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" En where the wave number or propagation constant kn of each harmonic is expressed as kn = (Φ + 2nπ) / p (-π < Φ < +π) 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 vn=ω/kn has the same sign as the group velocity vg=dω/dkn), synchronism condition for backward interaction is at point B, intersection of the line of slope ve - 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. M-type BWO The M-type carcinotron, or M-type backward wave oscillator, uses crossed static electric field E and magnetic field B, similar to the magnetron, for focussing an electron sheet beam drifting perpendicularly to E and B, along a slow-wave circuit, with a velocity E/B. Strong interaction occurs when the phase velocity of one space harmonic of the wave is equal to the electron velocity. Both Ez and Ey components of the RF field are involved in the interaction (Ey parallel to the static E field). Electrons which are in a decelerating Ez electric field of the slow-wave, lose the potential energy they have in the static electric field E and reach the circuit. The sole electrode is more negative than the cathode, in order to avoid collecting those electrons having gained energy while interacting with the slow-wave space harmonic. O-type BWO The O-type carcinotron, or O-type backward