1. Introduction

Inter-resonance interactions in pure photonic structures or photon-exciton hybrid structures have aroused great interest in the past few decades as these systems provide imitative research platforms for investigating ubiquitous resonances and interactions in the physical world [1,2]. At the weak and intermediate coupling regimes, the energy exchanging rates between resonances (e.g., light-light, light-matter) can hardly exceed their dissipations. Pure plasmonic nanostructures could take advantage of the asymmetries resulting from substrates, [3,4] particles’ materials, sizes, [5] arrangements [6] to tune the asymmetry parameter q (phase shift) [2,7]. Rich Fano resonance features from asymmetric lineshapes to Fano dips and electromagnetically induced transparencies (EIT) could be observed. EIT is a special case of Fano interference where only the continuum (a broad bright mode) could exchange energy with the far field, while the discrete mode (a narrow dark mode) could only transfer energy with the bright mode. The depth of the transparencies (dips) could be determined by the coupling strength between the two modes, and the linewidths of the dark plasmons. These features could also be explored in plasmon-exciton coupled structures where the excitons are usually qualified as a subradiant (dark) mode compared with the superradiant plasmons. This exciton-induced transparency indicates that the coupling between plasmons and excitons exceeds the weak Purcell regime and enters the intermediate coupling [8]. With high dip-to-peak contrast and narrow linewidth, Fano structures could find their wide applications in light filtering, slowing, and switching, [9–12] as well as refractive index (RI) sensing and plasmonic ruler (displacement sensing) [13]. Moreover, multiband EIT has also been intensively investigated in multi-resonator systems, e.g., microcavities, [9,14] photonic crystals, [15] metamaterials, [16,17] etc., due to their potential for modulating light over multiple frequency bands.

As the coupling strength exceeds the total linewidths of two bare modes, two new eigenmodes will emerge in the spectra presenting the Rabi splitting in the strong coupling regime. Newly-emerged dressed modes could inherit and combine the characteristics of their parental bare modes, manifesting the newly hybridized modes as a good compromise in many realistic applications [18]. Presenting similar “two-peak” features in lineshapes, strong coupling (i.e., Rabi splitting) is another viable strategy for generating transparency windows, although with more rigorous coupling conditions. For pure photonic nanostructures, strong coupling [19,20], and even ultrastrong coupling [21] were realized. For the plasmon-exciton hybrids, plexcitons namely, strong Rabi splittings were observed in many structures, paving the road for cavity quantum electrodynamic (cQED) realizations. Strong coupling between diverse emitters and photonic structures has experimentally observed single [18,22] and multiband [23] strong couplings. Yet, they usually have a relatively lower dip-to-peak contrast for plasmonic structures [18,23] due to the large damping of the plasmons.

Simplification and miniaturization are two significant targets in designing and fabricating nanostructures. From the former aspect, structural complexity is always of great consideration because, among the aforementioned strategies, multiband EIT devices may require a combination of various elements, e.g., numerous microcavities and waveguides, [9,14] designed metamaterials, [24] multiple photonic crystals [15]. Multiple photonic elements are required because each component generally supports a single resonance. As for miniaturization, the sizes of the aforementioned conventional photonic devices are mostly limited by the diffraction limit. With deep subwavelength footprints, the plasmonic nanostructures are the solution for both ultrasmall and ultrafast nanophotonic devices at room temperature [25]. And the hybridized plasmonic structures could support rich plasmonic resonances in a single-particle-level structure, [26–29] benefiting the demanded multiband EIT coupling. However, multiband EIT in a single-particle level configuration was less studied in the past, making the ultracompact and ultrasmall multiband EIT devices an open question.

In this paper, we explored a strong, self-induced EIT window in a nanocube-hexagonal nanoplate heterodimer which is proved to be chemically feasible by experiment with a high yield [28,29]. We will first reveal and stress the physics within the couplings by a simplification that the structure situates in a uniform dielectric environment n = 1, and further prove its validity with realistic experimental conditions (e.g., in water, or on dielectric substrates). Unlike the previous paper [29] that viewed the nanocube as a suppression for the induced transparency, we achieved deep transparency by utilizing the coupling between two lowest-order modes – bonding dipole-quadrupole mode (i.e., subradiant, transverse cavity mode) [30,31] and bonding dipole-dipole mode (i.e., superradiant antenna mode) [31]. These lower-order modes are supposed to be more sensitive than the higher-order modes, and generally have a larger interaction with light. The coupling is believed to situate in the intermediate coupling regime, as the fitted maximum 4 g/(Γ₁ + Γ₂) = 0.97 is right close to the strong coupling criteria (g means coupling strength, and Γ₁, Γ₂ are the linewidths of two resonances forming the coupling). Though the Fano interference and transparency are ubiquitous in many hybrid structures, [27,29,32] a strong Fano interference adopting the excellences of the fundamental cavity and antenna modes is generally hard to achieve due to the resonance mismatch [27,32]. Here, by modifying the size of the heterodimer, one could easily match the energy of two lowest-energy modes and present their evident and optically-accessible features in the scattering spectra. This could help fully realize the ultra-sensitivities of the nanocavity modes formerly only observed in absorption spectra, [30,32] featuring this EIT as highly tunable.

