Dispersion relations and sum rules in nonlinear optics

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Abstract
We prove that dispersion relations similar to the Kramers-Kronig equations of linear optics can be obtained for the nonlinear-response function to all orders in the electric field. When energy dissipation is involved the dispersion relations obtained here concern the case in which nonlinearity on a probe beam is produced by external radiaton beams of any given frequency. Using the superconvergence theorem, we find a set of nonlinear sum rules. Some of them imply that the already known sum rules of linear optics—in particular, the Thomas-Reiche-Kuhn and the Alatarelli-Dexter-Nussenzweig-Smith sum rule—are true to all orders because all the nonlinear contributions vanish. Others do not have a linear counterpart and are specific to nonlinear optics. Implications of these results and possibilities of anomalous emission effects are dicussed.