Surface Electromagnetic Waves

Abstract

As will be shown in the subsequent sections, reflection and emission of electromagnetic waves from surfaces of solids significantly depend on the presence of localized surface modes, which include surface electromagnetic waves. This particular type of waves exists at the interface between two different media. An electromagnetic surface wave propagates along the interface and decreases exponentially in the perpendicular direction.

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As will be shown in the subsequent sections, reflection and emission of electromagnetic waves from surfaces of solids significantly depend on the presence of localized surface modes, which include surface electromagnetic waves. This particular type of wave exists at the interface between two different media. An electromagnetic surface wave propagates along the interface and decreases exponentially in the perpendicular direction. Surface waves due to a coupling between the electromagnetic field and a resonant polarization oscillation in the material are called surface polaritons. From a microscopic point of view, the surface waves at the interface of a metal is a charge density wave or plasmon. They are therefore called surface plasmon polaritons. At the interface of a dielectric, the surface wave is due to the coupling of an optical phonon with the electromagnetic field. It is thus called surface phonon polariton. Plasmon polaritons and phonon polaritons can also exist in the whole volume of the material and are called polaritons. More details on this subject can be found in textbooks such as Kittel [173, 174] and Ziman [175]. In what follows, we will focus our attention on surface polaritons propagating along a plane interface. Excellent reviews of the subject can be found in [176–179].

2.1 Surface Polaritons

Let us now study the nature of surface polaritons in the case of a plane interface separating two linear, homogeneous and isotropic media, with different dielectric constants. The system considered is depicted in Fig. 2.1.

Fig. 2.1

A plane interface separating medium 1 (dielectric constant $\varepsilon _1$, magnetic constant $\mu _1$) and medium 2 (dielectric constant $\varepsilon _2$, magnetic constant $\mu _2$)

The medium 1 (dielectric constant $\varepsilon _1$ and magnetic constant $\mu _1$) fills the upper half-space $z>0$ whereas medium 2 (dielectric constant $\varepsilon _2$ and magnetic constant $\mu _2$) fills the lower half-space $z<0$. The two media are supposed to be local and dispersive so that their complex dielectric and magnetic constants only depend on $\omega$.

The three directions x, y, z shown in Fig. 2.1 are characterized by their unit vectors $\hat{\mathbf x}, \hat{\mathbf y}, \hat{\mathbf z}$. A point in space will be denoted $\mathbf {R}=(x, y, z) = (\mathbf {r}, z)$. Similarly, a wave vector $\mathbf {k}=(k_x, k_y, k_z)= (\mathbf {q}, k_z)$ where $\mathbf {q}$ is the component parallel to the interface and $k_z$ is the component in the z direction.

A surface wave is a particular solution of Maxwell’s equations, which propagates along the interface and decreases exponentially in the perpendicular directions. Because of the translational invariance of the system, it can be written in the form

$$\begin{aligned} \mathbf {E}_1(\mathbf {R}, z) = \left( \begin{array}{c}E_{x, 1}\\ E_{y, 1}\\ E_{z, 1} \end{array}\right) \exp {[i(\mathbf {q\cdot r} +k_{z1}z)]}, \end{aligned}$$

(2.1)

$$\begin{aligned} \mathbf {E}_2(\mathbf {R}, z) = \left( \begin{array}{c}E_{x, 2}\\ E_{y, 2}\\ E_{z, 2} \end{array}\right) \exp {[i(\mathbf {q\cdot r} -k_{z2}z)]} \end{aligned}$$

(2.2)

where

$$\begin{aligned} k_{z1}=\sqrt{\varepsilon _1\mu _1\left( \frac{\omega }{c}\right) ^2 - q^2}, \quad \quad \mathrm {Im}k_{z1}>0, \end{aligned}$$

(2.3)

$$\begin{aligned} k_{z2}=\sqrt{\varepsilon _2\mu _2\left( \frac{\omega }{c}\right) ^2 - q^2}, \quad \quad \mathrm {Im}k_{z2}>0, \end{aligned}$$

(2.4)

where c is the speed of light in vacuum. We now look for the existence of surface waves with s (TE: transverse electric field) or p (TM: transverse magnetic field) polarization. In what follows, we shall assume that the wave propagates along the y-axis.

