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Building blocks for space-time non-separable pulses

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.