volume | PIER Journals

Abstract

Potential-based integral equations are being explored to develop numerical methods that avoid low frequency breakdown issues and are better suited to couple to quantum physics computations. Important classes of quantum electrodynamics problems are typically formulated in the radiation gauge, leading to interest in efficient numerical solutions able to be performed directly in this gauge. This work presents time domain integral equations for penetrable regions that are developed in the radiation gauge. An appropriate marching-on-in-time discretization scheme is developed that fully conforms to the spatial and temporal Sobolev space properties of the integral equations. It is shown that following this approach leads to a discrete system with improved stability properties that produces accurate results down to very low frequencies. The accuracy and stability of this formulation at low frequencies are shown through numerical results.

1. Cohen-Tannoudji, C., J. Dupont-Roc, and G. Grynberg, Photons and Atoms: Introduction to Quantum Electrodynamics, Wiley Interscience, 1997.
doi:10.1002/9783527618422

2. Walls, D. F. and G. J. Milburn, Quantum Optics, Springer Science & Business Media, 2007.

3. Liu, A. Y. and W. C. Chew, "Dressed atom fields and dressed states in waveguide quantum electrodynamics," IEEE Journal on Multiscale and Multiphysics Computational Techniques, Vol. 2, 58-65, 2017.
doi:10.1109/JMMCT.2017.2698341

4. Rodriguez, A. W., A. P. McCauley, J. D. Joannopoulos, and S. G. Johnson, "Casimir forces in the time domain: Theory," Physical Review A, Vol. 80, No. 1, 012115, 2009.
doi:10.1103/PhysRevA.80.012115

5. Gregersen, N., P. Kaer, and J.Mørk, "Modeling and design of high-efficiency single-photon sources," IEEE Journal of Selected Topics in Quantum Electronics, Vol. 19, No. 5, 1-16, 2013.
doi:10.1109/JSTQE.2013.2255265

6. Kandala, A., A. Mezzacapo, K. Temme, M. Takita, M. Brink, J. M. Chow, and J. M. Gambetta, "Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets," Nature, Vol. 549, No. 7671, 242-246, 2017.
doi:10.1038/nature23879

7. Barends, R., J. Kelly, A. Megrant, A. Veitia, D. Sank, E. Jeffrey, T. C. White, J. Mutus, A. G. Fowler, B. Campbell, et al. "Superconducting quantum circuits at the surface code threshold for fault tolerance," Nature, Vol. 508, No. 7497, 500-503, 2014.
doi:10.1038/nature13171

8. Shanker, B., A. A. Ergin, M. Lu, and E. Michielssen, "Fast analysis of transient electromagnetic scattering phenomena using the multilevel plane wave time domain algorithm," IEEE Transactions on Antennas and Propagation, Vol. 51, No. 3, 628-641, 2003.
doi:10.1109/TAP.2003.809054

9. Yilmaz, A. E., J.-M. Jin, and E. Michielssen, "Time domain adaptive integral method for surface integral equations," IEEE Transactions on Antennas and Propagation, Vol. 52, No. 10, 2692-2708, 2004.
doi:10.1109/TAP.2004.834399

10. Chen, N.-W., K. Aygun, and E. Michielssen, "Integral-equation-based analysis of transient scattering and radiation from conducting bodies at very low frequencies," IEE Proceedings — Microwaves, Antennas and Propagation, Vol. 148, No. 6, 381-387, 2001.
doi:10.1049/ip-map:20010827

11. Cools, K., F. P. Andriulli, F. Olyslager, and E. Michielssen, "Time domain Calder´on identities and their application to the integral equation analysis of scattering by PEC objects Part I: Preconditioning," IEEE Transactions on Antennas and Propagation, Vol. 57, No. 8, 2352-2364, 2009.
doi:10.1109/TAP.2009.2024460

12. Qian, Z.-G. and W. C. Chew, "Fast full-wave surface integral equation solver for multiscale structure modeling," IEEE Transactions on Antennas and Propagation, Vol. 57, No. 11, 3594-3601, 2009.
doi:10.1109/TAP.2009.2023629

13. Taskinen, M. and P. Yla-Oijala, "Current and charge integral equation formulation," IEEE Transactions on Antennas and Propagation, Vol. 54, No. 1, 58-67, 2006.
doi:10.1109/TAP.2005.861580

14. Liu, Q. S., S. Sun, and W. C. Chew, "A potential based integral equation method for low-frequency electromagnetic problems," IEEE Transactions on Antennas and Propagation, Vol. 66, No. 3, 1413-1426, 2018.
doi:10.1109/TAP.2018.2794388

15. Li, J., X. Fu, and B. Shanker, "Decoupled potential integral equations for electromagnetic scattering from dielectric objects," IEEE Transactions on Antennas and Propagation, Vol. 67, No. 3, 1729-1739, 2018.
doi:10.1109/TAP.2018.2883636

16. Roth, T. E. and W. C. Chew, "Development of stable A-Φ time domain integral equations for multiscale electromagnetics," IEEE Journal on Multiscale and Multiphysics Computational Techniques, Vol. 3, 255-265, 2018.
doi:10.1109/JMMCT.2018.2889535

17. Roth, T. E. and W. C. Chew, "Stability analysis and discretization of A-Φ time domain integral equations for multiscale electromagnetics," Journal of Computational Physics, 109102, 2019.

