  Piezoresponse Force Microscopy (PFM) has emerged as a primary tool for
imaging, domain engineering, and switching spectroscopy on ferroelectric
materials. Quantitative interpretation of PFM data including measurements of
the intrinsic width of the domain walls, geometric parameters of the domain
below the tip in local hysteresis loop measurements, as well as interpretation
of switching and coercive biases in terms of materials properties and switching
mechanisms, requires reliable knowledge on electrostatic field structure
produced by the tip. Using linear imaging theory, we develop a theoretical
approach for interpretation of these measurements and determination of tip
parameters from a calibration standard. The resolution and object transfer
functions in PFM are derived and effect of materials parameters on resolution
is determined. Closed form solutions for domain wall profiles in vertical and
lateral PFM and signal from cylindrical domain in transversally isotropic
piezoelectric are derived for point-charge and sphere-plane geometry of the
tip.
