the output of the TWT for $s = 1.27$ : (a) test beam ($I_b=2$ nA) without electrostatic wave, (b) with helix mode (trapping domain shown by red line with phase velocity marked by a square) and beam mode (trapping domain shown by blue line with phase velocity marked by a triangle) at 30 MHz, (c) with an additional controlling wave at $60$ MHz and phase velocity given by grey circle and dotted line.[]{data-label="fig:127"}](Fig10c.eps "fig:"){width="7.5cm" height="7.2cm"}

![Measure of helix mode coupling constant $C_h$ calculated from the measured upper bound velocity of trapping domain (circles) at frequency $ f $ equal to (a) 30[MHz]{} and (b) 60[MHz]{}. Black line parabola is obtained using the average value of $ C_h $. Gray parabolas are given by the average error. The parabolic fit is only valid beyond the vertical segment indicating the wave amplitude beyond which beam trapping occurs over the length of the TWT.[]{data-label="fig:const3060"}](Fig11.eps){width="7.5cm"}

![Measure of coupling constant $C_b$, at frequency equal to 30[MHz]{} for a beam mode at $2.5\times 10^6$[m/s]{}. Black line parabola is obtained using the average value of $ C_b
$, calculated from the upper bound velocity of the trapping domain between the vertical segments where the beam become trapped 