Abstract

The widely used split-step Fourier method has difficulties when solving partial differential equations with saturable gain. Here, we describe a modified split-step Fourier method, and we compare it to several different algorithms for solving the Haus mode-locking equation and related equations that are used to model mode-locked lasers and other optical oscillators and amplifiers with saturable gain. These equations all include the product of a scalar nonlinearity and a stiff nonlinear operator. We find that a modified split-step method is the easiest to program with the same level of reliability and accuracy as the other methods that we investigated.

© 2013 Optical Society of America

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Corrections

Shaokang Wang, Andrew Docherty, Brian S. Marks, and Curtis R. Menyuk, "Comparison of numerical methods for modeling laser modelocking with saturable gain: erratum," J. Opt. Soc. Am. B 31, 1807-1807 (2014)
https://www.osapublishing.org/josab/abstract.cfm?uri=josab-31-8-1807

References

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  1. O. V. Sinkin, R. Holzlöhner, J. Zweck, and C. R. Menyuk, “Optimization of the split-step Fourier method in modeling optical-fiber communications systems,” J. Lightwave Technol. 21, 61–68 (2003).
    [CrossRef]
  2. S. A. Diddams, “The evolving optical frequency comb [invited],” J. Opt. Soc. Am. B 27, B51–B62 (2010).
    [CrossRef]
  3. H. A. Haus, “Theory of mode locking with a fast saturable absorber,” J. Appl. Phys. 46, 3049–3058 (1975).
    [CrossRef]
  4. T. Kapitula, J. N. Kutz, and B. Sandstede, “Stability of pulses in the master mode-locking equation,” J. Opt. Soc. Am. B 19, 740–746 (2002).
    [CrossRef]
  5. C. R. Menyuk, J. K. Wahlstrand, J. Willits, R. P. Smith, T. R. Schibli, and S. T. Cundiff, “Pulse dynamics in mode-locked lasers: relaxation oscillations and frequency pulling,” Opt. Express 15, 6677–6689 (2007).
    [CrossRef]
  6. M. Shtaif, C. R. Menyuk, M. L. Dennis, and M. C. Gross, “Carrier-envelope phase locking of multipulse lasers with an intracavity Mach–Zehnder interferometer,” Opt. Express 19, 23202–23214 (2011).
    [CrossRef]
  7. A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. II. Normal dispersion,” Appl. Phys. Lett. 23, 171–172 (1973).
    [CrossRef]
  8. A. Ablowitz and P. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, London Mathematical Society Lecture Note Series (Cambridge University, 1991).
  9. B. Fornberg and T. A. Driscoll, “A fast spectral algorithm for nonlinear wave equations with linear dispersion,” J. Comput. Phys. 155, 456–467 (1999).
    [CrossRef]
  10. L. Trefethen, Spectral Methods in MATLAB, Software, Environments and Tools Series (Cambridge University, 2000).
  11. R. J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations (Society for Industrial and Applied Mathematics, 2007).
  12. A. Iserles, A First Course in the Numerical Analysis of Differential Equations, Cambridge Texts in Applied Mathematics (Cambridge University, 1996).
  13. G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, “Suppression of spurious tones induced by the split-step method in fiber systems simulation,” IEEE Photon. Technol. Lett. 12, 489–491 (2000).
    [CrossRef]
  14. C. A. Kennedy and M. H. Carpenter, “Additive Runge–Kutta schemes for convection-diffusion-reaction equations,” Appl. Numer. Math. 44, 139–181 (2003).
    [CrossRef]
  15. S. Cox and P. Matthews, “Exponential time differencing for stiff systems,” J. Comput. Phys. 176, 430–455 (2002).
    [CrossRef]
  16. G. Strang, “On the construction and comparison of difference schemes,” SIAM J. Numer. Anal. 5, 506–517 (1968).
    [CrossRef]
  17. K. Bagrinovskii and S. Godunov, “Difference schemes for multidimensional problems,” Doklady Akademii Nauk 115, 431–433 (1957).
  18. H. Yoshida, “Construction of higher order symplectic integrators,” Phys. Lett. A 150, 262–268 (1990).
    [CrossRef]
  19. J. Boyd, Chebyshev and Fourier Spectral Methods, Dover Books on Mathematics (Dover Publications, 2001).
  20. G. Agrawal, Nonlinear Fiber Optics, Optics and Photonics (Elsevier, 2010).
  21. L. F. Richardson, “The approximate arithmetical solution by finite differences of physical problems involving differential equations, with an application to the stresses in a masonry dam,” Proc. R. Soc. A 210, 307–357 (1911).
  22. J. Kutz, “Mode-locked soliton lasers,” SIAM Rev. 48, 629–678 (2006).
    [CrossRef]
  23. U. M. Ascher, S. J. Ruuth, and B. T. R. Wetton, “Implicit–explicit methods for time-dependent partial differential equations,” SIAM J. Numer. Anal. 32, 797–823 (1995).
    [CrossRef]
  24. U. Ascher and L. Petzold, Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, Miscellaneous Titles in Applied Mathematics Series (Society for Industrial and Applied Mathematics, 1998).
  25. M. O. Williams, J. Wilkening, E. Shlizerman, and J. N. Kutz, “Continuation of periodic solutions in the waveguide array mode-locked laser,” Physica D 240, 1791–1804 (2011).
    [CrossRef]
  26. J. Nocedal and S. Wright, Numerical Optimization, Springer Series in Operations Research (Springer, 2006).
  27. A. Taflove and S. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, Artech House Antennas and Propagation Library (Artech House, 2005).
  28. D. R. Mott, E. S. Oran, and B. van Leer, “A quasi-steady-state solver for the stiff ordinary differential equations of reaction kinetics,” J. Comput. Phys. 164, 407–428 (2000).
    [CrossRef]
  29. A. Kassam and L. Trefethen, “Fourth-order time-stepping for stiff PDEs,” SIAM J. Sci. Comput. 26, 1214–1233 (2005).
    [CrossRef]
  30. H. Berland, B. Skaflestad, and W. M. Wright, “EXPINT-a MATLAB package for exponential integrators,” ACM Trans. Math. Softw. 33, 4 (2007).
    [CrossRef]

