  We describe the use of a near-field scanning microwave microscope to image
the permittivity and tunability of bulk and thin film dielectric samples on a
length scale of about 1 micron. The microscope is sensitive to the linear
permittivity, as well as to nonlinear dielectric terms, which can be measured
as a function of an applied electric field. We introduce a versatile finite
element model for the system, which allows quantitative results to be obtained.
We demonstrate use of the microscope at 7.2 GHz with a 370 nm thick barium
strontium titanate thin film on a lanthanum aluminate substrate. This technique
is nondestructive and has broadband (0.1-50 GHz) capability. The sensitivity of
the microscope to changes in relative permittivity is 2 at permittivity = 500,
while the nonlinear dielectric tunability sensitivity is 10^-3 cm/kV.
