Onset of electron-seeded cascades in generic electromagnetic fields

QED cascades in a strong electromagnetic field of optical range and arbitrary configuration are considered. A general expression for short-time dependence of the key electron quantum dynamical parameter is derived, allowing one to generalize the effective threshold condition of QED cascade onset. The generalized theory is applied to self-sustained cascades in a single focused laser pulse. According to numerical simulations, if a GeV electron bunch is used as a seed, an ordinary cascade can be converted into a self-sustained one. As an application, it would be also possible to produce this way bright collimated photon beams with up to GeV photon energies.