2011 (2)

M. O. Williams, J. Wilkening, E. Shlizerman, and J. N. Kutz, “Continuation of periodic solutions in the waveguide array mode-locked laser,” Physica D 240, 1791–1804 (2011).
[CrossRef]

M. Shtaif, C. R. Menyuk, M. L. Dennis, and M. C. Gross, “Carrier-envelope phase locking of multipulse lasers with an intracavity Mach–Zehnder interferometer,” Opt. Express 19, 23202–23214 (2011).
[CrossRef]

2010 (1)

2007 (2)

2006 (1)

J. Kutz, “Mode-locked soliton lasers,” SIAM Rev. 48, 629–678 (2006).
[CrossRef]

2005 (1)

A. Kassam and L. Trefethen, “Fourth-order time-stepping for stiff PDEs,” SIAM J. Sci. Comput. 26, 1214–1233 (2005).
[CrossRef]

2003 (2)

C. A. Kennedy and M. H. Carpenter, “Additive Runge–Kutta schemes for convection-diffusion-reaction equations,” Appl. Numer. Math. 44, 139–181 (2003).
[CrossRef]

O. V. Sinkin, R. Holzlöhner, J. Zweck, and C. R. Menyuk, “Optimization of the split-step Fourier method in modeling optical-fiber communications systems,” J. Lightwave Technol. 21, 61–68 (2003).
[CrossRef]

2002 (2)

T. Kapitula, J. N. Kutz, and B. Sandstede, “Stability of pulses in the master mode-locking equation,” J. Opt. Soc. Am. B 19, 740–746 (2002).
[CrossRef]

