Guided Waves and Cavities
~8 min read · Tier toggle changes depth (T1 plain · T2 standard · T3 rigour)
You should be comfortable with vector calculus identities, specifically the curl and divergence of vector fields, and have a working knowledge of solving second-order differential equations with boundary conditions.
Why this matters
Confining electromagnetic energy is the fundamental principle behind lasers, microwave ovens, and particle accelerators.
While free-space plane waves are idealised mathematical constructs that extend to infinity, real-world applications require electromagnetic fields to be contained within finite regions. This lecture explores how boundary conditions imposed by conducting walls transform the continuous spectrum of free-space waves into a discrete set of allowed modes. By analysing these guided waves and resonant cavities, we bridge the gap between abstract wave theory and the practical design of high-frequency electronic devices, revealing how geometry dictates the possible frequencies and field distributions of electromagnetic radiation.
The Physical Picture
Imagine a rectangular box made of perfect electric conductors. If you inject electromagnetic energy into this box, the waves reflect off the walls. Unlike a string on a guitar, which only vibrates along its length, electromagnetic waves in a cavity can oscillate in three dimensions. The conducting walls act as rigid boundaries: the electric field must vanish at the surface, while the magnetic field must be tangent to it. This constraint forces the wave to form standing wave patterns, much like the harmonics of a musical instrument. Only specific frequencies, determined by the dimensions of the box, are permitted to exist stably within the cavity. These discrete frequencies are called resonant modes, and they represent the natural "fingerprint" of the cavity's geometry.
Boundary Conditions at Conductors
To solve for the fields inside a cavity, we must first establish the boundary conditions at the interface with a perfect conductor. Inside a perfect conductor, the electric field is zero because any non-zero field would drive infinite currents. Consequently, the tangential component of the electric field just outside the conductor must also be zero because the tangential component of \(\mathbf{E}\) is continuous across any surface (Faraday's law applied to a thin loop). Mathematically, if \(\hat{\mathbf{n}}\) is the unit normal vector pointing out of the conductor, the condition is \(\hat{\mathbf{n}} \times \mathbf{E} = 0\) at the surface.
For the magnetic field, Gauss's law for magnetism, \(\nabla \cdot \mathbf{B} = 0\), makes the normal component of \(\mathbf{B}\) continuous across the boundary. A perfect conductor admits no time-varying field inside (any change would be cancelled instantly by induced currents), so the oscillating normal component just outside must also vanish. Thus, \(\hat{\mathbf{n}} \cdot \mathbf{B} = 0\) at the surface. These two conditions—tangential \(\mathbf{E}\) is zero and normal \(\mathbf{B}\) is zero—are the keys to determining the allowed modes.
Deriving the Cavity Modes
Consider a rectangular cavity with dimensions \(a\), \(b\), and \(d\) along the \(x\), \(y\), and \(z\) axes, respectively. We seek solutions to the wave equation \(\nabla^2 \mathbf{E} - \frac{1}{c^2} \frac{\partial^2 \mathbf{E}}{\partial t^2} = 0\) that satisfy the boundary conditions. Separation of variables gives each component of \(\mathbf{E}\) as a product of sines and cosines, for example \(E_x = E_{0x} \cos(k_x x) \sin(k_y y) \sin(k_z z)\, e^{-i\omega t}\). The sine factors make \(E_x\) vanish on the walls \(y = 0, b\) and \(z = 0, d\), where it is tangential; \(E_y\) and \(E_z\) are built the same way.
For the field to vanish at the boundaries, the arguments of the sine functions must be integer multiples of \(\pi\). This quantises the wavevectors: \(k_x = \frac{n_x \pi}{a}\), \(k_y = \frac{n_y \pi}{b}\), and \(k_z = \frac{n_z \pi}{d}\), where \(n_x, n_y, n_z\) are non-negative integers. Not all combinations are allowed: if two of the three integers are zero, every field component vanishes, so at most one mode number may be zero. The resonant frequency for a given mode \((n_x, n_y, n_z)\) is determined by the dispersion relation for a vacuum cavity:
This equation shows that the frequency is discrete, depending on the integer mode numbers and the physical dimensions of the cavity. The lowest frequency mode, known as the fundamental mode, corresponds to the smallest non-zero combination of integers that satisfies the transversality condition \(\mathbf{k} \cdot \mathbf{E} = 0\).
Energy Storage and Quality Factor
In an ideal cavity with perfect conductors, the electromagnetic energy oscillates between the electric and magnetic fields without loss. The total energy \(U\) stored in the cavity is the sum of the electric and magnetic energy densities integrated over the volume \(V\). For a sinusoidal steady state, the time-averaged energy is constant. However, in real cavities, the walls have finite conductivity, leading to resistive losses. The rate of energy loss is characterised by the quality factor \(Q\), defined as \(Q = \omega \frac{\text{Energy Stored}}{\text{Power Lost}}\).
A high \(Q\) factor indicates that the cavity can store energy for many cycles before it decays significantly. This is crucial in applications like atomic clocks and particle accelerators, where long coherence times are required. The skin depth \(\delta\) from the previous lecture plays a critical role here, as the power loss is proportional to the surface resistance, which depends on \(\delta\). Minimising the surface area-to-volume ratio and using high-purity conductors are standard techniques to maximise \(Q\).
