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Linking classical optics and contemporary photonics via the Hill-Floquet framework
1State Key Laboratory on Integrated Optoelectronics, College of Electronic Science and Engineering, International Center of Future Science, Jilin University, 2699 Qianjin Street, Changchun 130012, People's Republic of China.
Summary
We unify the analysis of electromagnetic waves in layered optical media using the Hill operator and transfer matrix methods. This framework reveals connections between topology, geometric phases, and topological interface states in photonic systems.
Area of Science:
- Physics
- Optics
- Condensed Matter Physics
Background:
- One-dimensional stratified optical media offer a clear framework for studying wave propagation, spectral theory, and topology.
- Existing models often lack a unified approach to analyze diverse phenomena in these structures.
Purpose of the Study:
- To present a unified framework for analyzing electromagnetic waves in layered optical media.
- To connect topological properties, such as Zak phases, with real-space characteristics like Wannier functions.
- To extend the analysis to time-modulated and space-time periodic photonic systems.
Main Methods:
- Utilizing the Hill operator, Floquet theory, and the monodromy (transfer) matrix formalism.
- Employing operator methods and transfer-matrix techniques.
- Applying modern topological concepts for characterization.
Main Results:
- Established the monodromy matrix as a key representation linking dispersion, band formation, and geometric phases.
- Demonstrated a direct correspondence between geometric phases and Wannier functions, explaining topological interface states.
- Extended the formalism to time-modulated and space-time periodic media, revealing phenomena like frequency conversion and nonreciprocity.
Conclusions:
- Stratified optical media provide a versatile platform for studying wave physics across static, driven, and periodic systems.
- The unified framework offers an intuitive understanding of topological phenomena in photonic structures.
- The approach bridges concepts from spectral theory, topology, and wave propagation.
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