  Ultrafast optical excitations trigger in materials a range of dynamics. An
interesting range of dynamics is the ones that unfold within the picosecond
timescale, corresponding to the THz frequency range. Within this short
timescale, the material's properties, and therefore its response to
electromagnetic fields, dynamically change. For this reason, any THz radiation
used to probe the material will interact with the multilayer undergoing
time-dependent modifications of the dielectric responses within a single
optical cycle (a very similar situation arises for THz radiation produced in
spintronics THz emitters). Such interaction goes beyond typical quasistatic
approaches. It is, therefore paramount to be able to describe accurately all
the interaction of THz radiation with out-of-equilibrium multilayers. We
develop here a theoretical framework called PTMM (Perturbative Transfer Matrix
Method) to model the production of THz accurately and its interaction with
multilayers undergoing ultrafast changes in their dielectric properties.
  That analysis allows us to propose two novel ways to utilize Terahertz
time-domain spectroscopy. We will first show that a simple analysis of the time
resolved Optical-Pump-Terahertz-Probe spectra can provide not only the time
scale of processes that are slower than the THz pulse time-width, but it can
accurately measure the timescale of sub-picosecond and faster processes as
well. Further, we will apply our method to THz probed spintronics THz emitters,
and compute the interference between the produced THz and the THz probe. We
will show that an analysis of the delay-resolved transmission spectra allows
for a direct measurement of the time distance between the laser excitation and
the peak of the spin current, allowing an unprecedented insight into the
ultrafast spin-to-charge conversion mechanism.
