1. Introduction

It has been shown that surface plasmon (SP) coupling with a light emitter, including that in a semiconductor quantum well (QW), can enhance the emission efficiency of the light emitter [1–18]. The SP modes involved in dipole coupling include the localized surface plasmons (LSPs) induced on metal nanoparticles for coupling the nearby quantum-dot-like light emitters [1–5]. They also include the surface plasmon polaritons (SPPs) generated on a metal/dielectric grating or rough interface for coupling with the nearby quantum dots or QWs [6–17]. In particular, the grating-assisted SPPs on a metal grating structure for the coupling process and their feasibility of enhancing the emission efficiency of a light emitter or QW have been widely discussed [9–15]. However, with the groove structure in a metal grating, LSPs can also be generated for effective coupling with a nearby light emitter. Numerical studies show that with a properly designed grating structure, an LSP coupling feature may result in more efficient emission enhancement when compared with most grating-assisted SPP features [14,15]. In such an SP coupling process, an LSP or SPP mode is excited by a radiation dipole placed near a metal/dielectric interface through energy transfer from the dipole into the SP mode. After the SP-dipole coupling energy system is built, it can emit photons with the emission efficiency determined by the spatial field (intensity and phase) distribution of the coupled system around the dipole near the metal/dielectric interface, which can be regarded the secondary source distribution for far-field radiation. Besides emission, the energy of the coupled system can be relaxed through other channels, including metal dissipation and SP propagation from the dipole position along the metal/dielectric interface. Understanding the relative importance of those energy relaxation mechanisms is useful for developing an effective coupling system for emission enhancement.

On the other hand, the SP-dipole coupling process can be an effective mechanism for pumping an SP mode in the effort of implementing a system of SP amplification by stimulated emission of radiation (SPASER) [19–22]. In realizing a SPASER system, a metal/dielectric interface nanostructure is also needed for effectively cavity-confining SP energy. It has been shown that a metal/dielectric grating structure can be a good choice for this purpose [23]. In the application of SP-dipole coupling to the implementation of a SPASER system, low emission efficiency is preferred for achieving a high Q value of the effective cavity. For the application either to the emission enhancement of a light-emitting device or to the implementation of a SPASER system, the understanding of those energy relaxation mechanisms of an SP-dipole coupling system is important. For this purpose, time-resolved study is useful for us to calibrate the decay time constants of those energy relaxation mechanisms and to further understand the SP-dipole coupling process.

In this paper, we report the numerical simulation results on the transient behaviors of such an SP-dipole coupling process for understanding the dominating decay mechanisms of different coupling features. Also, the different characteristics between two SP-coupling features of almost the same energy with one in a metal grating structure and the other on the flat metal/dielectric interface can be distinguished. For these goals, the time-response functions of a pulsed radiation dipole in the structures of a metal/dielectric grating interface and a flat interface are evaluated. Based on the transient behaviors, the time constants of various energy decay mechanisms of different SPP and LSP coupling features can be computed. In Section 2 of this paper, the problem geometry and parameters are defined. Also, the used numerical technique for simulation is briefly reviewed. Then, in Section 3, the SP-dipole coupling behaviors in the spectral domain are discussed. Next, the time-resolved phenomena of various SP coupling features are presented in Section 4. The evaluated decay time constants are also reported in this section. Based on the transient behaviors described in Section 4, the major decay mechanisms of various SP coupling features are studied in Section 5. Further discussions, particularly about the difference in SP coupling behavior between the grating and flat-interface structures, are given in Section 6. Finally, conclusions are drawn in Section 7.

