Origin of Dispersive Line Shapes in Plasmonically Enhanced

Article pubs.acs.org/JPCC

Origin of Dispersive Line Shapes in Plasmonically Enhanced Femtosecond Stimulated Raman Spectra Aritra Mandal,†,§ Shyamsunder Erramilli,‡,§ and L. D. Ziegler*,†,§ †

Department of Chemistry, ‡Department of Physics, and §The Photonics Center, Boston University, Boston, Massachusetts 02215, United States S Supporting Information *

ABSTRACT: A theoretical model is described to explain the observed dispersive-like vibrational line shapes reported in previous studies of plasmonically enhanced (PE) femtosecond stimulated Raman spectroscopy (FSRS). These Raman line shapes are rationalized in terms of interference or heterodyne terms between the PE stimulated Raman response from the molecules located in the plasmonically enhancing regions of Au nanoparticles and the nonlinear emission/scattering due to the resonant surface plasmon resonance (SPR) of these nanoparticles. The treatment of the FSRS signal response follows the standard perturbative polarization approach apart from the familiar g4 field enhancement factor found in spontaneous SERS description, where g represents the local electromagnetic field enhancement due to proximity with the plasmonic nanoparticles. The SPR contribution is modeled as a PE resonant two-level system. The heterodyne cross-term between these two complex third-order responses is large because both of the components are plasmonically enhanced and results in the observed dispersive-like line shape and plasmon resonance detuning dependence. The corresponding calculated spontaneous SERS spectrum for this system only exhibits a normal Lorentzian vibrational resonance, apart from a very weak broad baseline, in agreement with experimental observations.

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INTRODUCTION Femtosecond stimulated Raman spectroscopy (FSRS) has emerged as a powerful technique in the past decade to investigate dynamical processes on the time scale of tens of femtoseconds with vibrational specificity in a wide range of molecular systems including proteins and biomolecules, molecule−nanoparticle conjugates, transition metal complexes, and small conjugated polyenes.1−14 FSRS is a third-order nonlinear spectroscopic technique that uses a combination of a narrow-band picosecond Raman pump pulse and a temporally overlapped broad-band femtosecond probe pulse to generate stimulated Raman spectra of the system.1,4 Owing to the large bandwidth of the probe pulse, FSRS allows measurement of stimulated Raman spectra over a broad spectral range. Because vibrational dephasing timescale(s) are of the same order as the pulse width of the Raman pump pulse, the homogeneous line widths of the vibrational features are effectively determined by their inherent dephasing timescale(s) and the relative time delay between the Raman pump and probe pulses.9,14,15 Consequently, high spectral resolution (∼10 cm−1 or better) is usually obtained with this technique following dispersed detection of the signal. To date, the most valuable application of FSRS has been the observation of excited electronic state stimulated Raman spectra, following electronically resonant excitation by a femtosecond actinic pump pulse. As a result, FSRS can reveal vibrationally specific dynamics on excited electronic state surfaces as a function of time delay between the © XXXX American Chemical Society

actinic pulse and the picosecond−femtosecond pulse combination with a time resolution of kBT. The corresponding response function for this time evolution history defined by the three-level system shown in Figure 2 is given by (3) RFSRS (τ3′, τ2′, τ1′) =

e −t

/2σP2

(7)

where Nm is the number of molecular scatters contributing to this signal, Δv1 = (ωe0 − ΩP), Δv2 = (ω10 − ΩP + ΩF), and Θ(t − τ) is the Heaviside step function that insures causality (see the Supporting Information). The stimulated Raman signal field (E(3) FSRS(t,τ)) is thus temporally peaked at the center of the probe pulse, has a carrier frequency of (ΩP − ω10), and decays with the dephasing time scale of the vibrational coherence (Γ−1 10 ). Fourier transformation of this Raman resonant response yields the complex FSRS signal field spectrum, Ẽ FSRS(ω,τ), shown in Figure 3, for a 1000 cm−1 Raman-active vibrational mode with a typical 1 ps vibrational coherence decay time (Γ−1 10 = 1 ps) and an off-resonant tuning parameter (Δv1) of 20 000 cm−1. The real part of this complex amplitude spectrum is Lorentzian-like, and the imaginary part is dispersive in nature, as shown in Figure 3. The observed dispersed FSRS signal results from heterodyning the complex stimulated Raman signal field spectrum, Ẽ FSRS(ω,τ), with the probe pulse spectrum (eq 5). For transform-limited pulses, the resultant FSRS spectrum exhibits vibrational line shapes resembling the real part of the stimulated Raman amplitude spectrum, Re(Ẽ FSRS(ω,τ)), because the Raman resonance is much narrower than the femtosecond probe pulse bandwidth and the probe pulse electric field is all real. This result is completely analogous to previous treatments of FSRS in the absence of electronic resonance.5 However, we note, as pointed out previously as well, that the exact FSRS line shape will exhibit some dependence on the delay between the

