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Theoretical analysis of anharmonic coupling and cascading Raman signals observed with femtosecond stimulated Raman spectroscopy
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Image of FIG. 1.
FIG. 1.

Schematic of 2D-FSRS experiment. (a) The experimental diagram indicates the relative pulse durations and delays of the impulsive pump, Raman pump, and probe. (b) In the absence of any anharmonic coupling or coherent cascades, the impulsive pump simply drives a low-frequency mode into coherence, and the Raman pump and probe together take a stimulated Raman spectrum of the sample. (c) The detected spectrum of the probe measures the interaction between the impulsively driven modes and the modes probed by FSRS. In the presence of anharmonic coupling between the driven and probed modes or via a coherent cascade, sidebands are observed in the stimulated Raman spectrum at Raman shifts of and . The phase of the sideband field varies with changes in time delay and produces a time-dependent lineshape.

Image of FIG. 2.
FIG. 2.

Feynman diagrams (top) and WMEL diagrams (bottom) for various FSRS signals. (a) The standard diagram for an allowed fundamental transition. (b) The diagram for a combination band transition. There is a similar diagram describing the “difference frequency” transition from the (1,0) state to the (0,1) state via a coherence. “” refers to the frequency of the resulting FSRS field, i.e., the frequency of a narrow peak on top of the broad probe spectrum in Fig. 1(c). “” is the frequency of the high-frequency vibration, which corresponds to the energy difference between the (0,0) and (0,1) states. Likewise, “” is the frequency of the low-frequency vibration, or the frequency of the transition. “” is the anharmonicity of the effective Hamiltonian and quantifies the frequency shift of the combination band, i.e., the , transition from the sum of the fundamental transitions, i.e., the and .

Image of FIG. 3.
FIG. 3.

Feynman diagrams (top) and WMEL diagrams (bottom) for various predicted fifth-order 2D-FSRS signals. (a) Diagrams for the upshifted sideband occurring at the combination frequency, . (b) Diagrams for the downshifted sideband occurring at the difference frequency, . [(c) and (d)] Diagrams for the phase-shifted signal generated at approximately the fundamental frequency, occurring at and , respectively.

Image of FIG. 4.
FIG. 4.

Feynman diagrams (top) and WMEL diagrams (bottom) for two parallel cascading signals. The diagram in (a) is the cascade that gives rise to the downshifted sideband, termed . Here, a CARS transition produces a field, , at frequency , which then participates as the third field in a separate FSRS transition. The diagram in (b) gives rise to the upshifted sideband via a process termed . Here, a CSRS field is initially produced at , which then takes part in a separate FSRS transition.

Image of FIG. 5.
FIG. 5.

Feynman diagrams (top) and WMEL diagrams (bottom) for two sequential cascading signals. Both (a) and (b) produce a phase-shifted FSRS signal at the fundamental transition frequency. The physical processes are the same as the parallel cascades described in Fig. 4, except that the CARS or CSRS fields now act as the first field in the FSRS transitions.

Image of FIG. 6.
FIG. 6.

(a) Simulated 1D FSRS spectrum of . (b) Simulated time-dependent fifth-order difference spectra of showing the time-dependent sideband lineshapes generated by FM of the CD stretch by the bend, the bend, and the C–Cl stretch. The spectrum in (a) is simulated using Eq. (35) and the parameters in Tables I and II. The spectra in (b) are simulated using Eqs. (38) and (40) and the parameters in Tables I and II.

Image of FIG. 7.
FIG. 7.

Simulated time-dependent spectra of in the region around the CD stretch at , calculated using the parallel and sequential cascade mechanism with molecular and laser parameters from Tables I and II in Eqs. (51) and (55). The region near has been divided by a factor of 5 for clarity.

Image of FIG. 8.
FIG. 8.

(a) Simulated 2D-FSRS spectrum using the frequency modulated signal. (b) Simulated 2D-CARS/FSRS spectrum using only the cascading signals. The diagonal lines indicate the series of 2D peaks that occur as sidebands up- and downshifted from the C–Cl stretch (dashed) or the C–D stretch (dot-dashed). The horizontal slice at a pumped frequency of , indicates that sidebands due to the C–Cl bend vibration are expected in the cascading signal (b), but not in the fifth-order frequency modulated signal (a). Realistic parameters of the molecular polarizabilities and field strengths, as shown in Tables I and II, were used.

Image of FIG. 9.
FIG. 9.

(a) 1D and (b) 2D-FSRS spectra of . The diagonal lines indicate lines along which the up- and downshifted sidebands appear that are coupled to a single high-frequency mode, i.e., the -intercept of the lines is . The various lines indicate sidebands shifted from: the bend (dotted), the C–Cl stretch (dashed), and the C–D stretch (dot-dashed). All the sidebands that originate from the same low-frequency pump-driven frequency lie along horizontal lines with -intercepts at . Sideband peaks that are due to the -type low-frequency mode at coupling to -type high-frequency modes are highlighted with boxes.

Image of FIG. 10.
FIG. 10.

(a) 1D and (b) 2D FSRS spectra of 50:50 mixture of . In (a) the parentheses indicate the assignment to either (H) or (D). The symmetric C–Cl stretch of occurs at and that of at . As shown in the boxed areas of the 2D spectrum (b) and expanded below, the presence of sidebands ±650 and shifted from both the C–D stretch [, dot-dashed lines, regions (c) and (e)] and the C–H stretch [, dot-dot-dashed lines in regions (d) and (f)] indicate that the signal cannot be attributed to anharmonic coupling between the driven and probed modes. The C–Cl bending modes of and are at 262 and for both isotopomers and so the FM and cascade mechanisms cannot be differentiated in the sets of peaks along these two horizontal slices.


Generic image for table
Table I.

Parameters used in the simulations.

Generic image for table
Table II.

Molecular parameters for used in the simulations.


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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Theoretical analysis of anharmonic coupling and cascading Raman signals observed with femtosecond stimulated Raman spectroscopy