2. Problem Definition and Numerical Technique

The metal grating interface structure under study is shown in Fig. 1(a). Here, a periodical corrugation structure with a period a = 100 nm and the single-groove shape governed by y = h[1− exp(− ∣x/d∣^m)] with h = 10 nm, m = 6.73, and d = 23.75 nm. The grating interface separates a half-space of Ag and a half-space of dielectric. For numerical computation, the permittivity of Ag is assumed to follow the Drude model [24] with the angular plasma frequency set at ω_p = 1.19×10¹⁶ (rad·s^-1) and various damping factors of γ = 4γ ₀, 2γ ₀, γ ₀, γ ₀/2, and γ ₀/4 with γ ₀ = 1.33×10¹⁴ (rad·s^-1). Here, γ ₀ is a widely used damping factor for Ag in simulation study with the Drude model. The purpose of considering various damping factors is for understanding the role of metal dissipation in the energy decay process of an SP-dipole coupling system. The definition of the plasma frequency leads to the SPP resonance energy of 2.92 eV (~430 nm in wavelength) at a flat Ag/dielectric interface with the refractive index of the dielectric fixed at 2.5, which means to simulate GaN (~3.5 eV in band gap) in the concerned wavelength range of 400–700 nm. The coordinate system is shown in Fig. 1(a). A radiation dipole, oriented along the x axis and labeled by J_x, is placed at a position 10 nm below a grating valley. The dipole and the grating structure are assumed to be infinitely extended along the +z and −z directions for defining a two-dimensional problem. To compare the SP-dipole coupling behaviors in a grating structure with those in a flat-interface structure, we also evaluate the time-resolved response of the SPP-dipole coupling near the resonant energy. The problem geometry of the flat-interface SPP coupling is also shown in Fig. 1(a), in which the dotted line separating Ag and dielectric represents the flat interface. Here, a dipole is placed below the flat Ag/dielectric interface by 10 nm. It is noted that the two-dimensional problem geometry is reasonable for our study as we are mainly concerned with the grating-assisted SPP propagating along the x-axis and the z-invariable LSP features. The transient behaviors of the SP-coupling features in the defined two-dimensional problem geometry can provide us with sufficient information for achieving the aforementioned research goals.

 

Fig. 1. (a) Two-dimensional Ag/dielectric grating structure in the x-y plane. The dipole, J_x, is located 10 nm right below the center of a grating groove, which is defined as the origin, O, of the coordinate system. The flat interface structure is depicted by the dotted line along the x axis. (b) The dipole radiation power spectrum (continuous curve), the dielectric-side emission spectrum of the SP-dipole coupling system (dashed curve), the radiation power spectrum of the control case (dotted line near the bottom), and three source spectra (dashed Gaussian-like curves) for the three SP-dipole coupling features (A-C) in the grating structure. (c) The dipole radiation power spectrum (continuous curve), the dielectric-side emission spectrum of the SP-dipole coupling system (dashed curve), and the source spectrum (dashed Gaussian-like curve) for the SP-dipole coupling feature D in the flat-interface structure.

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To simulate the transient field response of a dipole coupling with an LSP or SPP at a metal/dielectric interface, we first calculate the electromagnetic field distribution in the frequency domain, based on a boundary integral-equation method (BIEM) [25], and then take the inverse Fourier transform to obtain the time-resolved response. The BIEM is based on the Stratton-Chu formula [26], in which the unknowns are the equivalent electric and magnetic surface currents on the metal/dielectric interface. In numerical computation, we first divide the interface into segments at which the unknown equivalent surface currents are expanded with the local linear bases. Then, the Galerkin testing procedure [27] is used to transform the boundary integral equation into a matrix equation. Once the equivalent surface currents are obtained through matrix inversion, the complex electromagnetic field at any position can be readily calculated through the boundary integral. After the complex electromagnetic field spectrum is calculated, it is multiplied by the source spectrum function before inverse Fourier transform and demodulation. It is noted that compared with other numerical techniques, such as the finite-difference time-domain method, in which the staircase partition scheme is usually needed when applied to a problem of a smoothly curved interface, the BIEM shows its advantage of higher accuracy for the problem defined in this research.

