In this chapter we discuss a class of solutions to a set of linear field equations on Minkowski spacetime \(\mathcal {M}\) of relevance to spacetime physics. The solutions are all constructed from: The coderivative \(\delta \equiv \star ^{-1}d\star \eta \) ( 8.22 ), and hence the Hodge-de Rham operator \(\Box \equiv \delta d\) ( 8.25 ), are defined with respect to the Minkowski metric \(\mathtt {g}_{\text{Mink}}\) . We call the complex scalar field \(\alpha \) a “wave profile” and use it to construct a complex solution \(\mathcal {F}\) to the free-space vacuum Maxwell equations and define \(F \,\equiv \, \text{Re}(\mathcal {F})\) to be the associated real Maxwell 2-form. The equations under consideration here include the vacuum Maxwell system for the Minkowski-Maxwell field tensor, one of its modifications proposed by Bopp-Landé-Podolsky [146–148], the tensor equation for perturbations of Einstein’s gravitational field equation in a matter-free background Minkowski spacetime and the electromagnetically-neutral massless Dirac equation. It will be shown how all these linear partial differential systems can be formulated in terms of certain differential tensor or spinor operators that render them amenable to analysis. The class of particular solutions derived from these offer useful models of interest in laser physics and astrophysics. In particular, we indicate how to construct both Maxwell free-space models describing propagating finite-energy, multi-chiral laser pulses that are bounded in all three spatial dimensions, and propagating gravitational wave pulses with similar characteristics. The former may offer new channels for quantum encryption and the latter may play a role in the formation of astrophysical jets observed in X-ray spectroscopy. The details of these models are outlined here since a full account can be found in the indicated references.