  Scattering-assisted synthesis of broadband optical pulses is recognized to
have a cross-disciplinary funda-mental and application importance. Achieving
full-waveform synthesis generally requires means for assessing the
instantaneous electric field, i.e. the absolute electromagnetic phase. These
are generally not accessible to established methodologies for
scattering-assisted pulse envelope and phase shaping. The lack of field
sensitivity also results in complex indirect approaches to evaluate the
scattering space-time properties. The terahertz frequency domain potentially
offers some distinctive new possibilities thanks to the availability of methods
to perform absolute measurements of the scattered electric field, as opposed to
optical intensity-based diagnostics. An interesting conceptual question is
whether this additional degree of freedom can lead to different types of
methodologies toward wave shaping and to a direct field-waveform control. In
this work, we theoretically investigate a deterministic scheme to achieve
broadband, spatio-temporal waveform control of terahertz fields mediated by a
scattering medium. The direct field access via Time-Domain Spectroscopy enables
a process in which the field and scattering matrix of the medium are assessed
with minimal experimental efforts. Then, the illumination conditions for an
arbitrary targeted output field waveform are deterministically determined. In
addition, the complete field knowledge enables reconstructing field
distributions with complex phase profiles, as in the case of phase-only masks
and optical vortices, a significantly challenging task for traditional
implementations at optical frequencies based on intensity measurements aided
with interferometric techniques.
