Exciton-scattering mechanism for enhanced nonlinear response of molecular nanostructures

We study the nonlinear optical response of molecular assemblies consisting of coupled two-level molecules with arbitrary geometry including inhomogeneous broadening and exciton dephasing. An analytical expression for the third-order optical susceptibility is derived based on a certain closure of the equation-of-motion hierarchy. An exact expression for the third-order response of small crystalline aggregates with exciton population relaxation is derived, and used to show that for low relaxation rate Γ the cooperative part of the nonlinear response scales as ${\mathrm{\Gamma }}^{\mathrm{-}1/2}$ in d=1, and up to logarithmic corrections, as ${\mathrm{\Gamma }}^{\mathrm{-}1}$ in d=2 and 3. We show that cooperative enhancement is caused by exciton scattering. This is to be contrasted to other cooperative effects such as superradiance, which are related to the size of molecular coherence domains and therefore scale as ${\mathrm{\Gamma }}^{\mathrm{-}\mathit{d}/2\mathrm{}}$. In a numerical study for dipole-dipole interactions in d=1, 2, and 3 dimensions we find that the local-field approximation is valid only in selected cases, and in general fails to reproduce the magnitude and the resonance structure of ${\mathrm{\chi }}^{\mathrm{}\left(3\right)\mathrm{}}$.