Questions: Transverse Magnetic (TM) Modes

Question 1

In a rectangular waveguide, why is TM₁₁ the lowest-order TM mode, while TE₁₀ is the lowest-order TE mode?

• A. TM modes inherently require higher frequencies than TE modes, so their mode indices must both be at least 1
• B. The boundary condition Ez = 0 at the walls forces both m and n ≥ 1, since setting either to zero makes Ez vanish everywhere; TE modes have no such constraint on Hz
• C. TM₁₀ exists but has a higher cutoff frequency than TE₁₀, so TM₁₁ is only the second-lowest TM mode
• D. The transverse magnetic field in TM modes prohibits a zero index in the y-direction only, requiring n ≥ 1

Answer: (B)

For TM modes, Ez ∝ sin(mπx/a)sin(nπy/b). Setting m = 0 or n = 0 forces a sine argument to zero, making Ez identically zero everywhere — a trivial solution with no actual mode. For TE modes, Hz ∝ cos(mπx/a)cos(nπy/b), and cos(0) = 1, so setting m = 0 (TE₁₀) still gives a non-trivial field distribution. The difference lies in the Dirichlet (sine) vs. Neumann (cosine) nature of the two boundary conditions.

Question 2

A rectangular waveguide operates at a frequency between the cutoff frequencies of TM₁₁ and TM₂₁. Which TM modes propagate?

• A. None — at a frequency between two mode cutoffs, all TM modes are evanescent
• B. Both TM₁₁ and TM₂₁ — once the guide is above any mode's cutoff, all modes propagate simultaneously
• C. Only TM₁₁ — its cutoff is below the operating frequency, so it propagates; TM₂₁ is above cutoff and cannot propagate
• D. Only TM₁₁ — it is above its cutoff frequency, so it propagates; TM₂₁ is below its cutoff and is evanescent

Answer: (D)

A mode propagates if and only if the operating frequency exceeds that mode's cutoff frequency. Between fc(TM₁₁) and fc(TM₂₁), the frequency is above the lower cutoff (TM₁₁ propagates) and below the upper cutoff (TM₂₁ is evanescent — exponentially decaying). Option C reverses the logic: 'above cutoff' means the mode propagates, not the reverse.

Question 3

In a TM waveguide mode, both the phase velocity and the group velocity exceed the speed of light in vacuum.

• T. True
• F. False

Answer: False

Only the phase velocity (v_p = ω/β) exceeds c; the group velocity (v_g = dω/dβ), which carries energy and information, is always less than c. Their product satisfies v_p · v_g = c². This apparent superluminal phase velocity is not a violation of special relativity because phase fronts do not carry energy or information — only the group velocity does.

Question 4

For TM modes in a rectangular waveguide, all four transverse field components (Ex, Ey, Hx, Hy) can be derived entirely from the longitudinal field Ez once the boundary conditions are satisfied.

• T. True
• F. False

Answer: True

This is a fundamental structural property of the TE/TM decomposition. For TM modes (Hz = 0), the 2D Helmholtz equation determines Ez, and the waveguide equations then give all transverse components algebraically in terms of Ez and its derivatives. Ez is the 'generating function' for the entire mode — knowing it completely specifies the field distribution.

Question 5

Explain why TM modes in a rectangular waveguide cannot have either mode index equal to zero, and what this implies about TM cutoff frequencies compared to TE modes.

Model answer: For TM modes, Ez ∝ sin(mπx/a)sin(nπy/b). If m = 0 or n = 0, the corresponding sine is identically zero, making Ez = 0 everywhere — no mode exists. So both m, n ≥ 1, and the lowest TM mode is TM₁₁. TE modes use cosines for Hz, so cos(0) = 1 ≠ 0 allows m = 0 or n = 0, giving the TE₁₀ fundamental mode. Since TM₁₁ has a higher cutoff frequency than TE₁₀, the dominant mode in a standard rectangular waveguide is always TE₁₀.

The underlying reason is the difference in boundary conditions: TM modes satisfy a Dirichlet condition (Ez = 0 at the wall), leading to sine eigenfunctions that vanish at zero index; TE modes satisfy a Neumann condition (∂Hz/∂n = 0 at the wall), leading to cosine eigenfunctions that do not vanish at zero index. This has major practical consequences — microwave engineers always design single-mode waveguides around TE₁₀, not any TM mode.