Abstract

Photonic bound states in the continuum (BICs) are protected eigenstates in optical systems with infinite lifetimes. This unique property, which translates in infinite Q-factor resonances, makes BICs extremely interesting not only from a fundamental perspective but also for various applications such as lasing and sensing. General means to achieve robust BICs are, however, elusive. Here we demonstrate analytically that BICs emerge in metasurfaces formed by arrays of detuned resonant dipolar dimers as a universal behavior occurring regardless of both dipole position within the unit cell and lattice constant in the nondiffracting regime. These resonances evolve continuously from a Fano resonance into a symmetry-protected BIC as the dipole detuning vanishes. We have experimentally verified this very robust response at terahertz frequencies through dimer rod arrays with different rod sizes by simultaneously measuring the reduction of linewidth and the increase of lifetime before the BIC is formed, as it is impossible to couple to it from the continuum. Similar configurations can be straightforwardly envisioned throughout the electromagnetic spectrum, enabling a simple geometry that is easy to fabricate with resonances of arbitrarily high Q factors.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. INTRODUCTION

Bound states in the continuum (BICs) have attracted much interest lately in physics for their (theoretically) infinite Q factor. These states are leaky modes that in a certain limit of some parameter space cannot couple to any radiation channel [1]. In the absence of material losses, BICs have infinite lifetimes and zero linewidths. An approach to trap light in such remarkable electromagnetic modes is to exploit metasurfaces [29], i.e., subwavelength arrays (in the nondiffractive region) where only the specular reflection/transmission channels are allowed by symmetry, wherein outgoing specular channels can be suppressed by tuning the parameters of the system in various manners, leading to symmetry-protected BICs. Achieving robust BICs and quasi-BICs will be of great interest for the realization of cavities with arbitrarily high Q-factor and for applications such as lasing and sensing [1012]. Nonetheless, there are no fundamental works describing how to achieve robust BICs on a general basis, except for well-known symmetry-protected and accidental degeneracies appearing in photonic crystals [3] and asymmetric metasurfaces [7]. A variety of resonant phenomena, such as surface plasmon lattice resonances in the optical domain [1319], Fano resonances [2026], and electromagnetically induced transparency at optical and THz frequencies [2736], have been reported in metasurfaces; however, robust symmetry-protected BICs remain unexplored. Metasurfaces are especially interesting for this purpose due to their enhanced collective response [21,26,3740].

In this paper, we will show that a simple and easily experimentally accessible configuration based on metasurfaces of tunable dipoles supports robust BICs that emerge in the nondiffractive regime. The symmetry protection of these BICs is not broken or lost by displacements of the dipoles within the unit cell, which increase the tolerance of these systems to imperfections. In addition, we will unequivocally demonstrate experimentally that such BICs emerge at terahertz (THz) frequencies in gold-rod dimer metasurfaces formed by two rods per unit cell by observing the vanishing linewidth in the THz transmission spectra accompanied by the pronounced increase of the resonance lifetime, when rods are made identical. The negligible Ohmic losses of metals at THz frequencies and the dipolar-like = λ/2 resonances of metallic rods at these frequencies are responsible for the appearance of BICs in this system. Apart from full numerical calculations in agreement with the experimental results, a simple model for arrays of detuned-resonant-dipole dimers is developed to yield physical insight. Hereby, we show analytically a very general condition for the emergence of symmetry-protected BICs in the parameter space of dipole detuning, from a dark Fano resonance (hybridized dipole lattice mode) that becomes an infinite-Q-factor BIC for zero detuning. These calculations fully explain the experimental results in this wider theoretical context of detuned-dipole arrays.

2. THZ TIME-DOMAIN SPECTROSCOPY MEASUREMENTS

Using optical lithography, metal deposition, and lift-off, we have fabricated samples containing 2D periodic lattices of gold rods on top of a 1.5 mm thick amorphous quartz substrate. These samples were clamped against another 1.5 mm substrate with an index matching liquid in between to suppress unwanted reflections, making the total sample thickness 3 mm. Before the thermal evaporation of the 100 nm gold layer, a 3 nm Ti adhesion layer was deposited. All the samples consist of 2D periodic lattices of two gold rods per unit cell. The lattice has a square symmetry with a pitch of a=b=300μm. One of the rods has a fixed dimension of 200μm×40μm, while the dimensions of the other rods are different for each sample that is investigated. The long dimensions of the rods are aligned with the y axis, and the rods inside the unit cell are separated by a distance dx along the x axis. Both rods support λ/2 resonances in the THz range given by their length. The resonance frequency is controlled by the long axis of the rods, so we classify each lattice by the length of the second rod L2, which covers the range L2=125–250μm in steps of 25 μm while keeping L1=200μm constant. The width of the rods is scaled such that surface area covered with gold is comparable between the rods and therefore between all samples.

Two different sets of samples have been measured, with separation between rods fixed at dx=120 and 150 μm. THz transmittance spectra of these samples have been measured at normal incidence using a 4f far-field THz time-domain spectrometer (Menlo TeraK15). The transmittance of the lattices for different L2 is shown in Fig. 1(a). Numerical simulations calculated through SCUFF [41,42] (an open-source software package for analysis of electromagnetic scattering problems using the method of moments) are shown in Fig. 1(b), corresponding to transmittance spectra at normal incidence for the same geometrical parameters, but considering gold rods as planar perfectly conducting rectangles embedded in a uniform medium with n=1.55 (the average of those of air and the supporting quartz substrate [43]). There is a shift between the Fano resonances simulated and the measurements, which can be corrected by slightly adjusting this refractive index. Good qualitative and nearly quantitative agreement is observed between the measurements and the simulations.

 figure: Fig. 1.

Fig. 1. (a) Measured transmittance spectra for square lattices (a=300μm) of two gold rods per unit cell, deposited on a quartz substrate, with two different rod separations: dx=a/2 (solid curves) and dx=2a/5 (dashed curves). One of the rods has fixed dimensions of L1=200μm and w1=40μm, while the other varies as shown in the center insets, with L2(μm)=125,150,175,200,225,250, while keeping the surface area fixed: L2w2=L1w1. All rod thicknesses are t=0.1μm. (b) Transmittance spectra numerically calculated through SCUFF [41,42] for the same geometrical parameters but considering gold rods as planar perfectly conducting rectangles embedded in a uniform medium (see text). Curves are offset by 1 for each different L2.

