I first present the design and use of a near-field permeability imaging microwave microscope to measure local permeability and ferromagnetic resonant fields. This microscope is then modified as a near-field nonlinear microwave microscope to quantitatively measure the local nonlinearities in high-Tc superconductor thin films of YBa2Cu3O7-d (YBCO). The system consists of a coaxial loop probe magnetically coupling to the sample, a microwave source, some low- and high-pass filters for selecting signals at desired frequencies, two microwave amplifiers for amplification of desired signals, and a spectrum analyzer for detection of the signals. When microwave signals are locally applied to the superconducting thin film through the loop probe, nonlinear electromagnetic response appearing as higher harmonic generation is created due to the presence of nonlinear mechanisms in the sample. It is expected that the time-reversal symmetric (TRS) nonlinearities contribute only to even order harmonics, while the time-reversal symmetry breaking (TRSB) nonlinearities contribute to all harmonics. The response is sensed by the loop probe, and measured by the spectrum analyzer. No resonant technique is used in this system so that we can measure the second and third harmonic generation simultaneously. The spatial resolution of the microscope is limited by the size of the loop probe, which is about 500 mm diameter. The probe size can be reduced to ~ 15 mm diameter, to improve the spatial resolution.

To quantitatively address the nonlinearities, I introduce scaling current densities JNL(T) and JNL'(T), which measure the suppression of the super-fluid density as , where J is the applied current density. I extract JNL(T) and JNL'(T) from my measurements of harmonic generation on YBCO bi-crystal grain boundaries, and a set of variously under-doped YBCO thin films. The former is a well-known nonlinear source which is expected to produce both second and third harmonics. Work on this sample demonstrates the ability of the microscope to measure local nonlinearities. The latter is proposed to present doping dependent TRS and TRSB nonlinearities, and I use my nonlinear microwave microscope to measure the doping dependence of these nonlinearities.