  Improving the photon-spin conversion efficiency without polarization
dependence is a major challenge in realizing quantum interfaces gate-defined
quantum dots (QDs) for polarization-encoded photonic quantum network systems.
Previously, we reported the design of an air-bridge bull's-eye cavity that
enhances the photon absorption efficiency of an embedded gate-defined QD
regardless of the photon polarization. Here, we numerically demonstrate that a
further 1.6 times improvement in efficiency is possible by simply adjusting the
distance of the substrate from the semiconductor slab where the bull's-eye
structure is formed. Our analysis clarifies that the upward-preferred coupling
and narrow far-field emission pattern realized by substrate-induced asymmetry
enable the improvement.
