## Abstract

Studying the interaction between molecules and surface plasmon polaritons (SPPs) is of great important in understanding surface-enhanced Raman scattering (SERS). While it is known that SERS consists of excitation and emission enhancements, each of them is manifested by several sub-steps which individually also deserve attention. For example, for emission enhancement, the energy from the excited molecules is first coupled to SPPs, which then radiatively scatter to far-field. To understand these two sequential processes completely, differentiating them one by one is necessary. Here, we decouple them and determine the coupling efficiency of molecules to SPPs by using a phenomenological rate equation model. We find the coupling efficiency, defined as the ratio of the coupling rate from molecules to SPPs to the direct Raman decay rate, can be expressed as the SERS intensity ratio and the SPP absorption and radiative decay rates, which all can be determined by polarization- and angle-dependent Raman and reflectivity spectroscopy. As a demonstration, the coupling efficiencies of 6-mercaptopurine to SPPs propagating in Γ-X direction on Ag nanohole array are measured for several Raman emission wavelengths.

© 2013 OSA

## 1. Introduction

Raman scattering provides fingerprint-like signature of molecules and thus is a promising technique for high specific chemical identification [1]. Unlike fluorescence, Raman scattering is a label-free technique in which tagging to targets is not necessary [2]. Unfortunately, the Raman cross-section is usually very small, around 12-14 orders of magnitude lower than that of fluorescence. In other words, for every 10^{10-12} photons incident on a molecule, only one or less undergoes Raman scattering [1,3,4]. Therefore, Raman scattering does not generate strong signal and is not very applicable when the concentration of target analyte is very low. However, when the molecules are placed in close proximity to a metal surface, much stronger Raman signal can be produced due to the excitation of surface plasmon polaritons (SPPs) [3,4]. Molecules can be excited and dissipated via SPPs, which extraordinarily enhance both Raman excitation and emission [3,4]. This electromagnetic mediated Raman process leads to the well-known surface-enhanced Raman scattering (SERS). Since the sensitivity is greatly improved, SERS can now be used as practical molecular probe with detection limit possibly down to one single molecule [5].

As a result, the ability to generate desired enhancement is of crucial in SERS and this requires in-depth understanding of the entire process. Although the first order E^{4} approximation provides good estimation of the total Raman enhancement for practical use, it reveals no details of SERS other than the contributions from excitation and emission enhancements [3,4]. In particular, it is known that several sequential steps are involved in determining the overall SERS [6,7]. As shown in Fig. 1, the excitation photon at ηω_{exc} is first resonantly coupled to the metal surface to produce ingoing SPPs for excitation. Then, theses SPPs, which possess much stronger field strength than that of the incident field, induce large dipole moments of the molecules, resulting in the excitation enhancement [8]. When the molecules depolarized, they radiate at several frequencies including the non-resonant Rayleigh and the Stokes and anti-Stokes Raman scattering. For the Stokes emission, for example, the energy is coupled into outgoing SPPs at ηω_{scatt} before radiatively scattering into far-field, giving rise to emission enhancement. Therefore, the whole Raman enhancement depends on several coupling and decay processes and each contributes distinctively to SERS. To study them in greater detail, differentiating them one by one is essential. However, to date, effort on this issue is rarely seen primarily because of the difficulty in decoupling the process.

In this work, we propose a simple method to determine the coupling efficiency of energy from excited molecules to outgoing SPPs in periodic array by using a phenomenological rate equation model. For SPPs with a given propagating wavevector, we find the coupling rate ratio can be determined by measuring the absorption and radiative decay rates of the outgoing SPPs and the Raman intensity ratio. As a proof of concept, we determine the coupling efficiency of 6-mercaptopurine to SPPs propagating in Γ-X direction supported on Ag nanohole array.

