Surface electromagnetic waves in lossy conductive media: tutorial

Igor I. Smolyaninov; Igor I. Smolyaninov

doi:10.1364/JOSAB.463692

1. INTRODUCTION

From the macroscopic electrodynamics point of view, all non-magnetic materials can be divided into two broad classes depending on their frequency-dependent conductivity (or the imaginary part of their dielectric permittivity ${\varepsilon }$). One class contains a relatively small number of transparent dielectric (or non-conductive) media, which transmit electromagnetic waves, and another class encompasses the vast majority of non-transparent (conductive) media, which are commonly believed to disallow electromagnetic wave propagation below their plasma frequency. It is typically assumed that electromagnetic waves quickly decay inside lossy conductive media over short length scales of the order of so-called skin depth. The goal of this tutorial is to prove that this common belief may be incorrect in some special situations in which the conductive medium is stratified.

It is well established in the scientific literature that sharp interfaces separating media having different electromagnetic properties [see Fig. 1(a)] may support low-loss propagation of surface electromagnetic waves. The most well-known examples of such surface waves include surface plasmon polaritons (SPPs), which propagate along a sharp interface between a good metal and a dielectric [1], and the so-called Zenneck surface wave [2], which may exist at an interface between a highly lossy conductive medium (such as graphite or seawater) and a good dielectric (such as air). The wave vector $k$ of these surface waves along the interface is defined as [1,2]

(1)$$k = \frac{\omega}{c}{\left({\frac{{{\varepsilon _1}{\varepsilon _2}}}{{{\varepsilon _1} + {\varepsilon _2}}}} \right)^{1/2}},$$

where ${\varepsilon _1}$ and ${\varepsilon _2}$ are the dielectric permittivities of the adjacent media. Besides the trivial case when both ${\varepsilon _1}$ and ${\varepsilon _2}$ are mostly real and positive, Eq. (1) may produce an almost purely real answer if ${\rm{Re}}({{\varepsilon _1}})$ and ${\rm{Re}}({{\varepsilon _2}})$ have opposite signs, ${\rm{Re}}({{\varepsilon _1} + {\varepsilon _2}}) \lt 0$, and the imaginary parts of both dielectric permittivities are small. This case corresponds to a SPP wave at a metal–dielectric interface [note that in the $({{\varepsilon _1} + {\varepsilon _2}}) \to 0$ limit, the $k$ vector of the wave may become very large, which is responsible for nanophotonics applications of SPPs]. Another possibility may be the case when ${\rm{Im}}({{\varepsilon _1}}) \gg {\rm{Re}}({{\varepsilon _2}})$, which corresponds to the Zenneck wave at an interface between a highly lossy conductive medium and a good dielectric. However, in the latter case, the resulting wave vector appears to be smaller than the wave vector of regular photons in the dielectric, which leads to the “leaky” character of this surface wave: while it should be able to propagate over a perfectly smooth interface, surface imperfections must strongly scatter the Zenneck waves into photons propagating inside the dielectric.

Fig. 1. Geometries of the problems of interest. The dielectric permittivity of the medium depends only on the $z$ coordinate, which is illustrated by the halftones: (a) usually considered step-like distribution of $\varepsilon(z)$. (b) Gradual interface between two media. The transition layer thickness equals $\xi$.

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The goal of this tutorial is to consider a more general situation in which the dielectric permittivity of a medium changes gradually across some real or imaginary planar surface, as shown in Fig. 1(b). I will demonstrate that a low-loss propagating electromagnetic wave may be sent along such a planar surface inside a lossy medium, even if the imaginary part of medium permittivity remains very high on both sides of the surface. This new surface wave solution of the macroscopic Maxwell equations appears when the interface between two media is no longer considered to be abrupt. This surprising result is applicable to any portion of the electromagnetic spectrum, from extremely low radio frequencies (ELFs) up to the visible and UV ranges.

2. GENERAL THEORETICAL FRAMEWORK

Let us consider solutions of the macroscopic Maxwell equations in a geometry in which the medium is non-magnetic ($B = H$), the dielectric permittivity of a medium is continuous, and it depends only on the $z$ coordinate: $\varepsilon = \varepsilon (z)$, as illustrated in Fig. 1(b). Under such conditions, the spatial variables in the Maxwell equations separate, and without loss of generality, we may assume electromagnetic mode propagation in the $x$ direction, leading to field dependencies proportional to ${e^{i({kx - \omega t})}}$. The macroscopic Maxwell equations may be written as

