  We present a detailed theoretical investigation of the linear and non-linear
optical response in a model system for a disordered d-wave superconductor,
showing that for both quantities the gap symmetry considerably changes the
paradigm of the optical response based on the conventional s-wave case. For
what concerns the linear response our findings agree with previous work showing
that in strongly-disordered d-wave superconductors a large fraction of
uncondensed spectral weight survives below Tc, making the optical absorption
around the gap-frequency scale almost unchanged with respect to the normal
state. Our numerical results are in excellent quantitative agreement with
experiments in overdoped cuprates. In the non-linear regime we focus on the
third-harmonic generation (THG), finding that, as already established for the
s-wave case, in general a large THG is triggered by disorder-activated
paramagnetic processes. However, in the d-wave case the BCS response is
monotonously increasing in frequency, loosing any signature of THG enhancement
when the THz pump frequency $\omega$ matches the gap maximum $\Delta$, a
hallmark of previous experiments in conventional s-wave superconductors. Our
findings, along with the mild polarization dependence of the response, provides
an explanation for recent THG measurements in cuprates, setting the framework
for the theoretical understanding of non-linear effects in unconventional
cuprates.
