1. Introduction

Since the discovery of extraordinary transmission (EOT) from periodic metallic arrays in 1998, surface plasmon polaritons (SPPs) have revived interest in science and engineering [1]. In almost two decades, SPPs have gone beyond EOT to other areas such as fluorescence enhancement [2], energy harvesting [3], sensing [4], and nonlinear optics [5]. For fluorescence enhancement, SPPs take part in both excitation and emission processes and they together define the surface plasmon mediated fluorescence in which the overall intensity can be enhanced by more than an order of magnitude [2,6]. As SPP mediated emission is useful in developing high brightness light-emitting diodes, lasers, and fluorescence probes, much effort has been devoted lately in understanding these two processes in order to gain better control [7–9]. Other than intensity enhancement, polarization and direction are the other two quantities that are also of interest [10,11]. While polarization control has been realized in chiral plasmonic systems for twisting the emission [12], only a few works are seen from the angular perspective. The success of angular control would allow us to beam the radiation for effective routing and collection as well as to diverge it for better viewing [13].

It is known that under weak coupling the spontaneous emission rate of light emitters in an optical cavity is governed by the Purcell factor, which strongly depends on the density-of-states (DOS) of the system [14]. Larger DOS lead to faster emission rate and therefore higher intensity. For inhomogeneous structure where the DOS are spatially non-uniform, the local DOS, or LDOS, take part and they define the emission properties at each single location in the structure [15]. In fact, by measuring the local emission rate of emitters, the LDOS across the structure can be probed accordingly [16]. On the other hand, for directional emission where the propagation vector of the radiation is important, the momentum-dependent DOS, or spectral DOS (SDOS) should be considered instead [17,18]. For example, Barth et al have studied the angle-resolved emission near the band gap of a photonic crystal and they suggest the redistribution of the DOS in momentum space is important in modifying the radiation probabilities at different angles [19]. However, how the SDOS and other parameters affect the angular profile of surface plasmon mediated emission has not yet been studied and knowledge of it would shed light in proper control of directional emission in plasmonic systems.

Here, we have studied the angular surface plasmon assisted emission from styryl 8 dyes deposited on a two-dimensional (2D) metallic nanohole array by reflectivity spectroscopy and Fourier space photoluminescence (PL) microscopy. By using the rate equation model and temporal coupled mode theory (CMT), we determine the coupling rate of the light emitters to the Bloch-like (−1,0) SPPs propagating at different directions. While tracing the SPP resonances, we find the rate increases gradually when the SPPs move away from the Γ-X direction, and then split into two when the dark and bright modes are formed along the Γ-M direction. The coupling rate is formulated as the interplay between the SDOS and the momentum-dependent plasmonic field energy. The behavior of the energies is found to be consistent with the finite-difference time-domain (FDTD) simulations. As a result, our work unravel the underlying physics of directional emission while at the same time provides a means for studying the plasmonic field properties in momentum space without involving the near-field techniques.

2. Formulations of the momentum-dependent coupling rate, SDOS, and plasmonic field energy

We first formulate the momentum-dependent coupling rate in a periodic metallic array capped with a layer of light emitters and then show its dependence on the SDOS and the plasmonic field energy. The determination of the coupling rate between the light emitters and the Bloch-like SPPs along the Γ-X direction has been described previously and here we extend it to cover the first Brillouin zone up to the Γ-M direction [20]. In general, for emission enhancement, the excited emitters with energy [EM] at emission photon energy ℏω_em can decay via three channels, namely the nonradiative and radiative decays and the coupling to the SPPs [20]. Their rates are defined as Γ_nr, Γ_r, and Γ_c. While the Γ_r and Γ_nr specify the direct emission and absorption of emitters, the Γ_c characterizes the energy being transferred from the emitters to the SPPs. The SPPs have wavevector ${\stackrel{\rightharpoonup }{k}}_{SPP}$ with the same magnitude$\left|{\stackrel{\rightharpoonup }{k}}_{SPP}\right|=\frac{{\omega }_{em}}{c}\sqrt{\frac{n^2{\epsilon }_m}{{\epsilon }_m+n^2}}$, where n and ε_m are the dielectric constants of the emitting layer and the metal, but propagate in different directions. In fact, the Γ_c is expected to be strongly momentum and position dependent since it is associated with the momentum and spatial local DOS. Therefore, the Γ_c for SPPs with different ${\stackrel{\rightharpoonup }{k}}_{SPP}$ is likely to be different.

