Modes: TE, TM, and TEM

In a hollow conducting structure (waveguide or cavity), the electromagnetic field decomposes into modes classified by which field component is parallel to the chosen longitudinal direction — here the z (= c) axis.

• TE (transverse electric): the electric field has no longitudinal component (E_z = 0); only the magnetic field has a non-zero H_z. The transverse E lines lie in the xy plane. Example: TE₁₀₁ has E pointing along y, varying as one half-cosine along x and one half-cosine along z.
• TM (transverse magnetic): the magnetic field has no longitudinal component (H_z = 0); only the electric field has a non-zero E_z. Example: TM₁₁₀ has E pointing along z, varying as one half-sine along x and one half-sine along y, with H circulating in the xy plane.
• TEM (transverse electromagnetic): both E_z = 0 and H_z = 0 — the field is purely transverse, like a plane wave. TEM solutions require two or more separated conductors (coax, parallel plate, stripline) so that a static-like transverse potential pattern can be set up between them. A closed rectangular cavity has a single connected conductor (its walls), so it cannot support TEM — only TE and TM appear in the table.

The indices (m, n, p) count the number of half-wavelengths of the standing-wave pattern along (x, y, z). TE₁₀₁ thus has one half-wave along x, none along y, one along z.

Formulas

• Resonant frequency (TE_mnp and TM_mnp, same formula): fmnp = (c0 / 2√(εr μr)) · √((m/a)² + (n/b)² + (p/c)²)
• Phase velocity in the medium: v = c0 / √(εr μr); free-space wavelength λ0 = c0 / f.
• Mode allowed indices (with c as the longitudinal z-axis):
  • TE_mnp: m, n ≥ 0 (not both zero), p ≥ 1. E_z = 0.
  • TM_mnp: m, n ≥ 1, p ≥ 0. H_z = 0.
• When all three indices are nonzero, TE_mnp and TM_mnp are two independent modes at the same frequency.
• TEM modes do not exist in a closed rectangular cavity (single conductor).

Constant: c0 = 2.99792458 × 108 m/s.