We present a coupled wave semiclassical theory to describe plasmonic enhancement effects in surface-enhanced femtosecond stimulated Raman scattering (SE-FSRS). A key result is that the plasmon enhanced fields which drive the vibrational equation of motion for each normal mode results in dispersive lineshapes in the SE-FSRS spectrum. This result, which reproduces experimental lineshapes, demonstrates that plasmon-enhanced stimulated Raman methods provide unique sensitivity to a plasmonic response. Our derived SE-FSRS theory shows a plasmonic enhancement of $|gpu|2Im\chi R(\omega )gst2/Im\chi R(\omega )$, where |*g _{pu}*|

^{2}is the absolute square of the plasmonic enhancement from the Raman pump,

*χ*(

_{R}*ω*) is the Raman susceptibility, and

*g*is the plasmonic enhancement of the Stokes field in SE-FSRS. We conclude with a discussion on potential future experimental and theoretical directions for the field of plasmonically enhanced coherent Raman scattering.

_{st}## I. INTRODUCTION

### A. Stimulated Raman scattering

Vibrational spectroscopy is a versatile technique for understanding chemistry at the physical limits of space, time, and molecular concentration. By understanding the vibrational spectrum as a function of space and time, the evolution of a single chemical species can be elucidated. Raman scattering is a common vibrational spectroscopy technique used to discern molecular vibrational information. A powerful form of Raman spectroscopy is stimulated Raman scattering (SRS). In contrast to normal Raman scattering (NRS) where the optical resonance is a spontaneous measurement of the system polarization, SRS coherently drives an optical resonance resulting in a stimulated measurement of the system polarization.

SRS was initially investigated by Eckhardt, Woodbury, and Ng at Hughes Research in the early 1960s. Early work on the theory of SRS was performed by Shen and Bloembergen.^{1,2} SRS has matured as a field and has been of significant recent interest as a vibrational imaging technique.^{3,4} The original theory of SRS by Shen and Bloembergen gave a great physical insight into the SRS process by using semiclassical coupled wave equations to describe the system polarization. Coupled wave theory of SRS, while non-quantum mechanical, reproduces many experimental observables.^{2} Due to the success of the coupled wave theory of SRS, it was used as the first approximation in the next advancement in stimulated Raman scattering: femtosecond stimulated Raman scattering (FSRS).

### B. Femtosecond stimulated Raman scattering

FSRS was developed by Mathies *et al.* as a method to use the advantages of stimulated Raman (i.e., heterodyne detection, greater Stokes, and/or anti-Stokes scattering) with the spectral and temporal advantages of femtosecond spectroscopy (i.e., multiplex vibrational spectroscopy and high temporal resolution).^{5–7} FSRS is a vibrationally resonant four-wave mixing spectroscopy that is characterized by the third order nonlinear optical susceptibility, *χ*^{(3)}. Fig. 1 shows the four-wave mixing energy level diagram for FSRS.

Briefly, a ground state FSRS experiment can be described by four field-matter couplings: the first three induce the third order nonlinear polarization, *ω _{pu}*,

*ω*, and

_{st}*ω*, which leads to the heterodyned FSRS signal. First, a picosecond bandwidth pulse, the Raman pump, induces a polarization of the media. Coincident in time with the Raman pump is a femtosecond bandwidth, spectrally broad pulse—the Stokes field. The Stokes field creates a coherence between vibrational states of the ground and virtual electronic states in the polarization of the media by the Raman pump. As the coherence evolves in time, a second Raman pump field interaction initiates decoherence of vibrational states between the ground and virtual electronic states. The decoherence then leads to stimulated emission of photons in the Stokes field. This process results in a heterodyned signal where the Stokes field serves as a local oscillator.

_{pu}As the experiment is an intensity modulation in the Stokes field, FSRS spectra are obtained by performing a division of Stokes field intensity while the Raman pump is coincident with the Stokes field intensity without the Raman pump,

This results in a Raman spectrum that is analogous to a spontaneous Raman scattering spectrum but is measured in terms of amplitude gain in the Stokes field instead of the traditional counts × mW^{−1} × s^{−1} of a spontaneous Raman scattering experiment.