Once the SPPs receive energy from the emitters, they subsequently undergo Ohmic absorption and radiation losses defined by the rates as Γ_abs and Γ_rad, respectively. It is noted that the Γ_abs and Γ_rad are also ${\stackrel{\rightharpoonup }{k}}_{SPP}$ dependent [21]. More importantly, both the ${\stackrel{\rightharpoonup }{k}}_{SPP}$ dependent Γ_c and Γ_rad together govern the angular SPP mediated emission [20]. For SPPs propagating with a given ${\stackrel{\rightharpoonup }{k}}_{SPP}$, the outgoing radiations follow the phase-matching equation expressed as $\frac{2\pi }{{\lambda }_{em}}\sin \theta \left(\cos \phi \widehat{x}+\sin \phi \widehat{y}\right)={\stackrel{\rightharpoonup }{k}}_{SPP}-\frac{2\pi }{P}\left(n_x\widehat{x}+n_y\widehat{y}\right)$, where (n_x,n_y) is the Bragg scattering order, P is the period of the array, and θ and φ are the polar and azimuthal emission angles defined with respect to the surface normal and the Γ-X direction [20]. Therefore, the SPPs are Bragg scattered so that their wavevector is matched with that in the free space depending on (n_x,n_y) and the emissions will then exit at well-defined θ and φ. The possible combination of (n_x,n_y) specifies the number of the emission port, m ≥ 1. In steady rate, the rate equation model gives [20]:

Likewise, the total coupling rate of an ensemble of N emitters to SPPs is given as [17]:

3. Experimental

The 2D Au nanohole array is fabricated by using interference lithography as described previously [20]. The plane-view scanning electron microscopy (SEM) image of the array is displayed in the inset of Fig. 1(a), showing it has a square lattice with period P = 510 nm, hole radius and depth = 85 and 280 nm. Since the Au film thickness is above 100 nm, or the skin-depth, the array has no optical transmission. After the sample fabrication, a thin layer of styryl 8 dye dissolved in polyvinyl alcohol polymer is then spin coated on the array. The emission spectrum of styryl 8 spans from 650 to 750 nm [20]. Two types of measurements are then performed. First, the sample is mounted on a computer-controlled goniometer for angle- and polarization-resolved specular reflectivity spectroscopy. A quartz lamp is used as the white light source. A pair of polarizer and analyzer is inserted after the light source and the sample for polarization-dependent measurements. The reflections are then collected by a CCD detector attached to a spectrometer. By contour measuring the reflectivity mappings of the sample as a function of wavelength and incident angle θ_inc at different sample azimuthal angles φ_inc, we obtain the dispersion relations for mode identification and decay rate determination [21]. The reflectivity is the power ratio taken between the sample and the incidence and θ_inc and φ_inc are defined with respect to the surface normal and Γ-X direction. Second, a wide field microscope is used for Fourier space PL measurements [23]. The microscope is equipped with a high numerical aperture NA = 0.9, magnification = 100X objective lens. An Ar ion laser at 514 nm is used to excite the sample at angles defined by the NA of the objective and the PL is collected by the same objective lens and is fed to an EMCCD camera via a series of lenses for Fourier space imaging. A bandpass filter at emission wavelength, λ_em, = 700 nm with FWHM = 12 ± 2 nm is placed before the camera, yielding the fluorescence isofrequency surface. For calibrating the angular PL, the transmission function of the objective lens in Fourier space is first corrected by taking the Fourier space PL image of a flat Au film coated with styryl 8 assuming an isotropic emission [24].

 

Fig. 1 (a) The p-polarized reflectivity mapping of the dye/nanohole array taken along the Γ-X direction. The dash line is calculated by using the phase-matching equation, indicating the (−1,0) SPP mode is excited. Inset: the plane-view SEM image of the array. (b) The p-polarized reflectivity spectra (solid lines) for different φ_inc and the best fits (dash lines) using CMT. They all have dips at λ = 700 nm. At φ_inc = 45° the spectra taken under p- and s-polarizations are shown. All the spectra are vertical shifted for visualization. (c) The plot of fitted total (Γ_tot) and radiative decay (Γ_rad) rates of the plasmonic modes as a function of φ_inc.

