Waves in Bounded Regions, Waveguides & Cavity Resonators

Derivation of Boundary Conditions from Maxwell's Integral Equations

When an electromagnetic wave encounters an interface separating two distinct physical media (characterized by $\epsilon_1, \mu_1, \sigma_1$ and $\epsilon_2, \mu_2, \sigma_2$), the fields must satisfy fundamental boundary conditions derived directly from the integral forms of Maxwell's equations.

Let $\hat{\mathbf{n}}$ be the unit normal vector pointing from medium 2 into medium 1.

1. Normal Component of Electric Displacement Field ($\mathbf{D}$)

Construct a Gaussian pillbox of cross-sectional area $\Delta A$ and height $h$ straddling the interface. Applying Gauss's law $\oint_S \mathbf{D} \cdot d\mathbf{a} = Q_{f,\text{enc}}$ in the limit as $h \to 0$:

where $\sigma_f$ is the free surface charge density on the interface. For linear dielectrics without free surface charge ($\sigma_f = 0$):

2. Normal Component of Magnetic Induction Field ($\mathbf{B}$)

Applying Gauss's law for magnetism $\oint_S \mathbf{B} \cdot d\mathbf{a} = 0$ over the same pillbox as $h \to 0$:

Result: The normal component of the magnetic induction $\mathbf{B}$ is always continuous across any interface without exception.

3. Tangential Component of Electric Field ($\mathbf{E}$)

Construct an Amperian loop of width $\Delta l$ parallel to the interface and height $h$ perpendicular to it. Applying Faraday's law $\oint_C \mathbf{E} \cdot d\mathbf{l} = -\frac{d}{dt} \int_S \mathbf{B} \cdot d\mathbf{a}$: As $h \to 0$, the magnetic flux through the loop vanishes ($h \Delta l \to 0$), leaving:

Result: The tangential component of the electric field is always continuous across any boundary.

4. Tangential Component of Magnetic Field Intensity ($\mathbf{H}$)

Applying the Ampère-Maxwell law $\oint_C \mathbf{H} \cdot d\mathbf{l} = I_{f,\text{enc}} + \frac{d}{dt} \int_S \mathbf{D} \cdot d\mathbf{a}$: As $h \to 0$, the displacement current flux vanishes, but a surface free current density $\mathbf{K}_f$ perpendicular to the loop can contribute:

For non-conducting dielectric media where no surface free currents exist ($\mathbf{K}_f = 0$):

Consider a monochromatic plane wave traveling along $+z$ in medium 1 ($z < 0$, parameters $\epsilon_1, \mu_1, n_1$) normally incident upon a flat boundary at $z = 0$ with medium 2 ($z > 0$, parameters $\epsilon_2, \mu_2, n_2$).

Field Formulations

1. Incident Wave:

2. Reflected Wave: (travels along $-z$)

3. Transmitted Wave: (travels along $+z$ in medium 2)

Matching Boundary Conditions at $z = 0$

1. Continuity of Tangential $\mathbf{E}$:

2. Continuity of Tangential $\mathbf{H}$ (assuming $\mu_1 \approx \mu_2 \approx \mu_0$):

Amplitude Reflection and Transmission Coefficients

Solving this $2 \times 2$ algebraic system:

Notice that if $n_2 > n_1$ (denser medium, e.g., air into glass), $r < 0$, corresponding to a $\pi$ phase reversal upon reflection.

Energy Conservation: Reflectance ($R$) and Transmittance ($T$)

The time-averaged power reflected and transmitted per unit area are:

Summing the two:

Energy flux is conserved across the interface.

Phase Matching and the Laws of Reflection & Refraction

Let a plane wave with wavevector $\mathbf{k}_i$ strike the interface ($z=0$) at angle of incidence $\theta_i$ relative to the normal $\hat{\mathbf{z}}$. For boundary conditions to hold at all spatial positions $(x, y)$ on the interface and at all times $t$, the spatial phases must match:

Since $k_i = k_r = n_1 \frac{\omega}{c}$ and $k_t = n_2 \frac{\omega}{c}$:

1. Law of Reflection: $\theta_r = \theta_i$

2. Snell's Law of Refraction: $n_1 \sin\theta_i = n_2 \sin\theta_t$

The Two Orthogonal Polarizations

Any arbitrarily polarized plane wave can be decomposed into two fundamental linear modes:

1. $s$-Polarization (TE / Perpendicular): Electric field $\mathbf{E}$ is perpendicular to the plane of incidence.

2. $p$-Polarization (TM / Parallel): Electric field $\mathbf{E}$ lies entirely within the plane of incidence.

The Fresnel Equations (for $\mu_1 = \mu_2 = \mu_0$)

1. Perpendicular Polarization ($s$ / TE):

2. Parallel Polarization ($p$ / TM):

Derivation of Brewster's Angle ($\theta_B$)

Observe the parallel reflection coefficient:

If the denominator approaches infinity, $r_p$ drops to identically zero. This occurs when:

Substitute this into Snell's law:

This unique angle of incidence is called Brewster's Angle (or the polarizing angle):

For an air-to-glass interface ($n_1 = 1.00, n_2 = 1.50$):

Microscopic Physical Mechanism

When incident light excites atomic bound electrons in medium 2, they oscillate parallel to the transmitted electric field vector $\mathbf{E}_t$. Oscillating electric dipoles radiate with an intensity proportional to $\sin^2\phi$, where $\phi$ is the angle between the dipole axis and the radiation direction. Because $\theta_i + \theta_t = 90^\circ$, the direction of reflected ray $\mathbf{k}_r$ lies precisely along the axis of oscillation of the dipoles! An oscillating dipole radiates zero electromagnetic power along its axis of oscillation. Hence, no reflected wave can be generated for $p$-polarization.