⎛ i ⎞3 2 2 −iωe0τ1′ −Γe0τ1′ −iω10τ2′ −Γ10τ2′ ⎜ ⎟ |μ | |μ | e e e e ⎝ ℏ ⎠ e0 e1 e iωe1τ3′e−Γe1τ3′

2

(6)

where 0 and 1 are the initial and final ground vibrational levels and e is a nonresonant electronic state that represents all of the C

DOI: 10.1021/acs.jpcc.6b03303 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C

Figure 3. Real (solid lines) and imaginary (dashed lines) components of the signal field amplitude spectra of the stimulated Raman (Ẽ FSRS(ω,τ)) and third-order plasmonic emission (Ẽ PLE(ω)) signals. Each spectrum is normalized to the maximum of the real component of the corresponding signal field. In this calculation, the pump and the probe pulses are temporally overlapped (τ = 0). The plasmon resonance energy is varied between 12 200 cm−1 (Im(KPLE) > 0) and 10 300 cm−1 (Im(KPLE) < 0) to obtain positive real components for the third-order plasmonic emission signal field spectrum with oppositely signed imaginary components of the same.

Raman pump and probe pulses (τ) because the vibrational dephasing time scale (Γ−1 10 ) is of the same order as the pulsewidth of the Raman pump (2 ln 2 ·σP).15 PE-FSRS Spectrum. A. Overview. In contrast to the observed FSRS signals from an electronically nonresonant molecular species described above, in PE-FSRS, two different plasmonically enhanced third-order polarization sources may contribute to the overall response viewed along the femtosecond probe pulse direction, a narrow-band stimulated Raman signal from the molecules near plasmonically enhancing nanoparticle regions and a broad third-order nonlinear emission/scattering signal due to the nanoparticle SPR. Here, the SPR is modeled as a resonant electronic two-level system (Figure 4). Thus, the PE-FSRS spectrum, SPE‑FSRS(ω,τ), in analogy to eq 4, is given by SPE‐FSRS(ω , τ ) =

Figure 4. Double-sided Feynman diagrams and WMEL diagrams for the time evolution histories contributing to the third-order nonlinear emission/scattering signal from the surface plasmons for the system parameters described in the text. The blue and red arrows represent interaction with the picosecond Raman pump and femtosecond probe pulses, respectively. The signal electric field in each of the diagrams is represented by a curly arrow. The dotted and solid arrows in the WMEL diagrams represent bra side and ket side interactions, respectively.

̃ (ω , τ ) + g 4 · E PLE ̃ (ω)|2 |E F̃ (ω) + g 4 · E FSRS |E F̃ (ω)|2

evident from eq 8, there are potentially a number of plasmonically enhanced spectral contributions to the PEFSRS spectrum for strongly enhancing nanostructures including terms that scale as g8 in this model. B. Plasmonically Enhanced Stimulated Raman, ẼFSRS(ω,τ), and Third-Order Plasmonic Emission Spectra, ẼPLE(ω). As indicated above (eq 8), the Raman contribution of an electronically nonresonant molecule near a plasmonic structure to the calculated PE-FSRS spectrum is described in this treatment by g4Ẽ FSRS(ω,τ), where g is the electric field enhancement factor and Ẽ FSRS(ω,τ) is identical to the result given above in the absence of the plasmonic structure obtained numerically by Fourier transformation of E(3) FSRS(t,τ) (eq 7). The perturbative polarization framework used to describe the stimulated Raman component for the defined picosecond and femtosecond incident pulses is analogously employed for the derivation of the PLE signal contribution for this sample system. The third-order, far-field response of the PLE signal, (3) EPLE (t,τ), is modeled as resulting from a plasmonically enhanced, resonant two-level system. Eight possible time evolution histories can contribute to the third-order PLE signal generated along the phase-matched direction of the probe pulse