### C. Plasmonically enhanced coherent Raman scattering

To improve the sensitivity of vibrational Raman spectroscopy, only one linear optical method has shown enhancements of up to 10^{12}: surface-enhanced Raman spectroscopy (SERS).^{8–11} The concept of electromagnetic field enhancement (EM) that lies at the heart of SERS has grown in importance since the initial observation^{8} of what was later recognized as the new physical phenomenon of SERS.^{9,10} It has been applied to the coherent Raman scattering techniques of CARS and FSRS, resulting in surface-enhanced coherent anti-Stokes Raman scattering (SECARS)^{12,13} and surface-enhanced femtosecond stimulated Raman scattering (SE-FSRS).^{14–16} Surface-enhancement in Raman scattering techniques primarily comes from the plasmonically enhanced optical fields used in the vibrational spectroscopy.^{17}

Previous SE-FSRS experiments displayed dispersive lineshapes in the Raman gain features of coupled molecule-plasmonic nanoparticle systems.^{14,15} The dispersive lineshapes were previously attributed to Fano-like interferences of the discrete molecular vibrational states on the continuum-like plasmon resonance.^{18,19} When the energy of the aggregate plasmon oscillation was varied with respect to the Raman pump and Stokes fields, the dispersive lineshapes changed phase.^{15} The behavior of resonance dependent dispersive lineshapes in SE-FSRS is qualitatively similar to lineshapes encountered in surface-enhanced infrared absorption (SEIRA) spectroscopy. A key difference however is that in SEIRA the sharp absorption profile of the molecular oscillator is perturbed by coupling to the infrared resonant broad absorption of the plasmon, while in SE-FSRS the molecule-plasmon coupling is observed by a stimulated scattering event.^{20}

Research into the lineshapes observed in surface-enhanced coherent Raman scattering has focused primarily on SECARS.^{21,22} In SECARS, dispersive lineshapes in the vibrational spectrum are observed due to interference of the vibrationally resonant signal with electronically resonant four-wave mixing signals, similar to the lineshapes seen in CARS. One paper has investigated the plasmonic-enhancement in SE-SRS; however, the theory only considered single vibrational modes and resulted in Lorentzian lineshapes.^{23}

More recently, there has been a paper describing SE-FSRS that used a perturbative polarization expansion.^{24} This paper modeled a stimulated Raman response that is convoluted with a stimulated resonant surface plasmon light emission (PLE). When the Raman gain signal is observed in the far-field, the cross terms between the heterodyned SE-FSRS signal and the PLE signal lead to dispersive lineshapes with phase-dependence depending on the plasmon resonance energy. Using this time-dependent picture of the SE-FSRS process, and a two state model to evaluate the PLE contribution, numerical lineshapes derived by Fourier transformation of the time-dependent field showed general agreement with previously published experiments for a single vibrational mode.^{15} While the previous study used a time-domain picture for SE-FSRS theory and resulted in plasmon energy resonance dependent dispersive lineshapes, they did not retrieve an analytic lineshape equation in the frequency domain which is an important goal for comparing to previous frequency domain experiments.^{14,15}

While more exotic theories of plasmonic enhancement in SERS by optomechanical cavity coupling have been explored recently,^{25,26} the electromagnetic mechanism has consistently reproduced and predicted experimental observables in SERS.^{27,28} Thus, the focus of this paper is to develop a theory of SE-FSRS that uses the coupled wave theory treatments of Lee,^{29} Shen,^{1} and Bloembergen^{2} with the electromagnetic mechanism of SERS to reproduce experimental observables in SE-FSRS. The SE-FSRS theory is presented in Part II of this paper, then in Part III we present an application of this theory to recent SE-FSRS measurements, showing the origin of the asymmetric lineshapes and how they are changed by modifying the plasmon wavelength. We note that our theory provides an explicit formula for analytical lineshape which involves a different dependence on the electromagnetic enhancement factors than was found in Ref. 24.

