Calculation of the brilliance of advanced accelerator-based x-ray sources

In regime of diffraction-limited electron beams at x-ray wavelengths, the spectral width of fundamental spontaneous undulator radiation is at ~ 1% level, and down to 0.01% for free-electron lasers. Consequently, the traditional characterization of brilliance in units of 0.1% bandwidth tends to overestimate the actual brilliance of longitudinally coherent sources. An update version of the brilliance plot in units of 0.01% relative bandwidth is proposed. It is shown that, in order to match experimental and simulated data, the calculation has to take into consideration a non-uniform electron current profile in self-amplified spontaneous emission, and the scaling of the light pulse duration with the harmonic jump in seeded architectures. More accurate and general brilliance expressions are therefore provided. A closed-form is found for the saturation length, providing an exact value in place of well-known but semi-quantitative estimations reported in the literature.