1. Introduction

Metallic structures can be used to manipulate light at the subwavelength scale. This opens an avenue to designing artificial metamaterials with optical properties new [1–3 ] or analogous to those in natural materials [4–6 ]. Due to easiness in fabrication and availability of various resonance elements (“meta-atoms”), such metamaterials have been mostly studied in the low-frequency regime (THz ~microwave), in which surface currents support the resonances, mimicking surface plasmons (SP) in the optical regime. But usually, metamaterial resonances have low quality (Q) factors, due to metal absorption loss in the optical regime or dominantly radiation loss in the low-frequency regime, which limit performances of the metamaterials. In the past few years, metamaterials based on coupling of different resonance modes in unit cells (“meta-molecules”) have attracted a lot of attention [7–15 ]. By designing appropriate molecular-like resonance elements and their intra-coupling parameters, electromagnetic (EM) waves can be more efficiently trapped in the near-field supporting high-Q resonances, so as to improve the metamaterial performances [7,8,14,16 ]. In this context, here we intend to design plasmonic meta-molecules in the optical regime to achieve superb optical properties, e.g., optical trapping, for a broad range of potential applications.

Resonance coupling in the molecular-metamaterials is generally considered belonging to the Fano resonances [17,18 ], resulting from the interference between a broader-band resonance mode and a narrower-band resonance mode, or interference of two neighbored overlapping resonance modes in general [6,19 ]. In literatures, there are mainly two strategies implemented to design coupled resonance modes with high-Q. One is based on coupling of two asymmetric bright resonance modes [7,20 ], which can be directly excited to form a symmetrically-coupled bright mode and an anti-symmetrically-coupled dark mode. Since the dark mode is subradiant, EM waves can be efficiently trapped to maintain high-Q resonances. Note that this is usually implemented in the low-frequency regime. When dealing with plasmon structures in the optical regime, it will be frustrated by the metal losses [7]. The other is based on coupling of a bright resonance mode and a dark resonance mode [8,21 ], in which the dark mode is indirectly excited by its coupling with the bright one to achieve high-Q resonances. Due to destructive interference of the bright and dark resonance modes, the phenomenon in analogy to electromagnetically induced transparence (EIT) in atomic systems [18,22 ] was demonstrated, showing a narrow transparency feature in a broader absorption spectrum band. Recently, conductive coupling of two dark resonance modes was reported for high-Q resonances, but its reduction of the radiation loss is achieved at the expense of its low excitability [14].

There have also been some other methods emphasizing on control of the resonance coupling in molecular-metamaterials, which generally involves delicate designing of material properties or inter-coupling parameters of the unit-cell elements. Particularly, parity-time (PT) symmetric metamaterials [23–25 ], based on balanced coupling of gain and loss in unit cells, were studied, whose properties can be potentially applied for active lasing, coherent perfect absorption and asymmetric transmission. But due to stringent requirement of balanced gain and loss, it is experimentally difficult to realize. Only in [25], ideal PT-symmetry was established in a passive metamaterial by balancing the scattering and dissipative losses with an effective gain coming from incident microwaves of the open system; and it demonstrated the phenomenon of coherent perfect absorption. Additionally, coherent control of the retardation phase in intermediate-regime coupling of a bright resonance mode and a dark resonant mode was reported in the optical regime, which can be used to control the excitation of individual plasmon resonance modes [15], or to achieve optical trapping for enhanced absorption [11].

Here we show that, by mediating dissipative losses in coupled anti-antisymmetric cavity resonance modes, plasmonic molecular-metamaterials can also be designed to efficiently trap the light, demonstrating high-Q resonances for perfect absorption. In this report, we designed a structure consisting of asymmetric binary nanogrooves, as shown in the inset of Fig. 1(a) , to implement the scheme. For this structure, incidence light can be trapped in the individual nanogrooves, resonating with anti-symmetric phases in a coupled state. By controlling the nanogroove sizes, the resonance modes and their internal losses can be tuned such that, at certain conditions, antiphase wavelets radiated from the nanogrooves are completely cancelled via destructive interferences. Hence, by reducing the radiation loss, quality factor of the cooperatively-coupled resonance mode is highly improved for narrow-band optical trapping and/or perfect absorption (at critical conditions).

