# 2. Wave propagation in different media

In the previous sections we looked at the basic properties of idealized EM-waves and their mathematical description. Here we look at the propagation behavior of these waves in different media. The propagation behavior can be described entirely in terms of the electric permittivity ($\epsilon = \epsilon' + i \epsilon''$) and the magnetic permeability ($\mu = \mu' + i \mu''$), which are both complex functions of the frequency and can be related to the refractive index ($n = n' + i n''$) and the impedance ($Z = Z' + i Z''$). All of these quantities can take a huge range of values for different materials and different frequency values. These differences are responsible for the effect of frequency dispersion: different propagation behavior of EM-waves for different frequencies. Dispersion gives rise to the splitting up of a ray of white light into its different frequency components in a prism and the formation of a rainbow.

Figure 1: Two examples of dispersion and refraction. (a) A beam of white light is split up into its spectral components by a prism. (b) The formation of a rainbow.

Both phenomena originate from frequency dispersion in the refractive index. For each of the wavelengths in the visible spectrum, the refractive index of the prism and the collection of rain droplets is slightly different. At the interface between two media, the angle between the normal and the transmitted wave differs from the angle between the incident wave and the normal, according to Snell’s law.

(1) $\frac{\sin{\theta_i}}{\sin{\theta_r}} = \frac{\nu_1}{\nu_2} = \frac{n_2}{n_1}$

This change in the transmitted angles is known as refraction and originates from the wavelength dependence of the refractive index. In order to formally derive this behavior, we consider the polarization dependent properties of electromagnetic waves that are incident on the interface between two different media. We consider two different perpendicular polarization states of a homogeneous plane wave: one with the electric field perpendicular to the plane of incidence (figure 2(a)) and one with the electric field parallel to the plane of incidence (figure 2(b)). The former is S-polarized or a transverse electric (TE) wave, while the latter is P-polarized or a transverse magnetic (TM) wave.

Figure 2: Illustration of Snell’s law and definition of S-polarized (a) and P-polarized (b) waves

For both polarization states the appropriate boundary conditions need to be applied in order to derive the propagation behavior of the incident wave at the interface. Three different waves are involved in this process, which depend on the material properties of both media: the incident wave (equation 2-3), the refracted wave (equation 4-5) and the reflected wave (equation 6-7).

(2) ${\bf E_1} = {\bf E_{1,0}} e^{i {\bf k_1 \cdot r} - i \omega t}$

(3) ${\bf B_1} = \sqrt{\mu_1 \epsilon_1} \frac{{\bf k_1} \times {\bf E_1}}{k_1}$

(4) ${\bf E_2} = {\bf E_{2,0}} e^{i {\bf k_2 \cdot r} - i \omega t}$

(5) ${\bf B_2} = \sqrt{\mu_2 \epsilon_2} \frac{{\bf k_2} \times {\bf E_2}}{k_2}$

(6) ${\bf E_{1'}} = {\bf E_{1',0}} e^{i {\bf k_{1'} \cdot r} - i \omega t}$

(7) ${\bf B_{1'}} = \sqrt{\mu_1 \epsilon_1} \frac{{\bf k_{1'}} \times {\bf E_{1'}}}{k_{1'}}$

The magnitudes of the different wave numbers are given by

(8) $|{\bf k_1}| = |{\bf k_{1'}}| = k = \omega \sqrt{\mu_1 \epsilon_1}$

(9) $|{\bf k_2}| = k_2 = \omega \sqrt{\mu_2 \epsilon_2}$

At the interface between medium 1 and medium 2 ($z = 0$) the boundary conditions have to be satisfied for all points. The spatial and time variation of all fields must be the same there, which implies that all phase vectors should be equal, independent of the boundary conditions.

(10) $({\bf k_1 \cdot r})_{interface} = ({\bf k_2 \cdot r})_{interface} = ({\bf k_{1'} \cdot r})_{interface}$

From equation 10 it is clear that all 3 wave vectors lie in the same plane. As $k_1 = k_{1'}$, it follows that the angle between the incident and refracted wave should be equal, and we also can derive Snell’s law directly.

(11) $k_1 \sin{\theta_i} = k_2 \sin{\theta_r} = k_{1'} \sin{\theta_i}$

The boundary conditions that have to be satisfied can now be written in terms of the field values (equations 2-7). The normal components of ${\bf D}$ and ${\bf B}$ and the tangential components of ${\bf E}$ and ${\bf H}$ have to be continuous at the interface, which can be expressed as follows

(12) $[\epsilon_1 ({\bf E_{1,0}} + {\bf E_{1',0}}) - \epsilon_2 {\bf E_{2,0}}] {\bf \cdot n} = 0$

(13) $[{\bf k_1 \times E_{1,0}} + {\bf k_{1'} \times E_{1',0}} - {\bf k_2 \times E_{2,0}} {\bf \cdot n} = 0$

(14) $[{\bf E_{1,0}} + {\bf E_{1',0}} - {\bf E_{2,0}})] \times {\bf n} = 0$

(15) $[\frac{1}{\mu_1} ({\bf k_1 \times E_{1,0}} + {\bf k_{1'} \times E_{1',0}}) - \frac{1}{\mu_2} ({\bf k_2 \times E_{2,0}})] \times {\bf n} = 0$

in which ${\bf n}$ is a unit vector perpendicular to the interface. From this point onwards, it is useful to split up the derivation in 2 cases for S-polarized (figure 2(a)) and P-polarized (figure 2(b)) waves, as any other polarization state can be constructed based on these two polarization states.