3. Spectral-domain Coupling Behaviors

The continuous and dashed curves (in black) in Fig. 1(b) show the radiation power spectrum of the dipole itself and the emission power spectrum of the coupling system, respectively, when γ = γ ₀. In this section, the γ value is fixed at γ0. The dipole radiation power is obtained by integrating the out-going Poynting vectors from a square boundary of 2.5 nm in dimension with the dipole at the center. The system emission (dielectric-side emission) power is obtained by integrating the Poynting vectors in the −y direction along a horizontal line 1 μm (far field) below the dipole. There is no net power flow in the +y direction. The dipole radiation power level is sensitive to its environment. Therefore, under the SP-coupling condition, the dipole radiation power can be enhanced. The coupling system emission power indicates the detectable power in application. It is lower than the dipole radiation power due to loss through metal dissipation and in-plane propagation of SP energy along the metal/dielectric interface. The propagating SP energy along the interface is eventually absorbed and not received at a far-field point. In Fig. 1(b), an almost-horizontal dotted line near the bottom is drawn to represent the dipole radiation level when the Ag/dielectric grating interface is replaced by an air/dielectric flat interface 10 nm above the dipole position (the control case). With the metal grating structure, the dipole radiation is always higher than that of the control case, indicating that through the dipole-SP coupling process, dipole radiation or spontaneous emission is enhanced. Also, below ~600 nm in wavelength, the system emission is also stronger than the control level, indicating that a metal grating structure in a light-emitting device can be useful for enhancing the device emission efficiency.

Three major dipole radiation features in Fig. 1(b) can be observed, as labeled by A, B, and C. Feature A is located at almost the same spectral position as the SPP resonance energy of the flat Ag/dielectric interface. Thus, to differentiate the different dipole-coupling properties between this SP feature in the grating structure and the resonant SPP feature on the flat interface, which is designated as feature D, the continuous and dashed curves are plotted in Fig. 1(c) to show the spectra of dipole radiation power and coupling system emission power, respectively, of the resonant SPP feature. Here, one can see the strong coupling-induced radiation intensity near 435 nm. However, the coupling system emission is quite weak. Note that the system of coupling a dipole with an SP can always emit photon, either strong or weak, even though this SP is momentum mismatched with plane wave.

The dark stripes in Fig. 2 show the dispersion curves of all possible SP features, which can be excited by incident plane waves in the grating structure shown in Fig. 1(a). These dispersion curves are obtained based on the reflection calculation of incident plane wave using the plane-wave-assisted BIEM [25]. Here, three vertical straight dark stripes at 405, 430, and 520 nm can be recognized with the one at 430 nm being faint. A vertical straight dispersion curve in Fig. 2 means an SP feature of zero velocity and corresponds to an LSP mode. Its coverage of all possible wavenumbers means that such an LSP can absorb photon from and emit photon into all the directions. The observation of those LSP features means that they can be excited by plane wave. An SP feature, which can be excited by plane wave, can also be excited by a nearby dipole. However, the excitation efficiencies by plane wave and dipole near field can be quite different. The LSP feature at 405 nm is strongly excited by plane wave leading to the dark vertical straight stripe in Fig. 2. However, it is weakly excited by dipole near field such that it shows itself as a small shoulder in Fig. 1(b). The LSP feature at 430 nm is weakly excited by plane wave, resulting in a faint vertical stripe in Fig. 2. Nevertheless, its strong coupling with the dipole near field leads to the strong radiation feature (A) in Fig. 1(b). Next, the LSP feature at 520 nm (B) is strongly excited by both plane wave and dipole near field. Regarding the faint stripe around 455 nm, which corresponds to a minor peak in Fig. 1(b), it is unclear whether it is an LSP or SPP feature. In Fig. 2, the curved dark stripe corresponds to an SPP, which originates from the folded dispersion relation of the flat-interface dispersion curve by grating diffraction. The interception of this dispersion curve with the dielectric light line around 590 nm coincides with feature C in Fig. 1(b). Hence, feature C results from the coupling of the dipole with a grating-assisted SPP mode. Regarding feature D in Fig. 1(c), because such a resonant SPP is momentum mismatched with any plane wave, no dispersion curve is expected in a figure like Fig. 2. However, the resonant SPP feature can be excited by the dipole near field. Four excitation source spectra are defined for the SP features A-D, as depicted by the four dashed Gaussian-like curves in Figs. 1(b) and 1(c), for evaluating their transient behaviors. Those source spectral shapes and intensities are identical in the frequency domain. However, their spectral peak positions follow the feature locations of individual SP-dipole coupling features.