 

Fig. 1 (a) Reflection spectrum of an array of asymmetric binary nanogrooves (d _1,2 = 172, 200 nm) showing perfect absorption at f = 220 THz. The spectra of arrays of symmetric binary nanogrooves (d ₁ = d ₂ = 200 nm) and single nanogrooves (d = 172 or 200 nm) are also plotted for reference. (b) Dependences of the enhanced resonant absorption frequency and its reflectance value on the depth of groove 1 (d ₁) for a fixed depth of groove 2 (d ₂ = 200 nm). Perfect absorption is obtained only at d ₁ = 172 nm and 232 nm. Resonance frequencies of an array of single nanogrooves with various depths are also plotted.

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2. Optical trapping in asymmetric binary nanogroove arrays

Figure 1(a) shows a typical reflection spectrum of the binary plasmonic nanostructures for normal incidence of transverse-magnetically (TM) polarized light, calculated with the finite-difference time-domain simulation method. The asymmetric nanogrooves are in a subwavelength array (p = 400 nm), with an equal groove width (w) of 60 nm and separation (s, edge-to-edge) of 100 nm. When depths (d ₁ and d ₂) of the nanogrooves are 172 nm and 200 nm, a sharp reflection dip appears at the frequency of f = 220 THz, ascribed to resonance of a coupled-mode hybridized from resonances in the individual binary nanogrooves. Compared to those of structures with a single nanogroove in each period, reflectance at the coupled-mode resonance position is strongly reduced, demonstrating a perfect absorption phenomenon. Q-factor of the resonance mode is also highly increased to ~20 from that of the single nanogroove arrays (~5). It is noted that the resonance dip position for the binary nanogroove structure is nearly the same (at 220 THz) as that of the single nanogroove structure with the larger depth of 200 nm. It is also shown that, if we have symmetric (i.e., identical) binary nanogrooves, the reflection dip becomes even shallower, implying weakened trapping of the incidence light.

In the study, we calculated reflection spectra of the binary nanogroove structures with various depths of one groove (d ₁), while the other is fixed (d ₂ = 200 nm), in each period. In Fig. 1(b), dependences of the coupled-mode resonance frequency and the dip reflectance value on the groove depth are plotted, together with that of the resonance frequency of a single-nanogroove counterpart. It is found that, for arrayed asymmetric binary nanogrooves that have overlapping intrinsic resonances, the coupled resonance mode generally exhibits enhanced absorption at a frequency nearly equal to that of the intrinsic resonance mode in the deeper nanogrooves that has a lower frequency. But perfect absorption appears only when difference of the binary nanogroove depths is at certain values, e.g. at d ₁ = 172 nm and 232 nm for d ₂ = 200 nm. It can be shown that, as separation of the binary nanogrooves is much larger than the optical skin depth (e.g., s>60 nm), coupling between the resonances in the individual nanogrooves is mainly via SPs at the top metal surface, instead of tunneling through ridges of the neighboring nanogrooves; therefore, the individual resonance modes have a weak coupling, and their intrinsic resonances are hardly modified. As shown later that coherent coupling of the resonance modes in the binary nanogrooves requires resonance frequency of the coupled mode to be close to that of the intrinsically lower-frequency mode in the deeper nanogrooves. Hence, the coupled-mode resonance is demonstrated as on-resonance of SPs in the deeper nanogrooves, while the shallower ones are in a near-resonance condition. To achieve perfect absorption, we need further to balance the on-resonance and near-resonance states for complete cancellation of radiation losses (i.e., reflection).

