Propagation in helical and narrow-band slow wave structures

A variational analysis is developed for a large class of helical slow-wave structures and illustrated for a narrow tape helix. A 2 × 2 determinantal equation accurately describes the dispersion of the wave with predominant n = 1 mode component, both in the ordinary and anomalous regions. A 3 × 3 determinantal equation would be necessary to describe well the dispersion of the ordinary TWT wave at low frequencies. The analysis is modified to include the presence of a filamentary electron beam. It is shown how the equation for the incremental propagation constant takes the Pierce form, provided the frequency does not lie too near cutoff. A variational analysis is also made of the propagation in a general narrow-band cavity chain interacting with a longitudinally-confined beam. It is shown how the equation for the incremental phase shift per cavity is formally related to Pierce´s equation for the incremental propagation constant in a helix TWT. Provided certain necessary (but not sufficient) conditions are met, an appropriately-defined C³parameter exists for the cavity chain. The resonant-slot coupled cavity chain is treated variationally and it is shown how a very accurate dispersion relation obtains. The waves in this structure interacting with a beam are also described formally, under certain conditions, by the Pierce equation for the TWT.