The modified single-cell cavity is theoretically analyzed by using FDTD method. From the single-cell cavity structure, the six nearest neighbor holes (NNHs) are made small and pushed away from the center of symmetry. The modified cavity still has the hexagonal symmetry. Structural variable parameters of the cavity are the radius $(r_n)$ of NNH and the distance $(c_n)$ between the cavity center and the NNH center. The thickness of slab and the radius of air hole constituting the triangular-lattice PhC slab are fixed to 0.5a and 0.25a, respectively, where the a is the lattice constant. The refractive index of the slab material is 3.46, which corresponds to silicon at λ=1500μm. Varying the structural parameters $c_n$ and $r_n$, the frequency and the Q-factor are investigated. As the size of the modified single-cell cavity increases, the hexapole mode is pulled down from the air-band of PhC and its frequency is monotonically decreased. When $c_n=1$.18a and $r_n=0.23a$, the hexapole mode has the highest Q of $3.3 ＼times 10^6$. This Q-factor value is much higher than those of previous works. This high-Q hexapole mode is to be coupled with PhC waveguide modes.
The Γ-K directional PhC waveguide, the coupling counterpart to the hexapole mode cavity, is formed by filling in one row of air holes along the Γ-K direction. Dispersion characteristics of the PhC waveguide are investigated. To avoid the modal dispersion in waveguide structure, an optical signal should propagate in a single-mode waveguide. And to send a photon over a long distance, the waveguide mode should have low propagation losses. The dispersion curve of the Γ-K PhC waveguide shows that waveguide modes in the frequency range of 0.262~0.275a/λ satisfy the above requirements. However, the hexapole mode is out of the frequency range. Therefore, we try to tune the dispersion curve of the waveguide by modifying the first side-hole at both sides of the waveguide. When the radius of the first side-hole is slightly increased to 0.30a from the original value of 0.25a, the two requirements are satisfied. The waveguide mode to be couple with the hexapole mode has even symmetry and its wave vector is 0.31a/λ.
Coupling characteristics between the high-Q hexapole mode of $Q=3.3 \times 10^6$ and the Γ-K directional waveguide mode is studied by the FDTD method. Based on the hexagonal symmetry of the hexapole mode, three representative types of coupling geometries (shoulder-couple, butt-couple and side-couple structures) are selected and tested. The shoulder-couple structure shows best coupling characteristics among the three tested structures. For example, the resonant-tunneling-filter composed by two shoulder-couple waveguides and a hexapole cavity shows a high performance with Q of $9.7 \times 10^5$ and transmittance of 0.48. In the butt-couple structure, the energy transfer from the cavity to the waveguide is prohibited because of the symmetry mismatch and no coupling is observed. In the side-couple structure, the coupling strength is much weaker than that of the shoulder-couple structure. And, most energy of the hexapole mode flows into the air as vertical loss.
The side-couple is an important geometric configuration to achieve functional devices in PhC platform. We try to understand the origin of vertical loss caused by the side-couple. Three resonant modes (donor-type hexapole, linear-3 and acceptor-type hexapole mode) are selected and coupled with the Γ-K directional waveguide in the side-couple type. In the case of donor-type hexapole mode, large vertical loss originates from the waveguide structure. In contrast, there is no vertical loss that originates from waveguide structure in the case of linear-3 mode. We focus on the different vertical-loss features and try to find the origin. Base on the vertical confinement mechanism in PhC slab waveguides, we can account for the vertical-loss features by using the Fourier-transform argument. From the 1-dimensional Fourier-transform analysis for resonant mode profiles, the vertical-loss features in side-couple structures can be qualitatively predicted. This analysis would be a useful tool to design in-plane-type optical devices based on 2-dimensional PC slabs.