(8)

where Ẽ F(ω) and Ẽ FSRS(ω,τ) are defined above and Ẽ PLE(ω) is the complex amplitude spectrum of the third-order nonlinear signal field due to the stimulated resonant surface plasmon emission/scattering (PLE). Ẽ PLE(ω) does not depend on the delay between the pump and the probe pulse, τ, as shown below. Analogous to the treatment of spontaneous SERS,30 g represents the local electric field enhancement factor of the incident/scattering optical radiation due to the SPR. For simplicity, the enhancement factor g is taken to be identical for the picosecond Raman pump and femtosecond probe pulse frequencies contributing to the stimulated Raman signal. We also explicitly let each field−matter interaction for the PLE signal scale with g. Hence, both third-order nonlinear signal contributions are plasmonically enhanced by a factor on the order of g4, as indicated in eq 8.31 No additional enhancement is explicitly considered for the transmitted femtosecond pulse field, Ẽ F(ω), striking the array detector given the small spatial region (>10 nm) of the strongest plasmonic interaction with the Au nanoparticle dimers in the focal volume relative to the focal diameter of the incident probe beam (∼100 μm). As is D


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⎛ E2 E ⎞ 2 2 ⎛N ⎞ ̃ (ω) = A⎜ PL ⎟|μ |4 2π σF⎜⎜ P,0 F,0 ⎟⎟KPLEe−σF (ω −ΩF) /2 E PLE E 0 3 ⎝ ℏ ⎠ 8 ⎝ ⎠

due to a two-level system excited by two different resonant optical frequencies within the homogeneous line width of the plasmon resonance. These PLE time evolution histories, corresponding to the pulse sequences −κP + κP + κF (I and II), κP − κP + κF (III and IV), −κP + κF + κP (V and VI), and κF − κP + κP (VII and VIII), are summarized in Figure 4. I, III, V, and VII may be categorized as resonance emission-type and II, IV, VI and VIII as resonance (Rayleigh) scattering-type time evolution histories based on whether excited level population is created after two resonant field interactions. Taking the resonant surface plasmon absorption to be dominated by homogeneous broadening, the total third-order response function contributing to the PLE signal is the sum of the response functions for each of the time evolution histories shown in Figure 4 and are given by

(12)

The real and imaginary components of the complex PLE field spectrum, Ẽ PLE(ω), in the probe pulse direction are both effectively proportional to the probe pulse spectrum. Their relative magnitudes are determined by the detuning of the plasmon resonance from the pump and probe center PL frequencies (ΔPL 1 and Δ2 ), the difference in center frequency of the two pulses (ς), and the plasmonic relaxation rates (ΓE0, ΓEE). For example, these relative real and imaginary PLE spectral components are plotted in Figure 3 for two different plasmon resonance frequencies (ωE0), 12200 and 10300 cm−1, that correspond to the approximate plasmon resonances of the nanoparticles employed in the previous PE-FSRS studies.20,21 As seen in this figure, these SPR frequencies result in positive and negative imaginary components of the PLE spectrum, respectively. The real part of the complex prefactor, KPLE, is always positive, and the sign of its imaginary component for a specific plasmonic system is determined by the sign of the PL detuning parameters, ΔPL 1 and Δ2 , for a given set of parameters for the pulses. The presence of this PLE signal field on the array detector offers the possibility of a complex heterodyning field not present in standard FSRS experimental configuration. When the probe pulse duration is shorter than the SPR dephasing time, this same analysis will apply and dispersive vibrational resonances will be observed, although the exact details of the distorted line shape effects can exhibit a different detuning dependence. We also note that unlike the Ẽ FSRS(ω,τ) spectrum, the PLE signal field (eq 10) and consequential spectral line shape (eq 12) do not depend explicitly on the exact time delay between the Raman pump and probe pulse (τ), as long as they are temporally overlapping, due to the rapid dephasing/population relaxation of the plasmonic resonance relative to the duration of the interacting optical pulses. C. Effects on Raman Line Shape in PE-FSRS. When the expression for the PE-FSRS spectrum (eq 8) is expanded, there are multiple contributions to the nonlinear signal along the phase-matched detection direction that may contribute to the observed dispersed spectrum, including terms that scale as g8 in this treatment