 

Fig. 2. Dispersion curves of a grating-assisted SPP feature and three LSP features evaluated with the plane-wave-assisted BIEM.

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4. Time-domain Coupling Behaviors

The field strength (absolute value of H_z) distributions of SP-dipole coupling features A-D at chosen delay times are shown in Figs. 3(a)–3(d), respectively. For obtaining the data in Fig. 3, the damping factor γ is set at γ ₀. The delay time for an SP-dipole coupling feature is chosen to show its broadest field distribution along the x axis. The delay times relative to the source peak of those features are shown in the corresponding figures. Here, one can see that the coupling field distributions are highly localized in features A and B, confirming that they correspond to LSPs. On the other hand, features C and D show broad field distributions in the x-y plane, confirming that they belong to the category of SPP. The field pattern of an individual SP coupling system determines its far-field intensity and emission efficiency. Those field patterns of features C and D lead to weak emission, as already demonstrated in Figs. 1(b) and 1(c). The detailed transient behaviors at the bright point on the Ag/dielectric interface right above the dipole, which was designated as the origin of the coordinate system and was assigned as point O in Fig. 1(a), will be further studied.

Figure 4 shows the time-resolved intensity profiles at the observation point O of the four SP-dipole coupling systems when the Gaussian-like pulsed dipole sources, corresponding to the source spectra shown in Figs. 1(b) and 1(c), are used. Here, the source profile is also shown and labeled by S for reference. The results are obtained by assuming that the damping factor of Ag is γ = γ ₀. The intensity profiles are normalized to the peak level of feature A, which is normalized to its source peak level. To better demonstrate the response delays, the intensity profiles in the linear scale and normalized to individual peak levels are shown in the insert. The response peak delay times relative to the source peak are shown in Table 1. Here, one can see the fast coupling response of feature C, followed by feature B. The coupling responses of features A and D are slightly slower. The intensity peaks at point O of all the coupling features appear within a few fs. The nature that features B and C can be excited by plane wave, as shown in Fig. 2, is supposed to be related to their fast excitations by the dipole.

 

Fig. 3. Field strength (absolute value of Hz) distributions of the dipole-coupling features A-D in parts (a)-–(d), (Media 1) (Media 2) (Media 3) (Media 4), respectively, at the individually chosen delay times for showing the broadest field distributions along the x axis when γ = γ ₀.

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Fig. 4. Time-resolved field intensity profiles at point O of the four SP-dipole coupling features. The source profile is shown and labeled by S. The fitting lines for calibrating the decay times are plotted. The insert shows the linear-scale profiles for demonstrating the temporal peak positions. The damping factor γ is set at γ ₀.

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Table 1. Delay times and decay times under the assumptions of various metal damping factors

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All the response profiles decay in single-exponential manners except feature C, in which two decay stages can be clearly seen. A close examination of the time-resolved behaviors at the bright points of the neighboring grating grooves leads to the observation that the energy coupled to pint O is transported via in-plane SPP propagation from the dipole position along the interface. The calibrated propagation speed is about 1.99 × 10⁷ m/s, which is consistent with that calculated from the slope of the dispersion curve, as indicated by the dashed line in Fig. 2. The first decay stage of feature C describes the intensity decay at point O through SPP in-plane propagation. This stage stops when the counter-propagation through grating diffraction balances the intensity distribution between point O and those in the neighboring grating grooves. Then, just like the single-stage decays of other features, the second-stage decay describes the energy relaxation process of the dipole-SP coupling system. The slight oscillating behaviors of the decay profiles are attributed to the energy exchanging nature during the coupling process. The fitting lines of those decay profiles are also plotted in Fig. 4. The calibrated decay times of those features are shown in the fifth column (the case of γ = γ ₀) of Table 1, including the two values of feature C. Here, one can see the fast decay (4.11 fs) of the first stage of feature C due to SPP propagation and the slow system decay (18.95 fs) in the second stage.

