Defect modes in a triangular lattice photonic crystal (PC) and the coupling characteristics of them have been investigated. As a PC of triangular lattice of air holes has large transverse electric (TE) band gap, the triangular lattice PC slab of air holes with in-plane photonic band gap and index-confinement in the third dimension is expected to serve as a good platform for compact optical devices. A point defect in the triangular lattice PC slab of air holes act as a micro-cavity and a line-defect as a waveguide because the defect modes are strongly localized about the local defects.
The coupling between defect modes causes the splitting of energy bands and the power transfer between the modes. The field of localized photons in defects of PC's oscillates with an exponentially decaying envelope because the localization in a PC is due to the strong multiple scattering. Due to the oscillatory nature of the evanescent waves of localized photons in PC's, the symmetries of the defect modes split by coupling are not conserved, and the coupling strength is significantly influenced by not only the distance between the defects but also the field profiles of the individual mode and the relative locations of two defects. The oscillatory nature of the evanescent waves of localized photons distinguishes the coupling between defect modes in a PC from that of electronic states in a solid crystal.
We have investigated channel drop tunneling processes in triangular lattice PC's based on the coupling between defect modes. For channel drop tunneling processes, the resonant cavity system must have a mirror symmetry plane perpendicular to the bus and drop waveguides, and support even and odd resonant modes with respect to the plane. Accidental degeneracy must be forced between the two resonant modes so that the frequency and width of the resonant modes must be equal. The energy transfer rates of channel drop tunneling processes in PC slabs have been theoretically analyzed by the coupled-mode theory in time. The $Q_{in}/Q_{v}$ value of a cavity system mainly determines the energy transfer rates.
A donor-type three-missing-holes cavity in a triangular lattice PC slab of air holes is made by filling three holes in line along the Γ - K direction with the same dielectric material as the slab, which is appropriate for the design of the channel drop tunneling system with two identical cavities. Channel drop tunneling processes utilizing two high-Q three-missing-holes cavities in a 2D triangular lattice PC show about 100% drop efficiency.
We have presented channel drop filters (CDF's) using channel drop tunneling processes in 2D triangular lattice PC slabs of air holes. A CDF composed of a bus waveguide, a drop waveguide, and two three-missing-holes cavities separated by 5a have been presented, where a is the lattice constant of a PC. For accidental degeneracy, the distance between the cavities and the waveguides is chosen to be $2 \sqrt{3} a $ and some additional adjustments are made in the structure. The filter responses of the CDF show that the power transferred to the drop and the transmission in the bus are about 61% and 5%, respectively. The forward drop of 61% with the 5% transmission corresponds to $Q_{in}$/$Q_{v}$ = 0.28. The total Q factor is about 1000.
The indirect coupling strength of two cavities through waveguides is directly related to $1/Q_{in}$ of a CDF, that is, the filtering resolution. In order to improve the Q factor of a CDF, the between the two cavities is increased to decrease the direct coupling between two cavities. The weak direct coupling requires the weak indirect coupling for the cancellation of the direct coupling, resulting in the higher Q factor. A CDF with two high-Q three-missing-holes type cavities separated by 8a and appropriate structural design shows the Q factor of about 5000, which is about five times the Q factor of the CDF with two cavities separated by 5a. The transferred power to the forward drop and the transmission in the bus for the CDF are found to be about 79% and 3%, respectively. The forward drop of 79% corresponds to $Q_{in}$/$Q_{v}$ of about 0.12.
We have analyzed theoretically the efficiencies of surface-emitting CDF's with the channel drop tunneling processes in PC slab's and showed that the drop efficiencies can be easily improved to the values much higher than 50%, the maximum output efficiency with one single-mode cavity. The output efficiencies are over 90% at the $Q_{in}/Q_{v}$ values which range from 0.52 to 1.92. When $Q_{in}/Q_{v}=1$, the theoretically obtained output efficiency with channel drop tunneling processes is 100%, and thus all the power of input electromagnetic (EM) waves can be dropped to the air. In order to demonstrate the theoretical results, we have presented a surface emitting CDF with channel drop tunneling processes. The CDF is composed of one waveguide and two donor-type three-missing-holes cavities in a 2D triangular lattice PC slab of air holes. The output efficiency is found to be about 94% in good agreement with the theoretical analysis.