By further applying emitters to the heterodimer (J-aggregates excitons chosen here), we realized multiband exciton-induced transparencies where the J-excitons strong couple with the dark modes and form two separated absorptive bands. And the redundant “uncoupled” excitons (i.e., those not forming strong coupling with the dark mode) situated near the weak plasmonic field area give rise to an extra transparency window to the bright mode obeying the pure exciton-induced regulation. Overall, by taking advantage of the intermediate coupling between plasmonic modes, and accessing the strong coupling by introducing excitons, we could generate the multiband EIT in a single-particle-level structure. Free of complicated designed structures, these multiband EIT windows are highly tunable depending on the RI and geometry separations, thanks to the outstanding properties of the bonding dipole-quadrupole mode [13,30]. Apart from modulating light over multi-frequency bands, this structure bearing rich and multiscale inter-resonance couplings (both intermediate and strong) could also serve as a basic cQED testbed for investigating many-body interactions.

2. Structure and theoretical model

Asymmetric nanostructures and material components have long been exploited to generate strong Fano features in the spectra. Here, to take advantage of the low-loss ultrasensitive dark cavity plasmons [30] in the interaction, we theoretically devised an asymmetric hybridized structure consisting of a Ag nanocube (AgNC) on a Au hexagonal nanoplate (AuNH). As sketched in Fig. 1(a), this heterodimer is composed of a 60 nm large AgNC (L_c = 60 nm) with a 150 nm large (L_h = 150 nm), 80 nm thick (H = 80 nm) AuNH. The heterodimer’s modes are hybridized from the individual AgNC and AuNH resonances (see charges distribution in Figs. 1(b1), (b2)) depending on the interparticle gap G. Here, the finite-element method was used to simulate the structure. To avoid the calculation singularities and non-convergencies at the sharp corners of the NC and NH, rounding curvatures of 5 nm and 10 nm radius were applied to the NC and NH, respectively. The surrounding media was the air with a RI of n = 1. It has to be stressed that we here consider n = 1 for computational convenience to better reveal the physics in the coupling, and more realistic cases with heterodimers in water (n = 1.33) and on a substrate (MgF₂ n = 1.37 or SiO₂ n = 1.5) were computed and compared later in the main text and SI. With only some energy differences, they all show very similar coupling responses with the same physics inside. Therefore, we believe that this simplification of the surrounding media would be valid and acceptable. On the other hand, the experimentally-accessible parameters could guide future experiments. The permittivity of Ag and Au was taken from the Drude model, $\varepsilon (\omega ){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = {\kern 1pt} {\kern 1pt} {\varepsilon _\infty }{\kern 1pt} - {\kern 1pt} {\kern 1pt} {\kern 1pt} \omega {\kern 1pt} _\textrm{p}^2{\kern 1pt} /({\omega ^2} + i\omega \Gamma {\kern 1pt} {\kern 1pt} {\kern 1pt} )$, with the high-frequency limit of Ag (Au) ɛ_∞= 4.039 (9.5), bulk plasma energy ω_p = 9.172 eV (8.95 eV), and the damping Γ = 0.0207 eV (0.069 eV), respectively. By neglecting the interband transition through a Drude model for the sake of clarity, those higher-order plasmons usually overwhelmed by prominent interband damping (e.g., resonances below 400 nm) could be carefully analyzed in our calculations.

 

Fig. 1. (a) The sketch of the heterodimer consisting of a AgNC and a AuNH. The modes of single NC and NH resonances were labeled according to the charge distribution shown in (b1, b2). Here we show six dipolar modes $\mathrm {D^{1\sim 6}_c}$ and two quadrupolar modes $\mathrm {Q^{1\sim 2}_c}$ for an individual NC. Similarly, major modes of individual NH were shown: dipolar $\mathrm {D^{1\sim 2}_h}$, and quadrupolar $\mathrm {Q^{1\sim 2}_h}$. (c) Calculated scattering (blue line) and absorption (red line) cross-sections of an individual AgNC (upper, multiplied by 3 times), AuNH (lower), and their coupled AgNC-AuNH heterodimer (d) with an interparticle gap G = 4 nm. An evident induced-transparency window appears around 607 nm. Blue shades in (c) and (d) mark the less-coupled region where resonances in heterodimer (d) could generally be linked to the resonances of single AgNC (c).

