Exorcizing ghost waves

TL;DR

This paper clarifies that electromagnetic ghost waves are actually nonuniform plane waves existing in isotropic dielectric materials, emphasizing their known association with propagating and evanescent fields.

Contribution

It provides a clear explanation that ghost waves are not new phenomena but are well-understood nonuniform plane waves in dielectric media.

Findings

Ghost waves are nonuniform plane waves.

They are associated with both propagating and evanescent fields.

This understanding applies to isotropic dielectric materials.

Abstract

The so-called electromagnetic ghost waves are simply electromagnetic nonuniform plane waves, whose association with both propagating and evanescent fields has long been known, even for isotropic dielectric materials that are non-dissipative.

Equations

$$\underline{k}={\underline{k}}_r+i{\underline{k}}_i,$$

$$\left.\begin{array}[]{l}\underline{E}(\underline{r})=\underline{E}_{\text{o}}\exp\left(i\underline{k}_{r}\mbox{ \tiny{${}^{\bullet}$} }\underline{r}\right)\exp\left(-\underline{k}_{i}\mbox{ \tiny{${}^{\bullet}$} }\underline{r}\right)\vspace{8pt}\\ \underline{H}(\underline{r})=\underline{H}_{\text{o}}\exp\left(i\underline{k}_{r}\mbox{ \tiny{${}^{\bullet}$} }\underline{r}\right)\exp\left(-\underline{k}_{i}\mbox{ \tiny{${}^{\bullet}$} }\underline{r}\right)\end{array}\right\}$$

$$\left.\begin{array}[]{l}\underline{k}\times\underline{E}_{\text{o}}=\omega\mu_{\scriptscriptstyle 0}\underline{H}_{\text{o}}\vspace{8pt}\\ \underline{k}\times\underline{H}_{\text{o}}=-\omega\varepsilon_{\scriptscriptstyle 0}\underline{\underline{\varepsilon}}_{\cal A}\mbox{ \tiny{${}^{\bullet}$} }\underline{E}_{\text{o}}\end{array}\right\},$$

$$k_z=\sqrt{k_0^2{\epsilon }_{\mathcal{A}}-k_x^2-k_y^2},$$

$${\underline{\underline{\epsilon }}}_{\mathcal{A}}={\epsilon }_{\mathcal{A}}^t\underline{\underline{I}}+\left({\epsilon }_{\mathcal{A}}^z-{\epsilon }_{\mathcal{A}}^t\right){\hat{\underline{u}}}_z{\hat{\underline{u}}}_z,$$

$$k_{z}=\left\{\begin{array}[]{l}\sqrt{k_{\scriptscriptstyle 0}^{2}\varepsilon_{\cal A}^{t}-k^{2}_{x}-k^{2}_{y}}\vspace{8pt}\\ \sqrt{\varepsilon_{\cal A}^{t}\left(\displaystyle{k_{\scriptscriptstyle 0}^{2}-\frac{k^{2}_{x}+k^{2}_{y}}{\varepsilon^{z}_{\cal A}}}\right)}\end{array}\right..$$

$${\underline{\underline{\epsilon }}}_{\mathcal{A}}={\epsilon }_{\mathcal{A}}^x{\hat{\underline{u}}}_x{\hat{\underline{u}}}_x+{\epsilon }_{\mathcal{A}}^y{\hat{\underline{u}}}_y{\hat{\underline{u}}}_y+{\epsilon }_{\mathcal{A}}^z{\hat{\underline{u}}}_z{\hat{\underline{u}}}_z.$$

$$k_{z}=\left\{\begin{array}[]{l}\displaystyle{\sqrt{\frac{-\beta\pm\sqrt{\beta^{2}-4\varepsilon^{z}_{\cal A}\gamma}}{2\varepsilon^{z}_{\cal A}}}}\\[12.0pt] \displaystyle{-\sqrt{\frac{-\beta\pm\sqrt{\beta^{2}-4\varepsilon^{z}_{\cal A}\gamma}}{2\varepsilon^{z}_{\cal A}}}}\end{array}\right.\,,$$