## 2. Theory

First, we describe how one can determine the coupling efficiency in periodic array based on a rate equation model. From Fig. 1, for emission enhancement, we assume, other than the Rayleigh and anti-Stokes emission decays, the excited molecules with energy [M] can dissipate via several radiative decay channels at the Stokes frequency ω_{scatt} to produce the overall SERS response. For example, they can scatter at rate Γ_{r} without involving SPPs, resulting in non-SPP related or direct emission. On the other hand, the rest of the energy can be coupled into outgoing SPPs at coupling rate ${\Gamma}_{c}^{{\stackrel{\rightharpoonup}{k}}_{SPP}}$. These SPPs then propagate with vectorial wavevectors ${\stackrel{\rightharpoonup}{k}}_{SPP}$ that have the same magnitude $\left|{\stackrel{\rightharpoonup}{k}}_{SPP}\right|={n}_{eff}\frac{\omega}{c}$, where ${n}_{eff}$is the effective refractive index, but different directions. While propagating, they lose their energy either by intrinsic Ohmic loss or radiation damping [9]. The radiation damping in fact yields the SPP mediated Raman emission. Therefore, under the conservation of energy, the rate equation of the SPPs with energy $[SP{P}^{{\stackrel{\rightharpoonup}{k}}_{SPP}}]$ and ${\stackrel{\rightharpoonup}{k}}_{SPP}$ can be written as $\frac{d[SP{P}^{{\stackrel{\rightharpoonup}{k}}_{SPP}}]}{dt}={\Gamma}_{c}^{{\stackrel{\rightharpoonup}{k}}_{SPP}}[M]-{\Gamma}_{tot}^{{\stackrel{\rightharpoonup}{k}}_{SPP}}[SP{P}^{{\stackrel{\rightharpoonup}{k}}_{SPP}}]$, where the first term denotes the power coupled to the SPPs from the molecules whereas the second one indicates the total SPP power losses. The total decay rate is the addition of absorption and radiative decay rates ${\Gamma}_{tot}^{{\stackrel{\rightharpoonup}{k}}_{SPP}}={\Gamma}_{abs}^{{\stackrel{\rightharpoonup}{k}}_{SPP}}+{\Gamma}_{rad}^{{\stackrel{\rightharpoonup}{k}}_{SPP}}$ and the radiative decay rate sums up the individual rates from all emission ports ${\Gamma}_{rad}^{{\stackrel{\rightharpoonup}{k}}_{SPP}}={\displaystyle \sum _{m}{\Gamma}_{rad}^{{\stackrel{\rightharpoonup}{k}}_{SPP}\text{,}m}}$. The number of m emission ports is governed by the phase-matching equation given as ${\stackrel{\rightharpoonup}{k}}_{SPP}=\left(\frac{\omega}{c}\mathrm{sin}\theta \mathrm{cos}\phi +\frac{2{n}_{x}\pi}{a}\right)\widehat{x}+\left(\frac{\omega}{c}\mathrm{sin}\theta \mathrm{sin}\phi +\frac{2{n}_{y}\pi}{a}\right)\widehat{y}$, where a is the period and n_{x} and n_{y} are the integers defining the order of Bragg scattering. From the equation, we see the SPP mediated Raman emissions emerge at well-defined polar θ and azimuthal φ angles depending on the combination of n_{x} and n_{y} [6,10–13]. As a result, the total Raman response is the summation of direct ${P}_{d}={\Gamma}_{r}[M]$ and total SPP mediated $\sum _{m}{P}_{SPP}^{{\stackrel{\rightharpoonup}{k}}_{SPP},m}\left(\theta \text{,}\phi \right)}={\displaystyle \sum _{m}{\Gamma}_{rad}^{{\stackrel{\rightharpoonup}{k}}_{SPP},m}}[SP{P}^{{\stackrel{\rightharpoonup}{k}}_{SPP}}]$. To determine ${\Gamma}_{c}^{{\stackrel{\rightharpoonup}{k}}_{SPP}}$ for each ${\stackrel{\rightharpoonup}{k}}_{SPP}$, we consider the steady state where $\frac{d[SP{P}^{{\stackrel{\rightharpoonup}{k}}_{SPP}}]}{dt}=0$ and the Raman power ratio between one of the SPP emission ports, e.g. the n^{th} port, and the direct channel is