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To demonstrate the optical properties of our hybridized structures, we respectively compared the absorption and scattering cross-sections of a single NC, single NH, (Fig. 1(c)), and their heterodimer (Fig. 1(d)), excited by an S-polarized, 53° oblique plane wave. The heterodimer shows multiple resonances which could be understood from the individual charge distributions and their hybridizations [33]. For clarity, we will start by identifying the eigenmodes of the individual AgNC [3] and AuNH in Fig. 1(c) by their charge distributions (Figs. 1(b1) and (b2)). By building up the correspondence between resonances and charge distributions, we found the AgNC majorly has six bright modes at 403 nm ($\mathrm {D^1_c}$), 342 nm ($\mathrm {D^2_c}$), 330 nm ($\mathrm {D^3_c}$), 310 nm ($\mathrm {D^4_c}$), 301 nm ($\mathrm {D^5_c}$), 292 nm ($\mathrm {D^6_c}$), and two dark modes at 376 nm ($\mathrm {Q^1_c}$), and 337 nm ($\mathrm {Q^2_c}$). As shown in Fig. 1(b1), the AgNC’s lower-order bright modes mainly have induced charges on the corners and edges, while higher-order modes (from $\mathrm {D^4_c}$) have few charges on the surface. Positive and negative charges reside on the opposite sides of the AgNC, ensuring a large electric dipole moment and well interaction with a free-space plane wave [3,32]. Besides, two dark modes observed have a quadrupole-like anti-symmetric charge distribution with a near-zero electric dipole moment. Dark modes in principle have zero dipole moment and could not interact with a plane wave. But for our realistic system with a finite size beyond quasi-static approximation, the dark modes could be weakly excited by phase retardation, which accounts for the observation of the higher-order dark modes in our excitation setup.

Likewise, we observed two bright modes at 580 nm ($\mathrm {D^1_h}$), 471 nm ($\mathrm {D^2_h}$), and two dark modes at 500 nm ($\mathrm {Q^1_h}$), and 445 nm ($\mathrm {Q^2_h}$) in a single NH (Fig. 1(b2)). These four modes majorly contribute to the scattering and absorption energy in the spectra, with charges distributed in a very similar manner to the NC.

The scattering (blue) and absorption (red) cross-sections of the heterodimer (G = 4 nm) in Fig. 1(d) show two distinct regions where a less-coupled region lies below around 450 nm (marked by blue shades). In this region, the spectra are the sum of NC and NH with almost no energy shifts (i.e., no mode hybridizations). Those fine features in the spectra of a heterodimer could have a one-to-one correspondence to the single NC resonances marked by $\mathrm {D^1_c}$–$\mathrm {D^6_c}$ and $\mathrm {Q^1_c}$–$\mathrm {Q^2_c}$. Moreover, those heterodimers’ charge distributions (see Supplement 1 Fig. S1) prove that almost no electron density was distributed on the AuNH, and only the AgNC has distinct charge distribution as the single NC case. Therefore, we name this part the less-coupled region. Beyond this region, mode hybridizations from NC and NH were observed, with a deep (scattering near zero) and wide (full width at half maximum, FWHM = 38 nm, corresponding to 146.8 meV) EIT window near 607 nm coming into being. This differs from the viewpoint [29] that they believed the NC would degrade the Fano transparency of the heterodimer. The reason for this deviation comes from the different modes and different polarized excitation we utilized in the structure.

To better reveal the role played by the mode coupling in this strong EIT, in Figs. 2(a) and (b), we calculated the scattering and absorption cross-sections of the AgNC-AuNH heterodimer with various interparticle gaps. Theoretically examining the gap dependence of the plasmonic resonances is a powerful tool for understanding the coupling-determined phenomena (e.g., plasmon hybridization) [3,32]. We studied this relationship for exploring the origin of the EIT here. Note that though nontrivial, one could still have various methods for experimentally tuning the gap distance. [34–36] The blue shadings mark the less-coupled region (same as Figs. 1(c), (d)) we would skip, and the coupled region is what we need to pay attention to. When the G = 20 nm, the interaction is small enough to be neglected, and the spectra would evolve back to the sum of the individual components (the uncoupled situation, i.e., G = ∞). Therefore, for comparison, we put individual (uncoupled, G = ∞) NC’s and NH’s mode name tags on the top of Figs. 2(a),(b) as a reference, which may help us clarify the origin of the heterodimer’s modes. As gradually shrinking down the gap G from 20 nm to 1 nm, we could observe a rapidly red-shifting EIT window (black markers) in Fig. 2(a), which corresponds to a sharp and intense absorption peak (e.g., FWHM = 60 meV for G = 2 nm) in Fig. 2(b). Meanwhile, a very broad and bright mode (blue markers) could be found in the scattering. These two modes constitute the two essential elements for a Fano interference, i.e., a broad continuum with a discrete level. As the interaction is strong enough (e.g., G ≤ 7 nm), this broad superradiant mode splits into two peaks following the EIT manner (Fig. 2(a)).

 

Fig. 2. Scattering (a) and absorption (b) cross-sections of a AgNC-AuNH heterodimer with different gap distances. We set stacking offsets for both spectra for a clearer demonstration: 1.3 × 10⁵ nm² for scattering, and 1.8 × 10⁴ nm² for absorption, respectively. The blue shades correspond to the less-coupled region. The dashed lines indicate that these heterodimer modes in the less-coupled region mostly originate from the individual uncoupled nanoparticles. (c), (d), and (e), (f) are the charge distributions and electric field enhancement of the $\mathrm {BD^1_c}$$\mathrm {Q^1_h}$ mode with G = 3 nm and $\mathrm {BD^1_c}$$\mathrm {D^1_h}$ mode with G = 15 nm, respectively. The geometries showing charge distributions are intentionally twisted for clearer illustration. The middle panels of the electric field, (e), and (f) are the central cut-plane of the gap region, while the lower panels track the field changes at y = 0.