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4. Results

We begin our measurements and analysis at φ_inc = 0° where the SPPs propagate along the Γ-X direction and then move on to other directions. Figure 1(a) shows the p-polarized angle-dependent specular reflectivity mapping, i.e. dispersion relation, of the array taken along the Γ-X direction. The dispersive dips outline the excitation of the (−1,0) Bloch-like SPPs as calculated by the phase-matching equation (dash line). The emission wavelength λ_em at 700 nm is given by the solid line and the reflectivity spectrum is then extracted in Fig. 1(b), showing a Fano-like reflection dip [22]. The radiative and total decay rates can be determined by fitting the spectrum with CMT as ${\left|r_p+\frac{{\kappa }^2{\Gamma }_{rad}^{{\stackrel{\rightharpoonup }{k}}_{SPP}}}{i\left(\omega -{\omega }_{em}\right)+{\Gamma }_{tot}^{{\stackrel{\rightharpoonup }{k}}_{SPP}}/2}\right|}^2$, where r_p is the nonresonant reflection coefficient, $\kappa =\kappa \text{'}\cos \alpha \text{=}\sqrt{\text{2}}$, and ${\stackrel{\rightharpoonup }{k}}_{SPP}$ = Γ-X for this case. κ is the multiplication of the coupling constant $\kappa \text{'}$ and the best coupling polarization angle α, which accounts for how much incident energy is being channeled to SPPs [25]. The best fit is also shown in Fig. 1(b) for reference, yielding r_p = 0.989, $\hslash {\Gamma }_{rad}^{\Gamma -X}$ = 16.8 and $\hslash {\Gamma }_{tot}^{\Gamma -X}$ = 24.8 meV. The decay rates are then plotted in Fig. 1(c). To find $P_{SPP}^{{\stackrel{\rightharpoonup }{k}}_{SPP}=\Gamma -X}/P_d$, we employ the Fourier-space PL imaging. At λ_em = 700 nm, the PL isofrequency surface is shown in Fig. 2(a). From the surface, we clearly see four emission arcs superimposing on a broad background. The arcs, as deduced from the phase-matching equation, arise from the ( ± 1, 0) and (0, ± 1) SPPs whereas the background is due to the direct emission. To find P_d, we integrate the background over the entire region, which gives 302 counts. On the other hand, the $P_{SPP}^{\Gamma -X}$, for example, is the integrated intensity covered by the arc at φ = 0° after removing the underlying background. We see from Fig. 2(a) that the arcs appear slightly asymmetric due to the misalignment of the optics in the microscope. Nevertheless, knowing the four-fold symmetry possessed by the square lattice array, we average the four arcs at φ = 0°, 90°, 180°, and 270° to determine $P_{SPP}^{\Gamma -X}/P_d$ to be 0.157 in Fig. 2(b). Once the decay rates and $P_{SPP}^{\Gamma -X}/P_d$ are available, we then determine ${\Gamma }_c^{\Gamma -X}/{\Gamma }_r$ to be 0.231 following Eq. (2) in Fig. 3(a).

 

Fig. 2 (a) The photoluminescence isofrequency surface of the dye/nanohole array taken at λ_em = 700 nm. Four arcs indicate the emissions from the ( ± 1, 0) and (0, ± 1) SPPs whereas the background arises from the direct emission. (b) The plot of power ratio $P_{SPP}^{{\stackrel{\rightharpoonup }{k}}_{SPP}}/P_d$ as a function of φ. The splitting of the power ratio, or the formation of the dark and bright modes, is seen at φ = 45°.

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Fig. 3 The plot of (a) coupling rate ratio ${\Gamma }_c^{{\stackrel{\rightharpoonup }{k}}_{SPP}}/{\Gamma }_r$, (b) SDOS and (c) plasmonic energy EF as a function of φ.

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We then follow the same procedures to determine ${\Gamma }_c^{{\stackrel{\rightharpoonup }{k}}_{SPP}}/{\Gamma }_r$ for other (−1,0) SPPs propagating in different directions. All the reflectivity mappings for different φ_inc are displayed in Fig. 4 for reference and their corresponding spectra taken at λ = 700 nm as well as the best fits are shown in Fig. 1(b). The $\hslash {\Gamma }_{rad}^{{\stackrel{\rightharpoonup }{k}}_{SPP}}$, $\hslash {\Gamma }_{tot}^{{\stackrel{\rightharpoonup }{k}}_{SPP}}$, $P_{SPP}^{{\stackrel{\rightharpoonup }{k}}_{SPP}}/P_d$, and ${\Gamma }_c^{{\stackrel{\rightharpoonup }{k}}_{SPP}}/{\Gamma }_r$ are summarized in Fig. 1(c), 2(b), and 3(a) as a function of φ.

 

Fig. 4 (a) – (r) The p-polarized reflectivity mappings taken at different φ_inc. At φ_inc = 45°, both p- and s-reflectivity spectra are shown. The dash lines are calculated by using the phase-matching equation, indicating (−1,0), (−1,-1) and (0,-1) SPPs are excited.

