Engineered Absorption Enhancement and Induced Transparency in Coupled Molecular and Plasmonic Resonator Systems Ronen Adato,†,‡,¶ Alp Artar,†,‡,¶ Shyamsunder Erramilli,§,∥ and Hatice Altug*,†,‡ †
Department of Electrical and Computer Engineering, ‡Photonics Center, §Department of Physics, and ∥Department of Biomedical Engineering, Boston University, Boston, Massachusetts 02215, United States S Supporting Information *
ABSTRACT: Coupled plasmonic resonators have become the subject of signiﬁcant research interest in recent years as they provide a route to dramatically enhanced light-matter interactions. Often, the design of these coupled mode systems draws intuition and inspiration from analogies to atomic and molecular physics systems. In particular, they have been shown to mimic quantum interference eﬀects, such as electromagnetically induced transparency (EIT) and Fano resonances. This analogy also been used to describe the surface-enhanced absorption eﬀect where a plasmonic resonance is coupled to a weak molecular resonance. These important phenomena are typically described using simple driven harmonic (or linear) oscillators (i.e., mass-on-a-spring) coupled to each other. In this work, we demonstrate the importance of an essential interdependence between the rate at which the system can be driven by an external ﬁeld and its damping rate through radiative loss. This link is required in systems exhibiting time-reversal symmetry and energy conservation. Not only does it ensure an accurate and physically consistent description of resonant systems but leads directly to interesting new eﬀects. Signiﬁcantly, we demonstrate this dependence to predict a transition between EIT and electromagnetically induced absorption that is solely a function of the ratio of the radiative to intrinsic loss rates in coupled resonator systems. Leveraging the temporal coupled mode theory, we introduce a unique and intuitive picture that accurately describes these eﬀects in coupled plasmonic/molecular and fully plasmonic systems. We demonstrate our approach’s key features and advantages analytically as well as experimentally through surface-enhanced absorption spectroscopy and plasmonic metamaterial applications. KEYWORDS: Plasmonics, SEIRA, vibrational spectroscopy, Fano resonance, coupling, critical coupling, EIT, EIA
parency (EIT).18,19 Finally, utilizing engineered resonators to increase the absorption cross sections of natural ones is the basis for the important surface-enhanced spectroscopy applications. This important application is also often understood via an analogy to Fano resonances and EIT. In general, coupled harmonic oscillators oﬀer an intuitive with which to treat a variety of quantum phenomena, and thus also their plasmonic analogues. The correspondence between the speciﬁc conﬁguration shown in Figure 1 and EIT (as well as the related Fano resonance eﬀect) is well established.18−21 Notably, the harmonic oscillator is the basis for the standard Lorentz pole dielectric function for a natural material.22 It therefore represents a pleasing starting point for the description of systems including (i) plasmonic meta-materials designed to exhibit EIT-like behavior,18 and (ii) coupled plasmonic and natural resonances that result in the surface-enhanced absorption eﬀect. Both of these systems can be modeled using two coupled harmonic oscillators, as shown in Figure 1.
ptical resonators are an essential element in the nanophotonic toolkit, and an immense body of work is dedicated to their design and realization. In particular, plasmonic structures are of special interest due to the extremely small mode volumes they support, their ability to signiﬁcantly enhance and localize ﬁelds at an interface1 and to direct the scattering of far-ﬁeld radiation.2−4 These are attractive to a wide range of applications ranging from nanophotonic sources,5 detectors and photovoltaics,6 to biomedical applications such as sensing, spectroscopy,7−9 and even treatment10 tools. Recently, coupled plasmonic resonators have become the subject of signiﬁcant research interest11−15 as they provide a route not only to dramatic improvements in these important properties but also enable unique implementations of directional antennas13 and novel biosensors.16 Often, the design of such coupled mode systems draws intuition and inspiration from analogies to atomic and molecular physics systems. This is the basis for the widely successful plasmon hybridization formalism,17 which draws parallels to the coupling of molecular or atomic orbital wave functions. Similarly, much recent work in metamaterial design is based on exploiting analogies to quantum eﬀects such as the Fano resonance14 and electromagnetically induced trans© XXXX American Chemical Society
Received: February 22, 2013 Revised: April 10, 2013
