Multi-level quantum electrodynamic calculation of spontaneous emission and small signal gain in high voltage free electron lasers

Abstract

Using the Weisskopf-Wigner technique, a self consistent quantum electrodynamic (SCQED) theory of spontaneous emission of radiation and single photon small signal gain is developed for high voltage free electron lasers (FEL). Excellent agreement is obtained simultaneously to our knowledge for the first time between the predictions and the experimental observations for lineshift, linewidth and gain. The SCQED theory predicts lineshift and broadening due to quantum mechanical effects for linear, helical and tapered undulator FELs which are not predicted by the classical/conventional FEL theories, but which have been observed ^{4,5,18,22,23,45,46}. Excellent agreement is obtained between the SCQED theory predicted spontaneous emission spectra and the 1980–81 ACO FEL^4,18, ACO Optical Klystron FEL^45,46, Stanford 10.6 μm FEL²² and Stanford 3.4 μm FEL²³ experimental spectra. This agreement is much better than the prediction from the classical/conventional FEL theory which gives errors of many tens of percent. We show that the spontaneous emission spectrum obtained from classical/conventional FEL theories is valid only in the limit of a short undulator containing a small number of periods. The small signal gain derived from the SCQED theory is shown to reduce to Colson's gain formula^12,34 in the classical limit. However, the SCQED theory predicts significant reductions in the small signal gain which agree well with the ACO gain data⁵, and are not predicted well by Colson's formula. Due to the non-neglible finite electron state lifetime, it is discovered that a fundamental physical gain limit exists which is universal to all types of FELs within the limits of the single photon transition scheme considered (i.e. if multiphoton effects are ignored). Finally, the implications of the theoretically obtained results are discussed for practical conditions of experimental interest. It is shown that under practical experimental conditions quantum effects can be quite important in the FEL.