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

TL;DR

This paper presents a direct, explicit method to construct Floquet-Bloch states from arbitrary solutions of the Hill equation, applicable to one-dimensional periodic systems like photonic crystals.

Contribution

It introduces a constructive, closed-form formulation that maps arbitrary fundamental solutions to Floquet-Bloch states using the monodromy matrix, applicable even in Jordan band-edge cases.

Findings

Explicit formulas relate fundamental solutions to Floquet-Bloch states.

Method applies to photonic crystals and other periodic systems.

Framework is suitable for analytical and numerical analysis.

Abstract

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.