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The more relevant feature in Fig. 1 is that for L2200μm all spectra present a strong Fano resonance with a narrow asymmetric line shape, with a high contrast approaching zero to total transmittance. The resonance moves toward lower frequencies as L2is increased and disappears for L2=200μm. In addition, as the rods become more similar in size, the resonance becomes narrower; it should be mentioned, though, that the resonance narrowing is not clearly visible in the measurements for L2=175,225μm due to the limited frequency resolution in the measurements. The Fano resonance can be understood in terms of the lattice resonances that the array supports: a bright, broad symmetric mode and a dark, narrow antisymmetric mode [33]. Incidentally, by lattice resonances we mean the resonances resulting from the lattice-induced modification of meta-atom resonances [44] rather than those emerging near Rayleigh anomalies [14]. The symmetry of these modes is preserved in the lattice, and they interfere either destructively or constructively, producing a characteristic dip/peak pair in the spectra. Moreover, the dark mode becomes inaccessible at L2=200μm and the interference disappears; as we will show below, the width of the resonance thus tends to zero, leading to a BIC that cannot be detected in the far field.

It should be noted that when dx=150μm, the BIC state is connected to the guided mode that the lattice supports. Indeed, for L2=200μm both rods are identical, so the array becomes a single rod lattice with the lattice constant halved along the x axis. We call this condition as the “half-period lattice.” The Brillouin zone is doubled when dx=a/4; thus, if one artificially assumes that the actual lattice constant is still a (instead of a/2), part of the lowest guided mode band (always below the light line of the true Brillouin zone) will be bent back into the propagating region (above the light line) of the artificially reduced Brillouin zone, indeed leading to an artificial BIC (stemming from a guided mode rather than from a leaky mode) at the Γ point. Furthermore, the number of resonances that the lattice supports can be associated with the number of particles per unit cell. For the identical half-period lattice, i.e., one particle per unit cell, there is only a bright (symmetric) mode. Therefore, for the half-period lattice it is difficult to relate the BIC with a real state.

Nonetheless, for a lattice with dx=120μm, where the half-period lattice is not recovered at L2=200μm for equal rods, the BIC is no longer directly connected to any guided mode, so the system has two well-defined lattice modes, and a true BIC emerges. This is further verified through numerical calculations at oblique incidence (see Supplement 1, Figure S1), which reveal a Fano resonance emerging for equal rods only in the case of the lattice with dx=2a/5; this is yet another evidence of the true BIC behavior.

Time-domain measurements [also with quasi-plane wave excitation as in Fig. 1(a)], shown in Fig. 2, yield further evidence of the transition to a BIC in the parameter space. Transmitted THz transients are plotted for lattices with a spacing of dx=120μm and asymmetric dimer rods L2=125,150,175μm. The transient for the BIC condition (L2=200μm) is subtracted from these measurements to remove the contribution of the broad bright mode. On long time scales, only this narrow, dark (anti-symmetric) resonance remains, which is in turn fitted to a single frequency-damped harmonic oscillator with lifetimes of (for increasing L2) 5.9, 10, and 20 ps, respectively, and oscillation frequencies of 0.4, 0.39, and 0.375 THz, respectively. The lifetimes thus extracted reasonably agree with those inferred from the widths of the numerical transmittance spectra at resonance in Fig. 1(b), the latter obviously overestimating the experimental lifetimes. More importantly, a clear increase of lifetime (resonance width becoming increasingly narrow) is demonstrated as the condition of the BIC emergence (infinite lifetime) is approached in parameter space: namely, as the detuning vanishes for |L2|200μm. We note that direct measurements of the lifetime of the BIC in symmetric dimers arrays is not possible with far-field techniques due to the complete suppression of the coupling to the continuum. Such a BIC emergence condition will be fully explored analytically below. Alternatively, as we show later, this observation could be achieved using near-field techniques.

 figure: Fig. 2.

Fig. 2. Measured transient response after background subtraction for square lattices (a=300μm) as in Fig. 1, consisting of two gold rods per unit cell, deposited on a quartz substrate, with rod separation dx=2a/5 (solid curves). One of the rods has fixed dimensions of L1=200μm and w1=40μm, while the other varies as shown in the figure legend, with L2(μm)=125 (blue), 150 (red), 175 (orange). In all cases, the transient for identical rod dimers (L2=200μm) is subtracted; the resulting differential transients are normalized to the maximum of L2(μm)=125 and offset by 1 for convenience. Recall that the rod surface area is fixed as L2w2=L1w1, and the rods have thicknesses of t=0.1μm. Dashed lines show fits to single frequency-damped harmonic oscillators with a lifetime of 5.9, 10, and 20 ps, respectively, and oscillation frequencies of 0.4, 0.39, and 0.375 THz, respectively.

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3. THEORETICAL MODEL: COUPLED DETUNED-DIPOLE PLANAR ARRAY

To shed light onto this rich phenomenology, we developed a simple coupled dipole-dimer model of an infinite array embedded in a homogeneous environment, shown in detail in Supplement 1. Dipoles are fully characterized by their polarizabilities along the y axis, namely, αy(1) and αy(2), where (1) and (2) account for each dimer dipole in the unit cell. The array is excited by an external plane wave, with ψ0 being its electric field polarization along the y axis. Upon imposing Bloch’s theorem and recalling that only normal incidence is considered, the local field at the position of the dipoles ψloc(i), with i=1,2, can be found through a self-consistent field equation

[ψloc(1)ψloc(2)]=[Ik2Gbα]1[ψ0(1)ψ0(2)],
where ψ0(i) is the incident field on dipole i=1,2, I is the identity matrix, α is the polarizability tensor, and Gb is the lattice “depolarization” dyadic (or return Green function)
α=[αy(1)00αy(2)],Gb=[GbyyGyy(12)Gyy(21)Gbyy].
Gbyy describes the self-interaction of each dipole array; Gyy(ij) is the interaction of the dipole array labeled as (i) over (j). Due to symmetry, at normal incidence Gyy(12)=Gyy(21).

Surface lattice resonances are the solutions of Eq. (1) in the absence of the external plane wave. To solve it, we diagonalize the system and find the condition at which the eigenvalues Λ are equal to zero. At real frequencies, the zeros of the real part of Λ give the resonance frequencies, whereas the imaginary parts define the resonance widths. Recall that the imaginary components of Gb and α (for lossless particles) are well defined and satisfy the following condition:

I[(1αyGbyy)+Gyy(12)]=0,
where we define the magnitudes
2αy=1k2(1αy(1)+1αy(2)),Δαy=1k2(1αy(1)1αy(2)).
Δαy thus represents the detuning between the two rods in the unit cell (introduced in the experiments by changing the size of one rod). For a small detuning Δαy2Gyy(12), the imaginary components of the eigenvalues can be approximated by
I[Λ+]=I[(Δαy)28Gyy(12)],
I[Λ]=I[2(1αyGbyy)(Δαy)28Gyy(12)].
Finally, the corresponding eigenvectors are given by
Λ±=0ν±=[ψloc(1)ψloc(2)]=[11].
The lattice resonances are associated with two modes in which the rods are out of/in phase (antisymmetric/symmetric). From Eq. (3), it follows that I[Λ+] goes to zero as the detuning is suppressed, i.e., as the two rods are made equal. Hence, as expected, the out-of-phase mode Λ+ is very narrow and becomes a BIC at zero detuning. On the other hand, I[Λ] is always larger than zero, so the in-phase mode Λ is broad. Therefore, for L1L2 (αy(1)αy(2)) we have a broad mode that interferes with a very narrow mode, leading to a Fano resonance. For L1=L2, the narrow mode converges into a BIC state, precluding any external coupling to it and the formation of the dip in transmission.

