1. Introduction

Decay is the key process in plasmonics since it defines many intriguing properties of surface plasmon polaritons (SPPs) [1]. The decay dynamics is defined by two parameters, which are the propagation length (L_D) and decay rate (Γ_tot). For example, L_D governs how far SPPs can travel before complete dissipation and a long L_D is favorable for waveguiding [2]. On the other hand, field enhancement has been indicated to hinge on the interplay between radiative (Γ_rad) and absorption (Γ_abs) decay rates [3, 4]. More importantly, the interplay between L_D and Γ_tot defines group velocity (v_g = L_DΓ_tot) [5, 6], which is related to the density-of-states (DOS) of the system [7]. Because of their significance, several methods based on spatial-, time-, and frequency-domains have been developed to study the decay process. Kim et al measure the propagating length of SPPs on 2D Au nanohole arrays by using near-field scanning optical microscopy and deduce the length to be 2-3 μm for (1,0) SPPs [8]. On the other hand, Ropers et al have used the time-resolved method to examine the transient of the electric field transmitted through the 1D Au grating and conclude the lifetime, which is 1/Γ_tot, is ~32 fs for −2 SPPs [9]. They also report the coupling of two degenerate −1 and + 1 SPP modes at normal incidence results in the formation of dark and bright modes due to the strong modification of the SPP radiation damping rates [9]. Dark mode is nonradiative with longer decay lifetime whereas bright mode is highly radiative and thus has shorter lifetime. We have deployed a frequency-domain method to examine the dependence of decay rate on wavelength and hole geometry in 2D nanohole arrays [10]. In particular, we use temporal coupled mode theory (CMT) to express the reflectivity spectrum explicitly on decay rates and have measured the absorption and radiative decay rates of (−1,0) SPPs [11]. More importantly, we find both the spectral profile and the field enhancement are associated with the interplay between the absorption and radiative decay rates [4].

To date, for periodic arrays, the decay rates or lengths have so far been measured in one single angle or direction, mostly along the Γ-X direction for different resonance wavelengths [4, 8–11]. However, it is known that SPPs can propagate in different directions and each could have different decay dynamics. Therefore, other than the spectral dependence, the knowledge of angle- or momentum-dependent L_D and Γ_tot, as well as v_g = L_DΓ_tot, is also of great importance. In addition, the angular dependence is expected to play a major role in controlling the directionality of SERS and fluorescence, which are essential directional emission [12–15]. Therefore, the understanding the angle-dependent decay process of SPPs presents not only a scientific interest but also a practical incentive.

In this paper, we have measured the momentum-dependent L_D and Γ_tot of ( ± 1,0) and (0, ± 1) Bloch-like SPPs in a 2D Au nanohole array by using angle- and polarization-resolved reflectivity spectroscopy and real- and Fourier-space microscopy. While reflectivity spectrum directly yields Γ_tot, we find the linewidth obtained from the equi-energy Fourier-space contour image is associated with L_D corresponding to a particular propagation direction, as verified by finite-difference time-domain (FDTD) simulation. Our results show two quantities, together with v_g, vary strongly with the propagation direction. In particular, when the direction is changing from Γ-X to Γ-M, v_g decreases due to the increase proportion of the standing wave character arising from the coupling of two SPPs. Along the Γ-M direction where two SPPs are degenerate, in addition to the formation of plasmonic gap, two dark and bright modes are formed, giving rise to two distinct velocities. We find although the dark mode has smaller Γ_tot, its v_g is higher than that of the bright mode. As two modes have different field symmetry, we elucidate the fact that the fields of the dark mode are mostly concentrated near the air hole region and thus produce a lower effective index n_eff and higher v_g. More importantly, the distinct v_g arising from the dark and bright modes imply different DOS, which could affect Raman and fluorescence emissions.

The paper is organized as follows. Section 2 discusses how one can determine L_D from the equi-energy contour by using spatial CMT. FDTD simulation on a simple 1D grating is provided for verifying CMT. In section 3, on an actual 2D nanohole array, we present the measurements of Γ_tot, L_D, and v_g as a function of wavelength by using angle-resolved reflectivity spectroscopy along the Γ-X direction. The results serve as a testbed for our approach. Section 4 reports L_D at λ = 633 nm as a function of SPP propagation direction by Fourier-space microscopy. Finally, in section 5, we summarize our findings on the dependences of Γ_tot and v_g on SPP propagation direction, with special attention along the Γ-M direction.

