  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.
