1. Introduction

When two optical systems are coupled together, their near- and/or far-fields can interact, manifesting the formation of band gap and a pair of new coupled states [1,2]. In addition, the decay rates of the coupled modes are also modified [3,4]. One such example is that their radiation damping could be enhanced and suppressed, resulting in the so-called bright/superradiant and dark/subradiant modes [3–5]. Specifically, bright mode is highly radiative whereas dark mode is weakly or sometimes totally nonradiative [3–5]. Due to the small radiation loss, dark mode exhibits narrow lineshape and high quality Q factor and is applicable in storing energy for waveguiding, plasmonic lasing, etc [6–10]. Likewise, bright mode is particularly useful in surface-enhanced Raman scattering (SERS) [11], surface plasmon mediated fluorescence [12], etc, where strong radiation decay from the resonant mode to far-field is required. Furthermore, the pair can complement with each other in which bright mode acts as an optical antenna in coupling energy from free space to the resonance while dark mode at the same time serves as a cavity for storing energy [13]. The energy transfer between them thus enables strong field enhancement in dark mode with minimal losses. Other than that, dark and bright modes can further interfere with each other in producing many intriguing phenomena such as Fano resonance [14], electromagnetically induced transparency (EIT) [15] and absorption (EIA) [16], and superscattering [17] that have been receiving much attention in photonics and plasmonics. Both Fano resonance and EIT exhibit many unique properties associated with their lineshape profile and have been used for biosensing due to their narrow linewidth, boosting the detection resolution [18]. Recently, EIT has also been exploited to render both strong field and high transparency, giving rise to large emission enhancement [19].

It is noted that for dissipative coupled system, not only the radiative decay rate but also the absorption rate should be modified as well. Bright and dark modes could be either more absorbing or less after coupling. For example, in free-standing metal thin film where both the top and bottom air/metal surface plasmon polaritons (SPPs) are identical, if the film thickness is sufficiently small to allow two degenerate SPPs to interact, asymmetric and symmetric modes are formed [20]. Since the asymmetric mode has smaller in-plane wavevector, its fields penetrate in the metal film at a smaller extent, thus resulting in smaller Ohmic absorption loss. This asymmetric mode, which is also known as long range SPPs [21], has found applications in waveguiding and biosensing due to its long propagation distance on metal surface [22]. On the other hand, the symmetric mode has deeper field penetration into the metal, yielding a higher absorption rate [20]. As a result, one would expect for other plasmonic systems both the absorption and radiative decay rates are modified simultaneously when under coherent coupling. However, other than thin films, no such measurements have been made on other systems due to the difficulty in differentiating two decay rates quantitatively.

In fact, measuring both the absorption and radiative decay rates in plasmonic systems is not a trivial task. The radiation damping of SPPs is usually addressed. For example, the spectrum acquired by the dark-field optical microscopy contains only the scattering loss with no information of the absorption loss provided [23]. Similarly, the leakage radiation microscopy (LRM) images only the radiative decay of SPPs after coupling energy from fluorescence dyes [24]. Only a few experiments attempt to study the absorption and scattering losses individually. Inagaki et al have used angle-resolved photoacoustic spectroscopy to selectively excite propagating SPPs and study their absorption in thin films and one-dimensional gratings [25]. Chang et al have imaged the scattering and absorption of localized SPPs arising from nanoparticle rings by using dark-field and photothermal microscopes and find the coupled nanoparticles do exhibit lower absorption loss when compared with that of isolated nanoparticle [26]. Husnik et al combine spatial modulation technique and common path interferometry to measure the absolute absorption and scattering cross-sections of metallic nanoantennas individually and the results compare favorably with theory [27].

Periodic metallic arrays are one of the most important plasmonic systems other than thin films and nanoparticles [28]. They can be fabricated by currently available lithographic methods to precisely tailor their lattice and basis with subwavelength resolution and thus are expected to produce more controllable SPPs and reliable performance. Periodic arrays have been widely used in making SERS [11,29] and surface plasmon resonance (SPR) biosensors [30], electrodes for enhancing emission from light-emitting diodes [12,31] and absorption in solar cells [32], optical tweezers [33], wave plates [34], etc. The plasmonic properties of periodic arrays are largely dependent on their decay rates. In particular, the electromagnetic field enhancement, scattering, and SPP phase change all depend on the interplay between the absorption and radiative decay rates [35,36]. Therefore, how one can determine two rates has become an important issue. We recently have employed temporal coupled mode theory (CMT) to measure the absorption and radiative decay rates of nondegenerate SPPs from nanohole arrays by orthogonal, or cross-polarized, reflectivity spectroscopy [37]. However, extension to degenerate or coupled SPP modes is not yet achieved although the knowledge of the modified absorption and radiative decay rates are essential in understanding and engineering of the coupling of SPPs.

