⇦Chinese Physics Letters, 2020, Vol. 37, No. 6, Article code 067802 Bound States in the Continuum in One-Dimensional Dimerized Plasmonic Gratings * Chen Huang (黄陈), Qian-Ju Song (宋前举), Peng Hu (胡鹏), Shi-Wei Dai (戴士为), Hong Xiang (向红)**, Dezhuan Han (韩德专)** Affiliations College of Physics, Chongqing University, Chongqing 401331, China Received 11 March 2020, online 26 May 2020 *Supported by the National Natural Science Foundation of China (Grant Nos. 11804288 and 91750102) and the Projects of President Foundation of Chongqing University (Grant No. 2019CDXZWL002).
**Corresponding author. Email: xhong@cqu.edu.cn; dzhan@cqu.edu.cn Citation Text: Huang C, Song Q J, Hu P, Dai S W and Xiang H et al 2020 Chin. Phys. Lett. 37 067802    Abstract A simple one-dimensional subwavelength plasmonic grating can support symmetry protected bound states in the continuum (BICs), but not necessarily for the non-symmetry protected BICs. By dimerizing the lattice, non-symmetry protected BIC can be supported on the dimerized grating and can be tuned readily. The mechanism for the BICs in the dimerized grating is interpreted in the viewpoint of interference between the electromagnetic multipoles. DOI:10.1088/0256-307X/37/6/067802 PACS:78.67.Pt, 42.79.Dj, 42.70.Qs © 2020 Chinese Physics Society Article Text Bound states in the continuum (BICs), as suggested by the name, are spatially localized non-radiative states with theoretically infinitely long lifetimes though their frequencies lie inside the continuous spectrum. Since the first proposal of this concept in the quantum system by von Neumann and Wigner in 1929,[1] BICs have been revealed as a general wave phenomenon in many physical systems from quantum to classic waves.[2–13] Different approaches such as vortex centers in the polarization direction of far-field radiation,[14] phase singularity of the quasi-mode coupling strength,[15,16] coherent perfect reflection of Bloch modes,[17] overlapping of multipolar singularities with open radiation channels[18,19] have been applied to explore the physical mechanism of BICs. BICs have been confirmed experimentally in different ways, including the disappearance of resonance features in the reflectivity spectrum,[20–22] momentum-space polarization vortices in two-dimensional plasmonic crystals,[23] radiation attenuation of the magnetic field in one-dimensional dimerized chains.[16] Due to the theoretically infinitely high $Q$ factor, BICs are of great interest in the ultrasensitive optical absorption[24] and strong light enhancement,[25,26] leading to numerous applications such as sensing,[27] lasing,[28] and filtering.[29] Usually, BICs can be classified into two types: symmetry protected BICs and non-symmetry protected (or accidental) BICs. The former ones arise from the symmetry incompatibility between the confined mode and the external field, while the latter ones come from the destructive interference between the leakage channels of the system.[21] Generally, optical BICs are realized in extended structures especially in periodic systems. As the simplest periodic system, one-dimensional (1D) arrays of dielectric rods or dielectric gratings have been extensively investigated.[12,20,22,25,26,30,31] Both types of BICs can be realized in these systems. On the other hand, scattering of light by a periodic array of metallic rods has been widely studied[32,33] for their versatile applications. However, research on BICs in such a metallic or plasmonic structure is quite limited due to the inherent material loss. In fact, BICs can still be achieved in a plasmonic system when the material loss is compensated for with gain from an active medium.[34] BICs have also been achieved in hybrid plasmonic-photonic systems based on avoided crossing of the plasmonic resonance with the photonic resonance.[35] Hence it is worthwhile to investigate the BICs supported on a plasmonic grating. Moreover, the simulation (see below) suggests that for a simple subwavelength plasmonic grating, symmetry protected BIC at the ${\it\Gamma}$ point can be achieved readily while the accidental BIC is not necessarily supported. It is necessary to find a new strategy to realize the accidental BIC on a 1D plasmonic grating and investigate the corresponding mechanism. On the other hand, dimerization of a simple periodic lattice can usually bring new features to the system. Compared to the original simple lattice, the two elements in the unit cell of the dimerized lattice can provide more degrees of freedom. For example, the dimerized polyacetylene chain, in which there are two carbon atoms in a unit cell, as described by the Su–Schrieffer–Heeger (SSH) model,[36] can support non-trivial topological phases and edge states. Analogs of the SSH model have been found in other quantum and classic wave systems.[37–41] Another example is the dimerized high contrast gratings.[42] Compared with the ordinary high contrast gratings, the dimerized ones can support modes accessible by free-space illumination with a long controllable photon lifetime and reduced lateral energy divergence, and hence offer a promising platform for engineering sharp spectral features in compact devices. Recently, BICs have been achieved in the dimerized periodic structures. By embedding the topological edge states in the complex structure of two indirectly coupled SSH chains in the continuum of the non-topological bands, BICs in an acoustic system is formed.