Threshold of a mirror-less photonic free-electron laser oscillator pumped by one or more electron beams

Transmitting electrons through a photonic crystal can result in stimulated emission and the generation of coherent Cerenkov radiation. Here we consider a photonic-crystal slab consisting of a two-dimensional, periodic array of bars inside a rectangular waveguide. By appropriately tapering the bars at both ends of the slab, we numerically show that an electromagnetic wave can be transmitted through the waveguide filled with the photonic-crystal slab with close to zero reflection. Furthermore, the photonic-crystal slab allows transmission of electrons in the form of one or more beams. By appropriately designing the photonic-crystal slab, we obtain a backward wave interaction at low electron-beam energy of around 15 kV, that results in distributed feedback of the radiation on the electrons without any external mirrors being present. Here we discuss the dynamics of the laser oscillator near threshold and numerically show that the threshold current can be distributed over multiple electron beams, resulting in a lower current per beam.