The progress in two-dimensional materials has led to rapid experimental
developments in quantum plasmonics, where light is manipulated using plasmons.
Although numerical methods can be used to quantitatively describe plasmons in
spatially inhomogeneous systems, they are limited to relatively small setups.
Here, we present a novel semi-analytical method to describe plasmons in
two-dimensional inhomogeneous media within the framework of the Random Phase
Approximation (RPA). Our approach is based on the semiclassical approximation,
which is formally applicable when the length scale of the inhomogeneity is much
larger than the plasmon wavelength. We obtain an effective classical
Hamiltonian for quantum plasmons by first separating the in-plane and
out-of-plane degrees of freedom and subsequently employing the semiclassical
Ansatz for the electrostatic plasmon potential. We illustrate this general
theory by considering scattering of plasmons by radially symmetric
inhomogeneities. We derive a semiclassical expression for the differential
scattering cross section and compute its numerical values for a specific model
of the inhomogeneity.Comment: 27 pages, 9 figure