Ultrashort pulses of electromagnetic radiation propagating through free space are used to perform coherent time-domain spectroscopy by probing the complex index of refraction of various materials, in particular thin films of high-critical- temperature superconductors and the microwave substrates that support them. The terahertz beam system utilizes Hertzian-dipole- like antennas consisting of a dc-biased photoconductive gap in a coplanar stripline as a transmitter, and an identical receiver with a photoconductive gap biased by the THz radiation. The transmitter is driven to produce the short radiation bursts by a 100-fs optical pulse from a Ti:sapphire self-mode-locked laser, while the receiver is synchronously gated by laser pulses split from the original beam. By performing measurements in the time domain and transforming data to the frequency domain, both the real and imaginary parts of the index of refraction of dielectrics and the conductivity of superconductors are determined over the entire range from approximately 200 GHz to several terahertz. This technique allows the direct broadband determination of these quantities in the mmw and sub-mmw regimes from the measurement of only a few time-domain waveforms and without the need for Kramers-Kronig analysis or complicated processing.