Short wavelength free electron laser operation in a periodic dielectric

A layered dielectric with periodic permittivity profile, ε=ε₀+ε₂ cos(k_0xx+k _0zz), is suitable for the generation of coherent radiation at short wavelengths with moderate energy electron beams. When an electron beam is passed through the dielectric with velocity v_bzˆ, along with a copropagating electromagnetic wave perturbation, E&oarr;~A&oarr; exp[-i(ωt-kz)], the displacement vector acquires a (ω,k&oarr;+k&oarr;₀) component, driving a space charge field E&oarr;₊ when ω=(k+k_0z)v_b. The space charge field couples withthe periodic permittivity to produce displacement vector at (ω,k&oarr;), driving the original electromagnetic perturbation. The frequency of radiation ω≃k_0zv_b/(1-v _bε^1/2₀/c) and the growth rate Γ≈[(k_0x²ω_pb² ωε₂²)/(8γ₀ ³ε₀³k_0z²)] ^1/3 where ω_pb is the beam plasma frequency and γ₀ is the relativistic gamma factor. The geometrical effects, however, limit full exploitation of the resonant enhancement of ω and Γ around v_b=c/ε^1/2₀