(3) RPLE(I,II,V,VI) (τ3′, τ2′, τ1′) ⎛ i ⎞3 = ⎜ ⎟ |μE0 |4 e−iωE0τ1′e−ΓE0τ1′e−ΓEEτ2′e iωE0τ3′e−ΓE0τ3′ ⎝ℏ⎠ (3) RPLE(III,IV,VII,VIII) (τ3′, τ2′, τ1′) ⎛ i ⎞3 = ⎜ ⎟ |μE0 |4 e iωE0τ1′e−ΓE0τ1′e−ΓEEτ2′e iωE0τ3′e−ΓE0τ3′ ⎝ℏ⎠

(9)

where μE0 and ωE0 are the transition dipole and transition frequency associated with the resonant surface plasmon EE excitation. ΓE0 = 1/TE0 2 and ΓEE = 1/T1 are the corresponding transition dephasing and excited plasmon population relaxation rates. The third-order signal field (eqs 1 and 2) resulting from the above-defined PLE response functions (eq 9) and the previously defined femtosecond and picosecond incident pulses is ⎛ E2 E ⎞ ⎛N ⎞ P,0 F,0 −(t − τ )2 /2σF2 iΩ F(t − τ ) (3) 4 ⎟|μ | ⎜ E PLE (t , τ ) = A⎜ PL e ⎜ 8 ⎟⎟KPLEe E 0 3 ⎝ ℏ ⎠ ⎝ ⎠ (10)

Here, KPLE is a complex prefactor defined by the system and radiation parameters ⎡⎛ ⎞ ⎞⎛ 1 ⎞ ⎛ 4Γ 2(−iς + 2ΓE0) ⎟ KPLE = ⎢⎜ PL 2 E0 2 ⎟⎜ ⎟ + ⎜ PL PL ⎢⎣⎝ (Δ1 ) + Γ E0 ⎠⎝ ΓEE ⎠ ⎝ (iΔ1 + ΓE0)(−iΔ2 + ΓE0) ⎠ ⎞ ⎛ ⎞⎤⎛ 1 1 ⎟ ⎜ ⎟⎥⎜ + Γ ⎝ −iς + ΓEE ⎠⎥⎦⎝ −iΔPL 2 E0 ⎠

SPE‐FSRS(ω , τ ) = 1 + g 4 ·

(11)

PL ΔPL 1 = (ωE0 − ΩP) and Δ2 = (ωE0 − ΩF) are the detunings of the picosecond and femtosecond pulse carrier frequencies from the peak of the plasmon resonance, respectively; ς = (ΩP − ΩF), and NPL is the number of plasmonic particles contributing to the observed signal (see the Supporting Information). The picosecond pump pulse and the 50 fs probe pulse are both slowly varying compared to the rapid dephasing and relaxation −1 dynamics of the plasmon resonance (Γ−1 E0 = 10 fs and ΓEE = 10 32 fs), enabling this analytical description for the PLE signal field (eq 10). As a consequence of these rapid plasmonic relaxation timescale(s), that is, when the SPR spectral width is greater than that of the resonant pulses, the time dependence of the PLE signal field is essentially that of the probe pulse envelope, as seen above (eq 10) and the following analytical expression for the complex amplitude-level PLE spectrum results

̃ * (ω , τ )) 2Re(E F̃ (ω)·E FSRS |E F̃ (ω)|2

+ g 4·

̃ * (ω)) 2Re(E F̃ (ω) ·E PLE |E F̃ (ω)|2

+ g 8·

̃ (ω , τ )·E PLE ̃ * (ω)) 2Re(E FSRS |E F̃ (ω)|2

+ g 8·

̃ (ω , τ )|2 |E FSRS |E ̃ (ω)|2 + g 8 · PLE 2 2 |E F̃ (ω)| |E F̃ (ω)| (13)