For the S-polarized case, equation 12 yields no result, as the electric fields are all perpendicular to the plane of incidence. From 14 and 15 we can derive equations 16-17, while equation 13 reproduces 14 when we apply Snell’s law.

(16) $E_{1,0} + E_{1',0} - E_{2,0} = 0$

(17) $\sqrt{\frac{\epsilon_1}{\mu_1}} (E_{1,0} - E_{1',0}) \cos{\theta_i} - \sqrt{\frac{\epsilon_2}{\mu_2}} (E_{2,0}) \cos{\theta_r} = 0$

From the obtained boundary conditions, the relative amplitudes of the refracted and reflected waves are obtained, which gives the following complex transmission and reflection coefficients:

(18) $t_s = \frac{E_{2,0}}{E_{1,0}} = \frac{2 n_1 \cos{\theta_i}}{n_1 \cos{\theta_i}+\frac{\mu_1}{\mu_2} \sqrt{{n_2}^2-{n_1}^2 \sin^2{\theta_i}}}$

(19) $r_s = \frac{E_{1',0}}{E_{1,0}} = \frac{n_1 \cos{\theta_i}-\frac{\mu_1}{\mu_2} \sqrt{{n_2}^2-{n_1}^2 \sin^2{\theta_i}}}{n_1 \cos{\theta_i}+\frac{\mu_1}{\mu_2} \sqrt{{n_2}^2-{n_1}^2 \sin^2{\theta_i}}}$

in which the square roots are introduced using Snell’s law to rewrite everything as function of the properties of the incident wave.

For the P-polarized case a similar derivation can be made, for which the tangential $E$ and $H$ have to be continuous, which results in

(20) $\cos{\theta_i} (E_{1,0} - E_{1',0}) - \cos{\theta_r} E_{2,0} = 0$

(21) $\sqrt{\frac{\epsilon_1}{\mu_1}} (E_{1,0} + E_{1',0}) - \sqrt{\frac{\epsilon_2}{\mu_2}} (E_{2,0}) = 0$

in this case the normal component of ${\bf D}$ should be continuous as well, which duplicates the second equation when we apply Snell’s law. Again, we can write the relative field amplitudes as function of the properties of the incident wave, which gives the following complex transmission and reflection coefficients:

(22) $t_p = \frac{E_{2,0}}{E_{1,0}} = \frac{2 n_1 n_2 \cos{\theta_i}}{\frac{\mu_1}{\mu_2} {n_2}^2 \cos{\theta_i} + n_1 \sqrt{{n_2}^2-{n_1}^2 \sin^2{\theta_i}}}$

(23) $r_p=\frac{E_{1',0}}{E_{1,0}}=\frac{\frac{\mu_1}{\mu_2} {n_2}^2 \cos{\theta_i}-n_1 \sqrt{{n_2}^2-{n_1}^2 \sin^2{\theta_i}}}{\frac{\mu_1}{\mu_2} {n_2}^2 \cos{\theta_i}+n_1 \sqrt{{n_2}^2-{n_1}^2 \sin^2{\theta_i}}}$

The propagation behavior of any type of EM-wave can be fully described by means of equations 18-19 and 22-23, which are often referred to as the Fresnel equations. In general the refractive index $n$, the electric permittivity $\epsilon$ and the magnetic permeability $\mu$ are all complex functions of the frequency. At optical frequencies in conventional materials it is often stated that $\mu_1 = \mu_2 = 1$, but as we’ll show later on, this is not valid in case of optical metamaterials which also exhibit a strong magnetic response.

The electric permittivity of a material depends primarily on the electric polarizability, which is closely related to the electron density. For this tutorial, we restrict ourselves to dielectrics and noble metals in the visible and NIR wavelength range of the EM spectrum. In case of dielectric materials (such as silica ($SiO_2$)), the refractive index $n$ and permittivity $\epsilon$ show almost no frequency dispersion and can be treated as positive real numbers (damping can be ignored). Metals show a totally different behavior, and can be described by the Drude-Sommerfeld model:

(24) $\epsilon_{Drude} ( \omega ) = \epsilon_{\infty} - \frac{{\omega_p}^2}{\omega^2 + i \gamma \omega}$

in which $\omega_p$ is the plasma frequency, and $\gamma$ a damping factor. The contribution of the bound electrons to the polarizability is included in $\epsilon_{\infty}$, which is 1 in case only the conduction band electrons contribute. This model gives a fairly good description for noble metals in the visible and NIR spectral range, although some modifications are needed at shorter wavelengths in order to compensate for inter-band transitions. Therefore in numerical calculations we use adapted models based on experimental data to obtain a more accurate description of the investigated structures. In general, the permittivity can be written as $\epsilon(\omega) = \epsilon' + i \epsilon''$. The real part is negative for frequencies below the plasma frequency and relates directly to the polarizability of the metal, while the imaginary part takes positive values and relates to the damping of propagating waves and the phase of the polarizability. An overview of the experimental values of the permittivity for gold (Au) is given in figure 3.

