Calculation of Band Structure for Square Lattice Photonic Crystals with Elliptical Cylinders Using Plane Wave Expansion Method

This paper discusses the computational approach for determining the band structure of square lattice photonic crystals with elliptical cylinders using the plane wave expansion (PWE) method. Photonic crystals are artificial structures with periodic dielectric constant distributions, whose unique photonic bandgap properties make them widely applicable in optical device design.

Calculating the photonic band structure forms the fundamental step in studying optical characteristics. The plane wave expansion method stands as one of the most prevalent numerical techniques, whose core algorithm involves expanding the electromagnetic wave equations in reciprocal lattice space, transforming them into eigenvalue problems for solution. From a coding perspective, this typically requires constructing a Hamiltonian matrix representation of Maxwell's equations in Fourier space, where dielectric Fourier coefficients are computed through numerical integration over the unit cell.

For square lattice elliptical cylinder photonic crystals, the computational procedure follows these key steps: First, establish the crystal's geometric model including basis vectors, elliptical cylinder parameters (major/minor axes, orientation), and dielectric constant distribution. In code implementation, this involves defining lattice constants and creating a function to generate the dielectric profile ε(r). Second, select an appropriate number of plane waves for Fourier expansion - this critical parameter balances computational accuracy against memory requirements, typically implemented through a wavevector cutoff criterion. Finally, solve the eigenvalue equation to obtain photonic band structure, which in practice involves diagonalizing a large sparse matrix using numerical libraries like LAPACK or ARPACK.

The introduction of elliptical cylinders provides higher design自由度 degrees of freedom for photonic crystals. By adjusting the aspect ratio between major and minor axes, programmers can implement parameter sweeps to systematically control the position and width of photonic bandgaps. This method establishes theoretical foundation for designing photonic crystal devices operating in specific frequency ranges, with potential code extensions including optimization algorithms for bandgap maximization.

Investigating band structure characteristics of such configurations holds significant importance for developing novel photonic devices including optical filters, waveguides, and laser cavities. The computational framework described enables researchers to prototype device designs through numerical simulation before fabrication.