A General Theory of Inhomogeneous Broadening for Nonlinear Susceptibilities: The Polarizability and Hyperpolarizability

While nonlinear optical spectroscopy is becoming more commonly used to study the excited states of nonlinear-optical systems, a general theory of inhomogeneous broadening is rarely applied in lieu of either a simple Lorentzian or Gaussian model. In this paper, we generalize all the important linear and second-order nonlinear susceptibility expressions obtained with sum-over state quantum calculations to include Gaussian and stretched Gaussian distributions of Lorentzians. We show that using the correct model to analyze experiments that probe a limited wavelength range can be critical and that this theory is better able to fit the subtle spectral features - such as the shoulder region of a resonance - when both models produce qualitatively similar responses.