In this work, we study the absorption and radiative decay rates of two degenerate SPP modes at and near coupling in periodic array. A functional form based on CMT has been developed to measure the decay rates of the coupled modes from reflectivity. We then study 1D Au gratings with different groove widths. It is found that not only the radiative decay rates of two modes are enhanced and suppressed to yield the bright and dark modes, but the absorption rates are modified as well, giving rise to “cold” and “hot” modes. In particular, for small groove widths, the dark mode becomes less absorbing whereas the bright mode tends to be more dissipative. However, the situation is upturned when the width increases. Our CMT and finite-difference time-domain (FDTD) analyses show that the real and imaginary parts of the coupling constant play a key role in determining the relative position and the absorption rates of the pair. They are then further examined by studying their field patterns. Since the interference of two counter-propagating SPPs yields two different standing waves, they exhibit different electric and magnetic field symmetries, surface charge distributions, and energy flows. We find different groove widths affect the surface charge distribution as well as the Poynting vector mapping significantly, rendering different extents of Ohmic absorption and radiation damping. In the end, we have measured the decay rates and the complex coupling constant of −1 and 1 degenerate SPP modes in Au grating by angle- and polarization-resolved reflectivity spectroscopy as a demonstration.

2. Coupled mode theory formulation

2.1 Coupling of two SPP modes

We first formulate the coupling of two SPP modes by using CMT. In particular, we derive the functional form of the far-field reflection spectrum which can be explicitly expressed in terms of the modified absorption and radiative decay rates. As an illustration, we consider an optically thick 1D periodic system [35]. Assume two SPP modes 1,2 are characterized by two complex eigenfrequencies ${\tilde{\omega }}_{1,2}={\omega }_{1,2}+i{\Gamma }_{\text{1},\text{2}}^{tot}/2$, where ${\omega }_{1,2}$ and ${\Gamma }_{\text{1},\text{2}}^{tot}$ define the resonant frequencies and the total decay rates. A 1/2 is introduced in the imaginary part to stand for the decay of energy but not field. For a dissipative system, ${\Gamma }_{\text{1},\text{2}}^{tot}$ is the summation of the absorption and radiative decay rates of SPPs given as ${\Gamma }_{1,2}^{abs}+{\Gamma }_{1,2}^{rad}$. We assume the system has a single input/output port for simplification but extension to multiple ports is possible after some minor modifications [38,39]. Two SPPs are excited simultaneously by a p-polarized light with the incident plane normal to the grooves. The incident angle θ is governed by the phase-matching equation given as${\left(\frac{2\pi }{\lambda }\sin \theta +m\frac{2\pi }{P}\right)}^2={\left(\frac{2\pi }{\lambda }\right)}^2\frac{{\epsilon }_m}{{\epsilon }_m+1}$, where ε_m is the dielectric constant of metal, P is the period of the array, and m is an integer defining the Bragg scattering [3]. Under time reversal symmetry, the time variants of the SPP mode amplitudes a_1,2 can be written in the matrix form as [38,39]:

2.2 Coordinate transformation

We notice that Eqs. (1) and (3) are expressed in terms of ${\tilde{\omega }}_{1,2}$, which will be modified when under coupling. Therefore, one needs to transform them into functions that explicitly contain the modified frequencies and rates. In other words, a transformation matrix T is required so that Eq. (1) can be rewritten as:

To find T for transformation, we need to determine the eigen values and vectors from the homogeneous counterpart of Eq. (2) [41]. The determinant of the homogeneous equation yields the complex eigenvalues ${\tilde{\omega }}_{+,-}$ are given as:

We then perform the coordinate transformation once the eigen values and vectors are ready. It is known that if Ω is diagonalizable into $T^{-1}\Omega T=\tilde{\Omega }.$, where $\tilde{\Omega }$ is displayed as $\left[\begin{array}{cc}i{\tilde{\omega }}_{\text{+}}& \text{0}\\ \text{0}& i{\tilde{\omega }}_{-}\end{array}\right]$, T can be expressed as $c\left[\begin{array}{cc}{\tilde{\omega }}_{+}-{\tilde{\omega }}_2& {\tilde{\omega }}_0\\ {\tilde{\omega }}_0& {\tilde{\omega }}_{-}-{\tilde{\omega }}_1\end{array}\right]$ and $T^{-1}=T^T$ provided that Ω is symmetric and each of the eigenvectors of Ω has an orthonormal basis [41]. Equations (2) and (3) can then be operated as $T^{-1}\left(i\omega I-\Omega \right)T{T}^{-1}A=T^{-1}B{s}_{+}$ and $s_{-}=r{s}_{+}+{\left(T^{-1}C\right)}^T{T}^{-1}A$, respectively. We define $\tilde{A}=T^{-1}A$, $\tilde{B}=T^{-1}B$, and $\tilde{C}=T^{-1}C$ and make use of $\Omega =T\tilde{\Omega }{T}^{-1}$, the equations can then be rewritten as $\left(i\omega I-\tilde{\Omega }\right)\tilde{A}=\tilde{B}{s}_{+}$ and $s_{-}=r{s}_{+}+i{\tilde{C}}^T\tilde{A}$. One sees the first equation is identical to Eq. (4) given that $\tilde{A}=\left[\begin{array}{c}{a}_{+}\\ a_{-}\end{array}\right]$ and $\tilde{B}=\left[\begin{array}{c}\sqrt{{\Gamma }_{+}^{rad}}{e}^{i{\phi }_{+}}\\ \sqrt{{\Gamma }_{-}^{rad}}{e}^{i{\phi }_{-}}\end{array}\right]$. As a result, the p-polarized reflectivity is now given as:

