  We present measurements of the superconducting coherence length \xi\ in thin
(d < 100 \AA) films of MoGe alloy and Nb using a combination of linear and
nonlinear mutual inductance techniques. As the alternating current in the drive
coil is increased at fixed temperature, we see a crossover from linear to
nonlinear coupling to the pickup coil, consistent with the unbinding of
vortex-antivortex pairs as the peak pair momentum nears \hbar\/\xi\ and the
unbinding barrier vanishes. We compare measurements of \xi\ made by this mutual
inductance technique to values determined from the films' upper critical
fields, thereby confirming the applicability of a recent calculation of the
upper limit on a vortex-free state in our experiment.
