Duality relation for the Maxwell system

Area of Science:

• Physics
• Materials Science
• Electromagnetism

Background:

• The behavior of light in periodic structures like photonic crystals is crucial for optical device design.
• Understanding the low-frequency electromagnetic response of these materials is a long-standing challenge.

Purpose of the Study:

• To establish a theoretical link between 3D and 2D finite photonic crystals within the low-frequency limit.
• To generalize existing solutions for periodic media to piecewise continuous permittivity profiles.

Main Methods:

• Generalization of classical results (Keller and Dykhne's chessboard problem) to periodic media.
• Application of homogenization theory for elliptic operators.
• Illustrative examples using integral equation and variational approaches (fictitious charges, finite-element method).

Main Results:

• A theorem is presented that connects the vector Maxwell system for 3D and 2D finite photonic crystals at low frequencies.
• The study extends Mendelson's findings on polycrystalline and multiphase media.
• Demonstration of the theoretical framework using computational methods.

Conclusions:

• The presented theorem provides a unified framework for analyzing photonic crystals.
• The methods offer a robust approach for studying electromagnetic properties of periodic materials.
• This work contributes to the theoretical understanding of photonic crystals and their applications.