Spontaneous emission in waveguide free-electron masers near waveguide cutoff

In this work spontaneous emission is investigated in a waveguide free-electron maser, taking into account previously untreated interaction effects in the vicinity of the waveguide cutoff frequency. Our study is based on the exact waveguide excitation equations, formulated in the frequency domain for a single electron moving in a planar magnetostatic wiggler. An analytical solution of the amplitude of the excited waveguide mode in the frequency domain was obtained using the Green function technique and allows us to calculate the spectral density of the radiated power and the time-dependent radiated field with good accuracy using a numerical inverse Fourier transform. The obtained solution shows that for TE-modes the spectral density of the radiated energy tends to infinity at the cutoff frequency of a lossless waveguide. The character of this singularity is, however, such that the total radiated energy is finite. The radiated electromagnetic field in the time domain has the form of very long (of the order of tens of characteristic times on the scale of L_w/c, where L_w is the wiggler length and c is the speed of light) pulse, lagging behind the electron, at the carrier of cutoff frequency, in addition to two finite wave packets, corresponding to the two synchronism frequencies. The results of a numerical calculation of the radiated energy spectral density and of the radiated electromagnetic field in the time domain are presented.