To visualize the resonances and their inter-coupling, we show distributions of the transverse magnetic field (H _y) and power flow in three types of binary nanogroove arrays at their coupled-mode resonance frequencies [Fig. 2 ]. It is noted that, for the dimensions of the structures in this work, we deal with zeroth-order standing-wave resonance modes in the nanogrooves. Thus, the maximum H _y-field locates at the bottom of the nanogrooves, while the corresponding E _x-field is maximal at the open ends of the nanogrooves; there is no zero-field node in the nanogrooves. As such, the nanogrooves behave as dipolar antennas, participating in exchange of energy between the neighboring nanogrooves (e.g. coupling) and radiating leakage power of the nanogrooves into the far field. As plasmon resonance modes in the binary nanogrooves are coupled, exchange and radiations of their energies become correlated, which provides us a degree of freedom to control the allocation of their power, to be either trapped or radiated, by mediating the coupling of the resonance modes. As shown in Figs. 2(a) and 2(b), for the symmetric binary nanogroove arrays, the field and power flow are symmetrically distributed, demonstrating in-phase resonances in the binary nanogrooves. But for the asymmetric structures in Figs. 2(c)-2(f), resonance fields in the binary nanogrooves are in antiphase, and the power flows more into the deeper nanogrooves. Recall that the coupled-mode resonance frequency is corresponding to that of the intrinsic resonance mode in the deeper nanogrooves; here it’s shown that the deeper nanogrooves are in an on-resonance state and the shallower ones are in a near-resonance state. In principle, the incidence light is more likely coupled into an on-resonance mode than an off-resonance one. Though the coupling of incidence light into the nanogrooves involves cooperative interaction of both resonance modes, overall it looks like that the shallower nanogrooves assist funneling of more incident power into the deeper ones, as demonstrated in Figs. 2(d) and 2(f). In this way, the fields in the asymmetric nanogrooves are strongly enhanced, compared to that of the symmetric case, shown in the figure. From the power flow charts, it is also observed that inter-coupling between the neighboring nanogrooves is via SPs at the top metal surface, if the groove-separation is relatively large [Figs. 2(b) and 2(d)]. But, as the separation becomes very small (~skin depth) [Fig. 2(f)], tunneling through the groove ridges becomes another important coupling channel, which will severely modify the intrinsic resonance modes in each individual nanogrooves and, consequently, shift the coupled-mode resonance frequencies.

 

Fig. 2 Distributions of the transverse magnetic field (H _y) (a, c, e) and power flow (b, d, f) at resonance frequencies of the symmetric and asymmetric binary nanogroove arrays. (a, b) d ₁ = d ₂ = 200 nm, s = 100 nm, f = 234 THz; (c, d) d ₁ = 172 nm, d ₂ = 200 nm, s = 100 nm, f = 220 THz; (e, f) d ₁ = 164 nm, d ₂ = 200 nm, s = 30 nm, f = 215THz; Perfect absorption is achieved for the asymmetric structures in (c)-(f).

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3. A harmoc oscillator model of the cooperative interaction processes

In Fig. 3(a) , we illustrate the near-field optical/plasmonic processes discussed above. For the asymmetric structure, antiphase resonance modes are excited in each unit cell of the binary nanogrooves by the incident optical field ($E_i$). We can assume that, initially, equal amounts of the incidence optical power are coupled into the respective nanogrooves, due to identical SP waveguiding mode in them. The final resonance modes are formed subjected to dissipative (${\gamma }_1$,${\gamma }_2$) and radiative (${\gamma }_{13}$,${\gamma }_{23}$) losses, in addition to inter-coupling (${\kappa }_{12}$) between the neighboring nanogrooves. For the structures with subwavelength period, reflection can be considered as a result of superposition of the fields ($E_r^{(\pm )}$) radiated from the nanogroove antennas following the Huygens’s principle. As the radiation fields are in antiphase at the near/on-resonance frequencies, they interfere destructively in the far field. Usually ${\gamma }_1\ne {\gamma }_2$ and ${\gamma }_{13}\ne {\gamma }_{23}$ due to asymmetry of the nanogrooves, the field cancellation in far field is only partial. But if the near- and on-resonance modes in the nanogrooves are tuned to be appropriate, their exchange of energy (${\kappa }_{12}$) can mediate the difference of dissipated power in the nanogrooves to result in equal magnitudes of the antiphase radiation fields, $E_r^{(+)}$ and $E_r^{(-)}$, leading to a complete cancellation of the reflection. Hence the incidence light is fully trapped in the nanogrooves, undergoing resonant perfect absorption.

