The particles form a self-assembled monolayer using dip coating. Thin gold layers are deposited on top and the particles are detached from the surface and diluted in solution.
The analytical solution to a sphere-shell system is available in bohren/huffman using Mie theory; however, for a half shell, you'd need to use FDTD or another brute-force maxwell equation solver to get an approximate solution to the optical responses. I recommend trying FDTD by Lumerical. The trial version is 30 days or so, and that may be all you need. The FDTD is straightforward, but setting up the geometry properly is where these industrial equation solvers really help out.