We present spatially resolved measurements of the electric field of terahertz pulses undergoing optical tunneling that show strong pulse reshaping in both time and space. This reshaping is shown to be a result of frequency and incidence-angle filtering of the complex amplitude of the plane-wave basis set that makes up the pulse. This filtering leads to spreading of the pulse in the time and space dimensions, as expected from linear dispersion theory. Measurement of the pulse shape after transmission through an optical tunneling barrier permits direct determination of the complex system transfer function in two dimensions. The transfer function, measured over both thin and thick barrier limits, contains a complete description of the tunneling barrier system from which the phase and loss times can be directly determined.
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