The full potential of the Kramers–Kronig relations and sum rules for nonlinear susceptibilities has unfortunately drawn relatively little attention in nonlinear optical spectra analysis. In this feature article a simple treatment of an anharmonic oscillator model in description of the nonlinear susceptibility of media and holomorphic properties of the nonlinear susceptibility were utilized. Using such concepts, conventional Kramers–Kronig, multiply-subtractive Kramers–Kronig, and generalized Kramers–Kronig dispersion relations can be derived. We demonstrate how in practice the variety of different Kramers–Kronig relations mentioned above, as well as various sum rules, can be applied in nonlinear optical spectra analysis. As an example we treat the third-harmonic wave generation spectrum from a polymer.
Kai-Erik Peiponen, Valerio Lucarini, Jarkko J. Saarinen, and Erik Vartiainen, "Kramers–Kronig Relations and Sum Rules in Nonlinear Optical Spectroscopy," Appl. Spectrosc. 58, 499-509 (2004)