2.3 Analytical simulations

We first demonstrate the feasibility of using Eq. (7) to determine the modified decay rates at on and off degeneracy as an illustration. We calculate various R_p spectra based on synthetic uncoupled parameters given as: ${\tilde{\omega }}_1=(1.5-k_{\parallel }\times {10}^{-5})+0.005ieV$ and ${\tilde{\omega }}_2=(1.5+k_{\parallel }\times {10}^{-5})+0.005ieV$, ${\Gamma }_{1,2}^{rad}=0.004eV$, and ${\tilde{\omega }}_o=-0.03+0.003ieV$, where k_|| is the in-plane k-vector varying from 0 to 6 ( × 10³) m⁻¹. All ${\omega }_{1,2}$, ${\Gamma }_{1,2}^{rad}$, ${\Gamma }_{1,2}^{abs}$, and ${\Gamma }_{1,2}^{tot}$ are plotted against k_|| in Fig. 1(a)-1(d) for illustration. One sees from Fig. 1(a) that we attempt to mimic the crossing of two m = −1 and 1 Bloch modes in 1D grating, covering the off-degeneracy when k_|| > 4 ( × 10³) m⁻¹, near degeneracy at 1 ( × 10³) m⁻¹ < k_|| ≤ 4 ( × 10³) m⁻¹, and the on-degeneracy when k_|| = 0 [3]. Following Ref [38,40], r_p ≈1 and φ_1,2 + δ_1,2 ≈π, the R_p spectra at several k_|| are then calculated by using Eq. (3) and displayed in Fig. 1(e). Apparently, two reflection dips are observed and they are far apart when k_|| is large but approach to each other when k_|| decreases, indicating the transition from off to on degeneracy occurs. In addition, their lineshapes change significantly with the high energy dip becoming smaller and narrower while the low energy dip becomes broadened and larger, implying strong modification in their decay rates. The high energy dip eventually disappears when two modes are degenerate at k_|| = 0.

 

Fig. 1 Plots of the synthetic (a) ${\omega }_{1,2}$, (b) ${\Gamma }_{1,2}^{rad}$, (c) ${\Gamma }_{1,2}^{abs}$, and (d) ${\Gamma }_{1,2}^{tot}$ (solid symbols ■ and ●) of two modes as a function of k_|| to mimic the coupling of −1 and 1 Bloch SPP modes in 1D grating. The CMT best-fitted (a) ${\omega }_{+,-}$, (b) ${\Gamma }_{+,-}^{rad}$, (c) ${\Gamma }_{+,-}^{abs}$, and (d) ${\Gamma }_{+,-}^{tot}$ (open symbols ∇ and Δ) are also overlaid for comparison. The ${\omega }_{+,-}^h$ and ${\Gamma }_{+,-}^h$calculated by solving the homogeneous equation are also plotted (crossed symbols + and × ). (e) The calculated reflectivity spectra (symbols) by using synthetic parameters

