An uncertainty principle in electromagnetic homogenization

Nontrivial magnetic response of periodic structures composed of intrinsically nonmagnetic constituents is now a well-established but still intriguing phenomenon; it plays a critical role in negative refraction, cloaking and other unusual effects. Much attention has been devoted to optimal (in some sense) design, whereby the magnetism would be enhanced and engineered to fall in a desirable range for specific applications - absorption, cloaking, lensing, etc. However, we show both analytically and numerically that this artificial magnetism has limitations: the stronger the magnetic response (as measured by the deviation of the effective permeability tensor from identity), the less accurate (“certain”) predictions of the effective medium theory are. We call this the uncertainty principle for the effective parameters of metamaterials. In practice, there is still room for engineering design, but the trade-offs between magnetic response and the accuracy of homogenization must be noted.