Linking classical optics and contemporary photonics via the Hill-Floquet framework

One-dimensional stratified optical media provide a remarkably rich platform for exploring wave propagation, spectral theory, and topology within a mathematically transparent setting. In this Perspective, we present a unified framework for analyzing electromagnetic waves in layered structures based on the Hill operator, Floquet theory, and the monodromy (transfer) matrix formalism. We emphasize the central role of the monodromy matrix as a finite-dimensional representation of the Floquet operator. This representation establishes a link between dispersion relations, band formation, Bloch waves, and geometric phases. Building on this foundation, we develop the topological characterization of one-dimensional photonic bands through Zak phases and their symmetry protection, and establish a direct correspondence between geometric phases and the real-space structure of Wannier functions. This correspondence provides an intuitive explanation for topological interface states in dimerized bilayer photonic structures, placing stratified optical media on equal footing with paradigmatic models such as the Su--Schrieffer--Heeger chain. We further extend the formalism to time-modulated and space-time periodic media, where temporal Floquet harmonics generate synthetic dimensions and promote the Hill operator to a matrix-valued Floquet--Hill operator. Within this extended setting, phenomena such as frequency conversion, nonreciprocity, and topologically protected Floquet interface states arise naturally from the geometry of the underlying Floquet spectrum. By combining analytic operator methods, transfer-matrix techniques, and modern topological concepts, this Perspective highlights stratified photonic media as a versatile and conceptually unifying platform for studying wave physics across static, driven, and space-time periodic systems.