S. Cox and P. Matthews, “Exponential time differencing for stiff systems,” J. Comput. Phys. 176, 430–455 (2002).
[CrossRef]

2000 (2)

G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, “Suppression of spurious tones induced by the split-step method in fiber systems simulation,” IEEE Photon. Technol. Lett. 12, 489–491 (2000).
[CrossRef]

D. R. Mott, E. S. Oran, and B. van Leer, “A quasi-steady-state solver for the stiff ordinary differential equations of reaction kinetics,” J. Comput. Phys. 164, 407–428 (2000).
[CrossRef]

1999 (1)

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

1995 (1)

U. M. Ascher, S. J. Ruuth, and B. T. R. Wetton, “Implicit–explicit methods for time-dependent partial differential equations,” SIAM J. Numer. Anal. 32, 797–823 (1995).
[CrossRef]

1990 (1)

H. Yoshida, “Construction of higher order symplectic integrators,” Phys. Lett. A 150, 262–268 (1990).
[CrossRef]

1975 (1)

H. A. Haus, “Theory of mode locking with a fast saturable absorber,” J. Appl. Phys. 46, 3049–3058 (1975).
[CrossRef]

1973 (1)

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. II. Normal dispersion,” Appl. Phys. Lett. 23, 171–172 (1973).
[CrossRef]

1968 (1)

G. Strang, “On the construction and comparison of difference schemes,” SIAM J. Numer. Anal. 5, 506–517 (1968).
[CrossRef]

1957 (1)

K. Bagrinovskii and S. Godunov, “Difference schemes for multidimensional problems,” Doklady Akademii Nauk 115, 431–433 (1957).

1911 (1)

L. F. Richardson, “The approximate arithmetical solution by finite differences of physical problems involving differential equations, with an application to the stresses in a masonry dam,” Proc. R. Soc. A 210, 307–357 (1911).

Ablowitz, A.

A. Ablowitz and P. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, London Mathematical Society Lecture Note Series (Cambridge University, 1991).

Agrawal, G.

G. Agrawal, Nonlinear Fiber Optics, Optics and Photonics (Elsevier, 2010).

Ascher, U.

U. Ascher and L. Petzold, Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, Miscellaneous Titles in Applied Mathematics Series (Society for Industrial and Applied Mathematics, 1998).

Ascher, U. M.

U. M. Ascher, S. J. Ruuth, and B. T. R. Wetton, “Implicit–explicit methods for time-dependent partial differential equations,” SIAM J. Numer. Anal. 32, 797–823 (1995).
[CrossRef]

Bagrinovskii, K.

K. Bagrinovskii and S. Godunov, “Difference schemes for multidimensional problems,” Doklady Akademii Nauk 115, 431–433 (1957).

Benedetto, S.

G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, “Suppression of spurious tones induced by the split-step method in fiber systems simulation,” IEEE Photon. Technol. Lett. 12, 489–491 (2000).
[CrossRef]

Berland, H.

H. Berland, B. Skaflestad, and W. M. Wright, “EXPINT-a MATLAB package for exponential integrators,” ACM Trans. Math. Softw. 33, 4 (2007).
[CrossRef]

Bosco, G.

G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, “Suppression of spurious tones induced by the split-step method in fiber systems simulation,” IEEE Photon. Technol. Lett. 12, 489–491 (2000).
[CrossRef]

Boyd, J.

J. Boyd, Chebyshev and Fourier Spectral Methods, Dover Books on Mathematics (Dover Publications, 2001).

Carena, A.

G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, “Suppression of spurious tones induced by the split-step method in fiber systems simulation,” IEEE Photon. Technol. Lett. 12, 489–491 (2000).
[CrossRef]

Carpenter, M. H.