(2)$$\begin{split}&\vec \nabla \cdot \vec D = 0,\,\vec \nabla \cdot \vec B = 0,\,\vec \nabla \times \vec E = i\frac{\omega}{c}\vec B,\quad{\rm{and}}\\& \vec \nabla \times \vec B = - i\frac{{\omega \varepsilon}}{c}\vec E,\end{split}$$

leading to the wave equation

(3)$$\vec \nabla \times \big({\vec \nabla \times \vec E} \big) = \frac{{{\omega ^2}\varepsilon}}{{{c^2}}}\vec E.$$

Since

(4)$$\vec \nabla \times \big({\vec \nabla \times \vec E} \big) = - {\nabla ^2}\vec E + \vec \nabla \big({\vec \nabla \cdot \vec E} \big)$$

and

(5)$$\vec \nabla \cdot \vec E = - {E_z}\frac{{\partial \varepsilon /\partial z}}{\varepsilon},$$

after straightforward transformations, we obtain

(6)$${-}{\nabla ^2}\vec E - \vec \nabla \left({{E_z}\frac{{\partial \varepsilon}}{{\varepsilon \partial z}}} \right) = \frac{{\varepsilon {\omega ^2}}}{{{c^2}}}\vec E.$$

For the ${E_z} = 0$ (TE) polarization, we obtain an effective Schrödinger equation:

(7)$${-}\frac{{{\partial ^2}{E_y}}}{{\partial {z^2}}} - \frac{{\varepsilon (z){\omega ^2}}}{{{c^2}}}{E_y} = - {k^2}{E_y},$$

while for the ${E_z} \ne 0$ (TM) polarization, the effective Schrödinger equation is

(8)$${-}\frac{{{\partial ^2}{E_z}}}{{\partial {z^2}}} - \frac{{\partial {E_z}}}{{\partial z}}\frac{{\partial \ln \varepsilon}}{{\partial z}} - \left({\frac{{\varepsilon (z){\omega ^2}}}{{{c^2}}} + \frac{{{\partial ^2}\ln \varepsilon}}{{\partial {z^2}}}} \right){E_z} = - {k^2}{E_z}.$$

In the latter equation, the wave function ψ may be introduced as ${E_z} = \psi /{\varepsilon ^{1/2}}$, leading to

(9)$$\begin{split}&{-}\frac{{{\partial ^2}\psi}}{{\partial {z^2}}} + \left({- \frac{{\varepsilon (z){\omega ^2}}}{{{c^2}}} - \frac{1}{2}\frac{{{\partial ^2}\varepsilon}}{{\varepsilon \partial {z^2}}} + \frac{3}{4}{{\frac{{\big({\partial \varepsilon /\partial z} \big)}}{{{\varepsilon ^2}}}}^2}} \right)\psi \\&\quad= - \frac{{{\partial ^2}\psi}}{{\partial {z^2}}} + V\psi = - {k^2}\psi .\end{split}$$

For both polarizations, ${-}{k^2}$ plays the role of effective energy in the corresponding Schrödinger equations. Let us study solutions of Eqs. (7) and (9), which have a propagating wave character $({{\rm{Im}}(k) \ll {\rm{Re}}(k)})$.

In the case of TE polarized light [see Eq. (7)], the effective potential energy is $V(z) = - \frac{{\varepsilon (z){\omega ^2}}}{{{c^2}}}$, and there are no surface wave solutions. Equation (7) admits only propagating solutions described by planar waveguide-like distributions of $\varepsilon (z)$, in which the dielectric permittivity is positive and almost purely real. On the other hand, the TM polarized solutions of Eq. (9) can be much more interesting.

First, let us consider the SPP-like surface electromagnetic wave solutions of Eq. (9). Let us note that by making the ${E_z} = {\psi /\varepsilon^{1/2}}$ substitution in transition from Eq. (8) to Eq. (9), and assuming continuity of $\psi$ (as it is typically done while solving Schrödinger equations), we lose the conventional SPP solution of Eq. (8) at an abrupt interface between an ideal metal and a lossless dielectric. Indeed, since ${D_z}$ must be continuous at such an interface, ${D_z} = {\varepsilon ^{1/2}}\psi$ cannot be continuous and non-zero while $\varepsilon (z)$ remains purely real, and while it abruptly changes sign from positive to negative (if the continuity of $\psi$ is assumed at the same time). In such a case, the $\psi = 0$ condition must be enforced, which leads to losing the SPP solution of Eq. (8). Nevertheless, it is easy to demonstrate that Eq. (9) remains quite general and reliable enough, since it allows the recovery of SPP-like solutions in the case of a gradual interface between a real (non-ideal) metal and a dielectric [see Fig. 1(b)]. In such a case, the effective potential energy $V(z)$ in Eq. (9) is dominated by the $\frac{{3{{({\partial \varepsilon /\partial z})}^2}}}{{4{\varepsilon ^2}}}$ term. This term becomes negative and strongly attractive whenever ${\rm{Re}}({\varepsilon (z)})$ passes through zero:

