Abstract

A split-step reconstruction technique is proposed to solve a nonlinear wave equation. A nonlinear wave equation can be segmented into a set of linear equations that can be solved in the time domain by use of the split-step reconstruction technique. With this technique, one can avoid the propagation errors that occur as a result of nonlinear wave equation splitting. We propose an adaptive mesh control to increase the speed with which nonlinear wave equations can be calculated.

© 2000 Optical Society of America

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References

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  1. F. Favre, D. Le Guen, M. L. Moulinard, M. Henry, P. L. Francois, F. Devaux, T. Georges, “Single-wavelength 40 Gbit/s, 8 × 100 km-span soliton transmission without any in-line control,” Electron. Lett. 33, 407–408 (1997).
    [CrossRef]
  2. V. E. Zakharov, A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).
  3. H. A. Haus, M. N. Islam, “Theory of the soliton laser,” IEEE J. Quantum Electron. QE-21, 1172–1188 (1985).
    [CrossRef]
  4. R. H. Hardin, F. D. Tappert, “Applications of the split-step Fourier method to the numerical solution of nonlinear and variable coefficient wave equations,” SIAM (Soc. Ind. Appl. Math.) Rev. 15, 423 (1973).
  5. H. Ghafouri-Shiraz, P. Shum, “Novel method for analysis of soliton propagation in optical fibers,” IEEE J. Quantum Electron. 31, 190–200 (1995).
    [CrossRef]
  6. P. Shum, S. F. Yu, “Improvement of Fourier series analysis technique by time-domain window function,” IEEE Photonics Technol. Lett. 18, 1364–1366 (1996).
    [CrossRef]
  7. P. Shum, S. F. Yu, “Numerical analysis of nonlinear soliton propagation phenomena using the fuzzy mesh analysis technique,” IEEE J. Quantum Electron. 34, 2029–2035 (1998).
    [CrossRef]
  8. G. P. Agrawal, Fiber-Optic Communication Systems, 2nd ed., Wiley Series in Microwave and Optical Engineering (Wiley-Interscience, New York, 1997).

1998

P. Shum, S. F. Yu, “Numerical analysis of nonlinear soliton propagation phenomena using the fuzzy mesh analysis technique,” IEEE J. Quantum Electron. 34, 2029–2035 (1998).
[CrossRef]

1997

F. Favre, D. Le Guen, M. L. Moulinard, M. Henry, P. L. Francois, F. Devaux, T. Georges, “Single-wavelength 40 Gbit/s, 8 × 100 km-span soliton transmission without any in-line control,” Electron. Lett. 33, 407–408 (1997).
[CrossRef]

1996

P. Shum, S. F. Yu, “Improvement of Fourier series analysis technique by time-domain window function,” IEEE Photonics Technol. Lett. 18, 1364–1366 (1996).
[CrossRef]

1995

H. Ghafouri-Shiraz, P. Shum, “Novel method for analysis of soliton propagation in optical fibers,” IEEE J. Quantum Electron. 31, 190–200 (1995).
[CrossRef]

1985

H. A. Haus, M. N. Islam, “Theory of the soliton laser,” IEEE J. Quantum Electron. QE-21, 1172–1188 (1985).
[CrossRef]

1973

R. H. Hardin, F. D. Tappert, “Applications of the split-step Fourier method to the numerical solution of nonlinear and variable coefficient wave equations,” SIAM (Soc. Ind. Appl. Math.) Rev. 15, 423 (1973).

1972

V. E. Zakharov, A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

Agrawal, G. P.

G. P. Agrawal, Fiber-Optic Communication Systems, 2nd ed., Wiley Series in Microwave and Optical Engineering (Wiley-Interscience, New York, 1997).

Devaux, F.

F. Favre, D. Le Guen, M. L. Moulinard, M. Henry, P. L. Francois, F. Devaux, T. Georges, “Single-wavelength 40 Gbit/s, 8 × 100 km-span soliton transmission without any in-line control,” Electron. Lett. 33, 407–408 (1997).
[CrossRef]

Favre, F.

F. Favre, D. Le Guen, M. L. Moulinard, M. Henry, P. L. Francois, F. Devaux, T. Georges, “Single-wavelength 40 Gbit/s, 8 × 100 km-span soliton transmission without any in-line control,” Electron. Lett. 33, 407–408 (1997).
[CrossRef]

Francois, P. L.