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We then fit the spectra by using Eq. (7) to extract the modified frequencies and rates. The best-fits are shown as the dash lines in Fig. 1(e) and the fitting parameters are plotted in Fig. 1(a)-1(d) for comparison. We see in Fig. 1(a) that the modified frequencies are almost identical with their original counterparts when k_|| is large but begin to deviate near degeneracy. More importantly, two modes level off when coupled, resulting in the formation of band gap. Likewise, the absorption and radiative decay rates in Fig. 1(b) and 1(c) are strongly modified. The radiative decay rates of the high and low energy modes are suppressed and enhanced, yielding the dark and bright modes. The dark mode is completely nonradiative at k_|| = 0, as expected from the coherent coupling of two identical modes. In fact, at degeneracy where ${\tilde{\omega }}_1={\tilde{\omega }}_2=\tilde{\omega }=\omega +i{\Gamma }^{tot}/2$ such that ${\tilde{\omega }}_{+,-}=\tilde{\omega }\pm {\tilde{\omega }}_o$, we see from $T^{-1}B$ that $\left[\begin{array}{c}\sqrt{{\Gamma }_{+}^{rad}}{e}^{i{\phi }_{+}}\\ \sqrt{{\Gamma }_{-}^{rad}}{e}^{i{\phi }_{-}}\end{array}\right]=\frac{\sqrt{2}}{2}\left[\begin{array}{cc}1& 1\\ 1& -1\end{array}\right]\left[\begin{array}{c}\sqrt{{\Gamma }_{\text{1}}^{rad}}{e}^{i{\phi }_1}\\ \sqrt{{\Gamma }_2^{rad}}{e}^{i{\phi }_2}\end{array}\right]=\left[\begin{array}{c}\sqrt{2}\sqrt{{\Gamma }_{}^{rad}}\\ 0\end{array}\right]$, indicating the ${\omega }_{-}$ mode, which in this case is at higher energy position, will be completely dark whereas the lower energy ${\omega }_{+}$ mode will double its original radiative decay rate. Other than that, the absorption rates are modified as well. Figure 1(c) indicates the dark mode becomes less absorbing while the bright mode is more dissipative. In other words, the dark mode is “cold” but the bright mode is “hot”. The modification of the absorption rates can be understood that the total rates are ${\Gamma }_{+,-}^{tot}={\Gamma }_{}^{tot}\pm {\omega }_o"$, the absorption rates thus are $\left[\begin{array}{c}{\Gamma }_{+}^{abs}\\ {\Gamma }_{-}^{abs}\end{array}\right]=\left[\begin{array}{c}{\Gamma }_{}^{abs}-\left({\Gamma }_{}^{rad}-{\omega }_o"\right)\\ {\Gamma }_{}^{abs}+\left({\Gamma }_{}^{rad}-{\omega }_o"\right)\end{array}\right]$. Therefore, for the dark mode, if ${\Gamma }_{}^{rad}-{\omega }_o"<0$ then ${\Gamma }_{-}^{abs}<{\Gamma }_{}^{abs}$, leading to smaller absorption loss. Finally, we plot the eigen frequencies ${\omega }_{+,-}^h$and total rates ${\Gamma }_{+,-}^h$ in Fig. 1(a) and 1(d) by solving Eq. (5) without involving any fitting and the results agree very well with the fitted results, validating our approach.

From the above example, we learn that the complex coupling constant ${\tilde{\omega }}_o={\omega }_o\text{'}+i{\omega }_o"/2$ is important in determining the behaviors of the coupled modes. From CMT, the ${\tilde{\omega }}_{+}$ and ${\tilde{\omega }}_{-}$ modes are always bright and dark but their relative spectral positions depend on the sign of ${\omega }_o\text{'}$ given as ${\omega }_{\pm }=\omega \pm {\omega }_o\text{'}$. Positive ${\omega }_o\text{'}$ yields high energy bright mode whereas negative one produces high energy dark mode. The size of the band gap always equals to $\left|{\omega }_{+}-{\omega }_{-}\right|=\left|2{\omega }_o\text{'}\right|$ [5]. On the other hand, the difference between the original radiative decay rate and ${\omega }_o"$ governs the resulting absorption rates. A bright and dark modes can be “cold” and “hot”, which are opposite to the above example, if ${\Gamma }_{}^{rad}-{\omega }_o"$ > 0. However, since ${\Gamma }_{+}^{abs}+{\Gamma }_{-}^{abs}=2{\Gamma }_{}^{abs}$, which is a constant, two modes cannot be both “hot” or “cold” at the same time. As a result, by manipulating ${\tilde{\omega }}_o$, one could control the dark and bright modes. Nevertheless, ${\tilde{\omega }}_o$ has been reported to be strongly dependent on geometry [3,5,42].

3. Numerical and experimental results

3.1 Finite-different time-domain simulations

We then move on to verify CMT by FDTD. We conduct simulations on a 1D Au groove array with period P = 750 nm, groove depth D = 40 nm and width W = 200 nm. The unit cell is given in Fig. 2(a) and the details of the simulation have been described previously [35,37]. Figure 2(b) shows the simulated in k_||-resolved p-polarized reflectivity mapping taken in the direction where the incident plane is normal to the grooves. The dispersive reflection dips indicate the excitation of + 1 and −1 SPPs as confirmed by the phase-matching equation (see the solid lines). The Wood’s anomalies are also observed as the dash lines. As we see near normal incidence, both + 1 and −1 SPPs lie below the ± 1 Wood’s anomalies, indicating the absence of higher order diffractions and only the specular reflections are present for both SPPs. Therefore, under coupling, the single port assumption in CMT remains valid and Eq. (7) applies. We extract the reflectivity spectra at different k_|| in Fig. 2(c) and they exhibit similar features as Fig. 1(e) in which a band gap and a pair of dark and bright modes are formed. Once the spectra are ready, we then best fit them to determine ${\Gamma }_{+,-}^{abs}$ and ${\Gamma }_{+,-}^{rad}$ and the results are plotted in Fig. 2(d) and 2(e) against k_||. They agree with the analytical predictions that both the absorption and radiative decay rates are modified when approaching to coupling. The high energy mode is totally nonradiative whereas the low energy mode becomes more and more radiative, forming the dark and bright modes. For the absorption rates, they slowly increase as k_|| decreases due to the nonconstant dependence of their original counterparts on wavelength, they reach two well separated values at degeneracy, signifying the dark mode is “cold” and the bright mode is “hot”.