Figure 3: Electric permittivity of gold

It is clear that also the propagation behavior of EM-waves will be influenced by the effects of dispersion. So far we only considered the propagation of idealized waves at a fixed frequency. In that case the permittivity defines the amplitude, phase and damping of a propagating wave. In that respect, we introduced the concept of phase velocity $\nu$, which defines the propagation velocity for a single frequency component. In practical applications even the most idealized source will contain more than one frequency component or wavelength, due to finite pulse durations or inherent broadening in the source. All of these frequency components propagate with slightly different phase velocities and as a consequence both the amplitudes and phases tend to show changes with respect to each other. This implies that in dispersive media the velocity of energy flow can differ largely from the phase velocity.

For dispersive media, the frequency depends on the wave vector ($\omega = \omega (k)$). In most spectral regions $\omega$ is a slowly varying function of $k$, but in certain spectral regions the variation is much more pronounced, for example in the vicinity of the plasma frequency of a metal. If we assume for now that $k$ and $\omega$ are real numbers, we can write a superposition of homogeneous plane waves  to construct a more general solution for a plane wave in a dispersive medium

(25) $u(z,t) = \frac{1}{\sqrt{2 \pi}} \int_{- \infty}^{+ \infty} A(k)e^{i k z - i \omega(k) t} dk$

where $A(k)$ defines the amplitude for the different components of the superposition of plane waves. The amplitude values are obtained by the Fourier transform of the spatial amplitude $u(z,t)$ at $t = 0$

(26) $A(k) = \frac{1}{\sqrt{2 \pi}} \int_{- \infty}^{+ \infty} u(z,0) e^{-i k z} dz$

In that way, we obtain homogeneous plane wave $u(z,t)=e^{i k_0 z - i \omega(k_0) t}$. If we consider a finite wave train $u(z,0)$ at $t = 0$ then $A(k)$ will not be a delta function but a peaked function with a finite width (determined by the length of the wave train) $\Delta k$ which is centered around the wave number $k_0$. If we assume a fairly sharp shape of this peaked function (i.e. the wave train is rather long), then the frequency $\omega(k)$ can be expanded around $k_0$.

(27) $\omega(k) = \omega_0 + \frac{\partial \omega}{\partial k} |_0 (k - k_0) + \ldots$

By inserting this expansion into equation 25, we can derive $u(z,t)$

(28) $u(z,t) \approx \frac{e^{i [ k_0 \frac{\partial \omega}{\partial k}|_0 - \omega_0 ] t}}{\sqrt{2 \pi}} \int_{- \infty}^{+ \infty} A(k)e^{i [ z - \frac{\partial \omega}{\partial k}|_0 t ] k} dk$

If we compare this with equation 26 then it follows that the integral in equation 28 describes $u(z',0)$ with $z' = z - \frac{\partial \omega}{\partial k}|_0 t$, from which we obtain

(29) $u(z,t) \approx u( z - t \frac{\partial \omega}{\partial k}|_0 , 0 ) e^{i [ k_0 \frac{\partial \omega}{\partial k}|_0 - \omega_0 ] t}$

This illustrates that apart from an overall phase factor, the pulse travels undistorted in shape with a group velocity $\nu_g$ which can differ significantly from the (average) phase velocity $\nu_p$ which was already introduced in section 1.

(30) $\nu_g = \frac{\partial \omega}{\partial k}|_0$

As the group velocity of a pulse describes its propagation velocity, it also determines at which rate the pulse is transporting energy in the direction of the Poynting vector. The relationship between $\omega$ and $k$ can be written in its most general shape

(31) $\omega(k) = \frac{c k}{n(k)}$

from which we obtain the general expressions for the phase velocity $\nu_p$ and group velocity $\nu_g$ as function of the refractive index $n$

(32) $\nu_p = \frac{\omega(k)}{k} = \frac{c}{n(k)}$

(33) $\nu_g = \frac{c}{n(\omega) + \omega (\partial n / \partial \omega)}$

In most conventional materials with normal dispersion ($(\partial n / \partial \omega) > 0$ and $n > 1$) the velocity of energy flow is smaller than the phase velocity and smaller than $c$. In regions of anomalous dispersion, huge differences between the phase and group velocity can occur.