  We present analytic expressions for the electronic contributions to the
linear conductivity $\sigma^{(1)}_{3d}(\omega)$ and the third order optical
conductivity $\sigma^{(3)}_{3d}(\omega_1,\omega_2,\omega_3)$ of three
dimensional massless Dirac fermions, the quasi-particles relevant for the low
energy excitation of topological Dirac semimetals and Weyl semimetals. Although
there is no gap for massless Dirac fermions, a finite chemical potential $\mu$
can lead to an effective gap parameter, which plays an important role in the
qualitative features of interband optical transitions. For gapless linear
dispersion in three dimension, the imaginary part of the linear conductivity
diverges as a logarithmic function of the cutoff energy, while the real part is
linear with photon frequency $\omega$ as $\hbar\omega>2|\mu|$. The third order
conductivity exhibits features very similar to those of two dimensional Dirac
fermions, i.e., graphene, but with the amplitude for a single Dirac cone
generally two orders of magnitude smaller in three dimension than in two
dimension. There are many resonances associated with the chemical potential
induced gap parameters, and divergences associated with the intraband
transitions. The details of the third order conductivity are discussed for
third harmonic generation, the Kerr effect and two-photon carrier injection,
parametric frequency conversion, and two-color coherent current injection.
Although the expressions we derive are limited to the clean limit at zero
temperature, the generalization to include phenomenological relaxation
processes at finite temperature is straightforward and is presented.

Materials Science;Optics