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By analyzing the charge distribution (Figs. 2(c),(d)) and tracking the mode evolvement (Figs. 2(a),(b)), we attribute this dark and sensitive resonance (black markers) to the bonding NC’s dipole-NH’s quadrupole ($\mathrm {BD^1_c}$$\mathrm {Q^1_h}$) mode (Fig. 2(c)), while the broad and insensitive resonance should come from the commonly studied bonding dipole-dipole ($\mathrm {BD^1_c}$$\mathrm {D^1_h}$) mode (Fig. 2(d)). Because the $\mathrm {BD^1_c}$$\mathrm {Q^1_h}$ mode is mostly dark and has nearly no energy exchanged with the free-space excitation (subradiant, no scattering), the Fano feature has a q ∼ 0 anti-resonance feature (EIT) [37]. Besides, this dark and sharp $\mathrm {BD^1_c}$$\mathrm {Q^1_h}$ mode is believed to have the lowest damping one could ever achieve because this mode has reached the quasi-static decay limit determined by the lossy material [30,38].

$\mathrm {BD^1_c}$$\mathrm {D^1_h}$ and $\mathrm {BD^1_c}$$\mathrm {Q^1_h}$ modes have the same modal symmetry but different charge distributions. The major difference between those two modes is that the $\mathrm {BD^1_c}$$\mathrm {Q^1_h}$ only has charges in the nanogap. Only a very slight charge distribution was observed on the top of the NC, lengths, and corners of the NH [27,30,31]. The compact and antisymmetric quadrupole-like charge distribution of $\mathrm {BD^1_c}$$\mathrm {Q^1_h}$ gives a cancelation of the dipole moment, revealing a dark plasmon performance as a cavity mode. Oppositely, in Fig. 2(d), the $\mathrm {BD^1_c}$$\mathrm {D^1_h}$ mode can be termed as an antenna mode and have the charges distributed all over the structure following the excitation polarization, thus leading to a giant dipole moment and bright mode nature.

We also plot the electric field enhancement of these two modes (Figs. 2(e) and (f)), where the $\mathrm {BD^1_c}$$\mathrm {Q^1_h}$ mode (G = 3 nm) holds the electromagnetic energy in a nanometer-scale gap with more than 160 times of E-field enhancement. Whereas the $\mathrm {BD^1_c}$$\mathrm {D^1_h}$ mode (G = 15 nm) has a strong dipolar signature imprinted by the large NH’s dipole, with the electric field being enhanced by around 10 folds. The reason for choosing this large separation (G = 15 nm) is to avoid spectral overlap between two modes when the gap becomes smaller. If they overlap, the intense $\mathrm {BD^1_c}$$\mathrm {Q^1_h}$ may overwhelm the distribution of $\mathrm {BD^1_c}$$\mathrm {D^1_h}$, hindering us to map out the charge and field distribution of $\mathrm {BD^1_c}$$\mathrm {D^1_h}$. The charge and field distributions (Figs. 2(c)-(f)) prove that $\mathrm {BD^1_c}$$\mathrm {Q^1_h}$ and $\mathrm {BD^1_c}$$\mathrm {D^1_h}$ have the same symmetry (mirror reflection), which accounts for the efficient interference and strong Fano transparency feature in the scattering.

In Fig. 2(a), we could also observe several small Fano dips in the scattering around 600 nm (G = 1 nm), whose dark plasmons could be correspondingly found in the absorption (Fig. 2(b)) as peaks. Those are higher-order nanocavity modes (details see Supplement 1 Fig. S2) that have the same symmetry as $\mathrm {BD^1_c}$$\mathrm {D^1_h}$ (having a dipole-like distribution). Note that there are also absorptive plasmons (e.g., $\mathrm {BQ^1_c}$$\mathrm {Q^1_h}$ mode, see Supplement 1 Fig. S2) that have no Fano effects on the scattering. It is due to the phase differences and inefficient interference between the modes (symmetry mismatch). As shown by Supplement 1 Fig. S2, the $\mathrm {BQ^1_c}$$\mathrm {Q^1_h}$ would have quadrupole-like distribution, which is orthogonal to the dipole mode and have weak interference. All in all, the mode (phase) matching between resonances takes charge of the coupling, which determines the spectra from a strong EIT (e.g., two symmetry-matched modes $\mathrm {BD^1_c}$$\mathrm {D^1_h}$ and $\mathrm {BD^1_c}$$\mathrm {Q^1_h}$) to a bare summation of two orthogonal modes (e.g., two symmetry-mismatched modes $\mathrm {BD^1_c}$$\mathrm {D^1_h}$ and $\mathrm {BQ^1_c}$$\mathrm {Q^1_h}$).