C. A. Kennedy and M. H. Carpenter, “Additive Runge–Kutta schemes for convection-diffusion-reaction equations,” Appl. Numer. Math. 44, 139–181 (2003).
[CrossRef]

Clarkson, P.

A. Ablowitz and P. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, London Mathematical Society Lecture Note Series (Cambridge University, 1991).

Cox, S.

S. Cox and P. Matthews, “Exponential time differencing for stiff systems,” J. Comput. Phys. 176, 430–455 (2002).
[CrossRef]

Cundiff, S. T.

Curri, V.

G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, “Suppression of spurious tones induced by the split-step method in fiber systems simulation,” IEEE Photon. Technol. Lett. 12, 489–491 (2000).
[CrossRef]

Dennis, M. L.

Diddams, S. A.

Driscoll, T. A.

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

Fornberg, B.

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

Gaudino, R.

G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, “Suppression of spurious tones induced by the split-step method in fiber systems simulation,” IEEE Photon. Technol. Lett. 12, 489–491 (2000).
[CrossRef]

Godunov, S.

K. Bagrinovskii and S. Godunov, “Difference schemes for multidimensional problems,” Doklady Akademii Nauk 115, 431–433 (1957).

Gross, M. C.

Hagness, S.

A. Taflove and S. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, Artech House Antennas and Propagation Library (Artech House, 2005).

Hasegawa, A.

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. II. Normal dispersion,” Appl. Phys. Lett. 23, 171–172 (1973).
[CrossRef]

Haus, H. A.

H. A. Haus, “Theory of mode locking with a fast saturable absorber,” J. Appl. Phys. 46, 3049–3058 (1975).
[CrossRef]

Holzlöhner, R.

Iserles, A.

A. Iserles, A First Course in the Numerical Analysis of Differential Equations, Cambridge Texts in Applied Mathematics (Cambridge University, 1996).

Kapitula, T.

Kassam, A.

A. Kassam and L. Trefethen, “Fourth-order time-stepping for stiff PDEs,” SIAM J. Sci. Comput. 26, 1214–1233 (2005).
[CrossRef]

Kennedy, C. A.

C. A. Kennedy and M. H. Carpenter, “Additive Runge–Kutta schemes for convection-diffusion-reaction equations,” Appl. Numer. Math. 44, 139–181 (2003).
[CrossRef]

Kutz, J.

J. Kutz, “Mode-locked soliton lasers,” SIAM Rev. 48, 629–678 (2006).
[CrossRef]

Kutz, J. N.

M. O. Williams, J. Wilkening, E. Shlizerman, and J. N. Kutz, “Continuation of periodic solutions in the waveguide array mode-locked laser,” Physica D 240, 1791–1804 (2011).
[CrossRef]

T. Kapitula, J. N. Kutz, and B. Sandstede, “Stability of pulses in the master mode-locking equation,” J. Opt. Soc. Am. B 19, 740–746 (2002).
[CrossRef]

LeVeque, R. J.

R. J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations (Society for Industrial and Applied Mathematics, 2007).

Matthews, P.

S. Cox and P. Matthews, “Exponential time differencing for stiff systems,” J. Comput. Phys. 176, 430–455 (2002).
[CrossRef]

Menyuk, C. R.

Mott, D. R.

D. R. Mott, E. S. Oran, and B. van Leer, “A quasi-steady-state solver for the stiff ordinary differential equations of reaction kinetics,” J. Comput. Phys. 164, 407–428 (2000).
[CrossRef]

Nocedal, J.

J. Nocedal and S. Wright, Numerical Optimization, Springer Series in Operations Research (Springer, 2006).

Oran, E. S.

D. R. Mott, E. S. Oran, and B. van Leer, “A quasi-steady-state solver for the stiff ordinary differential equations of reaction kinetics,” J. Comput. Phys. 164, 407–428 (2000).
[CrossRef]

Petzold, L.

