Eighteen selected two-photon absorption (TPA) transition line strengths with polarization angles θ = 0° and 45°, spanning several orders of magnitude, have been calculated for the Tb 3+ ion in the cubic host Cs 2 NaTbCl 6 . The results are in reasonable agreement with experimental results in the literature. The calculation utilized the crystal field (CF) wavefunctions for the initial and final states of the 4f 8 configuration, and utilized free ion or CF wavefunctions (with the corresponding energies) for 4f 7 core states of the whole intermediate 4f 7 5d configuration comprising 34 320 states. The intensities of certain transitions were found to be very sensitive to the inclusion of the CF interaction within the 4f 7 core. In contrast to previous fourth- or third-order calculations of the TPA transition line strength of the strong transition ( 7 F 6 )A 1g → ( 5 D 4 )A 1g using pure Russell–Saunders (RS) wavefunctions for the | 7 F 6 ⟩ initial and ⟨ 5 D 4 | final states, our second-order direct calculation shows that the admixed RS wavefunctions |[ 7 F 6 ]⟩ and ⟨[ 5 D 4 ]| must be used to account for its high intensity. The effects of CF interactions within the 4f 7 core, i.e. J -mixing and CF energy level splitting, upon the ( 7 F 6 )A 1g → ( 5 D 4 )E g TPA transition line strength have been separated, and the latter effect is shown to be more important for the transition investigated.
Recent observations of two-photon absorption within the configuration 4f 7 of Gd 3+ in LaF 3 have indicated an anomalously high line strength for the transition 8 S 7/2 to 6 P 7/2 compared with 8 S 7/2 to 6 P 5/2 and 8 S 7/2 to 6 P 3/2 . It is shown that the standard expression for the two-photon intensity is unusually small for all three transitions and that a higher-order correction, involving matrix elements of the spin-orbit interaction within the intermediate configuration 4f 6 5d, predominates. The principal term is scalar, implying (in the absence of J mixing by the crystal field) isotropic contributions to the various components of the Delta J=0 transition. The line strengths that have been observed can be accounted for reasonably well without having to invoke variations in the radial integral (4f mod r mod 5d) or inadequacies in the closure approximation. Expressions for the transition matrix elements are given for the general configuration l N .
A theoretical estimate of the two-photon absorption (TPA) cross section for the lowest 4f\ensuremath{\rightarrow}5d zero-phonon transition in
Ce3+:
CaF2 is presented. A point-charge crystal-field model is used to calculate the odd component of the
C4v crystal field which mixes states of opposite parity into the initial and final states participating in the transition. The sum over intermediate states is evaluated by assuming a constant energy denominator and using a closure approximation. The calculated magnitude and polarization anisotropy of the TPA cross section are compared with the experimentally measured values and methods to improve the calculation are discussed.
The two-photon transition intensity of the ${(}^{7}{F}_{0}){\ensuremath{\Gamma}}_{1g}\ensuremath{\rightarrow}{\ensuremath{\Gamma}}_{1g}{(}^{5}{D}_{0})$ transition of the
4f6-ion
Sm2+ in
SrF2 has been investigated both in Judd-Pooler and direct formalisms. Although the agreement between the different calculations may be regarded as satisfactory, the direct calculation provides additional information concerning the transition intensity mechanism. The dominant pathway is found to involve the
4f55d1 intermediate core state
6H. The effects of changing the intermediate state baricenter have been investigated for both calculations, and the Judd-Pooler calculation becomes closer in agreement with the direct calculation as the energy baricenter increases.
A comprehensive report and analysis of our observations of numerous parity-allowed direct two-photon transitions within the
4f7 configuration of the
Gd3+ ion is presented. The experimental technique is ultraviolet fluorescence detection following optical excitation with a dye-laser beam. Previous results and analyses are extended in three ways. (1) The intensities and polarization dependence of individual Stark components, in addition to integrated multiplet intensities, are reported and analyzed for
Gd3+:La
F3. In those cases ($^{8}S_{\frac{7}{2}}\ensuremath{\rightarrow}^{6}P_{\frac{5}{2}},^{6}D_{\frac{9}{2},\frac{3}{2},\frac{5}{2}}$) where a second-order theory of two-photon absorption adequately explains the integrated intensity, it also (with the exception of
6D29) explains the Stark-component intensities. The remaining transitions are anomalously strong, and in some cases violate the angular momentum selection rules $\ensuremath{\Delta}L,\ensuremath{\Delta}J\ensuremath{\le}2$ and exhibit strong anisotropies, in contradiction to the second-order theory. It is shown that introduction of third- and fourth-order contributions involving spin-orbit and/or crystal-field interactions among levels of the
4f65d configuration, which serve as intermediate states, can account for both the integrated and Stark-component intensities of these transitions. (2) The strong anisotropy observed for the transitions $^{8}S_{\frac{7}{2}}\ensuremath{\rightarrow}^{6}P_{\frac{3}{2}},^{6}I_{J},^{6}D_{\frac{1}{2}}$ in
Gd3+:La
F3 is explained quantitatively as a polarization-dependent interference between contributions to the intensity which are comparable in magnitude. Interference is destructive for $\stackrel{\ensuremath{\rightarrow}}{E}\ensuremath{\parallel}\stackrel{^}{z}$ and constructive for $\stackrel{\ensuremath{\rightarrow}}{E}\ensuremath{\perp}\stackrel{^}{z}$, creating an order-of-magnitude contrast in the line strength for the two polarizations. (3) A two-photon excitation spectrum of
Gd3+ in aqueous solution is reported. Linewidths comparable to those of room-temperature solids and Stark splittings which suggest a low-symmetry quasistatic environment are observed.
The density matrix resonant two-photon absorption (TPA) theory applicable to laser crystals doped with rare earth ions is described. Using this theory, resonant TPA cross sections for transitions from the ground state to the second excited state of the 4f5d configuration in Pr3+:Y3Al5O12 are calculated. The peak value of TPA cross section calculated is 2.75 × 10-50 cm4 s which is very close to the previous experimental value 4 × 10-50 cm4 s. The good agreement of calculated data with measured values demonstrates that the density matrix resonant TPA theory can predict resonant TPA intensity much better than the standard second-order perturbation TPA theory.