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It is noted that while there is only one single mode at φ ≠ 45° SPPs, two hybridized dark and bright modes form along the Γ-M direction at φ = 45° where the (−1,0) and (0,1) SPPs cross with each other [26]. As the dark and bright modes possess different field symmetries, they are excitable only by p- or s-incidence [26]. In particular, the dark mode is symmetric with respect to the incident plane and thus is excited by p-polarized light whereas the bright mode is excited by s-polarized light. For illustration, Figs. 4(p) and 4(o) show the dispersion relations of the array taken at φ = 45° under p- and s-excitations. We see two dark and bright modes are separated from each other due to the level repulsion, or the formation of a plasmonic gap. The spectra taken at θ_inc = 30° and 38° for p- and s-polarizations are shown in Fig. 1(b) and the best fits show the radiative and total decay rates are 6.21 and 9.40 meV for the dark mode and 9.74 and 15.88 meV for the bright mode, as plotted in Fig. 1(c). In fact, we see the radiative decay rate for the bright mode is larger than that of the dark counterpart, signifying stronger radiation damping. On the other hand, from the PL image in Fig. 2(a), the dark and bright modes exit at different emission angles, featuring distinct emission intensities. Apparently, the bright mode produces stronger emission than the dark mode. The $P_{SPP}^{\Gamma -M}/P_d$ and the corresponding ${\Gamma }_c^{\Gamma -M}/{\Gamma }_r$ for two modes are determined in Fig. 2(b) and 3(a).

From the ${\Gamma }_c^{{\stackrel{\rightharpoonup }{k}}_{SPP}}/{\Gamma }_r$plot, we see the rate increases gradually when the SPPs move away from the Γ-X towards the Γ-M direction and split in two along the Γ-M direction where the dark and bright modes are formed. After the splitting, it increases again. To elucidate such momentum dependence, we make use of Eq. (5) that the interplay between the field energy and the SDOS determines the coupling rate, i.e.${\Gamma }_c^{{\stackrel{\rightharpoonup }{k}}_{SPP}}\propto E{F}_{{\stackrel{\rightharpoonup }{k}}_{SPP}}\times SDO{S}_{{}_{{\stackrel{\rightharpoonup }{k}}_{SPP}}}$. As $SDO{S}_{{\stackrel{\rightharpoonup }{k}}_{SPP}}\propto 1/\hslash {\Gamma }_{tot}^{{\stackrel{\rightharpoonup }{k}}_{SPP}}$in Fig. 3(b), the behavior of EF is plotted accordingly in Fig. 3(c), showing the EF has a stronger influence on the coupling rate than the SDOS. Noted that as N is unknown in Eq. (5), we normalize EF to the value at φ_inc = 0°. We verify the EF by using FDTD simulation. The unit cell is shown in Fig. 5(a). It has period, hole depth and radius = 510, 280, and 85 nm to mimic our experiment. Bloch boundary condition is set at four sides and perfectly matched layers are located at the top and on the bottom of the cell. The dielectric constant of Au is obtained from Ref [27]. A cubic power monitor is placed above the Au surface to calculate the near-field patterns at upper half space. To take into consideration of the best polarization angle for exciting different propagating SPPs, a circularly polarized incidence is used instead [25]. For each φ, the reflectivity spectra as a function of θ_inc are calculated first to identify the excitation condition for (−1,0) SPPs at λ_em = 700 nm and the resulting spectra are plotted obtained in Fig. 5(b) where they all have the same spectral dip position. The corresponding near-field profiles are then simulated and integrated over the entire unit cell for determining the EF. Several field intensity patterns taken at different φ are displayed in Figs. 5(c)-5(f) for reference. The EF normalized with respect to φ_inc = 0° is also plotted in Fig. 5(g) as a function of φ_inc. We find the overall trends of the experiment and simulation agree well.

 

Fig. 5 (a) The unit cell for FDTD simulation. (b) The reflectivity spectra for different φ. All the spectra have the same spectral position at λ_em = 700 nm. Simulated field intensity pattern taken at φ = (c) 0°, (d) 45° (s incidence), (e) 45° (p incidence), and (f) 49.7° for λ_em = 700 nm. The dash lines are the incident planes. (g) The FDTD simulated EF as a function of φ.

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5. Conclusion

In summary, we have studied the surface plasmon mediated fluorescence from 2D Au nanohole array by angle-resolved reflectivity spectroscopy and Fourier-space photoluminescence microscopy. By using the rate equation model and coupled mode theory, we have determined the momentum-dependent coupling rate of the light emitters to the (−1,0) Bloch-like SPPs. We find the rate increases slowly when the SPPs propagate away from the Γ-X towards the Γ-M direction, splits distinctively into two for the hybridized dark and bright modes along the Γ-M direction, and then decreases again. To elucidate such dependence, the corresponding SDOS and plasmonic energies have been determined. The interplay between the SDOS and the energy is of importance in governing the momentum-dependent coupling rate. In contrast to the local density-of states where spatial information is provided, SDOS directly affect the directional emission of the surface plasmon mediated fluorescence.