Interestingly, if there is also an additional displacement along the y axis, given by dy, Eq. (3) still holds. Moreover, this identity is also valid for rectangular (non-square) lattices where ab. This is evidenced in the detailed formulation in Supplement 1, (Eq. S19), where the expressions of the imaginary parts of all three terms in Eq. (3) are shown to cancel out in the absence of diffraction orders despite the fact that both Gyy(12) and Gbyy depend on lattice parameters. Hence, Eq. (3) is universal and holds for: (i) any set of lattice constant parameters a and b and (ii) any relative displacement between dipoles inside the unit cell, not only along the x axis, but extensive to the full xy plane. Therefore, a major conclusion is that the BIC state is symmetry protected and robust against changes in the specific lattice parameters: a, b, dx, and dy. In this regard, bear in mind that this statement is purely theoretical (no experimental evidence is provided). Indeed, it is thus restricted to the domain of applicability of our coupled detuned-dipole formulation, namely, dimer meta-atoms accounted for by parallel detuned dipoles, with couplings among them fully reproduced by dipole–dipole interactions, and within the spectral regime where no diffraction orders are allowed other than the specular ones.

Let us now analyze the Fano–BIC transition using our coupled dipole-dimer model. We plot in Figs. 3(a) and 3(c) the spectra of the transmission coefficient (T=1R0) intensity and phase, for a square lattice (lattice constant a=b=300μm) with two dipoles per unit cell, separated by a distance of dx=120μm for varying L2; cuts for fixed lengths are shown in Figs. 3(b) and 3(d). The polarizability of the rods is calculated through SCUFF [41,42], considering the rods as perfect electric conductors [see Supplement 1, Fig. S2(b)].

 figure: Fig. 3.

Fig. 3. (a)–(d) Theoretical transmittance spectra calculated through coupled dipole theory for a square lattice (a=300μm) as in Fig. 1 but consisting of two detuned dipoles per unit cell, separated by dx=2a/5: (a), (b) transmittance; (c), (d) phase. Dipole polarizabilities are extracted from the numerically calculated scattering cross sections shown in Supplement 1, Fig. S2 (see text): one is fixed and corresponds to a rod with dimensions L1=200 μm and w1=40 μm, while the other one L2 (with identical areas L2w2=L1w1) varies continuously in the contour maps in (a), (c), whereas those cases corresponding to three of the experimental dimers, L2(μm)=150,200,250, are shown in (b), (d).

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Figure 3 shows that the coupled dipole-dimer model reproduces all the features exhibited in Fig. 1, fully extending the characterization of the BIC into the L2-parameter space. Remarkably, the contour map in Fig. 3(a) reveals the classical narrowing of the leaky resonance (Q factor tending to infinity) towards a BIC state, with the distinct feature that the resonant state [surface lattice dipole resonance with an abrupt π-phase jump shown in the contour map in Fig. 3(c)] manifests itself as a narrow Fano resonance instead. Transmittance spectra (intensity and phases) are shown in Figs. 3(b) and 3(d) for given L2, illustrating this Fano-like behavior that disappears at L2=L1 as a signature of the BIC.

Furthermore, we show in Fig. 4 the amplitudes (ψloc(1),ψloc(2)) and phases (ϕloc(1),ϕloc(2),Δϕ) of the local fields over the dipoles for the dipole dimer corresponding to L2=150μm. At low frequencies, both dipoles are driven in phase. At ν=0.38THz, coinciding with the zero of ψloc(1), Δϕ presents a discontinuity and becomes maximum (rods out of phase, with a high dispersion in both). The field amplitudes are enhanced and, indeed, exhibit a resonant line shape, corresponding to the dark lattice resonance, where the Fano asymmetric line shape emerges in the far field. Then at ν=0.48THz, upper-frequency end of the Fano resonance, Δϕ presents another discontinuity exactly at the zero of ψloc(2); for higher frequencies the phase difference vanishes and the dipoles are in phase again. Thus, at given frequencies at the lower/upper band of the Fano resonance, the fields at the long/short rods are strictly zero, manifesting the strong interaction that exists between the different rods near resonance. In addition, the local field is enhanced by more than a factor of three in both rods. At L2=L1, the anti-phase behavior is not allowed, so that the field at both rods is identical [cf. Figs. 4(a) and 4(b)], with no evidence whatsoever of a Fano resonance, and thus it becomes a BIC [as shown in Figs. 3(a) and 3(b)].

 figure: Fig. 4.

Fig. 4. Spectral variation of both local field (a) phases (including the relative phase) and (b) amplitudes, calculated through coupled dipole theory for a square lattice (a=300μm) as in Fig. 1, but consisting of two detuned dipoles per unit cell, separated by dx=2a/5 for L2=200 μm and L2=150 μm. The local amplitudes and phases at both rods for L2=200μm are identical.

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To reveal even more neatly the BIC behavior, we show a near-field map of the out-of-plane electric field component at the BIC frequency (0.357 THz) in Fig. 5. The field pattern is numerically calculated through FDTD simulations (Lumerical), for a broadband (center frequency around the BIC, ν=0.357THz) dipole source (located at the center of the image) inside both asymmetric and symmetric dimer rod arrays (dx=120μm) lying on a quartz substrate, over a large area of 2.5×2.5mm2, with rod dimer dimensions (as highlighted in the schematic insets) of: L1=200μm and L2=125μm, [Fig. 5(a)]; L1=L2=200μm [Fig. 5(b)]. In Fig. 5(a) the excitation with a point dipole is radiated to the far field, showing negligible coupling or propagation to the neighboring rods. By contrast, the near field for equal rod dimers is effectively trapped in the BIC mode as shown in Fig. 5(b): many dimers in the lattice are resonantly excited, with the expected opposite phase for each rod within the (dimer) unit cell that protects the BIC and precludes out-of-plane radiation losses.

 figure: Fig. 5.

Fig. 5. Near-field simulations showing the electric field component along the z direction for a broadband (center frequency around the BIC) dipole source (located at the center of the images indicated by an arrow) in a square lattice (a=300μm) of gold rod dimers on a quartz substrate with rod separation dx=2a/5. One of the rods has fixed dimensions of L1=200μm and w1=40μm, while the other is (a) L2=125μm (i.e., detuned dipoles) and (b) L2=200μm (i.e., equal dipoles), while keeping the surface area fixed: L2w2=L1w1. The dipole source emits at ν=0.357THz (BIC frequency) and is given 200 ps to propagate over the total simulated area, 2.5×2.5mm2, indeed much bigger than the area shown.

