  We have developed a near-field microwave microscope to locally apply
microwave frequency currents and fields to superconductors, and dielectric
substrates, and measure the locally generated 2nd and 3rd harmonic responses.
We measure the local nonlinear response of a Tl_2Ba_2CaCu_2O_y film grown on an
MgO substrate, and observe a large response due to the enhanced current density
near the edge. We also study the local nonlinear response of a YBa_2Cu_3O_7-d
thin film grown on a bi-crystal SrTiO_3 (STO) substrate, and spatially identify
the grain boundary through higher harmonic measurements. The spatial resolution
is determined by the magnetic loop probe size. A scaling current density JNL is
extracted to quantify the magnitude of the nonlinearity of the superconductor.
Preliminary results on the nonlinear properties of some commonly used
substrates, e.g. MgO and STO, have also been obtained
