2.1 Wave equation

Electromagnetic waves describe how electric and magnetic fields propagate through space and time. The wave equation, derived from Maxwell's equations, is the mathematical backbone of this propagation. Mastering it is essential before tackling reflection, transmission, guided waves, and everything else that builds on top of it in this course.

Wave equation in electromagnetics

The wave equation is a second-order partial differential equation that relates the electric and magnetic fields to their spatial and temporal derivatives. It governs how electromagnetic disturbances propagate, and it falls directly out of Maxwell's equations.

Derivation of wave equation

The derivation starts from two of Maxwell's equations: Faraday's law and Ampère's law (with Maxwell's correction). Here's the procedure for the electric field:

1. Start with Faraday's law in differential form: $\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}$. This tells you a time-varying magnetic field produces a curling electric field.

2. Take the curl of both sides: $\nabla \times (\nabla \times \vec{E}) = -\frac{\partial}{\partial t}(\nabla \times \vec{B})$

3. Substitute Ampère's law ($\nabla \times \vec{B} = \mu_0 \epsilon_0 \frac{\partial \vec{E}}{\partial t}$, assuming free space with no free currents) into the right-hand side.

4. Apply the vector identity $\nabla \times (\nabla \times \vec{E}) = \nabla(\nabla \cdot \vec{E}) - \nabla^2 \vec{E}$. In a source-free region, $\nabla \cdot \vec{E} = 0$, so the first term vanishes.

5. Result: $\nabla^2 \vec{E} - \mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2} = 0$

The magnetic field wave equation is derived analogously: take the curl of Ampère's law, substitute Faraday's law, and use $\nabla \cdot \vec{B} = 0$.

Wave equation for electric field

$\nabla^2 \vec{E} - \mu_0\epsilon_0\frac{\partial^2 \vec{E}}{\partial t^2} = 0$

• $\nabla^2$ is the Laplacian operator (sum of second spatial derivatives)
• $\vec{E}$ is the electric field vector
• $\mu_0$ is the permeability of free space ($4\pi \times 10^{-7}$ H/m)
• $\epsilon_0$ is the permittivity of free space ($8.854 \times 10^{-12}$ F/m)

This equation holds in free space with no charges or currents. The product $\mu_0 \epsilon_0$ sets the propagation speed.

Wave equation for magnetic field

$\nabla^2 \vec{B} - \mu_0\epsilon_0\frac{\partial^2 \vec{B}}{\partial t^2} = 0$

The structure is identical to the electric field equation. Both $\vec{E}$ and $\vec{B}$ satisfy the same wave equation, which means they propagate at the same speed and are coupled together as a single electromagnetic wave.

Solutions to wave equation

The solutions are electromagnetic waves propagating at the speed of light:

$c = \frac{1}{\sqrt{\mu_0\epsilon_0}} \approx 3 \times 10^8 \text{ m/s}$

The general solution can be decomposed into plane waves of the form:

$\vec{E}(\vec{r},t) = \vec{E}_0 \, e^{i(\vec{k}\cdot\vec{r} - \omega t)}$

$\vec{B}(\vec{r},t) = \vec{B}_0 \, e^{i(\vec{k}\cdot\vec{r} - \omega t)}$

• $\vec{E}_0$, $\vec{B}_0$ are the (complex) amplitudes
• $\vec{k}$ is the wave vector, pointing in the propagation direction with magnitude $k = \omega/c$
• $\omega$ is the angular frequency

You can verify these are solutions by substituting back into the wave equation; doing so yields the dispersion relation $k^2 = \mu_0 \epsilon_0 \, \omega^2$, which confirms $\omega/k = c$.

Plane electromagnetic waves

Plane waves are the simplest solutions to the wave equation. Their wavefronts (surfaces of constant phase) are infinite parallel planes perpendicular to $\vec{k}$. While no real wave is truly a plane wave, they're an excellent approximation in the far field of any source, where wavefront curvature is negligible.

Plane wave solutions

$\vec{E}(\vec{r},t) = \vec{E}_0 \, e^{i(\vec{k}\cdot\vec{r} - \omega t)}$

$\vec{B}(\vec{r},t) = \vec{B}_0 \, e^{i(\vec{k}\cdot\vec{r} - \omega t)}$

Key relationships:

• The wave vector magnitude is the wavenumber: $k = |\vec{k}| = 2\pi/\lambda$
• Angular frequency relates to ordinary frequency: $\omega = 2\pi f$
• The phase velocity is $v_p = \omega/k = f\lambda$
• $\vec{E}$, $\vec{B}$, and $\vec{k}$ are mutually perpendicular (this follows from $\nabla \cdot \vec{E} = 0$ and $\nabla \cdot \vec{B} = 0$)
• The amplitudes are related by $|\vec{B}_0| = |\vec{E}_0|/c$

Polarization of plane waves

Polarization describes the orientation of the electric field vector in the plane transverse to propagation. Since $\vec{B}$ is determined by $\vec{E}$ (via $\vec{B} = \hat{k} \times \vec{E}/c$), specifying the electric field polarization fully characterizes the wave.

