Fig. 4. Illustration of the finite dipole model for bulk and multilayered samples.

a The nano-FTIR tip is modeled as a prolate spheroid of length 2L and apex radius R, which is located at height H(t) above a bulk sample with permittivity ϵ and electrostatic reflection coefficient β = (ϵ − 1)/(ϵ − 1). The incident electric field E₀ induces the primary electric dipole p₀, which interacts with the sample via the point charge Q₀, yielding the near-field induced dipole p₁ (indicated by red curved arrows). The model accounts for far-field illumination and detection of the tip-scattered field E_scat via reflection at the sample surface, described by the Fresnel reflection coefficient r (indicated by red straight arrows). Illustration of (b) monopole field reflected at a multilayer sample, E_z,refl, and (c) monopole field E_z without sample. Both red dashed arrows have a length of 2z_a. The multilayer sample in b is characterized by the quasi-electrostatic reflection coefficient β(q), which is obtained from the single-interface electrostatic reflection coefficients β_ij and layer thicknesses t_j.