A three-layer coating matches free space to a substrate of εr = 12 over 8–12 GHz. Each layer is 5.5 mm (11 cells) thick; the three permittivities are the design variables and the objective is the band-mean power reflection. The transfer-matrix method gives the exact optimum, so the question is whether gradient descent through a time-domain solve lands there.
What happened
From the geometric-ladder start (1.861 / 3.464 / 6.447) the layer permittivities settle at 1.922 / 4.670 / 4.707 against the optimum 1.943 / 4.652 / 4.720, although the closed-form optimizer started from a different point. Evaluated with the transfer matrix, the design’s band-mean |R|² is 0.03434 against the optimum’s 0.03430, −14.6 dB, 3.0 dB below the start. FDTD and transfer-matrix |R|² of the final design differ by at most 1.5×10⁻³ over the 15 frequencies the cost uses (8.20–11.78 GHz).
How the gradient was checked
Against central differences at the start to 5.5×10⁻⁴, and against a 1.5× longer record to 4.8×10⁻⁴.
A one-dimensional stack at normal incidence; it shows the optimizer and the gradient work together, not the accuracy of a 3-D radome.
Downloads
Run it
python examples/inverse_design/multilayer_ar_coating.pyCPU, minutes; the showcase script scripts/showcase/ar_coating.py records every
iterate.