(10)$$V(z) \approx \frac{{3{{\left({\partial \varepsilon /\partial z} \right)}^2}}}{{4{\varepsilon ^2}}} \approx - \frac{3}{4}\frac{{{{\left({\partial {\mathop{\rm Re}\nolimits} (\varepsilon)/\partial z} \right)}^2}}}{{{\mathop{\rm Im}\nolimits} {{(\varepsilon)}^2}}}.$$

The deep potential well at the interface, described by Eq. (10), leads to the appearance of SPP-like solutions propagating along the metal–dielectric interface and having large and almost purely real $k$ vectors along the interface. Note that the Zenneck surface wave at a gradual interface between a highly lossy conductive medium and a good dielectric also arises due to the surface effective potential well described by Eq. (10).

Having recovered the conventional plasmonics-based nanophotonics within the scope of our newly developed gradient-index nanophotonics formalism, let us study what kind of other TM surface wave solutions we may obtain from Eq. (9). The effective potential energy near a gradual interface between two media shown in Fig. 1(b) may be re-written as

(11)$$V(z) = - \frac{{4{\pi ^2}\varepsilon (z)}}{{\lambda _0^2}} - \frac{1}{2}\frac{{{\partial ^2}\varepsilon}}{{\varepsilon \partial {z^2}}} + \frac{3}{4}{\frac{{\left({\partial \varepsilon /\partial z} \right)}}{{{\varepsilon ^2}}}^2},$$

where ${\lambda _0}$ is the free space wavelength. If $\varepsilon (z)$ of the medium changes on the spatial scale $\xi$, and this spatial scale is much smaller than ${\lambda _0}$, the second and third terms will dominate in Eq. (11). Moreover, if these terms are engineered (by either nanofabrication or suitable material choice) in such a way that ${\rm{Im}}(V) \ll {\rm{Re}}(V)$, and the resulting potential well is deep enough, the wave vector of the resulting surface wave solution will be very large $({k \sim l/\xi \gg 2\pi /{\lambda _0}}),$ and this surface wave will have a propagating character $({{\rm{Im}}(k) \ll {\rm{Re}}(k)})$. Surprisingly enough, unlike conventional plasmonics, according to Eqs. (9) and (11) a gradient-index medium that is used to support such a propagating surface wave solution does not need to be a low-loss medium. For example, a medium having purely imaginary dielectric permittivity $\varepsilon (z) = i{\varepsilon ^{{\prime \prime}}}(z) = i\sigma (z)/{\varepsilon _0}\omega$ [where ${\varepsilon _0}$ is the dielectric permittivity of vacuum, and the medium conductivity $\sigma (z)$ is expressed in practical SI units] will still result in ${\rm{Im}}(V) \ll {\rm{Re}}(V)$:

(12)$$V = - \frac{{i\sigma \omega}}{{{\varepsilon _0}{c^2}}} - \frac{1}{2}\frac{{{\partial ^2}\sigma}}{{\sigma \partial {z^2}}} + \frac{3}{4}{\frac{{\left({\partial \sigma /\partial z} \right)}}{{{\sigma ^2}}}^2}.$$

The second and third terms in Eq. (12) are real, and they are much larger than the first term, if once again we assume that $\xi \ll {\lambda _0}$. We should also note that the physical origins of the newly predicted surface wave solutions are different from the origins of surface plasmons. Indeed, these novel surface wave solutions may be traced back to the well-known effect of charge accumulation whenever there is a gradient of conductivity in a medium and a non-zero component of electric field parallel to it. In the low-frequency (electrostatic) limit, the corresponding volumetric charge density $\rho$ is obtained as

(13)$$\rho = - \frac{{{\varepsilon _0}\nabla \sigma \cdot \vec E}}{\sigma}$$

(see, for example, [3]). While this effect disappears ($\rho = {{0}}$) inside a homogeneous conductive medium, the charges may accumulate near a planar surface inside a lossy conductive medium, if the medium conductivity changes continuously across such a surface. This charge accumulation may start to vary over time, leading to the appearance of a relatively low-loss dynamic wave of charge density propagating along such a surface. In fact, charge density waves are well known in various fields of electromagnetism and solid-state physics, which range from superconductivity [4] to ELF underwater communication [5]. Most experimental observations of charge density waves occur in static situations, which basically correspond to the $\omega \to 0$ limit of Eq. (9), but propagating charge density waves have been observed too (see, for example, [6]).

In general, solutions of the effective Schrödinger Eq. (9) with an effective potential $V(z)$ given by Eq. (11) must be obtained numerically. However, these equations may be solved analytically for some simple spatial distributions of $\varepsilon (z)$.

