Exact Fundamental Limits of the First and Second Hyperpolarizabilities

Abstract

Nonlinear optical interactions of light with materials originate in the microscopic response of the molecular constituents to excitation by an optical field, and are expressed by the first (β) and second (γ) hyperpolarizabilities. Upper bounds to these quantities were derived seventeen years ago using approximate, truncated state models that violated completeness and unitarity, and far exceed those achieved by potential optimization of analytical systems. This letter determines the fundamental limits of the first and second hyperpolarizability tensors using Monte-Carlo sampling of energy spectra and transition moments constrained by the diagonal Thomas-Reiche-Kuhn (TRK) sum rules and filtered by the off-diagonal TRK sum rules. The upper bounds of β and γ are determined from these quantities by applying error-refined extrapolation to perfect compliance to the sum rules. The method yields the largest diagonal component of the hyperpolarizabilities for an arbitrary number of interacting electrons in any number of dimensions. The new method provides design insight to the synthetic chemist and nanophysicist for approaching the limits. This analysis also reveals that the special cases which lead to divergent nonlinearities in the many-state catastrophe are not physically realizable.PACS numbers: 42.65.An, 78.67.LtIntroduction-Nonlinear optics is the study of quantum systems with polarizations that are nonlinear functions of external electromagnetic fields. This letter solves the problem of determining the exact fundamental limits of nonlinear optics by delineating the first procedure for computing the first and second hyperpolarizabilities consistent with on-and off-diagonal quantum mechanical sum rules. In the process, we show that prior predictions of the limits using truncated sum rules are too high by nearly 30% for β[1, 2] and 40% for γ [2], that predictions of the many-state catastrophe [3] are spurious, and that predictions of the scaling of the hyperpolarizabilities with the strength of the ground-to-excited state transition moment are modified in a way that will direct molecular synthesists to make new design choices.The nonlinear optical response of a material is generated by the collective response of the basic elements comprising it. This letter concerns the maximum values of the nonlinear optical response of a molecular-scale structure, not a material. The nonlinear optics of an elemental structure is measured by the effect it has on the molecular polarization vector when perturbatively excited by an electric field E i , with i = x, y, z:

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