F. Favre, D. Le Guen, M. L. Moulinard, M. Henry, P. L. Francois, F. Devaux, T. Georges, “Single-wavelength 40 Gbit/s, 8 × 100 km-span soliton transmission without any in-line control,” Electron. Lett. 33, 407–408 (1997).
[CrossRef]

Georges, T.

F. Favre, D. Le Guen, M. L. Moulinard, M. Henry, P. L. Francois, F. Devaux, T. Georges, “Single-wavelength 40 Gbit/s, 8 × 100 km-span soliton transmission without any in-line control,” Electron. Lett. 33, 407–408 (1997).
[CrossRef]

Ghafouri-Shiraz, H.

H. Ghafouri-Shiraz, P. Shum, “Novel method for analysis of soliton propagation in optical fibers,” IEEE J. Quantum Electron. 31, 190–200 (1995).
[CrossRef]

Hardin, R. H.

R. H. Hardin, F. D. Tappert, “Applications of the split-step Fourier method to the numerical solution of nonlinear and variable coefficient wave equations,” SIAM (Soc. Ind. Appl. Math.) Rev. 15, 423 (1973).

Haus, H. A.

H. A. Haus, M. N. Islam, “Theory of the soliton laser,” IEEE J. Quantum Electron. QE-21, 1172–1188 (1985).
[CrossRef]

Henry, M.

F. Favre, D. Le Guen, M. L. Moulinard, M. Henry, P. L. Francois, F. Devaux, T. Georges, “Single-wavelength 40 Gbit/s, 8 × 100 km-span soliton transmission without any in-line control,” Electron. Lett. 33, 407–408 (1997).
[CrossRef]

Islam, M. N.

H. A. Haus, M. N. Islam, “Theory of the soliton laser,” IEEE J. Quantum Electron. QE-21, 1172–1188 (1985).
[CrossRef]

Le Guen, D.

F. Favre, D. Le Guen, M. L. Moulinard, M. Henry, P. L. Francois, F. Devaux, T. Georges, “Single-wavelength 40 Gbit/s, 8 × 100 km-span soliton transmission without any in-line control,” Electron. Lett. 33, 407–408 (1997).
[CrossRef]

Moulinard, M. L.

F. Favre, D. Le Guen, M. L. Moulinard, M. Henry, P. L. Francois, F. Devaux, T. Georges, “Single-wavelength 40 Gbit/s, 8 × 100 km-span soliton transmission without any in-line control,” Electron. Lett. 33, 407–408 (1997).
[CrossRef]

Shabat, A. B.

V. E. Zakharov, A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

Shum, P.

P. Shum, S. F. Yu, “Numerical analysis of nonlinear soliton propagation phenomena using the fuzzy mesh analysis technique,” IEEE J. Quantum Electron. 34, 2029–2035 (1998).
[CrossRef]

P. Shum, S. F. Yu, “Improvement of Fourier series analysis technique by time-domain window function,” IEEE Photonics Technol. Lett. 18, 1364–1366 (1996).
[CrossRef]

H. Ghafouri-Shiraz, P. Shum, “Novel method for analysis of soliton propagation in optical fibers,” IEEE J. Quantum Electron. 31, 190–200 (1995).
[CrossRef]

Tappert, F. D.

R. H. Hardin, F. D. Tappert, “Applications of the split-step Fourier method to the numerical solution of nonlinear and variable coefficient wave equations,” SIAM (Soc. Ind. Appl. Math.) Rev. 15, 423 (1973).

Yu, S. F.

P. Shum, S. F. Yu, “Numerical analysis of nonlinear soliton propagation phenomena using the fuzzy mesh analysis technique,” IEEE J. Quantum Electron. 34, 2029–2035 (1998).
[CrossRef]

P. Shum, S. F. Yu, “Improvement of Fourier series analysis technique by time-domain window function,” IEEE Photonics Technol. Lett. 18, 1364–1366 (1996).
[CrossRef]

Zakharov, V. E.

V. E. Zakharov, A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

Electron. Lett.

F. Favre, D. Le Guen, M. L. Moulinard, M. Henry, P. L. Francois, F. Devaux, T. Georges, “Single-wavelength 40 Gbit/s, 8 × 100 km-span soliton transmission without any in-line control,” Electron. Lett. 33, 407–408 (1997).
[CrossRef]

IEEE J. Quantum Electron.

