General properties of optical harmonic generation from a simple oscillator model

The nonlinear oscillator model allows a basic understanding of all nonlinear processes and can be adopted to analyse optical vibrational modes and electronic transitions in atoms, molecules and crystals, in order to derive general properties of harmonic generation to all orders. In particular, we obtain Kramers-Kronig relations and sum rules referred to all momenta of the susceptibility, and Miller's empirical rules to all orders. Since the above properties only depend on time causality and not on the specific microscopic model, they can also be adopted for the quantum-mechanical description, substituting in the classical expressions the derivatives of the potential with their expectation values.