  Space-time non-separable pulses hold promise for topological information
transfer, probing ultra-fast light-matter interactions and engaging toroidal
excitations in matter. Spurred by recent advances in ultra-fast and topological
optics, these exotic electromagnetic excitations are now becoming the focus of
growing experimental efforts. Many practical questions are yet to be answered
regarding their generation, detection and light matter interactions. In
particular, can these pulses be constructed from plane waves or other simple
but experimentally accessible waves? Here we demonstrate that they can and as
an example we provide analytical expressions for the characteristic case of the
flying doughnut pulse which allows the expansion and synthesis of the pulse
from sets of monochromatic beams, single-cycle pulses, and plane waves.
