Accurate formula for scattered-light intensity in scattering-type scanning near-field optical microscopy

Scattering-type scanning near-field optical microscopy (s-SNOM) is a powerful technique for characterizing the optical response of solid-material surfaces at the nanoscale. This method relies on the lock-in detection of scattered light modulated by an oscillating atomic force microscope cantilever. The s-SNOM approach curve, i.e., the lock-in signal as a function of the tip height relative to the sample, depends on the cantilever oscillation amplitude. This “modulation-broadening” effect is a fundamental feature of harmonic demodulation methods, which complicates the quantitative interpretation of the lock-in signal when using a large modulation amplitude. In this Letter, we report an accurate formula for determining the scattered-light intensity from this approach. The formula is identical in form to that used to convert the frequency shift measured in non-contact, frequency-modulation atomic force microscopy (FM-AFM) into the force acting on the tip. We apply this formula to convert approach curves recorded using the recently developed FM-AFM-based s-SNOM [Shiotari et al., Sci. Adv. 11, eadu1415 (2025)], characterizing the near-field component therein. This demonstrates that the scattered-light intensity can be recovered even when oscillation amplitudes differ by a factor of 50, effectively eliminating the modulation-broadening effect. Low-order harmonic lock-in detection with amplitudes below 1 nm offers advantages for the straightforward interpretation of the signal, providing a new experimental protocol for s-SNOM.