Explicit Construction of Floquet-Bloch States from Arbitrary Solution Bases of the Hill Equation

For the Hill equation describing one-dimensional periodic systems, a constructive formulation is developed for generating Floquet-Bloch states directly from arbitrary pairs of linearly independent solutions. One-dimensional photonic crystals are used as a concrete illustration. Explicit closed-form formulas map an arbitrary fundamental system to the corresponding Floquet-Bloch basis via the monodromy matrix, including the generic Jordan band-edge case, without reliance on canonically normalized solutions. The construction can be expressed directly in terms of the transfer matrix, making the residual representation freedom transparent and providing an implementation-ready framework for analytical and numerical studies of periodic systems.