Program for Calculating Photonic Crystal Band Structures Using Transfer Matrix Method

The Transfer Matrix Method (TMM) is an efficient approach for calculating photonic crystal band structures, which derives band characteristics by analyzing electromagnetic wave propagation in periodic dielectric media. This method decomposes the photonic crystal structure into multiple thin layers and utilizes transfer matrices to describe wave propagation through each layer. In MATLAB implementation, this typically involves constructing transfer matrices for individual layers and computing overall transmission properties through matrix multiplication operations.

The core implementation workflow includes: First, defining photonic crystal structural parameters such as refractive indices and layer thicknesses; then constructing single-layer transfer matrices that describe wave propagation within dielectrics and interface reflections; subsequently obtaining the unit cell's transfer matrix through sequential matrix multiplication; finally solving the eigenvalue problem using Bloch's theorem to derive the photonic band structure. Key MATLAB functions involved would include matrix operations (mtimes) for transfer matrix multiplication and eig() for eigenvalue solutions.

MATLAB's matrix computation capabilities are particularly suitable for TMM implementation, efficiently handling complex number operations and eigenvalue calculations. In programming design, special attention must be paid to handling transfer matrix differences between polarization modes (TE and TM waves), and implementing frequency sweeps to solve band structures. The final results are typically presented as band diagrams showing photonic bandgap and conduction band characteristics. The code would require separate matrix formulations for TE mode (involving electric field parallel to layers) and TM mode (magnetic field parallel to layers).

This method offers high computational efficiency and is applicable for band analysis of 1D and quasi-1D photonic crystal structures. By adjusting structural parameters through variable modifications in the code, researchers can investigate how different photonic crystal designs affect band structures, providing theoretical foundations for photonic device design. The program structure would typically include parameter initialization sections, matrix construction functions, and visualization routines for band diagram plotting.