Computation of Transmission and Reflection in One-Dimensional Photonic Crystals

Investigation of Transmission and Reflection Characteristics in One-Dimensional Photonic Crystals

In optical and electromagnetic wave research, photonic crystals have attracted significant attention due to their unique photonic bandgap properties. For computing transmission and reflection in one-dimensional photonic crystals, several critical factors must be considered in the numerical implementation.

The treatment of TM mode (Transverse Magnetic mode) requires special attention to field component selection. In this mode, the magnetic field component parallels the interface, while the electric field component remains perpendicular to the propagation direction. This polarization significantly influences the reflection and transmission characteristics of photonic crystals. Code implementation typically involves defining field components using matrix operations and applying appropriate boundary conditions for each polarization state.

Boundary condition handling employs PML (Perfectly Matched Layer) technology, an effective numerical boundary condition that absorbs outgoing waves without generating reflections. PML achieves this by introducing artificial absorbing materials in boundary regions, causing waves to gradually attenuate to zero, thereby preventing unphysical reflections. Implementation-wise, PML requires adding complex coordinate stretching terms to the wave equation and carefully tuning absorption parameters to match the computational domain.

The computation utilizes Total Field-Scattered Field (TFSF) separation technique, which clearly distinguishes between incident and scattered fields. The total field region contains both incident and scattered waves, while the scattered field region includes only waves generated by interaction with the structure. Algorithm implementation involves defining separate field update equations for each region and ensuring proper field injection at the TFSF boundary interface using carefully designed source terms.

The excitation source employs a Gaussian pulse, a broadband source with Gaussian envelope in the time domain. Its advantage lies in obtaining wideband frequency response through a single simulation, with well-defined and controllable spectral characteristics. By optimizing Gaussian pulse parameters (center frequency, bandwidth, amplitude), more accurate frequency domain responses can be obtained. Code implementation typically involves generating the pulse using mathematical functions like exp(-(t-t0)²/σ²) and applying it as a source term in the finite-difference time-domain (FDTD) update equations.

The computational process involves analyzing electromagnetic wave propagation characteristics in periodic dielectric structures, including calculation of Bloch wave vectors and investigation of bandgap properties. Through this method, transmission and reflection rates of one-dimensional photonic crystals at different frequencies can be accurately predicted. The algorithm typically implements periodic boundary conditions, computes dispersion relations through eigenvalue problems, and performs Fourier transforms to obtain frequency-domain results from time-domain simulations.