As seen above, the PE-FSRS spectrum potentially has contributions from both the stimulated Raman and PLE signals alone and heterodyned terms (eq 13). Three of these plasmonically enhanced terms scale as g8, and two scale as g4 in this PE-FRSR treatment. Furthermore, three of these terms contain the stimulated Raman spectrum and thus are central to controlling the observed line shape of the vibrational feature in E


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The Journal of Physical Chemistry C the PE-FSRS spectrum. Because the real and imaginary components of Ẽ PLE(ω) are essentially proportional to the probe spectrum, the PLE signal field heterodyned with the probe pulse (g4 · 2 Re[Ẽ F(ω) · Ẽ PLE * (ω)]), and the homodyne PLE signal (g8 · |Ẽ PLE(ω)|2), the third and sixth terms in eq 13 do not contribute to the stimulated Raman line shape of the PE-FSRS spectrum, apart from a g-dependent constant baseline offset. A priori calculation of the predicted PE-FSRS spectra (eq 13) requires estimates of several parameters governing the relative amplitude of the stimulated Raman and PLE signals (relative transition moments, number densities, etc.) in addition to an appropriate plasmonic enhancement factor (g). For a relative PLE/stimulated Raman amplitude factor of 2 corresponding to

4 NPL |μE0 | · , 2 Nm |μe0 | |μe1|2

Figure 6. Contribution of the homodyne FSRS signal (green trace), the heterodyne FSRS signal with the femtosecond probe pulse (blue trace), and the cross-term between FSRS and PLE signals (red trace) to the dispersed Raman line shape of the PE-FSRS spectrum (black trace) for a plasmonic enhancement factor g = 30. The plasmon resonance frequency is taken to be 12200 cm−1 for calculation of these τ = 0 spectra.

the PE-FSRS stimulated Raman

line shapes displayed in Figure 5 are obtained as a function of

heterodyned stimulated Raman signal with the femtosecond probe pulse; g8 · |Ẽ FSRS(ω,τ)|2, the stimulated Raman homodyne signal; and g8 · 2 Re[Ẽ FSRS(ω,τ) · Ẽ PLE * (ω)], the heterodyne spectrum resulting from the cross-term between the PLE and the stimulated Raman signal fields. As shown in Figure 6, for the defined system and radiation field parameters, the homodyne stimulated Raman signal term (g8 · |Ẽ FSRS(ω,τ)|2) is much smaller than both heterodyne terms due to the inherent weakness of the stimulated Raman signal despite the g8 scaling for this term. Thus, in this model, the two Ẽ FSRS(ω,τ) heterodyned terms determine the calculated vibrational line shape for g = 30. The plasmonically amplified “normal” FSRS, g4 · 2 Re[Ẽ F(ω) · Ẽ *FSRS(ω,τ)], and the cross-term between the stimulated Raman and PLE signals, g8 · 2 Re[Ẽ FSRS(ω,τ) · Ẽ PLE * (ω)], are on the same order of magnitude, but the latter exhibits a dispersive line shape, as shown in Figure 6. This dispersive line shape is a consequence of the complex heterodyning PLE signal field (Figure 3), allowing contributions from both the real and imaginary components of the stimulated Raman signal to appear in the PE-FSRS response. The imaginary, dispersive component of the stimulated Raman signal (Figure 3) does not contribute to the FSRS spectrum when it is heterodyned with the probe pulse whose electric field is all real. Furthermore, because the PLE-FSRS heterodyne spectral contribution (Re[Ẽ FSRS(ω,τ) · Ẽ *PLE(ω)]) is more strongly dependent on the field enhancement factor, g8 in this treatment, as compared to the g4 scaling of the Ẽ F(ω)dependent heterodyne term, the dispersive line shape character is anticipated to be more strongly dependent on g, as seen in Figure 5. Nondistorted Raman line shapes should be observed for very weakly enhancing plasmonic substrates. D. Dispersive Line Shape Phase Dependence. As summarized above, the phase of the dispersive stimulated Raman features in the observed PE-FSRS spectra were found to be dependent on the plasmon resonance frequency relative to the Raman pump/probe frequencies. The phase of this dispersive spectral line shape exhibits a nearly π flip, that is, changes sign, as the plasmon resonance is tuned from the red to the blue of the probe pulse frequency.20,21 Correspondingly, as shown in Figure 7, the phase of the calculated dispersive PEFSRS spectral line shape for g = 30 goes from positive to negative as the plasmon resonance energy is changed from 10300 to 12200 cm−1, where the center frequencies of the transform-limited pump and probe pulses are 12580 and 11800

