  We introduce two novel variants of the serpentine waveguide slow-wave
structure (SWS), often utilized in millimeter-wave traveling-wave tubes (TWTs),
with an enhanced interaction impedance. Using dispersion engineering in
conjunction with transfer matrix methods, we tune the guided wavenumber
dispersion relation to exhibit stationary inflection points (SIPs), and also
non-stationary, or tilted inflection points (TIPs), within the dominant TE10
mode of a rectangular waveguide. The degeneracy is found below the first upper
band-edge associated with the bandgap where neighboring spatial harmonics meet
in the dispersion of the serpentine waveguide (SWG) which is threaded by a beam
tunnel.
  The structure geometries are optimized to be able to achieve an SIP which
allows for three-mode synchronism with an electron beam over a specified
wavenumber interval in the desired Brillouin zone. Full-wave simulations are
used to obtain and verify the existence of the SIP in the three-way coupled
waveguide and fine-tune the geometry such that a beam would be in synchronism
at or near the SIP. The three-way waveguide SWS exhibits a moderately high
Pierce impedance in the vicinity of a nearly-stationary inflection point,
making the SWS geometry potentially useful for improving the power gain and
basic extraction efficiency of millimeter-wave TWTs. Additionally, the
introduced SWS geometries have directional coupler-like behavior, which enables
distributed power extraction at frequencies near the SIP frequency.
