Interpreting synchrotron infrared nano-spectra of hydroxyapatite through Kramers-Kronig relations and complex Lorentzian oscillators

This study introduces a hybrid numerical-analytical methodology for interpreting, for the first time, both the amplitude and phase of Synchrotron Infrared Nanospectroscopy (SINS) spectra of hydroxyapatite (HA) thin films. The approach combines the Self-Referenced Interferometry Model (SRIM) with the Kramers-Kronig relations (KKR). HA films with thicknesses of 100 nm and 600 nm were fabricated via magnetron sputtering and characterized using Fourier-transform infrared (FTIR) spectroscopy to identify vibrational modes of the phosphate group. These frequencies were used as initial parameters for Lorentzian-based SRIM fitting of the SINS amplitude spectrum. Both the integral and differential forms of KKR were applied to the fitted amplitude spectrum to reconstruct the phase, which was subsequently validated against experimental SINS data. The differential form further enabled analytical decomposition of the phase into individual oscillator and non-resonant contributions, revealing features not evident in the amplitude alone. The strong agreement between experimental and modeled spectra demonstrates the reliability of FTIR-guided SRIM fitting combined with KKR analysis. This methodology effectively bridges far-field and near-field infrared spectroscopy, showing that HA spectral responses comply with causality and analyticity. Moreover, it supports accurate phase reconstruction even under low-intensity signal conditions, indicating broad applicability to other nanostructured materials. Altogether, this work establishes a robust framework for interpreting SINS spectra and advances phase-resolved diagnostics and hybrid modeling strategies for nanoscale vibrational analysis.