Questions: Transverse Electric (TE) Modes

Question 1

A rectangular waveguide is excited at a frequency between the cutoff of TE₁₀ and TE₂₀. A higher-order mode like TE₂₀ is also launched into the guide. What happens to the TE₂₀ mode?

• A. It propagates but at a lower phase velocity than TE₁₀
• B. It reflects back toward the source and interferes with the incident TE₁₀ mode
• C. It decays exponentially along the guide, dying out within a few 'skin depths'
• D. It propagates at the speed of light c, unlike the TE₁₀ mode

Answer: (C)

Below its cutoff frequency, the longitudinal wavenumber of TE₂₀ becomes imaginary: kz = iγ. The mode field then goes as e^(−γz) — evanescent decay, not propagation. No power is transported by the evanescent mode; it simply dies out over a short distance. This is directly analogous to quantum tunneling below a potential barrier or total internal reflection in optics. Single-mode operation exploits this: the guide is designed so only the dominant TE₁₀ is above cutoff, keeping all higher modes evanescent and the signal clean.

Question 2

A TE mode in a rectangular waveguide is found to have a phase velocity of 1.8c at a particular frequency. Is this consistent with special relativity?

• A. No — special relativity forbids any velocity exceeding c, so this is physically impossible
• B. Yes — the phase velocity can exceed c because phase velocity does not carry information or energy; the group velocity remains below c
• C. Yes — Maxwell's equations in guided structures supersede special relativity
• D. No — a phase velocity above c means the mode is below its cutoff frequency and actually evanescent

Answer: (B)

Phase velocity vph = ω/kz > c is entirely consistent with special relativity. What special relativity forbids is transmitting *information* or *energy* faster than c. In a waveguide, energy travels at the group velocity vg = c²/vph, which is always less than c when vph > c. The phase velocity describes the rate at which a constant-phase surface moves — not the speed of any physical carrier. This is the same reason that the 'bright spot' from a spinning laser pointer can sweep a distant wall faster than c without violating relativity.

Question 3

Increasing the cross-sectional width (a) of a rectangular waveguide lowers the cutoff frequency of the dominant TE₁₀ mode.

• T. True
• F. False

Answer: True

The cutoff frequency for the TE_mn mode in a rectangular waveguide is ωc,mn = cπ√((m/a)² + (n/b)²). For TE₁₀ (m=1, n=0), this simplifies to ωc = cπ/a. Since ωc is inversely proportional to a, making the guide wider lowers the cutoff frequency. A larger guide supports lower frequencies in its dominant mode — which is why microwave waveguides for lower frequencies (e.g., the 1–2 GHz L band) are physically larger than those for millimeter waves.

Question 4

A hollow metallic rectangular waveguide can support TEM modes (with both Ez = 0 and Hz = 0) if operated at sufficiently high frequencies.

• T. True
• F. False

Answer: False

TEM modes require two separate conductors between which a static-like electric field can exist (as in coaxial cable). A hollow single-conductor waveguide cannot support TEM modes at any frequency. TEM modes require a zero cutoff frequency and a static solution between two conductors; a single hollow tube has no inner conductor for field lines to terminate on. This is why a hollow rectangular guide only supports TE and TM modes, never TEM — a fundamental topological constraint, not just an engineering limitation.

Question 5

Explain what cutoff frequency means for a TE mode in a waveguide and what physically distinguishes operation above cutoff from operation below cutoff.

Model answer: Cutoff frequency is the minimum frequency at which a TE mode can propagate as a traveling wave. It arises from the discrete eigenvalues of the transverse Helmholtz equation imposed by the boundary conditions. Above cutoff, the longitudinal wavenumber kz = √(ω²/c² − kc²) is real — the mode carries power as a traveling wave. Below cutoff, kz becomes imaginary (iγ), and the mode decays exponentially as e^(−γz) — it is evanescent and carries no net power along the guide.

The cutoff is not an arbitrary design choice but a mathematical consequence of the boundary conditions. Each mode has its own characteristic transverse wavenumber kc,mn (the eigenvalue), and propagation requires ω > c·kc,mn. Below this threshold, the guide acts like a high-pass filter for that mode. The physical picture: the mode is trying to bounce between the walls at an angle, and below cutoff the geometry simply does not allow a self-consistent traveling pattern to form — only a spatially decaying one.