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Finite-Difference Modeling of Nonlinear Phenomena in Time-Domain Electromagnetics: A Review

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Applications of Nonlinear Analysis

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 134))

Abstract

Nonlinearities are likely to emerge in a wide range of electromagnetic (EM) problems, commonly described by Maxwell’s equations, which can be encountered in several real-world applications, in areas such as optical communications, etc. The necessity for efficiently analyzing this type of problems has led to the development of suitable computational approaches, among which schemes based on the finite-difference time-domain (FDTD) method play a prominent role. Unlike other numerical methods that perform reliably only if specific approximations are valid (e.g. wave propagation along a dominant direction), FDTD-based techniques can be applied in more generalized frameworks, and are capable of computing credible outcomes without requiring very complex algorithmic implementations. In the present chapter, we report various key contributions presented over the years regarding the nonlinear FDTD analysis of EM problems, as the original algorithm is suitable for linear cases only, and describe their basic formulation, features, and range of applicability. In essence, this work aspires to provide an updated look on existing finite-difference models for nonlinear problems, and offer to those not familiar with the subject a solid starting point for studying the corresponding electromagnetic phenomena.

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Notes

  1. 1.

    In the simple—quite common—case of linear materials, it is P = 𝜖 0 χ (1) E.

  2. 2.

    The convention \(f\left ( {i\varDelta x,j\varDelta y,k\varDelta z,n\varDelta t} \right ) = \left . f \right |{ }_{i,j,k}^n\) is used in this work.

  3. 3.

    Second-harmonic generation is a phenomenon, according to which a wave within a nonlinear medium can produce another wave with twice the frequency of the former.

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Authors and Affiliations

  1. Department of Informatics and Telecommunications Engineering, University of Western Macedonia, Kozani, Greece

    Theodoros T. Zygiridis

  2. Department of Electrical and Computer Engineering, Aristotle University of Thessaloniki, Thessaloniki, Greece

    Nikolaos V. Kantartzis

Authors
  1. Theodoros T. Zygiridis
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  2. Nikolaos V. Kantartzis
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Correspondence to Theodoros T. Zygiridis .

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Editors and Affiliations

  1. Department of Mathematics, National Technical University of Athens, Athens, Greece

    Themistocles M. Rassias

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Zygiridis, T.T., Kantartzis, N.V. (2018). Finite-Difference Modeling of Nonlinear Phenomena in Time-Domain Electromagnetics: A Review. In: Rassias, T. (eds) Applications of Nonlinear Analysis . Springer Optimization and Its Applications, vol 134. Springer, Cham. https://doi.org/10.1007/978-3-319-89815-5_29

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