^{th}ports, the ratio between coupling and direct rates ${\Gamma}_{c}^{{\stackrel{\rightharpoonup}{k}}_{SPP}}/{\Gamma}_{r}$, defined as the coupling efficiency here, for ${\stackrel{\rightharpoonup}{k}}_{SPP}$ can be determined. ${\Gamma}_{c}^{{\stackrel{\rightharpoonup}{k}}_{SPP}}$can be estimated once Γ

_{r}is known.

While the power ratio can be obtained from angle-resolved Raman spectroscopy [10], the acquisition of SPP decay rates relies on polarization- and angle-dependent reflectivity spectroscopy and temporal coupled mode theory (CMT) [14–17]. In fact, following [16–18], we consider an optically thick periodic system supporting one resonance is excited at a fixed incident angle, the p-polarized diffraction coefficient of the p^{th} output port can be written as: ${c}_{p}+\frac{\sqrt{{\Gamma}_{rad}^{1}}\sqrt{{\Gamma}_{rad}^{p}}{e}^{i{\varphi}_{p}}}{i(\omega -{\omega}_{res})+{\Gamma}_{tot}/2}$, where ω_{res} is resonant frequency, c_{p} and ϕ_{p} are the nonresonant diffraction efficiency and the phase-shift of SPPs. The numerator also contains the multiplication of the square root of the in- and out-coupling radiative decay rates ${\Gamma}_{rad}^{1}$ and ${\Gamma}_{rad}^{p}$ provided the excitation is input from the 1st port. For the 0th order diffraction, i.e. specular reflection, since the 1st port is also the output port, the equation can be expressed as: ${\left|{c}_{1}+\frac{{\Gamma}_{rad}^{1}{e}^{i{\varphi}_{1}}}{i(\omega -{\omega}_{res})+{\Gamma}_{tot}/2}\right|}^{2}$, yielding Fano-like resonance [15]. In addition, by using Jones calculus, under orthogonal or cross polarization configuration, it is possible to eliminate c_{1} and attain Lorentzian lineshape given as $\frac{1}{4}\frac{{({\Gamma}_{rad}^{1})}^{2}}{{(\omega -{\omega}_{res})}^{2}+{\left({\Gamma}_{tot}/2\right)}^{2}}$ in which both Γ_{tot} and Γ_{rad}^{1} can be determined accordingly [16].

## 3. Experiment

We attempt to experimentally measure the coupling efficiency based on the above formulations. Interference lithography is used to prepare Ag nanohole array with period = 670 nm, hole depth and radius = 100 and 140 nm [10] and its scanning electron microscopy (SEM) image is shown in the inset of Fig. 2(a). The array is then immersed in an ethanolic solution of 6-mercaptopurine at concentration of 5 mM for 18 hours to form a self-assembled monolayer [19]. The sample is mounted on a goniometer for reflectivity and Raman measurements. A pair of incident polarizer and detection analyzer is used for polarization-dependent spectroscopy [15,16]. For reflectivity, a collimated white light is illuminated on the sample at different incident angles and the specular reflections are collected. The polarizer is placed at 45° with respect to the incidence plane and the analyzer is oriented at 45° and −45°, resulting in parallel and orthogonal polarizations. Given the incident and the reflection Jones vectors are $\left[\begin{array}{c}1\\ 1\end{array}\right]$ and $\left[\begin{array}{c}{r}_{p}+\frac{{\Gamma}_{rad}^{1}{e}^{i{\varphi}_{1}}}{i(\omega -{\omega}_{res})+{\Gamma}_{tot}/2}\\ {r}_{s}\end{array}\right]$, where r_{p} and r_{s} are the p- and s-nonresonant reflection coefficients, the two specular reflectivity spectra from CMT thus can be expressed as:

_{p}≈r

_{s}[16]. On the other hand, for Raman spectroscopy, a p-polarized 514 nm Ar laser with power ~20 mW is incident on the sample at 10° and the signals are collected at detection angles scanning from 15°-50° without the analyzer. All spectra are taken in Γ-X direction.