3. EXACTLY SOLVABLE ANALYTICAL MODEL

Let us confirm the qualitative arguments above by detailed analytical analysis. As an example, let us assume that ${\rm{Re}}(\varepsilon) \ll {\rm{Im}}(\varepsilon)$ and consider the following simple parabolic spatial distribution of the dielectric permittivity inside a single conductive stratum:

(14)$$\varepsilon (z) = A + B{z^2},$$

where both $A$ and $B$ are large imaginary coefficients, so that ${\alpha ^2} = A/B$ is real and positive. These assumptions are typically valid for graphite and silicon in the UV range [7] and for seawater and many biological tissues in the radio frequency range [8]. The resulting effective potential for the TM electromagnetic wave is

(15)$$V(z) \approx - \frac{{4{\pi ^2}A}}{{\lambda _0^2}}\left({1 + \frac{{{z^2}}}{{{\alpha ^2}}}} \right) - \frac{1}{{{z^2} + {\alpha ^2}}} + \frac{{3{z^2}}}{{{{\left({{z^2} + {\alpha ^2}} \right)}^2}}},$$

where ${\lambda _0}$ is the free space wavelength. We may also introduce a notional “wavelength” inside a conductive homogeneous medium with a dielectric permittivity $\varepsilon = A$ as $\lambda = {\lambda _0}/\sqrt {| A |}$. The plot of $V(z)$ in the limit $\lambda \gg \alpha$ [so that ${\rm{Re}}(V) \gg {\rm{Im}}(V)$] is shown in Fig. 2(b). We may cut off this potential at $z = \pm \alpha /\sqrt 2$ (and keep $\varepsilon = 1.5A$ constant at longer distances on both sides of the stratum) as illustrated in Fig. 2(a), so that the effective energy level inside such a potential well may be obtained analytically using the well-known shallow well approximation [9] as

(16)$$k = \frac{1}{2}\int {{\rm{d}}zV(z)} \approx \frac{1}{{2\sqrt 2 \alpha}}$$

(note that a 1D Schrödinger equation describing a potential well of arbitrary shape always has at least one eigenstate [9]). Let us demonstrate that the so-found TM solution of Maxwell equations inside a strongly conductive absorptive medium has a propagating character, and that the propagation length of such a wave may greatly exceed skin depth.

Fig. 2. (a) Spatial distribution of ${\varepsilon ^{{\prime \prime}}}$ inside a “parabolic” conductive stratum defined as $\varepsilon = A + B{z^2}$, where both $A$ and $B$ coefficients are large and imaginary, so that $A/B = {\alpha ^2}$ is positive and real. In this example, $A = 100i$ and $\alpha = 7$. The parabolic behavior is cut off at $z = \alpha /{2^{1/2}}$ since beyond these points, the wave function is exponentially small anyway. (b) Effective potential energy $V(z)$ inside such a “parabolic” conductive stratum. When the potential well is cut off at $z = \alpha /{2^{1/2}}$, as indicated by the red line, the shallow energy level (shown in green) may be obtained analytically as ${{k^2} \approx 1/8{\alpha ^2}}$.

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The conventional skin depth [10] inside a homogeneous medium having $\varepsilon \approx A$ equals

(17)$$\delta = \frac{1}{{\sqrt {\pi {\mu _0}\sigma \nu}}} = \frac{{{\lambda _0}}}{{\pi \sqrt {2\left| A \right|}}} = \frac{\lambda}{{\pi \sqrt 2}}$$

(recall that $A$ is imaginary). On the other hand, the propagation length $L$ of the newly found TM wave may be obtained based on the magnitude of ${\rm{Im}}(k)$ calculated using Eqs. (15) and (16) as follows:

(18)$${L^{- 1}} = {\mathop{\rm Im}\nolimits} (k) \approx \frac{1}{2}\int_{- \alpha /\sqrt 2}^{\alpha /\sqrt 2} {\frac{{4{\pi ^2}A}}{{\lambda _0^2}}} \left({1 + \frac{{{z^2}}}{{{\alpha ^2}}}} \right){\rm{d}}z = \frac{{14{\pi ^2}\alpha A}}{{3\sqrt 2 \lambda _0^2}}.$$

As a result, the ratio $L/\delta$ may be obtained as

(19)$$\frac{L}{\delta} = \frac{{3{\lambda _0}}}{{7\pi {{\left| A \right|}^{1/2}}\alpha}} \approx \frac{\lambda}{{7\alpha}},$$

which means that the TM wave propagation length may indeed be much longer than skin depth if $\lambda \gg \alpha$. Moreover, the obtained TM wave has a propagating character. Its wavelength ${\lambda _{{\rm{TM}}}}$, calculated as

