Theory of nonlinear polaritonics: chi⁽²⁾ scattering on a β-SiC surface

In this article we provide a practical prescription to harness the rigorous microscopic, quantum level descriptions of light-matter systems provided by Hopfield diagonalisation for quantum description of nonlinear scattering. A general frame to describe the practically important second-order optical nonlinearities which underpin sum and difference frequency generation is developed for arbitrarily inhomogeneous dielectric environments. Specific attention is then focussed on planar systems with optical nonlinearity mediated by a polar dielectric $\beta$-SiC halfspace. In this system we calculate the rate of second harmonic generation and the result compared to recent experimental measurements. Furthermore the rate of difference frequency generation of subdiffraction surface phonon polaritons on the $\beta$-SiC halfspace by two plane waves is calculated. The developed theory is easily integrated with commercial finite element solvers, opening the way for calculation of second-order nonlinear scattering coefficients in complex geometries which lack analytical linear solutions.