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plasmonic mode couples to propagating light, it loses energy to both external radiation as well as intrinsic material absorption, therefore its damping rate is signiﬁcantly larger than that of the weak absorber (i.e., dark mode; γA ≫ γP). The coupling between a bright and dark resonance is the key element of the Fano resonance and EIT and, indeed, calculating the absorption directly from this mass-on-a-spring picture reproduces the anticipated narrow spectral dip (antiresonance) at the common center frequency when the two resonances are aligned (ωA = ωP).19,21 Hence, surface-enhanced absorption, and SEIRA in particular, is often referred to as a Fano eﬀect. As discussed, the model of Figure 1 applies equally well to fully plasmonic metamaterials and thus provides an intuitive physical description of general Fano resonant systems and is regularly used to interpret data, ﬁt the functional line-shape form, and extract the magnitudes of coupling coeﬃcients.18−21 An important extension to the general coupled oscillator model can be obtained if one considers the possibility of reﬂected or scattered waves. In this case, the driving term is not an independent arbitrary parameter. Instead, it is directly related to the damping rate of the bright mode, γA which is in fact comprised of both losses due to coupling to external radiation (γAe) and intrinsic material absorption (γA0).Timereversal symmetry and energy conservation considerations then require that the former, γAe, obeys a direct relationship with the driving term.27,28 This is simply a statement of the fact that the rate of excitation of the system by external radiation should be directly related to the rate of the reverse process, the loss of energy by that system to external radiation. In this work, we demonstrate the important eﬀects that occur/can be predicted through resonant oscillator models that account for the presence of reﬂected and scattered waves. Here we show that, not only does the interdependence between the drive term and external damping rate (γAe) ensure an accurate physical description but also leads directly to interesting new
Figure 1. Traditional, damped harmonic oscillator conception of a coupled bright (A) and dark (P) mode. Mode A is driven by external radiation, via g·Einc, and experiences damping at a rate γA. Mode P is dark, hence does not interact directly with external radiation or the driving ﬁeld. Its damping rate, γP is typically much smaller than γA. The two modes are coupled by some rate, μ.
Considering surface-enhanced absorption, the key interaction is the coupling of a plasmonic mode to a natural absorption line (masses A and P in the ﬁgure). This coupling is particularly useful for probing a weak absorber, in which the natural resonance does not interact strongly with external radiation and is eﬀectively a dark mode. Plasmonic resonances, in contrast, can interact strongly with incident light and therefore act as bright modes. Coupling between the dark natural and bright plasmonic resonances allows the former to be driven eﬃciently. Applied to infrared vibrational modes, this concept has led to the development of surface-enhanced IR absorption (SEIRA) spectroscopy,7,8,23−25 which oﬀers an important means with which to study trace quantities of molecules and even protein conformational changes.24,25 An important aspect of this application (as well as general surface-enhanced absorption, sometimes referred to as plasmon resonance energy transfer at shorter wavelengths26), that only the plasmonic mode is driven, via the applied force, g·Einc, as shown. As a result, in terms of absorption the molecular resonance is not observed directly but rather by a modulation in the plasmonic spectra. Because the
Figure 2. Critical coupling eﬀects in scattering by an absorber-coated nanoparticle. (a) Absorption cross sections (Cabs, normalized to particle volume) for three diﬀerent length (semiaxis a) Ag prolate spheroidal particles (dashed curves). The solid curves show the same, but where the Ag particles are coated with a 5 nm thick shell of a model material exhibiting an absorption band aligned with the particle resonance (colored regions at the bottom of the panel). (b) Schematic of the physical system and its generalized cavity model (without the absorber present). (c) Absorption and scattering cross sections on resonance. (d) Peak absorption for the generalized cavity shown in (b), as calculated from (3) evaluated at ω = ωA, as a function of in external to intrinsic damping rate. The colored arrows indicate the relevant movements along the curve for the three diﬀerent ellipsoids in (a). B
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dolmen type systems and referred to as an analogue to EIA.35 The eﬀects in ref 35 were attributed to a phase change due to variations in the physical distance between the bright and dark modes. In contrast, the eﬀect observed here clearly cannot be of this form given that the geometry is kept identical for the three diﬀerent ellipsoids apart from the change in their long axis size. To explain this eﬀect, we turn to the generalized resonator model of the coupled mode theory, shown in Figure 2b. The diagram considers a single cavity, coupled to input and output traveling waves (s+ and s− respectively) through a single port. The cavity resonance is characterized in terms of its mode amplitude, A, center frequency, ωA, and damping rate, γA, which is the summation of the intrinsic and external loss rates (γA → γA0 + γAe) as previously discussed. The situation is described by the coupled mode equations27
