  Weyl semimetals (WSM) have been discovered in time-reversal symmetric
materials, featuring monopoles of Berry's curvature in momentum space. WSM have
been distinguished between Type-I and II where the velocity tilting of the cone
in the later ensures a finite area Fermi surface. To date it has not been clear
whether the two types results in any qualitatively new phenomena. Here we focus
on the shift-current response ($\sigma_{shift}(\omega)$), a second order
optical effect generating photocurrents. Surprisingly we find that up to an
order unity constant, $\sigma_{shift}(\omega)\sim
\frac{e^3}{h^2}\frac{1}{\omega}$ in Type-II WSM, diverging in the low frequency
$\omega\rightarrow 0$ limit. This is in stark contrast to the vanishing
behavior ($\sigma_{shift}(\omega)\propto \omega$) in Type-I WSM. In addition,
in both Type-I and Type-II WSM, a nonzero chemical potential $\mu$ relative to
nodes leads to a large peak of shift-current response with a width $\sim
|\mu|/\hbar$ and a height $\sim \frac{e^3}{h}\frac{1}{|\mu|}$, the latter
diverging in the low doping limit. We show that the origin of these divergences
is the singular Berry's connections and the Pauli-blocking mechanism. Similar
results hold for the real part of the second harmonic generation, a closely
related nonlinear optical response.
