1. Introduction

The interaction between light emitters and surface plasmon polaritons (SPPs) arising from metal nanostructures has aroused much interest recently. Its knowledge plays a major role in understanding the fundamental of light-matter interaction as well as in the advancement of fluorescence enhancement [1], energy harvesting [2], biosensing [3], and nonlinear optics [4]. For the fluorescence, SPPs enhance not only the spontaneous emission but also the excitation of the emitters [5,6]. The excitation and emission enhancements together yield the SPP mediated fluorescence [5,6]. To understand the entire process completely, differentiating the enhancements one by one is necessary for quantification. While studies on extracting the enhanced emission rate from the well-known Purcell factor have been conducted for years [7,8], little is known about the enhanced absorption. In fact, understanding how the absorption is affected by SPPs is of great importance not only in fluorescence but also in photovoltaics where an increase of the excited carrier concentration is always favorable for photocurrent generation [2,9]. Unfortunately, the exact determination of the excitation rate is not yet available and most of the studies reveal only the excitation efficiency ratio, which simply compares the emission with and without SPPs, in the photoluminescence excitation spectroscopy (PLE) and this method sometimes is very system dependent for quantification [10,11].

In this work, we combine the rate equation model and coupled mode theory (CMT) to determine the excitation rate of light emitters positioned on a metallic array which generates Bloch-like SPPs. The rate is analytically expressed in measurable quantities. By angle- and polarization-resolved reflectivity and photoluminescence (PL) spectroscopy, we have determined the excitation rate of CdSeTe quantum dots (QDs) on 2D Au nanohole array mediated by the (−1,0) SPPs propagating in different directions. At λ = 633 nm, it is found the rate is ~44 ps⁻¹ and is independent of the propagation direction when the SPPs move away from the Γ-X direction as well as along the Γ-M direction where two degenerate SPPs couple with each other to form the bright and dark modes. Our results are consistent with the finite-difference time-domain (FDTD) simulations and they indicate the enhanced absorption of the QDs depends only on the plasmonic field.

2. Excitation rate formulation

2.1 Rate equation model

When light emitters are placed near a periodic plasmonic system such as a 2D square lattice array, SPP mediated fluorescence occurs [12]. Physically, the emitters can be described as a simple three level system as given in Fig. 1(a) such that the excitation and emission wavelengths, λ_ex and λ_em, are slightly different. This difference occurs in many organic and inorganic emitters where the Stokes shift prevails [13]. The incident light at λ_ex with power P_o excites the electrons in the emitters from the ground state (S₀) to the second excited state (S₂) with excitation efficiency η and at the same time generates the incoming Block-like propagating SPPsⁱⁿ if the phase-matching condition is fulfilled [12]:

 

Fig. 1 (a) The schematic of surface plasmon mediated emission fluorescence. The emitters contain a ground state (S₀) and two excited states (S₁ and S₂). Excitation occurs from S₀ to S₂ by light source at λ_ex with power P_o whereas emission arises from the decay from S₁ to S₀ at λ_em. Both incoming and outgoing SPPs, SPPsⁱⁿ and SPPs^out, can take part in the process. They have their respective absorption and radiative decay rates, Γ_abs and Γ_rad. Direct excitation has efficiency η and direction emission has nonradiative and radiative emission rates Γ_nr and Γ_r. (b) The schematic for the best coupling condition. Inset: the SEM image of the array. (c) The absorption and emission spectra of the CdSeTe quantum dots.

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By using the rate equation model [12,16], the transient of [EM] in the presence of incoming SPPs can be expressed as:

2.2 Coupled mode theory

If only one input and output port is available, the incoming SPPsⁱⁿ can be described by the CMT as $\frac{da}{dt}=i{\omega }_{ex}a-\frac{{\Gamma }_{\text{tot}}^{\text{in}}}{2}a+\kappa \sqrt{{\Gamma }_{rad}^{\text{in}}}{s}_{p+}$ [14,17], where a is the SPPs mode amplitude, ω_ex is the resonant excitation frequency, Γ_totⁱⁿ is the total decay rate of SPPs $={\Gamma }_{abs}^{in}+{\Gamma }_{rad}^{in}+{\Gamma }_{ex}$, and ${\left|s_{\text{p}+}\right|}^2=P_o$ is the p-polarized incident power. κ is the complex coupling constant, which depends on the in-coupling phase shift [14,17] and the relationship between the incident polarization and the field of SPPs [20], as will be explained later. At ω_ex, $\frac{{\left|a\right|}^2}{P_o}\text{=}\frac{4{\kappa }^2{\Gamma }_{rad}^{in}}{{\Gamma }_{tot}^{in}{}^2}$. Therefore, Eq. (3) becomes:

