Fundamental limits of all nonlinear-optical phenomena that are representable by a second-order nonlinear susceptibility.

Applying the three-level ansatz and the sum rules to the new dipole-free sum-over-states expression, we develop a rigorous method for calculating the fundamental limits of the dispersion of the real and imaginary parts of the second-order electronic nonlinear-optical susceptibilities. These results can be applied to all orders of nonlinearity, hence can be used to study any nonlinear-optical phenomena at any wavelength. The theory can be used to understand how strongly light interacts with matter and can be applied to optimizing a material's properties for applications. In particular, we find that the resonant first hyperpolarizability peaks when the energy difference between excited states is small. In contrast, the maximal off-resonance hyperpolarizability requires the excited states to be well separated. Therefore, one molecular design strategy does not fit all applications.