Calculation of Band Structure for Square Lattice Elliptical Cylinder Photonic Crystals Using Plane Wave Method

We employ the plane wave expansion method to investigate the band structure of square lattice photonic crystals containing elliptical cylindrical elements. This computational approach utilizes Fourier space transformations to solve Maxwell's equations, where the periodic dielectric function is expanded in reciprocal space using plane wave basis sets. The implementation typically involves constructing the master equation for electromagnetic waves in periodic structures, which reduces to an eigenvalue problem solvable through numerical diagonalization techniques. Our methodology enables comprehensive analysis of photon propagation characteristics and bandgap formation mechanisms in these specialized photonic crystals. Key algorithmic components include: reciprocal lattice vector generation, dielectric Fourier coefficient calculation, and Hamiltonian matrix construction for TE/TM polarization modes. The code implementation often features frequency domain solvers with convergence optimization for accurate band structure computation. This research provides critical insights for future photonic crystal applications, offering valuable references and inspiration for advancements in photonics and nanotechnology. The plane wave method's flexibility allows for systematic investigation of geometric parameters' effects on bandgap properties, supporting device optimization for optical communication, sensing, and quantum computing applications.