2.3 Parametric determination for nondegenerate SPPs

We need to determine κ, η, and the decay rates in prior to solving Γ_ex. κ is expected to depend on both the incident angle and the polarization [20]. In fact, to elucidate the dependence completely, the concept of the best coupling polarization angle needs to be considered here. The best polarization angle accounts for how much incident energy can be channeled to SPPs when the electric field component of the incident polarization is not aligned with the field of SPPs [20,21]. In other words, the best condition for energy input occurs when the incident polarization and the SPP field is alike. For nondegenerate SPPs where the electric field ${\stackrel{\rightharpoonup }{E}}_{SPP}$is on the same plane as the propagation direction ${\widehat{k}}_{SPP}$, this condition can be mathematically simplified as: $\widehat{e}\cdot \left({\widehat{k}}_{SPP}\times \widehat{z}\right)=0$, where $\widehat{e}$ is the polarization unit vector and $\widehat{z}$is the unit vector normal to the sample surface [20]. Following Fig. 1(b), these unit vectors can be written as $\widehat{e}=\left(\cos \alpha \cos \theta ,sin\alpha ,\cos \alpha sin\theta \right)$, ${\widehat{k}}_{SPP}=\left(\cos \beta ,\sin \beta ,0\right)$, and $\widehat{z}=\left(0,0,1\right)$, where α is the best polarization angle α between $\widehat{e}$ and the incident plane and β is the angle between the SPPs propagation and the incident plane that can be estimated from the phase matching equation in Eq. (1). Substituting them into the vector product, we have:

Following [20], κ can now be written as $\kappa \text{'}\cos \alpha$, which contains a complex constant κ' for in-coupling phase shift and cosα for the dependence of energy channeling on α. On the other hand, for out-coupling, under reciprocal theorem, the polarization of the radiation damping for SPPs will also follow Eq. (5) [20,25]. As a result, for any arbitrary incident polarization angle, the reflectivity contains two components. While the first is the non-resonant reflection that has the same polarization as the incidence, the second is the radiation damping from the SPPs with the polarization angle defined by Eq. (5). Therefore, upon p-incidence, the p- and s-polarized (orthogonal) reflectivity, R_pp and R_ps, can be written as [14,20]:

2.4 Parametric determination for two coupled degenerate SPPs

Equation (6) is only valid for one traveling SPPs. For the coupling of two degenerate SPP modes along the Γ-M direction, the analysis is different but has been studied earlier for 1D and 2D arrays by CMT [23,24,26]. Under coordinate transformation, the dynamics of the bright and dark mode amplitudes, $a_{+}$ and $a_{-}$, can be written as $\frac{d}{dt}\left(\begin{array}{c}{a}_{\text{+}}\\ a_{-}\end{array}\right)=\left(\begin{array}{cc}i{\tilde{\omega }}_{+}& 0\\ 0& i{\tilde{\omega }}_{-}\end{array}\right)\left(\begin{array}{c}{a}_{\text{+}}\\ a_{-}\end{array}\right)+\left(\begin{array}{cc}0& {\kappa }_{+}\text{'}\sqrt{{\Gamma }_{+}^{r\text{ad}}}\\ {\kappa }_{-}\text{'}\sqrt{{\Gamma }_{-}^{r\text{ad}}}& 0\end{array}\right)\left(\begin{array}{c}{s}_{p+}\\ s_{s+}\end{array}\right)$, where${\tilde{\omega }}_{+,-}={\omega }_{+,-}\text{+}i{\Gamma }_{{}_{+,-}}^{tot}/2$are the complex frequencies for two modes, ${\Gamma }_{+,-}^{\text{rad}}$are the radiative decay rates, ${\kappa }_{+,-}\text{'}$ are the modified coupling constants, $a_{+}$ and $a_{-}$ are excited solely by s- and p-polarized lights [23]. The mode amplitudes are ${\text{a}}_{+}=\frac{{\kappa }_{+}\text{'}\sqrt{{\Gamma }_{+}^{\text{rad}}}}{i\left(\omega -{\omega }_{+}\right)+{\Gamma }_{+}^{tot}/2}{s}_{s+}$and $a_{-}=\frac{{\kappa }_{-}\text{'}\sqrt{{\Gamma }_{-}^{\text{rad}}}}{i\left(\omega -{\omega }_{-}\right)+{\Gamma }_{-}^{tot}/2}{s}_{p+}$. Therefore, at ${\omega }_{+,-}$, the excitation rates are ${\Gamma }_{\text{ex,bright}}=\eta \left({\left(\frac{P_d^w}{P_d^{w/o}}\right)}_s\right)\frac{{\left({\Gamma }_{+}^{\text{tot}}\right)}^2}{4{\kappa }_{+}{\text{'}}^2{\Gamma }_{+}^{\text{rad}}}$ and ${\Gamma }_{\text{ex,dark}}=\eta \left({\left(\frac{P_d^w}{P_d^{w/o}}\right)}_p\right)\frac{{\left({\Gamma }_{-}^{\text{tot}}\right)}^2}{4{\kappa }_{-}{\text{'}}^2{\Gamma }_{-}^{rad}}$, where ${\left(\frac{P_d^w}{P_d^{w/o}}\right)}_{s,p}$are the power ratios under s- and p-excitations. The parameters $\kappa {\text{'}}_{+,-}$, ${\Gamma }_{+,-}^{\text{rad}}$, and ${\Gamma }_{+,-}^{\text{tot}}$ again can be determined by fitting the s- and p-polarized reflectivity spectra with$R_{s\text{s}}={\left|r_s+\frac{{\kappa }_{+}{\text{'}}^2{\Gamma }_{+}^{rad}}{i\left(\omega -{\omega }_{+}\right)+{\Gamma }_{+}^{tot}/2}\right|}^2-\eta$, and $R_{\text{pp}}={\left|r_p+\frac{{\kappa }_{-}{\text{'}}^2{\Gamma }_{-}^{rad}}{i\left(\omega -{\omega }_{-}\right)+{\Gamma }_{-}^{tot}/2}\right|}^2-\eta$.

