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We present the first analysis of harmonic generation data where the fullpotential of the generalized nonlinear Kramers-Kronig (K-K) relations and sumrules is exploited. We consider two published sets of wide spectral rangeexperimental data of third harmonic generation susceptibility on differentpolymers, the polysilane (frequency range: $0.4-2.4$ $eV$) and thepolythiophene (frequency range: $0.5-2.0$ $eV$). We show that, independenttruncated dispersion relations connect the real and imaginary part of themoments of the third harmonic generation susceptibility$\omega^{2\alpha}\chi^{(3)}(3\omega; \omega, \omega ,\omega)$, with $\alpha$ranging from 0 to 3, in agreement with the theory, while there is noconvergence if we choose $\alpha=4$. We repeat the same analysis for$\omega^{2\alpha}[\chi^{(3)}(3\omega; \omega, \omega ,\omega)]^{2}$ and showthat a larger number of independent K-K relations connect the real and theimaginary part of the function under examination. We also compute the sum rulesfor the suitable moments of the real and imaginary parts, and observe that onlyconsidering higher powers of the susceptibility the correct vanishing sum rulesare more precisely obeyed. Sum rules providing explicit information aboutstructural properties of the material seem to require wider spectral range.These constraints are expected to hold for any material and provide fundamentaltests of self-consistency that any experimental or model generated data have toobey.Comment: 37 pages, 12 Figure

Verification of generalized Kramers–Kronig relations and sum rules on experimental data of third harmonic generation susceptibility on polymer