Absolute Instability near Band Edges in a Traveling Wave Tube

We re-examine the beam mode and its interaction with the circuit mode near the lower and upper band edges in a traveling wave tube. We find that an absolute instability may arise, according to the Briggs-Bers criterion, if the beam current is sufficiently high, even if the beam mode intersects with the circuit mode at a point in the (w, k)$=$(frequency, wavenumber) plane with a positive group velocity. This finding differs from the previous works [1, 2] for the lower band edge, and points to the vulnerability to absolute instabilities at both the upper and lower band edges of a TWT. When the threshold current is exceeded, the Green's function, at a fixed position, exponentiates in time as t**(1/3) initially, but as (wi*t) at later time, where wi is the imaginary part of w in the unstable pole-pinch root. [1] D. M. H. Hung, et al., \textit{Phys. Rev. Lett}. \textbf{115}, 124801 (2015). [2] A. P. Kuznetsov, et al., \textit{Sov. Radiophys. Electron.} \textbf{27}, 1575 (1984).