Nonasymptotic Homogenization of Periodic Electromagnetic Structures: Uncertainty Principles

TL;DR

This paper establishes fundamental limits on the accuracy of homogenized models for periodic electromagnetic structures, showing that strong artificial magnetism leads to inherent prediction inaccuracies and limits in reproducing physical properties.

Contribution

The paper introduces two uncertainty principles that constrain the effectiveness of homogenization in periodic electromagnetic media, regardless of the method used.

Findings

Homogenization accuracy decreases with stronger magnetic response.

Homogenization cannot simultaneously predict transmission, reflection, and power dissipation accurately.

Uncertainty principles hold even in highly isotropic structures.

Abstract

We show that artificial magnetism of periodic dielectric or metal/dielectric structures has limitations and is subject to at least two "uncertainty principles". First, the stronger the magnetic response (the deviation of the effective permeability tensor from identity), the less accurate ("certain") the predictions of any homogeneous model. Second, if the magnetic response is strong, then homogenization cannot accurately reproduce the transmission and reflection parameters and, simultaneously, power dissipation in the material. These principles are general and not confined to any particular method of homogenization. Our theoretical analysis is supplemented with a numerical example: a hexahedral lattice of cylindrical air holes in a dielectric host. Even though this case is highly isotropic, which might be thought as conducive to homogenization, the uncertainty principles remain valid.