Simulating Band Structure of 2D Photonic Crystals Using Plane Wave Expansion Method

To comprehensively analyze the properties of 2D photonic crystals, multiple simulation approaches have been developed. The plane wave expansion method serves as a fundamental technique for calculating photonic band structures, where electromagnetic fields are decomposed into Fourier series components. This method typically involves implementing eigenvalue solvers to handle large matrices representing Maxwell's equations in reciprocal space, with key computational steps including wave vector sampling across Brillouin zones and convergence testing for plane wave counts. Complementarily, the finite-difference time-domain (FDTD) method enables numerical modeling of energy propagation dynamics. This time-domain approach discretizes spatial domains using Yee's grid scheme and applies leapfrog time stepping to solve coupled electric and magnetic field components. Critical implementation aspects include Perfectly Matched Layer boundary conditions for absorption and spatial resolution optimization relative to photonic crystal lattice constants. These complementary methodologies - the frequency-domain plane wave expansion for band structure analysis and time-domain FDTD for propagation visualization - collectively provide multidimensional insights into light-matter interactions within periodic dielectric structures. The integration of these simulation techniques facilitates advanced photonic device design for applications in optical communications, sensors, and quantum photonics systems.