Multiphonon-assisted anti-Stokes and Stokes fluorescence of triply ionized rare-earth ions

Experimental demonstration of multiphonon-assisted anti-Stokes and Stokes excitation of rare-earth ions ($R^{3+}$) in different hosts, that is, in a weak-coupling ion-lattice interaction case, is obtained. An exponential behavior of excitation probabilities with respect to energy difference between excitation and electronic line is found, just as for multiphonon-assisted nonradiative decay and energy transfer. Exponential parameters ${\alpha }_{\mathrm{AS}}$ and ${\alpha }_{\mathrm{S}}$ for anti-Stokes and Stokes excitation are measured and linked to the Miyakawa-Dexter $\alpha$ parameter for decay, leading to estimates for Pekar-Huang-Rhys coupling parameter $S_0$ of about 0.04, in agreement with values obtained from one-phonon spectra or nonradiative-decay measurements. A semiempirical formula is deduced to obtain $\alpha$ from only one parameter, an effective phonon energy.