## II. THEORY

In a semiclassical picture, FSRS can be described similarly to the previous, non-ultrafast versions of stimulated Raman-like that derived by Bloembergen and Shen.^{1,2} An important limitation to the coupled-wave theory of stimulated Raman scattering includes the lack of proper treatment for the polarization when excited on molecular resonance. To treat the case of electronically resonant FSRS (FSRRS), quantum mechanical derivations including density matrix and quantum wavepacket simulations are necessary.^{29–32} While a quantum mechanical formulation is currently being pursued, all SE-FSRS experiments performed thus far have been far from molecular resonance.^{14,15}

Physically, the SE-FSRS process involves a sum of Raman pump and Stokes probe fields inducing a nonlinear response in the molecule-nanoparticle ensemble with total polarization given by $P\u2192(r\u2192,t)$. The ensemble with an induced polarization from the optical fields then modifies the incident optical fields as they propagate through the ensemble. The back-and-forth nature of the incident field effecting the polarization and the induced polarization effecting the propagating field is a classical, physical picture of the coupled wave description of SE-FSRS.

The derivation of SE-FSRS theory will proceed in two steps according to the coupled wave equations approach used in previous FSRS theory by Lee *et al.*,^{29,33} with a special note to issues that are unique to plasmonic enhancement of the optical fields. First, a derivation of the equation of motion for the vibrational coordinate *Q* by plasmonically enhanced optical fields will be presented. The resulting expression for *Q* describes the driven vibration in the infrared region of the electromagnetic spectrum. Once *Q* is derived the second portion of the SE-FSRS derivation analyzes how the driven oscillator described by *Q* perturbs the Stokes field. We analyze this perturbation of the Stokes field by *Q* by solving Maxwell’s equation. Once Maxwell’s equation is solved, a final expression for the SE-FSRS Raman gain is given and compared to experimental data.

### A. Solving the driven oscillator equations for SE-FSRS

For details on the derivation of the driven wave equation for FSRS, readers are directed to Refs. 29 and 33. Starting with the driven wave equation (Eq. (7) of Ref. 29), we investigate the correct driving fields,

In $|E\u2192(r\u2192,t)|2$ there are two different field products that could drive a vibration *ω*_{0}: $EpuEst\u2217$ and $Epu\u2217Est$. If we assume that the fields have the time dependence *E _{pu}* ∼

*e*

^{−iωput}and

*E*∼

_{st}*e*

^{−iωstt}, then if we are interested in Stokes Raman scattering, the product of $EpuEst\u2217$ leads to

*Q*∼

*e*

^{−iω0t}with the damping term in the homogeneous solution to the driven oscillator being divergent. Thus we will consider the driving term $Epu\u2217Est$, such that

*Q*∼

*e*

^{iω0t}is coherently driven, but damped,

To describe how the electric field is modified by the collection of oscillators, we use Maxwell’s equation,

To reduce the number of spatial variables present in $r\u2192$, we assume a linearly polarized field propagating along the z-axis. In addition, we assume that the driven dipoles are orientated along the polarization direction of the propagating fields. This assumption is identical to that made in the previous SRS and FSRS theory papers and can be modified by introducing orientational matrices.^{17}

Using the Placzek polarization the equation describing the change of the electric field by the collection of oscillators becomes

As 4*πNα*_{m,0} ≪ 1, we neglect the first term on the right of the equality in Eq. (5b) and keep the second term that drives the oscillating field (i.e., produces stimulated Raman gain). This leads to the following expression for Maxwell’s equation for the field perturbed by the medium:

Eq. (6) is the general form for the Maxwell’s equation that will be solved in part B of this section. At this point we turn our focus to the fields that drive the coupled wave equations.

We now add in the contribution of the metal nanoparticle to the field, here assuming that the external field induces a dipole in the particle, and then it is the field of this dipole that needs to be added. Here the particle is assumed to be located along the polarization direction.

