We study the emission behavior of an electric dipolar nano-emitter coupled with a disk–ring nanostructure (DRN) that sustains multiple plasmonic Fano resonances. The emitter–DRN electromagnetic coupling efficiency strongly depends on the relative position of the nano-emitter and the DRN, which determines whether the multiple Fano interactions are visibly activated. More specifically, for longitudinal polarization, the multiple Fano resonances are pronounced when the nano-emitter is at the outer apex of the disk or at the gap center of the DRN, observable in the far-field and/or near-field characteristics. However, no obvious Fano feature shows up when the nano-emitter is near the outer apex of the ring. For the case in which the nano-emitter oscillates vertically with respect to the DRN axis, Fano resonance is dramatic only when the nano-emitter is inside the gap of the DRN. We show that the cascading amplification of the dipole moment by the nanodisk is crucial for the excitation of the multiple Fano resonances. Our results are useful in engineering plasmon-modified optical spectroscopy and photon emission control, particularly in resonant plasmonic heterostructures.
© 2014 Optical Society of America
Optical single quantum emitter (or nano-emitter), such as excitation in J-aggregates, dye molecules, or quantum dots, is the key element in quantum information and sensing [1,2], and the modifiability of its emission properties by plasmonic nanostructures is an interesting subject [3,4]. Some single-molecule experiments that focus on the resonant coupling of emitters with plasmonic modes in metallic nanostructures have observed that the photoluminescence of a nano-emitter can be either enhanced or quenched, depending on the working frequency, the polarization, and the distance between the nano-emitter and metal [5,6], with concomitant changes in the excited state lifetime [7,8]. In addition to these fundamental breakthroughs, recent progress in nanotechnology enables the introduction of plasmon-enhanced luminescence in various applications, e.g., for enhancing the brightness of technologically important light emitters , or for enhancing emission in a polarization-selective way . Besides, plasmon-enhanced luminescence has great potential for sensing applications and for improving the light-harvesting capabilities of solar cells . Despite of the numerous potential applications, there are still many issues to be explored regarding the emission modification by exotic hybridized plasmon modes or collective surface plasmon polaritons in the weak interaction regime, even within the framework of classical electrodynamics. Emitter–plasmon coupling in the strong interaction regime requires a more comprehensive treatment such as the time-dependent density functional theory in quantum electrodynamics .
Resonance-modified spontaneous emission in plasmonic heterostructures is of particular interest. The resonant modes in such systems are usually hybridized, which results in symmetric or antisymmetric modes that are of super-radiative or subradiative properties . The spectrum and radiation patterns of such systems are dramatically different from their homogenous counterparts. This makes them essentially unique and useful in various plasmon-associated applications, such as high-sensitivity bio-sensing , sharp-band spectral selectivity , and beam filtering and steering .
Fano resonance is a type of resonance discovered by Fano in a quantum mechanical study of the autoionizing states of atoms . Different from Lorentz-type resonance, Fano resonance generally exhibits a distinctly asymmetric line shape. The microscopic origin of Fano resonance arises from the constructive and destructive interference of a narrow (discrete) resonance state with a broad spectral line or continuum . Over the past few years, Fano resonance has been observed in numerous plasmonic nanostructures, such as single metallic particles , metallic dimers , nanoparticle clusters [21,22], disk–ring nanostructures (DRNs) [23,24], periodic arrays [25,26], and nonlinear lattices . Fano resonance in plasmonic heterostructures has been widely explored and is expected to be useful in chemical and biological sensors , surface-enhanced Raman scattering (SERS) , and nonlinear optics (such as impulsive stimulated Raman scattering ), due to its fascinating optical characteristics . In most theoretical and experimental scenarios, the excitation source is often considered as plane wave (i.e., far-field illumination), and the Fano resonance features are generally illustrated by far-field (e.g., extinction, absorption, and/or scattering cross section) spectra for finite-sized plasmonic structures, or by transmission/reflection spectra for periodic plasmonic structures. The use of Fano resonance structures to tailor the emission properties of a nano-emitter shall be of both fundamental and application interests, but remains largely unexplored .
