  We investigate the anisotropic hybrid plasmon-SO phonon dispersion relations
in monolayer and double-layer phosphorene systems located on the polar
substrates, such as SiO2, h-BN and Al2O3. We calculate these hybrid modes with
using the dynamical dielectric function in the RPA by considering the
electron-electron interaction and long-range electric field generated by the
substrate SO phonons via Frohlich interaction. In the long-wavelength limit, we
obtain some analytical expressions for the hybrid plasmon-SO phonon dispersion
relations which represent the behavior of these modes akin to the modes
obtaining from the loss function. Our results indicate a strong anisotropy in
plasmon-SO phonon modes, whereas they are stronger along the light-mass
direction in our heterostructures. Furthermore, we find that the type of
substrate has a significant effect on the dispersion relations of the coupled
modes. Also, by tuning the misalignment and separation between layers in
double-layer phosphorene on polar substrates, we can engineer the hybrid modes.
