Circular waveguides and symmetry: Which modes exhibit circular symmetry (no azimuthal variation, i.e., m = 0)?

Introduction:
Modes in circular waveguides are labeled TE_mn or TM_mn, where m is the azimuthal index and n is the radial index. Circular symmetry means the fields do not depend on the azimuthal angle φ; this property is pivotal in certain low-loss transmission applications and in mode converters.

Given Data / Assumptions:

• Circular, air-filled, perfectly conducting waveguide.
• Mode labeling follows standard conventions.
• Circular symmetry corresponds to m = 0.

Concept / Approach:

For circular symmetry, m must be zero, so any TE_0n and TM_0n mode qualifies. The options that meet this requirement are TE01 and TM01 together. Modes with m ≥ 1 (e.g., TE11, TM11, TE21) vary with φ and are not circularly symmetric.

Step-by-Step Solution:

Verification / Alternative check:

Field plots for TE01/TM01 show concentric circles with purely radial variation; no angular lobes appear, confirming circular symmetry and explaining their use in low-loss long-distance waveguide systems (especially TE01 at high power).

Why Other Options Are Wrong:

• TE11 or TE11 and TE21: these have m ≠ 0, so they are not circularly symmetric.
• TE01 alone omits TM01, which is also symmetric.
• TM11 only: m = 1, not symmetric.

Common Pitfalls:

Confusing the dominant mode in circular guides (TE11) with circular symmetry. Dominance relates to lowest cutoff; symmetry relates to angular dependence.

Final Answer:

TE01 and TM01