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Model for quantitative tip-enhanced spectroscopy and the extraction of nanoscale-resolved optical constants

Near-field infrared spectroscopy by elastic scattering of light from a probe tip resolves optical contrasts in materials at dramatically sub-wavelength scales across a broad energy range, with the demonstrated capacity for chemical identification at the nanoscale. However, current models of probe-sample near-field interactions still cannot provide a sufficiently quantitatively interpretation of measured near-field contrasts, especially in the case of materials supporting strong surface phonons. We present a model of near-field spectroscopy derived from basic principles and verified by finite-element simulations, demonstrating superb predictive agreement both with tunable quantum cascade laser near-field spectroscopy of SiO$_2$ thin films and with newly presented nanoscale Fourier transform infrared (nanoFTIR) spectroscopy of crystalline SiC. We discuss the role of probe geometry, field retardation, and surface mode dispersion in shaping the measured near-field response. This treatment enables a route to quantitatively determine nano-resolved optical constants, as we demonstrate by inverting newly presented nanoFTIR spectra of an SiO$_2$ thin film into the frequency dependent dielectric function of its mid-infrared optical phonon. Our formalism further enables tip-enhanced spectroscopy as a potent diagnostic tool for quantitative nano-scale spectroscopy.