2. SYSTEM AND METHODS
In this paper, we study the modifications of radiative decay rate and nonradiative decay rate for a nano-emitter that is in near proximity to a silver heterodimer of a DRN that supports multiple Fano resonances . As shown in Fig. 1, the prototype DRN consists of a nanodisk with diameter and a nanoring with outer diameter and inner diameter . The gap width and the thickness are used in the calculation, respectively. In the visible, the fundamental dipolar mode in the disk and the higher-order mode in the ring strongly hybridize, enabling dipole–multipole Fano features that can result in multiple asymmetric spectrum, as well as a transitional optical binding force effect . The geometry, size, and gap of the DRN are crucial design parameters that can tune the resonance properties, which would certainly tailor the emission behaviors.
We employ the finite element method (Comsol Multiphysics) to investigate the responses of a nano-emitter near such a DRN that is assumed freestanding in vacuum. (Note that putting the DRN in other dielectric media such as water may lead to red-shift of the resonances but would not change the physics reported here.) The optical constants of silver are taken from Johnson and Christy . The nano-emitter (i.e., excitation source) is treated as a classical point dipole , at different locations (e.g., , , or in Fig. 1), where is the oscillating angular frequency. The point-like dipole is of no intrinsic losses. In response to the interaction with a neighboring structure, its radiative power can be calculated by the electric field and magnetic field ,32] 3) is carried over the whole disk exclusively, and it can be used to analyze the individual contributions of the disk in the DRN. It is noted that is dominated by its component in the polarization direction.
3. RESULTS AND DISCUSSION
A. Case of Plane Wave Illumination
When the distance between a nano-emitter and an object goes to infinity, the situation can be regarded as a plane wave illumination. To better understand the radiative and nonradiative processes of a nano-emitter in the presence of the DRN, we first study the case of a plane wave at normal incidence. The incident light propagates along the axis and is polarized in the axis (see Fig. 1). It is well known that the dipolar mode of the single disk can be efficiently excited by such a plane wave. The dipolar mode is often referred to as the “bright” mode in examining Fano phenomena [20,24]. The dashed line in Fig. 2(a) is the scattering spectrum of an isolated nanodisk (i.e., removing the ring in the DRN), with surface plasmon resonance (SPR) at . The induced dipole moment on the disk, which is dominantly contributed by the y component in this case, is plotted in Fig. 2(b), and its phase in Fig. 2(c) by the dashed line. When the wavelength goes through the SPR, the phase is changed by . Let us remark that is phase-advanced above the SPR frequency (shorter wavelengths) and phase-delayed below the SPR frequency. In fact, a coupled two-harmonic oscillator model shows that the dipole moment p might be written as 33].
For a single nanoring, we can excite its quadrupolar, hexapolar, and octupolar resonance modes, which are at , 583, and 497 nm, respectively, by a nearby nano-emitter [23,24]. These modes are referred to as “dark” because of their relatively higher quality factor and weak coupling to the external field. We note that these modes are the symmetric ones, while the antisymmetric modes in the ring fall in the shorter wavelength regime and are not considered here .
In the DRN, the bright mode of the nanodisk is excited by two pathways: one directly from the incident light, and the other from the near field generated by the ring’s dark modes, which are not directly excited by the plane wave, but by the induced dipole in the disk. It is commonly believed that the destructive interference of the two pathways results in Fano resonance, leading to dips in the scattering spectrum [solid curve in Fig. 2(a)]. It is very interesting to see that the Fano dips are pronounced in both the scattering spectrum [solid curve in Fig. 2(a)] and the amplitude spectrum [solid curve in Fig. 2(b)] of the dipole moment defined in Eq. (3). The phase of the y component is plotted as a solid curve in Fig. 2(c). Notice that the phase jumps abruptly at the dipole–octupole and dipole–hexaple Fano dips [nearly zero SCS in Fig. 2(a)] at around and 587 nm, respectively , quite similar to what happens in the classical model . For the Fano dip at in Fig. 2(b), however, the phase jump is relatively weak. This is partially because the dipolar bright plasmon resonance () of the disk is spectrally far from the quadrupolar mode () of the nanoring and the destructive Fano interaction becomes much weaker.