To demonstrate the time evolutions of field distributions of those four dipole-SP coupling features, four motion pictures are provided.

5. Decay Mechanisms of the Coupling Systems

In the simulations described above, the damping factor of Ag was set at γ = γ ₀ = 1.33×10¹⁴ (rad·s^-1). This number determines the metal dissipation level of an SP-dipole coupling system. To understand the role of damping factor or metal dissipation in the decay behavior of an SP-dipole coupling system, we consider four additional damping factors of γ = γ ₀/4, γ ₀/2, 2γ ₀, and 4γ ₀ for comparing the decay times of the coupling systems. The decay times at point O of the four SP-dipole coupling features with various γ values are listed in Table 1. The small variation of the first-stage decay time of feature C in varying γ implies that this decay stage is indeed dominated by the SPP propagation process and is weakly related to the system energy relaxation. In the second-stage decay of feature C and the one-stage decays of other SP-dipole coupling features, the energy relaxation mechanisms include the dielectric-side emission, in-plane SP energy propagation along the metal/dielectric interface, and metal dissipation. For further analysis, we define a time constant, τ_t, for describing the energy relaxation dynamics, which includes the dielectric-side emission and the in-plane SP energy propagation. This time constant is expected to be independent of metal dissipation. Therefore, for an SP-dipole coupling feature, we can use a rate equation to relate the time constant, τ_t, and dissipation time constant, τ_di, to the simulated system decay time, τ_d, which was given in Table 1 (the second-stage decay time in feature C), as

The dissipation time constant, τ_di, is supposed to be related to the metal damping factor, γ. Since the damping process is the microscopic origin of metal dissipation, it is reasonable to assume that τ_di is inversely proportional to the damping factor. With this assumption of inverse proportionality, for each SP-dipole coupling feature, we can use the two decay times of τ_d in the cases of γ = γ ₀/4 and γ ₀/2, as already given in Table 1, for evaluating the τ_t value (independent of γ) and the τ_di value for each case of γ based on Eq. (1). Once the τ_t value and the τ_di value in the case of γ = γ ₀/4 are obtained, one can reevaluate the system decay times of the cases of γ = γ ₀/2, γ ₀, 2γ ₀, and 4γ ₀ based on Eq. (1) for comparing with the similar data based on simulations. Such reevaluated system decay times are denoted by τ_ds and are listed in Table 2. In Table 2, one can see that the τ_ds values based on Eq. (1) in the cases of γ = γ ₀/4 and γ ₀/2 are exactly the same as the τ_d values, which were obtained directly from simulations and were repeated from Table 1, since they were used for starting the evaluations with Eq. (1). The values of τ_ds based on Eq. (1) in the cases of γ = γ ₀, 2γ ₀, and 4γ ₀ are quite similar to those of τ_d obtained directly from simulations, indicating that Eq. (1) represents a reasonable model in describing the energy relaxation behavior. In Table 2, the percentage deviation of τ_ds from τ_d in each γ case is shown. One can see the reasonably small deviations. The τ_di values of different γ cases show the inverse proportional relation with γ, as assumed. The analysis described above was performed for all the four SP-dipole coupling features. Among the four features, it is interesting to see that except feature C, in each γ case, the τ_di values are almost the same. Actually, all those τ_di values are about the same as the inverse of the corresponding y values, implying that the SP behaviors in features A, B, and D are similar to local charge oscillations. This consistency implies that the dissipation mechanisms of those three features are similar.

