A new boundary condition is introduced to calculate the effective impedance matrix of semi-infinite periodic structures such as photonic crystals and metamaterials, which leads to a reduction of the solution space. The obtained effective impedance matrix allows one to relate a matrix to a PC, which includes all of its properties in terms of reflection from its interface. For one-dimensional photonic crystals or multilayer films, it is shown that a closed-form equation can be found for the effective impedance. For two-dimensional photonic crystals the impedance is obtained using the scattering matrices by solving a unilateral quadratic matrix equation. Several examples are outlined to validate the developed scheme. In the examples, the goal is mainly the computation of the reflection from a semi-infinite periodic structure when a plane wave illuminates its boundary.
© 2009 Optical Society of America
Diffraction and Gratings
Original Manuscript: August 27, 2009
Revised Manuscript: October 28, 2009
Manuscript Accepted: November 5, 2009
Published: December 7, 2009
Arya Fallahi and Christian Hafner, "Analysis of semi-infinite periodic structures using a domain reduction technique," J. Opt. Soc. Am. A 27, 40-49 (2010)