B. Case for Longitudinal Polarization of a Nano-Emitter
It is worth noting that although the direct coupling between the dark mode and the far-field radiation is basically negligible in the case of plane wave illumination, it must be accounted for when considering dipole emission modification by the DRN. As a result, in the coupled oscillator model put forward in the Section 3.A, both of them are driven by the external field (electric dipole source), which is position dependent. In addition, the distance between the dipole source and the DRN affects their coupling efficiency dramatically . Similar to the situation of plane wave illumination, here we consider the longitudinal polarization. Figures 3(a) and 3(b) show, respectively, the radiative and nonradiative decay rate spectra when the nano-emitter is placed at (in the principle xoy plane) and 10 nm away from the disk boundary (see Fig. 1). There is discernible Fano profile appearing in the spectrum [Fig. 3(a)], as well as in the spectrum of the disk dipole moment [Fig. 3(c)]. In this configuration, the dipolar plasmon mode of the nanodisk can be directly and efficiently excited by the nano-emitter, thus acting as a “bright” mode in terms of Fano interference. Although the high-order plasmon modes of the nanoring can be excited by a nearby electric dipole, the efficiency is relatively weak due to the large distance () between the nano-emitter and the ring. It is observed that the disk’s nonradiative decay rate [red circles in Fig. 3(b)] is comparable to that of the ring [brown squares in Fig. 3(b)] in the DRN. The ring’s high-order plasmon modes are regarded as “dark,” with respect to the nano-emitter in this case. Nevertheless, the high-order plasmon modes of the nanoring can couple with the nanodisk’s bright mode (or induced dipole on it) via strong near-field interactions. As a result, in Fig. 3(d) one can see fairly clear Fano-resonance-induced phase jump of the disk dipole moment . The radiative decay rate spectrum [black solid curve in Fig. 3(a)] shows that three Fano dips are at approximately , 590, and 778 nm. The Fano dips are very close to the ring’s dark mode resonance wavelengths [green dashed curve in Fig. 3(a)], as predicted in the classical model . In addition, Fig. 3(b) plots the nonradiative decay rate spectrum (red solid curve) indicating that the hybridized plasmon resonance peaks are at around , 602, and 807 nm. The slight red-shift between the radiative decay rate dips and the nonradiative peaks is ascribed to the dissipation in the DRN [37–39].
Figure 4 shows the electric fields of the DRN corresponding to the four peaks in Fig. 3(c) at , 532, 619, and 820 nm (labeled by “,” “,” “,” and “”) and for the three Fano dips at , 590, and 778 nm (labeled by “,” “,” and “”). One can observe that the dipole moment on the disk is substantially reduced by Fano interference at the Fano dips [see Figs. 4(b), 4(d), and 4(f)]. Notice that the near field around the disk is visibly small for these three cases. This is one of the typical consequences of Fano resonance and corroborates the fact that strong multiple Fano resonances happen in the DRN in this case.