eﬀects. Signiﬁcantly, we demonstrate this dependence to predict a transition between EIT and electromagnetically induced absorption that is solely a function of the ratio of the radiative to intrinsic loss rates in coupled resonator systems. We analyze these eﬀects by utilizing the temporal coupled mode theory (TCMT)27−30 to derive an intuitive and powerful description of coupled plasmonic and Fano-resonant systems. We show that the interactions in and spectral response of these systems can be understood and predicted by considering what can be thought of as an impedance matching eﬀect. This approach is based on the idea of a critically coupled resonator for which maximal absorption occurs when it is impedance matched to its input transmission line.27 Within this picture, we view the addition of the dark mode to a plasmonic bright one as either driving the system toward the critical coupling condition, increasing absorption (EIA), or away from it, resulting in EIT. Broadly, this treatment is required to correctly describe the fundamental behavior of coupled dark-bright mode systems, which can be thought of as the building blocks for a wide range of physical nanophotonic systems. A notable example is the class of surface-enhanced spectroscopies,7,8,31−34 hence our results are of particular importance for these applications. Signiﬁcantly, not only does our approach oﬀer insight into the fundamental nature of the eﬀect; it also makes clear that the standard intuition of increased ﬁeld enhancement leading to increased signal7,8,23,34 is not necessarily the case. We verify these seemingly unexpected eﬀects through analytical and numerical calculations, as well as a SEIRA experiment. Finally, we conclude by noting that our critically coupled resonator model is quite general and applies equally well to fully plasmonic systems, such as the prototypical dolmen metamaterial. In a unifying manner, not only can the behavior of the dark meta-molecule be treated identically to a natural molecular absorption line, but we show that a transition between behaviors typically described as EIT and its converse, EIA, can occur as a straightforward consequence of the impedance matching eﬀect proposed here. The consequences of critical coupling and the essential features of our description are presented in the scattering calculations shown in Figure 2. In particular, we calculate the absorption cross section (Cabs) for a silver (Ag) prolate spheroid before and after coating with a dielectric shell with an absorption line. These are performed using the standard analytical quasi-static approximations with the radiation damping correction (see Methods). Beginning with a small particle, (long semiaxis, a = 30 nm, short semiaxis, b = 25 nm), resonant in the blue part of the visible spectrum, we increased a, holding b constant, so as to red shift the resonance as shown in Figure 2a,b. Throughout, the absorber dielectric function (εP) is chosen to result in an absorption line at the resonance frequency of the particle and maintain a constant ratio of the particle to absorption band line-width (colored regions at the bottom of the panel correspond to Im(εP), see also Methods). For the blue resonant particle (a = 30 nm), the calculation for the absorber-coated particle displays a narrow transparency window at the center frequencies of the two (plasmonic and molecular) resonances, which is in agreement with the standard EIT-based intuition. Simply increasing the length of the ellipsoid, however, completely reverses the observed behavior, producing for the largest particle (a = 90 nm) an obvious peak in Cabs at the position of the absorber line (red curves in the ﬁgure). This eﬀect, the increase in absorption upon coupling to a dark mode, has been previously observed in fully plasmonic,
dA = jωA A − (γA0 + γAe)A + κs+ dt
s− = −s+ + κA
where the mode and traveling wave amplitudes are normalized such that |A|2 and |s(+,−)|2 give the energy stored in the resonance and power carried by the (incoming, outgoing) waves respectively. The key feature of eqs 1and 2 is that the coupling rate to incident waves, κ, as opposed to being an arbitrary (or even geometrical18) factor, is constrained and related to γAe via κ = (2γAe)1/2.27,36 This link to the external damping rate is an essential consequence of time-reversal symmetry and energy conservation and one of the basic tenets of the coupled mode theory.27,28 As we will show, this provides a means for the observed behavior to arise in terms of an intuitive picture and leads to a number of interesting consequences. Equations 1 and 2 can be used to determine the reﬂection, R = |s−/s+|2, and absorption Abs. = 1 − R of the system.37 In terms of the cavity parameters 4γA0γAe Abs. = (ω − ωA )2 + (γA0 + γAe)2 (3) which, on resonance (ω = ωA) takes the form Abs. = 4f/(1 + f)2 with f = γAe/γA0. The amplitude of the absorption peak thus depends on the ratio of the external to intrinsic loss rates as shown in Figure 2d. Absorption by the cavity reaches a peak value of 1 when the two rates are equal and is reduced to either side when they are not. This eﬀect, known as critical-coupling (CC, i.e., peak