 

Fig. 3 (a) Schematic illustration of the near-field optical processes. (b) A coupled harmonic oscillator model of the interaction processes. In the model, two coupled damping oscillators (“1” and “2”) driven by antiphase forces can cooperatively make a free oscillator (“3”) maintaining stationary in spite of their interactive biases, subject to appropriate damping and inter-coupling conditions.

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To help understanding the mechanisms, we developed a coupled oscillator model [13,26–28 ] to interpret cooperative roles of the interaction processes for optical trapping. As illustrated in Fig. 3(b), the model consists of two coupled (${\kappa }_{12}$) harmonic oscillators (labeled “1” and “2”) with discrepant intrinsic frequencies (${\omega }_1$,${\omega }_2$) and an associated free oscillator (labeled “3”). The two bound oscillators are harmonically driven by a pair of external forces$F(t)=\pm F_0{e}^{-i\omega t}$in opposite phases, and damped by the environmental viscosity (${\gamma }_1$,${\gamma }_2$) and fricative (or electromagnetic) interactions with the free oscillator (${\gamma }_{13}$,${\gamma }_{23}$). Here the mechanical parameters in Fig. 3(b) are considered as analogies of the optical ones in Fig. 3(b), besides correspondence of the antiphase forces to the fields of the optical power coupled into the nanogrooves. And the perfect absorption (i.e., null-reflection) corresponds to a stationary state of the free oscillator “3”, though it is subjected to the uneven counter-forces coming from its interactions with the bound oscillators “1” and “2”. So, in the oscillator system, the energy accumulated from the work of the applied external forces is stored in the motion of the coupled damping oscillators, and fully dissipated eventually, which are analogies of the optical trapping and perfect absorption. Now, the problem is that what conditions are required for the two antiphase-driven oscillators to cooperatively make the free oscillator to keep stationary. For the purpose, the following coupled oscillator mode equations are used to describe the above interaction processes:

4. Effects of the periodicity

So far, we neglected discussion on periodicity of the structures, except its critical role on cancellation of the reflected light by destructive interferences. Actually, in previous discussion, effects of the periodicity has been included in the intrinsic resonance frequencies of individual nanogroove arrays. As shown in Fig. 4(a) , we calculated back-scattering spectra of non-periodic, isolated pairs of binary nanogrooves. The spectra show EIT-like phenomenon for asymmetric nanogrooves that have different depths, instead of enhanced perfect absorption. And in literatures, usually metamaterial EIT does not consider interactions between unit cells. But in this work, except its far-field interference effects eliminating radiation losses, near-field coupling via SP waves at the top metal surface also affects the individual and coupled resonance modes. Note that, for the subwavelength-period arrays, Bloch-wave-like resonance of the SP waves at the structured metal surface locates at much higher frequencies, far from resonances in the nanogrooves of our concern. So its effect is only on short-range near-field coupling between neighboring nanogrooves.

 

Fig. 4 (a) Back-scattering spectra of isolated pairs of coupled nanogrooves for various depth of one of the nanogrooves (d ₁: nm). The width and separation of the nanogrooves are w = 60 nm, s = 100 nm, and depth of the other nanogroove is d ₂ = 200 nm. (b) Reflection spectrum of a 800-nm-period structure (w = 60 nm, s = 100 nm, d ₁ = 172 nm, d ₂ = 200 nm), for which perfect absorption is achieved when its nanogroove separation is adjusted to be s = 300 nm, or its shallower nanogroove depth d ₁ = 182 nm.

