Immittance matching for multidimensional open-system photonic crystals

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 semiinfinite 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.

Coupling into the slow light mode in slab-type photonic crystal waveguides

“…Several recent publications examined theoretically the coupling efficiency of the strip/PhC interface for different terminations of the lattice 7,8,9 . Although the slow light regime was intentionally omitted from consideration, for frequencies far from the mode cut-off it has been shown that terminations around τ =0 are preferred over τ =0.…”

“…However, increase of the group index in the slow light regime prevents efficient coupling into the PhC waveguide due to increasingly large impedance mismatch. Several recent theoretical studies 7,8,9 indicate that the exact termination of the photonic lattice at the coupling interface is important for improving impedance matching. It has also been suggested that surface states localized at the PhC interface can play a significant role in the coupling process 4,5,6 .…”

Analysis and design of efficient coupling in photonic crystal circuits

“…On the other hand, propagation in conventional dielectric waveguides relies on index-contrast guiding. At the moment, coupling losses in PhC circuits have mainly been studied by means of simulations, while theoretical work remains very scarce, (Palamaru and Lalanne 2001;Botten et al 2003;Ushida et al 2003;Biswas et al 2004;Sanchis et al 2004). However, the knowledge of analytic expressions for the transmission and reflection at an interface between an external medium and a PhC circuit can be very useful when testing novel designs and studying the influence of different parameters on the coupling efficiency.…”

“…Interface between photonic crystal and dielectric waveguides Figure 1(a) shows the analyzed interface formed by a line defect PhC waveguide butt-coupled to a dielectric waveguide. The transmission and reflection matrices are calculated by using the well known mode matching technique (Zaki et al 1988). Therefore, we consider that an interface is placed at z ¼ 0 and a single Bloch mode with index p is incident from medium I.…”

Coupling techniques between dielectric waveguides and planar photonic crystals.

“…Only a few works have been focused in the modeling of the interface between photonic crystal circuits and external media such as free space of dielectric waveguides [Pal01,Bot03,Ush03,Bis04]. The modeling of photonic crystal circuits with efficient and accurate approaches may significantly reduce the computation time, which is usually very long in conventional numerical methods such as the FDTD method.…”