18. Jackson, J. D., Classical Electrodynamics, Wiley, 1999.

19. Stratton, J. A., Electromagnetic Theory, John Wiley & Sons, 2007.

20. Tai, C.-T., "Direct integration of field equations," Progress In Electromagnetics Research, Vol. 28, 339-359, 2000.
doi:10.2528/PIER99101401

21. Jin, J.-M., Theory and Computation of Electromagnetic Fields, John Wiley & Sons, 2011.

22. Chew, W. C., Waves and Fields in Inhomogeneous Media, IEEE Press, 1995.

23. Ha-Duong, T., "On retarded potential boundary integral equations and their discretisation," Topics in Computational Wave Propagation, 301-336, Springer, 2003.
doi:10.1007/978-3-642-55483-4_8

24. Cools, K., F. Andriulli, D. De Zutter, and E. Michielssen, "Accurate and conforming mixed discretization of the MFIE," IEEE Antennas and Wireless Propagation Letters, Vol. 10, 528-531, 2011.
doi:10.1109/LAWP.2011.2155022

25. van’tWout, E., D. R. van der Heul, H. van der Ven, and C. Vuik, "Stability analysis of the marchingon-in-time boundary element method for electromagnetics," Journal of Computational and Applied Mathematics, Vol. 294, 358-371, 2016.
doi:10.1016/j.cam.2015.09.002

26. Bachelot, A., L. Bounhoure, and A. Pujols, "Couplage elements finis-potentiels retardes pour la diffraction electromagnetique par un obstacle heterogene," Numerische Mathematik, Vol. 89, No. 2, 257-306, 2001.
doi:10.1007/PL00005468

27. Roth, T. E. and W. C. Chew, "Potential-based TDIEs for dielectric regions using magnetic currents," 2019 IEEE International Symposium on Antennas and Propagation and USNC-URSI Radio Science Meeting, 1443-1444, IEEE, 2019.
doi:10.1109/APUSNCURSINRSM.2019.8889161

28. Roth, T. E. and W. C. Chew, "Initial potential-based time domain surface integral equations for dielectric regions," 2019 PhotonIcs & Electromagnetics Research Symposium — Spring (PIERS — Spring), Rome, Italy, June 17–20, 2019.

29. Buffa, A. and S. Christiansen, "A dual finite element complex on the barycentric refinement," Mathematics of Computation, Vol. 76, No. 260, 1743-1769, 2007.
doi:10.1090/S0025-5718-07-01965-5

30. Rao, S., D. Wilton, and A. Glisson, "Electromagnetic scattering by surfaces of arbitrary shape," IEEE Transactions on Antennas and Propagation, Vol. 30, No. 3, 409-418, 1982.
doi:10.1109/TAP.1982.1142818

31. Dai, Q. I., W. C. Chew, L. J. Jiang, and Y. Wu, "Differential-forms-motivated discretizations of electromagnetic differential and integral equations," IEEE Antennas and Wireless Propagation Letters, Vol. 13, 1223-1226, 2014.
doi:10.1109/LAWP.2014.2332300

32. Chen, Q. and D. Wilton, "Electromagnetic scattering by three-dimensional arbitrary complex material/conducting bodies," International Symposium on Antennas and Propagation Society, Merging Technologies for the 90's, 590-593, IEEE, 1990.
doi:10.1109/APS.1990.115179

33. Walker, S., M. Bluck, and I. Chatzis, "The stability of integral equation time-domain scattering computations for three-dimensional scattering; similarities and differences between electrodynamic and elastodynamic computations," International Journal of Numerical Modelling: Electronic Networks, Devices and Fields, Vol. 15, No. 5-6, 459-474, 2002.
doi:10.1002/jnm.473

34. Bamberger, A., T. Ha-Duong, and J. C. Nedelec, "Formulation variationnelle espace-temps pour le calcul par potentiel retard´e de la diffraction d’une onde acoustique (I)," Mathematical Methods in the Applied Sciences, Vol. 8, No. 1, 405-435, 1986.
doi:10.1002/mma.1670080127

35. Terrasse, I., Resolution mathematique et numerique des equations de Maxwell instationnaires par une methode de potentiels retardes, Ph.D. dissertation, 1993.

36. Hsiao, G. C. and R. E. Kleinman, "Mathematical foundations for error estimation in numerical solutions of integral equations in electromagnetics," IEEE Transactions on Antennas and Propagation, Vol. 45, No. 3, 316-328, 1997.
doi:10.1109/8.558648

37. Desbrun, M., E. Kanso, and Y. Tong, "Discrete differential forms for computational modeling," Discrete Differential Geometry, 287-324, Springer, 2008.
doi:10.1007/978-3-7643-8621-4_16

Mr. Thomas Edgar Roth

Professor Weng Cho Chew