3. Experiment

3.1 Method

We will study the dependence of Γ_ex on φ, which reflects different SPP propagation directions. CdSeTe QDs with diameter of 21 nm (Life Technologies) are spin-coated on a 2D Au nanohole array prepared by interference lithography [12]. The array has period P, hole depth and radius = 450, 80, and 60 nm as illustrated in the scanning electron microscopy (SEM) image in the inset of Fig. 1(b). Before spin-coating, the concentration of QDs is further diluted in methanol from 2 µM to 50 nM to reduce dot-dot interaction. The absorption and emission spectra of the QDs in methanol are shown in Fig. 1(c). The sample is then mounted onto a computer-controlled goniometer as shown in Fig. 2(a) for angle- and polarization-resolved reflectivity and PL measurements. The goniometer has three rotation stages to independently control θ and φ [12,27]. A pair of incident polarizer and detection analyzer is placed after the light source and the sample for polarization measurements. A collimated quartz lamp is used for the reflectivity measurements whereas a 633 nm HeNe laser is used for the PL spectroscopy. The signals are collected by a spectrometer attached with a CCD detector.

 

Fig. 2 (a) The schematic of the angle- and polarization resolved reflectivity and photoluminescence spectroscopy. (b) The p-polarized specular reflectivity mapping of the array taken at φ = 0°. The dash line, obtained from the phase matching equation, indicates the excitation of (−1,0) SPPs. (c) The reflectivity spectrum (dot) extracted from θ = 18.5° and φ = 0° together with the best fit (solid line) for parametric determination.

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3.2 Reflectivity and photoluminescence measurements in one SPP direction