2. Analytical and numerical calculations

Both L_D and Γ_tot are two measurable quantities. While the measurement of Γ_tot has been discussed in earlier works, we focus on the relationship between L_D and the linewidth in the equi-energy contour here.

A simple case of 1D metallic periodic nanoslit array is considered for proof of the concept. The nanoslit array is illuminated by p-polarized plane wave with constant real in-plane momentum k_//, along the Γ-X direction. We assume the nanoslit array is optically thick and it supports only specular reflection. In analogy to temporal CMT, we write the spatial dynamics of a SPP mode amplitude, a, at a particular frequency ω as: $\frac{d{a}^{\omega }}{dx}=i{\tilde{k}}_{spp}{a}^{\omega }+{\kappa }^{\omega }{s}_{+}^{\omega }=i{k}_{spp}{a}^{\omega }-\frac{1}{2L_D}{a}^{\omega }+{\kappa }^{\omega }{s}_{+}^{\omega }$, where κ^ω is the in-coupling constant, $s_{+}^{\omega }$ is the momentum resolved amplitude of the incident wave power, ${\tilde{k}}_{spp}=k_{spp}+\frac{i}{2L_D}$ is the complex resonant momentum indicating the SPP mode decays exponentially with x. Here ${\tilde{k}}_{\text{spp}}$ is assumed a scalar form but in fact is a vector when we consider the actual nanohole array later. The momentum dependent $s_{+}^{\omega }$ is related to $k_{\text{spp}}$ by phase matching condition under empty lattice approximation given as $k_{spp}=\mathrm{Re}\left(\frac{2\pi }{\lambda }\sqrt{\frac{\epsilon }{\epsilon +1}}\right)=k_{\text{//}}+\frac{2n\pi }{P}$, where ε is metal permittivity, P is the period of the nanoslit array and n is integer characterizing Bragg scattering order. As a result, one can solve for a^ω to be: $a^{\omega }=\frac{{\kappa }^{\omega }{s}^{\omega }{}_{+}}{i(k_{//}+2n\pi /P-k_{spp})+(1/2L_D)}$. Under conservation of energy, the outgoing power $s_{-}^{\omega }=C{s}_{+}^{\omega }+{\chi }^{\omega }{a}^{\omega }$, where C is the nonresonant scattering constant and ${\chi }^{\omega }$is the outcoupling constant for SPPs. $s_{-}^{\omega }$ will undergo phase-matching condition $k_{//}=k_{spp}-\frac{2n\pi }{P}$ again for far-field radiation. Therefore, in momentum space at a single frequency, we are able to write k-solved reflectivity as:

We conduct FDTD simulations to verify the CMT result. For simplicity, we choose 1D grating as we believe the underlying physics is essentially identical to 2D cases. Two types of simulations have been performed and their simulation cells are shown in Figs. 1(a) and 1(c). The first type has unit cell with P = 510 nm, groove width and depth = 60 and 40 nm. Bloch boundary condition is used at two sides and perfectly matched layer (PML) is set on the top and at the bottom of the cell [4, 11]. We use a plane wave source to illuminate the grating at different k_// along the Γ-X direction. Therefore, the incident plane is always perpendicular to the slits. The source has wavelength centered at 725 nm and bandwidth = 150 nm to excite only the −1 SPP mode. The specular reflection is calculated by using the near-to-far-field projection method given earlier [4, 11]. On the other hand, the second type involves a supercell with total cell length = 100 μm, which contains 180 grooves. The geometry of each single groove is identical to that of the first cell and PML is applied on the whole structure. Instead of using a plane wave source, total-field/scattered-field approach is used to exclude the input excitation and record only the scattered field. A power line monitor is set at 4 nm above the metal surface to examine the near-field distribution across the entire surface. Assuming the cell is sufficiently large to allow the SPPs to decay, this real space near-field profile directly yields L_D. We first calculate the p-polarized reflectivity as a function of k_// = 2πsinθ/λ at several wavelengths from λ = 710 – 770 nm by using the first cell and the results are shown in Fig. 1(b). They all exhibit Fano-like reflection dips positioning at different k_//. Both ω and k_// match well with the phase-matching equation, revealing −1 SPP mode is excited. By best-fitting the graphs using Eq. (1), L_D is extracted and plotted in Fig. 1(e) against resonant wavelength, showing L_D increases steadily with wavelength from 5 to 9 μm. This increase is expected because longer wavelength experiences weaker radiative scattering and lower Ohmic absorption, thus resulting in longer lifetime and decay length [1,10]. After that, we then examine the spatial decay of SPPs for the corresponding resonant wavelengths based on the second type simulation. The incident momentum for each wavelength is determined from k_// in Fig. 1(b). The intensity profiles in logarithmic scale are shown in Fig. 1(d) and they all display exponential decrease with distance. L_D for different wavelengths are then determined from the inverse of the slopes and plotted in Fig. 1(e) for comparison. Apparently, two results agree very well with each other with discrepancy smaller than 2.3%, confirming the applicability of determining L_D in k-space.

