  Nonlinear optical responses are a crucial probe of physical systems including
periodic solids. In the absence of electron-electron interactions, they are
calculable with standard perturbation theory starting from the band structure
of Bloch electrons, but the resulting formulas are often large and unwieldy,
involving many energy denominators from intermediate states. This work gives a
Feynman diagram approach to calculating non-linear responses. This diagrammatic
method is a systematic way to perform perturbation theory, which often offers
shorter derivations and also provides a natural interpretation of nonlinear
responses in terms of physical processes. Applying this method to second-order
responses concisely reproduces formulas for the second-harmonic, shift current.
We then apply this method to third-order responses and derive formulas for
third-harmonic generation and self-focusing of light, which can be directly
applied to tight-binding models. Third-order responses in the semiclasscial
regime include a Berry curvature quadrupole term, whose importance is discussed
including symmetry considerations and when the Berry curvature quadrupole
becomes the leading contribution. The method is applied to compute third-order
optical responses for a model Weyl semimetal, where we find a new topological
contribution that diverges in a clean material, as well as resonances with a
peculiar linear character.
