Delayed Convergence of S-Parameter Sensitivities to Design Parameters in FDTD Simulations

In the finite-difference time-domain (FDTD) method, the sensitivities of S-parameters to design parameters come from discrete Fourier transforms of responses ending at a simulation time T. We show that, when a design parameter moves a resonance, the sensitivity converges later than the S-parameter. For a single mode with field decay time $\tau$, the truncation error of the sensitivity is $(1+T\mathrm{/}\tau )e^{-T\mathrm{/}\tau }$, against $e^{-T\mathrm{/}\tau }$ for the resonant part of the S-parameter, so a 1% sensitivity error needs $T\ge 6.6\tau$ instead of $4.6\tau$. Automatic differentiation, the discrete adjoint and finite differences on the same run carry this error, so their agreement cannot reveal it. In three solvers, with no fitted parameter, the estimate predicts the simulated error within 4% at cavity resonances with $Q$ of 1000-7000; at lower $Q$ the error is larger. On a cavity whose slowest mode is weakly coupled, common end criteria leave $\mathrm{\mid }{S}_{11}\mathrm{\mid }$ within 0.5 dB of a 1.3 dB dip but its sensitivity 10-72% in error. A matrix-pencil extrapolation inside the differentiable computation recovers the sensitivity on a cavity with field probes from a run 66 times shorter. In a single-pole filter loop with a symmetric objective, truncation leaves the maximum of the objective in place, and the optimizer reaches the resonance of a full run despite the wrong gradient. With two coupled poles, the truncated objective moves the optimum: on runs of $1\tau$ it over-couples by $\mathrm{\Delta }k\approx 1.24\mathrm{/}Q$, and an equal-ripple objective still shifts it by about $0.2\bar{\alpha }$ at $3\tau$, but not at $4.6\tau$.