2.1.1 s-Polarization (TE)

In s-polarization, the electric field is perpendicular to the yz plane. The electric field, $\mathbf {E}$, is thus parallel to the x direction

$$\begin{aligned} \mathbf {E}_1(\mathbf {R}, \omega )=E_{x, 1}\hat{\mathbf x}\exp [i(\mathbf {q\cdot r}+k_{z1}z)], \end{aligned}$$

(2.5)

$$\begin{aligned} \mathbf {E}_2(\mathbf {R}, \omega )=E_{x, 2}\hat{\mathbf x}\exp [i(\mathbf {q\cdot r}-k_{z2}z)]. \end{aligned}$$

(2.6)

The magnetic field is then derived from the Maxwell equation $\mathbf {H}=-i\mathbf {\nabla }\times \mathbf {E}/(\mu (\omega )\omega )$. The continuity conditions of the parallel components of the fields $\mathbf {E}$ and $\mathbf {H}$ across the interface yield the following equations:

$$\begin{aligned} E_{x, 1}-E_{x, 2}=0, \end{aligned}$$

(2.7)

$$\begin{aligned} \frac{k_{z1}}{\mu _1}E_{x, 1}+\frac{k_{z2}}{\mu _2}E_{x, 2}=0. \end{aligned}$$

(2.8)

The system of (2.7) and (2.8) has a non-trivial solution if the coefficient determinant is equal to zero. This gives

$$\begin{aligned} \mu _2k_{z1} + \mu _1k_{z2}=0. \end{aligned}$$

(2.9)

Taking into account (2.3) and (2.4), one obtains from (2.9) the surface wave dispersion relation for s-polarization:

$$\begin{aligned} q^2 = \frac{\omega ^2}{c^2}\frac{\mu _1\mu _2[\mu _2\varepsilon _1- \mu _1\varepsilon _2]}{\mu _2^2(\omega )-\mu _1^2(\omega )}. \end{aligned}$$

(2.10)

For the particular case when $\varepsilon _1 = \varepsilon _2 = \varepsilon$, the dispersion relation takes the simple form:

$$\begin{aligned} q^2 = \frac{\omega ^2}{c^2}\varepsilon \frac{\mu _1\mu _2}{\mu _2(\omega )+\mu _1(\omega )}. \end{aligned}$$

(2.11)

2.1.2 p-Polarization (TM)

For p-polarization, the electric field lies in the yz-plane and can be written in the form:

$$\begin{aligned} \mathbf {E}_1(\mathbf {R}, z) = \left( \begin{array}{c}0\\ E_{y, 1}\\ E_{z, 1} \end{array}\right) \exp {[i(\mathbf {q\cdot r} +k_{z1}z)]}, \end{aligned}$$

(2.12)

$$\begin{aligned} \mathbf {E}_2(\mathbf {R}, z) = \left( \begin{array}{c}0\\ E_{y, 2}\\ E_{z, 2} \end{array}\right) \exp {[i(\mathbf {q\cdot r} -k_{z2}z)]}. \end{aligned}$$

(2.13)

The continuity of the tangential electric field yields

$$\begin{aligned} E_{y, 1}-E_{y, 2}=0. \end{aligned}$$

(2.14)

The continuity of the z-component of displacement field yields:

$$\begin{aligned} \varepsilon _1E_{z, 1}=\varepsilon _2E_{z, 2}. \end{aligned}$$

(2.15)

The condition of transversality of the electromagnetic waves gives

$$\begin{aligned} qE_{y, 1}+k_{z1}E_{z, 1}=0, \quad qE_{y, 2}-k_{z2}E_{z, 2}=0. \end{aligned}$$

(2.16)

Using (2.14) and (2.15) in (2.16), we get

$$\begin{aligned} \varepsilon _2k_{z1} + \varepsilon _1k_{z2}=0. \end{aligned}$$

(2.17)

Taking into account (2.3) and (2.4), one obtains from (2.17) the surface wave dispersion relation for p-polarization:

$$\begin{aligned} q^2 = \frac{\omega ^2}{c^2}\frac{\varepsilon _1\varepsilon _2[\mu _1\varepsilon _2- \mu _2\varepsilon _1]}{\varepsilon _2^2-\varepsilon _1^2} \end{aligned}$$

(2.18)

For the particular case where $\mu _1 = \mu _2 = \mu$, the dispersion relation takes the simple form:

$$\begin{aligned} q^2 = \frac{\omega ^2}{c^2}\mu \frac{\varepsilon _1\varepsilon _2}{\varepsilon _1+\varepsilon _2}. \end{aligned}$$

(2.19)

2.1.3 Some Comments

• 1. When the media are non-magnetic, there are no surface waves in s-polarization. Indeed, the imaginary part of the z-components $k_{zi}$ is always positive, so that $k_{z1} + k_{z2}$ cannot be zero.

• 2. At a material–vacuum interface ($\varepsilon _1=\mu _1=1$), the dispersion relation for p-polarization has the form

  $$\begin{aligned} q = \frac{\omega }{c}\sqrt{\frac{\varepsilon _2}{\varepsilon _2+1}}. \end{aligned}$$

  (2.20)

  It follows that the wave vector becomes very large for a frequency such that $\varepsilon _2 +1=0$.

