  We present a novel high resolution contactless technique for thermal
conductivity determination and thermal field mapping based on creating a
thermal distribution of phonons using a heating laser, while a second laser
probes the local temperature through the spectral position of a Raman active
mode. The spatial resolution can be as small as $300$ nm, whereas its
temperature accuracy is $\pm 2$ K. We validate this technique investigating the
thermal properties of three free-standing single crystalline Si membranes with
thickness of 250, 1000, and 2000 nm. We show that for 2-dimensional materials
such as free-standing membranes or thin films, and for small temperature
gradients, the thermal field decays as $T(r) \propto ln(r)$ in the diffusive
limit. The case of large temperature gradients within the membranes leads to an
exponential decay of the thermal field, $T \propto exp[-A \cdot ln(r)]$. The
results demonstrate the full potential of this new contactless method for
quantitative determination of thermal properties. The range of materials to
which this method is applicable reaches far beyond the here demonstrated case
of Si, as the only requirement is the presence of a Raman active mode.