Bigelow et al. recently showed that the electron energy loss spectroscopy and cathodoluminescence excited by an electronic beam are very sensitive to the relative position of the nanostructure, which supports Fano interference . We expect similar characteristics in our system. We now consider the situation in which the nano-emitter is placed at , i.e., in the gap center of the DRN (see Fig. 1). In this configuration, the disk dipolar mode is still the bright mode because of the high coupling efficiency to the emitter source. Though the nonradiative decay rate is dominant in the DRN, it is seen that the quality factor , estimated from the linewidth of the resonance peaks of the ring’s higher-order plasmon modes (dark green dashed curve), is much larger than that of the disk dipolar mode (yellow dashed curve) in Fig. 5(a). The higher-order modes in the ring would thus be “dark” because of the direct but relatively weaker coupling to the nano-emitter. However, the higher-order modes can be affected by the induced dipole moment in the disk, which substantially amplifies that of the nano-emitter. Therefore, Fano dips (marked as “,” “,” and “”) in the dipole moment spectrum [solid curve in Fig. 5(c)] of the disk are obvious. It should be noted that the Fano dips and the Fano line shape are not as pronounced as in the previous case, reflected in both the radiative decay rate spectrum and the disk dipole moment spectrum; in this case the line profile appears to deviate greatly from the asymmetric Fano shape and resembles the Lorentzian symmetric shape . In other words, the degree of asymmetry is reduced, and the Fano parameter decreases .
Similar to the case of a nano-emitter at , Fig. 6 shows that the electric near field around the disk is heavily suppressed at the Fano dips at , 566, and 740 nm [see Figs. 6(b), 6(d), and 6(f)]. The near-field intensity is stronger for the dipole moment spectrum peaks at , 511, 606, and 808 nm, as shown in Figs. 6(a), 6(c), and 6(g), respectively. The situation is quite different, however, when the DRN is excited by a nano-emitter at (see Fig. 1). In this configuration, the direct coupling between the disk’s dipolar mode and the nano-emitter exponentially weakens due to the larger separation and the ring as a metallic obstacle between them. The ring’s higher-order modes, as usual, remain weakly active to the nano-emitter excitation. Therefore, they are to a certain extent both “dark,” and no obvious Fano resonance would be expected in this configuration. Figure 7(d) shows that the dipole moment phase basically does not jump sharply in a narrow band. Rather than that, the unwrapped phase can be regarded as continuous, indicating no (or extremely weak) Fano resonance associated behavior. Corresponding, the radiative and nonradiative decay rates [Figs. 7(a) and 7(b)], as well as the induced dipole moment spectra [Fig. 7(c)], all look quite symmetrical around the peaks. This is very much like the case in Ref. . Indeed, from the near-field distributions shown in Fig. 8, the electric field intensity near the disk is comparable for all peak and dip wavelengths, marked from “” to “” in Fig. 7(c). In this case, the disk remains nearly silent at all the wavelengths, as reflected by the amplitude of in Fig. 7(c). Notice that the amplitude is much smaller than that in Fig. 5(c).
C. Case for Transverse Polarization of a Nano-Emitter
Next, we study the emission behavior when the emitter is transversely polarized, i.e., with polarization directed along the axis. The results are shown in Fig. 9, which presents the optical spectra for the cases in which the nano-emitter is at , , and in the three columns, respectively. Similar to the longitudinal case, the disk can directly and efficiently couple to the source when the nano-emitter is at . However, the near-field enhancement in the gap is much smaller in this situation , and the transverse near-field coupling between the disk dipole and the ring higher-order modes is approximately half that in the longitudinal case and is negative, making them not able to induce obvious Fano resonance .
This can be further confirmed by the nonradiative decay rate [Fig. 9(a)], the dipole spectrum of the disk in the DRN, and the dipole spectra of an isolated disk [Fig. 9(b)]. The spectra associated with the disk in presence and in absence of the ring have very similar line shapes, indicating that the interaction between the nano-emitter and the DRN is dominated by the dipolar mode of the disk.
In the dipole moment amplitude spectrum of the disk for the case in which the nano-emitter is at the gap center of the DRN, we can see mutipolar modes with clear Fano dips, which is like in the longitudinal case [Fig. 9(d)]. The source and the near field are strongly confined in the gap region in both configurations . The electric field in the gap region is dramatically enhanced, and the nano-emitter couples to the disk dipolar mode more effectively than to the ring’s higher-order modes, making one “bright” and the others “dark.” As a result, Fano interference becomes pretty effective in this configuration.