Using notation from Mullin *et al.*^{34} and realizing the total field (either pump or Stokes) is the sum of the incident ($Est/pu0$) and scattered fields induced in the particle, we first derive the Stokes field,

where Λ is a parameter determined by the field of the induced dipole that has an inverse cube dependence on distance between dipoles, *μ _{P}* is the particle dipole, $\alpha \u0303P,st$ is the particle polarizability evaluated at the Stokes frequency, and

*μ*is the molecular dipole.

_{m}Now dropping the *μ _{m}* term as it leads to the image field effect

^{17}which has been shown to be important only in regions where quantum plasmonic effects dominate,

^{35}we let $gst=1+\Lambda \alpha \u0303P,st$ to obtain

Analogously for *E _{pu}*(

*z*,

*t*),

Note that *g _{pu}* and

*g*are the complex valued electromagnetic enhancement factors associated with the Raman pump and Stokes fields in SERS, with the overall SERS enhancement factor being |

_{st}*g*|

_{pu}^{2}|

*g*|

_{st}^{2}.

^{17,34,36}In the present derivation, we assume that these parameters are time-independent, and thus do not change the time profile of these pulses. We also assume that the frequency dependence is weak compared to the linewidth of the pump pulse, and the vibrational line associated with the Stokes probe. While this assumption is approximate for the Stokes pulse, it should be reasonable for order of magnitude estimates.

Finally to approximate experimental conditions we give a Gaussian envelope to the pulses by defining $Epu0(z,t)$ and $Est0(z,t)$ (as well as their Fourier transformed frequency domain definitions) as

Letting *t*′ = *t* − *z*/*c* and $(1/\tau pu2+1/\tau st2)\u22481/\tau st2$,

To obtain *Q*(*z*, *ω*) from *Q*(*z*, *t*) in Eqs. (9a) and (9b) we solve using Fourier transforms^{29,33} to get the following result:

Thus, *Q*(*z*, *ω*) is an equation of the coherently driven vibration that occurs in a Raman shifted frequency domain from the pump pulse. Here we note that in the limit of no plasmonic enhancement: $\alpha \u0303P=0$, Re{*g*} → 1, and Im{*g*} → 0; an identical equation of motion to that derived by Lee *et al.* is obtained.^{29} This driven vibrational coordinate, *Q*(*z*, *ω*), will be used to solve Maxwell’s equation in Sec. II B.

### B. Solving Maxwell’s equation to derive experimental observables in SE-FSRS

To solve for the effect of the driven vibration *Q*(*z*, *ω*) on the Stokes field, we use the result in Eq. (10) to solve Maxwell’s equation from Eq. (6),

In going from Eq. (11a) to Eq. (11b), we have added in a factor of *g _{st}* to include for the effect of the plasmonic particle on the emitted field that drives the Stokes probe. This is analogous to including for dipole re-radiation effects in the SERS enhancement factor calculation,

^{37}but here it is the driven oscillator

*Q*that generates the dipole field at the Stokes frequency (through the

*QE*term). Another issue in Eq. (11b) is that we have assumed that the dipole fields that are coherently emitted by all the molecules in the sample have been coherently summed to generate a polarization that drives a plane wave solution. This replacement (also used by Lee

_{pu}*et al.*

^{29}) is essential, as the driven vibration needs to generate a field that can interfere constructively with the incident Stokes plane wave field, and it arises from the fact that coherent sum of dipole fields generates a plane wave in the far field.

A key issue about Eq. (11b) is that the time dependence of *Q* (*e*^{iω0t}) combines with *E _{pu}* (

*e*

^{−iωput}) to drive a field with the Stokes shifted time dependence, as required with the rotating wave approximation (RWA). While this may seem like an obvious point, it means that the driving term in Eq. (11b) contains the product of

*g*from Eq. (11b) and

_{st}*g*from Eq. (10), or $gst2$ rather than an absolute square. This will play a crucial role in generating dispersive lineshapes later on, and here we see that it is a natural result of the RWA in which the Stokes field acts twice with the same phase. One physical interpretation of this approximation is that we assume the driving fields on the right hand side of Eq. (11a) are creating a radiating dipole. This dipole re-interacts with the plasmon gaining an additional

_{st}*local field effect*(LFE) enhancement as

*g*, before the collection of dipoles are treated as a plane wave as described by Eqs. (8b) and (8d). In contrast to this, the pump field enhancement appears as an absolute square as it applies first as a complex conjugate $Epu\u2217$ in Eq. (3) and then not as a complex conjugate within

_{st}*E*in Eq. (11b).

