Multiphonon sideband intensities in rare earth ions in crystal

Optical J --> J ' transitions involving emission (or absorption) of one pho ton and n phonons are theoretically investigated in trivalent rare earth (R E) ions doped crystals. The transitions are induced by joint action of elec tromagnetic field and high-order optical anharmonicity (M-process). Hence, this work extends Judd's study of one photon-one phonon transitions to the multiphonon case. The processes under consideration give rise to the Stokes and anti-Stokes multiphonon sidebands in emission and absorption. Besides, these processes can significantly increase the transition rate in presence of the stimulated emission field existing in an active material above thre shold. Here we obtained general expressions for transition probabilities. I n our calculations, the crystal field model taking into account both Coulom b and non-Coulomb interactions of 4f-electrons with ligands is considered. The contributions to multipthonon sideband due to inhomogeneous-dielectric and polarized-ligand mechanisms are considered too. As a result, all mechan isms considered by Judd are extended to the multiphonon case. The "electron ic part" of the end-result has the same form as the well-known Judd-Ofelt e xpression for radiative transition probability between multiplets. The simp lified form of the general expression is proposed. Other mechanisms of opti cal photon-phonon transitions are discussed.