771–777 (1962).

Zweck, John

Oleg V. Sinkin, Ronald Holzlöhner, John Zweck, and Curtis Menyuk, “Optimization of the Split-Step Fourier Method in Modeling Optical-Fiber Communications Systems” IEEE J. of Lightwave Technol. 21, 61–68 (2003).
[Crossref]

IEEE J. of Lightwave Technol. (1)

Oleg V. Sinkin, Ronald Holzlöhner, John Zweck, and Curtis Menyuk, “Optimization of the Split-Step Fourier Method in Modeling Optical-Fiber Communications Systems” IEEE J. of Lightwave Technol. 21, 61–68 (2003).
[Crossref]

IEEE Photon. Technol. Lett. (1)

Malin Premaratne, “Split-Step Spline Method fpr Modeling Optical Fiber Communications Systems” IEEE Photon. Technol. Lett. 16, 1304–1306 (2004).
[Crossref]

J. Comp. Phys. (2)

B. Fornberg and T.A. Driscoll, “A fast spectral algorithm for nonlinear wave equations with linear dispersion” J. Comp. Phys.,  155, 456–467 (1999).
[Crossref]

Q. Chang, E. Jia, and W. Suny, “Difference schemes for solving the generalized nonlinear Schrödinger equation” J. Comp. Phys.,  148, 397–415 (1999).
[Crossref]

J. Opt. Soc. Am. (1)

J. Van Roey, J. van der Donk, and P.E. Lagasse, “Beam propagation method: analysis and assessment” J. Opt. Soc. Am.,  71, 808–810 (1981).
[Crossref]

Sov. Phys. JETP (1)

N.N. Akhmediev, V.I. Korneev, and Yu.V. Kuz’menko, “Excitation of nonlinear surface waves by Gaussian light beams” Sov. Phys. JETP,  61, 62–67 (1985).

Other (7)

G.M. Muslu and H.A. Erbay,, “Higher-order split-step Fourier schemes for generalized nonlinear Schrödinger equation” Mathematics and Computers in Simulation (In press).

Xueming Liu and Byoungho Lee, “A Fast Method for Nonlinear Schrödinger Equation,” IEEE Photon. Technol. Lett.15, 1549–1551 (2003). See also Xueming Liu and Byoungho Lee, “Effective Algorithms and Their Applications in Fiber Transmission Systems” Japanese Journal of Applied Physics, 43, 2492–2500, (2004).
[Crossref]

Eq. (6) is a variant of the so-called Baker-Hausdorff formula. See, for example, G.H. Weiss, A.A. Maraudin, and J. Math. Phys, 3, 771–777 (1962).

E. Merzbacher, Quantum Mechanics (Wiley, New York, 1970).

Although N is a non-linear operator, it involves only a multiplication operation and is considered constant in each interval. Eq. (10) follows from applying the Schwartz inequality to the functions [D-<D>]A and [N-<N>]A. Actually, this rigorous derivation determines an even smaller upper bound than that stated by eq. (10): 〈14[D,N]2〉≤ΔDΔN−〈C〉2, where 〈C〉2, the “quantum covariance,” is given by 〈C〉 〈C〉≤12〈DN+ND〉−〈D〉〈N〉.

This is a straightforward consequence of Parseval’s theorem. In quantum mechanics the Fourier transform of an operator is nothing but the same operator expressed in the conjugate representation. Of course, the QM average value of an observable operator can not depend on the representation.

G. P. Agrawal, Nonlinear Fiber Optics (London, U.K. Academic, 1995).

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Figures (4)

Fig. 1.
Fig. 1.

Number of FFTs versus global relative error for a second order soliton propagating through ~81.2 km of fiber. Results form the UPM, LEM, and NPRM are shown. The lines joining the points are just aids to the eyes in identifying the general behavior of the methods.

Fig. 2.
Fig. 2.

Number of FFTs versus global relative error for a two first-order soliton collision propagating through 400 km of fiber. Results from the UPM, LEM, and NPRM are shown.

Fig. 3.
Fig. 3.

Number of FFTs versus global relative error for a WDM eight-channel system simulated using the UPM, LEM and WOM for propagating distances of (a) 10 km and (b) 50 km.

Fig. 4.
Fig. 4.

Global relative error as a function of the method parameter (ε or δ), for the UPM and the LEM, and linear fit of each curve. We show the results from the systems simulated in Section 3. The slopes given by the linear fits of the curves are shown in the labels.

Tables (1)

Tables Icon

Table 1. Main parameters used in the simulation of an eight-channel WDM system

Equations (21)

Equations on this page are rendered with MathJax. Learn more.

i A z = 1 2 β 2 2 A t 2 i 6 β 3 3 A t 3 + γ A 2 A
i A ( z , t ) z = ( D + N ) A ( z , t ) ,
D = 1 2 β 2 2 t 2 + i 6 β 3 3 t 3
N = γ A ( z , t ) 2
A ( z + h , t ) e i h D e i h N = F 1 e i h D ˜ F e i h N A ( z , t ) ,
A ( z + h , t ) = e i h ( D + N ) A ( z , t ) .
e D e N = e D + N + 1 2 [ D , N ] + 1 3 ! { ( N + 2 D ) [ D , N ] + [ D , N ] ( D + 2 N ) } + ,
δ A n A a A a ,
δ = ( 1 2 ) h 2 ( [ D , N ] A ( t ) 2 d t A ( t ) 2 d t ) 1 2 .
ε ( 1 2 ) h 2 [ D , N ] ( 1 2 ) h 2 A * ( z , t ) [ D , N ] A ( z , t ) d t A ( t ) 2 d t ,
[ D , N ] 2 Δ D Δ N ,
Δ B = ( B B ) 2 with B = A * ( z , t ) B A ( z , t ) d t A ( t ) 2 d t
h = ε Δ D Δ N
B = B ̃ = A ˜ * B A ˜ d ω A ˜ 2 d ω ,
Δ D 2 = Δ D ˜ 2 = [ 1 2 β 2 ( ω 2 ω 2 ) + 1 3 ! β 3 ( ω 3 ω 3 ) ] 2 .
i A z = ( L + N ) A ,
L ˜ ( z , ω ) = D ˜ ( z , ω ) 1 2 i α ( z , ω ) ,
ε = ( 1 2 ) h 2 [ L , N ] = ( 1 2 ) h 2 [ D , N ] 2 + [ ( 1 2 ) α , N ] 2 .
ε ( 1 2 ) h 2 Δ N Δ D 2 + ( 1 4 ) Δ α 2 ,
h = ε Δ D Δ N 1 + ( Δ α 2 Δ D ) 2
A ( z + h , t ) e i h D 2 e i h N e i h D 2 A ( z , t ) .

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