  We describe an optical method to directly measure position-dependent thermal
diffusivity of reflective single crystal samples across a broad range of
temperatures for condensed matter physics research. Two laser beams are used,
one as a source to locally modulate the sample temperature, and the other as a
probe of sample reflectivity, which is a function of the modulated temperature.
Thermal diffusivity is obtained from the phase delay between source and probe
signals. We combine this technique with a microscope setup in an optical
cryostat, in which the sample is placed on a 3-axis piezo-stage, allowing for
spatially resolved measurements. Furthermore, we demonstrate experimentally and
mathematically that isotropic in-plane diffusivity can be obtained when
overlapping the two laser beams instead of separating them in the traditional
way, which further enhances the spatial resolution to a micron scale,
especially valuable when studying inhomogeneous or multidomain samples. We
discuss in detail the experimental conditions under which this technique is
valuable, and demonstrate its performance on two stoichiometric bilayer
ruthenates: Sr3Ru2O7 and Ca3Ru2O7. The spatial resolution allowed us to study
the diffusivity in single domains of the latter, and we uncovered a
temperature-dependent in-plane diffusivity anisotropy. Finally, we used the
enhanced spatial resolution enabled by overlapping the two beams to measure
temperature-dependent diffusivity of Ti-doped Ca3Ru2O7, which exhibits a
metal-insulator transition. We observed large variations of transition
temperature over the same sample, originating from doping inhomogeneity, and
pointing to the power of spatially resolved techniques in accessing inherent
properties.