 

Fig. 2 (a) The cross-section view of the FDTD unit cell. Bloch boundary condition is used at two sides and perfectly match layer is set on the top and bottom. (b) The contour plot of the simulated p-polarized reflectivity spectrum mapping taken at different in-plane k-vector k_||. The solid lines show the + 1 and −1 SPP modes calculated by the phase-matching equation. The Wood’s anomalies are also shown as the dash lines. Both the dark and bright modes lie under the Wood’s anomalies. (c) Several reflectivity spectra (symbols) at different k_|| ( × 10³ m⁻¹) are extracted together with their CMT best-fits (dash lines). They are vertically shifted for visualization. The plots of best-fitted (d) ${\Gamma }_{+,-}^{abs}$ and (e) ${\Gamma }_{+,-}^{rad}$ as a function of k_|| for dark (●) and bright (■) modes. The ${\Gamma }_{+,-}^{abs,t}$ and ${\Gamma }_{+,-}^{rad,t}$ for dark (Δ) and bright (∇) determined by the FDTD time-domain method are also shown for comparison. The transients of the (f) bright and (g) dark modes taken at different in-plane k-vector k_|| ( × 10³ m⁻¹), showing exponential decays. The linear fits are given as dash lines.

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To double confirm our results, we perform time-domain method without involving CMT to determine the rates of two modes independently [35,37]. First, we calculate Γ_tot of the pair individually by placing the time monitor close to the metal surface to record the transient of the electric field intensity. Γ_tot can then be determined by extracting the slope from the decay curve. Since two modes are very close in frequency, we select the excitation source with a very narrow bandwidth down to 5 THz to avoid double excitation. A few examples for the transients of the bright and dark modes taken at different k_|| are illustrated in Fig. 2(f) and 2(g) for reference and they all exhibit single exponential decay behavior, revealing single mode excitation. For calculating Γ_rad, we construct several new dielectric functions of Au by reducing the imaginary part of Au to 50% and 20% so that the absorption of Au can be progressively decreased. The simulated decay rates in these cases thus have the same radiative decay rate but different contributions from the absorption loss. The radiative decay rate then is determined by extrapolating from the intercept of the plot of decay rate against the reduced imaginary part. The ${\Gamma }^{rad,t}$and ${\Gamma }^{abs,t}$ of two modes are plotted in Fig. 2(d) and 2(e) for comparison with the CMT method. Except the first (0 × 10³ m⁻¹) and last (85 × 10³ m⁻¹) data of ${\Gamma }_{-}^{rad}$, others agree with each other very well with discrepancies within 10%.

We study the effect of geometry on the rate modifications of the hybridized modes. Figure 3(a) shows the reflectivity spectra calculated at k_|| = 14 × 10³ m⁻¹ for gratings with the same period and depth but different groove widths W varying from 200 to 550 nm. They are then fitted by Eq. (7) and the obtained ${\omega }_{\pm }$, ${\Gamma }_{\pm }^{rad}$, and ${\Gamma }_{\pm }^{abs}$ and the calculated ${\tilde{\omega }}_o$ are plotted in Fig. 3(b)-3(e). In fact, one sees a strong correlation between the properties of the hybridized modes, the sign and magnitude of ${\tilde{\omega }}_o$, and the groove width. When the width increases from 200 to 350 nm, from Fig. 3(a) and 3(b), the size of the band gap diminishes, showing a reduction of the absolute value of ${\omega }_o\text{'}$ as indicated in Fig. 3(e). In fact, at W around 400 nm, we foresee ${\omega }_o\text{'}$ becomes zero, resulting in the disappearance of the band gap and the merging of two modes together. The ${\omega }_o\text{'}=0$ could result from the energies concentrated in both high and low dielectric constant regions of the grating are almost the same [43]. Beyond 400 nm, ${\omega }_o\text{'}$ increases again and the band gap reopens. However, as ${\omega }_o\text{'}$ changes sign from negative to positive, the dark mode is at higher energy position for smaller widths but red shifts below the bright mode for width larger than 400 nm. Consistent with our CMT prediction, as shown in Fig. 3(c), the ${\tilde{\omega }}_{-}$ mode is always completely dark regardless of the sign and magnitude of ${\tilde{\omega }}_o$. The bright mode shows a noticeable dependence on the width with the radiative decay rate peaks at ~W = 325 nm. The absorption rate of two modes also change quite substantially with the width due to the strong variation of ${\omega }_o"$. Although ${\omega }_o"$ remains as positive across the entire range, it reaches 14 meV at 300 nm but decreases rapidly to ~0.5 meV at 550 nm. By making use of ${\Gamma }_{+}^{rad}=2{\Gamma }_{}^{rad}$ for evaluating the original ${\Gamma }_{}^{rad}$, we find the drastic change of ${\omega }_o"$leads to ${\Gamma }_{}^{rad}-{\omega }_o"<0$ for smaller widths, i.e. “hot” dark mode, but becomes positive when width increases above 400 nm, forming “cold” dark mode.