 

Fig. 1 (a) The FDTD unit cell for 1D groove grating. (b) The simulated reflectivity as a function of incident wavevector for different wavelengths indicating (−1,0) SPP mode is excited at the reflection dips. (c) The FDTD supercell for 1D groove grating. (d) The spatial profile of the scattered near field for different wavelengths in logarithmic scale, showing exponentially decrease of intensity. (e) The plot of propagation length against resonant wavelength deduced from two methods.

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3. Testbed results

In this section, we begin with measuring Γ_tot, L_D, and v_g at different resonant wavelengths by using the angle- and polarization-resolved reflectivity spectroscopy. We have fabricated 2D Au nanohole array by interference lithography [4, 10, 11]. Its scanning electron microscopy image is shown in the inset of Fig. 2(a), showing the nanohole array has period P = 510 nm, hole depth and radius = 200 and 50 nm, respectively. After fabrication, the sample is mounted on a computer controlled goniometer for p-polarized angle-dependent reflectivity mapping [10]. For 2D nanohole array, the phase matching equation should be written in vectorial form as:

 

Fig. 2 (a) The p-polarized wavevector-resolved reflectivity mapping of 2D Au nanohole array taken in Γ-X direction. The dashed line is calculated by using phase-matching equation, indicating the excitation of (−1,0) SPP mode. Inset: the SEM image of the nanohole array. The scale bar is 500nm. (b) The corresponding reflectivity spectra for different incident wavevectors. The dashed lines are the best fits. (c) The corresponding reflectivity plots for different photon energies. The dashed lines are the best fits. (d) The deduced group velocities for different resonant wavelength. The dot line is ∂ω/∂k from the mapping.

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4. Real- and Fourier-space microscopy for L_D measurement

We then turn our attention to measure L_D as a function of SPP propagation direction. HeNe 633 nm laser and white light source are used separately for real- and Fourier-space imaging. The schematic of the microscope is shown in Fig. 3(a) [17]. For real-space imaging, the laser light is first coupled into a single-mode fiber before collimated by an achromatic lens. It is then focused onto the back focal plane of a 100X, numerical aperture NA = 0.9 air dry objective lens and finally is output from the lens at well-defined incident and azimuthal angles. The illumination optics is set on a motorized translation stage to vary θ from 0° to 63° with a step size of ~0.2° [17]. The sample is placed on a rotation stage to vary φ from 0° to 360°. A hexagon adjustable iris is positioned at the image plane in the incident path to reduce the beam size so that only part of the sample (i.e. 1/3 of the viewing area) is illuminated. As a result, SPPs could propagate to the un-illuminated region and radiate out to the far-field. The radiation from the sample is collected by the same objective lens and is either spatially imaged by an EMCCD camera or detected by a photodiode. On the other hand, for Fourier-space imaging, the white light is fed on the entire back focal plane so that the sample is fully illuminated at all possible θ and φ as defined by the NA of the objective lens. This configuration excites all allowable SPPs simultaneously with the same $\left|{\stackrel{\rightharpoonup }{k}}_{SPP}\right|$ but different directions as defined by the phase-matching equation (Eq. (2)). The reflections, including both resonant and non-resonant components, are collected and projected onto the Fourier-space image to yield the equi-energy contour in k_x and k_y space [18]. A 633 nm laser line filter, with bandwidth FWHM of 3 ± 0.5 nm, is inserted in the detection path to collect the desired wavelength so that a clear and sharp equi-energy contour can be observed for analysis. The technique is similar to those of Regan et al [19], Angelini et al [20], and Wagner et al [21] by using a leakage radiation microscopy to image the angular emission from 2D nanohole arrays and 3D photonic crystals. However, our technique relies only on the reflections from the sample and no emission from fluorescent dyes is required, which often suffers from photobleaching. As a result, for a particular SPP mode along the equi-energy contour, we measure the linewidths normal to the contour to determine the momentum-dependent L_D.