$$\left.\begin{array}[]{l}\beta=\displaystyle{k^{2}_{x}\left(\varepsilon^{x}_{\cal A}+\varepsilon^{z}_{\cal A}\right)+k^{2}_{y}\left(\varepsilon^{y}_{\cal A}+\varepsilon^{z}_{\cal A}\right)-k_{\scriptscriptstyle 0}^{2}\left(\varepsilon^{x}_{\cal A}+\varepsilon^{y}_{\cal A}\right)\varepsilon^{z}_{\cal A}\vspace{8pt}}\\ \gamma=\displaystyle{\left(k^{2}_{x}+k^{2}_{y}-k_{\scriptscriptstyle 0}^{2}\varepsilon^{z}_{\cal A}\right)\left(k^{2}_{x}\varepsilon^{x}_{\cal A}+k^{2}_{y}\varepsilon^{y}_{\cal A}-k_{\scriptscriptstyle 0}^{2}\varepsilon^{x}_{\cal A}\varepsilon^{y}_{\cal A}\right)}\end{array}\right\}.$$

Full text

**Exorcizing ghost waves **

Tom G. Mackay***E–mail: T.Mackay@ed.ac.uk.

*School of Mathematics and Maxwell Institute for Mathematical Sciences

University of Edinburgh, Edinburgh EH9 3FD, UK

*and

*NanoMM — Nanoengineered Metamaterials Group

Department of Engineering Science and Mechanics

Pennsylvania State University, University Park, PA 16802–6812, USA

Akhlesh Lakhtakia†††E–mail: akhlesh@psu.edu

*NanoMM — Nanoengineered Metamaterials Group

Department of Engineering Science and Mechanics

Pennsylvania State University, University Park, PA 16802–6812, USA*

Abstract

The so-called electromagnetic ghost waves are simply electromagnetic nonuniform plane waves, whose association with both propagating and evanescent fields has long been known, even for isotropic dielectric materials that are non-dissipative.

Keywords: nonuniform plane waves, ghost waves, surface waves, propagating, evanescent

In recent reports [1, 2, 3], the notion of “electromagnetic ghost waves” has been presented. These are claimed to represent a new form of planewave propagation, supported by non-dissipative (and non-active) biaxial dielectric materials. The key characteristic of these waves is that the Cartesian components of their wavevector are complex valued, despite the permittivity dyadic of the biaxial material being positive definite. Accordingly, these waves are associated with both propagating and evanescent fields.

In fact, the ghost waves described in Refs. 1, 2, 3 are simply nonuniform plane waves. The defining characteristic of a nonuniform plane wave is that the planes of constant phase do not coincide with the planes of constant amplitude (as they do in the case of a uniform plane wave). The existence of nonuniform plane waves is well established for both non-dissipative and dissipative materials, including isotropic dielectric materials, as is described in textbooks [4, 5].

The wavevector $\underline{k}$ of a nonuniform plane wave may be decomposed as

[TABLE]

with $\underline{k}_{r},\underline{k}_{i}\in\mathbb{R}^{3}$ and $\underline{k}_{r}\times\underline{k}_{i}\neq\underline{0}$. Thus, $\underline{k}$ has complex-valued components, even for non-dissipative materials. The corresponding electric and magnetic field phasors, written as

[TABLE]

with amplitudes $\underline{E}_{\text{o}},\underline{H}_{\text{o}}\in\mathbb{C}^{3}$, have a propagating aspect determined by $\underline{k}_{r}$ and an evanscent aspect determined by $\underline{k}_{i}$.

Three straightforward examples are presented here of nonuniform planewave propagation in dielectric materials. The following boundary-value problem is envisaged: A half-space $z>0$ is filled with a generally anisotropic dielectric material, characterized by the relative permittivity dyadic $\underline{\underline{\varepsilon}}_{\cal A}$. The half-space $z<0$ is filled with an isotropic dielectric material, characterized by the relative permittivity scalar $\varepsilon_{\cal B}$. The materials in both half-spaces are considered to be non-dissipative. A uniform plane wave is launched from a distant source in half-space $z<0$, which propagates towards the interface $z=0$. Since the material occupying half-space $z<0$ is non-dissipative and the plane wave is uniform, the Cartesian components of the wavevector in $z<0$ are real valued. Our attention is focused on the corresponding nonuniform planewave(s) in the half-space $z>0$, with wavevector(s) $\underline{k}=k_{x}\hat{\underline{u}}_{x}+k_{y}\hat{\underline{u}}_{y}+k_{z}\hat{\underline{u}}_{z}$. By invoking standard boundary conditions [4], it is inferred that $k_{x},k_{y}\in\mathbb{R}$. The nonuniform plane wave(s) in $z>0$ can have both propagating and evanescent aspects, as we now demonstrate.

