Perturbative transfer matrix method for optical-pump terahertz-probe spectroscopy of ultrafast dynamics in spintronic terahertz emitters

The rise of spintronic terahertz (THz) emitters has captured considerable attention owing to their potential for delivering high-intensity broadband THz pulses. The quest for enhancing their performance has largely concentrated on two key aspects: the out-coupling of THz radiation and its generation within the material system. A thorough understanding of THz generation and its interaction with photoexcited, out-of-equilibrium materials is pivotal for achieving further progress. Ultrafast optical excitations elicit a range of dynamic responses within the picosecond timescale, aligning with the THz frequency range. Although existing methodologies, such as optical-pump THz-probe experiments, offer valuable insights, the intricate details of spin current generation in spintronic THz emitters, including the possible delay between energy rise and spin current generation, continue to be somewhat obscure. To address this gap, we theoretically propose two approaches to extract the subpicosecond timescales involved in the complex process of converting a laser excitation to a THz pulse in spintronics THz emitter use. In doing so we leverage the interference of the THz-probe pulse interacting with time-varying material properties and the THz pulse generated by the spin-to-charge conversion process. To describe such processes and theoretically support the experimental approaches we suggest, we introduce the perturbative transfer matrix method to take into account both the interaction of a THz probe with a multilayer undergoing subpicosecond dynamics and the production of THz radiation within some of the layers.