• 3. The conditions (2.9) and (2.17) correspond to the poles of the Fresnel reflection amplitude. Finding these poles is an alternative and simple way to find the dispersion relation. This is a particularly useful approach for the multilayers system.

• 4. For non-lossy media, one can find a real q corresponding to a real $\omega$. This mode exists only if $\varepsilon _2 <-1$ in the case of an interface separating a vacuum from a material.

• 5. In the presence of losses, the dispersion relation yields two equations but both frequency and wavector can be complex, so that there are four parameters. Two cases are of practical interest: (i) a real frequency and a complex wavevector, (ii) a complex frequency and a real wavector. These two choices lead to different shapes of the dispersion relation as discussed in [18–21]. The imaginary part of the frequency describes the finite lifetime of the mode due to losses. Conversely, for a given real frequency, the imaginary part of the lateral component of wave vector yields a finite propagation length along the interface.

• 6. The dispersion relation (2.20) shows that for a real dielectric constant $\varepsilon _2<-1$, $q>\omega /c$. This mode cannot be excited by a plane wave whose wavevector is such that $q<\omega /c$. In order to excite this mode, it is necessary to increase the wavevector. One can use a prism [14, 22, 23] or a grating [176].

2.1.4 Dispersion Relation

In this subsection, we will consider two types of surface waves: surface plasmon polaritons and surface phonon polaritons. Surface plasmon polaritons are observed at surfaces separating a dielectric from a medium with a gas of free electrons such as a metal or a doped semiconductor. The dielectric function of the latter can in the simplest case be modeled by a Drude model:

$$\begin{aligned} \varepsilon (\omega ) = \varepsilon _{\infty } - \frac{\omega _p^2}{\omega ^2 + i\Gamma \omega }, \end{aligned}$$

(2.21)

where $\omega _p$ is the plasma frequency and $\Gamma$ accounts for the losses. Using this model and neglecting the losses, we find that the resonance condition $\varepsilon (\omega )+1=0$ yields $\omega =\omega _p/\sqrt{2}$. For most metals, this frequency lies in the near UV so that these surface waves are difficult to excite thermally. By contrast, surface phonon polaritons can be excited thermally because they exist in the infrared. They have been studied through measurements of emission and reflectivity spectra by Zhizhin and Vinogradov [179].

Let us study the dispersion relation of surface-phonon polaritons at a vacuum/Silicon Carbide (SiC) interface. SiC is a non-magnetic material whose dielectric constant is well described by an oscillator model in the 2–22 $\upmu$m wavelength range [180]:

$$\begin{aligned} \varepsilon (\omega )=\epsilon _\infty \left( 1+\frac{\omega _L^2-\omega _T^2}{\omega _T^2-\omega ^2-i\Gamma \omega }\right) , \end{aligned}$$

(2.22)

where $\epsilon _\infty =6.7$, $\omega _L=1.8\times 10^{14}$ s$^{-1}$, $\omega _T=1.49\times 10^{14}$ s$^{-1}$, and $\Gamma =8.9\times 10^{11}$ s$^{-1}$. The dispersion relation at a SiC/vacuum interface is shown in Fig. 2.2.

Fig. 2.2

Dispersion relation for surface phonon polariton at a SiC/Vacuum interface. The flat asymptote is situated at $\omega _{asym}=1.784\times 10^{14}$ s$^{-1}$. The dashed line represents the light cone, above which a wave is propagating and below which a wave is evanescent

This dispersion relation has been derived by assuming that the frequency, $\omega$ is complex and the parallel wavevector, $q=K$, is real. This choice is well-suited to analyze experimental measurements of spectra for fixed angles. The width of the resonance peaks observed is related to the imaginary part of the frequency of the mode. We note that the curve is situated below the light cone $\omega =cK$ so that the surface wave is evanescent. We also observe a horizontal asymptote for $\omega _{asym}=1.784\times 10^{14}$ s$^{-1}$ so that there is a peak in the density of electromagnetic states. We will see in the next sections that the existence of surface modes at a particular frequency plays a key role in many phenomena. Figure 2.3 shows the dispersion relation obtained when choosing a real frequency, $\omega$, and a complex wavevector, $q=K$.The x-axis is the real part of the complex wavevector. It can be seen that the shape of the dispersion relation is significantly changed and a backbending of the curve is observed. This type of behavior is observed experimentally when measurements are taken at a fixed frequency and the angle is varied. Observed resonances in reflection or emission experiments have an angular width, which is related to the imaginary part of the complex wavevector.

Fig. 2.3

Dispersion relation for surface phonon polariton at a SiC/vacuum interface. Real $\omega$ chosen so as to obtain a complex K. The x-axis is the real part of K. The horizontal asymptote is situated at $\omega _{asym}=1.784\times 10^{14}$ s$^{-1}$. The slanting dashed line represents the light line above which a wave is propagating and below which a wave is evanescent