Homogenization of periodic structures: One layer is “bulk”

¹ Department of Electrical and Computer Engineering, The University of Akron - OH 44325-3904, USA
² Department of Electrical, Computer, and Systems Engineering, Case Western Reserve University Cleveland, OH, USA 44106
³ Departments of Radiology and Bioengineering and the Graduate Group in Applied Mathematics and Computational Science, University of Pennsylvania - Philadelphia, PA 19104, USA

^(a) igor@uakron.edu (corresponding author)

Received: 7 September 2021
Accepted: 18 February 2022

Abstract

In homogenization theories of periodic electromagnetic structures, such as photonic crystals or metamaterials, it is commonly assumed that the accuracy of the effective medium representation is affected by the thickness of the sample in two different ways. The first factor is the optical length (OL) of wave propagation within the sample. For a large OL, minor deviations in the wave number may strongly affect transmission and reflection. The second factor is the putative convergence to bulk characteristics (CB), once the number of layers in the sample has approached or passed a certain threshold. The theoretical analysis and numerical examples of this letter show that, while the OL influence is real, CB is not actually relevant; in a sense, one layer is already “bulk”.