Performance evaluation of numerical methods for the Maxwell–Liouville–von Neumann equations

  • Michael Riesch
  • Nikola Tchipev
  • Sebastian Senninger
  • Hans-Joachim Bungartz
  • Christian Jirauschek
Article
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Part of the following topical collections:
  1. 2017 Numerical Simulation of Optoelectronic Devices

Abstract

The Maxwell–Liouville–von Neumann (MLN) equations are a valuable tool in nonlinear optics in general and to model quantum cascade lasers in particular. Several numerical methods to solve these equations with different accuracy and computational complexity have been proposed in related literature. We present an open-source framework for solving the MLN equations and parallel implementations of three numerical methods using OpenMP. The performance measurements demonstrate the efficiency of the parallelization.

Keywords

Quantum cascade lasers Maxwell–Bloch equations Liouville–von Neumann equation Parallelization 

Notes

Acknowledgements

This work was supported by the German Research Foundation (DFG) within the Heisenberg Program (JI 115/4-2) and under DFG Grant No. JI 115/9-1. Nikola Tchipev acknowledges the funding provided by Intel as part of the Intel Parallel Computing Center ExScaMIC-KNL. The authors gratefully acknowledge the Gauss Centre for Supercomputing e.V. (http://www.gauss-centre.eu) for funding this project by providing computing time on the GCS Supercomputer SuperMUC at Leibniz Supercomputing Centre (http://www.lrz.de). Finally, the authors thank Mariem Kthiri for her help in the development of the mbsolve project.

References

  1. Bidégaray, B.: Time discretizations for Maxwell–Bloch equations. Numer. Methods Partial Differ. Equ. 19(3), 284–300 (2003)MathSciNetCrossRefMATHGoogle Scholar
  2. Bidégaray, B., Bourgeade, A., Reignier, D.: Introducing physical relaxation terms in Bloch equations. J. Comput. Phys. 170(2), 603–613 (2001)MathSciNetCrossRefMATHADSGoogle Scholar
  3. Cartar, W., Mørk, J., Hughes, S.: Self-consistent Maxwell–Bloch model of quantum-dot photonic-crystal-cavity lasers. Phys. Rev. A 96(023), 859 (2017)Google Scholar
  4. Deinega, A., Seideman, T.: Self-interaction-free approaches for self-consistent solution of the Maxwell–Liouville equations. Phys. Rev. A 89(022), 501 (2014)Google Scholar
  5. Guennebaud, G., Jacob, B. et al.: Eigen v3 (2010) http://eigen.tuxfamily.org
  6. Hairer, E., Nørsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I, 2nd edn. Springer, Berlin (1993)MATHGoogle Scholar
  7. Jirauschek, C., Kubis, T.: Modeling techniques for quantum cascade lasers. Appl. Phys. Rev. 1(1), 011,307 (2014)CrossRefGoogle Scholar
  8. Krishnamoorthy, S., Baskaran, M., Bondhugula, U., Ramanujam, J., Rountev, A., Sadayappan, P.: Effective automatic parallelization of stencil computations. SIGPLAN Not. 42(6), 235–244 (2007)CrossRefGoogle Scholar
  9. Liu, Q.H.: The PSTD algorithm: a time-domain method requiring only two cells per wavelength. Microw. Opt. Technol. Lett. 15(3), 158–165 (1997)CrossRefGoogle Scholar
  10. Marskar, R., Österberg, U.: Multilevel Maxwell–Bloch simulations in inhomogeneously broadened media. Opt. Express 19(18), 16,784–16,796 (2011)CrossRefGoogle Scholar
  11. Riesch, M., Jirauschek, C.: mbsolve: an open-source solver tool for the Maxwell–Bloch equations. https://github.com/mriesch-tum/mbsolve (2017)
  12. Saut, O., Bourgeade, A.: Numerical methods for the bidimensional Maxwell–Bloch equations in nonlinear crystals. J. Comput. Phys. 213(2), 823–843 (2006)MathSciNetCrossRefMATHADSGoogle Scholar
  13. Slavcheva, G., Arnold, J.M., Wallace, I., Ziolkowski, R.W.: Coupled Maxwell-pseudospin equations for investigation of self-induced transparency effects in a degenerate three-level quantum system in two dimensions: finite-difference time-domain study. Phys. Rev. A 66(6), 63,418 (2002)CrossRefGoogle Scholar
  14. Sukharev, M., Nitzan, A.: Numerical studies of the interaction of an atomic sample with the electromagnetic field in two dimensions. Phys. Rev. A 84(043), 802 (2011)Google Scholar
  15. Taflove, A., Hagness, S. C.: Computational electrodynamics: the finite-difference time-domain method. Artech House (2005)Google Scholar
  16. Tzenov, P., Burghoff, D., Hu, Q., Jirauschek, C.: Time domain modeling of terahertz quantum cascade lasers for frequency comb generation. Opt. Express 24(20), 23,232–23,247 (2016)CrossRefGoogle Scholar
  17. Tzenov, P., Burghoff, D., Hu, Q., Jirauschek, C.: Analysis of operating regimes of terahertz quantum cascade laser frequency combs. IEEE Trans. THz Sci. Technol. 7(4), 351–359 (2017)CrossRefGoogle Scholar
  18. Wang, C.Y., Diehl, L., Gordon, A., Jirauschek, C., Kärtner, F.X., Belyanin, A., Bour, D., Corzine, S., Höfler, G., Troccoli, M., Faist, J., Capasso, F.: Coherent instabilities in a semiconductor laser with fast gain recovery. Phys. Rev. A 75(031), 802 (2007)Google Scholar
  19. 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)CrossRefMATHADSGoogle Scholar
  20. Ziolkowski, R.W., Arnold, J.M., Gogny, D.M.: Ultrafast pulse interactions with two-level atoms. Phys. Rev. A 52, 3082–3094 (1995)CrossRefADSGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Department of Electrical and Computer EngineeringTechnical University of Munich (TUM)MunichGermany
  2. 2.Department of InformaticsTechnical University of Munich (TUM)GarchingGermany

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