The properties and design concepts of air-defect symmetric and asymmetric directional couplers (DCs) made of photonic crystal slabs (PCSs) are investigated by the tight-binding theory. We give criteria in both cases for determining the specific frequency, named as the decoupling point, at which the even (-like) and odd (-like) parities of eigenmodes should switch. Two dispersion curves will cross in symmetric DCs but not in asymmetric DCs because eigenfrequencies of isolated photonic waveguides (PCWs) are different in asymmetric DCs. In the dielectric-rod PCS, the decoupling point of the DC is located almost at the same wave vector as a two-dimensional (2D) DC with only a blue shift in its frequency. Therefore, 2D simulation can give a primary result in designing a dielectric-rod DC. On reducing the radius of the defect rods, the increase in coupling coefficients leads to a faster group velocity and a longer coupling length. On the other hand, the dispersion curves of the air-hole PCS DC are no longer parallel to those of the 2D DC so that performing a 3D simulation is necessary in designing an air-hole PCS DC with enlarged air holes. Moreover, the dispersion curve of the single PCW is no longer located at the centre of the curves of the symmetric DC and the group velocity may become negative which is not observed in the dielectric-rod PCS. The simulation results indicate that the coupling length of an air-defect DC can be achieved as short as 5 lattice constants, which is much smaller than that made of dielectric defects in air-hole PCS DCs.
