## Comparison of numerical methods for modeling laser mode locking with saturable gain |

JOSA B, Vol. 30, Issue 11, pp. 3064-3074 (2013)

http://dx.doi.org/10.1364/JOSAB.30.003064

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### 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

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

(140.3280) Lasers and laser optics : Laser amplifiers

(140.4050) Lasers and laser optics : Mode-locked lasers

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: July 11, 2013

Revised Manuscript: October 1, 2013

Manuscript Accepted: October 1, 2013

Published: October 31, 2013

**Citation**

Shaokang Wang, Andrew Docherty, Brian S. Marks, and Curtis R. Menyuk, "Comparison of numerical methods for modeling laser mode locking with saturable gain," J. Opt. Soc. Am. B **30**, 3064-3074 (2013)

http://www.opticsinfobase.org/josab/abstract.cfm?URI=josab-30-11-3064

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### References

- 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]
- S. A. Diddams, “The evolving optical frequency comb [invited],” J. Opt. Soc. Am. B 27, B51–B62 (2010). [CrossRef]
- H. A. Haus, “Theory of mode locking with a fast saturable absorber,” J. Appl. Phys. 46, 3049–3058 (1975). [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]
- 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]
- 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]
- 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]
- A. Ablowitz and P. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, London Mathematical Society Lecture Note Series (Cambridge University, 1991).
- 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]
- 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).
- 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]
- C. A. Kennedy and M. H. Carpenter, “Additive Runge–Kutta schemes for convection-diffusion-reaction equations,” Appl. Numer. Math. 44, 139–181 (2003). [CrossRef]
- S. Cox and P. Matthews, “Exponential time differencing for stiff systems,” J. Comput. Phys. 176, 430–455 (2002). [CrossRef]
- G. Strang, “On the construction and comparison of difference schemes,” SIAM J. Numer. Anal. 5, 506–517 (1968). [CrossRef]
- K. Bagrinovskii and S. Godunov, “Difference schemes for multidimensional problems,” Doklady Akademii Nauk 115, 431–433 (1957).
- H. Yoshida, “Construction of higher order symplectic integrators,” Phys. Lett. A 150, 262–268 (1990). [CrossRef]
- J. Boyd, Chebyshev and Fourier Spectral Methods, Dover Books on Mathematics (Dover Publications, 2001).
- G. Agrawal, Nonlinear Fiber Optics, Optics and Photonics (Elsevier, 2010).
- 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).
- J. Kutz, “Mode-locked soliton lasers,” SIAM Rev. 48, 629–678 (2006). [CrossRef]
- 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]
- 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).
- 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]
- 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).
- 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]
- A. Kassam and L. Trefethen, “Fourth-order time-stepping for stiff PDEs,” SIAM J. Sci. Comput. 26, 1214–1233 (2005). [CrossRef]
- H. Berland, B. Skaflestad, and W. M. Wright, “EXPINT-a MATLAB package for exponential integrators,” ACM Trans. Math. Softw. 33, 4 (2007). [CrossRef]

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