Dispersion of n2 in Solids

Abstract

We recently performed an extensive series of measurements of the bound electronic nonlinear refractive index n2 of a variety of solids at several wavelengths. We found that as the photon energy approached Eg (the band-gap energy), that n2 changed from positive to negative. This observed wavelength dispersion of n2 can be well explained for wavelengths well below the fundamental absorption edge using a Kramers-Kronig transformation on the two-photon absorption coefficient β which we had previously studied. While this theory fit the data well for 0.1Eg<ℏω<0.8Eg (Eg is the band-gap energy), there was a significant deviation toward larger negative values of n2 near the fundamental absorption edge. We speculated that the AC Stark effect could account for this deviation. Here we extend the data to photon energies nearer the gap and redo the Kramers-Kronig calculation to include the AC Stark (virtual band-blocking), and electronic Raman contributions to the imaginary part of the third order susceptibility. Indeed the fit obtained for n2 as calculated by Kramers-Kronig is amazingly good for a five orders of magnitude change of n2 including a change in sign. The change in sign from positive to negative with increasing frequency occurs midway between the two-photon absorption edge and the fundamental absorption edge. Thus, we now have a comprehensive theory that allows prediction of n2 at any wavelength below the band edge given only Eg and the linear index of refraction. Such information is useful for a variety of applications including optical limiting, laser-induced damage, and all-optical switching.