Calculating Reflection and Transmission of Plane Waves Incident on One-Dimensional Layered Media Using FDTD, and Propagation in Ideal Media

The Finite-Difference Time-Domain (FDTD) method is a classical numerical technique for modeling electromagnetic wave propagation, particularly effective for analyzing interactions between plane waves and layered media. In one-dimensional implementations, spatial and temporal discretization enables iterative updates of electric and magnetic field components through staggered grid arrangements. Code implementations typically employ central-difference approximations to Maxwell's equations, with field updates following the leapfrog scheme where E and H fields are calculated at alternating time steps.

For plane wave incidence on layered media, the FDTD algorithm must enforce field continuity conditions at material interfaces through proper boundary handling. Reflection and transmission coefficients are computed by recording amplitudes of incident, reflected, and transmitted waves using field monitoring probes. When modeling perfect electric conductors (PEC), the reflection introduces a 180-degree phase reversal, while lossy media require incorporation of skin depth calculations - the penetration distance where field amplitude decays to 1/e of its surface value, determined by material conductivity and wave frequency. Implementation-wise, this involves adding conductivity terms to the update equations for field components within lossy layers.

Source excitation can be implemented using Gaussian pulses or continuous sinusoidal waves. Gaussian sources with adjustable bandwidth are suitable for broadband frequency analysis through Fourier transformation of time-domain results, while sinusoidal sources facilitate steady-state observation. Wave propagation in ideal dielectric media maintains constant velocity determined solely by material parameters (ε, μ) with preserved waveform shape. FDTD simulations visually demonstrate energy distribution and phase variations as waves traverse different media, with practical code implementations including absorbing boundary conditions (e.g., PML) to minimize reflections from computational domain edges.