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 longitudinal direction — here the cylinder axis z.

• TE (transverse electric): E_z = 0 everywhere; only the magnetic field has a longitudinal component. The wall boundary condition fixes the transverse pattern through the zeros of J'm(x).
• TM (transverse magnetic): H_z = 0 everywhere; only the electric field has a longitudinal component. The wall condition Ez(r=R) = 0 fixes the transverse pattern through the zeros of Jm(x). The famous TM₀₁₀ "pillbox" mode has E_z uniform along z and a J0(x0,1 r/R) radial profile, vanishing only at r = R.
• TEM (transverse electromagnetic): both E_z = 0 and H_z = 0 — the field is purely transverse. TEM requires 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 cylindrical cavity has a single connected conductor (its walls), so TEM does not exist here — only TE and TM appear in the table.

The three indices in a cylindrical cavity label mean:

• m: azimuthal index — the field varies as cos(mφ) or sin(mφ) around the axis (m = 0, 1, 2, …).
• n: radial index — selects the n-th positive zero of Jm (for TM) or J'm (for TE), n = 1, 2, 3, ….
• p: longitudinal index — counts half-wavelengths along the cylinder axis. TM allows p = 0 (no z-dependence — this is the pillbox case); TE requires p ≥ 1.

Formulas

Let xm,n be the n-th positive zero of the Bessel function Jm(x), and x'm,n the n-th positive zero of its derivative J'm(x).

• TM_mnp (E_z ≠ 0, H_z = 0): f = (c0 / 2π√(εrμr)) · √((xm,n/R)² + (pπ/d)²), indices m ≥ 0, n ≥ 1, p ≥ 0.
• TE_mnp (H_z ≠ 0, E_z = 0): same formula with xm,n replaced by x'm,n, indices m ≥ 0, n ≥ 1, p ≥ 1.
• For m ≥ 1 modes, the two orthogonal azimuthal orientations (cos mφ, sin mφ) are degenerate and counted once.
• Fundamental mode crossover: TM010 is lowest when d/R < π/√(x0,1² − x'1,1²) ≈ 2.03; otherwise TE111 is lowest.
• The classic "pillbox" mode TM010 has a frequency independent of d: f010 = c0 x0,1 / (2π R √(εrμr)) with x0,1 = 2.4048.
• TEM modes do not exist in a single-conductor closed cavity.

First few Bessel zeros (used here): x0,1 = 2.4048, x1,1 = 3.8317, x2,1 = 5.1356; x'1,1 = 1.8412, x'2,1 = 3.0542, x'0,1 = 3.8317.

Constant: c0 = 2.99792458 × 108 m/s.