(20)$${\lambda _{{\rm{TM}}}} = \frac{{2\pi}}{{{\mathop{\rm Re}\nolimits} (k)}} \approx 4\sqrt 2 \pi \alpha ,$$

appears to be much smaller than $L$ in the $\lambda \gg \alpha$ limit:

(21)$$\frac{L}{{{\lambda _{{\rm{TM}}}}}} \approx \frac{3}{{56{\pi ^2}}}{\left({\frac{\lambda}{\alpha}} \right)^2}.$$

We should also note that Eq. (20) may be used to provide an additional straightforward justification for the fact that the propagation length of the TM wave in such a geometry may greatly exceed skin depth. Indeed, based on Eq. (9), we may now write

(22)$${k^2} \approx i\frac{{4{\pi ^2}}}{{{\lambda ^2}}} + \frac{{4{\pi ^2}}}{{\lambda _{{\rm{TM}}}^2}} = \frac{{4{\pi ^2}}}{{\lambda _{{\rm{TM}}}^2}}\left({1 + i\frac{{\lambda _{{\rm{TM}}}^2}}{{{\lambda ^2}}}} \right),$$

where ${\lambda _{{\rm{TM}}}}\sim\alpha \ll \lambda$. As a result,

(23)$$k \approx \frac{{2{\pi ^{}}}}{{\lambda _{{\rm{TM}}}^{}}}\left({1 + i\frac{{\lambda _{{\rm{TM}}}^2}}{{2{\lambda ^2}}}} \right),$$

which leads to $L = {({{\mathop{\rm Im}\nolimits} k})^{- 1}} \approx \lambda \frac{\lambda}{{\pi {\lambda _{{\rm{TM}}}}}} \gt \gt \lambda \sim\delta$. Also note that $L{\lambda _{{\rm{TM}}}} \approx {\lambda ^2} \approx {\delta ^2}$, which makes this result quite natural from the point of view of electromagnetic energy conservation. Anisotropic dielectric properties of the absorptive medium are supposed to deform the shape of the volume in which electromagnetic energy is absorbed compared to the homogeneous and isotropic case, while keeping the effective volume in which the energy is absorbed approximately the same. The resulting shape is supposed to be “squeezed” along the $x$ direction. Since we were solving a two-dimensional propagation problem in the $xz$ plane, while disregarding the field behavior in $y$ direction, the effect of medium anisotropy leads to the effective mode area conservation in the $xz$ plane, which is exactly the result we obtained [if we note that based on Eq. (9), the field penetration in $z$ direction approximately equals ${\lambda _{{\rm{TM}}}}$, which is considerably smaller than $\delta$].

4. ANALYTICAL MODEL OF A LOSSY MULTILAYER METAMATERIAL

The fact that the newly discovered class of surface wave solutions is deeply subwavelength $({\lambda _{{\rm{TM}}}} \ll {\lambda _0})$ means that a periodically stratified conductive medium, in which the conductivity distribution in each individual layer looks like that in Fig. 2(a), should behave as a very high-refractive-index metamaterial. Indeed, based on Eq. (20), the effective refractive index in such a “metamaterial” in $x$ and $y$ directions may be estimated as

(24)$$n = \frac{{{\lambda _0}}}{{{\lambda _{{\rm{TM}}}}}} \approx \frac{{{\lambda _0}}}{{4\sqrt 2 \pi \alpha}}.$$

Since the properties of such periodically stratified media are of interest in many fields of science and engineering (e.g., in super-resolution optical microscopy, underground and underwater radio communication, plasma, etc.), let us study these properties in more detail. In particular, let us determine the transmission properties of such a periodically stratified conductive media along $z$ direction.

As an example, let us consider the following straightforward periodic extension of the parabolic distribution of the dielectric permittivity given by Eq. (14):

(25)$$\varepsilon (z) = A\left({1 + {{\sin}^2}\frac{z}{\alpha}} \right),$$

where $A \gg 1$ is an imaginary coefficient, and $\alpha \ll \lambda$ is real. The corresponding effective potential $V(z)$ for the TM wave is

(26)$$\begin{split}V(z)& \approx - \frac{{4{\pi ^2}A}}{{\lambda _0^2}}\left({1 + {{\sin}^2}\frac{{{z^{}}}}{{{\alpha ^{}}}}} \right) - \frac{{\cos \frac{{2z}}{\alpha}}}{{{\alpha ^2}{{\sin}^2}\frac{z}{\alpha} + {\alpha ^2}}} \\&\quad+ \frac{{3{{\sin}^2}\frac{{2z}}{\alpha}}}{{4{\alpha ^2}{{\left({1 + {{\sin}^2}\frac{z}{\alpha}} \right)}^2}}}.\end{split}$$

Once again, ${\rm{Re}}(V) \gg {\rm{Im}}(V)$ in the limit $\lambda \gg \alpha$. This effective potential is plotted in Fig. 3.