We present the reflectivity and PL results at one φ as an example to show how the parameters are extracted. By setting φ = 0°, the θ-dependent p-polarized (R_pp) reflectivity mapping, also known as the dispersion relation, of the sample is shown in Fig. 2(b) for mode identification. The orthogonal (R_ps) mapping for φ = 0° is zero and thus is not shown. Apparently, the dispersive reflection dips match well with the (−1,0) SPP mode calculated by Eq. (1) [12]. In contrast, the absorption band of QDs is not seen owing to the low concentration. We extract the reflectivity spectrum taken at θ = 18.5° where the λ_ex = 633 nm incoming SPPs are excited in Fig. 2(c) and then fit it with Eq. (6) to extract the parameters. The results, r_p, κ', α, η, Γ_totⁱⁿ, and Γ_radⁱⁿ, are tabulated in Table. 1. For determining $P_d^w/P_d^{w/o}$, angle-resolved PL scans are then performed. Two types of measurements are conducted and they are the detection and incident scans [12,27,28]. For the detection scan, the incident angle is kept fixed so that the excitation condition does not change but the detection angle is scanned for screening the angular emission at a certain emission plane [12,27,28]. As the radiation damping of the outgoing SPPs is highly emission angle dependent, detection scan probes the emission enhancement [12]. On the other hand, for the incident scan, while the detection angle is fixed with respect to the sample normal, the incident angle is varied continuously. This configuration allows us to selective excite the incoming SPPs while monitoring the emission at the same angle. In other words, it probes the excitation enhancement. As we will see, the interplay between the incident and detection scans assist us in identifying $P_d^w$ and $P_d^{w/o}$ and determining their ratio. A p-incident polarizer is used without the analyzer so that full emission is detected in these two scans. For illustration, Fig. 3(a) & (b) show the PL detection scans with φ = 0° and θ = 0° and 18.5°. An almost one to one correspondence is seen between the reflectivity mapping in Fig. 2(b) and the detection scans, confirming the emission enhancement from (−1,0) SPPs. One also sees a non-dispersive emission band at λ ~690 nm, which is attributed to the direct emission of QDs when comparing with Fig. 1(c). Except the intensities, two detection scans are almost the same, indicating the emission enhancement is identical with and without the incoming SPPs. We then perform the incident scan by keeping the detection angle at 0° in Fig. 3(c). Strong direct emission occurs at 18.5° is due to the excitation of the 633 nm incoming SPPs. The normalized emission intensity at λ_em = 690 nm is extracted in Fig. 3(d) for determining $P_d^w/P_d^{w/o}$. The emission shows an asymmetric “Fano-like” profile and the power ratio $P_d^w/P_d^{w/o}$is determined to be 6.9 in Table. 1. Once all the parameters are available, Γ_ex is then calculated to be 40.3 ps⁻¹ along the Γ-X direction.

Table 1. Parametric determination of nondegenerate and degenerate SPP modes for different φ.

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Fig. 3 The detection scans of the array taken at φ = 0° and θ = (a) 0° without the incoming 633 nm SPPs and (b) 18.5° with the incoming SPPs. (c) The incident scan of the array taken at φ = 0° and a fixed detection angle = 0°. The band at 690 nm is from the direct emission. (d) The line incident scan taken from the direct emission at 690 nm.

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3.3 Dependence of Γ_ex on propagation direction

The above procedures then apply to different φ. We measure the R_pp and R_ps reflectivity spectra and the PL incident scans at λ_em = 690 nm from φ = 5° to 35°, where only one single 633 nm incoming SPP mode is excited, in Fig. 4. The best fits yield r_p, κ, α, η, Γ_totⁱⁿ, Γ_radⁱⁿ, and $P_d^w/P_d^{w/o}$ in Table 1. We compare the fitted α with the values deduced from Eq. (5) in Fig. 5(a) and they agree with each other. Γ_ex are then determined in Fig. 5(b), showing it is weakly dependent on φ.

 

Fig. 4 The (a) p-polarized (R_pp) and (b) orthogonal (R_ps) reflectivity spectra with the nondegenerate incoming SPPs located at 633 nm taken from φ = 5° – 35°. The best fits are indicated by the dash lines for parametric determination. The spectra are vertically shifted for visualization. (c) The line incident scans taken from the direct emission at 690 nm at detection angle = 0° and the corresponding φ.

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Fig. 5 (a) The comparison between the fitted α and the values deduced from Eq. (5). (b) The plot of Γ_ex with φ, including the bright and dark modes.

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For φ = 45° where two (−1,0) and (0,-1) SPPs cross, the p- and s-polarized reflectivity spectra are taken at θ = 40° and 29.5° in Fig. 6(a) & (b) so that the dark and bright modes are both excited at 633 nm. The PL incident scans taken under p- and s-polarizations are plotted in Fig. 6(c) for λ_em = 690 nm. Consistent with the reflectivity, strong excitation enhancements are observed at 29.5° and 38° for the bright and dark modes. The emission from the bright mode is stronger than that of the dark mode. The parameters are extracted from the reflectivity spectra as well as the PL incident scans and they are then tabulated in Table 1. Γ_ex for the dark and bright modes are plotted in Fig. 5(b), showing they are almost the same as the nondegenerate counterparts.

 

Fig. 6 The (a) R_pp and (b) R_ss reflectivity spectra for the dark and bright modes taken at φ = 45° together with the best fits (dash lines). (c) The line incident scans taken from the direct emission at 690 nm for the dark and bright modes.

