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Accuracy of the Yee FDTD Scheme for Normal Incidence of Plane Waves on Dielectric and Magnetic Interfaces

Pavel A. Makarov, Vladimir I. Shcheglov

TL;DR

The paper analyzes the accuracy of the Yee FDTD scheme for normal incidence at planar interfaces between lossless media. It derives discrete Fresnel coefficients for two common interface models by translating boundary conditions through Yee update equations and demonstrates that the staggered grid inherently spreads the interface over a transition layer of width one grid cell, producing systematic errors that depend on the discretization and Courant number. A transition-layer model is developed to quantify these errors, with rigorous estimates for both weak and strong impedance contrasts and insights into how numerical dispersion and interface discretization interact. The results establish frequency- and discretization-dependent FDTD Fresnel coefficients that converge to the exact continuous values as Δx → 0, offering practical guidance for grid design and Courant-number selection in photonics and RF simulations.

Abstract

This paper analyzes the accuracy of the standard Yee finite-difference time-domain (FDTD) scheme for simulating normal incidence of harmonic plane waves on planar interfaces between lossless, linear, homogeneous, isotropic media. We consider two common FDTD interface models based on different staggered-grid placements of material parameters. For each, we derive discrete analogs of the Fresnel reflection and transmission coefficients by formulating effective boundary conditions that emerge from the Yee update equations. A key insight is that the staggered grid implicitly spreads the material discontinuity over a transition layer of one spatial step, leading to systematic deviations from exact theory. We quantify these errors via a transition-layer model and provide (i) qualitative criteria predicting the direction and nature of deviations, and (ii) rigorous error estimates for both weak and strong impedance contrasts. Finally, we examine the role of the Courant number in modulating these errors, revealing conditions under which numerical dispersion and interface discretization jointly influence accuracy.

Accuracy of the Yee FDTD Scheme for Normal Incidence of Plane Waves on Dielectric and Magnetic Interfaces

TL;DR

The paper analyzes the accuracy of the Yee FDTD scheme for normal incidence at planar interfaces between lossless media. It derives discrete Fresnel coefficients for two common interface models by translating boundary conditions through Yee update equations and demonstrates that the staggered grid inherently spreads the interface over a transition layer of width one grid cell, producing systematic errors that depend on the discretization and Courant number. A transition-layer model is developed to quantify these errors, with rigorous estimates for both weak and strong impedance contrasts and insights into how numerical dispersion and interface discretization interact. The results establish frequency- and discretization-dependent FDTD Fresnel coefficients that converge to the exact continuous values as Δx → 0, offering practical guidance for grid design and Courant-number selection in photonics and RF simulations.

Abstract

This paper analyzes the accuracy of the standard Yee finite-difference time-domain (FDTD) scheme for simulating normal incidence of harmonic plane waves on planar interfaces between lossless, linear, homogeneous, isotropic media. We consider two common FDTD interface models based on different staggered-grid placements of material parameters. For each, we derive discrete analogs of the Fresnel reflection and transmission coefficients by formulating effective boundary conditions that emerge from the Yee update equations. A key insight is that the staggered grid implicitly spreads the material discontinuity over a transition layer of one spatial step, leading to systematic deviations from exact theory. We quantify these errors via a transition-layer model and provide (i) qualitative criteria predicting the direction and nature of deviations, and (ii) rigorous error estimates for both weak and strong impedance contrasts. Finally, we examine the role of the Courant number in modulating these errors, revealing conditions under which numerical dispersion and interface discretization jointly influence accuracy.

Paper Structure

This paper contains 26 sections, 13 theorems, 70 equations, 11 figures, 1 table.

Table of Contents

  1. Introduction
  2. Short review of problem
  3. Geometry of the Problem and Electrodynamics Foundations
  4. Problem Geometry
  5. Governing Equations
  6. Reflection and Transmission at the Interface
  7. Boundary Conditions and Exact Solution
  8. Space-time discretization and finite-differences approach
  9. Yee Algorithm
  10. Generation of the Incident Wave Field
  11. FDTD Analogs of Incident, Reflected and Transmitted Wave Fields
  12. FDTD Dispersion
  13. FDTD boundary conditions
  14. Necessary definitions and lemmas
  15. Dielectrics
  16. ...and 11 more sections

Key Result

Lemma 1

The dispersion relation for the Yee grid can be represented in the form

Figures (11)

  • Figure 1: Orientation of electromagnetic field components in incident, reflected and transmitted waves
  • Figure 2: Yee grid
  • Figure 3: FDTD-model of a planar interface between two dielectrics
  • Figure 4: FDTD-model of a planar interface between two magnetics
  • Figure 5: FDTD Fresnel coefficients for reflection $\widetilde{r}$ (two images in the upper row) and transmission $\widetilde{t}$ (lower row) in the case of the interface between two dielectrics (left column) and magnetics (right column) at slightly different impedances of the two media $\eta_1/\eta_2=\sqrt{4/3}\approx1.16$ and $\eta_1/\eta_2=\sqrt{3/4}\approx0.87$ (the specific values $\varepsilon_\mathrm{r}$ and $\mu_\mathrm{r}$ at which the curves are plotted are shown in the corresponding parts of the figure)
  • ...and 6 more figures

Theorems & Definitions (40)

  • Remark 1
  • Remark 2
  • Remark 3
  • Lemma 1: see Statement 1 in Makarov2024a
  • proof
  • Remark 4
  • Definition 1
  • Example 1
  • Definition 2
  • Lemma 2: see Schneider
  • ...and 30 more