Fig. 3. Periodic conductive multilayer configuration. The periodic dielectric constant is defined as $\varepsilon (z) = A({1 + \sin^{2}({z/\alpha})})$. Similar to Fig. 2(b), the positive sections of the periodic potential well are cut off, as indicated by the red line.

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Fig. 4. (a) Real and imaginary parts of the dielectric permittivity of graphite (based on data reported in [13]). (b) Plot of an assumed $\xi = 10\;{\rm{nm}}$ thick planar transition layer between graphite and a graphite-based composite material. The magnitude of ${\varepsilon ^{{\prime \prime}}}(z)$ is shown at ${\lambda _0} = 275\;{\rm{nm}}$. (c) Corresponding effective potential energy (both real and imaginary parts) at the graphite interface defined by Eq. (11) (for TM light) plotted at ${\lambda _0} = 275\;{\rm{nm}}$. The numerically obtained effective energy level is shown in green. (d) Effective $\varepsilon$ inside the surface waveguide defined by Eq. (34).

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Similar to Fig. 2(b), the regions of positive $V(z)$ have been cut off, so that we may use the same approximate energy level given by Eq. (6) in our analysis. According to the Bloch theorem [11], the solution of the effective Schrödinger Eq. (9), when the potential is periodic, can be written as

(27)$$\psi (z) = {e^{i{k_z}z}}u(z),$$

where $u(z)$ is a periodic function that satisfies $u(z) = u({z + \pi \alpha})$; see [12]. Using the Born–von Karman boundary conditions $y(0) = y({M\pi \alpha})$, where $M$ is the number of layers, results in the following quantization for ${k_z}$:

(28)$${k_z} = \frac{2}{{M\alpha}}m,\quad{\rm{where}}\; m = 0, \pm 1,\ldots, \pm \frac{M}{2}.$$

In the limit $M\rightarrow\infty$, a continuous transmission band is formed for $0 \le kz \le 1/\alpha$. The periodic function $u(z)$ may be expressed as a Fourier series:

(29)$$u(z) = \sum\nolimits_m {{U_m}} {e^{i2mz/\alpha}}.$$

On the other hand, the periodic potential $V(z)$ may also be expanded as a Fourier series as

(30)$$V(z) = \sum\nolimits_m {{V_m}} {e^{i2mz/\alpha}}.$$

As a result, the Schrödinger equation may be re-written as

(31)$$\left[{{{\left({{k_z} + \frac{{2m}}{\alpha}} \right)}^2} + {k^2}} \right]{U_m} + \sum\nolimits_{{m^\prime}} {{V_{{m^\prime}}}} {U_{m - m^\prime}} = 0.$$

Using the tight binding approximation, its solution may be written approximately as

(32)$$k^2 = k_0^2 - 2S\cos \left({\pi \alpha {k_z}} \right),$$

where ${k_0} \approx 1/2\sqrt 2 \alpha$ is the energy level of a single potential well given by Eq. (16), and $S$ is the hopping integral $S \approx - {k_0}^2\langle {{\psi _0}|{\psi _1}}\rangle$, which is calculated using the wave functions of the original potential well [Eq. (15)] centered at $z = 0$ and $z = \pi \alpha$ planes, respectively. Using the approximations ${\psi _0}\sim{e^{- |{k_0}z|}}$ and ${\psi _1}\sim{e^{- |{k_0}(z - \pi \alpha)|}}$, the hopping integral may be estimated as

Fig. 5. (a) Simulations of surface wave excitation and scattering in a graphite-based gradient waveguide at ${\lambda _0} = 275\;{\rm{nm}}$. The UV light field in the waveguide is scattered by a 4 nm diameter metal nanowire placed near the waveguide, which leads to a pattern of standing surface waves. To demonstrate deep subwavelength resolution of surface wave scattering maps, the metal nanowire position has been changed by 20 nm between the top and the bottom images, indicated by the dashed lines. This 20 nm shift in nanocylinder position leads to a drastic change in UV light distribution: the “guided mode” is almost completely scattered by the nanowire beyond this position. (b) Numerical simulations of TM wave propagation away from a point dipole source in a homogeneous graphite at ${\lambda _0} = 275\;{\rm{nm}}$.

