Competition between coherent emission and broadband spontaneous emission in the quantum free electron laser

We extend previous analyses of spontaneous emission in a quantum free electron laser (QFEL) and competition between spontaneous and coherent QFEL emission to include a broad distribution of photon frequencies and momenta appropriate for spontaneous undulator radiation. We show that although the predictions of monochromatic and broadband models predict different electron momentum distributions for the quantum regime due to spontaneous emission alone after many photon emissions, the inclusion of broadband spontaneous emission has a negligible effect on the competition between spontaneous and coherent emission in the QFEL. Numerical results from both models are well described by the same condition for the threshold/critical value of spontaneous emission rate.