Maximum Cross Sections for Excited State and Two-Photon Absorption

There is a maximum value for any nonlinear process, which can be found on the basis of fundamental quantum mechanics. With regard to single photon processes, there is a maximum absorption cross section, σa, which any molecule may reach. The maximum is 1.13 × 10-17 λ is the wavelength of the light, n is the refractive index of the host medium, and Q is the quality factor ascribed to the absorption line width. Thus large cross sections can only be obtained at the expense of a narrow operating range. The maximum is set by fundamental physics. Since nonlinear phenomena are highly resonant, the maxima of higher-order processes are also a strong function of the width of the resonance and the peak wavelength. In the case of twophoton absorption the peak value of the maximum cross section, σ2N or δ, is roughly 3.5 × 10-32 λ4nQ cm4/GW, or about 7 × 10-57 λ3 nQ cm4/photon. No value for a nonlinear coefficient is meaningful in the absence of the width of the resonance, the wavelength, and the refractive index of the medium at the wavelength of the measurement.The basic quantity of interest is the linear electronic polarizability. This fundamental quantity determines not only the absorption for the various processes of interest, but also the surface tension and the closely related solubility parameter. All scale with the polarizability a. The polarizability has a maximum value of e2/mω2; consequently the absorption cross sections have a corresponding ceiling. The maximum value for the peak in a resonant polarizability curve is set by a balance between the Coulomb potential energy of the electron and its kinetic energy. Hence it is an inherent property of the electron itself, irrespective of the molecular orbital in which it finds itself. All that the particular quantum configuration of the molecule does is partition the line strength for absorption among various possible transitions. The extreme case is when virtually all the strength is concentrated in a single transition, an inherently resonant condition.