Saturation effects in coherent anti-Stokes Raman scattering (CARS) spectroscopy are discussed. The discussion is limited to Raman-resonant CARS (${\ensuremath{\omega}}_{l}\ensuremath{-}{\ensuremath{\omega}}_{s}\ensuremath{\simeq}{\ensuremath{\omega}}_{21}$, where ${\ensuremath{\omega}}_{l,s}$ are the frequencies of the pump fields with powers
Pl,s, and ${\ensuremath{\omega}}_{21}$ is the frequency of the Raman transition $|1〉\ensuremath{\rightarrow}|2〉$) with the possible addition of a one-photon resonance (${\ensuremath{\omega}}_{l}\ensuremath{\simeq}{\ensuremath{\omega}}_{31}$ is the frequency of the electronic transition $|1〉\ensuremath{\rightarrow}|3〉$). For these cases we show that the CARS polarization
pCARS is proportional to the off-diagonal density-matrix element ${\ensuremath{\rho}}_{21}$. In order to determine ${\ensuremath{\rho}}_{21}$, we use Laplace transforms to solve the Bloch equations for the effective two-level system
∣1〉 and
∣2〉 when ${\ensuremath{\omega}}_{l}$ is far from resonance, or for the three-level system
∣1〉,∣2〉, and
∣3〉, when ${\ensuremath{\omega}}_{l}\ensuremath{\simeq}{\ensuremath{\omega}}_{31}$. The steady-state expression for
pCARS in the former case gives ${\mathcal{p}}_{\mathrm{CARS}}\ensuremath{\propto}{P}_{l}{P}_{s}^{\frac{1}{2}}$ at low powers and ${\mathcal{p}}_{\mathrm{CARS}}\ensuremath{\propto}{P}_{l}^{0}{P}_{s}^{\frac{\ensuremath{-}1}{2}}$ at high powers. In the three-level system, we show that when the pressure is low and at least one field is weak, the slow time dependence of ${\ensuremath{\rho}}_{21}$ must be considered. When one field is strong and the other weak, the CARS spectrum is Stark split. When
Pl is high, for example, ${\mathcal{p}}_{\mathrm{CARS}}\ensuremath{\propto}{P}_{l}^{0}{P}_{s}^{\frac{1}{2}}$ for ${\ensuremath{\omega}}_{s}\ensuremath{\simeq}{\ensuremath{\omega}}_{32}$ and ${\mathcal{p}}_{\mathrm{CARS}}\ensuremath{\propto}{({P}_{l}{P}_{s})}^{\frac{1}{2}}$ when ${\ensuremath{\omega}}_{s}\ensuremath{\simeq}{\ensuremath{\omega}}_{32}\ifmmode\pm\else\textpm\fi{}{V}_{13}$ where
V13 is the one-photon Rabi frequency for the $|1〉\ensuremath{\rightarrow}|3〉$ transition. The Wilcox-Lamb approximation is used to reduce the three-level Bloch equations to rate equations containing one- and two-photon terms. When the fields are so weak that both one- and two-photon terms are small compared to the decay terms, the usual expression for
pCARS is reproduced. If only the direct two-photon processes are important, the effective-two-system results are reproduced. When both fields are intense and nearly resonant, the steady state is rapidly achieved. The results for the case where one field is much stronger than the other are essentially the same as those for one strong and one weak field. When the fields are of comparable strength, ${\mathcal{p}}_{\mathrm{CARS}}\ensuremath{\propto}{P}_{l}^{\frac{1}{2}}{P}_{s}^{0}$ for ${\ensuremath{\omega}}_{l}\ensuremath{\simeq}{\ensuremath{\omega}}_{31}$ and ${\ensuremath{\omega}}_{s}\ensuremath{\simeq}{\ensuremath{\omega}}_{32}$, and the CARS spectrum is split into five components.
We have measured coherent anti-Stokes Raman spectra between crystal-field levels of the
2F5/2 and
2F7/2 manifolds of the
Ce3+ ion diluted in
LuPO4 at \ensuremath{\sim}10 K. The anti-Stokes signal is nearly resonant with the lowest crystal-field level of the 5
d1 configuration of
Ce3+ at 30 460 ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}$. For a
LuPO4 crystal doped with 0.06 mol %
Ce3+, we found enhancements of the third-order susceptibility \ensuremath{\Vert}${\mathrm{\ensuremath{\chi}}}^{(3)\mathrm{R}}$/${\mathrm{\ensuremath{\chi}}}^{(3)\mathrm{NR}}$\ensuremath{\Vert} as high as 11.6. This enhancement is over 20 times larger than that seen previously in singly resonant experiments with pure rare-earth crystals.
The coherent anti-Stokes Raman spectroscopy (CARS) of electronic states of the Ce3+ ion in a CeF3 crystal at 3.8 K is reported. CARS resonances involving the ground state of the 2F5/2 manifold and the Stark components of the 2F7/2 manifold at 2161 and 2239 cm−1 have been detected. Measurements were made with both visible (λ1 ≈ 476 nm, λ2 = 532 nm) and near-ultraviolet (λ1 = 355 nm, λ2 ≈ 385 nm) lasers. The enhancement of the third-order susceptibility owing to the electronic transitions, |χ(3)R/χ(3)NR|max, is as great as 4.8. This represents an order-of-magnitude increase over that in singly resonant electronic CARS experiments in other rare-earth crystals. The CARS data also yielded accurate measurements of the Ce3+ electronic Raman cross sections. In the visible region, σzz(2161 cm−1) = (5.0 ± 1.1) × 10−30 cm2 and σzz(2239 cm−1) = (1.9 ± 0.7) × 10−30 cm2. One could calculate both the absolute magnitude and the dispersion of |χ(3)R|max by modeling the virtual intermediate states as a single degenerate 5d state at 45 000 cm−1.
Coherent Raman scattering can generate Stokes and anti-Stokes fields of comparable intensities. When the Raman shift is due to a magnetic resonance transition (usually in the MHz to GHz range), the Raman fields are generally detected by optical heterodyne detection, using the excitation laser as the local oscillator. In this case, the two sidebands generate beat signals at the same frequency and are therefore indistinguishable. Separation of the two contributions becomes possible, however, by superheterodyne detection with a frequency-shifted optical local oscillator. We compare the two scattering processes, and show how the symmetry between them can be broken in
Pr3+:YAlO3.