Complex envelope Faber polynomial method for the solution of Maxwell’s equations

  • Hendrik KleeneEmail author
  • Dirk Schulz


A complex envelope approach for the numerical solution of Maxwell’s equations based on Faber polynomial expansions is investigated. The Faber polynomial expansion used for the approximation of the exponential time propagator offers a highly accurate and efficient calculation while allowing the application of large time steps. The complex envelope approach incorporates only the envelope around a carrier frequency. This is especially beneficial when bandlimited source field distributions are investigated as it is the case for many applications from terahertz technology or photonics.


Time-domain methods Complex envelope Finite-difference time-domain method (FDTD) Computational modeling 



This work was supported by the German research funding association Deutsche Forschungsgemeinschaft under Grant SCHU 1016/6-1.


  1. Al-Mohy, A.H., Higham, N.J.: Computing the action of the matrix exponential, with an application to exponential integrators. SIAM J. Sci. Comput. 33(2), 488–511 (2011)MathSciNetCrossRefGoogle Scholar
  2. Borisov, A.G., Shabanov, S.V.: Wave packet propagation by the Faber polynomial approximation in electrodynamics of passive media. J. Comput. Phys. 216(1), 391–402 (2006)ADSMathSciNetCrossRefGoogle Scholar
  3. Busch, K., Niegemann, J., Pototschnig, M., Tkeshelashvili, L.: A Krylov-subspace based solver for the linear and nonlinear Maxwell equations. Physica Status Solidi (b) 244(10), 3479–3496 (2007)ADSCrossRefGoogle Scholar
  4. De Raedt, H., Michielsen, K., Kole, J., Figge, M.: Solving the Maxwell equations by the Chebyshev method: a one-step finite-difference time-domain algorithm. IEEE Trans. Antennas Propag. 51(11), 3155–3160 (2003)ADSMathSciNetCrossRefGoogle Scholar
  5. Ellacott S.:  A survey of Faber methods in numerical approximation. Comput. Math. Appl. 12(5-6), 1103–1107 (1986)MathSciNetCrossRefGoogle Scholar
  6. Fahs, H.: Investigation on polynomial integrators for time-domain electromagnetics using a high-order discontinuous Galerkin method. Appl. Math. Model. 36(11), 5466–5481 (2012)MathSciNetCrossRefGoogle Scholar
  7. Gedney, S.D., Zhao, B.: An auxiliary differential equation formulation for the complex-frequency shifted PML. IEEE Trans. Antennas Propag. 58(3), 838–847 (2010)ADSMathSciNetCrossRefGoogle Scholar
  8. Helfert, S., Pregla, R.: A finite difference beam propagation algorithm based on generalized transmission line equations. Opt. Quantum Electron. 32(6–8), 681–690 (2000)CrossRefGoogle Scholar
  9. Hochbruck, M., Ostermann, A.: Exponential integrators. Acta Numer. 19(May), 209–286 (2010)ADSMathSciNetCrossRefGoogle Scholar
  10. Huisinga, W., Pesce, L., Kosloff, R., Saalfrank, P.: Faber and Newton polynomial integrators for open-system density matrix propagation. J. Chem. Phys. 110(12), 5538–5547 (1999)ADSCrossRefGoogle Scholar
  11. Kleene H., Schulz D.:Concept of a complex envelope faber polynomial approach for the solution of Maxwell’s equations. In: 2018 IEEE MTT-S Int. Conf. Numer. Electromagn. Multiphysics Model. Optim., IEEE, 6, pp 1–3 (2018a)Google Scholar
  12. Kleene, H., Schulz, D.: On the evaluation of sources in highly accurate time domain simulations on the basis of faber polynomials. In: Progress in Electromagnetics Research Symposium (PIERS), p 2A03 (2018b)Google Scholar
  13. Kleene, H., Schulz, D.: Time domain solution of Maxwell’s equations using Faber polynomials. IEEE Trans. Antennas Propag. 66(11), 6202–6208 (2018c)ADSMathSciNetCrossRefGoogle Scholar
  14. Ma, F.: Slowly varying envelope simulation of optical waves in time domain with transparent and absorbing boundary conditions. J. Light Technol. 15(10), 1974–1985 (1997)ADSCrossRefGoogle Scholar
  15. Ma, C., Chen, Z.: Stability and numerical dispersion analysis of CE-FDTD method. IEEE Trans. Antennas Propag. 53(1), 332–338 (2005)ADSMathSciNetCrossRefGoogle Scholar
  16. Namiki, T.: A new FDTD algorithm based on alternating-direction implicit method. IEEE Trans. Microw. Theory Technol. 47(10), 2003–2007 (1999)ADSCrossRefGoogle Scholar
  17. Novati, P.: Solving linear initial value problems by Faber polynomials. Numer. Linear Algebra Appl. 10(3), 247–270 (2003)MathSciNetCrossRefGoogle Scholar
  18. Pursel, J., Goggans, P.: A finite-difference time-domain method for solving electromagnetic problems with bandpass-limited sources. IEEE Trans. Antennas Propag. 47(1), 9–15 (1999)ADSMathSciNetCrossRefGoogle Scholar
  19. Taflove, A., Hagness, S.C.: Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd edn. Artech House, Norwood (2005)zbMATHGoogle Scholar
  20. Taflove, A., Oskooi, A., Johnson, S.G.: Advances in FDTD Computational Electrodynamics: Photonics and Nanotechnology, 1st edn. Artech House, Norwood (2013)Google Scholar
  21. Yee, K.: Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media. IEEE Trans. Antennas Propag. 14(3), 302–307 (1966)ADSCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Technische Universität DortmundDortmundGermany

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