absorption when γAe = γA0) is well-known for a resonator coupled to a single input transmission line38 and, given the generality of the coupled mode theory framework, persists for other resonator types. The CC eﬀect has important consequences for the response of the resonator when used to drive a second, dark, resonance that is not directly coupled to the incident and outgoing waves. Qualitatively, the dark mode or weak absorber can be then thought of as adding a small amount of additional intrinsic damping to the bright mode. In terms of Figure 2d, this results in a leftward movement along the curve. A signiﬁcant implication is then immediately obvious. The addition of the absorber does not necessarily lead to increased absorption in accordance with a naive expectation, nor does it always result in a decrease, as would be predicted by analogy with EIT. Instead, depending on the initial properties of the bright mode and the magnitude of the interaction with the absorber, either eﬀect may occur. If the damping rates of the bright mode alone are C
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such that γAe > γA0, it is overcoupled, and coupling to the dark mode drives the system toward the CC condition. If γAe < γA0, then the system begins in the undercoupled state, and the converse is true. While the scattering ellipsoid couples to a continuum of ports, rather than a single one,29,30 the CC condition persists for an arbitrary number of ports with the only quantitative diﬀerence (depending on the number of ports) being a directivity dependent scaling factor for the peak absorption value.29,30 Considering now Figure 2c, the origin of the reversal in the direction of the absorption feature in the spectra for the coated particle can be clearly understood in terms of this eﬀect. Plotting the variation in the scattering, Csca, and absorption, Cabs, cross sections for the bare particle with increasing length, a, shows the former to increase while the latter decreases as the particle resonance red-shifts. Since Csca/ Cabs = γAe/γA0, (see Methods and Supporting Information, Notes S2−3), this indicates a transition from the undercoupled (γAe < γA0) to the overcoupled (γAe > γA0) regime (not to be confused with the under/over damped regimes for a generic harmonic oscillator).38 The dashed vertical lines indicate the resonance frequencies of the three spectra show in Figure 2a. For the blue resonant particle (a = 30 nm), Csca/Cabs = γAe/γA0 < 1. A small increase in the intrinsic absorption term, resulting from adding the absorbing ﬁlm, therefore pushes the resonator left along the curve in Figure 2d, away from the CC peak. In contrast, for the a = 90 nm particle, the reverse is true, the external loss rate for the bare particle is larger than its intrinsic damping term, hence, adding the absorber ﬁlm drives the system toward the CC condition, causing an overall increase in absorption. For the middle particle, the spectral feature associated with the dark mode is relatively small in comparison with the other two cases. This is because the bright mode is characterized initially by an external damping rate that is only slightly larger than the intrinsic loss. As a result, the path traveled along the CC curve is through the peak, and to the exact opposite side of it (see Supporting Information, Note S2). This observation is particularly relevant to surface-enhanced absorption eﬀects. First, it serves to emphasize that what is observed or measured in these applications is the perturbation of the plasmonic resonance by the absorber ﬁlm. Second, it indicates that the magnitude of the associated feature is not necessarily a monotonically increasing function of, for example, ﬁeld enhancement, as is commonly assumed, but rather depends critically on the impedance matching eﬀect we have described here. Signiﬁcantly, beyond a qualitative description, we note that the formula for the absorption on resonance derived from eq 3 is highly accurate in its prediction of the change in absorption cross section. As shown in Table 1, using the cross sections (Cabs and Csca)) for the bare and coated particles to compute f and therefore the change in Abs. via eq 3 gives results within 3% of the data in Figure 2. Furthermore, these eﬀects can be obtained rigorously within the coupled mode theory framework by adding the second resonator, P, such that the interaction between it and A is described by dA = jωA A − (γA0 + γAe)A + jμP + κs+ dt
dP = jωPP − γPP + jμA dt
Table 1. Quantitative Comparison of Absorption Change Between the Exact Analytical Calculaton and the TCMTa Blue particle (a = 30 nm, ωA = 26 300 cm−1) Cabs(ωA) bare coated
0.01402 0.00575 0.4103 1.3598 1.3611 0.01031 0.00236 0.2289 Green particle (a = 60 nm, ωA = 21 400 cm−1) Cabs(ωA)
0.02453 0.03870 1.5777 0.9646 0.9923 0.02543 0.01671 0.6571 Red particle (a = 90 nm, ωA = 17 600 cm−1) Cabs(ωA)
% err 0.09
% err 2.87
In the table, f is computed for the bare and coated ellipsoids via f = Cabs(ωA)/Csca(ωA) and substituted into eq 3 to compute the relative change in absorption, bare/coated = Abs0/Abs1 = ( f 0/(1 + f 0)2)/( f1/ (1 + f1)2). This is compared with the values taken from the exact spectra, Cabs,0/Cabs,1.
