A model of optical pump-terahertz probe by ultrafast terahertz near-field microscopy.

Terahertz scattering-type scanning near-field optical microscopy (THz s-SNOM) has emerged as a powerful tool for capturing information beyond the diffraction limit. Adding an external pumping laser enables nanoscale optical pump-THz probe (OPTP) spectroscopy, extending ultrafast dynamics studies to the nanoscale. However, unlike the thin-film approximation in far-field transmission, the relation between transient permittivity and the near-field OPTP signal remains computationally demanding to solve. Here, we introduce a generalized evolutionary model (GEM) derived from the physical independence between the ultrafast pump delay and the mechanical tip oscillation. By assuming a multiplicative evolution of the spectral response, GEM establishes a direct analytical link that decouples the transient permittivity from the complex near-field scattering integral. This approach allows for the rapid prediction of permittivity evolution without the need for full-time-domain spectral acquisition at every delay step. Our findings demonstrate that GEM significantly accelerates the extraction of transient permittivity during the pumping process, as validated on typical semiconductor samples.