The perturbation theory of stimulated Raman scattering (SRS), with Raman pump on minus pump off and heterodyne detection along the probe direction, is reviewed. It has four third-order polarization terms, labeled as SRS or inverse Raman scattering (IRS): SRS(I), SRS(II), IRS(I), and IRS(II). These four polarizations have a wave packet interpretation. The polarizations, with homogenous and inhomogeneous broadening included, can be written as integrals over four-time correlation functions, and analytic formulas are derived for the latter for multidimensional harmonic potential surfaces with Franck-Condon displacements in the modes which facilitates the calculation of the SRS cross sections. The theory is applied to understand recent experimental results on the femtosecond SRS (FSRS) of a fluorescent dye, rhodamine 6G (R6G), where the Raman pump pulse is about 1 ps long, and the probe pulse is about 10 fs. The calculations compared very well with the R6G experimental results for off-resonance and resonance FSRS spectra spanning both Stokes and anti-Stokes bands, and for negative and positive pump-probe delay times on resonance.

The dispersive lines observed in time-resolved femtosecond stimulated Raman spectroscopy ͑FSRS͒, using a pair of 809 nm, 3 ps Raman pump, and 840-960 nm ultrashort probe pulse, for the first 500 fs photoisomerization dynamics in the excited state of bacteriorhodopsin, BR ‫ء‬ ͑S 1 ͒, created by a prior 500 nm, 35 fs actinic pump pulse, have previously been attributed to Raman initiated by nonlinear emission ͑RINE͒. We used four-wave mixing energy level diagrams to describe the FSRS process, which include RINE as a subset, and a 29-mode harmonic oscillator model for BR 568 in the calculations. Our calculations showed that FSRS of BR ‫ء‬ effectively occurs from the ground vibrational state of each of the observed 800-1800 cm −1 modes of S 1 . The lifetime on S 1 determines the linewidth and decay of the dispersive lines, and is estimated to be ϳ600 fs, comparable to the stimulated emission decay time. The FSRS dipole couplings are from the ground vibrational state of S 1 to high energy vibrational states on BR ͑S 0 ͒, and we place a fast decay lifetime of ϳ100 fs on S 0 which can be attributed to the correlation function from the many unobserved low frequency modes. The FSRS dispersive lines are shown to be due to the inverse Raman scattering term with ͉0͗͘1͉ vibrational coherence on the S 1 surface, and are not due to RINE with vibrational coherence on the S 0 surface. Our calculations show that the RINE process gives rise to broad featureless spectra.

Ultrafast Raman loss spectroscopy (URLS) is equivalent to anti-Stokes femtosecond stimulated Raman spectroscopy (FSRS), using a broadband probe pulse that extends to the blue of the narrow bandwidth Raman pump, and can be described as inverse Raman scattering (IRS). Using the Feynman dual time-line diagram, the third-order polarization for IRS with finite pulses can be written down in terms of a four-time correlation function. An analytic expression is obtained for the latter in the harmonic approximation which facilitates computation. We simulated the URLS of crystal violet (CV) for various resonance Raman pump excitation wavelengths using the IRS polarization expression with finite pulses. The calculated results agreed well with the experimental results of S. Umapathy et al., J. Chem. Phys. 133, 024505 (2010). In the limit of monochromatic Raman pump and probe pulses, we obtain the third-order susceptibility for multi-modes, and for a single mode we recover the well-known expression for the third-order susceptibility, χ(IRS) ((3)), for IRS. The latter is used to understand the mode dependent phase changes as a function of Raman pump excitation in the URLS of CV.

Femtosecond stimulated Raman spectroscopy (FSRS), using an overlapping pair of narrow band Raman pump and broadband probe pulses with heterodyne detection along the probe pulse direction, is a new nonlinear spectroscopic technique to record vibrational spectra of even highly fluorescent molecules and to study vibrational dynamics on excited electronic states of molecules, as in photoisomerization. FSRS is described by diagrammatic third-order perturbation theory with wave packet analysis. The phase matching condition gives rise to forty-eight terms for FSRS, but the resonant condition reduces it to just eight terms, which can be depicted by Feynman dual time-line diagrams, or closed time path loop diagrams, or the complementary four-wave mixing energy level diagrams. The eight terms fall into four sets-SRS(I), SRS(II), IRS(I), IRS(II)-where SRS stands for stimulated Raman scattering and IRS stands for inverse Raman scattering. The SRS(I) set can also account for spontaneous Raman scattering, but the remaining SRS(II), IRS(I) and IRS(II) terms are only present in stimulated scattering with the presence of a probe field. The SRS(I) set accounts for the Stokes Raman lines while the IRS(I) term accounts for the anti-Stokes lines, relative to the Raman pump frequency, in the FSRS spectrum. The remaining SRS(II) and IRS(II) terms give rise to broad baselines. Using a harmonic oscillator model, analytic results are obtained for the four-time correlation functions in the third-order polarizations. The issue of high time and high frequency resolution in time-resolved FSRS spectra is discussed. Calculations are made with the theory to compare with experimental results for: (a) resonance FSRS of fluorescent Rhodamine 6G and (b) 2D-FSRS from a coherent vibrational state that has been prepared by an impulsive, off-resonant pump pulse on CDCl 3 . The calculated results compared well with experimental results, and in the case of 2D-FSRS on CDCl 3 there is a dominant cascade effect contributing to the FSRS spectra.femtosecond stimulated Raman spectroscopy, temporal and spectral resolution, cascade effect