mode, γP corresponds entirely to intrinsic loss and eq 2 still holds. Assuming a time harmonic dependence for both modes, eq 4 can be written as jωA = jωA A − (γA0 + γAe)A − γμA + jωμA + κs+
Here μ is the coupling rate corresponding to direct energy exchange between the two modes. As P is assumed to be a dark D
where γμ = μ γP/[(ω − ωP) + and ωμ = μ (ω − ωP)/[(ω − ωP)2+ γ2P] are eﬀective damping constants and frequency shifts respectively, resulting from the interaction with P. In general, this set of equations describes an asymmetric Fano-type lineshape, but when the two resonances are aligned (ωA = ωP) and on resonance with the driving ﬁeld (ω = ωP = ωA) we have γμ = μ2/γP while ωμ = 0. This corresponds exactly to the previous description and clearly illustrates that the relative values of the three damping rates γA0, γAe, and γμ, determine whether dark mode’s signature will manifest itself as an EIT-like dip in or EIA-like peak superimposed on the bright mode spectrum. Signiﬁcantly, the quantities, γAe, γA0, μ can all be computed from energy considerations, such that a predictive model can be applied to our physical system. Performing these calculations for the ellipsoid scattering cases in Figure 2 we ﬁnd that the paths traced out along the CC curve are exactly those discussed (Supporting Information, Notes S1−S2). Finally, we note that the TCMT formalism has also previously been used to predict superscattering phenomena, which can achieve scattering cross sections above the theoretical maxima for a single resonance.39 Despite the similarity between the line-shapes observed,30,39 superscattering is concerned with the constructive or destructive addition of the scattering cross sections of multiple resonant modes via far-ﬁeld interference whereas, the CC eﬀect here relies on a direct coupling mechanism and requires only a single bright mode. While the variations in damping rates as a function of operating wavelength observed in these model calculations can be thought of as a natural, exogenous eﬀect, subject to the speciﬁcs of an application or experimental setup, the plasmonic toolkit oﬀers ample means with which to engineer the external damping rate of a nanostructure.14,40−42 Indeed, a large body of recent work is devoted to this subject and demonstrates the ability to control radiation damping through a variety of means 2
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size, hence γAe/γA0 decreases with gap thickness, as shown in Figure 3b. On the basis of the picture presented to explain the results of Figure 2, if one were to coat the structure with an absorbing ﬁlm, depending on the gap size, either the opening of an EIT-like transparency window or the appearance of a superimposed peak (EIA) is possible. We demonstrate these eﬀects through FDTD, by coating the PA structures with a thin (10 nm thick) ﬁlm having a narrow absorption band at ∼1686 cm−1. The absorption spectra (Abs. = 1 − R) shown in Figure 3c for four diﬀerent gap sizes (see vertical dashed lines in Figure 3b) clearly demonstrate this correspondence to the scattering calculations in terms of our CC picture. The 80 nm gap now represents the undercoupled system with its spectrum upon coating presenting the appropriate dip at the ﬁlm’s absorption line. For the 400 nm thick gap for which the PA system is overcoupled, the complete reverse eﬀect is apparent. The 200 nm gap corresponds to the usual implementation of a PA in the CC state (such that Abs. = 1 for the bare structure). Here, adding absorption again drives the system away from the CC condition and the resultant reduction in total absorption occurs, which is in agreement with previous studies.34 Finally, we ﬁnd that the absorption line feature in the spectrum for the 280 nm gap PA is negligible compared to the other structures. The eﬀect observed here is analogous to that for the a = 60 nm spheroid (in Figure 2), where the increase in intrinsic damping upon coupling to the absorbing ﬁlm (γA0 → γA0 + γμ) drove the system from one side of the CC peak to the exact opposite position. Signiﬁcantly, for this gap size the ﬁeld enhancement is comparable to the two smaller ones and even higher than for the 400 nm gap structure (Supporting Information, Note S3). Given that the structures are otherwise virtually identical, this eﬀect is unexpected from the standard intuition behind, for example, SEIRA spectroscopy, which predicts absorption signal to scale with the near-ﬁeld enhancement generated by a plasmonic resonance. In future applications, leveraging plasmonic enhancement and coupling between bright and dark modes, it is thus evident that exactly what quantity is in fact being measured must be considered carefully and the eﬀects predicted by the description here be taken into account. In addition to the FDTD simulations, we demonstrate these eﬀects experimentally, as shown in Figure 4. As in ref 34, the structures are fabricated as a 100 nm thick Au particle separated from an Au ﬁlm of the same thickness by a thin magnesium ﬂuoride (MgF2) spacer layer. The particles are formed by electron beam lithography and a standard lift-oﬀ process. We used a thin (∼8 nm) poly(methyl methacrylate) (PMMA, see Methods) ﬁlm as our dark resonance (P). Speciﬁcally, the vibrational mode associated with the CO stretch provides a clean, narrow (∼20 cm−1 full width at half-maximum) absorption band at ∼1730 cm−1. To control the center frequency of our bright resonance, Au particles of a rod geometry, as in Figure 3, were used such that their length could be adjusted to keep ωA ∼ ωP for the diﬀerent gaps. It was also necessary to account for the expected red shift due to the increase in refractive index (constant real term) surrounding the nanoantenna upon coating with PMMA (n = 1.53), apparent in comparing the before and after (red and blue curves) data in the ﬁgure. For comparison, we also include FDTD simulations of the same structures coated with the PMMA ﬁlms (right column). The ﬁeld intensity is expected to follow the same trend as for the similar PA examined in Figure 3. The spatial proﬁle throughout is similar to that of a typically
such as dipole cancellation,14 lattice modes7,40,41 and magnetic dipole generation.34,42 Given that one important motivation for these eﬀorts is to generate high quality factor resonances and, correspondingly, large ﬁeld enhancement factors for applications such as surface-enhanced absorption, the concepts presented in conjunction with Figure 2 are extremely signiﬁcant. They imply, ﬁrst, that depending on the plasmonic resonator design EIT or EIA like features may be observed. Second, the magnitude of this feature, and therefore the success of the plasmonic structure design, does not necessarily scale with the standard ﬁeld enhancement metric. An implementation of the well-known perfect absorber (PA) conﬁguration, using a dipolar rod antenna as shown in Figure 3a, oﬀers a convenient means with which to tune the ratio of
Figure 3. Absorption enhancement and induced transparency in a perfect absorber coupled to an absorbing ﬁlm. (a) Schematic of the PA structure. (b) Varying the gap thickness tunes external (γAe) and intrinsic (γA0). (c) Spectra of four diﬀerent PA structures, before (dashed lines) and after (solid lines) coating with a 10 nm thick ﬁlm with an absorption line at ∼1686 cm−1. Gap sizes of 80, 200, and 400 nm correspond to under-, critically-, and overcoupled structures. The gap size of 280 nm is selected to provide the minimum modulation.
the external to intrinsic loss rates and thereby demonstrate these eﬀects. The ﬁgure presents FDTD simulations of the structure, where the gap between the nanorod antenna and metal ﬁlm is varied (see Methods and Supporting Information, Note S3). The eﬀect of placing the dipolar antenna in close proximity to the metal ﬁlm is to induce image currents in the ﬁlm that are antiparallel to those in the antenna.42,43 The resultant circulating currents yield a magnetic dipole character response that is associated with a reduction in radiation damping.42 The strength of this eﬀect varies inversely with gap E
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Figure 5. A uniﬁed description of the transition from plasmonic EIA to EIT in terms of critical coupling. (a) Schematic of the plasmonic metamaterial structure. (b) Paths traced out along the CC curve upon introducing the quadrupole antenna at diﬀerent gap distances (the same dipolar antenna is used, hence the starting point is always the same). (c) Resultant absorption spectra for the three diﬀerent gap sizes.
Figure 4. (a) Experimental demonstration of under-, critically-, and overcoupled systems and their behavior upon coupling to a dark molecular absorption band. Experimental data for the undercoupled (UC), critically coupled (CC), and overcoupled (OC) cases are shown. In addition to the data, ﬁts to the theoretical model based on coupled mode theory are included (black lines). Corresponding FDTD simulations are shown in the right panel. (b) SEM image of the critically coupled sample. The scale bar is 2 mm. (c) FDTD ﬁeld intensity (|E/E0|2) enhancement for the CC structure.
natural dark mode is that the former allows for signiﬁcantly greater control over the coupling parameter, μ. First, by varying the gap between the quadrupolar and dipolar antennas the coupling rate, μ, can be directly controlled with a high degree of precision. Second, the larger inherent dipole moments associated with the plasmonic “meta-molecule” allow for much larger values of μ to be obtained. This dramatically higher modulation interval can itself lead the system through EIA to EIT regime without altering the decay rates of the bright mode. For the three sets of spectra plotted, the dipole antenna is kept constant, such that the system begins in the over coupled regime, while the gap is reduced from 350 to 100 nm. Thus, instead of selecting systems with diﬀerent values of γAe/ γA0, as in Figures 2−4, here we are ﬁxing γAe/γA0 and varying γμ (through its dependence on μ) and therefore the distance traveled along the CC curve, as shown in Figure 5b. This procedure results in, as shown in the spectra in Figure 5c, ﬁrst an EIA-like increase in absorption associated with a movement along the path in Figure 5b toward the CC peak (350 nm gap). Reducing the gap further increases γμ until the system is critically coupled, γAe = γA0 + γμ, and absorption is maximized at Abs. = 1 (240 nm gap). After this