Photonic Crystal Bandgap Calculation Program Using the Plane Wave Expansion Method

In photonic crystal research, the Plane Wave Expansion Method (PWEM) serves as a classical numerical approach for calculating photonic bandgap properties. This method is particularly suitable for analyzing photonic crystal structures with periodic dielectric constant distributions.

The core principle utilizes Bloch's theorem for electromagnetic waves, expanding the electromagnetic fields in periodic media as linear combinations of plane waves. During program implementation, two critical steps are primarily handled: First, constructing the Hamiltonian matrix in reciprocal lattice space - where the matrix dimension depends on the number of truncated plane waves; then obtaining the photonic band structure by solving the eigenvalue problem. Code implementation typically involves matrix diagonalization algorithms like Arnoldi iteration or full diagonalization for smaller systems.

A typical program contains core modules including reciprocal space grid sampling, Fourier series expansion of dielectric constants, and constructing eigenmatrices for Maxwell's equations. The computational accuracy directly correlates with the plane wave expansion order, but requires balancing computational resource consumption. Separate eigenvalue equations must be established for TE and TM polarizations, involving different matrix formulations in the code. The dielectric Fourier coefficients are typically computed using fast Fourier transform (FFT) algorithms for efficiency.

The advantage of this method lies in its ability to visually demonstrate how photonic bandgaps vary with geometric parameters, providing theoretical basis for designing photonic crystals for specific frequency ranges. Common applications include analyzing photonic bandgap characteristics for different structures such as 2D triangular lattices and square lattices, where lattice vectors and basis definitions are crucial input parameters.