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4. CONCLUSIONS

We have shown experimentally through the THz transmission spectra and transients that metasurfaces consisting of sub-wavelength gold-rod dimers support BICs when the rods are identical and that such BICs emerge from strong Fano resonances, which become narrower (higher Q factor) as the dimensions of the rods approach each other. Such experimental results have been fully explained in the theoretical context of detuned-dipole arrays as a universal condition for the emergence of symmetry-protected BICs and vanishing dipole detuning. Remarkably, such BIC condition is shown to be independent of both the relative position between dipoles/rods inside the unit cells and the lattice constants (provided that no diffractive orders come into play). Hence, these properties make an array of detuned dipoles a general scenario to engineer robust and versatile metasurfaces supporting bound states in the continuum throughout the electromagnetic spectrum, with appealing implications in sensing, lasing, and related phenomenology. Finally, bear in mind that the formalism and phenomenology of our coupled detuned-dipole model could be extrapolated to other fields of wave physics with arrays of dipolar scatterers, such as acoustic, elastic, seismic, and even atom waves.

Funding

Ministerio de Economía, Industria y Competitividad, Gobierno de España (FIS2015-69295-C3-2-P, FIS2017-91413-EXP); Ministerio de Educación, Cultura y Deporte (MECD) (FPU15/03566); Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO) (680-47-628); Ministerio de Ciencia, Innovación y Universidades (PGC2018-095777-B-C21).

Acknowledgment

We thank M. Ramezani for fruitful discussions.

 

See Supplement 1 for supporting content.

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31. Z. Zhu, X. Yang, J. Gu, J. Jiang, W. Yue, Z. Tian, M. Tonouchi, J. Han, and W. Zhang, “Broadband plasmon induced transparency in terahertz metamaterials,” Nanotechnology 24, 214003 (2013). [CrossRef]  

32. M. Manjappa, Y. K. Srivastava, and R. Singh, “Lattice-induced transparency in planar metamaterials,” Phys. Rev. B 94, 161103 (2016). [CrossRef]  

33. M. C. Schaafsma, A. Bhattacharya, and J. G. Rivas, “Diffraction enhanced transparency and slow THz light in periodic arrays of detuned and displaced dipoles,” ACS Photon. 3, 1596–1603 (2016). [CrossRef]  

34. A. Halpin, C. Mennes, A. Bhattacharya, and J. Gómez Rivas, “Visualizing near-field coupling in terahertz dolmens,” Appl. Phys. Lett. 110, 101105 (2017). [CrossRef]  

35. A. Halpin, N. van Hoof, A. Bhattacharya, C. Mennes, and J. Gomez Rivas, “Terahertz diffraction enhanced transparency probed in the near field,” Phys. Rev. B 96, 85110 (2017). [CrossRef]  

36. T. C. Tan, Y. K. Srivastava, M. Manjappa, E. Plum, and R. Singh, “Lattice induced strong coupling and line narrowing of split resonances in metamaterials,” Appl. Phys. Lett. 112, 201111 (2018). [CrossRef]  

37. C. Wu, A. B. Khanikaev, R. Adato, N. Arju, A. A. Yanik, H. Altug, and G. Shvets, “Fano-resonant asymmetric metamaterials for ultrasensitive spectroscopy and identification of molecular monolayers,” Nat. Mater. 11, 69–75 (2011). [CrossRef]  

38. R. Singh, W. Cao, I. Al-Naib, L. Cong, W. Withayachumnankul, and W. Zhang, “Ultrasensitive terahertz sensing with high-Q Fano resonances in metasurfaces,” Appl. Phys. Lett. 105, 171101 (2014). [CrossRef]  

39. Z.-J. Yang, T. J. Antosiewicz, R. Verre, F. J. García de Abajo, S. P. Apell, and M. Käll, “Ultimate limit of light extinction by nanophotonic structures,” Nano Lett. 15, 7633–7638 (2015). [CrossRef]  

40. S. Yuan, X. Qiu, C. Cui, L. Zhu, Y. Wang, Y. Li, J. Song, Q. Huang, and J. Xia, “Strong photoluminescence enhancement in all-dielectric Fano metasurface with high quality factor,” ACS Nano 11, 10704–10711 (2017). [CrossRef]  

41. M. Homer Reid and S. Johnson, “Efficient computation of power, force, and torque in BEM scattering calculations,” arXiv:1307.2966 (2013).

42. M. T. H. Reid, https://github.com/HomerReid/scuff-em.

43. R. Yu, L. M. Liz-Marzán, and F. J. García de Abajo, “Universal analytical modeling of plasmonic nanoparticles,” Chem. Soc. Rev. 46, 6710–6724 (2017). [CrossRef]  

44. D. R. Abujetas, J. A. Sánchez-Gil, and J. J. Sáenz, “Generalized Brewster effect in high-refractive-index nanorod-based metasurfaces,” Opt. Express 26, 31523–31541 (2018). [CrossRef]  