H. A. Haus, M. N. Islam, “Theory of the soliton laser,” IEEE J. Quantum Electron. QE-21, 1172–1188 (1985).
[CrossRef]

H. Ghafouri-Shiraz, P. Shum, “Novel method for analysis of soliton propagation in optical fibers,” IEEE J. Quantum Electron. 31, 190–200 (1995).
[CrossRef]

P. Shum, S. F. Yu, “Numerical analysis of nonlinear soliton propagation phenomena using the fuzzy mesh analysis technique,” IEEE J. Quantum Electron. 34, 2029–2035 (1998).
[CrossRef]

IEEE Photonics Technol. Lett.

P. Shum, S. F. Yu, “Improvement of Fourier series analysis technique by time-domain window function,” IEEE Photonics Technol. Lett. 18, 1364–1366 (1996).
[CrossRef]

SIAM (Soc. Ind. Appl. Math.) Rev.

R. H. Hardin, F. D. Tappert, “Applications of the split-step Fourier method to the numerical solution of nonlinear and variable coefficient wave equations,” SIAM (Soc. Ind. Appl. Math.) Rev. 15, 423 (1973).

Sov. Phys. JETP

V. E. Zakharov, A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

Other

G. P. Agrawal, Fiber-Optic Communication Systems, 2nd ed., Wiley Series in Microwave and Optical Engineering (Wiley-Interscience, New York, 1997).

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Figures (4)

Fig. 1
Fig. 1

Flow chart of the split-step reconstruction technique.

Fig. 2
Fig. 2

Flow chart for solving the finite difference equation.

Fig. 3
Fig. 3

Soliton pair propagation characteristics that were obtained with the SSRT.

Fig. 4
Fig. 4

Soliton pair propagation characteristics that were obtained with the FMAT.

Equations (28)

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

ux=j 122uT2+j|u|2unonlinear term,
12ux=j|u|2u,
12ux=j 122uT2,
un+1=un expj2|un|2h,
un+1-un2h=j|un+1|2un+1j|un|2un+1 linearized.
un+1-un2h=j 122un+1T2.
un+1-un=j h22un+1T2+jh|un|2un+1 linearized.
R=i=1m Ri,
Ri=aiai+12NT2 + 2|M|2N + j 2hN-M2dT,
M=i=1m Mi,
N=i=1m Ni,
Mi=S4i-3yi+S4i-2yi+S4i-1yi+1+S4iyi+1,
Ni=S4i-3zi+S4i-2zi+S4i-1zi+1+S4izi+1,
S4i-3=ai+12ai+1-3ai+6aiai+1T-3ai+ai+1T2+2T3ai+1-ai3,
S4i-2=-aiai+12+2ai+ai+1ai+1T-2ai+1+aiT2+T3ai+1-ai2,
S4i-1=ai23ai+1-ai-6aiai+1T+3ai+ai+1T2-2T3ai+1-ai3,
S4i=-ai2ai+1+aiai+2ai+1T-2ai+ai+1T2+T3ai+1-ai2,
Rzi=Rzi=Rzi+1=Rzi+1=0
Rzi=ziaiai+1 F4i-32dT+zi aiai+1 F4i-3F4i-2dT+zi+1aiai+1 F4i-3F4i-1dT+zi+1 aiai+1 F4i-3F4idT-j 2haiai+1 F4i-3MdT,
Fi=Si+j 2h Si+2|M|2Si.
Az=jh d,
A=I1,11I1,21I1,41I1,41I1,21I2,21I2,31I2,41I1,31I2,31I3,31+I5,52I3,41+I5,62I5,72I5,82I1,41I2,41I3,41+I5,62I4,41+I6,62I6,72I6,82I5,72I6,72I7,72+I9,93I7,82+I9,103I9,112+I9,123I5,82I6,82I7,82+I9,103I8,82+I10,103I10,112+I10,123 ,
z=z1z1z2z2zm+1zm+1,  d=I1,M1I2,M1I3,M1+I5,M2I4,M1+I6,M2I7,M2+I9,M3I4m,Mm,
Ii,ki=aiai+1 FiFkdT,
Ii,Mi=aiai+1 FiMdT.
ux=0, T=sechT-3.5+sechT+3.5.
Er=1100m=1m=100 |unm-un+1m|,
Em=1101T=-15T=15 |ucx, T-uex, T|x=π/2,

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