The source-free Maxwell curl postulates in the half-space $z>0$ yield

[TABLE]

with $\varepsilon_{\scriptscriptstyle 0}$ and $\mu_{\scriptscriptstyle 0}$ being the permittivity and permeability of free space, and $\omega$ being the angular frequency. The wavevectors $\underline{k}$ are deduced by solving the dispersion relation that follows from seeking non-trivial solutions to Eqs. (3) for a specific relative permittivity dyadic $\underline{\underline{\varepsilon}}_{\cal A}$.

First, we consider the case of nonuniform planewave propagation in an isotropic dielectric material characterized by the relative permittivity $\varepsilon_{\cal A}$, i.e., $\underline{\underline{\varepsilon}}_{\cal A}=\varepsilon_{\cal A}\underline{\underline{I}}$. The corresponding dispersion relation yields [4]

[TABLE]

wherein $k_{\scriptscriptstyle 0}=\omega\sqrt{\varepsilon_{\scriptscriptstyle 0}\mu_{\scriptscriptstyle 0}}$ is the free-space wavenumber. Thus, $k_{z}$ is purely imaginary provided that $k^{2}_{x}+k^{2}_{y}>k_{\scriptscriptstyle 0}^{2}\varepsilon_{\cal A}$; otherwise $k_{z}$ is purely real valued. 2. 2.

Second, we consider nonuniform planewave propagation in a uniaxial dielectric material with relative permittivity dyadic [5]

[TABLE]

whose optic axis is directed along the unit vector $\hat{\underline{u}}_{z}$, and we assume that $\varepsilon_{\cal A}^{t},\varepsilon_{\cal A}^{z}>0$. The corresponding dispersion relation yields the following two solutions [5]:

[TABLE]

The upper solution holds for an ordinary plane wave and is identical in character to the solution (4) for an isotropic dielectric material. For the lower solution, which holds for an extraordinary plane wave, $k_{z}$ is purely imaginary provided that $k^{2}_{x}+k^{2}_{y}>\varepsilon^{z}_{\cal A}k_{\scriptscriptstyle 0}^{2}$; otherwise, $k_{z}$ is purely real valued. 3. 3.

Third, we consider nonuniform planewave propagation for a biaxial dielectric material with relative permittivity dyadic [5]

[TABLE]

The corresponding dispersion relation yields the following four solutions [5]:

[TABLE]

where

[TABLE]

Only two solutions from Eqs. (8) are physically acceptable, namely those two solutions yielding $\mbox{Im}\left[k_{z}\right]>0$. Unlike the isotropic dielectric and the uniaxial dielectric solutions represented by Eqs. (4) and (6), respectively, the biaxial dielectric solution (8) can deliver a complex-valued $k_{z}$ with non-zero real and imaginary parts. This is most easily illustrated numerically: For example, suppose that $k_{x}=\sqrt{2}k_{\scriptscriptstyle 0}$ and $k_{y}=k_{\scriptscriptstyle 0}$, which is possible if $\varepsilon_{\cal B}>3$. Also, suppose that $\varepsilon^{x}_{\cal A}=1.45$, $\varepsilon^{y}_{\cal A}=1.65$, and $\varepsilon^{z}_{\cal A}=1.5$. Then the two physically acceptable solutions emerging from Eqs. (8) are $k_{z}=(\pm 0.032988+1.21151i)/k_{\scriptscriptstyle 0}$. These signify nonuniform plane waves that have both propagating and evanescent aspects.

Parenthetically, characteristics of the type often associated with certain metamaterials can arise via nonuniform planewave propagation. For example, a simple homogeneous isotropic dielectric material can support nonuniform planewave propagation with negative phase velocity (and orthogonal phase velocity), without the need for a complex micro- (or nano-) structure of the type often associated with certain metamaterials [6]. A richer palette of characteristics arises when nonuniform plane waves are considered in anisotropic and bianisotropic materials [6].

Lastly, planewave solutions are used to construct surface waves guided by the planar interfaces of two dissimilar mediums. Accordingly, the so-called ghost surface waves, including ghost Dyakonov waves [1] and ghost surface phonon polariton waves [7], are simply manifestations of nonuniform surface waves.

Acknowledgments. TGM acknowledges the support of EPSRC grant EP/S00033X/1. AL thanks the Charles Godfrey Binder Endowment at the Pennsylvania State University for ongoing support of his research, with partial funding from US NSF grant number DMS-1619901.