Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A general framework for the study of electrostatic point charges in multilayer planar structures

George Fikioris, Theodoros T. Koutserimpas, Elias N. Glytsis

Abstract

We develop a general framework for the electrostatic analysis of point charges in multilayer planar structures with arbitrary layer thicknesses and material parameters. Starting from a Hankel-transform analysis, we derive alternative representations of the solution and establish a Stokes-like formulation based on ``generalized reflection coefficients,'' yielding a systematic and physically transparent treatment of multilayer media. This approach extends classical image theory to parameter regimes in which the conventional image-charge series (which has an infinite number of terms) diverges. The formulation applies to arbitrary permittivity values, including negative permittivities, where overscreening effects and plasmon-resonant conditions may occur. In these regimes, we show that the boundary-value problem no longer has a unique solution because homogeneous (source-free) modes appear; and we derive Cauchy-principal-value integral representations for the particular solution. We also introduce an asymptotic ``phantom-image'' method that replaces a divergent infinite image series by a finite set of effective sources, thus providing a computationally efficient approximation in large-reflection regimes. These results furnish both practical computational tools and additional mathematical insight into the structure of electrostatic image theory in layered media.

A general framework for the study of electrostatic point charges in multilayer planar structures

Abstract

We develop a general framework for the electrostatic analysis of point charges in multilayer planar structures with arbitrary layer thicknesses and material parameters. Starting from a Hankel-transform analysis, we derive alternative representations of the solution and establish a Stokes-like formulation based on ``generalized reflection coefficients,'' yielding a systematic and physically transparent treatment of multilayer media. This approach extends classical image theory to parameter regimes in which the conventional image-charge series (which has an infinite number of terms) diverges. The formulation applies to arbitrary permittivity values, including negative permittivities, where overscreening effects and plasmon-resonant conditions may occur. In these regimes, we show that the boundary-value problem no longer has a unique solution because homogeneous (source-free) modes appear; and we derive Cauchy-principal-value integral representations for the particular solution. We also introduce an asymptotic ``phantom-image'' method that replaces a divergent infinite image series by a finite set of effective sources, thus providing a computationally efficient approximation in large-reflection regimes. These results furnish both practical computational tools and additional mathematical insight into the structure of electrostatic image theory in layered media.

Paper Structure

This paper contains 19 sections, 90 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Hankel-Transform Solution for Multilayer Structures
  3. Stokes-like generalized reflection coefficients
  4. Two Layers ($N=1$); Meaning of Reflection and Transmission Coefficients
  5. Three Layers ($N=2$)
  6. Application of algorithm; the $\Psi$-function
  7. Validity of integral solutions; non-uniqueness when $R>1$
  8. The homogeneous solution explicitly
  9. Two Layers above a Ground
  10. On-axis values and Lerch's transcedent
  11. Image-series solution; $R$-regions
  12. Phantom images
  13. Finte sum of phantom images
  14. Choice of $M$; numerical accuracy
  15. Homogeneous solution for more layers
  16. ...and 4 more sections

Figures (7)

  • Figure 1: Multilayer planar structure with electric charge $q$ located at $z = {d_q}$ in the top layer. The top and bottom layers, which are numbered 1 and $N+1$, extend to infinity
  • Figure 2: Two-layer problem ($N=1$), with origin placed at the boundary
  • Figure 3: Three-layer problem ($N=2$), with origin placed at the 2-3 boundary
  • Figure 4: The solution $V^R$ in Section V.A holds for $R$ on the half line of the real axis $R<1$. In Section V.B, we analytically continue this solution into the unphysical region $\mathbb{C\setminus R}$; and then, for any $R_0$ on the cut, linearly combine the solutions $V^{R_0+i0}$ and $V^{R_0-i0}$ to obtain a one-parameter family of solutions to the boundary-value problem $\mathrm{BVP}^{R_0}$.
  • Figure 5: Two layers above a ground plane
  • ...and 2 more figures