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(33)$$S \approx - \pi \alpha k_0^3{e^{- {k_0}\pi \alpha}}.$$

5. NUMERICAL EXAMPLES

Let us confirm the qualitative arguments above by detailed simulations. While the losses in a gradient optical medium do not need to be extreme to manifest the newly found surface waves, just to illustrate the point, let us consider a gradient medium based on such highly lossy material as graphite, which has almost purely imaginary dielectric permittivity in the 200–300 nm UV range [13] [see Fig. 4(a)]. Let us assume that we have engineered a planar $\xi = 10\;{\rm{nm}}$ thick gradual transition layer between bulk graphite and some graphite-based compound that has lower but also purely imaginary $\varepsilon$, as illustrated in Fig. 4(b). The resulting effective potential for TM light is shown in Fig. 4(c) at ${\lambda _0} = 275\;{\rm{nm}}$. If desired, based on Eq. (11), it may be represented using an “effective dielectric permittivity” distribution ${\varepsilon _{{\rm{eff}}}}(z)$ [see Fig. 4(d)], defined as

(34)$${\varepsilon _{{\rm{eff}}}} = - \frac{{\lambda _0^2V(z)}}{{4{\pi ^2}}} = \varepsilon + \frac{{{\lambda ^2}\left({\frac{{{\partial ^2}\varepsilon}}{{\partial {z^2}}}} \right)}}{{8{\pi ^2}\varepsilon}} - \frac{{3\lambda {}^2{{\left({\frac{{\partial \varepsilon}}{{\partial z}}} \right)}^2}}}{{16{\pi ^2}{\varepsilon ^2}}}.$$

In agreement with the qualitative arguments above, we have obtained that the effective potential well near the interface is rather deep, and that ${\rm{Im}}(V) \ll {\rm{Re}}(V)$. This means that similar to any other 1D Schrödinger equation, such a potential well will always have at least one bound state. Such a bound state will give rise to at least one solution having almost a purely real wave vector $k$, which corresponds to a surface mode with a long propagation length. For the effective potential well $V(z)$ shown in Fig. 4(c), the long-propagating-range eigenstate may be approximately determined using the virial theorem [9] as

(35)$${k^2} \approx - \frac{{\int_{- \infty}^{+ \infty} {\psi V(z)\psi *{\rm{d}}z}}}{{2\int_{- \infty}^{+ \infty} {\psi \psi *{\rm{d}}z}}}$$

$[$due to the almost ${\sim}1/z$ functional behavior of $V(z)$ near the potential barrier located in the vicinity of the graphite surface; see Fig. 4(c)]. Alternatively, the effective Schrödinger equation may be solved numerically using the Numerov method [14]. The numerically obtained effective energy level is shown schematically in green in Fig. 4(c). The wavelength of the resulting surface wave solution is $\lambda = 2\pi /k$, and $L = {\rm{Im}}{(k)^{- 1}}$ defines the propagation distance of the wave. Based on the numerical solution, it appears that at ${\lambda _0} = 275\;{\rm{nm}}$, the surface wave propagation distance reaches about 500 nm, which considerably exceeds the surface wave’s wavelength computed numerically as $\lambda = 2\pi /{\rm{Re}}(k)\sim 60\;{\rm{nm}}$. This clearly indicates the “propagating” character $(\lambda \ll L)$ of the surface wave. The newly found long-range propagating surface wave is tightly localized near the interface. Based on Eq. (9), far from the interface, its electric field attenuates as $Ez\sim{e^{- kz}}$ away from the interface inside both media. In the particular case shown in Fig. 4(c), this means that the attenuation distance away from the interface equals $l = 1/k\sim 11\;{\rm{nm}}$. The field configuration in this wave is partially longitudinal, since the electric field component along the propagation direction is non-zero. Based on Eq. (2), far from the interface, ${E_x}\sim i{E_z}$, while the only nonzero component of the magnetic field is $By\sim - ({2kc/\omega}){E_z}\sim - ({2{\lambda _0}/\lambda}){E_z}$.

Potential applicability of this novel surface wave in super-resolution microscopy and nanolithography techniques is illustrated in Fig. 5(a), which depicts numerical simulations of surface wave excitation and scattering in a graphite-based gradient waveguide at ${\lambda _0} = 275\;{\rm{nm}}$ using the commercial COMSOL Multiphysics solver. In these simulations, the UV light field in the graphite-based gradient surface waveguide is scattered by a 4 nm diameter metal nanowire. A 20 nm shift in nanowire position leads to a drastic change in UV light distribution, which is consistent with the nanometer-scale wavelength of the surface wave. For comparison, Fig. 5(b) shows numerical simulations of TM wave propagation away from a point dipole source in a homogeneous graphite sample. As expected, the latter is totally defined by skin depth at the frequency of excitation

Fig. 6. Difference between the effective dielectric permittivity ${\varepsilon _{{\rm{eff}}}}$ (red) calculated using Eq. (34) and the step-like multilayer structure representation (green) for the locally linear and locally quadratic gradient structures.

