Short note: Bunching effect in reflex klystron

In a reflex klystron, bunching happens in the repeller region. The electrons pass the single cavity gap, are turned back by a negative repeller, and return to the gap as bunches.

 cathode  anode/cavity gap        repeller (−Vr)
 |>====== [ gap ] ------------------>  |
            ^ <---- returning bunch ----|
            |   fast e: goes deeper, returns later
         RF out slow e: turns back sooner

Process

1. Velocity modulation: an electron crossing the gap at time t₁ leaves with v = v₀[1 + (β_i V₁/2V₀) sin ωt₁], where v₀ = √(2eV₀/m).
2. Reflection: in the retarding field between gap and repeller, an electron with higher velocity travels farther and takes longer to return. A slower electron returns sooner. The round-trip time is

T' = 2 m L v(t1) / [e (Vr + V0)]

3. Bunch formation: the bunch forms around the electron that crossed when the gap voltage was going from positive to negative (the reference electron). Electrons that left a little earlier (sped up, so they return later) and those that left a little later (slowed down, so they return sooner) arrive back at the gap at the same time as the reference electron.
4. Energy transfer: for oscillation, the bunch must return when the gap field most strongly retards it:

ω T' = 2πn − π/2,  n = 1, 2, 3, ...
modes: N = n − 1/4 = 3/4, 1¾, 2¾, ... cycles

The returning bunch gives kinetic energy to the cavity, and the oscillation is sustained.

Key points

• The bunching parameter is X' = β_i V₁ θ₀' / (2V₀), and the output is largest near X' ≈ 2.41.
• Changing V_r changes the transit time, so the frequency can be electronically tuned a little.
• The efficiency is at most about 22.8% in theory and much lower in practice. Output is in mW. Used as local oscillators and in lab signal sources.