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For periodic structures, we calculated reflection spectra of the binary nanogroove structures with various periods, as structure dimensions of the nanogrooves (w = 60 nm, s = 100 nm, d ₁ = 172 nm, d ₂ = 200 nm) and their separation (s = 100 nm) are fixed (not shown here). It is observed that, with increase of the period, the resonance position shifts towards low frequencies and the absorption is no longer to be perfect, shown as an example in Fig. 4(b) for p = 800 nm. But it’s also shown that perfect absorption can still be established by adjusting the separation of the binary nanogrooves or depths of the nanogrooves. It is generally found that, if the nanogrooves distribute evenly in an array (e.g., s is roughly in a scale of p/4~p/2) and their dimensions are properly designed, the perfect absorption can be well maintained for various periods. For example, given d ₁ = 172 nm, d ₂ = 200 nm, perfect absorption is achieved both at (p = 400 nm, s = 100 nm) and (p = 800 nm, s = 300 nm), though their resonance positions are different due to influence of the period on the intrinsic resonance frequencies in individual nanogrooves. Besides, near-field scattering of SPs at the periodically-structured top metal surface influences coupling between resonances in the neighboring nanogrooves, and thus affects the perfect absorption condition.

5. Optical trapping in asymmetric ternary nanogroove arraysp

We further studied the phenomenon in asymmetric ternary nanogroove arrays. The asymmetric nanogrooves may be arranged in the form of …AAB…, …ABA… or …ABC…, where A, B and C refers to nanogrooves of different depths or intrinsic resonance frequencies. It was verified that, regardless of the nanogrooves’ arrangement, perfect absorption can all be obtained when differences of the nanogrooves are optimized. In Fig. 5 , we demonstrate an ABC-type ternary nanogroove array with period p = 400 nm, groove separations s = 60 nm, groove widths w = 60 nm, and groove depths of d _1,2,3 = 172, 200 and 235 nm. Perfect absorption of the ternary structure is shown in Fig. 5(a) at the frequencies of f = 220 and 192 THz, corresponding to the intrinsic resonance frequencies of the two deeper nanogrooves d _2,3 = 200 and 235 nm. The field distributions in Figs. 5(b) and 5(d) show that, at the two resonance frequencies of the coupled modes, the fields are more highly confined in the nanogrooves which are intrinsically resonant at the frequencies, i.e., groove 2 in Fig. 5(b) and groove 3 in Fig. 5(d). And phase of the fields in the nanogrooves that is shallower than the resonant nanogroove is opposite to that of the fields in the resonant and deeper nanogrooves. Figures 5(c) and 5(e) demonstrate that, at the resonant frequencies, the incidence power is funneled mainly into the resonant nanogrooves, assisted by the shallower nanogroove(s). The mechanisms are believed similar to that of the binary structures, except that here it involves three nanogroove resonance modes to cooperate for optical trapping and perfect absorption.

 

Fig. 5 (a) Reflection spectrum of an array of asymmetric ternary nanogrooves (d _1,2,3 = 172, 200, 235 nm) showing perfect absorption at nearly the intrinsic resonance frequencies of the deeper grooves 2 and 3 (f = 192, 220 THz). (b-e) Distributions of the transverse magnetic field (H _y) and power flow at the resonance frequencies of 192 THz (b, c) and 220 THz (d, e).

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6. Asymmetric binary nanogroove arrays for optical trapping of nanoparticles

The asymmetric binary nanogroove arrays are shown able to efficiently trap the incidence light in the deeper nanogrooves, in which the fields are strongly enhanced. This phenomenon may be used for self-induced back-action (SIBA) trapping of nano-objects in the nanogrooves in biomedical applicatioins [29]. For the purpose, we verified that, when a nano-object is trapped in the nanogrooves, strong field enhancement can be maintained in the nanogrooves; i.e., the perfect absorption conditions can still be satisfied, but with a slight adjust of the depth of the deeper nanogrooves. In Fig. 6 , we calculated reflection spectra of the metallic binary nanogroove arrays (p = 400 nm, w = 60 nm, s = 100 nm, d ₂ = 200 nm) with dielectric nanoparticles (refractive index = 1.5, diameter = 30 nm) in the deeper nanogroove for various depth of the shallower nanogrooves (d ₁). It is shown that, for fixed depth of d ₂, perfect absorption is achieved at f = 219.6 THz when d ₁ = 173 nm, indicating a high-efficiency trapping of light in the deeper nanogrooves. Generally, inclusion of the nanoparticles in the deeper nanogrooves is in fact equivalent to an increase of the resonance mode volume, which needs just an appropriate adjustment of the asymmetry for optical trapping. Besides, it is worth to mention that presence of the nanoparticles also induces another resonance mode in the deeper nanogrooves, which appears even for the structure with single nanogrooves in each period. The reflection dip at f = 193THz is a fingerprint of the resonance mode, whose position is shown to be invariant for the various structures.