point, subsequent increases in coupling (100 nm gap) correspond to a movement over the top of the curve (panel b again), and the EIT eﬀect is observed. In conclusion the broad range of examples and analysis presented here demonstrate the critical need to appropriately treat the interaction with external radiation in coupled resonator systems. We show clearly that the relation between the excitation eﬃciency of a resonator and its external damping rate leads to widely diﬀerent spectral responses. Signiﬁcantly, completely contrary outcomes may be observed depending on experimental parameters (e.g., wavelength) and also the design of the nanophotonic resonator. Utilizing a model that links the drive term with the radiative decay rate via time reversal symmetry and energy conservation, we develop an intuitive
isolated rod-antenna, with the intensity conﬁned primarily to the tip ends as shown in Figure 4c. The data and simulations clearly reproduce the trends observed in Figure 3c with reﬂectance increasing (Abs. = 1 − R, decreases) for the under and critically coupled cases (gap thickness, 75 and 185 nm respectively). When the gap is increased to 314 nm, however, a clear decrease in reﬂectance (increase in absorption) is observed. Moreover, the curves generated by ﬁtting our theoretical model (using eqs 3 and 6, see Supporting Information, Note S4) are in close agreement (see black lines in the ﬁgure) with the measured and simulated data. In addition to serving as an experimental validation of our predictions, these results thus demonstrate that in optimizing resonators for surface-enhanced spectroscopy applications, our approach oﬀers new insight into their design and the subsequent interpretation of measured spectra. While Figures 2−4 focused on the coupling between an engineered plasmonic resonance and a natural molecular one, the TCMT framework, and hence the intuition and predictions derived from it are completely general and should apply to an arbitrary physical coupled resonator system. We conclude by illustrating the occurrence of identical eﬀects in a wholly plasmonic system, the prototypical dolmen structure, shown in Figure 5. In this case, the bright resonator remains a dipolar nanoantenna but the dark mode now corresponds to a quadrupolar antenna formed by two closely spaced identical Au bars (see Figure 5a). Although this system is also described by eq 6, an important diﬀerence between a plasmonic and F
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We selected parameters for the absorbing ﬁlm to be representative of J-aggregates, as in ref 32. The representative values are ω0 = 15 000 cm−1, Γ = 420 cm−1, and S = (4200)2 cm−2. Since, however, we are interested in the eﬀects of changing wavelength while keeping ωA = ωP, as the resonance frequency of the ellipsoid was varied, we altered ω0 such that the two resonances were aligned. While varying the frequency of our model absorber in this manner, we additionally varied Γ so as to hold Γ/γA, and therefore the relative line-widths of the two resonances constant. In conjunction with keeping the thickness of the absorbing ﬁlm constant at 5 nm, this kept the theoretical value of the coupling rate, μ (see Supporting Information, Note S2) roughly ﬁxed throughout. In order to calculate the relative damping rates described in conjunction with Figure 2, we note that in the coupled mode theory, γAe and γA0 are proportional to the power dissipated by scattering and absorption, respectively. Since the absorption and scattering cross sections are also proportional to the power dissipated by their respective mechanisms, γAe/γA0 = Csca/Cabs. The coupling parameter μ can be calculated on the basis of an overlap integral as described in Supporting Information, Notes S1−2. Numerical Simulations. FDTD calculations were performed using a commercial software package (FDTD Solutions ver. 5.5, Lumerical). The refractive index of the MgF2 taken to be a constant 1.38 throughout, while the material properties of Au were taken from ref 47. The Au ﬁlm and dipolar antennas were simulated as 100 nm thick throughout (the nominal thickness in our experiments) without any chromium (Cr) adhesion layer. All simulations were performed on a single unit cell of a periodic array (i.e., using periodic boundary conditions) of a size 3 × 3 μm. The incident source was simulated as a normally incident plane wave. Sample Fabrication. Our PA samples were fabricated via electron beam lithography (EBL) and a standard lift-oﬀ process. Silicon chips were ﬁrst coated with a 100 nm thick Au ﬁlm (5 nm thick chromium, Cr, adhesion layer) with in an electron beam evaporator. Various spacer layer thicknesses were then formed by subsequent evaporation of MgF2. A spectroscopic ellipsometer (JA Woolham) was used to measure the thicknesses of the deposited dielectric throughout. The dipolar antenna resonators were formed by EBL on these substrates, using a scanning electron microscope (Supra 40VP, Zeiss), NPGS pattern generation system (NPGS, Nabity) and a standard lift-oﬀ process. PMMA (950 A5, Michrochem) was used as resist. Prior to EBL, the resist was subjected to a hardbake on a hot-plate. Nominal dimensions for the three cases examined were as