Figures 2(a) and 2(b) show the parallel and orthogonal polarized angle-dependent reflectivity mappings of 6-mercaptopurine coated Ag array. The mappings, also known as dispersion relations, identify the excitation of different SPP modes by using the phase-matching equation. The dash lines are labeled near the reflectivity dips in the parallel mapping indicating (−1,0), (0, ± 1), (1,0), (−1, ± 1), (1, ± 1), and (−2,0) Ag/air SPPs are excited. Likewise, the orthogonal mapping displays almost identical features as the parallel counterpart except the dips are now flipped into peaks. Here, we only focus on the decay rates of (1,0) SPPs since they will be used for determining the coupling efficiency in positive Γ-X direction. We extract the (1,0) reflectivity spectra from λ = 500 – 600 nm and incident angle = 15°-25° and display them in Fig. 2(c) and 2(d). While the parallel spectra exhibit Fano-like profiles, the orthogonal spectra show Lorentzian lineshapes, which both agree with the predictions from CMT. We fit both spectra simultaneously by using Eq. (2) and the best fits are shown as the dash lines for reference. The deduced (1,0) SPP Γ_{rad} and Γ_{tot} are plotted in logarithmic scale against wavelength in Fig. 2(e) showing they decrease with increasing wavelength. In fact, the radiative decay rates display a λ^{-6.95} dependence as indicated by the dash line, revealing a Mie-like scattering behavior [9]. It has been shown that the wavelength dependence is very sensitive to hole geometry such as size and shape [9,20]. In particular, within the range of geometries and wavelengths considered here, the behavior of Γ_{rad} can be understood as the result of the scattering of SPPs by single isolated holes. Our results [9] show Γ_{rad} exhibits close to λ^{−4} dependence when the hole size is small [16] but becomes more *λ*^{-4 to −8} when hole size increases.

After determining the decay rates, we turn our attention to the Raman scattering. Figure 3(a) shows the detection angle-dependent Raman mapping taken at an off-resonant incident angle of 10^{o}, which does not generate any SPPs at 514 nm, thus minimizing excitation enhancement. The mapping features two strong dispersive bands which bare close resemblance with the p-polarized reflectivity mapping as shown in Fig. 3(b) in the same scale. Therefore, one can conclude the strong emissions originate from the (1,0) and (−2,0) SPPs. However, it is reminded that although both (1,0) and (−2,0) are observed in Γ-X direction, they arise from SPPs propagating with different vectorial wavevectors. The SPPs that give rise to (1,0) emissions propagate in positive Γ-X direction whereas the (−2,0) counterpart propagates in negative Γ-X direction. Therefore, their coupling efficiencies could/could not be the same. In addition, the SPPs that generate (1,0) Bragg scattering can at the same time produce other (n_{x},n_{y}) emissions at different angles. Nevertheless, from Eq. (1), we see only the information of one single port is sufficient to calculate the coupling efficiency in positive Γ-X. We plot the Raman spectra taken at several detection angles in Fig. 3(c). We see a number of peaks superimposed on a broad background and their spectral positions are at 537, 542, 550, and 558 nm in consistent with the Stoke Raman signatures of 6-mercaptopurine [10]. No peak is found at 592 nm indicating the absence of S-H bond, confirming a single monolayer is formed on the metal surface [19]. The board emission background is of fluorescence origin and several speculations have been proposed [21,22]. We believe it is not SERS relevant even though its intensity also varies closely with the dispersion relation very well. Therefore, to extract the Raman emissions, we angularly plot the peak intensities in Fig. 3(d)-3(f) for λ = 537, 550, and 558 nm after subtracting them from the fluorescence background. The angular p-polarized reflectivity plots are also shown for reference, indicating that the SERS peaks are closely related to the (1,0) and (−2,0) SPPs. Finally, for comparison with the array, the background subtracted angular intensities obtained from 6-mercaptopurine coated on flat Ag film are also plotted and they simply display weak direct emissions. We assume a uniform coverage of monolayer on the array and the film and a factor of $1-\pi {R}^{2}/{P}^{2}$ is used to roughly correct the intensities the flat surface so that two sets of data contain the same number of molecules.

