Circular waveguides and symmetry: Which of the following circular waveguide modes are circularly symmetric (no azimuthal variation)?

Introduction:
Modes in circular waveguides are labeled TE_mn or TM_mn, where the first index m corresponds to azimuthal variations and the second index n corresponds to radial order. Recognizing which modes are circularly symmetric helps in low-loss transmission and mode conversion design.

Given Data / Assumptions:

• Circular metallic waveguide with air filling.
• TE_0n modes have m = 0 (no dependence on azimuthal angle φ).
• We are considering TE01 and TE02 specifically.

Concept / Approach:

Azimuthal symmetry means the fields do not vary with φ. In circular guides, m = 0 satisfies this, so any TE_0n or TM_0n mode is circularly symmetric. Therefore, TE01 and TE02 are both circularly symmetric modes.

Step-by-Step Solution:

Verification / Alternative check:

Field patterns for TE01 and TE02 show concentric rings with no angular variation, confirming circular symmetry useful for certain low-loss TE01 long-haul systems.

Why Other Options Are Wrong:

• Neither or a single mode only contradict m = 0 symmetry criterion.
• TM11 only: TM11 has m = 1, hence not circularly symmetric.

Common Pitfalls:

Confusing dominant mode (TE11) with symmetry; dominance relates to cutoff, not azimuthal variation.

Final Answer:

both (a) and (b)