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Finite-difference time-domain methods

Nature Reviews Methods Primers volume 3, Article number: 75 (2023) Cite this article

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Abstract

The finite-difference time-domain (FDTD) method is a widespread numerical tool for full-wave analysis of electromagnetic fields in complex media and for detailed geometries. Applications of the FDTD method cover a range of time and spatial scales, extending from subatomic to galactic lengths and from classical to quantum physics. Technology areas that benefit from the FDTD method include biomedicine — bioimaging, biophotonics, bioelectronics and biosensors; geophysics — remote sensing, communications, space weather hazards and geolocation; metamaterials — sub-wavelength focusing lenses, electromagnetic cloaks and continuously scanning leaky-wave antennas; optics — diffractive optical elements, photonic bandgap structures, photonic crystal waveguides and ring-resonator devices; plasmonics — plasmonic waveguides and antennas; and quantum applications — quantum devices and quantum radar. This Primer summarizes the main features of the FDTD method, along with key extensions that enable accurate solutions to be obtained for different research questions. Additionally, hardware considerations are discussed, plus examples of how to extract magnitude and phase data, Brillouin diagrams and scattering parameters from the output of an FDTD model. The Primer ends with a discussion of ongoing challenges and opportunities to further enhance the FDTD method for current and future applications.

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Fig. 1: Details for common finite-difference time-domain grid arrangements.
Fig. 2: Advanced finite-difference time-domain modelling approaches.
Fig. 3: Example approaches for obtaining results from finite-difference time-domain simulations.
Fig. 4: The finite-difference time-domain method applied to the biomedical field.
Fig. 5: The finite-difference time-domain method applied to electromagnetic propagation in the Earth–ionosphere waveguide.
Fig. 6: The finite-difference time-domain method applied to a metamaterial: a negative index lens.
Fig. 7: The finite-difference time-domain method applied to plasmonics.
Fig. 8: Results from a quantum finite-difference time-domain model.

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Acknowledgements

F.L.T. acknowledges support from the US Department of Energy Grant No. DE-SC0022982 through the NSF/DOE Partnership in Basic Plasma Science and Engineering. Part of the material from J.J.S. is based on work supported by the National Science Foundation under Grant No. 1662318. C.S. acknowledges support from the Natural Sciences and Engineering Research Council of Canada (NSERC) through a Discovery Grant. The authors acknowledge the help of K. Niknam in the generation of Fig. 5e using input data provided by T. Reichler.

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Authors and Affiliations

  1. Department of Electrical and Computer Engineering, The Ohio State University, Columbus, OH, USA

    F. L. Teixeira

  2. Department of Electrical and Computer Engineering, University of Toronto, Toronto, Ontario, Canada

    C. Sarris

  3. Department of Electrical and Computer Engineering, University of Utah, Salt Lake City, UT, USA

    Y. Zhang & J. J. Simpson

  4. Department of Electrical Engineering, Pohang University of Science and Technology, Pohang, Korea

    D.-Y. Na

  5. Department of Electrical and Electronic Engineering, The University of Manchester, Manchester, UK

    J.-P. Berenger

  6. Biomedical Engineering Department, Northwestern University, Evanston, IL, USA

    Y. Su & V. Backman

  7. Department of Electrical and Software Engineering, University of Calgary, Calgary, Canada

    M. Okoniewski

  8. Department of Electrical and Computer Engineering, Purdue University, West Lafayette, IN, USA

    W. C. Chew

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Contributions

Introduction (F.L.T., Y.Z. and J.J.S.); Experimentation (F.L.T., C.S., J.-P.B., M.O. and J.J.S.); Results (F.L.T., C.S. and J.J.S.); Applications (F.L.T., C.S., Y.Z., D.-Y.N., J.-P.B., Y.S., V.B., J.J.S. and W.C.C.); Reproducibility and data deposition (C.S., Y.Z. and J.J.S.); Limitations and optimizations (F.L.T., Y.Z. and J.J.S.); Outlook (C.S., Y.Z., D.-Y.N., J.J.S. and W.C.C.); Overview of the Primer (all authors).

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Correspondence to J. J. Simpson.

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Nature Reviews Methods Primers thanks Theodoros Zygiridis, Jiefu Chen, Jun Shibayama, Stephen Gedney and Zhizhang Chen for their contribution to the peer review of this work.

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Message passing interface: https://www.open-mpi.org/

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Glossary

Boundary conditions

Description of the behaviour of the solutions at certain points in space, usually along the outer edges of the finite-difference time-domain grid.

Courant–Friedrichs–Lewy (CFL) condition

A constraint that must be satisfied to achieve convergence and maintain numerical stability in a simulation.

Iterative Stencil Loops simulator

A numerical data processing solution where an array of elements is updated according to a fixed pattern called a stencil.

Maxwell’s equations

Two coupled partial differential equations that govern the propagation of electromagnetic waves.

Monte Carlo method

When applied to the finite-difference time-domain method, the Monte Carlo method involves rerunning a finite-difference time-domain simulation numerous times, often thousands or millions of times, to obtain a range of possible electric or magnetic field outcomes for an uncertain modelling scenario.

Scattering parameters

S-parameters provide a relationship between the input and output of an electrical network.

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Teixeira, F.L., Sarris, C., Zhang, Y. et al. Finite-difference time-domain methods. Nat Rev Methods Primers 3, 75 (2023). https://doi.org/10.1038/s43586-023-00257-4

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