U. Ascher and L. Petzold, Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, Miscellaneous Titles in Applied Mathematics Series (Society for Industrial and Applied Mathematics, 1998).

Poggiolini, P.

G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, “Suppression of spurious tones induced by the split-step method in fiber systems simulation,” IEEE Photon. Technol. Lett. 12, 489–491 (2000).
[CrossRef]

Richardson, L. F.

L. F. Richardson, “The approximate arithmetical solution by finite differences of physical problems involving differential equations, with an application to the stresses in a masonry dam,” Proc. R. Soc. A 210, 307–357 (1911).

Ruuth, S. J.

U. M. Ascher, S. J. Ruuth, and B. T. R. Wetton, “Implicit–explicit methods for time-dependent partial differential equations,” SIAM J. Numer. Anal. 32, 797–823 (1995).
[CrossRef]

Sandstede, B.

Schibli, T. R.

Shlizerman, E.

M. O. Williams, J. Wilkening, E. Shlizerman, and J. N. Kutz, “Continuation of periodic solutions in the waveguide array mode-locked laser,” Physica D 240, 1791–1804 (2011).
[CrossRef]

Shtaif, M.

Sinkin, O. V.

Skaflestad, B.

H. Berland, B. Skaflestad, and W. M. Wright, “EXPINT-a MATLAB package for exponential integrators,” ACM Trans. Math. Softw. 33, 4 (2007).
[CrossRef]

Smith, R. P.

Strang, G.

G. Strang, “On the construction and comparison of difference schemes,” SIAM J. Numer. Anal. 5, 506–517 (1968).
[CrossRef]

Taflove, A.

A. Taflove and S. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, Artech House Antennas and Propagation Library (Artech House, 2005).

Tappert, F.

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. II. Normal dispersion,” Appl. Phys. Lett. 23, 171–172 (1973).
[CrossRef]

Trefethen, L.

A. Kassam and L. Trefethen, “Fourth-order time-stepping for stiff PDEs,” SIAM J. Sci. Comput. 26, 1214–1233 (2005).
[CrossRef]

L. Trefethen, Spectral Methods in MATLAB, Software, Environments and Tools Series (Cambridge University, 2000).

van Leer, B.

D. R. Mott, E. S. Oran, and B. van Leer, “A quasi-steady-state solver for the stiff ordinary differential equations of reaction kinetics,” J. Comput. Phys. 164, 407–428 (2000).
[CrossRef]

Wahlstrand, J. K.

Wetton, B. T. R.

U. M. Ascher, S. J. Ruuth, and B. T. R. Wetton, “Implicit–explicit methods for time-dependent partial differential equations,” SIAM J. Numer. Anal. 32, 797–823 (1995).
[CrossRef]

Wilkening, J.

M. O. Williams, J. Wilkening, E. Shlizerman, and J. N. Kutz, “Continuation of periodic solutions in the waveguide array mode-locked laser,” Physica D 240, 1791–1804 (2011).
[CrossRef]

Williams, M. O.

M. O. Williams, J. Wilkening, E. Shlizerman, and J. N. Kutz, “Continuation of periodic solutions in the waveguide array mode-locked laser,” Physica D 240, 1791–1804 (2011).
[CrossRef]

Willits, J.

Wright, S.

J. Nocedal and S. Wright, Numerical Optimization, Springer Series in Operations Research (Springer, 2006).

Wright, W. M.

H. Berland, B. Skaflestad, and W. M. Wright, “EXPINT-a MATLAB package for exponential integrators,” ACM Trans. Math. Softw. 33, 4 (2007).
[CrossRef]

Yoshida, H.

H. Yoshida, “Construction of higher order symplectic integrators,” Phys. Lett. A 150, 262–268 (1990).
[CrossRef]

Zweck, J.

