Deterministic spatiotemporal focusing of terahertz waves through scattering media

In photonics, the study of wave propagation in scattering media is limited by the inherent challenges in detecting the time-varying electric field oscillations, as detectors and cameras are only sensitive to the optical intensity. In the Terahertz (THz) domain, however, Time-Domain Spectroscopy (TDS) is an established technique capable of directly measuring the electric field and provide full-wave characterisation (amplitude and optical delay) in time. The combination of THz time-resolved measurements with the theory of wave propagation in scattering media is a rapidly emerging subject. It opens a promising route to gain coherent control over multiple-scattering phenomena in ways not previously accessible in optics [1] . Besides, the combination of broadband excitation and coherent, field-sensitive detection could allow investigating the transmission properties of a scattering medium in unprecedented detail and retrieve its coherent transfer-matrix representation, a task out-of-reach at optical frequencies. In this work, we theoretically demonstrate spatiotemporal refocusing of THz waves following a direct measurement of the transfer matrix of the scattering medium. Our approach combines the advantages offered by field-sensitive detection with the nonlinear wavefront shaping of THz waves. Wavefront shaping represents the state-of-the-art in the control of complex wave propagation at optical frequencies, as it allows to sequentially retrieve the transfer matrix elements of the scatterer [2] . To overcome the limited availability of THz Spatial Light Modulator (SLM) devices, we employ a recently proposed technique based on the nonlinear conversion of structured optical beams to THz patterns [3] . Specifically, we take inspiration from nonlinear Ghost-Imaging techniques and project a complete set of orthogonal Hadamard patterns on the scatterer [4] . This technique allows us to retrieve the complex-valued elements of the transfer matrix by numerical inversion ( Fig. 1a,b ). We apply the direct knowledge of the transfer matrix elements to identify the optimal focusing pattern in a deterministic fashion ( Fig. 1c,d ). Unlike iterative optimisation techniques [2] , our deterministic approach takes advantage of the full amount of spatiotemporal information encoded in the transfer matrix elements. Under this view, the task of identifying the optimal input wavefront is rewritten into an inverse problem that can be solved via constraint least-square optimisation methods. An exemplary result is shown in Fig. 1c,d , where our technique is employed to achieve spatiotemporal refocusing (i.e., focusing in space and compression in time) of a single cycle THz pulse.