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Numerical Algorithms

, Volume 82, Issue 1, pp 223–243 | Cite as

A new FDTD scheme for Maxwell’s equations in Kerr-type nonlinear media

  • Hongen Jia
  • Jichun LiEmail author
  • Zhiwei Fang
  • Ming Li
Original Paper
  • 100 Downloads

Abstract

In this paper, we develop a totally new direct finite difference solver for solving the Maxwell’s equations in Kerr-type nonlinear media. The direct method is free of iteration error and more efficient than the classic iteration method. We also prove the continuous stability for the Kerr model and the discrete stability for the proposed scheme. Extensive numerical results are presented to validate our analysis.

Keywords

Maxwell’s equations FDTD method Kerr medium Soliton Nonlinear media 

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References

  1. 1.
    Berenger, J.P.: A perfectly matched layer for the absorption of electromagnetic waves. J. Computat. Phys. 114, 185–200 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bloembergen, N.: Nonlinear optics world scientific (1996)Google Scholar
  3. 3.
    Bokil, V.A., Cheng, Y., Jiang, Y., Li, F.: Energy stable discontinuous Galerkin methods for Maxwell’s equations in nonlinear optical media. J. Comput. Phys. 350, 420–452 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chen, Z., Du, Q., Zou, J.: Finite element methods with matching and nonmatching meshes for Maxwell equations with discontinuous coefficients. SIAM J. Numer. Anal. 37, 1542–1570 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Ciarlet, P. Jr, Zou, J.: Fully discrete finite element approaches for time-dependent Maxwell’s equations. Numer. Math. 82, 193–219 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Gao, L., Zhang, B.: Optimal error estimates and modified energy conservation identities of the ADI-FDTD scheme on staggered grids for 3D Maxwell’s equations. Sci. China Math. 56(8), 1705–1726 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Fathy, A., Wang, C., Wilson, J., Yang, S.: A fourth order difference scheme for the Maxwell equations on Yee grid. J. Hyperbolic Differ. Equ. 5(3), 613–642 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Fisher, A., White, D., Rodrigue, G.: An efficient vector finite element method for nonlinear electromagnetic modeling. J. Comput. Phys. 225, 1331–1346 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Fujii, M., Russer, P.: A nonlinear and dispersive APML ABC for the FD-TD methods. IEEE Microwave Wireless Compon. Lett. 12(11), 444–446 (2002)CrossRefGoogle Scholar
  10. 10.
    Fujii, M., Tahara, M., Sakagami, I., Freude, W., Russer, P.: High-order FDTD and auxiliary differential equation formulation of optical pulse propagation in 2-D Kerr and Raman nonlinear dispersive media. IEEE J. Quantum Electron. 40, 175–182 (2004)CrossRefGoogle Scholar
  11. 11.
    Gedney, S. D.: An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD lattices. IEEE Trans. Antennas Propag. 44(12), 1630–1639 (1996)CrossRefGoogle Scholar
  12. 12.
    Huang, J., Shu, C.-W.: A second-order asymptotic-preserving and positivity-preserving discontinuous Galerkin scheme for the Kerr-Debye model. Math. Models Methods Appl. Sci. 27, 549–580 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Joseph, R.M., Taflove, A.: Spatial soliton deflection mechanism indicated by FD-TD Maxwell’s equations modeling. IEEE Photon. Technol. Lett. 6(10), 1251–1254 (1994)CrossRefGoogle Scholar
  14. 14.
    Li, J., Hesthaven, J. S.: Analysis and application of the nodal discontinuous Galerkin method for wave propagation in metamaterials. J. Comput. Phys. 258, 915–930 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Li, J., Huang, Y.: Time-domain Finite Element Methods for Maxwell’s Equations in Metamaterials. Springer Series in Computational Mathematics, vol. 43. Springer (2013)Google Scholar
  16. 16.
    Li, J., Shields, S.: Superconvergence analysis of Yee scheme for metamaterial Maxwell’s equations on non-uniform rectangular meshes. Numer. Math. 134(4), 741–781 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Li, P., Zheng, W.: An Hψ formulation for the three-dimensional eddy current problem in laminated structures. J. Differential Equations 254(8), 3476–3500 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Li, W., Liang, D., Lin, Y.: A new energy-conserved s-FDTD scheme for Maxwell’s equations in metamaterials. Int. J. Numer. Anal. Model 10, 775–794 (2013)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Monk, P.: Finite element methods for maxwell’s equations. Oxford University Press, Oxford (2003)CrossRefzbMATHGoogle Scholar
  20. 20.
    Monk, P., Süli, E.: A convergence analysis of Yee’s scheme on nonuniform grids. SIAM J. Numer. Anal. 32, 393–412 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Shen, Y.-R.: The Principles of Nonlinear Optics. Wiley Interscience (1994)Google Scholar
  22. 22.
    Taflove, A., Haguess, S.C.: Computational Electrodynamics: the Finite-Difference Time-Domain Method, 3rd edn. Artech House (2005)Google Scholar
  23. 23.
    Wu, H., Zou, J.: Finite element method and its analysis for a nonlinear Helmholtz equation with high wave numbers. SIAM J. Numer. Anal. 56, 1338–1359 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Xu, Z., Bao, G.: A numerical scheme for nonlinear Helmholtz equations with strong nonlinear optical effects. J. Opt. Soc. Am. A 27, 2347–2353 (2010)CrossRefGoogle Scholar
  25. 25.
    Yuan, L., Lu, Y.: Robust iterative method for nonlinear Helmholtz equation. J. Comput. Phys. 343, 1–9 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Zhang, Y., Nguyen, D.D., Du, K., Xu, J., Zhao, S.: Time-domain numerical solutions of Maxwell interface problems with discontinuous electromagnetic waves. Adv. Appl. Math. Mech. 8(3), 353–385 (2016)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Ziolkowski, R.W., Judkins, J.B.: Full-wave vector Maxwell equation modeling of the self-focusing of ultrashort optical pulses in a nonlinear Kerr medium exhibiting a finite response time. J. Optical Soc. Amer. B 10, 186–198 (1993)CrossRefGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.College of MathematicsTaiyuan University of TechnologyTaiyuanChina
  2. 2.Department of Mathematical SciencesUniversity of Nevada Las VegasLas VegasUSA

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