ACM Trans. Math. Softw. (1)

H. Berland, B. Skaflestad, and W. M. Wright, “EXPINT-a MATLAB package for exponential integrators,” ACM Trans. Math. Softw. 33, 4 (2007).
[CrossRef]

Appl. Numer. Math. (1)

C. A. Kennedy and M. H. Carpenter, “Additive Runge–Kutta schemes for convection-diffusion-reaction equations,” Appl. Numer. Math. 44, 139–181 (2003).
[CrossRef]

Appl. Phys. Lett. (1)

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. II. Normal dispersion,” Appl. Phys. Lett. 23, 171–172 (1973).
[CrossRef]

Doklady Akademii Nauk (1)

K. Bagrinovskii and S. Godunov, “Difference schemes for multidimensional problems,” Doklady Akademii Nauk 115, 431–433 (1957).

IEEE Photon. Technol. Lett. (1)

G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, “Suppression of spurious tones induced by the split-step method in fiber systems simulation,” IEEE Photon. Technol. Lett. 12, 489–491 (2000).
[CrossRef]

J. Appl. Phys. (1)

H. A. Haus, “Theory of mode locking with a fast saturable absorber,” J. Appl. Phys. 46, 3049–3058 (1975).
[CrossRef]

J. Comput. Phys. (3)

D. R. Mott, E. S. Oran, and B. van Leer, “A quasi-steady-state solver for the stiff ordinary differential equations of reaction kinetics,” J. Comput. Phys. 164, 407–428 (2000).
[CrossRef]

S. Cox and P. Matthews, “Exponential time differencing for stiff systems,” J. Comput. Phys. 176, 430–455 (2002).
[CrossRef]

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

J. Lightwave Technol. (1)

J. Opt. Soc. Am. B (2)

Opt. Express (2)

Phys. Lett. A (1)

H. Yoshida, “Construction of higher order symplectic integrators,” Phys. Lett. A 150, 262–268 (1990).
[CrossRef]

Physica D (1)

M. O. Williams, J. Wilkening, E. Shlizerman, and J. N. Kutz, “Continuation of periodic solutions in the waveguide array mode-locked laser,” Physica D 240, 1791–1804 (2011).
[CrossRef]

Proc. R. Soc. A (1)

L. F. Richardson, “The approximate arithmetical solution by finite differences of physical problems involving differential equations, with an application to the stresses in a masonry dam,” Proc. R. Soc. A 210, 307–357 (1911).

SIAM J. Numer. Anal. (2)

U. M. Ascher, S. J. Ruuth, and B. T. R. Wetton, “Implicit–explicit methods for time-dependent partial differential equations,” SIAM J. Numer. Anal. 32, 797–823 (1995).
[CrossRef]

G. Strang, “On the construction and comparison of difference schemes,” SIAM J. Numer. Anal. 5, 506–517 (1968).
[CrossRef]

SIAM J. Sci. Comput. (1)

A. Kassam and L. Trefethen, “Fourth-order time-stepping for stiff PDEs,” SIAM J. Sci. Comput. 26, 1214–1233 (2005).
[CrossRef]

SIAM Rev. (1)

J. Kutz, “Mode-locked soliton lasers,” SIAM Rev. 48, 629–678 (2006).
[CrossRef]

Other (9)

U. Ascher and L. Petzold, Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, Miscellaneous Titles in Applied Mathematics Series (Society for Industrial and Applied Mathematics, 1998).

J. Boyd, Chebyshev and Fourier Spectral Methods, Dover Books on Mathematics (Dover Publications, 2001).

G. Agrawal, Nonlinear Fiber Optics, Optics and Photonics (Elsevier, 2010).

A. Ablowitz and P. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, London Mathematical Society Lecture Note Series (Cambridge University, 1991).

L. Trefethen, Spectral Methods in MATLAB, Software, Environments and Tools Series (Cambridge University, 2000).

R. J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations (Society for Industrial and Applied Mathematics, 2007).

A. Iserles, A First Course in the Numerical Analysis of Differential Equations, Cambridge Texts in Applied Mathematics (Cambridge University, 1996).

