Efficient matrix-based formulation for coherent scattering from rough multilayers via an effective layer approximation

We introduce the effective layer-transfer matrix method (EL-TMM) to efficiently model coherent scattering from rough multilayered structures. The EL-TMM simplifies the complex N-layer problem by replacing the buried substructure with an equivalent homogeneous half-space. Using a matrix-based recursive scheme, we derive a closed-form, polarization-dependent effective permittivity that accounts for all coherent interactions. Validated against numerical benchmarks and an independent full-wave FEM reference and further compared with independent experimental terahertz measurements, EL-TMM demonstrates high numerical accuracy and consistency with the measured specular reflection while maintaining robust numerical stability. Furthermore, it offers high computational efficiency with a reduced marginal cost per layer, providing a fast forward model for inverse scattering and material characterization in rough multilayer media.