 

Fig. 3 (a) The schematic setup for real- and Fourier-space microscopy. (b) The plot of reflectivity against incident angle from 2D Au nanohole array for λ = 633 nm obtained from the microscopy. The dashed line is extracted from the goniometer for comparison. Three crosses are located at off- and on- SPP resonance regions labeled as (c)-(e). (c) & (e) The off-resonance real images. The hexagon indicates the region illuminated by light. The solid line indicates the region extracted for examining the spatial decay of light. (d) The on-resonance real image featuring long light tails extending around the hexagon in a particular direction. The arrows define the directions of incident light and SPP propagation. (f) The extracted spatial decay of light from (c)-(e) in logarithmic scale. Only the on-resonance decay shows exponential behavior. (g) The measured propagation length as a function of azimuthal angle.

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In analogy to the FDTD simulation, we use the real-space imaging to determine L_D for (−1,0) SPP mode at several φ. In principle, if a collimated laser beam is incident on the nanohole array at θ and φ according to the phase-matching equation, SPPs propagate at a well-defined direction and subsequently undergo dissipation. The spatial distribution of the leakage radiation can be imaged for L_D measurement [22]. We first identify the desired incident angle for exciting SPPs. Figure 3(b) shows the p-polarized angular reflectivity of the nanohole array at λ = 633 nm taken by CCD-based spectrometer along the Γ-X direction. The reflectivity from Fig. 2(b) is also extracted and superimposed on the plot as the dashed line for comparison. Two agree with each other, indicating (−1,0) SPP mode is excited at θ = 9.3°. We then select three incident angles marked by the crosses in Fig. 3(b), spanning from off- to on-resonance, and capture their images in Figs. 3(c)-3(e) for comparison. Apparently, one sees the image taken at on-resonance is completely different from the off-resonance counterparts. For the off-resonance images in Figs. 3(c) and (e), the hexagons, which are the opening of the iris, are bright due to the strong non-resonant reflection background. In contrast, at on-resonance, the hexagon is dark, corresponding to the low reflection dip (Fig. 3(d)). More importantly, we see from the on-resonance image that the edges of the hexagon are not sharp but featured with long light tails on both sides pointing in one distinct direction [22]. In fact, the direction of the tail indicates the direction where the (−1,0) SPP mode is propagating. In this case, the mode is traveling in the –x direction, which is opposite to the incident direction as indicated by the arrow in Fig. 3(d). The radiation decreases in intensity with increasing distance. The off-resonance edges are sharp and do not show such behavior. We then extract the intensity profiles along the incident plane from the figures and show them in Fig. 3(f) in logarithmic scale. We see only the resonance decay curve exhibits a single exponential decrease with distance and the slope yields L_D = 3 μm. We then repeat the same procedure for azimuthal angles φ = 10° and 20° and their corresponding L_D for (−1,0) SPPs are plotted in Fig. 3(g). We find L_D varies weakly with φ.

For Fourier-space imaging, an unpolarized light is incident on the nanohole array and the reflection image at λ = 633 nm is shown in Fig. 4(a) displaying four low reflection arcs at well-defined angles [14]. The nanohole array is rotated such that its Γ-X direction is parallel with k_x. A four-fold symmetry is expected as the nanohole array is square lattice. The background is due to the non-resonant reflection whereas the arcs arise from the excitation of SPPs. In fact, they can be traced by the phase-matching equation and the calculated curves are shown in Fig. 4(a) as solid lines identifying (1,0), (−1,0), (0,1), and (0,-1) as well as (1,1), (−1,1), (1,-1), and (−1,-1) SPP modes are excited [10, 11, 14]. Wood’s anomalies are also observed and plotted as dashed lines [23]. In fact, the image defines the plasmonic equi-energy contour with the first Brillouin zone given as the dot square with length = 12.32 μm⁻¹ [24]. In analogy to electronic crystal, owing to the square lattice, only 1/8 of the Brillouin zone is required with the reduced Brillouin zone defined by the triangle which indicates the Γ-X-M region [25]. Therefore, the decay lengths measured in this region are sufficient to represent the entire system. To confirm our measurement, we also have mapped the equi-energy contour by using the angle-resolved goniometry and the region taken at λ = 633 nm is extracted in Fig. 4(b) for comparison. In fact, two contours are very similar and they share common features.