 

Fig. 6 Reflection spectra of asymmetric binary nanogroove arrays (p = 400 nm, w = 60 nm, s = 100 nm, d ₂ = 200 nm) with dielectric nanoparticles (index = 1.5, diameter = 30 nm) locating in the deeper nanogrooves. The spectrum of an arrays of singlet nanogrooves (p = 400 nm, w = 60 nm, d = 200 nm) with such nanoparticles is also plotted for reference (dashed curve).

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7. Asymmetric binary nanogroove arrays with round edges

Practically, in fabrication of the structure, edges of the nanogrooves would be rounded. To verify the tolerance of the phenomenon to the rounded nanogroove geometry, we simulated the structures of asymmetric binary nanogroove arrays (p = 400 nm, w = 60 nm, s = 100 nm, d ₂ = 200 nm) with round edges, as shown in Fig. 7 . It is noted that, when the nanogroove edges are rounded, the cavity resonances and their interactions are correspondingly modified though the modification may be minor for small round curvature radius. Thus the perfect absorption condition should also be shifted, and so for the resonance position, compared to those of its unrounded counterpart. It is demonstrated in Fig. 7 that, when the radius of round curvature (R) is increased from R = 0 nm (unrounded structure) to R = 10, 20 and 30 nm, optimized depth of the shallower nanogrooves for perfect absorption varies slightly from d ₁ = 172 nm to d ₁ = 172, 171 and 173 nm respectively (note that resolution of d ₁ in simulations is 1 nm); and the resonance position at the reflection dip correspondingly shifts from f = 220 THz to f = 226, 231 and 239 THz. Thus, rounding the edges of the nanogroove structures in fabrication does not change the phenomenon and its nature in mechanisms. Anyway, in practice, one need to experimentally optimize the structure dimensions to satisfy the perfect absorption conditions.

 

Fig. 7 Reflection spectra of asymmetric binary nanogroove arrays (p = 400 nm, w = 60 nm, s = 100 nm, d ₂ = 200 nm) with round edges. For the round curvature radii R = 10, 20 and 30 nm at fixed value of d ₂ = 200 nm, perfect absorption is achieved when d ₁ = 172, 171 and 169 nm respectively; and the corresponding resonance positions are f = 226, 231 and 239 THz. The spectrum of unrounded structure (R = 0 nm) is plotted for reference.

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8. Conclusions

In conclusion, we demonstrated high-Q optical trapping and resonant perfect absorption of light in asymmetric binary (and ternary) metallic nanogroove arrays. It provides a way for coupled resonance modes in the subwavelength scale to cooperatively trap the incidence light by appropriate mediation of their resonance parameters to satisfy a matching condition. A harmonic oscillator model is proposed to elaborate their cooperative interactions. The metamaterials of arrayed binary/ternary asymmetric nanostructures are analogies of covalence-bond binary/ternary polar molecules in crystalline solid materials, not only in their elementary compositions but also in their internal interactions between the “atoms” of periodic “molecules”. As the outmost-shell electron clouds in atomic materials concentrates more around the more-electronegative atoms to maintain a stable hybridized state, here, near-field optical energy in the metamaterial is confined more in the low-frequency deeper nanogrooves to maintain a trapped-mode state. The proposed scheme may also be implemented with other asymmetric plasmonic resonance structures for optical trapping. Such high-Q plasmonic metamaterial structures can find potential applications in optical sensing, nonlinear optics and novel photonic devices.