References

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  1. C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljačić, “Bound states in the continuum,” Nat. Rev. Mater. 1, 16048 (2016).
    [Crossref]
  2. D. C. Marinica, A. G. Borisov, and S. V. Shabanov, “Bound states in the continuum in photonics,” Phys. Rev. Lett. 100, 183902 (2008).
    [Crossref]
  3. C. W. Hsu, B. G. DeLacy, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Theoretical criteria for scattering dark states in nanostructured particles,” Nano Lett. 14, 2783–2788 (2014).
    [Crossref]
  4. E. N. Bulgakov and D. N. Maksimov, “Topological bound states in the continuum in arrays of dielectric spheres,” Phys. Rev. Lett. 118, 267401 (2017).
    [Crossref]
  5. A. Taghizadeh and I.-S. Chung, “Quasi bound states in the continuum with few unit cells of photonic crystal slab,” Appl. Phys. Lett. 111, 031114 (2017).
    [Crossref]
  6. H. M. Doeleman, F. Monticone, W. den Hollander, A. Alù, and A. F. Koenderink, “Experimental observation of a polarization vortex at an optical bound state in the continuum,” Nat. Photonics 12, 397–401 (2018).
    [Crossref]
  7. K. Koshelev, S. Lepeshov, M. Liu, A. Bogdanov, and Y. Kivshar, “Asymmetric metasurfaces with high-Q resonances governed by bound states in the continuum,” Phys. Rev. Lett. 121, 193903 (2018).
    [Crossref]
  8. S. I. Azzam, V. M. Shalaev, A. Boltasseva, and A. V. Kildishev, “Formation of bound states in the continuum in hybrid plasmonic-photonic systems,” Phys. Rev. Lett. 121, 253901 (2018).
    [Crossref]
  9. L. Cong and R. Singh, “Symmetry-protected dual bound states in the continuum in metamaterials,” Adv. Opt. Mater. 7, 1900383 (2019).
    [Crossref]
  10. A. Kodigala, T. Lepetit, Q. Gu, B. Bahari, Y. Fainman, and B. Kanté, “Lasing action from photonic bound states in continuum,” Nature 541, 196–199 (2017).
    [Crossref]
  11. S. T. Ha, Y. H. Fu, N. K. Emani, Z. Pan, R. M. Bakker, R. Paniagua-Domínguez, and A. I. Kuznetsov, “Directional lasing in resonant semiconductor nanoantenna arrays,” Nat. Nanotechnol. 13, 1042–1047 (2018).
    [Crossref]
  12. S. Han, L. Cong, Y. K. Srivastava, B. Qiang, M. V. Rybin, W. X. Lim, Q. Wang, Y. S. Kivshar, and R. Singh, “All-dielectric active photonics driven by bound states in the continuum,” arXiv:1803.01992 (2018).
  13. F. J. García de Abajo, “Colloquium: light scattering by particle and hole arrays,” Rev. Mod. Phys. 79, 1267–1290 (2007).
    [Crossref]
  14. V. Giannini, G. Vecchi, and J. Gómez Rivas, “Lighting up multipolar surface plasmon polaritons by collective resonances in arrays of nanoantennas,” Phys. Rev. Lett. 105, 1–4 (2010).
    [Crossref]
  15. A. B. Evlyukhin, S. I. Bozhevolnyi, A. Pors, M. G. Nielsen, I. P. Radko, M. Willatzen, and O. Albrektsen, “Detuned electrical dipoles for plasmonic sensing,” Nano Lett. 10, 4571–4577 (2010).
    [Crossref]
  16. S. R. K. Rodriguez, A. Abass, B. Maes, O. T. A. Janssen, G. Vecchi, and J. Gómez Rivas, “Coupling bright and dark plasmonic lattice resonances,” Phys. Rev. X 1, 021019 (2011).
    [Crossref]
  17. N. Meinzer, W. L. Barnes, and I. R. Hooper, “Plasmonic meta-atoms and metasurfaces,” Nat. Photonics 8, 889–898 (2014).
    [Crossref]
  18. S. Baur, S. Sanders, and A. Manjavacas, “Hybridization of lattice resonances,” ACS Nano 12, 1618–1629 (2018).
    [Crossref]
  19. M. Habib, M. Gokbayrak, E. Ozbay, and H. Caglayan, “Electrically controllable plasmon induced reflectance in hybrid metamaterials,” Appl. Phys. Lett. 113, 221105 (2018).
    [Crossref]
  20. A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, “Fano resonances in nanoscale structures,” Rev. Mod. Phys. 82, 2257–2298 (2010).
    [Crossref]
  21. B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9, 707–715 (2010).
    [Crossref]
  22. V. Giannini, A. I. Fernández-Domínguez, S. C. Heck, and S. A. Maier, “Plasmonic nanoantennas: fundamentals and their use in controlling the radiative properties of nanoemitters,” Chem. Rev. 111, 3888–3912 (2011).
    [Crossref]
  23. P. Fan, Z. Yu, S. Fan, and M. L. Brongersma, “Optical Fano resonance of an individual semiconductor nanostructure,” Nat. Mater. 13, 471–475 (2014).
    [Crossref]
  24. N. Verellen, F. López-Tejeira, R. Paniagua-Domínguez, D. Vercruysse, D. Denkova, L. Lagae, P. Van Dorpe, V. V. Moshchalkov, and J. A. Sánchez-Gil, “Mode parity-controlled Fano- and Lorentz-like line shapes arising in plasmonic nanorods,” Nano Lett. 14, 2322–2329 (2014).
    [Crossref]
  25. D. R. Abujetas, M. A. G. Mandujano, E. R. Méndez, and J. A. Sánchez-Gil, “High-contrast Fano resonances in single semiconductor nanorods,” ACS Photon. 4, 1814–1821 (2017).
    [Crossref]
  26. M. F. Limonov, M. V. Rybin, A. N. Poddubny, and Y. S. Kivshar, “Fano resonances in photonics,” Nat. Photonics 11, 543–554 (2017).
    [Crossref]
  27. S.-Y. Chiam, R. Singh, C. Rockstuhl, F. Lederer, W. Zhang, and A. A. Bettiol, “Analogue of electromagnetically induced transparency in a terahertz metamaterial,” Phys. Rev. B 80, 153103 (2009).
    [Crossref]
  28. S. I. Bozhevolnyi, A. B. Evlyukhin, A. Pors, M. G. Nielsen, M. Willatzen, and O. Albrektsen, “Optical transparency by detuned electrical dipoles,” New J. Phys. 13, 023034 (2011).
    [Crossref]
  29. J. Gu, R. Singh, X. Liu, X. Zhang, Y. Ma, S. Zhang, S. A. Maier, Z. Tian, A. K. Azad, H.-T. Chen, A. J. Taylor, J. Han, and W. Zhang, “Active control of electromagnetically induced transparency analogue in terahertz metamaterials,” Nat. Commun. 3, 1151 (2012).
    [Crossref]
  30. S. R. K. Rodriguez, O. T. A. Janssen, G. Lozano, A. Omari, Z. Hens, and J. G. Rivas, “Near-field resonance at far-field-induced transparency in diffractive arrays of plasmonic nanorods,” Opt. Lett. 38, 1238–1240 (2013).
    [Crossref]
  31. Z. Zhu, X. Yang, J. Gu, J. Jiang, W. Yue, Z. Tian, M. Tonouchi, J. Han, and W. Zhang, “Broadband plasmon induced transparency in terahertz metamaterials,” Nanotechnology 24, 214003 (2013).
    [Crossref]
  32. M. Manjappa, Y. K. Srivastava, and R. Singh, “Lattice-induced transparency in planar metamaterials,” Phys. Rev. B 94, 161103 (2016).
    [Crossref]
  33. M. C. Schaafsma, A. Bhattacharya, and J. G. Rivas, “Diffraction enhanced transparency and slow THz light in periodic arrays of detuned and displaced dipoles,” ACS Photon. 3, 1596–1603 (2016).
    [Crossref]
  34. A. Halpin, C. Mennes, A. Bhattacharya, and J. Gómez Rivas, “Visualizing near-field coupling in terahertz dolmens,” Appl. Phys. Lett. 110, 101105 (2017).
    [Crossref]
  35. A. Halpin, N. van Hoof, A. Bhattacharya, C. Mennes, and J. Gomez Rivas, “Terahertz diffraction enhanced transparency probed in the near field,” Phys. Rev. B 96, 85110 (2017).
    [Crossref]
  36. T. C. Tan, Y. K. Srivastava, M. Manjappa, E. Plum, and R. Singh, “Lattice induced strong coupling and line narrowing of split resonances in metamaterials,” Appl. Phys. Lett. 112, 201111 (2018).
    [Crossref]
  37. C. Wu, A. B. Khanikaev, R. Adato, N. Arju, A. A. Yanik, H. Altug, and G. Shvets, “Fano-resonant asymmetric metamaterials for ultrasensitive spectroscopy and identification of molecular monolayers,” Nat. Mater. 11, 69–75 (2011).
    [Crossref]
  38. R. Singh, W. Cao, I. Al-Naib, L. Cong, W. Withayachumnankul, and W. Zhang, “Ultrasensitive terahertz sensing with high-Q Fano resonances in metasurfaces,” Appl. Phys. Lett. 105, 171101 (2014).
    [Crossref]
  39. Z.-J. Yang, T. J. Antosiewicz, R. Verre, F. J. García de Abajo, S. P. Apell, and M. Käll, “Ultimate limit of light extinction by nanophotonic structures,” Nano Lett. 15, 7633–7638 (2015).
    [Crossref]
  40. S. Yuan, X. Qiu, C. Cui, L. Zhu, Y. Wang, Y. Li, J. Song, Q. Huang, and J. Xia, “Strong photoluminescence enhancement in all-dielectric Fano metasurface with high quality factor,” ACS Nano 11, 10704–10711 (2017).
    [Crossref]
  41. M. Homer Reid and S. Johnson, “Efficient computation of power, force, and torque in BEM scattering calculations,” arXiv:1307.2966 (2013).
  42. M. T. H. Reid, https://github.com/HomerReid/scuff-em .
  43. R. Yu, L. M. Liz-Marzán, and F. J. García de Abajo, “Universal analytical modeling of plasmonic nanoparticles,” Chem. Soc. Rev. 46, 6710–6724 (2017).
    [Crossref]
  44. D. R. Abujetas, J. A. Sánchez-Gil, and J. J. Sáenz, “Generalized Brewster effect in high-refractive-index nanorod-based metasurfaces,” Opt. Express 26, 31523–31541 (2018).
    [Crossref]

