Simulating 3D periodic structures at oblique incidences with discontinuous Galerkin time-domain methods: theoretical and practical considerations
The SMAI journal of computational mathematics, Volume 5 (2019) , pp. 131-159.

In this work, we focus on the development of the use of Periodic Boundary Conditions (PBC) with sources at oblique incidence in a nanophotonics context. In particular, we concentrate on the field transform technique used for time dependent electromagnetic wave propagation problems. We especially supplement the existing references with an analysis of the continuous model equations. Furthermore, we propose to use a Discontinuous Galerkin Time Domain (DGTD) discrete framework and study stability issues. In order to consider realistic test cases, we also provide additional details about sources, observables (reflectance, transmittance and diffraction efficiency), and the use of Complex Frequency-Shifted Perfectly-Matched Layers (CFS-PMLs). Finally, after academic numerical validations, two engineering relevant test cases are considered in the precise physical context of nanophotonics with the Diogenes DGTD solver (

Published online:
Classification: 35Q61,  65M60,  65M12
Keywords: computational electromagnetics, time-domain Maxwell equations, discontinuous Galerkin method, periodic structures, oblique incidence sources, nanophotonics
     author = {Jonathan Viquerat and Nikolai Schmitt and Claire Scheid},
     title = {Simulating {3D} periodic structures at oblique incidences with discontinuous {Galerkin} time-domain methods: theoretical and practical considerations},
     journal = {The SMAI journal of computational mathematics},
     pages = {131--159},
     publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
     volume = {5},
     year = {2019},
     doi = {10.5802/smai-jcm.45},
     language = {en},
     url = {}
Jonathan Viquerat; Nikolai Schmitt; Claire Scheid. Simulating 3D periodic structures at oblique incidences with discontinuous Galerkin time-domain methods: theoretical and practical considerations. The SMAI journal of computational mathematics, Volume 5 (2019) , pp. 131-159. doi : 10.5802/smai-jcm.45.

[1] F. Assous; Jr Ciarlet P.; S. Labrunie Mathematical Foundations of Computational Electromagnetism, Springer, 2018 | Article | Zbl 06811152

[2] M. H. Carpenter; C. A. Kennedy Fourth-order 2N-storage Runge-Kutta schemes (1994) (Technical report)

[3] M. A. Green Self-consistent optical parameters of intrinsic silicon at 300 K including temperature coefficients, Solar Energy Materials and Solar Cells, Volume 92 (2008), pp. 1305-1310 | Article

[4] H. Hotter; H. Steyskal Broadband FDTD analysis of infinite phased arrays using periodic boundary conditions, Electronics Letters, Volume 35 (1999), p. 758-759

[5] T. Kokkinos; C. D. Sarris; G. V. Eleftheriades Periodic Finite-Difference Time-Domain Analysis of Loaded Transmission-Line Negative-Refractive-Index Metamaterials, IEEE Transactions on Microwave Theory and Techniques, Volume 53 (2005), pp. 1488-1495 | Article

[6] M. König Discontinuous Galerkin Methods in Nanophotonics (2011) (Ph. D. Thesis)

[7] M. Kuzuoglu; R. Mittra Frequency dependence of the constitutive parameters of causal perfectly matched layers, IEEE Microwave and Guided Wave Letters, Volume 6 (1996), pp. 447-449 | Article

[8] S. Lanteri; C. Scheid; J. Viquerat Analysis of a Generalized Dispersive Model coupled to a DGTD method with application to Nanophotonics, SIAM J. Sci. Comput., Volume 39 (2017), A831–A859 pages | Article | MR 3654126 | Zbl 1368.78140

[9] K.-L. Lee; P.-K. Wei Surface Plasmon Resonance Sensing: Periodic metallic nanostructures for high-sensitivity biosensing applications, IEEE Nanotechnology Magazine, Volume 10 (2016), pp. 16-23 | Article

[10] R. T. Lee; G. S. Smith A Conceptually Simple Method for Incorporating Periodic Boundary Conditions Into the FDTD Method, Antennas and Propagation Society International Symposium (2005), pp. 114-117

[11] N. C. Miller; A. D. Baczewski; J. D. Albrecht; B. Shanker A Discontinuous Galerkin Time Domain Framework for Periodic Structures Subject to Oblique Excitation, IEEE Transactions on Antennas and Propagation, Volume 62 (2014), pp. 4386-4391 | Article | Zbl 1371.78315

[12] C. Palmer Diffraction Grating Handbook, Newport Corporation, 2005

[13] C.-S. Park; V. R. Shrestha; W. Yue; S. Gao; S.-S. Lee; E.-S. Kim; D.-Y. Choi Structural Color Filters Enabled by a Dielectric Metasurface Incorporating Hydrogenated Amorphous Silicon Nanodisks, Nature Scientific Reports, Volume 7 (2017), 2556 pages

[14] J. A. Roden; S. D. Gedney; M. P. Kesler; J. G. Maloney; P. H. Harms Time-Domain Analysis of Periodic Structures at Oblique Incidence: Orthogonal and Nonorthogonal FDTD Implementations, IEEE Transactions on Microwave Theory and Techniques, Volume 46 (1998), pp. 420-427 | Article

[15] J. S. Shang Characteristic Based Methods for the Time-Domain Maxwell Equations, 1993 (29th Aerospace Sciences Meeting,

[16] A. Taflove; S. Hagness Computational Electrodynamics: The Finite-Difference Time- Domain Method, Artech House, Boston, 2005

[17] I. Valuev; A. Deinega; S. Belousov Implementation of the iterative finite-difference time-domain technique for simulation of periodic structures at oblique incidence, Computer Physics Communications, Volume 185 (2014), pp. 1273-1281 | Article | Zbl 1344.65084

[18] J. Viquerat Simulation of electromagnetic waves propagation in nano-optics with a high-order discontinuous Galerkin time-domain method (2015) (Ph. D. Thesis)

[19] Y. Zeng Modeling and design of near-field antennas with periodic structures (2017) (Ph. D. Thesis)

[20] L. Zou; W. Withayachumnankul; C. Shah; A. Mitchell; M. Bhaskaran; S. Sriram; C. Fumeaux Dielectric resonator nanoantennas at visible frequencies, Optics Express, Volume 21 (2013), pp. 1344-1352 | Article