• Linear polarization: $\vec{E}$ oscillates along a fixed direction. The two components of $\vec{E}$ are in phase (or exactly out of phase).
• Circular polarization: $\vec{E}$ traces a circle. The two orthogonal components have equal amplitude and a $\pm 90°$ phase difference.
• Elliptical polarization: The general case. Unequal amplitudes or an arbitrary phase difference between components produces an elliptical trace.

Any polarization state can be decomposed into two orthogonal linear polarizations with appropriate amplitudes and phases.

Poynting vector and energy flux

The Poynting vector gives the power per unit area carried by an electromagnetic wave:

$\vec{S} = \vec{E} \times \vec{H}$

where $\vec{H} = \vec{B}/\mu_0$ in free space. For a plane wave, $\vec{S}$ points along $\vec{k}$.

The time-averaged intensity (what you'd measure with a detector) is:

$\langle |\vec{S}| \rangle = \frac{|\vec{E}_0|^2}{2\eta}$

where $\eta = \sqrt{\mu_0/\epsilon_0} \approx 377 \, \Omega$ is the intrinsic impedance of free space. This impedance relates the electric and magnetic field amplitudes: $|\vec{E}_0| = \eta |\vec{H}_0|$.

Boundary conditions at interfaces

When an electromagnetic wave hits an interface between two media, part of the wave reflects and part transmits. The boundary conditions require continuity of the tangential components of $\vec{E}$ and $\vec{H}$ across the interface. These conditions, combined with the plane wave solutions in each medium, lead to the Fresnel equations.

Reflection and transmission coefficients

The reflection coefficient $r$ and transmission coefficient $t$ are amplitude ratios:

$r = \frac{E_r}{E_i}, \qquad t = \frac{E_t}{E_i}$

These depend on:

• The refractive indices $n_1$ and $n_2$ of the two media
• The angle of incidence $\theta_i$
• The polarization of the incident wave (s or p)

Note that $r$ and $t$ are amplitude ratios, not power ratios. For power, you need the reflectance $R = |r|^2$ and transmittance $T = 1 - R$ (in lossless media).

Fresnel equations

For s-polarization (TE, electric field perpendicular to the plane of incidence):

$r_s = \frac{n_1\cos\theta_i - n_2\cos\theta_t}{n_1\cos\theta_i + n_2\cos\theta_t}$

$t_s = \frac{2n_1\cos\theta_i}{n_1\cos\theta_i + n_2\cos\theta_t}$

For p-polarization (TM, electric field parallel to the plane of incidence):

$r_p = \frac{n_2\cos\theta_i - n_1\cos\theta_t}{n_2\cos\theta_i + n_1\cos\theta_t}$

$t_p = \frac{2n_1\cos\theta_i}{n_2\cos\theta_i + n_1\cos\theta_t}$

Here $\theta_t$ is found from Snell's law: $n_1 \sin\theta_i = n_2 \sin\theta_t$.

Brewster's angle

At Brewster's angle, the p-polarized reflection coefficient vanishes ($r_p = 0$). This happens when the reflected and refracted rays are perpendicular to each other.

$\tan\theta_B = \frac{n_2}{n_1}$

At this angle, only s-polarized light is reflected. For an air-glass interface ($n_2 \approx 1.5$), Brewster's angle is about $56.3°$. This is the principle behind polarizing filters and glare reduction.

Total internal reflection

When light goes from a denser medium to a less dense one ($n_1 > n_2$), there's a critical angle beyond which all light is reflected:

$\sin\theta_c = \frac{n_2}{n_1}$

For angles $\theta_i > \theta_c$, no propagating transmitted wave exists. Instead, an evanescent wave appears in the second medium: its amplitude decays exponentially with distance from the interface, carrying no time-averaged power across it.

Total internal reflection is the operating principle behind optical fibers and certain prism designs. For glass-to-air ($n_1 = 1.5$, $n_2 = 1.0$), the critical angle is about $41.8°$.

Electromagnetic wave propagation

How a wave behaves as it travels depends on the electromagnetic properties of the medium: permittivity $\epsilon$, permeability $\mu$, and conductivity $\sigma$. These parameters determine the wave speed, attenuation, and dispersion.

Propagation in lossless media

In a lossless medium ($\sigma = 0$, no absorption), the wave equation becomes:

$\nabla^2 \vec{E} - \mu\epsilon\frac{\partial^2 \vec{E}}{\partial t^2} = 0$

The wave propagates without attenuation. The phase velocity is:

$v_p = \frac{1}{\sqrt{\mu\epsilon}} = \frac{c}{n}$

where the refractive index is $n = \sqrt{\mu_r \epsilon_r}$ (with $\mu_r$ and $\epsilon_r$ the relative permeability and permittivity). Vacuum, air at most frequencies, and ideal dielectrics fall into this category.

