Upconversion of sub-band-gap photons constitutes a promising way for improving the efficiency of silicon-based solar cells beyond the Shockley-Queisser limit. 1500 to 980 nm upconversion by trivalent erbium ions is well-suited for this purpose, but the small absorption cross section hinders real-world applications. We employ tailored gold nanostructures to vastly improve the upconversion efficiency in erbium-doped TiO$_2$ thin films. The nanostructures are found using topology optimization and parameter optimization and fabricated by electron beam lithography. In qualitative agreement with a theoretical model, the samples show substantial electric-field enhancements inside the upconverting films for excitation at 1500 nm for both s- and p-polarization under a wide range of incidence angles and excitation intensities. An unprecedented upconversion enhancement of 913(51) is observed at an excitation intensity of 1.7 Wcm$^{-2}$. We derive a semi-empirical expression for the photonically enhanced upconversion efficiency, valid for all excitation intensities. This allows us to determine the upconversion properties needed to achieve significant improvements in real-world solar-cell devices through photonic-enhanced upconversion.

9 pages (main text), 4 figures, 1 supporting information of 15 pages