 

Fig. 3 (a) The FDTD simulated reflectivity spectra of 1D Au gratings with different groove widths taken at k_|| = 14 ( × 10³ m⁻¹) (symbols). They are vertically shifted for visualization. The CMT best-fits are given as the dash lines. Inset: the zoomed reflectivity spectra of the dark mode. The obtained (b) ${\omega }_{+,-}$, (c) ${\Gamma }_{+,-}^{rad}$, and (d) ${\Gamma }_{+,-}^{abs}$, for the bright (■) and dark (●) modes as a function of groove width. (e) The plot of the calculated real and imaginary parts of the complex coupling constant as a function of groove width.

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To further explain the rate modifications, we examine the near field patterns of the hybridized modes. Physically, the hybridization arises from the interference between two counter-propagating SPP waves traveling in ± x directions [44]. This superposition results in two standing waves corresponding to the dark and bright modes. Therefore, as an approximation, the in-plane magnetic fields $\stackrel{\rightharpoonup }{H}$ of the bright and dark modes can be expressed as the sum of two SPP waves $A_d{e}^{-k_{d}z}{e}^{i2\pi x/P}\widehat{y}$ and$B_d{e}^{-k_{d}z}{e}^{-i2\pi x/P}\widehat{y}$ in the dielectrics and $A_m{e}^{k_{m}z}{e}^{i2\pi x/P}\widehat{y}$ and $B_m{e}^{k_{m}z}{e}^{-i2\pi x/P}\widehat{y}$ in the metal, where A_d,m and B_d,m are the amplitude constants, ε_d and ε_m and k_d and k_m are the dielectric constants and the penetration constants for the dielectrics and metal. Their sum can be reorganized as: $A_d{e}^{-k_{d}z}{e}^{i2\pi x/P}\widehat{y}+B_d{e}^{-k_{d}z}{e}^{-i2\pi x/P}\widehat{y}=\left(A_d-B_d\right)i{e}^{-k_{d}z}\sin \frac{2\pi x}{P}\widehat{y}+\left(A_d+B_d\right)e^{-k_{d}z}\cos \frac{2\pi x}{P}\widehat{y}$ in the dielectric and $A_m{e}^{k_{m}z}{e}^{i2\pi x/P}\widehat{y}+B_m{e}^{k_{m}z}{e}^{-i2\pi x/P}\widehat{y}=\left(A_m-B_m\right)i{e}^{k_{m}z}\sin \frac{2\pi x}{P}\widehat{y}+\left(A_m+B_m\right)e^{k_{m}z}\cos \frac{2\pi x}{P}\widehat{y}$ in the metal, forming two distinctive standing waves $\sin \frac{2\pi x}{P}$ and $\cos \frac{2\pi x}{P}$ with respect to the groove center [43]. The electric fields $\stackrel{\rightharpoonup }{E}$ can then be calculated by$\frac{\nabla \times \stackrel{\rightharpoonup }{H}}{-i\omega \epsilon }$, which gives$\frac{\left(A_d-B_d\right)e^{-k_{d}z}}{-\omega {\epsilon }_d}\left(k_d\sin \frac{2\pi x}{P}\widehat{x}+\frac{2\pi }{P}\cos \frac{2\pi x}{P}\widehat{z}\right)$and$\frac{\left(A_d+B_d\right)e^{-k_{d}z}}{-i\omega {\epsilon }_d}\left(k_d\cos \frac{2\pi x}{P}\widehat{x}-\frac{2\pi }{P}\sin \frac{2\pi x}{P}\widehat{z}\right)$ as well as $\frac{\left(A_m-B_m\right)e^{k_{m}z}}{-\omega {\epsilon }_m}\left(-k_m\sin \frac{2\pi x}{P}\widehat{x}+\frac{2\pi }{P}\cos \frac{2\pi x}{P}\widehat{z}\right)$ and $\frac{\left(A_m-B_m\right)e^{k_{m}z}}{-i\omega {\epsilon }_m}\left(-k_m\cos \frac{2\pi x}{P}\widehat{x}-\frac{2\pi }{P}\sin \frac{2\pi x}{P}\widehat{z}\right)$ in the dielectrics and metal for the dark and bright modes, respectively. Once the fields and the boundary conditions are available, we estimate the surface charge distribution at z = 0 for the hybridized modes. The charge density for the pair can be calculated from the difference between the normal components of the fields in the air and metal at z = 0 and it shows $\cos \frac{2\pi }{P}x$ and $\sin \frac{2\pi }{P}x$ dependences. Therefore, the charge distribution for the dark mode always possesses mirror symmetry with respect to the groove center while that of the bright modes has inversion symmetry.

