Spectral Properties of Photonic Crystal Double-Heterostructure Resonant Cavities

Spectral properties of photonic crystal double-heterostructure resonant cavities were investigated using the three-dimensional finite-difference time-domain method. Bound state formation associated with dispersion minima is observed as well as Fabry-Perot resonances associated with the waveguide cladding


I. INTRODUCTION
Two-dimensional planar photonic crystals (PCs) are a candidate for optical integrated circuits due to their versatility and small size. Recently the PC double-heterostructure (DH) cavity has received attention due to its very large quality factor (Q) and its potential for integrability with PC components [1][2][3][4] . In this report, we investigate the spectral properties of several PCDH devices using the threedimensional finite-difference time-domain method and compare the results where possible with our experimental data.
Specifically, the identification of bound state frequencies near points of zero group velocity in the PC waveguide (PCWG) dispersion is made. Furthermore, Fabry-Perot resonances associated with PCWG dispersion adjacent to the corresponding group velocity zeros (GVZs) are observed. In addition, it was observed that out of plane radiation can be reduced by arranging the DH such that an odd number of PC holes are adjacent to the waveguide core.
A typical PCDH cavity is depicted in Fig 1. From the figure, it can seen that the double heterostructure is formed by perturbing the lattice constant by 2.5% of an otherwise single line defect (W1) triangular PCWG. For the device shown, the air-clad membrane has a thickness to lattice constant ratio, d/a, of 0.6 and a hole radius to lattice constant ratio, r/a, of 0.3. The lattice constant for a device operating near 1550 nm typically falls in the range of 380nm -420nm. Material parameters used were consistent with the InGaAsP material system.
The computational domain is especially large due to the need for long unperturbed waveguide sections on either side of the DH. In this study 40 unperturbed PCWG periods were placed on either side of the perturbed 2-3 periods making up the DH. The algorithm is parallelized on 192 processors running for 16 hours. The resulting time sequence is Fourier transformed, and the Pade interpolation is employed to measure spectral peak widths and center frequencies accurately. Time-domain filtering is used on subsequent program runs to analyze spatial field profiles associated with different resonances.  When the GVZs are concave up (down), increasing (decreasing) the lattice constant in the defect region creates the bound states. This is expected, because the PCWG dispersion diagram scales inversely with the the lattice constant, so that increasing (decreasing) the lattice constant shifts the corresponding frequencies down (up). Figure 3a displays the spatial Fourier transform of the PCDH bound state at normalized frequency 0.287 and illustrates Fourier components near the Brillouin zone boundary in the xdirection. In Fig 3b, the Fourier components of a PCDH with y x -2.5% defect are located at 0.6

II. BOUND STATE FORMATION
x a π β = consistent with the GVZ with downward curvature in the dispersion diagram. In this case the calculated Q is 1049. This lower value is a result of its proximity to the light cone.  Fig 4a. The shift between the group indices obtained from the heterostructure spectrum and those obtained from a finiteelement PCWG calculation is attributed to slight differences in the material indices of refraction used in the two calculations. Fig 4b shows a comparison between a calculated spectrum and data obtained experimentally. Lasing was observed at the frequency associated with the bound state at the normalized frequency of 0.297.

III. SPATIAL FIELD DISTRIBUTION
Because the spatial electromagnetic field distribution associated with a bound state consists of the product of the periodic part of a Bloch waveguide field and a localized exponentially decaying envelope, the peaks and zeros of the fields are determined by the particular orientation of the PC air holes. Fig 1 displays schematic diagrams of two PCDH cavities both created by perturbing 2-3 PC air holes. The left side of Fig 1 shows an odd number of PC air holes adjacent to the PCWG core, whereas on the right, there are an even number of air holes adjacent to the core. The WG field consists of nulls at air holes and peaks between air holes. This suggests the ability to control the placement of nulls and peaks in the cavity which could be beneficial for reducing out of plane radiation when double-heterostructure cavities are clad above and/or below with materials with higher indicies of refraction than air. We have obtained numerical Q's a factor 5 larger for cavities with a null in the center over cavities with peaks in the center for PCDHs placed on high index posts.

IV. CONCLUSION
In summary, we have analyzed spectral properties of air-clad PC double-heterostructure resonant cavities and identified bound state formation near group velocity zeros, identified Fabry-Perot resonances and illustrated controllability of peaks and nulls in the spatial field profile.