Differentiable Transfer-Matrix Method for Gradient-Based Inverse Design of Multilayer Electromagnetic Shields

Inverse design of multilayer electromagnetic shields is a challenging problem due to nonlinear, frequency-dependent interactions between lossy conductors and dispersive magnetic materials. Existing approaches rely on derivative-free metaheuristics that require thousands of function evaluations, limiting scalability to many-layer stacks. This work presents a differentiable transfer-matrix method that enables gradient-based optimization of multilayer shield designs. The method combines a scattering matrix (S-matrix) recursion—unconditionally stable for arbitrarily thick lossy layers—with forward-mode automatic differentiation to provide exact shielding effectiveness (SE) gradients, and a softmax reparameterization that satisfies total-thickness budget constraints exactly at every iteration. Comprehensive validation confirms agreement with analytical solutions, independent finite-difference (FD) time-domain simulation, and FD gradient verification. The gradient-based optimizer limited-memory Broyden–Fletcher–Goldfarb–Shanno (L-BFGS) converges in one to two orders of magnitude fewer evaluations than differential evolution (DE) on 2–20 layer shield design problems, and reaches at least comparable SE; the residual gap reflects stagnation of the DE baseline in local optima of the multimodal SE landscape under the stated hyperparameter settings, rather than a fundamental performance ceiling of the baseline method. Including frequency-dispersive permeability in the differentiable model reveals that nondispersive permeability assumptions can significantly overestimate achievable broadband shielding performance.