2019 (1)

L. Cong and R. Singh, “Symmetry-protected dual bound states in the continuum in metamaterials,” Adv. Opt. Mater. 7, 1900383 (2019).
[Crossref]

2018 (8)

H. M. Doeleman, F. Monticone, W. den Hollander, A. Alù, and A. F. Koenderink, “Experimental observation of a polarization vortex at an optical bound state in the continuum,” Nat. Photonics 12, 397–401 (2018).
[Crossref]

K. Koshelev, S. Lepeshov, M. Liu, A. Bogdanov, and Y. Kivshar, “Asymmetric metasurfaces with high-Q resonances governed by bound states in the continuum,” Phys. Rev. Lett. 121, 193903 (2018).
[Crossref]

S. I. Azzam, V. M. Shalaev, A. Boltasseva, and A. V. Kildishev, “Formation of bound states in the continuum in hybrid plasmonic-photonic systems,” Phys. Rev. Lett. 121, 253901 (2018).
[Crossref]

S. T. Ha, Y. H. Fu, N. K. Emani, Z. Pan, R. M. Bakker, R. Paniagua-Domínguez, and A. I. Kuznetsov, “Directional lasing in resonant semiconductor nanoantenna arrays,” Nat. Nanotechnol. 13, 1042–1047 (2018).
[Crossref]

S. Baur, S. Sanders, and A. Manjavacas, “Hybridization of lattice resonances,” ACS Nano 12, 1618–1629 (2018).
[Crossref]

M. Habib, M. Gokbayrak, E. Ozbay, and H. Caglayan, “Electrically controllable plasmon induced reflectance in hybrid metamaterials,” Appl. Phys. Lett. 113, 221105 (2018).
[Crossref]

T. C. Tan, Y. K. Srivastava, M. Manjappa, E. Plum, and R. Singh, “Lattice induced strong coupling and line narrowing of split resonances in metamaterials,” Appl. Phys. Lett. 112, 201111 (2018).
[Crossref]

D. R. Abujetas, J. A. Sánchez-Gil, and J. J. Sáenz, “Generalized Brewster effect in high-refractive-index nanorod-based metasurfaces,” Opt. Express 26, 31523–31541 (2018).
[Crossref]

2017 (9)

S. Yuan, X. Qiu, C. Cui, L. Zhu, Y. Wang, Y. Li, J. Song, Q. Huang, and J. Xia, “Strong photoluminescence enhancement in all-dielectric Fano metasurface with high quality factor,” ACS Nano 11, 10704–10711 (2017).
[Crossref]

R. Yu, L. M. Liz-Marzán, and F. J. García de Abajo, “Universal analytical modeling of plasmonic nanoparticles,” Chem. Soc. Rev. 46, 6710–6724 (2017).
[Crossref]

A. Kodigala, T. Lepetit, Q. Gu, B. Bahari, Y. Fainman, and B. Kanté, “Lasing action from photonic bound states in continuum,” Nature 541, 196–199 (2017).
[Crossref]

E. N. Bulgakov and D. N. Maksimov, “Topological bound states in the continuum in arrays of dielectric spheres,” Phys. Rev. Lett. 118, 267401 (2017).
[Crossref]

A. Taghizadeh and I.-S. Chung, “Quasi bound states in the continuum with few unit cells of photonic crystal slab,” Appl. Phys. Lett. 111, 031114 (2017).
[Crossref]

D. R. Abujetas, M. A. G. Mandujano, E. R. Méndez, and J. A. Sánchez-Gil, “High-contrast Fano resonances in single semiconductor nanorods,” ACS Photon. 4, 1814–1821 (2017).
[Crossref]

M. F. Limonov, M. V. Rybin, A. N. Poddubny, and Y. S. Kivshar, “Fano resonances in photonics,” Nat. Photonics 11, 543–554 (2017).
[Crossref]

A. Halpin, C. Mennes, A. Bhattacharya, and J. Gómez Rivas, “Visualizing near-field coupling in terahertz dolmens,” Appl. Phys. Lett. 110, 101105 (2017).
[Crossref]

A. Halpin, N. van Hoof, A. Bhattacharya, C. Mennes, and J. Gomez Rivas, “Terahertz diffraction enhanced transparency probed in the near field,” Phys. Rev. B 96, 85110 (2017).
[Crossref]

2016 (3)

M. Manjappa, Y. K. Srivastava, and R. Singh, “Lattice-induced transparency in planar metamaterials,” Phys. Rev. B 94, 161103 (2016).
[Crossref]

M. C. Schaafsma, A. Bhattacharya, and J. G. Rivas, “Diffraction enhanced transparency and slow THz light in periodic arrays of detuned and displaced dipoles,” ACS Photon. 3, 1596–1603 (2016).
[Crossref]

C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljačić, “Bound states in the continuum,” Nat. Rev. Mater. 1, 16048 (2016).
[Crossref]

2015 (1)

Z.-J. Yang, T. J. Antosiewicz, R. Verre, F. J. García de Abajo, S. P. Apell, and M. Käll, “Ultimate limit of light extinction by nanophotonic structures,” Nano Lett. 15, 7633–7638 (2015).
[Crossref]

