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Nonlinear electromagnetic response functions have reemerged as a crucial toolfor studying quantum materials. Most attention has been paid to responses tospatially uniform electric fields, relevant to optical experiments inconventional materials. However, magnetic and magnetoelectric phenomena arenaturally connected by responses to spatially varying electric fields due toMaxwell's equations. Furthermore, in moir\'{e} materials, characteristiclattice scales are much longer, allowing spatial variation of optical electricfields to potentially have a measurable effect in experiments. To address theseissues, we develop a formalism for computing linear and nonlinear responses tospatially inhomogeneous electromagnetic fields. Starting with the continuityequation, we derive an expression for the second-quantized current operatorthat is manifestly conserved and model independent. Crucially, our formalismmakes no assumptions on the form of the microscopic Hamiltonian and so isapplicable to model Hamiltonians derived from tight-binding or ab initiocalculations. We then develop a diagrammatic Kubo formalism for computing thewavevector dependence of linear and nonlinear conductivities, using Wardidentities to fix the value of the diamagnetic current order-by-order in thevector potential. We apply our formula to compute the magnitude of the Kerreffect at oblique incidence for a model of a moir\'{e} Chern insulator anddemonstrate the experimental importance of spatially inhomogeneous fields inthese systems. We further show how our formalism allows us to compute the(orbital) magnetic multipole moments and magnetic susceptibilities ininsulators. We next use our formalism to compute the second-order transverseresponse to spatially varying transverse electric fields in our moir\'{e} Cherninsulator model, with an eye towards the next generation of experiments inthese systems.

Spatially Inhomogeneous Linear and Nonlinear Electromagnetic Response in Periodic Solids: A General Approach