J. Nocedal and S. Wright, Numerical Optimization, Springer Series in Operations Research (Springer, 2006).

A. Taflove and S. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, Artech House Antennas and Propagation Library (Artech House, 2005).

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Figures (4)

Fig. 1.
Fig. 1.

Solving the HME starting from a small pulse u0(t)=0.25exp[(t/5)2] using set 1 in Table 1 with a step size h=0.04: (a) The pulse evolves to a mode-locked pulse using the asymmetric splitting of Eq. (24) and (b) the pulse energy grows exponentially using the symmetric splitting scheme given in Eq. (17), where the nonlinear integration is evaluated using a second-order Runge–Kutta method.

Fig. 2.
Fig. 2.

(a) Relative error versus the step size and (b) the relative error versus the computational time. The HME of the Eq. (2) computation was solved with an initial small pulse up to z=300 using set 1 in Table 1.

Fig. 3.
Fig. 3.

(a) Relative error versus the step size and (b) the relative error versus the computational time. The HME of the Eq. (2) computation was solved with an initial small pulse up to z=300 using set 2 in Table 1.

Fig. 4.
Fig. 4.

Stability regions of the split-step method applying to Eq. (B1). The white regions indicate stability. (a)–(f) show the cases in which N(u,z) is handled explicitly and (g)–(l) show the cases in which N(u,z) is handled analytically.

Tables (1)

Tables Icon

Table 1. Two Sets of Normalized Parameters Used for Computing the HME Numerically

Equations (65)

