4-1 Conclusion
By applying the extended tight-binding theory (TBT), which includes the coupling effects up to the next-nearest neighboring defects, we successfully derive the analytical equations used to describe the dispersion behaviors and the mode distribution of the symmetric directional couplers (DCs) based on the coupled resonant optical waveguides (CROWs) in the photonic crystal slab (PCS). We choose the structures created by a square lattice of dielectric rods in air and a triangular lattice of air holes in a dielectric slab, and the defects are formed by reducing the effective refraction index, which means to reduce the radius of dielectric rods or to enlarge the radius of air holes.
Additionally, for the 3D PWEM simulations, there may exist the fake guiding modes called the boundary guiding modes (BGMs) in the photonic bandgap (PBG) due to the finite size of the simulation super cell. These modes can interfere with the defect modes and cause the discontinuities to the dispersion curves. By proper choosing the non-integral times of the lattice constant as the cell boundaries, we can shift the BGMs outward the PBG and the dispersion relation will free from the simulation errors.
In a single PCS-CROW, the eigenfrequency ω0 of a single point defect will cause the frequency shift to the dispersion curves, and the coupling coefficients C1 and C2 lead to the sinusoidal modulation. In the opposite-type PCS-DCs, the nearest neighboring coupling coefficient α causes the splitting and the next-nearest neighboring coupling coefficient β leads to the sinusoidal modulation of the dispersion curves, which will cross at a point under the condition |β/α| > 1/2. On the other hand, in the alternating-type PCS-DCs, both α and β lead to the sinusoidal modulation to the dispersion relations, and the curves will have a decoupling
point under the condition |β/α| > 1/3.
In the dielectric-rod structures, the coupling coefficients are dominated by the electric fields parallel to the rod axis (z-polarization) owing to the TM-like polarization propagating waves, and ω0 will shift toward the lower frequency by increasing the radius of the defect rods. For a single PCS-CROW, C2 is much smaller than C1 and can be neglected, and C1 can be estimated being negative. The dispersion curves of that will move toward the lower frequency by increasing the defect radius, and decrease at the higher values of the wavevector.
In addition, for the opposite-type PCS-DCs, we find that α is much larger than β and they possess the opposite sign. Therefore, the dispersion curves never have a crossing point, and the frequency difference between them is nearly a constant for all wavevectors, which can be used to separate the waves with the frequencies in a certain range from a broadband optical signal. On the other hand, for the alternating-type PCS-DCs, the magnitude of α is much smaller than that in the opposite-type PCS-DCs and the ratio |β/α| increases as the defect radius increases due to the decrease of α. Consequently, the dispersion curves of the structures with the larger defect rods may have a crossing point, which will shift to the smaller wavevector by increasing the defect radius. At this so-called decoupling point, the EM waves will propagate in one waveguide without leaking to another one, and the coupling length becomes infinite. This feature can be used to separate the waves with a certain frequency from a wideband optical signal by properly choosing the parameters of the device.
On the other hand, in the air-hole structures, the coupling coefficients are dominated by the electric fields parallel to the slab plane (x-polarization and y-polarization) due to the TE-like polarization propagating waves, and the enlarging of defect holes will cause the blueshift to the eigenfrequency ω0. For a single PCS-CROW, C1 and C2 of the y-polarization waves are too small to be taken into consideration, and these coupling coefficient can be just slightly influenced by the defect radius. The dispersion curves of that will increase and then decrease as the wavevector becoming larger due to the cosine modulation, and just shift to the
higher frequency by enlarging the defect radius. For the opposite-type PCS-DCs, we find that the magnitude of α is about twice the magnitude of β and they have the opposite sign, and both of them are nearly independent from the size of defects. Therefore, there exists a decoupling point of all the dispersion relations, and the wavevector position of this point is almost a constant. In addition, the frequency difference between the curves increases at the larger wavevector, and reaches the maximum value at the edge of the first Brillouin zone.
Furthermore, because this structure can support only one even mode or one odd mode at a certain frequency, that can be used to form the beam splitters with the slow-light propagation near the zone edge. On the other hand, for the alternating-type PCS-DCs, α and β are small and almost unaffected by the defect radius as in the opposite-type PCS-DCs. Consequently, the dispersion relations also have a decoupling point with a certain wavevector. Moreover, the frequency difference between the curves would increase and then decrease as increasing of the wavevector. However, owing to the multi-mode propagation, this structure is hard to be used due to the complexity of designing. In conclusion, these features discussed by the TBT are confirmed by the PWEM simulations, and this analytical method provide valuable design concepts on the PCS-DCs in integrated photonic circuits, which can be controlled by only few parameters related to the coupling effects.
4-2 Perspectives
In this thesis, we have discussed the coupling effects and dispersion behaviors in different types of PCS-DCs, and the characteristics of the signal couplings in these structures have been obtained. However, we consider only the linear optical effects in these structures.
The other advantages of the PCS-DCs as the slow-light propagation and nonlinear interaction are also valuable topics, which can make the DCs become the active optical components.
For example, the incident EM waves with different intensities can be separated by the
propagating conditions tuned by nonlinear effects. Therefore, the properties of the nonlinear PCS-DCs should be further investigated as the future works, and the devices can be designed to examine whether this kind of structures have practical values.
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