2014 (5)

R. Singh, W. Cao, I. Al-Naib, L. Cong, W. Withayachumnankul, and W. Zhang, “Ultrasensitive terahertz sensing with high-Q Fano resonances in metasurfaces,” Appl. Phys. Lett. 105, 171101 (2014).
[Crossref]

P. Fan, Z. Yu, S. Fan, and M. L. Brongersma, “Optical Fano resonance of an individual semiconductor nanostructure,” Nat. Mater. 13, 471–475 (2014).
[Crossref]

N. Verellen, F. López-Tejeira, R. Paniagua-Domínguez, D. Vercruysse, D. Denkova, L. Lagae, P. Van Dorpe, V. V. Moshchalkov, and J. A. Sánchez-Gil, “Mode parity-controlled Fano- and Lorentz-like line shapes arising in plasmonic nanorods,” Nano Lett. 14, 2322–2329 (2014).
[Crossref]

C. W. Hsu, B. G. DeLacy, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Theoretical criteria for scattering dark states in nanostructured particles,” Nano Lett. 14, 2783–2788 (2014).
[Crossref]

N. Meinzer, W. L. Barnes, and I. R. Hooper, “Plasmonic meta-atoms and metasurfaces,” Nat. Photonics 8, 889–898 (2014).
[Crossref]

2013 (2)

S. R. K. Rodriguez, O. T. A. Janssen, G. Lozano, A. Omari, Z. Hens, and J. G. Rivas, “Near-field resonance at far-field-induced transparency in diffractive arrays of plasmonic nanorods,” Opt. Lett. 38, 1238–1240 (2013).
[Crossref]

Z. Zhu, X. Yang, J. Gu, J. Jiang, W. Yue, Z. Tian, M. Tonouchi, J. Han, and W. Zhang, “Broadband plasmon induced transparency in terahertz metamaterials,” Nanotechnology 24, 214003 (2013).
[Crossref]

2012 (1)

J. Gu, R. Singh, X. Liu, X. Zhang, Y. Ma, S. Zhang, S. A. Maier, Z. Tian, A. K. Azad, H.-T. Chen, A. J. Taylor, J. Han, and W. Zhang, “Active control of electromagnetically induced transparency analogue in terahertz metamaterials,” Nat. Commun. 3, 1151 (2012).
[Crossref]

2011 (4)

S. I. Bozhevolnyi, A. B. Evlyukhin, A. Pors, M. G. Nielsen, M. Willatzen, and O. Albrektsen, “Optical transparency by detuned electrical dipoles,” New J. Phys. 13, 023034 (2011).
[Crossref]

V. Giannini, A. I. Fernández-Domínguez, S. C. Heck, and S. A. Maier, “Plasmonic nanoantennas: fundamentals and their use in controlling the radiative properties of nanoemitters,” Chem. Rev. 111, 3888–3912 (2011).
[Crossref]

S. R. K. Rodriguez, A. Abass, B. Maes, O. T. A. Janssen, G. Vecchi, and J. Gómez Rivas, “Coupling bright and dark plasmonic lattice resonances,” Phys. Rev. X 1, 021019 (2011).
[Crossref]

C. Wu, A. B. Khanikaev, R. Adato, N. Arju, A. A. Yanik, H. Altug, and G. Shvets, “Fano-resonant asymmetric metamaterials for ultrasensitive spectroscopy and identification of molecular monolayers,” Nat. Mater. 11, 69–75 (2011).
[Crossref]

2010 (4)

A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, “Fano resonances in nanoscale structures,” Rev. Mod. Phys. 82, 2257–2298 (2010).
[Crossref]

B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9, 707–715 (2010).
[Crossref]

V. Giannini, G. Vecchi, and J. Gómez Rivas, “Lighting up multipolar surface plasmon polaritons by collective resonances in arrays of nanoantennas,” Phys. Rev. Lett. 105, 1–4 (2010).
[Crossref]

A. B. Evlyukhin, S. I. Bozhevolnyi, A. Pors, M. G. Nielsen, I. P. Radko, M. Willatzen, and O. Albrektsen, “Detuned electrical dipoles for plasmonic sensing,” Nano Lett. 10, 4571–4577 (2010).
[Crossref]

2009 (1)

S.-Y. Chiam, R. Singh, C. Rockstuhl, F. Lederer, W. Zhang, and A. A. Bettiol, “Analogue of electromagnetically induced transparency in a terahertz metamaterial,” Phys. Rev. B 80, 153103 (2009).
[Crossref]

2008 (1)

D. C. Marinica, A. G. Borisov, and S. V. Shabanov, “Bound states in the continuum in photonics,” Phys. Rev. Lett. 100, 183902 (2008).
[Crossref]

2007 (1)

F. J. García de Abajo, “Colloquium: light scattering by particle and hole arrays,” Rev. Mod. Phys. 79, 1267–1290 (2007).
[Crossref]

Abass, A.

S. R. K. Rodriguez, A. Abass, B. Maes, O. T. A. Janssen, G. Vecchi, and J. Gómez Rivas, “Coupling bright and dark plasmonic lattice resonances,” Phys. Rev. X 1, 021019 (2011).
[Crossref]

Abujetas, D. R.

D. R. Abujetas, J. A. Sánchez-Gil, and J. J. Sáenz, “Generalized Brewster effect in high-refractive-index nanorod-based metasurfaces,” Opt. Express 26, 31523–31541 (2018).
[Crossref]

D. R. Abujetas, M. A. G. Mandujano, E. R. Méndez, and J. A. Sánchez-Gil, “High-contrast Fano resonances in single semiconductor nanorods,” ACS Photon. 4, 1814–1821 (2017).
[Crossref]

Adato, R.

C. Wu, A. B. Khanikaev, R. Adato, N. Arju, A. A. Yanik, H. Altug, and G. Shvets, “Fano-resonant asymmetric metamaterials for ultrasensitive spectroscopy and identification of molecular monolayers,” Nat. Mater. 11, 69–75 (2011).
[Crossref]

Albrektsen, O.

S. I. Bozhevolnyi, A. B. Evlyukhin, A. Pors, M. G. Nielsen, M. Willatzen, and O. Albrektsen, “Optical transparency by detuned electrical dipoles,” New J. Phys. 13, 023034 (2011).
[Crossref]

A. B. Evlyukhin, S. I. Bozhevolnyi, A. Pors, M. G. Nielsen, I. P. Radko, M. Willatzen, and O. Albrektsen, “Detuned electrical dipoles for plasmonic sensing,” Nano Lett. 10, 4571–4577 (2010).
[Crossref]

Al-Naib, I.

R. Singh, W. Cao, I. Al-Naib, L. Cong, W. Withayachumnankul, and W. Zhang, “Ultrasensitive terahertz sensing with high-Q Fano resonances in metasurfaces,” Appl. Phys. Lett. 105, 171101 (2014).
[Crossref]

Altug, H.

