Short note: TE and TM modes

In a rectangular waveguide (a × b, a > b, wave travelling along +z) the fields cannot be TEM, so they are split into two families defined by which longitudinal field is zero.

TE (transverse electric) modes

The electric field has no component along the direction of travel: E_z = 0, H_z ≠ 0.

H_z = H₀ cos(mπx/a) cos(nπy/b) e^{−jβz}

• m = 0, 1, 2, ...; n = 0, 1, 2, ... (not both zero).
• TE₁₀ is the dominant mode of the guide.
• Wave impedance Z_TE = η/√(1 − (f_c/f)²), which is greater than η.

TM (transverse magnetic) modes

The magnetic field has no component along z: H_z = 0, E_z ≠ 0.

E_z = E₀ sin(mπx/a) sin(nπy/b) e^{−jβz}

• m = 1, 2, ...; n = 1, 2, ... (both at least 1, since E_z must be zero at the walls).
• TM₁₁ is the lowest TM mode.
• Wave impedance Z_TM = η√(1 − (f_c/f)²), which is less than η.

Common relations

f_c = (c/2)√[(m/a)² + (n/b)²]
β = β₀ √(1 − (f_c/f)²)
λ_g = λ₀/√(1 − (f_c/f)²)
v_p = c/√(1 − (f_c/f)²),   v_g = c√(1 − (f_c/f)²)

where η = 377 Ω in air and v_p v_g = c². TE_mn and TM_mn with the same m, n have the same cut-off frequency. Below cut-off, β is imaginary and the mode does not propagate.