Frequency-dependent optical dephasing and the nature of inhomogeneous broadening in crystals

We present a theory of frequency-dependent optical dephasing as measured by narrow-band photon-echo experiments. We consider a substitutionally disordered crystal with inhomogeneous broadening (diagonal disorder) and electronic interactions between occupied sites. We include the possibility of site-energy correlations. We find that the frequency dependence of the dephasing rate 1/$T_2$ is very sensitive to the ratio ξ/a, where ξ is the correlation length and a is the lattice spacing. When ξ/a→∞, and there are macroscopic domains of resonant ions, 1/$T_2$ becomes frequency independent. In the other limit when ξ/a→0, and the site energies are uncorrelated (microscopic broadening), 1/$T_2$ is proportional to the inhomogeneous line shape. We have compared our results to experiments by Macfarlane and Shelby on ${\mathrm{Y}}_2$ ${\mathrm{O}}_3$:${\mathrm{Eu}}^{3+}$ (2 at. %) and ${\mathrm{EuP}}_5$ ${\mathrm{O}}_{14}$. We find reasonable agreement with experiment when ξ is less than a few lattice sites. Thus, for these systems the model of microscopic broadening is more appropriate than that of macroscopic broadening.