Tunability is a fundamental requirement for a high-performance EIT device. Due to the intermediate coupling [39] (proven later) between $\mathrm {BD^1_c}$$\mathrm {D^1_h}$ and $\mathrm {BD^1_c}$$\mathrm {Q^1_h}$, the coupled EIT feature would inherit the outstanding optical performance from $\mathrm {BD^1_c}$$\mathrm {Q^1_h}$. It is almost, to our best knowledge, the most sensitive mode against the gap distance and surrounding environment (RI) [30,40]. This property would be both capable of helping build up actively or passively tunable EIT devices by inserting suitable materials [41] into the nanogap (e.g., phase-change, [42] photochromic [43] material). To prove and optimize this tunability, in Fig. 3(a), we investigated the gap distance sensitivity and studied its dependence on the material (Au/Ag, the two most used noble metals). The gap distance could be tuned following various methods. [34–36]

 

Fig. 3. (a) Peak positions of the $\mathrm {BD^1_c}$$\mathrm {Q^1_h}$ mode of the heterodimers constructed by different metals, as a function of gap distance. (b) The coupling strength g, 4 g/(Γ₁+ Γ₂) as a function of gap distance.

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The peak positions of the $\mathrm {BD^1_c}$$\mathrm {Q^1_h}$ mode in AuNC-AuNH, AgNC-AuNH, and AgNC-AgNH heterodimer are extracted in Fig. 3(a) as a function of gap distance. In addition, the red dashed line is the BDQ mode of a 36 nm AgNC dimer as a Ref [30]. As the gap decreases, the modes would rapidly redshift following a near-exponential law [44]. To better evaluate its sensitivity to the gap distance, we compared the slope k = dλ/dG at G = 1 nm (inset of Fig. 3(a)). From our previous work, [45] we analytically understand that the gap sensitivity would be much dependent on the material according to its bulk plasma frequency ω_p in the Drude model. Therefore, Ag is supposed to show a better performance than Au due to the larger ω_p. Here, we found that the sensitivity slopes were 32.5 nm/Å for AuNC-AuNH, 33.5 nm/Å for AgNC-AuNH, and 36 nm/Å for AgNC-AgNH (the best), accordant with our speculation. As a comparison, for a nanocavity mode in nanowire-on-mirror systems, the experimental sensitivity was 14 nm/Å at G = 1 nm in the scattering spectrum, [40] and that of a AgNC dimer was 50.5 nm/Å theoretically (only in absorption spectrum due to its total darkness in scattering) [30]. So far, such high gap sensitivity has less been reported in a scattering setup yet, the $\mathrm {BD^1_c}$$\mathrm {Q^1_h}$ in heterodimer on one hand provides an efficient way to tune the EIT band, and on the other hand, manifests itself as an ultrasensitive plasmonic ruler [46] compatible with the scattering apparatus setup.

The coupling strength is an essential, quantitative ingredient in Fano transparency, which masters and evaluates the performance of EIT. By fitting the scattering of heterodimers following the temporal coupled mode theory (Eq. (1)), [47]

 

Fig. 4. (a) The scattering cross-section of the AgNC-AuNH heterodimers (G = 4 nm) with the changing RI. (b) The coupling strength g, 4 g/(Γ₁+ Γ₂) as a function of the RI [n = 1.0 (air), 1.3260 (methanol), 1.3970 (butanol), 1.4458 (chloroform), and 1.5 (oil)].

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As another tunable parameter that determines the EIT energy band, we hereby investigate the influence of the RI on the heterodimers (Fig. 4(a)). Since BDQ mode in AgNC dimer structure has shown great RI sensitivity, [30] we expect our heterodimer $\mathrm {BD^1_c}$$\mathrm {Q^1_h}$ should be sensitive enough for a highly tunable EIT. Here we chose five representative experimentally accessible material for demonstration: air n = 1, methanol n = 1.326, butanol n = 1.397, chloroform n = 1.4458, oil n = 1.5. As RI increases, the EIT window does rapidly redshift and FWHM expands due to the dielectric screening effect. Due to the different RI sensitivity of $\mathrm {BD^1_c}$$\mathrm {D^1_h}$ and $\mathrm {BD^1_c}$$\mathrm {Q^1_h}$ modes, the two modes would lose spectral overlap (off-resonance) as the RI increases. As a result, the coupling strength g (blue markers) gradually decreases with the increasing RI in Fig. 4(b). The decreasing g and increasing damping rate concurred to bring about the persistently dropping of 4 g/(Γ₁+Γ₂) in Fig. 4(b), revealing a transition of intermediate coupling to a weaker situation (red markers).

