  The soliton-like superradiant regime of free-electron lasers (FEL) offers a
promising path towards ultrashort pulses, beyond the natural limit dictated by
the bandwidth of the high-gain FEL instability. In this work we present a
three-dimensional theory of the superradiant regime, including the effects of
beam emittance and energy spread. Our work takes advantage of recent
developments in non-linear FEL theory to provide a fully analytical description
of soliton-like superradiance. Our theory proves the existence of a
diffraction-dominated steady-state regime in which the superradiant peak power
grows indefinitely while leaving the pulse duration and on-axis intensity
almost unchanged. These results are in excellent agreement with
three-dimensional simulations and are supported by recent experimental results
at the Linac Coherent Light Source. This work advances non-linear FEL theory
and provides a theoretical framework for the next generation of attosecond
x-ray FELs.
