Backward Wave Oscillator - The Slow-wave Structure

The Slow-wave Structure

The needed slow-wave structures must support a Radio Frequency (RF) electric field with a longitudinal component; the structures are periodic in the direction of the beam and behave like microwave filters with passbands and stopbands. Due to the periodicity of the geometry, the fields are identical from cell to cell except for a constant phase shift Φ. This phase shift, a purely real number in a passband of a lossless structure, varies with frequency. According to Floquet's theorem (see Floquet theory), the RF electric field E(z,t) can be described at an angular frequency ω, by a sum of an infinity of "spatial or space harmonics" E_n

E(z,t)=

where the wave number or propagation constant k_n of each harmonic is expressed as:

k_n=(Φ+2nπ)/p (-π<Φ<+п)

z being the direction of propagation, p the pitch of the circuit and n an integer.

Two examples of slow-wave circuit characteristics are shown, in the ω-k or Brillouin diagram:

• on figure (a), the fundamental n=0 is a forward space harmonic (the phase velocity v_n=ω/k_n has the same sign as the group velocity v_g=dω/dk_n), synchronism condition for backward interaction is at point B, intersection of the line of slope v_e - the beam velocity - with the first backward (n = -1) space harmonic,

• on figure (b) the fundamental (n=0) is backward

A periodic structure can support both forward and backward space harmonics, which are not modes of the field, and cannot exist independently, even if a beam can be coupled to only one of them.

As the magnitude of the space harmonics decreases rapidly when the value of n is large, the interaction can be significant only with the fundamental or the first space harmonic.