Calculating Band Structure of 1D Photonic Crystals Using Transfer Matrix Method

The Transfer Matrix Method (TMM) is an effective numerical approach for studying optical properties of one-dimensional photonic crystals, particularly suitable for calculating their bandgap structures. This method abstracts photonic crystals as multilayer dielectric models and constructs transmission matrices based on electromagnetic wave propagation characteristics at dielectric interfaces, ultimately deriving the system's overall transmission properties.

In practical implementation, the program first defines basic parameters of the photonic crystal, including refractive indices of two dielectric materials, thickness of each layer, and the wavelength range of incident light. The core of TMM lies in establishing the characteristic matrix for a single dielectric layer, which comprises two key components: phase change during wave propagation and handling of reflection/refraction at interfaces. By sequentially multiplying characteristic matrices of all layers, we obtain the total transfer matrix for the entire photonic crystal structure.

During computation, the program scans incident light across a specified wavelength range. For each wavelength point, it solves the eigenvalue problem of the total transfer matrix to determine whether the structure permits (conduction band) or prohibits (bandgap) light propagation. The final results are typically presented as band structure diagrams, with wave vector as horizontal coordinate and frequency or energy as vertical coordinate, clearly displaying the distribution of allowed and forbidden bands in photonic crystals.

MATLAB is particularly suitable for such calculations due to its powerful matrix operation capabilities that efficiently handle consecutive matrix multiplications, while built-in plotting functions can directly generate intuitive band structure diagrams. By adjusting dielectric parameters, researchers can observe changes in bandgap positions and widths, providing crucial references for photonic crystal design optimization. Key MATLAB functions involved include matrix multiplication operations, eigenvalue computation (eig function), and visualization tools for plotting dispersion relationships.