Figure 4(a)-4(f) show the FDTD calculated ${\left|E_x\right|}^2$, ${\left|E_z\right|}^2$, and ${\left|H_y\right|}^2$ of the dark and bright modes taken at k_|| = 57 × 10³ m⁻¹ for W = 200 nm. The calculations are not performed at normal incidence since the dark mode is completely nonradiative, making the calculation impossible. In fact, from the figures, the symmetries of the field patterns all agree very well with our predictions. The corresponding surface charge density are then calculated and displayed in Fig. 5(a) after removing the influence from the nonresonant background [45]. It shows, for the bright mode, the charge distribution is asymmetric at the groove edges, revealing strong dipolar character and thus higher radiative decay rate [3]. In contrast, the dark mode is nonradiative due to its almost zero net dipole moment arising from the symmetric charge distribution. We also have calculated the Poynting vector mappings in Fig. 6(a) for two modes. For the dark mode, the Poynting vectors indicate energy flows almost across the groove without much dissipation into it. However, a net flow of energy into the groove is observed for the bright mode along the surface, leading to stronger Ohmic absorption loss. Figure 5(b), 5(c), 6(b) and 6(c) show the surface charge distributions and the Poynting vector mappings of the pair for W = 350 and 550 nm for comparison. We see all dark modes consistently have an overall symmetric charge distribution across the unit cell, i.e. weak dipole moment, but the bright counterparts exhibit asymmetric profiles or are thus strongly dipolar. For the Poynting vectors, the net energy flowing into the groove again determines the absorption rates. For W = 550 nm where the dark mode is has high absorption rate, its net energy flow is now larger than that of the bright mode. Near the surface, we have calculated the net energy flux normalized by corresponding SPPs amplitude in the direction normal to the surface for two modes. For W = 200 nm, the net energy flux normalized by ${\left|a\right|}^2$ are 6.73 × 10⁻¹² s⁻¹ and 1.64 × 10⁻¹¹ s⁻¹ for dark and bright modes, respectively, flowing into the groove. On the other hand, for W = 350 nm, the absorption rates are 1.11 × 10⁻¹¹ s⁻¹ and 1.34 × 10⁻¹¹ s⁻¹ for dark and bright modes and for W = 550 nm, the normalized net energy flux are 3.49 × 10⁻¹¹ s⁻¹ and 6.29 × 10⁻¹² s⁻¹ for dark and bright modes. In fact, from the values, one sees the mode with higher absorption rate has larger net energy flowing into the metal.

 

Fig. 4 The ${\left|E_x\right|}^2$, ${\left|E_z\right|}^2$, and ${\left|H_y\right|}^2$ patterns of (a)-(c) dark and (d)-(f) bright modes taken at λ = 758.7 and 804.2 nm and k_|| = 57 × 10³ m⁻¹. The field patterns display different symmetries with respect to the groove center. All the field patterns are normalized with respect to their own maximum values.

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Fig. 5 The surface charge distributions at the interface of (top) dark and (bottom) bright modes for W = (a) 200 nm, (b) 350 nm, and (c) 550 nm.

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Fig. 6 The Poynting vector maps of (top) dark and (bottom) bright modes for W = (a) 200 nm, (b) 350 nm, and (c) 550 nm.

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3.2 Experiment

Finally, as a demonstration, we have fabricated a 1D Au groove grating by interference lithography [35,37]. The plane-view and cross-section atomic force microscopy images of the grating is given in Fig. 7(a) and its inset, showing it has period = 756 nm, groove depth and width = 30 and 300 nm, respectively.The sample is then mounted on a home-built goniometer for angle- and polarization-resolved reflectivity spectroscopy and the optical setup is shown in Fig. 7(b). A collimated white light from a quartz lamp is used for illumination and a light guide connected to CCD based spectrometer is mounted on a rotation stage to capture the specular reflections. It is noted that traditional goniometer configuration could not measure reflectivity at and near normal incidence since the detection unit will block the incidence light. Therefore, in our setup, two rotation stages are not concentric but are offset. The sample (i.e. grating) is mounted on the first rotation stage whereas the detection unit, the light guide, is mounted on the second. The rotation axes of the two rotation stages are always positioned at the mirror image to each other with respect to the beam splitter. The first dash line D1 aligns with the normal of the sample. As a result, by trigonometry, when light is illuminated on the sample at incident angle θ, the specular reflection will be fully captured by the light guide as long as the rotation arm is placed at 2θ from the dash line D2, which aligns at the zero position of the rotation arm. A pair of polarizer and analyzer is placed in the optical path to ensure only p-polarized excitation and detection. We have measured the divergence of the beam after collimation and it is found to be less than 0.027°. All reflection spectra are calculated by normalizing them with respect to the incident light spectra.