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Here, we should note that numerical simulations of spatial distribution of TM fields in strongly lossy gradient media need to be done very carefully and with great caution. As indicated by Eq. (34), the effective dielectric permittivity ${\varepsilon _{{\rm{eff}}}}$ in such structures may differ greatly from the actual local magnitude of $\varepsilon$. For example, if the local distribution of $\varepsilon$ may be approximated with a linear function $\varepsilon = A + Bz$, the corresponding effective dielectric permittivity for the locally defined TM polarization may be obtained as

(36)$${\varepsilon _{{\rm{eff}}}} = A + Bz - \frac{{3{\lambda ^2}}}{{16{\pi ^2}}}\frac{{{B^2}}}{{{{(A + Bz)}^2}}},$$

while in the case of local quadratic dependence $\varepsilon = A + B{z^2}$, the corresponding effective permittivity is

(37)$${\varepsilon _{{\rm{eff}}}} = A + B{z^2} + \frac{{{\lambda ^2}}}{{4{\pi ^2}}}\frac{B}{{(A + B{z^2})}} - \frac{{3{\lambda ^2}}}{{4{\pi ^2}}}\frac{{{B^2}{z^2}}}{{{{(A + B{z^2})}^2}}}.$$

If during numerical simulations these gradient structures are approximated with multilayer step-like structures, the effective potential $V(z)$ and hence the effective dielectric permittivity ${\varepsilon _{{\rm{eff}}}}$ in these structures will be grossly misrepresented (see Fig. 6), leading to considerable numerical errors.

6. CONCLUSION: NANOPHOTONIC APPLICATIONS OF THE NOVEL SURFACE WAVES

In conclusion, I have demonstrated that surface electromagnetic waves in stratified lossy conductive media may have a propagating character, and the propagation length of such waves may be considerably longer than skin depth. The comparison of these novel surface wave solutions with the properties of the more well-known surface waves, such as surface plasmons [1] and Zenneck waves [2], is summarized in Table 1. Similar to surface plasmons, the wavelength of this wave may be considerably shorter than the light wavelength in free space, which may enable its applications in super-resolution microscopy and nanolithography techniques. However, unlike plasmonics-based nanophotonic devices, which are typically built using a very limited number of low-loss optical materials, the newly found class of surface waves may be supported by a much broader range of lossy media.

Table 1. Comparison of Basic Properties of Different Kinds of Surface Electromagnetic Waves

View Table

Utilization of the newly discovered class of surface electromagnetic waves that propagate along gradual interfaces of lossy optical media should bring about quite a few developments in linear and nonlinear nanophotonics. As illustrated in Fig. 5, these novel surface waves may bring the spatial resolution of 2D microscopes [15] down to 10 nm scale and beyond. Unlike plasmonics-based nanophotonic devices, which are typically built using a very limited number of low-loss optical materials, the newly found class of surface waves may be supported by a much broader range of lossy media. Such materials as graphite and silicon seems to be ideal in UV nanophotonics applications where classical plasmonic materials are not operational.

Indeed, such a classic CMOS material as silicon appears to be highly suitable for gradient-index nanophotonics applications. Similar to graphite, silicon exhibits very large and purely imaginary dielectric permittivity $\varepsilon \sim 50i$ around 290 nm [16], which strongly depends on the doping level. Therefore, silicon-based gradient-index nanowaveguides may potentially be fabricated using CMOS technology, which would greatly extend various silicon photonics applications. As illustrated in Fig. 5, fabrication of deeply subwavelength UV nanoresonators may become possible, leading to numerous applications in sensing and UV nanolasers. In addition, such applications as deep UV high-harmonic generation in these nanostructures may also become possible. Such developments have a potential to completely revolutionize nanophotonics.

Disclosures

The authors declare no conflicts of interest.

Data availability

No data were generated or analyzed in the presented research.

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Surface plasmon	Abrupt	Low-loss metal–dielectric	From free space to deep subwavelength
Surface Wave Type	Interface Type	Bounding Media	Wavelength Range
Zenneck wave	Abrupt	Lossy conductor–dielectric	Similar to free space
Interfacial wave	Continuous	Lossy conductor–lossy conductor	Deep subwavelength

Surface electromagnetic waves in lossy conductive media: tutorial

Abstract

1. INTRODUCTION

2. GENERAL THEORETICAL FRAMEWORK

3. EXACTLY SOLVABLE ANALYTICAL MODEL

4. ANALYTICAL MODEL OF A LOSSY MULTILAYER METAMATERIAL

5. NUMERICAL EXAMPLES

6. CONCLUSION: NANOPHOTONIC APPLICATIONS OF THE NOVEL SURFACE WAVES

Disclosures

Data availability

REFERENCES

Data availability

Cited By

Figures (6)

Tables (1)

Equations (37)

Journal of the Optical Society of America B