The derivations
Structure of Monochromatic Plane Waves
In a source-free, linear, homogeneous, isotropic, non-dispersive medium (vacuum unless stated), every monochromatic plane-wave solution of Maxwell's equations, E(r,t) = E0 ei(k·r − ωt) and B(r,t) = B0 ei(k·r − ωt), is strictly transverse: k·E = k·B = 0. The triad (E, B, k) is mutually orthogonal and right-handed, the two fields oscillate in phase, their amplitudes obey |E| = c|B|, and the complex vector E0 encodes the polarization state — linear, circular, or elliptical.
Waves in Conductors and Skin Depth
For a monochromatic plane wave of angular frequency \( \omega \) propagating in a linear, homogeneous, ohmic medium (permittivity \( \varepsilon \), permeability \( \mu \), conductivity \( \sigma \), with \( \vec{J}_f = \sigma \vec{E} \)), Maxwell's equations require the wavevector to be complex, \( \tilde{k} = k + i\kappa \), with \( \tilde{k}^2 = \mu\varepsilon\omega^2 + i\,\mu\sigma\omega \). The fields decay as \( e^{-\kappa z} \) while oscillating as \( e^{i(kz-\omega t)} \). In the good-conductor regime \( \sigma \gg \varepsilon\omega \) this reduces to \( k = \kappa = 1/\delta \) with the skin depth \( \delta = \sqrt{2/(\mu\sigma\omega)} \), and the magnetic field lags the electric field by \( \pi/4 \) with magnitude ratio \( |B_0|/|E_0| = \sqrt{\mu\sigma/\omega} \).
A common misconception is that the electric field must be zero everywhere on the conducting surface. In reality, only the tangential component is zero; the normal component can be non-zero, corresponding to surface charge density. Confusing these components leads to incorrect mode structures, particularly in cavities with non-rectangular geometries where the normal direction varies across the surface. C
Worked example
Problem. A rectangular cavity has dimensions \(a = 10\ \mathrm{cm}\), \(b = 5\ \mathrm{cm}\), and \(d = 5\ \mathrm{cm}\). Determine the resonant frequency of the fundamental mode, assuming the mode is TE\(_{101}\) (the electric field points along \(y\) only, \(E_y \propto \sin(\pi x/a)\sin(\pi z/d)\)).
- Identify the mode numbers for TE\(_{101}\): \(n_x = 1\), \(n_y = 0\), \(n_z = 1\).
- Calculate the wavevector components: \(k_x = \frac{\pi}{a} = \frac{\pi}{0.1\ \mathrm{m}} = 10\pi\ \mathrm{m^{-1}}\), \(k_y = 0\), \(k_z = \frac{\pi}{d} = \frac{\pi}{0.05\ \mathrm{m}} = 20\pi\ \mathrm{m^{-1}}\).
- Compute the magnitude of the wavevector \(k = \sqrt{k_x^2 + k_y^2 + k_z^2} = \sqrt{(10\pi)^2 + 0 + (20\pi)^2} = \pi\sqrt{100 + 400} = 10\pi\sqrt{5}\ \mathrm{m^{-1}}\).
- Use the dispersion relation \(\omega = c k\) with \(c = 3.00 \times 10^8\ \mathrm{m/s}\).
- Calculate \(\omega = 3.00 \times 10^8 \times 10\pi\sqrt{5} \approx 3.00 \times 10^8 \times 70.25 \approx 2.11 \times 10^{10}\ \mathrm{rad/s}\).
- Convert to frequency \(f = \frac{\omega}{2\pi} \approx \frac{2.11 \times 10^{10}}{6.28} \approx 3.35 \times 10^9\ \mathrm{Hz}\).
Units check. The wavevector \(k\) is in \(\mathrm{m^{-1}}\), speed of light \(c\) is in \(\mathrm{m/s}\), so \(\omega\) is in \(\mathrm{s^{-1}}\) (radians per second). Dividing by \(2\pi\) (dimensionless) gives frequency in \(\mathrm{Hz}\) (\(\mathrm{s^{-1}}\)). The result is physically reasonable for a cavity of this size.
Because \(b = d\), the TE\(_{110}\) mode has exactly the same frequency: the two modes are degenerate. Making \(b\) and \(d\) slightly different splits them, which is how practical cavities keep a single clean resonance.
Three quick tests
- Why must the tangential component of the electric field be zero at the surface of a perfect conductor?
Answer
If it were non-zero, it would exert an infinite force on the free charges in the conductor, causing them to move until the field is cancelled. In the ideal limit, the field must vanish to maintain equilibrium. - How does the resonant frequency of a cavity change if the length \(a\) is doubled?
Answer
The frequency is inversely proportional to the dimensions. Doubling \(a\) reduces the \(k_x\) term by half, generally lowering the total wavevector magnitude and thus the resonant frequency. - What is the physical significance of a high quality factor \(Q\)?
Answer
A high \(Q\) indicates low energy loss per cycle, meaning the cavity can store electromagnetic energy for a long time, leading to sharp resonance peaks and high spectral selectivity.
Problem set for PU-204 →
Each derivation in this unit ends with five graded problems with full solutions.
Lecture notes checked line by line by the editor; every number in the worked example was recomputed.
Derivations cited: D-165 · D-173 · Threads: force, energy, light