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4. Numerical simulation

The weak dependence of Γ_ex on φ shows the excitation rate is an intrinsic parameter and is not affected by the environment. It depends only on the dipole moment of the emitters [29,30]. The plasmonic absorption therefore is caused by ${\left|a\right|}^2/P_o$, which is related to the ${\left|{\stackrel{\rightharpoonup }{E}}_{SPP}\right|}^2$. As a result, if ${\Gamma }_{\text{ex}}/\eta$ is a constant, we expect $\frac{P_d^w}{P_d^{w/o}}=\frac{\left[EM\right]}{{\left[EM\right]}^{w/o}}=\frac{{\Gamma }_{\text{ex}}{\left|a\right|}^2}{P_o\eta }\propto \frac{{\left|{\stackrel{\rightharpoonup }{E}}_{SPP}\right|}^2}{{\left|{\stackrel{\rightharpoonup }{E}}^{w/o}\right|}^2}$, where ${\left|{\stackrel{\rightharpoonup }{E}}^{w/o}\right|}^2$is the field strength without the incoming SPPs. To verify it, we perform FDTD simulations. The unit cell of the QDs/array for FDTD simulations is shown in the inset of Fig. 7(a). Following the SEM image, the nanohole array has P = 450 nm, hole depth and radius = 80 and 60 nm. The QDs layer is modeled as a 21 nm thick dielectric layer with the complex refractive index of 1.12 + 0.05i capped on the optically thick Au array. While the thickness of the QDs layer is measured by a profilometer, the refractive index is obtained by fitting the experimental p-polarized reflectivity spectrum taken at θ = 18.5° and φ = 0° with the FDTD data, as shown in Fig. 7(a). Bloch boundary condition is used at four sides whereas perfectly matched layer is used at the top and the bottom. A power monitor is placed at 5 nm above the surface to record the electric field intensity, which is then averaged across the entire simulation cell. The field intensity at different wavelengths are then simulated and plotted in Fig. 7(b) at different θ and φ under p-incidence for exciting the (−1,0) SPPs at 633 nm. For the bright and dark modes, s- and p-incidences are used at φ = 45° and θ = 29.5° and 38°. An intensity spectrum taken at both θ and φ = 0° so that no 633 nm SPPs is excited is also simulated in Fig. 7(b) for comparison. The intensity ${\left|{\stackrel{\rightharpoonup }{E}}_{SPP}\right|}^2/{\left|{\stackrel{\rightharpoonup }{E}}^{w/o}\right|}^2$ and the experimental $P_d^w/P_d^{w/o}$ ratios are then plotted in Fig. 7(c) for comparison. Their overall trend including the dark and bright modes agrees well between the simulation and experiment, verifying that Γ_ex is a constant. Importantly, the decrease of the intensity of the nondegenerate SPPs as a function of φ is mostly due to the mismatch between the incident and the best polarizations, thereby reducing the energy coupling from the incidence to the SPPs, i.e. ${\left|{\stackrel{\rightharpoonup }{E}}_{SPP}\right|}^2$. By plotting ${\cos }^2\alpha$ in Fig. 7(c), all are found to be consistent with each other.

 

Fig. 7 (a) The FDTD simulated p-polarized reflectivity spectra with and without the dielectric layer calculated at θ = 18.5° and φ = 0° together with the experimental spectrum. Inset: the unit cell for the FDTD simulation. (b) The electric field intensity spectra calculated at 633 nm for φ = 0° - 45°. (c) The comparison between the direct emission power ratio $P_d^w/P_d^{w/o}$(solid squares), the calculated electric field intensity enhancement ${\left|{\stackrel{\rightharpoonup }{E}}_{SPP}\right|}^2/{\left|{\stackrel{\rightharpoonup }{E}}^{w/o}\right|}^2$ (open circles), and cos²α (open triangles).

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5. Conclusion

In summary, we have determined φ-dependent Γ_ex of CdSeTe QDs deposited on 2D Au nanohole array by using angle- and polarization-resolved reflectivity and photoluminescence spectroscopy. Based on the rate equation model and coupled mode theory, we express Γ_ex in terms of measurable quantities that can be determined sequentially. We find Γ_ex at 633 nm ~44 ps⁻¹ and is independent of the propagation direction and the coupling of SPPs. Our results indicate Γ_ex is an intrinsic parameter and depends only on the emitters themselves but not the environment. The enhanced absorption of the emitters is due solely to the plasmonic field enhancement. The experiment is found to agree very well with the FDTD simulations.