 

Fig. 4 (a) The unpolarized Fourier reflection image taken at λ = 633 nm from 2D Au nanohole array taken by microscopy. The solid and dashed lines indicate the SPP modes and Wood’s anomalies calculated from the phase-matching equations. The dot line defines the first Brillouin zone together with the reduced zone given by Γ-X-M triangle. (b) The unpolarized k-space mapping at λ = 633 nm from 2D Au nanohole array taken by angle-resolved goniometry. (c) Several (−1,0) SPP cuts taken at different azimuthal angles are extracted from the unpolarized Fourier image. The dashed lines are the best-fits for determining the corresponding propagation lengths. (d) The deduced (■) propagation length as a function of azimuthal angle. The positions of dark and bright modes and the Wood’s anomaly are also indicated. The (●) L_D obtained from the real-space imaging in Fig. 3(g) is overlaid for comparison.

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When tracing along the (−1,0) SPP mode, we find the reflectivity contrast changes noticeably. In particular, we note from Figs. 4(a) and 4(b) that at the point where (−1,0) and (0,-1) SPP modes cross, a small plasmonic gap is found at $\stackrel{\rightharpoonup }{k}=\left(0.2\widehat{x}+0.2\widehat{y}\right)k_0$ in which propagation of SPPs is forbidden. At the same time, two new coupled modes are formed at different $\stackrel{\rightharpoonup }{k}$ vectors along the Γ-M direction [9, 26]. In fact, this hybridized pair signifies the formation of dark and bright modes [9, 26, 27]. Specifically, dark mode is weakly radiative whereas bright mode is highly radiative and they thus have different damping rates as well as propagation lengths. Therefore, the contrast clearly identifies the dark and bright modes are located at $\stackrel{\rightharpoonup }{k}=\left(0.213\widehat{x}+0.213\widehat{y}\right)k_0$and$\left(0.174\widehat{x}+0.174\widehat{y}\right)k_0$, respectively. One also sees the reflectivity changes quite abruptly around the Wood’s anomalies in which the number of diffraction orders changes [23]. All these observations suggest that L_D is not a constant with φ but changes continuously along the (−1,0) SPP resonance. We then evaluate L_D as a function of φ or ${\stackrel{\rightharpoonup }{k}}_{SPP}$ by extracting cuts perpendicular to the (−1,0) SPP contour at different azimuthal angles and several of them are displayed in Fig. 4(c). We fit them by using Eq. (2) and plot L_D against φ in Fig. 4(d). The L_D obtained from the real-space imaging in Fig. 3(g) is also overlaid for comparison. From the plot, one sees although L_D remains almost unchanged for small azimuthal angles, it gradually decreases when approaching to the band gap region, yielding 2.77 and 1.33 μm for the dark and bright modes, respectively. L_D increases again afterwards.

5. Dependences of Γ_tot and v_g on SPP propagation direction

Before the determination of k-dependent v_g, we measure the corresponding Γ_tot. Again, we measure the Γ_tot of (−1,0) SPP mode at λ = 633 nm by using goniometer. For different φ, we extract the reflectivity spectra at the corresponding θ and fit them by temporal CMT to determine Γ_tot [11] and the results are plotted in Fig. 5(a). We see Γ_tot varies strongly with SPP propagation direction. It remains almost constant at ~40-44 meV between φ = 0° - 30°, splits into two bright and dark values of 56 and 37 meV when under coupling at φ = 45°, and finally increases again at larger φ. To explain these variations, we find from φ = 0° - 30° that ${\stackrel{\rightharpoonup }{k}}_{SPP}$ varies from $-1.067\widehat{x}{k}_0$ to $\left(-1.063\widehat{x}+0.1\widehat{y}\right)k_0$ indicating the propagation direction of (−1,0) SPPs rotates only by 5.4° from the Γ-X direction. Such small directional change does not alter the damping of SPPs much and thus leads to an almost constant Γ_tot. However, when φ approaches to 45°, two (−1,0) and (0,-1) SPP modes begin to interfere and the resulting standing waves significantly alter the propagation direction. More importantly, the formation of dark and bright modes gives rise to strong modification of the decay rates, with the rates of dark and bright modes lower and higher than the nondegenerate counterparts. In fact, the splitting can be explained within the framework of CMT. Consider two SPPs with mode amplitudes a₁ and a₂, their time variants can be written as [9, 28]:

 

Fig. 5 (a) The experimental total decay rate Γ_tot of (−1,0) SPPs plotted as a function of azimuthal angle φ. The solid line indicates the emergence of Wood’s anomaly. (b) The FDTD simulated (▲) total decay Γ_tot, (■) radiative decay Γ_rad, and (●) absorption Γ_abs rates of (−1,0) SPPs as a function of azimuthal angle φ. At φ = 45°, two (−1,0) and (0,-1) SPP modes couple, leading to the splitting of the rates.

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After discussing the decay rates, we move on to determine v_g = L_DΓ_tot and the results are shown in Fig. 6(a). We see v_g gradually decreases from 0.7c to 0.6c from φ = 0° - 30° as the SPP propagation direction remains almost unchanged. However, above 30° v_g decreases dramatically as the standing wave begins to take off. More interestingly is at the Γ-M direction where two dark and bright modes form, the dark mode has higher v_g despite its lower Γ_tot. To confirm it, we display in Fig. 6(b) the unpolarized incident momentum-resolved reflectivity mapping taken at φ = 45°. The dark and bright modes are clearly seen and their dispersions are weakly dependent on momentum. As a result, two modes are expected to have slow group velocities compared to the (−1,0) nondegenerate SPPs. The v_g of dark and bright modes are then determined to be 0.43c and 0.47c, respectively, which are consistent with the k-space measurement.

 

Fig. 6 (a) The plots of v_g of (−1,0) SPPs as a function of azimuthal angle φ. At φ = 45°, two (−1,0) and (0,-1) SPP modes couple, leading to the splitting of v_g. (b) The unpolarized incident angle-resolved reflectivity mapping taken at φ = 45°, indicating the presence of dark and bright modes. The dispersions are weak.

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To address the distinct v_g for the dark and bright modes, we use FDTD to simulate the plane-view field patterns superimposed with the Poynting vector maps of the bright and dark modes in Figs. 7(a) and 7(b). The Poynting vectors unambiguously indicate the energy flows in the Γ-M direction for two cases. From the field patterns, the crests and troughs do not represent the wavefronts of a traveling plane wave but the profile of a standing wave. While two coupled SPP coupled modes travel in the Γ-M direction, they are standing perpendicular to it. In fact, two modes display different field symmetry in the y-direction. The dark mode is asymmetric whereas the bright mode is symmetric [9, 29]. This difference results in different field localizations. As the fields of the dark mode are more localized near the air hole region, it produces lower n_eff, which gives rise to higher v_g. On the other hand, the bright mode has larger n_eff and thus lower v_g.

 

Fig. 7 The plane-view field patterns |E| and the Poynting vector maps for (a) bright and (b) dark modes simulated by FDTD for λ = 633 nm at φ = 45°.

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6. Conclusion

In summary, we have employed angle- and polarization-resolved reflectivity spectroscopy and real- and Fourier-space microscopy to study the momentum-dependent Γ_tot and L_D, and v_g, of Bloch-like SPPs in 2D Au nanohole array. The Γ_tot has been determined by reflectivity spectroscopy at different SPP propagating direction. On the other hand, by using spatial coupled mode theory and FDTD simulations, we show the L_D is associated with the linewidth of the reflectivity profile in Fourier-space. Therefore, by mapping out the equi-energy contour for a given wavelength, the momentum-deponent L_D can be determined accordingly. We find both the decay rate and length are strongly dependent on the propagation direction of SPPs. In particular, they are strongly modified when two degenerate modes undergoing coherent coupling, yielding nontrivial and distinct v_g for two dark and bright modes. We believe v_g is mostly governed by n_eff. Dark and bright modes experience different n_eff and thus v_g due to their field symmetry.