_{pu}Ignoring the homogenous solution to Maxwell’s equation due to averaging over all possible initial phases of the oscillators,^{29} we solve using Fourier transforms,

and focusing on the convolution integral,

To solve the convolution integral we make identical approximations as used by Lee *et al.*:^{29} substitution of $\omega \u0303=\u2212\omega 0$ in the $Q(z,\omega \u0303)$ terms, evaluation of the integrand denominator at $\omega \u0303=\omega \u2212\omega pu$, evaluation of the resulting Gaussian integral, and finally substitution of *ω* = *ω _{pu}* −

*ω*

_{0}to regain the original unperturbed Stokes field from Eq. (8d). These approximations and evaluations lead to the following solution for the convolution integral in Eq. (13b):

Returning to Eq. (12) and evaluating Maxwell’s equation more fully,

where the Raman susceptibility, *χ _{R}*(

*ω*), is given by

To solve Eq. (15), we make the ansatz that can be verified by substitution,

From this, the Raman gain in SE-FSRS can be expressed as:

For small gain this can be approximated as

Finally, the experimental observable for stimulated Raman gain (SRG) SE-FSRS is an optical density,

or, analogously,

From this expression we can see that if only the pump field were enhanced, it would contribute a factor |*g _{pu}*|

^{2}to the enhancement of SE-FSRS over FSRS. Analyzing Eqs. (20a) and (20b) further we can observe a few more points:

An additional complex plasmonic enhancement comes from the competition of the real and imaginary portions of the LFE from the Stokes field interacting with the plasmon. Rigorously, the total plasmonic enhancement is given as $|gpu|2Im\chi R(\omega )gst2/Im\chi R(\omega )$.

In the case of no plasmonic enhancement, the real and imaginary portions of

*g*go to 1 and zero, respectively. This limiting case of SE-FSRS matches identically with the results of Lee*et al.*^{29,33}SE-FSRS examines an optical process with competing real and imaginary components of the Raman susceptibility that FSRS does not.

Probing both the real and imaginary portions of

*χ*(_{R}*ω*) will lead to interference and dispersive lineshapes as seen in SECARS (but with a different mechanism)^{21,22}and previous SE-FSRS experiments.^{14,15}

To fully understand the Raman gain optical density, we re-evaluate Eqs. (20) to separate the full Raman gain optical density expression into Fano lineshapes.

From Eq. (20b),

and from Eq. (16),

Recognizing *ω _{pu}* −

*ω*

_{0}=

*ω*,

_{st} and letting $A=\u22124\pi |gpu|2|Epu0|2N\alpha m\u20322(\omega 0)\u22121$, *D _{R}*(

*ω*) becomes

Separating $gst2$ into real and imaginary parts leads to

To look for a form of this equation that can be compared to a Fano profile, we complete the square in terms of an expansion of the term in square brackets in powers of (*ω* − *ω _{st}*),

Then we can rewrite *D _{R}*(

*ω*) as

Letting $q=\u22122Re{gst2}Im{gst2}\u22121$ as the Fano profile parameter similar to Frontiera *et al.*,^{15}

Thus we get that the lineshapes in SE-FSRS are the sum of two Fano profiles where the Fano asymmetry parameter of one of the lineshapes is exactly zero while the other has a Fano parameter that depends on the ratio of $Re{gst2}$ to $Im{gst2}$.

## III. DISCUSSION

We now investigate the lineshapes of the optical density in SE-FSRS with our theoretical model developed herein by simulating a model system corresponding to published experiments.

