  In this tutorial, we aim to directly recreate some of our "aha" moments when
exploring the impact of heat diffusion on the spatial resolution limit of
photothermal imaging. Our objective is also to communicate how this physical
limit can nevertheless be overcome and include some concrete technological
applications. Describing diffusion as a random walk, one insight is that such a
stochastic process involves not only a Gaussian spread of the mean values in
space, with the variance proportional to the diffusion time, but also temporal
and spatial fluctuations around these mean values. All these fluctuations
strongly influence the image reconstruction immediately after the short heating
pulse. The Gaussian spread of the mean values in space increases the entropy,
while the fluctuations lead to a loss of information that blurs the
reconstruction of the initial temperature distribution and can be described
mathematically by a spatial convolution with a Gaussian thermal
point-spread-function (PSF). The information loss turns out to be equal to the
mean entropy increase and limits the spatial resolution proportional to the
depth of the imaged subsurface structures. This principal resolution limit can
only be overcome by including additional information such as sparsity or
positivity. Prior information can be also included by using a deep neural
network with a finite degrees of freedom and trained on a specific class of
image examples for image reconstruction.
