Experimental Observation of Nonlinear Synchronization due to a Single Wave

A test electron beam is propagated in a specially designed traveling wave tube. It interacts with a nonresonant wave, and its energy distribution is recorded at the tube output. We report the direct experimental observation of the spatially periodic electron velocity bunching, and of a nonlinear effect on the electron velocity modulation: the synchronization of the particles with the wave responsible for Landau damping in plasma physics. The results are explained by second order perturbation theory in the wave amplitude.

Figure 1

Traveling wave tube rendering: (1) helix, (2) electron gun, (3) trochoidal analyzer, (4) antenna, (5) glass vacuum tube, (6) slotted rf ground cylinder, and (7) magnetic coil.

Figure 2

Measured velocity distribution function in a single wave at $v_{\varphi }=4.06\times {10}^6\mathrm{m}/\mathrm{s}$ with increasing amplitude: (a) 3D plot, (b) 2D plot with first order estimates of modulation (lines) and trapping (half parabola) domains.

Figure 3

Exact value (dashed curve) and second order estimate (continuous curve) of the velocity averaged over initial position vs time, for $ϵ=0.04$, $u_0=1$.

Figure 4

Measured velocity distribution function for a single wave with phase velocity $4.06\times {10}^6\mathrm{m}/\mathrm{s}$ and amplitude 18 mV vs emitting antenna position $z$ from the gun end; second order estimate of mean test beam velocity (continuous curve).

Figure 5

Mean test beam velocity measured from Fig. 4 (shaggy curve joining data), estimated by second order perturbation theory (smooth curve).