follows. Overcoupled: Length, L = 1600 nm. Critically coupled: L = 1500 nm. Undercoupled: L = 1500 nm. All were 200 nm wide and set in a periodic square lattice with a unit cell size of 3 μm. Sample development was carried out via immersion in a 3:1 isopropanol (IPA) to methyl-iso-butyl ketone (MIBK) mixture, then quenched with IPA. To form particles, a 5 nm thick Cr adhesion layer followed by 100 nm of Au was deposited on the samples via electron beam evaporation (CHA industries). Lift-oﬀ was then performed by immersing samples in acetone and IPA in conjunction with gentle sonication. Following lift-oﬀ an O2 plasma clean was used to remove any small residual amount of PMMA. To couple our PA resonators to an absorber a thin PMMA ﬁlm was cast on the substrates. PMMA (same as used in EBL) was diluted to 0.25% (weight/volume) in toluene and spun
view of the interaction between plasmonic bright and dark modes based on the idea of a critically coupled resonator. This picture allows these previously unaccounted for eﬀects to be treated in an accurate and predictive manner. In addition, the developed model can be instrumental in certain applications such as surface-enhanced spectroscopy experiments, where a signiﬁcant signal increase can be achieved by a careful engineering of the plasmonic resonators’ damping rates. Finally, in addition to the consequences demonstrated here the fundamental nature of the critical coupling eﬀect implies the potential to form the basis for a wide range of new predictions. While we have focused here primarily on a single port, single bright/dark resonator system, the concepts examined and demonstrated here can readily be extended to various combinations of multiple ports and resonators. Additionally, the generality of the TCMT formalism makes it applicable to numerous diﬀerent physical implementations of resonators. These may motivate new experimental studies, applications, and approaches to designing plasmonic, metamaterial and other resonator-based devices. Methods. Analytical Scattering Calculations. The analytical scattering calculations presented for the prolate spheroids in Figure 2 were obtained via a quasi-static approximation, corrected for radiation damping. In particular, the polarizability of a prolate spheroid (c = b) is given by44 α=
ε1 − εm 4πabc , 3 εm + L(ε1 − εm)
1 − e2 ⎡ 1 ⎛⎜ 1 + e ⎞⎟⎤ ln ⎢−1 + ⎥, 2 2e ⎝ 1 − e ⎠⎦ e ⎣
e2 = 1 −
for the E ﬁeld polarized along its long (a) semiaxis, such that the induced dipole moment is given by, p⃗ = ε0εmαE⃗ 0, where E0 is the driving ﬁeld. Equation 7 does not, however, account for radiation damping, which is a key component of our analysis. This is corrected for by introducing the radiation reaction ﬁeld, E⃗ self = jk 3 /(6πε 0 ε m )p⃗ , (k = 2π/λ is the wave-vector amplitude).22 Noting that the induced dipole moment is driven by the total ﬁeld, p⃗ = ε0εmα(E⃗ 0 + E⃗ self) we obtain, p ⃗ = ε0εm
1 − α 6π
Equating eq 9 to p⃗ = ε0εmαeffE⃗ 0, an eﬀective polarizability, accounting for radiation damping eﬀects is obtained. From this polarizability, extinction (Cext), scattering (Csca), and absorption (Cabs) cross sections can be obtained. In particular, Cabs = Cext − Csca = k Im(αeff) − k4/6π|αeff|2.45,46 In Figure 2, spheroids with dimensions b = 25 nm and a = 30, 60, and 90 nm were used to scan the resonant frequency from 26 000, 21 000 and 18 000 cm−1 (385, 476, and 556 nm wavelength, respectively). For the calculation of spectra of the coated ellipsoid, the analytical formula extending (eq 7) to a multilayered structure in ref 44 was used and eq 9 used to again account for radiation damping. For the material properties, the permittivity of Ag was taken from a Lorentz+Drude model ﬁt to the data in ref 47. The absorber was modeled as a Lorentz oscillator with permittivity ε(ω) = 1 +
S − ω 2 − jω Γ
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onto the samples. The thickness was measured using a spectroscopic ellipsometer (JA Woolham) to be ∼8 nm. Data Collection. All data were collected using a Fourier transform infrared (FTIR) spectrometer (IFS 66/s, Bruker Optics) and IR microscope (Hyperion 1000, Bruker Optics). All data were taken with a mercury−cadmium telluride (MCT) detector under identical acquisition settings speciﬁcally a mirror velocity of 40 kHz, 256 scans coadded, and 4 cm−1 resolution. A plastic enclosure around our microscope was purged with dry and CO2 ﬁltered air (Parker Hanniﬁn purge generator) to limit interference from water vapor lines. An IR polarizer was used to polarize the incident E-ﬁeld along the nanoantenna length.
S Supporting Information *
Additional information, ﬁgures, and tables. This material is available free of charge via the Internet at http://pubs.acs.org.
*E-mail: [email protected]
. Author Contributions ¶
R.A. and A.A. contributed equally.
The authors declare no competing ﬁnancial interest.
ACKNOWLEDGMENTS This work is supported in part by NSF CAREER Award (ECCS-0954790), ONR Young Investigator Award, and NSF Engineering Research Center on Smart Lighting (EEC0812056). The authors would like to acknowledge Professor Onuttom Narayan of the University of California, Santa Cruz for useful discussions.
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