Figure 5. PE-FSRS spectra as a function of the plasmonic enhancement factor, g. The spectra looks increasingly dispersive with an increase in the enhancement factor. The constant baseline is subtracted off of these spectra, and they are vertically shifted by 0.7 counts for clarity in presentation. The plasmon resonance frequency is taken to be 12200 cm−1 for calculation of these τ = 0 spectra.

the field enhancement factor, g. Assuming that the Ramanactive molecules with an area of ∼1 nm2 are present in the interstitial region of the 60−90 nm Au nanoparticle dimers,33 the value of the relative nanoparticle to molecule number densities NPL may be at least on the order of 10−4. If we use a Nm

ratio of Rayleigh to Raman scattering intensity to estimate the value of

|μE0 |4 |μe0 |2 |μe1|2

, then this factor is at least on the order of 104.34

Hence, an overall lower limit for the product of these terms is on the order of unity. A larger value of this product will only magnify the effect of plasmonic enhancement on the observed dispersive line shape in PE-FSRS spectra at lower values of g. As is evident in this figure, the PE-FSRS spectrum acquires increasing dispersive character as the plasmonic enhancement factor, g, attributable to the nanoparticle structure increases. For g = 30, corresponding to an enhancement factor (g4) of ∼106, the observed stimulated Raman line shape exhibits a dispersive character that resembles the stimulated Raman lines shapes reported for the observed PE-FSRS spectra.20,21 The three terms containing the Raman resonance (eq 13) contributing to the dispersive-shaped vibrational resonance in Figure 5 for g = 30 are separately plotted in Figure 6 in order to show the relative importance of each of these contributions to the total PE-FSRS line shape. These contributions are the spectrum resulting from g4 · 2 Re[Ẽ F(ω) · Ẽ FSRS * (ω,τ)], the F


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Figure 7. PE-FSRS spectra for two different detunings of the SPR from the pump/probe pulse, where the plasmon resonance energy (ωE0) is chosen to be 12200 (blue trace) and 10300 cm−1 (red trace), respectively. The constant baseline in these spectra is subtracted, and the Raman pump and probe pulses are considered to be overlapped in time (τ = 0).

Figure 8. PE-FSRS spectra as a function of time delay between the Raman pump and the probe pulses (τ). The background constant baselines in these spectra are subtracted, and they are artificially offset by 0.3 counts for clear presentation purposes. The plasmon resonance frequency is taken to be 12200 cm−1 with a plasmonic enhancement factor g = 30 for calculation of these spectra.

cm−1, respectively. As seen in Figure 6, the dispersive line shape in PE-FSRS spectra is attributed here to the heterodyne crossterm between the stimulated Raman and PLE signals (Re[Ẽ FSRS(ω,τ) · Ẽ *PLE(ω)]). The relative phasing of the real and imaginary Ẽ FSRS(ω,τ) components (Figure 3) are unaffected by the tuning of plasmon resonance energy. However, the sign of the imaginary component of the nonlinear PLE signal (Im(Ẽ PLE(ω))) depends on the detuning terms, ΔPL 1 and ΔPL 2 (eqs 11 and 12), which in turn induces a ∼π phase shift of the resultant PE-FSRS spectral line shape. Furthermore, from eq 11, it can be predicted that for some value of the plasmon resonance energy, the imaginary component of the PLE signal (Im(KPLE) = 0) will vanish, resulting in a fully absorptive PE-FSRS spectral line shape, at least within this simplified model of homogeneous dephasing and transformlimited pulses. The value of plasmon resonance energy in this model for which the PE-FSRS spectral line shape is fully absorptive is 11480 cm−1, slightly red-shifted from the probe pulse center frequency. It is also worth noting that the relative amplitude ratio of the imaginary and real components of the PLE signal, Im(KPLE)/Re(KPLE), is larger for probe pulses to the red of the plasmon resonance peak than that for probes on the blue side of the plasmon peak. Thus, dispersion line shape effects are predicted to be more noticeable on the red of the plasmon resonance peak for the same magnitude of SPR detuning (ΔPL 2 ). Phenomenologically, the experimental PE-FSRS spectra fits well to a Fano line shape, as demonstrated in the previous PEFSRS study, because the asymmetry parameter (q) in the Fano line shape function (eq 1 in ref 21) can be related to the relative amplitude of the plasmonically enhanced PLE signal field and the probe pulse electric field. The sign of this parameter determines the phase of the resultant PE-FSRS spectra. E. τ Dependence of PE-FSRS Line Shapes. The dependence of the time delay between the Raman pump and the probe pulses, τ, was reported in the previous PE-FSRS study.21 Although described as τ-independent, a small systematic change in line shape seemed discernible in the experimental spectra as this delay was tuned from −τ to +τ values. The calculated τ dependence of the dispersive PE-FSRS spectral line shapes is shown in Figure 8 for the set of system parameters used in the PE-FSRS calculations discussed above (Figures 5−7) for an enhancement factor of g = 30. The widths of the vibrational