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

uz=Lu+g(u)Ku+N(u,z),
uz=l2uiβ22ut2+g(u)2(u+1Ωg22ut2)+(iγ+δ)|u|2u,
g(u)=g01+Pav(u)/Psat,
Pav(u)=1TRTR/2TR/2|u(z,t)|2dt,
L=12(l+iβ2t2),
K=12(1+1Ωg22t2),
N(u,z)=(iγ+δ)|u|2u.
L˜=(iβω2l)/2,
K˜=[1(ω/Ωg)2]/2,
dudz=Lu+g(u)Ku+N(u,z),
τ2=TR/2TR/2(ttc)2|u(z,t)|2dtTR/2TR/2|u(z,t)|2dt,
tc=TR/2TR/2t|u(z,t)|2dtTR/2TR/2|u(z,t)|2dt.
g(u)=g01+uHu/E0,
uz=2ut2,
u˜z=ω2u˜.
du(j)dz=u(j+1)2u(j)+u(j1)(Δt)2,
du˜(j)dz=sin2(ωjΔt)(Δt)2u˜(j),
uz=i2ut2,
u˜z=iω2u˜,
uk+1=exp(hL/2)exp(zkzk+1N[u(z),z]dz)exp(hL/2)uk,
uk+1=exp(h/2[L+g(uk)K])exp(zkzk+1N[u(z),z]dz))exp(h/2[L+g(uk)K])uk,
uk+1=2uk+1h/2uk+1h,
uk+1=exp(hL/2)exp(zkzk+1{g[u(z)]K+N[u(z),z]}dz)exp(hL/2)uk,
exp(zkzk+1{g[u(z)]K+N[u(z),z]}dz)uFN{exp(zkzk+1g[u(z)]Kdz)FN(u)},
exp(zkzk+1g(u)Kdz)uIFFT{[1+hg(u)(1ω2/Ωg2)/2]FFT(u)},
uk+1=exp(hL/2+{zk+1/2zk+1g[u(z)]dz}K)exp(zkzk+1N[u(z),z]dz)exp(hL/2+{zkzk+1/2g[u(z)]dz}K)uk,
uk+1=exp(Lh/2+[g(zk+1/2)+g(zk+1/2)h/4]Kh/2)exp(zkzk+1N(u(z),z)dz)exp(Lh/2+[g(zk)+g(zk)h/4]Kh/2)uk.
u(zk+δ)u(zk)+δdudz|z=zk.
g(zk+δ)g(uk)+δJgH(uk)dukdz+O(δ2),
uk+1=exp(Lh/2+[g(uk+1/2)+g2(uk)h/4]Kh/2)exp(zkzk+1N[u(z),z]dz)exp(Lh/2+[g(uk)+g2(uk)h/4]Kh/2)uk,
g2(u)=JgH(u)dudz=2[g(u)]2g0E0Re{uH[L+g(u)K]u},
g2(uk)=1h/2[g(u^k+1/2)g(uk)],
u^k+1/2=exp[Lh/2+g(uk)Kh/2]uk.
uk+1=121(32uk+1h/412uk+1h/2+uk+1h),
u(z,t)=A0sech1+iβ(t/τ)exp(iϕz),
exp[0h(δ+iγ)|u(ζ,t)|2u(ζ,t)dζ]=u(0,t)[12hδ|u(0,t)|2](δ+iγ)/(2δ).
uki=IFFT{exp[h(L˜+g(uk)K˜)/2+h2g2K˜/8]FFT[uk]},
ukii=uki(12hδ|uki|2)(δ+iγ)/(2δ),
uk+1=IFFT{exp[h(L˜+g(ukii)K˜)/2+h2g2K˜/8]FFT[ukii]},
L˜jj=(iβωj2l)/2,
K˜jj=[1(ωj/Ωg)2]/2,
g2=2g2(uk)g0NPsatRe{[FFT[uk]]H[L˜+g(uk)K˜]FFT[uk]}.
k1=uk,
ki=uk+hj=1i1[aijsFs(kj)+aijnsFns(kj)]+hγrkFs(ki),i2,
uk+1=uk+hi=1mbisFs(ki)+hi=1mbinsFns(ki),
hj=1i1[aijsFs(qj)+aijnsFns(qj)]=qihγrkFs(qi),
Fs(q)=[L^q+g^(q)K^q]Fns(q)=[Re{(δ+iγ)|u|2u}Im{(δ+iγ)|u|2u}],
L^=[Re(L)Im(L)Im(L)Re(L)],K^=[Re(K)Im(K)Im(K)Re(K)],
g^(q)=g01+qTq/E0.
Jext=Ihγrk[L^+g^(q)K^2g^(q)2g0E0K^qqT].
(A+abT)1=A1A1abTA11+bTA1a.
A=L^+g^(q)K^,a=2g^(q)g0E0K^q,andb=q.
uz={L+g(uk)K}u+{[g(u)g(uk)]Ku+N(u,z)}.
ak=exp(L^h/2)uk+L^1[exp(L^h/2)I]N^(uk,zk),
bk=exp(L^h/2)uk+L^1[exp(L^h/2)I]N^(ak,zk+h/2),
ck=exp(L^h/2)ak+L^1[exp(L^h/2)I][2N^(bk,zk+h/2)N^(uk,zk)],
uk+1=exp(L^h)uk+h2L^3{[4IL^h+exp(L^h)(4I3L^h+(L^h)2)]N^(uk,zk)+2[2I+L^h+exp(L^h)(2I+L^h)][N^(ak,zk+h/2)+N^(bk,zk+h/2)]+[4I3L^h(L^h)2+exp(L^h)(4IL^h)]N^(ck,zk+h)},
L^=L+g(uk)K,
N^=[g(u)g(uk)]Ku+N(u,z).
Erel(h)=uh(z)ur(z)2ur(z)2,
dudz=cu+(δ+iγ)|u|2u,
σN=uk+1uk.
uk+1=exp(C)[1+(A+iB)|D|2Dexp(C)]uk,
σN=exp(C)[1+(A+iB)|D|2Dexp(C)].
σN=exp{CA+iB2Alog[12Aexp(C)]},

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