Table 2. Comparisons between simulated and calibrated decay times

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The four τ_t values are quite different from each other implying that their energy relaxation mechanisms are different. To differentiate the individual contributions of in-plane energy propagation and the dielectric-side emission to the time constant of τ_t, we evaluate the outgoing power in the +x and −x direction along the Ag/dielectric interface from a rectangular region confined by x = +100, -100 nm and y = +5, -5 nm. Such in-plane propagating power is obtained by integrating the Poynting vectors toward the +x and -x directions on the right and left boundaries, respectively. Figure 5(a) shows the in-plane propagation power spectra of the grating structure, which is labeled by “Grating (x)”, and the flat-interface structure, which is labeled by “Flat (x)”, in the case of γ = γ ₀. For comparison, the dielectric-side emission spectra, as already shown in Figs. 1(b) and 1(c), are repeated in Fig. 5(a), as labeled by “Grating (-y)” and “Flat (-y)”. Here, one can see that the in-plane propagation powers along the interface of features A and B are much smaller than the individual dielectric-side emission powers, confirming the LSP-coupling features. In other words, τ_t in these two features mainly describe the dielectric-side emission behaviors. On the other hand, in feature D at 435 nm, the in-plane propagation power along the interface is slightly weaker than the dielectric-side emission power level, implying that besides metal dissipation, the coupling system releases energy through both dielectric-side emission and in-plane propagation along the metal/dielectric interface. The propagation power diminishes through dissipation along propagation.

 

Fig. 5. Comparison of the spectra of in-plane propagation energy with the dielectric-side emission spectra in the case of γ = γ ₀. The label (x) means the energy propagation along the +x and −x directions. The label (-y) implies dielectric-side emission. (a) In-plane propagation evaluated at x = +100 and -100 nm; (b) in-plane propagation evaluated at x = +500 and -500 nm.

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In Fig. 5(a), one can see that the in-plane propagation power of feature C at 590 nm is comparable to its dielectric-side emission level even though it is small. However, when we extend the x boundary for evaluating the outgoing power from ±100 nm to ±500 nm, this power level is significantly reduced, as shown in Fig. 5(b). With grating diffraction, which bounces the energy back to the dipole location, the dipole-coupled SPP energy is well confined in this coupling system of low emission efficiency and low dissipation rate (see Table 2). Therefore, when the metal damping factor is set at γ = γ ₀, in all the four dipole-coupling features, the energy releases mainly through metal dissipation even though their dielectric-side emission contributions are different from each other. It is noted that although metal dissipation may dominate the energy relaxation process of an SP-dipole coupling system, efficient emission of such a coupling feature can still guarantee its useful application to emission enhancement in a light-emitting device. For instance, by comparing the τ_t and τ_di values of feature B in Table 2 in the case of γ = γ ₀, we can estimate that about 27 % coupled system energy is emitted. In other words, under the condition of the assumed metal/dielectric interface geometry, the emission enhancement application is useful for a light emitter, which has internal quantum efficiency lower than 27 %. By considering other factors, such as light extraction efficiency of a device, this number can be even higher. On the other hand, the large decay times of feature C implies that a grating-assisted SPP-coupling system can be a good candidate used for implementing a SPASER system.

6. Discussion

For feature C, which clearly shows the SPP nature, we have numerically evaluated the flat-interface SPP attenuation coefficient, α, which is equal to the imaginary part of the wavenumber in the x direction multiplied by the group velocity. Feature C can be regarded as a flat-interface SPP under grating diffraction. Figure 6(a) shows the α spectra of the five γ cases. Here, the data are truncated on the short-wavelength side to avoid the unreasonable results around the SPP resonance. Figure 6(b) shows the a values normalized by its levels in the case of γ = γ ₀/4 at individual wavelengths. Here, one can see the proportionality of α to the metal damping factor in the spectral range beyond 500 nm, confirming the reasonable assumption for calibrating the time constants of feature C, as shown in Table 2. By reading the α values at 590 nm, one can obtain the reasonable consistency between the 1/α and τ_di values of feature C. Figure 6 does not help in interpreting the calibrated τ_di values of feature D due to data truncation on the short-wavelength side. However, the similarity of the τ_di values of feature D to those of features A and B implies that the damping behavior of a dipole-coupled resonant SPP feature on a flat interface is similar to that of an LSP due to its characteristics of near-field excitation and minimized group velocity.