 

Fig. 7 (a) The plane-view atomic force microscopy image of the 1D Au grating. Inset: the corresponding cross-section view. (b) The schematic setup of p-polarized angle-dependent reflectivity measurement. (c) The measured p-polarized angle-resolved reflectivity contour mapping. The solid lines show the + 1 and −1 SPP modes calculated by the phase-matching equation. The Wood’s anomalies are also shown as the dash lines. (d) Several reflectivity spectra (symbols) taken at different incident angles are extracted together with their CMT best-fits (dash lines). They are vertically shifted for visualization. The plots of best-fitted (e) ${\omega }_{+,-}$, (f) ${\Gamma }_{+,-}^{rad}$, and (g) ${\Gamma }_{+,-}^{abs}$, as a function of incident angle for dark (■) and bright (●) modes.

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Figure 7(c) shows the angle-resolved specular reflectivity mapping of the sample where the incident plane in normal to the grooves. Reflection dips are observed typically for the propagating SPP modes and they are consistent with the FDTD simulations. Both + 1 and −1 SPPs are readily identified together with the presence of band gap and two dark and bright modes. Several spectra are extracted and plotted in Fig. 7(d), exhibiting features similar to the analytical and numerical results. In fact, the dark and bright modes become weaker and stronger when the incident angle is approaching to zero. We use Eq. (7) to best fit the spectra to determine ${\omega }_{+,-}$, ${\Gamma }_{\pm }^{rad}$, and ${\Gamma }_{\pm }^{abs}$ and the results are plotted in Fig. 7(e)-7(g). For the modified frequencies, we clearly see the + 1 and −1 SPPs begin to level at θ smaller than 0.2°, indicating the formation of plasmonic band gap. However, at the same time, the dark mode becomes so weak that makes the determination of its frequency impossible. For ${\Gamma }_{\pm }^{rad}$, and ${\Gamma }_{\pm }^{abs}$, we see the rates for + 1 and −1 modes begin to diverge when θ is smaller than 0.5°. In particular, for ${\Gamma }_{\pm }^{rad}$, the lower energy bright mode is more radiative but the dark mode becomes less radiative. After extrapolating the dark mode to θ = 0°, it is completely nonradiative, following very well with the analytical and numerical calculations. The radiative decay rate of the bright mode increases to 1.5 meV. On the other hand, for ${\Gamma }_{\pm }^{abs}$, while the rate of dark mode consistently decreases to ~6 meV when approaching to θ = 0°, that of bright mode increases to 8.8 meV. We also have determined${\tilde{\omega }}_o$ based on our experimental data. Knowing the size of the plasmonic band gap is equal to 5.8 meV whereas the total decay rate difference between the dark and bright modes is 2.2 meV, we estimate ${\tilde{\omega }}_o$ = −0.0029 + i0.0011 eV. It is noted that the experimental ${\tilde{\omega }}_o$ deviates from the theory (for W = 300 nm and D = 40 nm, ${\tilde{\omega }}_o$ = −0.032 + i0.014 eV) considerably primarily due to geometrical imperfections, such as surface roughness, irregular shapes, nonuniformity, etc, of our grating. In fact, instead of a perfect rectangular shape, our grooves are more like trapezoid, as indicated from the AFM cross-section in Fig. 7(a).

4. Conclusion

To summarize, we have deduced a simple analytical method to directly measure the absorption and radiative decay rates of dark and bright SPP modes in periodic system based on temporal coupled mode theory. The analytical results compare favorably with the FDTD simulations. From the study of 1D gratings with different groove widths, we find the dark and bright modes can be at high or lower energy position and more or less absorbing depending on the interplay between the real and imaginary parts of the complex coupling constant. The coupling constant is strongly geometry dependent and has been shown to vary significantly with groove width. The modification of the rates can also be explained by the symmetry of the field patterns. The distinctive symmetries of the electric and magnetic fields of the hybridized modes result in different surface charge density distributions and Poynting vector profiles for the dark and bright modes. The dark mode always has weak dipole moment but the bright mode is strongly dipolar. The net flow of energy into the groove region determines their Ohmic absorption losses. A 1D Au grating has been fabricated and its + 1 and −1 SPP absorption and radiative decay rates have been measured at degeneracy. The experimental results are found to be consistent with the analytical and numerical calculations. As both the absorption and radiative decay rates play a crucial part in controlling many intriguing properties of SPPs, our study provides a direct means for measuring them, which is useful for rationally designing the plasmonic systems for given applications.