peaks broaden by ∼30% as τ increases from −400 to 400 fs. This result is expected because increasing the value of τ allows the picosecond Raman pump pulse to truncate the vibrational coherence earlier, resulting a broader spectrum. This change, however, is small and solely contributed by the stimulated Raman signal and has been described in previous treatments of FSRS in the absence of plasmonic enhancement.15 The PLE signal does not depend on the interpulse time delay, as discussed above, due to the rapid decay of the plasmon coherence. As a result, the phase of the dispersive line shape is predicted to change only slightly as a function of the Raman pump and probe pulse delay via this PE-FSRS calculation model, in seeming accordance with the previously reported experimental data.21 F. Corresponding Spontaneous Raman Line Shape. Surface-enhanced spontaneous Raman spectra, in general and specifically for BPE on the Au dimers used for PE-FSRS, do not show such dispersive line shapes, as is evident in the SERS spectrum excited at 795 nm for this sytem.20 The predicted spontaneous SERS emission spectrum can be calculated for the model system employed here with the same set of molecular and plasmonic parameters used to calculate the PE-FSRS spectra (Figures 5−8). The corresponding total spontaneous emission excited by the Raman pump pulse for this twocomponent system, an electronically nonresonant molecule with a Raman active-mode and a SPR resonant nanoparticle, is given by35 ⎛ 1 ⎞⎛ 1 ⎞ Spon SRaman (ΩS) = g 4 Nm |μe0 |2 |μe1|2 ⎜⎜ v ⎟⎟⎜⎜ v ⎟⎟ ⎝ ℏΔ1,S ⎠⎝ ℏΔ3,S ⎠ ⎞ 2ΩS4 ⎛ Γ10 ⎜ ⎟⎟ I P⎜ v 2 πc 2 ⎝ (Δ2,S ) + (Γ10)2 ⎠ Spon SPlasmon (ΩS) = g 4 NPL |μE0 |4

(14a)

4 ⎞ 1 2ΩS ⎛⎜ 1 ⎟ I 2 2 P⎜ PL 2 2⎟ ℏ πc ⎝ (Δ1,S) + (ΓE0) ⎠

⎡⎛ ̂ ⎞⎛ ⎤ ⎞ ΓE0 ⎢⎜ 2Γ ⎟⎜ ⎥ ⎟ ( ) + πδ Ω − Ω S P 2 2⎟ ⎢⎣⎝ ΓEE ⎠⎜⎝ (ΔPL ⎥⎦ 2,S) + (ΓE 0) ⎠ (14b)

where, ΩS is the frequency of the spontaneous emission field and IP is the intensity of the incident excitation. Δv1,S = (ωe0 − G