Previous SE-FSRS experiments analyzed systems of 1,2-trans(bis-4-pyridyl)ethylene (BPE) adsorbed on gold nanoparticles. The gold nanoparticles were either 60 or 90 nm diameter spherical particles where the 60 nm diameter cores had broad aggregate localized surface plasmon resonances (LSPR) from 750 to 1000 nm. The 90 nm diameter core particles had broad aggregate LSPRs from 825 to 1200 nm. Using a picosecond Raman pump centered at 795 nm and an ∼100 fs spectrally broad (820-930 nm) Stokes field, the far-field stimulated Raman spectrum of BPE in the near-field of two different plasmon resonances was observed in separate experiments. Further experimental details are listed in Appendix C and in the work of Frontiera *et al.*^{15}

Using a free-electron Drude model for the particle polarizability (see Appendix A), in addition to static molecular polarizabilities from density functional theory (see Appendix B), we can simulate the optical response of the molecule-nanoparticle system to different field interactions.

Shown in Fig. 2 is the structure of the nanoparticle-molecule system used for molecular polarizabilities in the simulations to be discussed: a single BPE molecule symmetrically bridging two Au_{8} structures to mimic a real molecule-nanoparticle system.

Using the Au_{8}-BPE-Au_{8} structure for the molecular polarizability parameters, we simulate the full optical response observed in SE-FSRS, the optical density (*D _{R}*(

*ω*)) in Eq. (20).

First we simulate a single BPE molecule near an 825 nm LSPR, shown in Fig. 3.

The simulated signal in Fig. 3 shows frequency dependent asymmetry that qualitatively matches the SE-FSRS gain observed in the experiment of BPE on 60 nm Au nanoparticles aggregates. The SE-FSRS gain observed in BPE on 90 nm Au nanoparticle aggregates is given in Fig. 4.

Here we again see a qualitative agreement between experiment and theory, showing a reversal in the sign of the dispersive SE-FSRS gain lineshape.

In previous SE-FSRS experiments, the dispersive lineshapes were attributed to Fano-like resonances resulting from interference of discrete molecular vibrational states on the broad continuum of the plasmon resonance.^{15} The lineshapes observed in simulating the SE-FSRS Raman gain from Eq. (20) (Figs. 3 and 4) are related to this; however, here we see that it is the plasmon enhancement associated with the Stokes probe coherently acting twice that dominates that lineshape asymmetry.

Generally, Fano resonances occur when there are two weakly coupled oscillators where one oscillator is resonantly driven. The resonantly driven oscillator then exhibits a force on the coupled oscillator which results in either constructive or destructive interference effects in the driven oscillator resonance. This argument is also applicable in the present case, where the driven vibrational oscillator subsequently drives the Stoke’s field, resulting in a gain that includes interference of the real and imaginary components of the Raman susceptibility with the square of the Stokes field LFE (Eq. (20)).

To further study the effects of plasmon resonance frequency on SE-FSRS lineshapes, Appendix D shows the lineshapes associated with Eq. (20) in which the LSPR frequencies are allowed to vary. In Fig. 9, we observe an abrupt transition from positive to negative SE-FSRS gain signals near 875 nm. At even longer LSPR wavelengths (around 950 nm), another transition occurs where the lineshapes have a mixture of positive and negative characters. Also noteworthy is that the intensities of the spectra can vary by orders of magnitude depending on choice of LSPR. These effects are probably exaggerated by the partially unphysical nature of the analytic particle polarizability as described in Appendix A, so it will be important to relax these approximations in further work.

## IV. CONCLUSIONS

Using a semiclassical coupled wave theory for SE-FSRS, we have developed an expression for the lineshape which qualitatively reproduces experimental data, including the dispersive lineshapes. In addition, the theory makes important predictions concerning the plasmonic enhancement factor, which is given by $|gpu|2Im\chi R(\omega )gst2/Im\chi R(\omega )$, where |*g _{pu}*|

^{2}is the absolute square of the plasmonic enhancement from the Raman pump,

*χ*(

_{R}*ω*) is the Raman susceptibility, and

*g*is the plasmonic enhancement of the Stokes field in SE-FSRS. While the dispersive nature of the lineshapes was originally hypothesized to be purely Fano-like interferences,

_{st}^{15}our analysis suggests that they are a combination of two Fano-like contributions. Similar to previously observed dispersive lineshapes that originate from multiple physical interference processes,

^{21,22}the lineshapes in this work are attributed to interference of the real and imaginary components of the Raman susceptibility with the LFE of the Stokes field interacting with a plasmonic particle.

