Dipole modelling for a robust description of subdiffractional polariton waves

The nanophotonics of van der Waals (vdW) materials relies critically on the electromagnetic properties of polaritons defined on sub-diffraction length scales. Here, we use a full electromagnetic Hertzian dipole antenna (HDA) model to describe the hyperbolic phonon polaritons (HP²s) in vdW crystals of hexagonal boron nitride (hBN) on a gold surface. The HP² waves are investigated by broadband synchrotron infrared nanospectroscopy (SINS) which covers the type I and type II hyperbolic bands simultaneously. Basically, polariton waves, observed by SINS, are assigned to the resultant electric field from the summation over the irradiated electric fields of dipoles distributed along the crystal edge and at the tip location and a non-propagating field. The values of polariton momenta and damping extracted from the HDA model present excellent agreement with theoretical predictions. Our analysis shows that the confinement factor of type I HP²s exceeds that of the type II ones by up to a factor of 3. We extract anti-parallel group velocities (v_g) for type I (v_g,typeI = -0.005c, c is the light velocity in a vacuum) in relation to type II (v_g,typeII = 0.05c) polaritonic pulses, with lifetimes of ∼0.6 ps and ∼0.3 ps, respectively. Furthermore, by incorporating consolidated optical-near field theory into the HDA model, we simulate real-space images of polaritonic standing waves for hBN crystals of different shapes. This approach reproduces the experiments with a minimal computational cost. Thus, it is demonstrated that the HDA modelling self-consistently explains the measured complex-valued polariton near-field, while being a general approach applicable to other polariton types, like plasmon- and exciton-polaritons, active in the wide range of vdW materials.