To calculate the Raman power ratio ${P}_{SPP}^{\Gamma -X,(1,0)}/{P}_{d}$as well as the coupling efficiency ${\Gamma}_{c}^{\Gamma -X}/{\Gamma}_{r}$, we first fit the angular plots in Fig. 3(d)-3(f) to obtain the (1,0) SPP and the flat Raman emissions. The best fits for (1,0) SPPs are shown as the dash lines in the figures and the results are plotted in Fig. 4(a). However, the background powers in Fig. 4(a) do not represent the complete direct emission since they provide only a portion of it from θ = 0°-55° at φ = 0°. The entire half space is required for the direct emission. To roughly estimate them, we find our detection system supports a solid angle of Δφ ≈2° and thus P_{d} is simply assumed to be equal to P_{background} × 360°/2° considering that direct emission should possess a full circular symmetry over φ and the power ratios are plotted in Fig. 4(b). The decay rate ratio, ${\Gamma}_{rad}^{{\stackrel{\rightharpoonup}{k}}_{SPP},(1,0)}/{\Gamma}_{tot}^{{\stackrel{\rightharpoonup}{k}}_{SPP},(1,0)}$, can be readily obtained from Fig. 2(e) and is shown in Fig. 4(c). Therefore, by using Eq. (1), the calculated coupling efficiencies for three wavelengths are plotted in Fig. 4(d). Since the knowledge of Γ_{r} for 6-mercaptopurine is not yet available, we do not attempt to estimate the coupling rate. However, once Γ_{r} is being measured, we believe the coupling rate can be determined accordingly.

Finally, we briefly comment on the dependence of coupling efficiency on wavelength. The coupling efficiency remains almost constant with wavelength although there is a slight increase at 550 nm. The reason behind this is still unknown but is unlikely due to the electromagnetic origin as the decay rates and the rate ratio given in Fig. 2(e) and 4(c) show a monotonic variation with wavelength, suggesting the field strength would follow similarly. It is noted that Le Ru et al have studied the relative enhancement ratios of different SERS peaks for different molecular probes and find they behave differently [23]. While rhodamine 6G exhibits a gradual decrease of enhancement with increasing Raman shift that can be attributed to the SPP effect due to the different degrees of excitation and emission enhancements, 3-methoxy-4-(5′-azobenzotriazolyl)phenylamine and crystal violet (CV) display a more complicated behavior. In particular, for CV, the enhancement is found similar to our case in which the ratio increases with Raman shift initially but decrease afterwards. They attribute this trend to the surface selection rule where the geometry of molecules absorbed on the surface, the induced image charges, etc, play important roles in governing the resulting enhancement [24]. Nevertheless, more study is required for quantification.

## 4. Conclusion

In summary, by combining the rate equation model and the temporal coupled mode theory, we measure the coupling efficiency of 6-mercaptopurine to SPPs in Γ-X direction supported on periodic Ag nanohole array for several Raman emission wavelengths. This simple method resolves the coupling process and the radiative decay of SPPs in the excitation enhancement. We expect the coupling efficiency could provide much information about the underlying physics of SERS and shed light in controlling it in the future.

## Acknowledgments

This research was supported by the Chinese University of Hong Kong through the RGC Competitive Earmarked Research Grants (402908, 402909 and 403310), the Shun Hing Institute of Advanced Engineering (BME-p3-11), UGC special equipment grant (SEG_CUHK07), and CRF CUHK1/CRF/12G.

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