However, when the nano-emitter is shifted to the apex near the ring (i.e., at in Fig. 1), we do not observe clear Fano dips in the dipole moment spectrum [Fig. 9(f)]. In this case, it is anticipated that the disk dipolar mode and the ring’s higher-order modes are both “dark.” The excitation pathway from emitter bright mode dark mode bright mode does not exist, which prohibits constructive and/or destructive interference. We shall stress that the enhancement of decay rates for the dipole source being parallel to the axis of the DRN (longitudinal) is much larger than those in the transverse case .
D. Effect by z-Offset of the Nano-Emitter
To have a better physical understanding for the above results and to study the effect of the emitter-structure distance in more detail, we turn to the eigenmode pattern of the DRN. Without loss of generality, the first plasmon peak at in the longitudinal polarization is used as an example. Figures 10(a) and 10(b) show the electric field of this resonant mode. It is easy to understand that the decay rate enhancement maximizes when the nano-emitter is at the position of the strongest local field. For example, Fig. 10 intuitively tells that the enhancement for a nano-emitter at is larger than in the case at , while both of them are smaller than in the case at . These results are correlated to the local density of states (LDOS), and we can find optimal position in terms of enhancing the overall decay rate and Purcell factor.
Finally, we study the effect of emitter-structure distance, particularly in the yoz plane in view of the finite thickness of the DRN. The longitudinal movement of the emitter basically leads to monotonically varied decay rate. This characteristic is reported in the literature  and is also observed in our case (figures not shown). However, the transverse shift of the nano-emitter is of different consequences. Figure 11 shows the effects when the nano-emitter is shifted vertically off the DRN principle axis with the offset –200 nm. We use , , , and to label the peaks of the decay rate and plot them as a function of in Figs. 11(b), 11(c), 11(e), 11(f), 11(h), and 11(i), corresponding to the cases in Figs. 11(a), 11(d), and 11(g). Notice that the peaks in the calculated decay rate nearly do not shift when increases (see insets in each panel for cases of and ). However, the decay rate enhancement is highly dependent on the vertical shift , and their maximum values emerge when the emitter is close to the DRN upper edge, i.e., at . This is ascribed to the hot spots at this position, caused by the finite thickness of the DRN.
To understand that visually, the electric field pattern of the corresponding eigenmode (same as in Fig. 10) on the yoz plane is shown in Fig. 12(a). More specifically, the electric field strength as a function of is plotted in Fig. 12(b) for cut lines at , , and . All the three curves have their peaks near , reasonably in accordance with the data in Fig. 11. The coincidence of the nano-emitter position for maximum decay rate and the maximum eigenmode field position is as expected in view of the LDOS, but should be carefully considered when finding the optimal situation in molecular fluorescence or single photon emission, particularly when the planar nanostructures are of finite thickness.
In summary, we have studied the interaction between a single nano-emitter and a prototype plasmonic DRN that sustains multiple Fano resonances. Depending on the direct coupling strength of the emitter to either the fundamental dipolar mode of the disk or the higher-order modes in the ring, the plasmonic hybridization between them is crucially dependent on the emitter position. Besides, since the polarization direction also influences the coupling strength, it also affects the emitters’ radiative and nonradiative decay rates. We explicitly demonstrate that when the nano-emitter is at the hot spot (which resides inside the gap, but near the upper and lower edges of the DRN) of the corresponding eigemode field pattern, the decay rate enhancement reaches maximum. These results may be useful in engineering plasmon-modified fluorescence spectroscopy and in single photon emission control.
This work was supported by the NSFC (11274083), and the Shenzhen Municipal Science and Technology Plan (Nos. KQCX20120801093710373, JCYJ20120613114137248, and 2011PTZZ048). We acknowledge help from the National Supercomputer Shenzhen Center.
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