TM11 lowest mode and cross-field amplifier

Proof that TM₁₁ is the lowest TM mode

For TM modes in a rectangular waveguide (a × b, a > b), H_z = 0 and E_z must satisfy the wave equation with E_z = 0 on all four walls (tangential E vanishes on a perfect conductor). Solving by separation of variables:

E_z(x,y,z) = E0 sin(mπx/a) sin(nπy/b) e^(-jβz)
kc² = (mπ/a)² + (nπ/b)²
fc  = (c/2) √[(m/a)² + (n/b)²]

• If m = 0, then sin(0) = 0 and E_z = 0 everywhere. All other transverse fields come from E_z (E_t, H_t ∝ ∇_t E_z), so the whole field is zero and no TM₀ₙ mode exists.
• If n = 0, likewise E_z = 0, so no TMₘ₀ mode exists.
• So both m ≥ 1 and n ≥ 1. The smallest values are m = n = 1, which give the lowest TM cut-off:

fc(TM11) = (c/2) √[(1/a)² + (1/b)²]

Example: for WR-90 (a = 2.286 cm, b = 1.016 cm), fc(TM₁₁) ≈ 16.2 GHz, while fc(TE₁₀) = 6.56 GHz. So TM₁₁ is the lowest TM mode, but TE₁₀ is still the dominant mode of the guide.

Circular-beam cross-field amplifier (reentrant CFA / amplitron)

A cross-field amplifier (CFA) is an M-type tube in which DC E and B fields are at right angles. The circular, reentrant type (the amplitron) is built like a magnetron, but the anode is a slow-wave structure of coupled cavities (vanes) with an RF input and an RF output.

        RF in       RF out
          \           /
       _/ \_/ \_/ \_/ \_   cavity (vane)
      /   circular beam   \  slow-wave anode
     |  ( cathode / sole ) |  B axial
      \_   spokes  ->     _/
        \_/ \_/ \_/ \_/
     drift space between in and out

Working

1. A cylindrical cathode (or sole) and the anode give a radial E field, and a magnet gives an axial B field. Electrons drift around in a circular path at the E × B drift velocity v = E/B.
2. The RF input fed into the cavity structure travels around as a slow wave. Its phase velocity is set close to the electron drift velocity (synchronous condition).
3. As in a magnetron, phase focusing makes the electrons in retarding RF fields give potential energy to the wave and move toward the anode. Electrons in accelerating fields are returned to the cathode. Spokes form and grow with the wave.
4. The wave grows as it travels from input to output, and the amplified RF is taken out. A drift region without the slow-wave structure stops the RF from feeding back, so the tube amplifies instead of oscillating.

Features: high efficiency (about 40–70%), moderate gain (about 10–20 dB), broadband (about 10%), high peak power. Used as power amplifiers in radar and satellite transmitters.