 

Fig. 6. (a) Attenuation coefficients, a, as functions of wavelength for the five γ cases. Part (b) shows the normalized attenuation coefficients.

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With the grating structure, feature A shows the nature of an LSP even though its spectral location coincides with the resonant SPP of a flat interface. To further manifest their differences, Fig. 7(a) shows the H_z intensity at point O of the five γ cases in a spectral range between 380 and 500 nm in the grating structure. Here, one can see that although the dipole radiation power level decreases with increasing γ, the spectral positions of maximum dipole radiation power is essentially fixed around 430 nm. On the other hand, as shown in Fig. 7(b), the H_z magnitude at point O of feature D (in the flat-interface structure) shows the red shift trend of the maximum radiation power as damping factor increases. Different from the resonant SPP feature, feature A of fixed spectral position does manifest itself as an LSP.

The charge distribution at the metal/dielectric interface can also reveal the properties of a dipole-SP coupling feature. In Figs. 8(a)–8(d), we show the contours of the electrical field and the charge distributions of features A-D, respectively, in the case of γ = γ ₀. The + (-) signs for the positive (negative) charges are marked at those interface positions of outgoing (incoming) electrical fields. In those figures, the curved arrows depict roughly the electric-line directions. In Figs. 8(a) and 8(b), one can see the charge distributions of the two LSP features, in which the charge separations are non-uniform. The distance between a positive (negative) and a neighboring negative (positive) charge distribution is related to an SP wavenumber. Thus, the non-uniform charge distributions in features A and B once again confirm their LSP nature. Feature A is excited mainly by the dipole near field (see the faint feature in Fig. 2), which corresponds to larger wavenumbers. Consequently, its charge distribution separation is generally smaller, when compared with that of feature B, which can be excited by plane waves of smaller wavenumbers. The charges of the two SPP features show periodic distributions, as shown in Figs. 8(c) and 8(d). The half-period of 18 nm of feature D is consistent with the theoretical wavenumber calculated from the SPP dispersion relation. The half-period of charge distribution in feature C is about 90 nm, which can be calculated from the SPP dispersion relation as well as the grating-diffraction formula, illustrating the characteristics of a grating-assisted SPP. It is noted that the charge distribution in the flat portion of a grating groove of feature A is similar to that of feature D, implying that feature A may have certain property of SPP in this region. However, because of the groove structure, the SPP energy is localized to become an LSP.

 

Fig. 7. H_z field intensities as functions of wavelength at point O with different γ values in the grating structure (a) and flat-interface structure (b).

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7. Conclusions

In summary, we have demonstrated the transient behaviors of the dipole couplings with SP features in an Ag/dielectric grating structure in order to understand the characteristics of those dipole-coupling features. In particular, from the transient behaviors, we could work out the major decay mechanisms of those coupling features. For comparison, the transient behaviors of the resonant SPP coupling feature on the Ag/dielectric flat interface were also demonstrated. Two grating-induced SP-dipole coupling features were identified to be LSPs. The other one was a grating-assisted SPP, which manifested two decay components, corresponding to the first stage of SPP in-plane propagation and the second stage of coupling system decay. The system decays of all the four SP coupling features included three mechanisms: metal dissipation, dielectric-side emission, and in-plane energy propagation along the metal/dielectric interface. All the dissipation rates were proportional to the assumed damping factor in the Drude model of the metal. The dissipation rates of the LSP and resonant SPP features were about the same as the damping rate, implying their local electron oscillation natures. The dissipation rates of the grating-assisted SSP feature were consistent with numerical calculations based on the theory. In all the concerned dipole coupling features, dissipation could dominate the system energy relaxation, depending on the assumption of damping factor. In the LSP features under study, dielectric-side emission was prominent. The coupled energy in the grating-assisted SPP feature could be most efficiently stored in the coupling system due to its low emission efficiency and energy confinement through grating diffraction.

 

Fig. 8. Contours of electric-field line and charge distributions of the four SP-dipole coupling features A-D in parts (a)–(d), respectively, when γ = γ ₀.

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