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The excellent spectral fitting of the observed PE-FSRS spectra achieved by the use of the so-called Fano line shape function plus a broad polynomial baseline component reported earlier21 is fully consistent with the line shapes predicted here by this multiple nonlinear polarization interference treatment. As shown in Figure 6, the observed PE-FSRS signal has some Lorentzian contribution from the pulse heterodyned contributions and a dispersive-like component due to the plasmonically enhanced PLE heterodyned Raman contribution. The relative amplitudes of these contributions are controlled by the best-fit determined q Fano asymmetry parameter in prior phenomenological analysis. Thus, the good fits of the PE-FSRS spectra to the Fano line shape are fully consistent with the model described here for this effect. Interestingly, interference effects from plasmon-based third-order polarization signals were ruled out as the cause of these distorted vibrational PE-FSRS spectra in this prior work,21 although, as shown here, all features of these experimental spectra appear to be captured and understood via such a model. This two-nonlinear signal model also predicts that not all plasmonically enhanced experimental time domain Raman approaches will exhibit the interference effects in evidence for PE-FSRS. For example, in plasmonically enhanced ultrafast pump−probe experimental studies, the PLE emission/scattering signal will be greatly diminished for interpulse separations beyond a few tens of femtoseconds, whereas the pump pulse impulsively excited vibrational coherence decays will persist on the time scale of picoseconds, the vibrational dephasing timescale, and thus, heterodyning cross-terms due to these contributions should not be observed.37 Hence, in such ultrafast pump−probe studies of molecule-coated nanoparticle samples, Fourier transform of the observed responses will not show the dispersive vibrational line shapes as seen in PE-FSRS, even for the most strongly enhancing plasmonic particles. We anticipate that further PEFSRS experimental observations can confirm the modeling framework described here.

ΩP), Δv2,S = (ω10 − ΩP + ΩS), and Δv3,S = (ωe1 − ΩS). PL ̂ Furthermore, ΔPL 1,S = (ωE0 − ΩP), Δ2,S = (ωE0 − ΩS), and Γ is the pure dephasing rate given by Γ̂ = ΓE0 − ΓEE/2 (see the Supporting Information). The spontaneous emission spectrum is plotted in Figure 9 for this model system. The detected signal

Figure 9. SERS spectra of molecules on plasmonic nanoparticle surfaces. The weak baseline of the spectra is resulted by the spontaneous plasmonic emission signal, where the plasmon resonance energy is 10300 cm−1.

is just the sum of the spontaneous SERS intensity and the intensity of the resonant PLE emission because there is no phase relationship between the electric fields of these two spontaneous emission components. The SERS narrow-band vibrational Lorentzian spectral feature overlaps a very weak plasmon emission due to the small but finite SPR pure dephasing contribution in this model, resulting. in a very slightly sloped baseline to the SERS spectrum at this excitation wavelength (795 nm). In agreement with observations, only Lorentzian Raman line shapes (inhomogeneous and spectral diffusion effects neglected) result for the plasmonically enhanced Raman spectra via this treatment. Broad baselines in SERS spectra of plasmonically enhanced vibrations have been recently explained as a consequence of broad plasmon emission signal overlapping the SERS signal.36

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ASSOCIATED CONTENT

* Supporting Information

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S

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.6b03303. Derivations of eqs 7, 10, 11 and 14 (PDF)

CONCLUSION A simple theoretical model is described here that offers a plausible explanation for the origin of the dispersive line shapes reported in previous PE-FSRS experiments.20,21 An interference between plasmonically enhanced third-order nonlinear signals due to the molecular Raman response and the resonant plasmon emission/scattering can account for the dispersive line shapes and the plasmon detuning dependence observed for the PE-FSRS spectra, as well as the absorptive line shapes in the corresponding spontaneous SERS spectra of the BPE on Au nanoparticle dimers. Line shapes in solution-phase FSRS experiments can exhibit similar dispersive features when the incident pulses are weakly or nearly resonant with an electronic excitation of the Raman-active molecule and attributable to a similar interference effect.25 However, when the FSRS pump/ probe pulses are tuned into exact electronic resonance, the Raman component of the FSRS signal is masked by the stronger stimulated luminescence emission and no longer observable for Stokes scattering.25 Whether such effects are also evident for PE-FSRS as the fs/ps pulse pairs are tuned throughout the SPR may depend on the Raman scatterer and nanoparticle system details and will be explored in future experimental studies.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: +1-617-353-8663. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The support of NSF Grants CHE 1152797 and CHE 1609952 is gratefully acknowledged. REFERENCES

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Origin of Dispersive Line Shapes in Plasmonically Enhanced

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