Phase reconstruction and causality in nano-FTIR signals based on the finite dipole model with Kramers–Kronig relations

Fourier-transform infrared nanospectroscopy (nano-FTIR), a technique based on scattering-type scanning near-field optical microscopy (s-SNOM), enables the characterization of materials’ vibrational properties with nanometric spatial resolution. This capability arises from highly confined broadband infrared radiation near the curvature apex of a metallic tip, allowing the detection of amplitude and phase signals associated with the local tip–sample interaction. Consequently, the physical interpretation of s-SNOM and nano-FTIR signals critically relies on understanding the tip–sample interaction. Theoretical models, such as the Finite Dipole Model (FDM), have provided valuable insights into this interaction, offering analytical formulations for the near-field amplitude response. However, the phase signal, often associated with infrared absorptive contributions in weak oscillator regimes, remains less direct from a theoretical standpoint. To address this issue, we explore a method based on the Kramers–Kronig relations (KKR) within the FDM framework, yielding analytical approximations for normalized amplitude–phase relations that agree well with simulated data under the assumptions considered. Furthermore, the proposed method is consistent with the causality requirements imposed by the KKR. This relation allows the reconstruction of the normalized phase from normalized amplitude data within the FDM approximation, providing an additional physical interpretation in terms of complex polarizability. To evaluate the FDM–KKR framework, we reconstructed the normalized phase and normalized amplitude of the effective polarizability of the PMMA polymer in simulated spectra. Our findings indicate that the proposed FDM–KKR framework offers a complementary, self-consistent analytical approach for interpreting normalized near-field optical signals in terms of measurable quantities, thereby bridging analytical modeling and experimental observations.