The slow-wave structure Backward-wave oscillator

(a) forward fundamental space harmonic (n=0), (b) backward fundamental

the needed slow-wave structures must support radio frequency (rf) electric field longitudinal component; structures periodic in direction of beam , behave microwave filters passbands , stopbands. due periodicity of geometry, fields identical cell cell except constant phase shift Φ. phase shift, purely real number in passband of lossless structure, varies frequency. according floquet s theorem (see floquet theory), rf electric field e(z,t) can described @ angular frequency ω, sum of infinity of spatial or space harmonics en

e(z,t) =




∑

n
=
−
∞


+
∞




e

n




e

j
(

ω

t
−


k

n



z
)




{\displaystyle \sum _{n=-\infty }^{+\infty }{e_{n}}e^{j({\omega }t-{k_{n}}z)}}

where wave number or propagation constant kn of each harmonic expressed as:

kn = (Φ + 2nπ) / p (-π < Φ < +п)

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

two examples of slow-wave circuit characteristics shown, in ω-k or brillouin diagram:

on figure (a), fundamental n=0 forward space harmonic (the phase velocity vn=ω/kn has same sign group velocity vg=dω/dkn), synchronism condition backward interaction @ point b, intersection of line of slope ve - beam velocity - first backward (n = -1) space harmonic,
on figure (b) fundamental (n=0) backward

a periodic structure can support both forward , backward space harmonics, not modes of field, , cannot exist independently, if beam can coupled 1 of them.

as magnitude of space harmonics decreases rapidly when value of n large, interaction can significant fundamental or first space harmonic.