The expression obtained from our evaluation is significantly different from that recently published by Ziegler and co-workers.^{24} Their expression involves the interference between the Raman susceptibility and a stimulated resonant surface plasmon optical field, for which the enhancement factor is given as being proportional to |*g*|^{4}, or in our notation is |*g _{pu}*|

^{2}|

*g*|

_{st}^{2}. Thus unlike our expression, where it is the phase of the square of the Stokes field enhancement factor $gst2$ that is essential in determining the lineshape asymmetry, in the Ziegler derivation it is the interference of two independent scattering mechanisms that results in asymmetry. Ziegler presents a detailed analysis of the relative sizes of these scattering mechanisms and argues that they might be close enough to provide meaningful interference. In our derivation the asymmetry is built-in to a single mechanism, and the size of the contributing terms in the asymmetry is easily estimated. While it is possible that the interference involved in the Ziegler derivation could also arise in our work by adding in the PLE contribution, this would be subject to the same ambiguous evaluation as in the Ziegler analysis.

Future directions for SE-FSRS theory will involve treating the case of electronically resonant pulses in the SE-FSRS experiment by using a density matrix formulation. By pursuing a density matrix formulation for SE-FSRS, the resulting theory will include the effects of hot luminescence and other Feynman diagrams to the third order optical response that has been observed in non-plasmonically enhanced resonant FSRS (FSRRS). Deriving a density matrix theory for SE-FSRS, the theory will predict the optical response of as yet unrealized experiments.

## Acknowledgments

The authors acknowledge helpful discussions with Dr. Nicolas Large and Dr. Lindsey Madison as well as experimental help on SE-FSRS timing and acquisition parameters from Dr. Bogdan Negru and Dr. Jon Dieringer. We also thank Eric Smoll for his contributions to an early version of this theory. All the authors gratefully acknowledge the financial support from the CaSTL Center of the National Science Foundation (Grant No. CHE-141466). J.M.M. acknowledges startup support from Washington State University and the Department of Physics and Astronomy thereat. M.O.M. also acknowledges support from the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE-0824162.

### APPENDIX A: VARYING *χ* TO DETERMINE $\alpha \u0303P(\omega )$ IN THE FREE-ELECTRON DRUDE MODEL

We have two cases of plasmon resonances that we wish to compare the SE-FSRS response in Eq. (20). The first case is with a plasmon resonance at 825 nm and the second case is at 900 nm. The general method we will use is to solve the shape-dependent spheroidal extinction equation from the work of Schatz *et al.*^{17} by varying a parameter *χ* that corresponds to the aspect ratio of a spheroidal particle, then substituting in values for the desired plasmon resonance and the Drude model parameters for the plasma frequency (*ω _{p}*) and plasma decay (

*γ*). In effect this will approximate the dimer type structures in the experiment as a spheroidal particle with an effective plasmon resonance determined by tuning a shape parameter (

*χ*) as shown in Fig. 5.

Starting with the extinction spectrum in frequency,

We observe that tuning the *χ* parameter allows for changing the aspect ratio of a plasmonic spheroid and thus the resulting plasmon resonance energy. Substituting in values for *χ* of the desired plasmon resonances and the Drude model parameters for the plasma frequency (*ω _{p}* = 8.55 eV) and plasma decay (

*γ*= 0.0184 eV), we get frequency dependent polarizabilities for the two choices of particle dimers.

For a plasmon resonance analogous to the experimental 60 nm dimer aggregate plasmon at *ω* = 825 nm (Fig. 6),

For a plasmon resonance analogous to the experimental 90 nm dimer aggregate plasmon at *ω* = 900 nm (Fig. 7),

where *ϵ _{i}* is given by the Drude model,

Note that this results in a complex, frequency dependent polarizability.

