  We review a technique for solving a class of classical linear partial
differential systems of relevance to physics in Minkowski spacetime. All the
equations are amenable to analysis in terms of complex solutions in the kernel
of the scalar Laplacian and a complexified Hertz potential. The
complexification prescription ensures the existence of regular physical
solutions with chirality and propagating, non-singular, pulse-like
characteristics that are bounded in all three spatial dimensions. The technique
is applied to the source-free Maxwell, Bopp-Land\'e-Podolsky and linearised
Einstein field systems, and particular solutions are used for constructing
classical models describing single-cycle laser pulses and a mechanism is
discussed for initiating astrophysical jets. Our article concludes with a brief
introduction to spacetime Clifford algebra ideals that we use to represent
spinor fields. We employ these to demonstrate how the same technique used for
tensor fields enables one to construct new propagating, chiral, non-singular,
pulse-like spinor solutions to the massless Dirac equation in Minkowski
spacetime.
