Measurement of the violation of the Madey theorem induced by a diverging wave

Simultaneous measurements have been conducted of the spectra of the gain and the forward spontaneous radiation in the Orsay free-electron laser so that the relative shift of these two curves can be extracted. The results are in agreement with the Madey theorem in the low divergence limit, and demonstrate the violation of the theorem produced as the Rayleigh range of the wave becomes comparable to or smaller than the length of the interaction region. The single-particle theory is extended to account for the divergence of the wave to first order. The predictions from this linearized model are in reasonable agreement with the experimental results in view of the fundamental limitations on this kind of experiment.