Dispersion relation for a large-orbit gyrotron

The interaction between a beam of large-orbit axis encircling electrons and a waveguide is studied by linearizing the fluid equations. A dispersion relation is derived for an annular layer of relativistic electrons of small thickness guided by a uniform magnetic field along the waveguide axis. It is shown in the general case that the dispersion relation couples the TE and TM modes of the waveguide at the two resonances omega -k_z upsilon _0z=(l+or-1) omega _c and omega -k_z upsilon _0z=l omega _c. The resonance at omega -k_z upsilon _0z=(l+or-1) omega _c which is associated with the finite velocity shear (( delta nu _0z/ delta rho ) not=0) and finite k_z is shown to give the dominant contribution to the dispersion relation.