For a standard rectangular waveguide supporting TEm0 modes, which mode between TE20 and TE30 has the lower cutoff frequency, and why?

Introduction / Context:
Rectangular waveguides support discrete transverse electric (TE) and transverse magnetic (TM) modes. Each mode begins to propagate only when the operating frequency exceeds its cutoff frequency. Understanding which modes cut on first is fundamental for single-mode operation and low-dispersion design.

Given Data / Assumptions:

• Standard rectangular waveguide with dimensions a (broad wall) and b (narrow wall).
• Comparing TEm0 modes: TE20 and TE30.
• Same physical guide; material losses do not change cutoff ordering.

Concept / Approach:

For a rectangular guide, the cutoff frequency for TEmn is f_c = (c / 2) * sqrt( (m/a)^2 + (n/b)^2 ). For TEm0 specifically (n = 0), this simplifies to f_c = (m * c) / (2a). Thus cutoff scales linearly with m for these modes.

Step-by-Step Solution:

Verification / Alternative check:

Waveguide mode charts and measured S21 vs frequency show TE10 first, followed by TE20, then TE30 for increasing frequency in a fixed guide, consistent with f_c ∝ m for TEm0.

Why Other Options Are Wrong:

• Same cutoff: False because m differs (2 vs 3) for TEm0.
• TE30 lower: Opposite of the linear dependence.
• Material dependence: Material affects loss, not geometric cutoff order.
• Independent of indices: Contradicts the mode formula.

Common Pitfalls:

Confusing TE and TM formulas; forgetting that for TEm0, only the broad-wall dimension a sets cutoff; assuming dielectric filling changes ordering (it scales all cutoffs similarly).

Final Answer:

TE20 has lower cutoff frequency than TE30