Ultracold neutrons (UCNs) are widely used in the physics of elementary particles and fundamental interactions, and they can potentially be used in neutron scattering. However, most of these studies are limited by the available UCN densities and fluxes. One way to increase them is to use peak fluxes in pulsed neutron sources that are orders of magnitude higher than average. In the present work, a concept of UCN sources is proposed that makes it possible to implement this idea. We propose producing very cold neutrons (VCNs) in converters located in neutron sources and extracting and slowing them down to UCN energies by an escaping decelerating material or a magnetic trap. For both pulsed and permanent neutron sources, this method can provide a high conversion efficiency of VCNs to UCNs with low losses of phase space density.