Computational Projection Band Diagrams for Photonic Crystals

Projected band diagrams serve as essential tools for analyzing the optical properties of photonic crystals, playing particularly critical roles in waveguide mode identification and photonic crystal slab design. Understanding this computational process requires clarifying how periodic dielectric structures influence electromagnetic wave propagation.

Band structure calculations for photonic crystals typically employ numerical methods like Plane Wave Expansion (PWE) or Finite-Difference Time-Domain (FDTD). These approaches discretize and solve Maxwell's equations to obtain eigenfrequencies corresponding to different wave vectors (k), enabling band diagram generation. In code implementation, PWE methods often utilize Fourier space discretization while FDTD employs Yee's algorithm for spatial and temporal field updates.

Key considerations for computing projected band diagrams include: Periodic boundary condition handling: The periodicity necessitates implementing Bloch boundary conditions to ensure phase matching of wave functions at lattice boundaries. This typically involves modifying standard eigenvalue solvers to incorporate complex phase factors. High-symmetry point path selection: Calculations usually follow paths through high-symmetry points (e.g., Γ-X-M-Γ path) within the Brillouin zone to capture complete dispersion relationships. Code implementations often include symmetry-path generation algorithms. Mode analysis: Electric or magnetic field distributions help identify guided modes, leakage modes, and other modal characteristics - particularly crucial for waveguide design. Post-processing routines typically include field visualization and mode classification algorithms.

For photonic crystal slabs, additional vertical confinement conditions must be introduced to analyze evanescent wave effects on band structures. The resulting projected band diagrams not only reveal photonic bandgap locations but also provide theoretical foundations for device design (such as filters or laser cavities). Modern implementations often combine these calculations with optimization algorithms for inverse design applications.