### APPENDIX B: DENSITY FUNCTIONAL THEORY CALCULATIONS OF NANOPARTICLE-MOLECULE-NANOPARTICLE SYSTEM POLARIZABILITIES

Static polarizabilities of the nanoparticle-molecule-nanoparticle system shown in Fig. 2 were calculated using the Amsterdam Density Functional (ADF) computational chemistry package.^{38} Frequency and polarizability calculations for the symmetric Au_{8}-BPE-Au_{8} complex were completed using the Becke-Perdew (BP86) generalized gradient approximation (GGA) exchange correlation functional and a triple-*ζ* polarized (TZP) Slater orbital basis set with zeroth order regular approximation for relativistic effects.

Static Raman polarizabilities (*ω* = 0) were calculated in the RESPONSE package by two-point numerical differentiation using the RAMANRANGE keyword. SERS scattering intensities were determined by the following equation:^{39}

where the scattering factor (*S*) is defined as $45\alpha \u0304j\u20322+7\gamma j\u20322$ and $\alpha \u0304j\u2032$, $\gamma j\u2032$ are the isotropic and anisotropic polarizability tensors with respect to the *j*th vibrational mode.

For SE-FSRS simulations the $\alpha m\u2032$ used was defined as $\alpha m\u2032\u2261S$, where *S* is the scattering factor defined previously. The SERS line for each mode was broadened to a Lorentzian lineshape with full-width at half-maximum (FWHM) of 20 cm^{−1} for comparison to experimental data.

### APPENDIX C: EXPERIMENTAL METHODS OF SE-FSRS

The experimental methods for collecting data in SE-FSRS have been described previously.^{14,15} Briefly, the 1 W, 800 nm output of a 100 kHz regenerative amplifier (Coherent RegA) is split into two paths for the FSRS experiment. Approximately 500 mW of the amplified output is used to generate a picosecond bandwidth Raman pump field by passing through two identical angle tuned bandpass filters (CVI optics) at 795 nm. A portion of the remainder of the amplified output is used for white light continuum generation in a sapphire plate which is then temporally compressed by a prism pair. The continuum is then filtered using a set of short- and longpass filters to create a Stokes field in the spectral region of interest. The pulses are overlapped spatiotemporally in a collinear geometry in a 2 mm cell equipped with a micro-stir bar to keep the sample viable in the focal overlap volume. The pump pulse is mechanically chopped so that sequential pump-on and pump-off spectra are obtained in a home-built LabView program. The heterodyned SE-FSRS signal is detected with the Stokes field using a spectrograph (Princeton Instruments SP2538) and CCD (PIXIS:100F). Average powers were 25 *μ*W and 10 *μ*W for the Raman pump and Stokes fields respectively. Each experimental SE-FSRS spectrum shown has a total acquisition time of 6 min.

The key difference between the experimental spectra shown in this paper compared to previous SE-FSRS papers^{14,15} of identical substrates is the correction of a chopper phase issue. Previous papers published SE-FSRS spectra as pump-off divided by pump-on while this paper discusses spectra with the issue corrected, showing proper pump-on divided by pump-off spectra.

### APPENDIX D: VARIABLE PLASMON RESONANCE EFFECTS ON SE-FSRS RAMAN GAIN

To examine the effects of plasmon resonance wavelength on SE-FSRS Raman gain of a BPE molecule, we simulated Eq. (20) with plasmon resonance wavelengths varying between 800 nm and 1125 nm. Varying *χ* in Eq. (A1), the plasmon resonance is tuned, and different particle polarizability functions are used in Eq. (20).

By tuning the plasmon resonance with respect to the Raman pump and Stokes fields, the SE-FSRS lineshapes show high sensitivity to the plasmon resonance. In Figs. 8–10 we see an abrupt transition from positive to negative gain signals with different mixtures of positive and negative characters to the dispersive lineshapes. In addition, the SE-FSRS Raman gain can vary by up to orders of magnitude depending on the plasmon resonance wavelength.