C. Wu, A. B. Khanikaev, R. Adato, N. Arju, A. A. Yanik, H. Altug, and G. Shvets, “Fano-resonant asymmetric metamaterials for ultrasensitive spectroscopy and identification of molecular monolayers,” Nat. Mater. 11, 69–75 (2011).
[Crossref]

Alù, A.

H. M. Doeleman, F. Monticone, W. den Hollander, A. Alù, and A. F. Koenderink, “Experimental observation of a polarization vortex at an optical bound state in the continuum,” Nat. Photonics 12, 397–401 (2018).
[Crossref]

Antosiewicz, T. J.

Z.-J. Yang, T. J. Antosiewicz, R. Verre, F. J. García de Abajo, S. P. Apell, and M. Käll, “Ultimate limit of light extinction by nanophotonic structures,” Nano Lett. 15, 7633–7638 (2015).
[Crossref]

Apell, S. P.

Z.-J. Yang, T. J. Antosiewicz, R. Verre, F. J. García de Abajo, S. P. Apell, and M. Käll, “Ultimate limit of light extinction by nanophotonic structures,” Nano Lett. 15, 7633–7638 (2015).
[Crossref]

Arju, N.

C. Wu, A. B. Khanikaev, R. Adato, N. Arju, A. A. Yanik, H. Altug, and G. Shvets, “Fano-resonant asymmetric metamaterials for ultrasensitive spectroscopy and identification of molecular monolayers,” Nat. Mater. 11, 69–75 (2011).
[Crossref]

Azad, A. K.

J. Gu, R. Singh, X. Liu, X. Zhang, Y. Ma, S. Zhang, S. A. Maier, Z. Tian, A. K. Azad, H.-T. Chen, A. J. Taylor, J. Han, and W. Zhang, “Active control of electromagnetically induced transparency analogue in terahertz metamaterials,” Nat. Commun. 3, 1151 (2012).
[Crossref]

Azzam, S. I.

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Luk’yanchuk, B.

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J. Gu, R. Singh, X. Liu, X. Zhang, Y. Ma, S. Zhang, S. A. Maier, Z. Tian, A. K. Azad, H.-T. Chen, A. J. Taylor, J. Han, and W. Zhang, “Active control of electromagnetically induced transparency analogue in terahertz metamaterials,” Nat. Commun. 3, 1151 (2012).
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ACS Nano (2)

S. Baur, S. Sanders, and A. Manjavacas, “Hybridization of lattice resonances,” ACS Nano 12, 1618–1629 (2018).
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ACS Photon. (2)

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Supplementary Material (1)

NameDescription
» Supplement 1       Numerical simulations of off-normal transmittance spectra; scattering efficiencies of isolated rods and dimers, along with polarizibilities; coupled detuned-dipole formulation.

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Figures (5)

Fig. 1.
Fig. 1. (a) Measured transmittance spectra for square lattices (a=300μm) of two gold rods per unit cell, deposited on a quartz substrate, with two different rod separations: dx=a/2 (solid curves) and dx=2a/5 (dashed curves). One of the rods has fixed dimensions of L1=200μm and w1=40μm, while the other varies as shown in the center insets, with L2(μm)=125,150,175,200,225,250, while keeping the surface area fixed: L2w2=L1w1. All rod thicknesses are t=0.1μm. (b) Transmittance spectra numerically calculated through SCUFF [41,42] for the same geometrical parameters but considering gold rods as planar perfectly conducting rectangles embedded in a uniform medium (see text). Curves are offset by 1 for each different L2.
Fig. 2.
Fig. 2. Measured transient response after background subtraction for square lattices (a=300μm) as in Fig. 1, consisting of two gold rods per unit cell, deposited on a quartz substrate, with rod separation dx=2a/5 (solid curves). One of the rods has fixed dimensions of L1=200μm and w1=40μm, while the other varies as shown in the figure legend, with L2(μm)=125 (blue), 150 (red), 175 (orange). In all cases, the transient for identical rod dimers (L2=200μm) is subtracted; the resulting differential transients are normalized to the maximum of L2(μm)=125 and offset by 1 for convenience. Recall that the rod surface area is fixed as L2w2=L1w1, and the rods have thicknesses of t=0.1μm. Dashed lines show fits to single frequency-damped harmonic oscillators with a lifetime of 5.9, 10, and 20 ps, respectively, and oscillation frequencies of 0.4, 0.39, and 0.375 THz, respectively.
Fig. 3.
Fig. 3. (a)–(d) Theoretical transmittance spectra calculated through coupled dipole theory for a square lattice (a=300μm) as in Fig. 1 but consisting of two detuned dipoles per unit cell, separated by dx=2a/5: (a), (b) transmittance; (c), (d) phase. Dipole polarizabilities are extracted from the numerically calculated scattering cross sections shown in Supplement 1, Fig. S2 (see text): one is fixed and corresponds to a rod with dimensions L1=200 μm and w1=40 μm, while the other one L2 (with identical areas L2w2=L1w1) varies continuously in the contour maps in (a), (c), whereas those cases corresponding to three of the experimental dimers, L2(μm)=150,200,250, are shown in (b), (d).
Fig. 4.
Fig. 4. Spectral variation of both local field (a) phases (including the relative phase) and (b) amplitudes, calculated through coupled dipole theory for a square lattice (a=300μm) as in Fig. 1, but consisting of two detuned dipoles per unit cell, separated by dx=2a/5 for L2=200 μm and L2=150 μm. The local amplitudes and phases at both rods for L2=200μm are identical.
Fig. 5.
Fig. 5. Near-field simulations showing the electric field component along the z direction for a broadband (center frequency around the BIC) dipole source (located at the center of the images indicated by an arrow) in a square lattice (a=300μm) of gold rod dimers on a quartz substrate with rod separation dx=2a/5. One of the rods has fixed dimensions of L1=200μm and w1=40μm, while the other is (a) L2=125μm (i.e., detuned dipoles) and (b) L2=200μm (i.e., equal dipoles), while keeping the surface area fixed: L2w2=L1w1. The dipole source emits at ν=0.357THz (BIC frequency) and is given 200 ps to propagate over the total simulated area, 2.5×2.5mm2, indeed much bigger than the area shown.

Equations (7)

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[ψloc(1)ψloc(2)]=[Ik2Gbα]1[ψ0(1)ψ0(2)],
α=[αy(1)00αy(2)],Gb=[GbyyGyy(12)Gyy(21)Gbyy].
I[(1αyGbyy)+Gyy(12)]=0,
2αy=1k2(1αy(1)+1αy(2)),Δαy=1k2(1αy(1)1αy(2)).
I[Λ+]=I[(Δαy)28Gyy(12)],
I[Λ]=I[2(1αyGbyy)(Δαy)28Gyy(12)].
Λ±=0ν±=[ψloc(1)ψloc(2)]=[11].

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