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Computational Photonic Sensors pp 153–175Cite as

FDTD in Cartesian and Spherical Grids

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Abstract

The numerical dispersion relation is derived for the finite-difference time-domain method when implemented on spherical grids using Maxwell’s equations in spherical coordinates. Derivation is appropriately based on elementary spherical functions which renders the resulting numerical dispersion relation valid for all spherical FDTD space including near the singular regions at the origin and along the z-axis. Accuracy of this relation is verified through convergence tests to the continuous-space limit and the Cartesian FDTD limit far from the origin. Numerical dispersion analyses are carried out to demonstrate numerical wavenumber error bounds and their dependence on absolute position as well as on spherical solutions’ modes. The chapter is concluded by visiting the existing challenges of designing absorbing boundary conditions for spherical FDTD when the grid truncation is in the near vicinity of the origin. Such a design challenge can be effectively studied in the future with the aid of the derived spherical FDTD numerical dispersion relation.

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References

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Authors and Affiliations

  1. Colorado School of Mines, Golden, CO, USA

    Mohammed Hadi & Atef Elsherbeni

  2. University of Colorado at Boulder, Boulder, CO, USA

    Ravi Bollimuntha & Melinda Piket-May

Authors
  1. Mohammed Hadi
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  2. Atef Elsherbeni
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  3. Ravi Bollimuntha
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  4. Melinda Piket-May
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Corresponding author

Correspondence to Atef Elsherbeni .

Editor information

Editors and Affiliations

  1. Center for Photonics and Smart Materials and Nanotechnology Engineering Program, Zewail City of Science and Technology, Giza, Egypt

    Mohamed Farhat O. Hameed

  2. Center for Photonics and Smart Materials, Zewail City of Science and Technology, Giza, Egypt

    Salah Obayya

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Hadi, M., Elsherbeni, A., Bollimuntha, R., Piket-May, M. (2019). FDTD in Cartesian and Spherical Grids. In: Hameed, M., Obayya, S. (eds) Computational Photonic Sensors. Springer, Cham. https://doi.org/10.1007/978-3-319-76556-3_7

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  • DOI: https://doi.org/10.1007/978-3-319-76556-3_7

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-76555-6

  • Online ISBN: 978-3-319-76556-3

  • eBook Packages: EngineeringEngineering (R0)

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