Asymptotics for metamaterials and photonic crystals

Area of Science:

• Optics and Photonics
• Materials Science
• Electromagnetics

Background:

• Metamaterials and photonic crystals are crucial in modern optics, built from repeating elementary cells.
• Analyzing wave propagation in these complex, periodic structures at high frequencies is challenging.

Purpose of the Study:

• To develop a continuum theory for modeling metamaterial and photonic crystal structures.
• To capture the behavior of high-frequency waves propagating through periodic media.
• To recover classical homogenization coefficients in the low-frequency limit.

Main Methods:

• Asymptotic replacement of discrete structures with a continuum model.
• Development of theory for two-dimensional square lattices with arbitrary holes and Neumann boundary conditions.
• Numerical implementation for cylinders and split-ring resonators.

Main Results:

• The high-frequency homogenization method successfully models wave propagation.
• Demonstrated applications include all-angle negative refraction, omni-directional antennas, endoscopes, and cloaking.
• Highlighted the importance of selecting the correct Brillouin zone for accurate physical effect prediction.

Conclusions:

• The continuum approach provides a powerful tool for understanding wave phenomena in periodic media.
• This method simplifies the analysis of complex optical structures, enabling novel applications.
• Careful consideration of the Brillouin zone is essential for comprehensive analysis.