Practical Application: If unpolarized sunlight strikes a glass window or water puddle at Brewster's angle, the reflected light is 100% linearly polarized perpendicular to the plane of incidence ($s$-polarized). Polarized sunglasses with vertical transmission axes completely filter out this blinding horizontal glare.

The Critical Angle ($\theta_c$)

When an electromagnetic wave travels from an optically denser medium into a rarer medium ($n_1 > n_2$, e.g., glass to air):

As $\theta_i$ increases, $\theta_t$ reaches $90^\circ$ before $\theta_i$ does. The angle of incidence for which $\theta_t = 90^\circ$ is the critical angle:

For water into air ($n_1 = 1.333, n_2 = 1.00$): $\theta_c = \arcsin(1/1.333) \approx 48.6^\circ$. For glass into air ($n_1 = 1.50, n_2 = 1.00$): $\theta_c = \arcsin(1/1.50) \approx 41.8^\circ$.

Total Internal Reflection ($\theta_i > \theta_c$)

When $\theta_i > \theta_c$, $\sin\theta_t = \frac{n_1}{n_2} \sin\theta_i > 1$. The cosine of the transmitted angle becomes purely imaginary:

Substituting into the transmitted spatial wave term $e^{i\mathbf{k}_t \cdot \mathbf{r}} = e^{i(k_{tx} x + k_{tz} z)}$:

where $\kappa = \frac{\omega}{c} \sqrt{n_1^2 \sin^2\theta_i - n_2^2}$.

The transmitted electric field is therefore:

Properties of the Evanescent Wave

1. Exponential Decay: The field amplitude decays exponentially into medium 2 with distance $z$ from the interface.

2. Characteristic Penetration Depth ($d_p$):

3. Zero Time-Averaged Power Transport: The time-averaged Poynting vector normal to the boundary is identically zero: $\langle S_z \rangle = 0$. All energy is reflected back into medium 1 ($R = |r|^2 \equiv 1.00$).

4. Frustrated Total Internal Reflection (FTIR): If a third medium is placed within a distance $z < d_p$, photons tunnel through the gap via quantum-like electromagnetic barrier penetration, transmitting energy into the third medium.

When an electromagnetic wave strikes a good conductor or metal (conductivity $\sigma$), Snell's law and the Fresnel equations generalize by introducing the complex refractive index:

where $\kappa = \frac{c\alpha}{\omega}$ is the extinction coefficient.

At normal incidence from air ($n_1 = 1$) into metal:

The reflectance is:

For good conductors at microwave and radio frequencies, $\kappa \approx n \gg 1$:

This relation is the Hagen-Rubens formula. For polished metals (such as gold, silver, and copper), $R > 0.99$ for infrared and microwave radiation, making metals ideal mirrors and waveguide boundaries.

Boundary Value Problem in Rectangular Waveguides

Consider a hollow rectangular metallic pipe of inner width $a$ along $x$ ($0 \le x \le a$) and height $b$ along $y$ ($0 \le y \le b$), with walls made of an ideal conductor (conductivity $\sigma \to \infty$). The wave propagates along $+z$:

Because the walls are perfect conductors, boundary conditions dictate that tangential $\mathbf{E}$ and normal $\mathbf{B}$ must vanish at the boundaries:

Decomposition into Transverse Electric (TE) and Transverse Magnetic (TM) Modes

1. Transverse Electric (TE) Modes ($E_z = 0$):

The longitudinal magnetic field $H_z$ satisfies the 2D Helmholtz equation:

Applying the boundary condition $\frac{\partial H_z}{\partial n} = 0$:

2. Transverse Magnetic (TM) Modes ($H_z = 0$):

Applying $E_z = 0$ on the walls:

Cutoff Frequencies ($f_{c,mn}$)

The transverse cutoff wavenumber is:

The cutoff frequency for mode $(m, n)$ is:

Propagation Constant ($k_z$), Guide Wavelength ($\lambda_g$), and Velocities

• If $f < f_c$: $k_z$ is imaginary. The mode is evanescent and attenuates exponentially; no wave propagation occurs.
• If $f > f_c$: $k_z$ is real and propagation proceeds.

The guide wavelength $\lambda_g$ inside the pipe is:

The phase velocity exceeds the speed of light:

The group velocity (energy velocity) is strictly subluminal:

Notice that:

The Dominant Mode ($\text{TE}_{10}$)

In standard rectangular waveguides with $a > b$, the lowest cutoff frequency belongs to the $\text{TE}_{10}$ mode ($m=1, n=0$):

For this dominant mode, the fields are:

Resonant Microwave Cavities

Closing both ends of a rectangular waveguide with conducting plates at $z = 0$ and $z = d$ forms a closed metallic enclosure called a cavity resonator.

Boundary conditions force standing waves along all three spatial axes:

where $m, n, p$ are mode integers.

Quality Factor ($Q$)

Due to the non-zero surface resistance $R_s$ of the metallic cavity walls, electromagnetic energy stored in the cavity is dissipated as thermal Joule losses. The Quality Factor $Q$ measures the sharpness of the cavity resonance:

For microwave cavities constructed from copper or silver, $Q$ typically ranges from $10^4$ to $10^5$, and in superconducting niobium RF cavities used in particle colliders, $Q$ exceeds $10^{10}$.