The origin of the supercollimation (SC) phenomenon in two-dimensional photonic crystals (PhCs), which is generated by a weak periodic modulation of the dielectric constant, is studied by the plane-wave expansion (PWE) method. We find that the smallest order of PWE required in order to obtain the SC phenomenon is the second order. The distance in k-space and the frequency difference between the SC point and the M point are proportional to the square of the periodic modulation strength. These results strongly imply that the SC phenomenon originates from second-order Bragg scattering. In addition, we rotate the coordinates and study the SC phenomenon using a model which is physically the same as the original model but has a unit cell two times larger and a folded Brillouin zone. In such a rotated model, the mathematical form of the eigenequation becomes much simpler and the SC phenomenon can be obtained with the first-order expansion alone. By neglecting the smaller off-diagonal terms, we can block diagonalize the matrix, which then corresponds to a layered model. Since the layered model is strictly solvable, we derive that a curvature-reversion (CR) point must exist for equi-frequency contours because of the robust asymptotic behavior in the small region. Theoretically and numerically, we also prove that the CR point of the layered model has similar properties to those of the SC point of a square-lattice PhC and infer that the physical origin of these properties is second-order Bragg scattering.

二维光子晶体 (PhC) 中超准直 (SC) 现象的起源是通过平面波展开 (PWE) 方法研究的，该现象由介电常数的弱周期调制产生。我们发现为了获得 SC 现象所需的 PWE 的最小阶数是二阶。k空间中的距离和 SC 点与M点之间的频率差点与周期调制强度的平方成正比。这些结果强烈暗示 SC 现象源于二阶布拉格散射。此外，我们旋转坐标并使用一个模型研究 SC 现象，该模型在物理上与原始模型相同，但具有两倍大的晶胞和折叠的布里渊区。在这样的旋转模型中，特征方程的数学形式变得简单得多，SC现象可以单独通过一阶展开来获得。通过忽略较小的非对角项，我们可以对矩阵进行块对角化，然后对应于分层模型。由于分层模型是严格可解的，我们推导出等频轮廓必须存在曲率反转 (CR) 点，因为在小范围内具有稳健的渐近行为。地区。从理论上和数值上，我们还证明了分层模型的 CR 点具有与方格 PhC 的 SC 点相似的特性，并推断这些特性的物理起源是二阶布拉格散射。