Relevance of the light line in planar photonic crystal waveguides with weak vertical confinement

P Kaspar,H Jäckel,R Kappeler,D Erni

doi:10.1364/oe.19.024344

Abstract

The concept of the so-called light line is a useful tool to distinguish between guided and non-guided modes in dielectric slab waveguides. Also for more complicated structures with 2D mode confinement, the light lines can often be used to divide a dispersion diagram into a region of a non-guided continuum of modes, a region of discrete guided modes and a forbidden region, where no propagating modes can exist. However, whether or not the light line is a concept of practical relevance depends on the geometry of the structure. This fact is sometimes ignored. For instance, in the literature on photonic crystal waveguides, it is often argued that substrate-type photonic crystal waveguides with a weak vertical confinement are inherently lossy, since the entire bandgap including the line defect modes is typically located above the light line of the substrate. The purpose of this article is to illustrate that this argument is inaccurate and to provide guidelines on how an improved light line concept can be constructed.

Highlights

Line defect planar photonic crystal (PhC) waveguides have become popular in the context of slow light applications
The general interest in slow light was boosted by the demonstration of extreme slow-down factors in a Bose Einstein condensate [1] and by rapid advances made in PhC fabrication technology, which led to the experimental verification of various photonic bandgap effects, such as PhC mirrors [2, 3], the enhancement of nonlinear effects [4], high-Q cavities [5, 6], and slow light in PhC waveguides [7, 8], to mention but a few
We show the fundamental modes of odd parity [11] with respect to the x-z mirror plane (Ex = Hy = Ez = 0 in the symmetry plane), computed by Lumerical, a commercially available 2D finite difference frequency domain (FDFD) eigenmode solver [12]

Summary

Introduction

Line defect planar photonic crystal (PhC) waveguides have become popular in the context of slow light applications. One of them is based on the so-called “light line” concept: membrane-type PhC waveguides can be designed to have line defect modes below the light line of the cladding (e.g. air), whereas in the first Brillouin zone of substrate-type structures the entire bandgap is typically situated above the light line of the substrate. There is a flaw in this line of arguments, which can be boiled down to the fact that the geometry for lateral confinement is not taken into account when the light line concept is applied in a too simplistic manner It is the purpose of this article to substantiate this statement and to replace the light line argument by a more accurate model to predict the separatrix between high-loss and low-loss modes in a dispersion diagram.

The buried rectangular waveguide

The role of total internal reflection

Substrate-type photonic crystal waveguides

The background line of PhC waveguides

Outlook: vertical current injection for substrate-type PhC devices

Conclusion