Masiello, David JRossi, Andrew W2026-04-202026-04-202026-04-202026Rossi_washington_0250E_29325.pdfhttps://hdl.handle.net/1773/55463Thesis (Ph.D.)--University of Washington, 2026As atomic and nanoscale materials continue to advance and reveal novel physical properties, the development of reliable and precise characterization techniques has become increasingly crucial. Motivated by recent experimental advances in electron energy loss spectroscopy (EELS) performed inside electron microscopes, this dissertation presents a general theoretical framework to describe the momentum-resolved inelastic electron scattering of wide-field electrons from two-dimensional materials, spanning from atomic crystals to nanophotonic arrays. The formalism incorporates fully-retarded electromagnetic interactions, which are required to accurately model nanophotonic material responses. By accounting for the energy–momentum dependence of the probing electron’s polarization and relativistic kinematics, the theory establishes selection rules for scattering processes within and beyond the first Brillouin zone, reveals optically dark transitions outside the light cone, and illustrates the roles of electron velocity and scattering geometry. Building on this theory and recent advances in structuring the transverse phase of electron beams, pinwheel free electron states carrying well-defined pseudoangular momentum (PAM) are also introduced. These beams enable direct detection of chiral phonons, a long-standing challenge for conventional characterization techniques. Taken together, this work establishes a pathway for tailoring the polarization and symmetry of free electron probes to selectively couple to targeted material excitations, enabling precise investigation of emergent material properties.application/pdfen-USnoneElectron energy loss spectroscopyElectron microscopyNanophotonicsPhononsPlasmonsQuantum materialsChemistryMaterials ScienceCondensed matter physicsChemistryThesis