Propagation in lossy media

When $\sigma \neq 0$, the medium absorbs energy from the wave. The wave equation picks up a first-order time derivative:

$\nabla^2 \vec{E} - \mu\epsilon\frac{\partial^2 \vec{E}}{\partial t^2} - \mu\sigma\frac{\partial \vec{E}}{\partial t} = 0$

The plane wave solution now has a decaying envelope: $\vec{E} \propto e^{-\alpha z} e^{i(\beta z - \omega t)}$, where $\alpha$ is the attenuation constant and $\beta$ is the phase constant.

$\alpha = \omega\sqrt{\frac{\mu\epsilon}{2}}\left(\sqrt{1 + \left(\frac{\sigma}{\omega\epsilon}\right)^2} - 1\right)^{1/2}$

The ratio $\sigma/\omega\epsilon$ is called the loss tangent. It determines whether a material behaves more like a dielectric ($\sigma/\omega\epsilon \ll 1$) or a conductor ($\sigma/\omega\epsilon \gg 1$). In a good conductor, the fields penetrate only a short distance called the skin depth $\delta = 1/\alpha \approx \sqrt{2/\omega\mu\sigma}$.

Dispersion in electromagnetic waves

Dispersion means the phase velocity depends on frequency. In a dispersive medium, different frequency components of a pulse travel at different speeds, causing the pulse to spread out.

Dispersion arises when $\epsilon$, $\mu$, or $\sigma$ are frequency-dependent. The dispersion relation $\omega(k)$ encodes this: if $\omega$ is not simply proportional to $k$, the medium is dispersive. Glass, water, and plasmas are all dispersive.

Phase and group velocity

These two velocities describe different aspects of wave propagation:

• Phase velocity $v_p = \omega/k$: the speed at which a single-frequency wavefront moves. In dispersive media, this can exceed $c$.
• Group velocity $v_g = d\omega/dk$: the speed at which the envelope of a wave packet (and thus energy/information) propagates.

In a non-dispersive medium, $v_p = v_g$. In dispersive media they differ. A phase velocity greater than $c$ doesn't violate relativity because no energy or information travels faster than $c$; the group velocity (or more precisely, the signal velocity) remains at or below $c$.

Guided electromagnetic waves

Guided waves are confined to propagate along a structure such as a waveguide, transmission line, or optical fiber. The guiding structure imposes boundary conditions that restrict the fields to discrete modes, each with its own field pattern and cutoff frequency.

Waveguides and modes

A waveguide is typically a hollow metallic tube (rectangular or circular cross-section). The conducting walls force the tangential electric field to zero at the boundary, which quantizes the allowed field patterns.

• Each mode is labeled by integers $(m, n)$ that count the number of half-wavelength variations across the waveguide dimensions.
• Every mode has a cutoff frequency $f_c$: below this frequency, the mode cannot propagate and instead decays exponentially (evanescent mode).
• The dominant mode has the lowest cutoff frequency: $\text{TE}_{10}$ for rectangular waveguides, $\text{TE}_{11}$ for circular waveguides.
• Modes are classified as TE (transverse electric, $E_z = 0$) or TM (transverse magnetic, $H_z = 0$). TEM modes (both $E_z = 0$ and $H_z = 0$) cannot exist in a single-conductor hollow waveguide.

Transverse electric (TE) modes

In TE modes, the electric field has no component along the propagation direction ($E_z = 0$), while the magnetic field does ($H_z \neq 0$). The longitudinal magnetic field component $H_z$ acts as a generating function from which all transverse components are derived.

For a rectangular waveguide with dimensions $a \times b$ ($a > b$):

$H_z = H_0 \cos\!\left(\frac{m\pi x}{a}\right)\cos\!\left(\frac{n\pi y}{b}\right) e^{i(k_z z - \omega t)}$

The transverse field components ($E_x, E_y, H_x, H_y$) are obtained by applying the transverse gradient operators to $H_z$, divided by $k_c^2$. The cutoff wavenumber is:

$k_c = \sqrt{\left(\frac{m\pi}{a}\right)^2 + \left(\frac{n\pi}{b}\right)^2}$

and the propagation constant along $z$ is $k_z = \sqrt{k^2 - k_c^2}$, where $k = \omega\sqrt{\mu\epsilon}$. When $k < k_c$, $k_z$ becomes imaginary and the mode is evanescent.

Transverse magnetic (TM) modes

In TM modes, the magnetic field is entirely transverse ($H_z = 0$), while the electric field has a longitudinal component ($E_z \neq 0$). Now $E_z$ is the generating function:

$E_z = E_0 \sin\!\left(\frac{m\pi x}{a}\right)\sin\!\left(\frac{n\pi y}{b}\right) e^{i(k_z z - \omega t)}$

Note the sine functions (vs. cosine for TE), which ensure $E_z = 0$ at the conducting walls. For TM modes, both $m$ and $n$ must be nonzero ($m \geq 1, n \geq 1$), so the lowest TM mode in a rectangular waveguide is $\text{TM}_{11}$, which always has a higher cutoff frequency than $\text{TE}_{10}$.

The cutoff wavenumber and propagation constant use the same formulas as for TE modes. The transverse components are again derived from $E_z$ via the transverse gradient operators divided by $k_c^2$.