Circular waveguide TE mode field equations

A circular waveguide is a hollow metal tube of inner radius a. Fields are written in cylindrical coordinates (ρ, φ, z) with propagation e^(−jβz). In a TE mode, Ez = 0 and Hz ≠ 0.

          .-----.
        /    ρ    \
       |  o---->   |   radius a
        \         /    wave along z
          '-----'

Wave equation and solution

 ∂²Hz/∂ρ² + (1/ρ)∂Hz/∂ρ + (1/ρ²)∂²Hz/∂φ² + kc² Hz = 0
 kc² = k² - β²

Separation of variables, Hz = R(ρ)·Φ(φ), gives Φ = A sin nφ + B cos nφ and Bessel's equation for R, whose finite solution at ρ = 0 is Jn(kc ρ):

 Hz = (A sin nφ + B cos nφ) Jn(kc ρ) e^(-jβz)

Boundary condition: Eφ = 0 at ρ = a. Since Eφ ∝ J'n(kc ρ):

 J'n(kc a) = 0  ->  kc = p'nm / a

where p'nm is the m-th root of J'n(x).

TEnm field equations

 Hz = (A sin nφ + B cos nφ) Jn(kcρ) e^(-jβz)
 Eρ = -(jωμn/(kc²ρ)) (A cos nφ - B sin nφ) Jn(kcρ) e^(-jβz)
 Eφ =  (jωμ/kc) (A sin nφ + B cos nφ) J'n(kcρ) e^(-jβz)
 Hρ = -(jβ/kc) (A sin nφ + B cos nφ) J'n(kcρ) e^(-jβz)
 Hφ = -(jβn/(kc²ρ)) (A cos nφ - B sin nφ) Jn(kcρ) e^(-jβz)
 Ez = 0

Characteristic equations

• Cut-off frequency: fc = p'nm·c/(2πa) (air-filled)
• Cut-off wavelength: λc = 2πa/p'nm
• Phase constant: β = √(k² − (p'nm/a)²)
• Guide wavelength: λg = λ0/√(1 − (fc/f)²)
• Phase and group velocity: vp = c/√(1 − (fc/f)²), vg = c√(1 − (fc/f)²)
• Wave impedance: Z_TE = ωμ/β = η/√(1 − (fc/f)²)

Roots of J'n(x):

Important modes

• TE11 has the smallest root, so it is the dominant mode of the circular waveguide: fc = 1.841c/(2πa), λc = 3.41a. Example: a = 1 cm gives fc = 8.79 GHz.
• TE01 has circularly symmetric fields and its wall loss falls as frequency rises, so it is used for low-loss long-distance waveguide links.