To quantitatively assess the performance of our EIT against environment RI, we could define a figure of merit (FOM) analogous to plasmonic sensing [49]. The FOM of the resonance is defined as the ratio of the peak shift per unit RI change (S = dλ/dn) to the FWHM, [49] $\textrm{FOM}{\kern 1pt} = {{S(\textrm{nm} \cdot \textrm{RI}{\textrm{U}^{\textrm{ - 1}}})} / {\textrm{FWHM}}}$. To retrieve the FWHM of an asymmetric Fano resonance, the FWHM is defined as the difference between the EIT window (dip) and the longer wavelength peak [50]. Likewise, we could examine the heterodimers with three different material combinations: AuNC-AuNH, AgNC-AuNH, and AgNC-AgNH. As shown in Fig. S5 in Supplement 1, the AgNC-AgNH heterodimer has the highest FOM among those three geometries, ranging from 25 (FWHM calculated from n = 1.5) to 42 (FWHM calculated from n = 1.0). These sensitivities against surrounding RI provide a toolbox for tuning the EIT band in the energy diagram with various materials. On the other hand, this Fano-induced feature in the scattering spectrum is suitable in RI sensing (maximum FOM ∼ 42), which is comparable to other state-of-the-art single-particle sensors [13].

Since experimentally acquiring the spectra usually requires nanoparticles in solution or on a substrate, we checked the validity of our proposal in realistic conditions (Fig. S6 in Supplement 1). In Fig. S6, the AgNC-AuNH heterodimer with a 2 nm molecular gap (n = 1.4) was deposited in the air, water, and on the MgF_2­, SiO₂ substrate. We could find the redshifting of the plasmon resonances, but they all show distinct EIT bands, verifying the validity of our proposal. Since we have a tremendously large variety of emitters covering the range from ultraviolet to infrared, one may demand a methodology for tuning the plasmons to match with the emitters of choice. As shown in Supplement 1 Fig. S7, we could intuitively tune the EIT band by tuning the size of the heterodimer. The resonance of the $\mathrm {BD^1_c}$$\mathrm {Q^1_h}$ mode (EIT frequency) is determined by the size of the NC, while the superradiance $\mathrm {BD^1_c}$$\mathrm {D^1_h}$ mode depends on the NH’s size (details see Supplement 1 S5, Fig. S7). All in all, we proved that evident and deep EIT could be realized in realistic configurations, and it is highly tunable.

3. Experimentally realistic design allowing multiband EIT

Strong coupling is a versatile method for generating new eigenmodes which could maintain the excellence of their parental components [18]. By introducing an excitonic material (J-aggregate) to our already optically hybridized system and accomplishing the strong coupling, we “duplicate” the strong EIT band window into multiple EIT windows (two or three according to the exciton density). Apart from previous tuning methods, the energies of the multiband EIT here could be tunable by varying the density of the excitons and the plasmon-exciton detuning, which is beneficial for EIT-based on-chip nanophotonic devices, especially for multi-frequency modulating.

In this section, we adopted parameters and configurations which are usually adopted in experiments. The AgNC and AuNH were covered by 1 nm J-aggregate with dielectric function following the Lorentzian oscillator model: $\varepsilon (\omega ){\kern 1pt} {\kern 1pt} {\kern 1pt} = {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\varepsilon _\infty } - {\kern 1pt} {\kern 1pt} {{f{\omega _\textrm{e}}^\textrm{2}} / {({{\omega^\textrm{2}} - {\omega_\textrm{e}}^\textrm{2} + {\kern 1pt} i{\gamma_\textrm{e}}\omega } )}}$, where ɛ_∞ = 1 is the high-frequency permittivity, ω_e = 1.79 eV (693 nm) is the excitonic frequency. f ranging from 0 to 0.2 here is the oscillation strength which may be linked with the exciton density, and γ_e = 52 meV is the damping [51]. The nanogap between NC and NH is 2 nm, and the J-aggregate covered heterodimer was deposited on a MgF₂ substrate. To match the J-band of the excitons, we could tune the size of the heterodimer following the designing rule demonstrated in Fig. S7 in Supplement 1. The length of the AuNH and AgNC used here is 170 nm and 55 nm, respectively. There are a lot of parameter combinations for realizing multiband EIT, we only adopt a general case for demonstration.

From Fig. 5(a), we could conclude that three transparency windows ($|{{\omega_ + }} \rangle$, $|{{\omega_ - }} \rangle $, and $|{{\omega_{\mathrm{e}}}} \rangle $) in scattering (dashed lines) are all accordant with the absorption peaks (Fig. 5(b), dashed lines), revealing a Fano-like coupling mechanism. And considering the further Rabi splitting behavior brought by the increasing f of the excitons, these multiband transparency windows could be attributed to so-called exciton-induced-transparencies. When the oscillation strength f = 0 (without excitons), only one single window will appear located near 690 nm due to the pure plasmonic interference between the aforementioned $\mathrm {BD^1_c}$$\mathrm {D^1_h}$ and $\mathrm {BD^1_c}$$\mathrm {Q^1_h}$ modes. But as f becomes non-zero, the window gradually splits into two, and finally to three. This could be explained by the coupling mechanism shown in the energy diagram Fig. 5(g), where the J-aggregate excitons are strongly coupled with the dark $\mathrm {BD^1_c}$$\mathrm {Q^1_h}$ mode and form two discrete branches ($|{{\omega_ + }} \rangle $ and $|{{\omega_ - }} \rangle $). Inheriting the subradiant property from the dark plasmons and J-excitons, these two modes are all dark and sharp. Thus they brought about two transparencies on the superradiant continuum ($\mathrm {BD^1_c}$$\mathrm {D^1_h}$), at the energy ω₊ and ω_-. In this case, two windows were presented in the spectrum.

