Foundations of Mathematical Methods in Nanophotonics

How do we control light on the scale of a single wavelength? To design modern optical devices, you must understand the mathematical framework that governs electromagnetism in structured materials. This course provides a clear, accessible entry point into the physics and mathematics of nanophotonics, bridging the gap between classical electromagnetism and solid-state physics. You will transition from a basic understanding of wave equations to analyzing complex optical phenomena like photonic band gaps, slow light, and waveguiding. By studying the mathematical structures behind these behaviors, you will gain the theoretical tools needed to model, analyze, and predict how light behaves in nanostructured media. What you'll learn: Understand the foundational concepts of electromagnetism, Maxwell's equations, and wave propagation in structured media; Apply linear algebra and eigensystem formulation to solve electromagnetic wave equations; Use Bloch's theorem to analyze periodic structures and calculate photonic band gaps; Explore group theory and symmetry principles to predict optical behavior and conservation laws; Analyze optical systems using perturbation methods and coupled-mode theory; Study modern computational formulations and inverse design concepts used in contemporary nanophotonics research. This course begins with essential terminology and the basic physics of light-matter interaction before guiding you through the algebraic and perturbation methods used to solve real-world optical problems. Through structured written explanations, mathematical proofs, and conceptual exercises, you will build a solid theoretical foundation. This program is designed for beginners in applied mathematics, physics, or engineering who want to explore nanophotonics without requiring advanced prior knowledge. Start reading today to unlock the mathematical secrets of structured light.