Analyzing the propagating waves in the two-dimensional photonic crystal by the decoupled internal-field expansion method

We propose the decoupled internal-field expansion (IFE) method to discuss refractions in the photonic crystal (PhC). This method decouples the full wave in the PhC and classifies them into two categories including the forward-propagating and backward-propagating waves denoted by the index n. A triangular-PhC case is demonstrated and both positive and negative refractions are discussed by this method. The incident angle of 10° results in the positive refracted wave with the refracted angle about 8°, which approximately corresponds to the forward wave of n=0 order. The negative refracted waves, which exist in the left and right edge regions, propagate almost parallel to the interfaces between the PhC and outside media. Meanwhile, due to the interaction between the negative refracted wave and the nearest few rows of air cylinders, the reflected wave and another weaker negative refracted wave are created. Finally, the weaker negative refracted waves from both edge regions interfere with each other in the middle region. It is found out that the negative refracted waves in edge regions as well as the interfered wave in the middle region can be constructed by two n=-1 forward and backward waves. On the other hand, the positive refracted wave is composed of the n=0 forward wave dressed other n≠0 forward waves, the propagation angle is affected by these dressed waves, especially near the edge region. Finally, another case proves this point of view explicitly.