 

Fig. 5. The total scattering (a) and absorption (b) of the heterodimer composed of a AgNC and a AuNH (both covered by 1 nm thick J-aggregates) with different oscillation strengths f. The heterodimer was deposited on a MgF₂ substrate. The total absorption could be decomposed into the absorption in metal (c) and J-aggregates (d), respectively. For analyzing the three bands in absorption, we calculated the normalized loss density j · E at a strongly coupled (e) and a weakly coupled (f) point in J-aggregates, respectively. Strongly and weakly coupled points are at the side and central points under the cube, respectively. (g) Energy diagram explaining the coupling mechanism of the multiband EIT. (h) The scattering cross-section (f = 0.06) with different NC angles and positions shows the robustness.

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And when the exciton density became even larger (larger f), as in many other works, [48,52] we also found an extra branch gradually coming out from the energy of the excitons. Because the plasmonic field is inhomogeneous, the excitons covering the whole geometry would have coupled and uncoupled excitons according to their vacuum E-field intensity. The coupled excitons give birth to the splitting $|{{\omega_ + }} \rangle $ and $|{{\omega_ - }} \rangle $, while the redundant uncoupled excitons [52] won’t split and will couple with the $\mathrm {BD^1_c}$$\mathrm {D^1_h}$ in pure exciton-induced transparency [8] manner. That’s to say, the uncoupled (or weakly coupled) excitons at the central point under the NC (e.g., excitons at “a” point in Fig. 2(e)) fail to strongly couple with the dark $\mathrm {BD^1_c}$$\mathrm {Q^1_h}$. It is, in fact, directly coupled with the $\mathrm {BD^1_c}$$\mathrm {D^1_h}$ to form exciton-induced transparency. This could account for the middle branch of transparency (Fig. 5(g)). However, the strongly coupled excitons near the corner of the NC (e.g., excitons at “b” point in Fig. 2(e)) would majorly contribute to the hybridized branches (i.e., $|{{\omega_{\mathrm \pm }}} \rangle $ in Fig. 5(g)).

To prove this coupling mechanism, we separated the total absorption by the contribution from metal (Fig. 5(c)) and J-aggregates (Fig. 5(d)) by only integrating the Ohmic loss density j · E in their specific domains. From Fig. 5(c) we understand that the middle branch $|{{\omega_{\mathrm{e}}}} \rangle $ is truly related to the excitons due to its absence in the absorption in the metal, whereas this branch $|{{\omega_{\mathrm{e}}}} \rangle $ together with two branches of dressed plexcitons ($|{{\omega_{\mathrm \pm }}} \rangle $) are pretty much evident in the J-aggregate absorption (Fig. 5(d)).

To further prove the origin of the three branches and decompose them, in Figs. 5(e), (f), we calculated the normalized Ohmic loss densities j · E at the strongly coupled point (e.g., “b” point at the corner of the AgNC, see Fig. 2(e) lower panel) and weakly coupled point (e.g., “a” at the central gap below AgNC, see Fig. 2(e) lower panel). At the strongly coupled point (Fig. 5(e)), two evident dressed plexcitons ($|{{\omega_ + }} \rangle $ and $|{{\omega_ - }} \rangle $) were observed, and there were no contributions from the excitonic bare modes ($|{{\omega_{\mathrm{e}}}} \rangle $). For the weakly coupled situation (Fig. 5(f)), we could observe a mainly strong absorption by the excitons, and weak contributions from the dressed $|{{\omega_ + }} \rangle $ and $|{{\omega_ - }} \rangle $. The reason for observing hybridized $|{{\omega_ + }} \rangle $ and $|{{\omega_ - }} \rangle $ in weakly coupled situations is due to the enhancing effect from the plasmonic field. The Ohmic loss density is proportional to Im(ɛ)|E|², where the field enhancement |E|² could reshape the Lorentzian lineshape of Im(ɛ) by introducing two shoulders. Previous works [48,52] have also observed such a mysterious third branch dependent on the excitons, they recognized it as the excitonic effect when f is larger enough so that the Re(ɛ) < 0, and the absorbance Im(ɛ) becomes dominant over metal. Or when the Re(ɛ) remains larger than zero, the volume of the excitonic material is large enough that the unhybridized excitons form this band. All in all, this middle branch $|{{\omega_{\mathrm{e}}}} \rangle $ is believed to be uncoupled-exciton related. The combination of the absorption from the coupled and uncoupled excitons (Figs. 5(e),(f)) gives rise to a three-band structure in the total absorption (Fig. 5(d)).

