Immittance Matching for Multi-dimensional Open-system Photonic Crystals

TL;DR

This paper proves that the immittance of electromagnetic Bloch waves in photonic crystals is real on reflection planes, simplifying analysis and enabling qualitative evaluation of wave reflection at various interfaces in photonic crystal structures.

Contribution

It provides a general proof that the immittance is real on reflection planes in 1D, 2D, and approximates this in 3D photonic crystals, simplifying wave analysis methods.

Findings

Immittance is real on reflection planes in 1D and 2D PCs.

Approximate real immittance in 3D PCs based on numerical calculations.

A new method for qualitative reflection analysis using immittance matching.

Abstract

An electromagnetic (EM) Bloch wave propagating in a photonic crystal (PC) is characterized by the immittance (impedance and admittance) of the wave. The immittance is used to investigate transmission and reflection at a surface or an interface of the PC. In particular, the general properties of immittance are useful for clarifying the wave propagation characteristics. We give a general proof that the immittance of EM Bloch waves on a plane in infinite one- and two-dimensional (2D) PCs is real when the plane is a reflection plane of the PC and the Bloch wavevector is perpendicular to the plane. We also show that the pure-real feature of immittance on a reflection plane for an infinite three-dimensional PC is good approximation based on the numerical calculations. The analytical proof indicates that the method used for immittance matching is extremely simplified since only the real part of the immittance function is needed for analysis without numerical verification. As an application of the proof, we describe a method based on immittance matching for qualitatively evaluating the reflection at the surface of a semi-infinite 2D PC, at the interface between a semi-infinite slab waveguide (WG) and a semi-infinite 2D PC line-defect WG, and at the interface between a semi-infinite channel WG and a semi-infinite 2D PC slab line-defect WG.