Therefore, as shown in Fig. 5(g), we summarize that the three transparency windows could be attributed to two processes: (1) the coupling between a bright broad $\mathrm {BD^1_c}$$\mathrm {D^1_h}$ and hybridized dressed states $|{{\omega_ + }} \rangle $ and $|{{\omega_ - }} \rangle $, (2) the coupling between the uncoupled excitonic state $|{{\omega_{\mathrm{e}}}} \rangle $ (“uncoupled” means uncoupling with the $\mathrm {BD^1_c}$$\mathrm {Q^1_h}$) with the $\mathrm {BD^1_c}$$\mathrm {D^1_h}$. This coupling mechanism is beyond the description of the conventional Jaynes-Cummings model because these three peaks are basically the sum of two unrelated processes. Whereas the Jaynes-Cummings model is for a coherent collective emitter-photon coupling system. For a coherent $\mathrm {BD^1_c}$$\mathrm {Q^1_h}$ and exciton coupled system, the Hamiltonian having 2 × 2 dimensions only gives birth to two new eigenmodes (i.e., $|{{\omega_ + }} \rangle $ and $|{{\omega_ - }} \rangle $) after diagonalization, rather than the three presented here. As a result, this system could be reconsidered as the linear combination of two separate processes [52].

Apart from the tunability, robustness is another key issue for such a multiband EIT system. Because, in principle, the relative displacement and angle between NC and NH could be hard to control in a chemical assembly method. They are likely to be randomly composed. Here we calculate several representative structures with different angles (blue lines) and displacements (red lines) in Fig. 5(h). As shown by the inset sketch of each structure, the multiband transparencies are supposed to be robust against structural deviation [29]. No matter where the NC is and how much the NC is rotated, the spectra remain almost the same. Orientation-independence of the nanocube could be accounted for by the degenerated $\mathrm {BD^1_c}$$\mathrm {Q^1_h}$ modes supported by two sides. No matter what polarization is applied, it could always be decomposed into these two degenerated plasmonic modes. And the overall responses would remain the same. The independence of position could be explained by the origin of the hybridized modes. The $\mathrm {BD^1_c}$$\mathrm {Q^1_h}$ modes here could be accounted for by a plasmonic nanocavity mode truncated from an infinite MIM waveguide [45,53]. Therefore, no matter where the AgNC is, it will always have the same overlapping area forming a nanocavity. Thus the $\mathrm {BD^1_c}$$\mathrm {Q^1_h}$ mode would be independent of the position. On the other hand, the $\mathrm {BD^1_c}$$\mathrm {D^1_h}$ mode could be largely attributed to the AuNH due to the significant size difference. The dipole mode would be insensitive to the small coupling variation brought by the position difference. Therefore, the overall multiband EIT is robust against the relative position. In addition, we also calculated the scattering spectra of this heterodimer consisting of a single AgNC and AuNH with different thicknesses H and radius of rounding curvature (see Supplement 1 Fig. S9). They all showed robustness against geometry modification. This robustness test increases the possibilities of our configurations in fulfilling the potential in photon manipulating and switching.

4. Conclusions

In this paper, different from conventional multiband configurations involving nontrivial combinations of optical elements, we have achieved a single-particle-level, subwavelength, tunable, multiband EIT prototype using a heterodimer structure consisting of a AgNC and a AuNH, assisted by J-aggregate excitons. Both intermediate coupling and strong coupling were utilized to pursue a multiband high peak-to-dip contrast. The “parental” pure photonic EIT window was realized through the intermediate coupling between the subradiant $\mathrm {BD^1_c}$$\mathrm {Q^1_h}$ and superradiant $\mathrm {BD^1_c}$$\mathrm {D^1_h}$ mode. Extra strong coupling involved by J-aggregate excitons helps the hybrid system inherit the properties of the narrow $\mathrm {BD^1_c}$$\mathrm {Q^1_h}$ peak, and duplicates the EIT windows into three bands. The heterodimers within different dielectric environments (in air and water, on MgF₂ or SiO₂ substrates) were checked. They all present a similar EIT behavior, revealing that the robust system is compatible with various experimental conditions.

Thanks to the outstanding sensitivity of the $\mathrm {BD^1_c}$$\mathrm {Q^1_h}$, those EITs are highly tunable against the geometry (e.g., gap distance) and the RI of the surrounding media. From our calculations, we proved that the optimized tunability against the nanogap is 36 nm/Å at G = 1 nm, and the RI FOM is 42. Additionally, the multiband EIT could also be tuned by the oscillation strength f (exciton density). When f is getting larger, we observed two gradually splitting EIT branches following the Rabi regulation, and one additionally emerging EIT window linked to the excitons. At last, the robustness of the configuration is tested towards the possible experimental randomness faced by the chemical synthesis. This deep-subwavelength, highly tunable, but also robust plasmonic multiband EIT prototype could have the potential for light slowing, and switching over multi-frequency bands. And on the other hand, it also acts as a basic platform for studying multiscale plasmon-exciton coupling at the nanoscale.