[43] In an optical system, BICs are identified by the singularities of effective mode polarizability of a dimerized chain of plasmonic spheres.[16] In view of this background, dimerizing the simple plasmonic grating may bring new features to the formation of BICs. In this Letter, we construct a one-dimensional dimerized plasmonic grating and investigate BICs in this system. First a simple periodic plasmonic grating is studied for comparison and only the symmetry protected BIC at the ${\it\Gamma}$ point is observed in the frequency range of interest. Then the dimerized grating is studied, and both types of BICs are observed in this system. The symmetric protected BICs are supported in the high-order bands and the accidental BIC can be supported in the lowest band above the light line. The corresponding $Q$-factors and field profiles are calculated to characterize the BICs. It is demonstrated that the accidental BIC can be tuned by adjusting the period of the system or by adjusting the gap between the two bars in the unit cell. In the end, the mechanisms for the formation of the BICs are discussed based on the symmetry analysis and the multipole decomposition. At the accidental BIC point, the multipole moments are calculated to obtain the total radiation pattern of a unit cell. At certain direction the radiation drops to zero, and the zero-radiation direction overlaps with the direction of the radiation channel of the plasmonic grating.
cpl-37-6-067802-fig1.png
Fig. 1. Schematic diagram of the system studied. The structure is periodic in the $x$ direction and uniform in the $y$ direction. The background is the free space. The system is illuminated by a p-polarized plane wave with an incident angle of $\theta$.
The dimerized plasmonic grating is composed of periodic square bars with infinite length in $y$ direction, and the cross section of the grating is schematically shown in Fig. 1. The grating is placed in vacuum and illuminated by a p-polarized plane waves with an incidence angle $\theta$. The geometry parameters (periodicity $a$, side length $L$ and gap $g$) are marked in Fig. 1. The permittivity of the plasmonic bars is described by the Drude model: $\varepsilon (\omega)=1-\omega_{\rm p}^{2}/(\omega^{2}+i\gamma \omega)$, where $\omega_{\rm p}$ is the plasma frequency and $\gamma$ is the electron scattering rate. In this work, $\omega_{\rm p}$ is set as 6.18 eV and the loss is neglected. The optical response of the gratings is numerically calculated with the finite element method by the commercial software Comsol Multiphysics. In the simulation, an open boundary condition is employed. Before discussing the BICs on the dimerized grating, let us briefly illustrate the optical response of the corresponding simple grating which can be obtained simply by replacing the bar dimer by a single bar in each unit cell. The calculated transmission spectrum of the simple grating is shown in Fig. 2(a), in which only one band can be observed in the frequency of interest. The disappearance of the resonance feature at the ${\it\Gamma}$ point indicates a BIC here. The band diagram of this structure, obtained by solving the eigenvalue problem numerically, is shown Fig. 2(b). The corresponding $Q$-factor shown in Fig. 2(c) diverges at the ${\it\Gamma}$ point, in accordance with the transmission spectra. The near-field profile of the electric field corresponding to the BIC point, shown by an inset in Fig. 2(c), exhibits a mirror symmetry with respect to the $y$–$z$ plane.
cpl-37-6-067802-fig2.png
Fig. 2. (a) The simulated transmission spectrum. The gray area corresponds to the first order diffraction region. (b) Band diagram of the simple grating. The dashed line indicates the light line. (c) The $Q$ factors correspond to the band in (b). The inset in (c) is the electric field profile corresponding to the ${\it\Gamma}$ point in the band, as marked by A in (b). Here $L_{} = 80$ nm, $a = 360$ nm. The inset on the top illustrates the grating under illumination.
The dimerized grating exhibits a different optical response compared with the simple grating. The simulated transmission spectrum is shown in Fig. 3(a). Here the gap is set as $g = 50$ nm, and the other parameters are the same as those in Fig. 2. In the transmission spectrum, four bands are observed in the interested frequency range. In the upper two and the lowest branches, the disappearance of resonance feature indicates the existence of BICs. The transmission corresponding to the normal incidence ($\theta = 0$) is presented in Fig. 3(b) and two resonant peaks corresponding to the lower two bands are observed. The upper peak exhibits a broad profile while the lower one is sharp and exhibits a Fano line shape. The near-field profiles of the electric field corresponding to the two resonant peaks are presented in Fig. 3(c). The upper one shows a symmetric distribution while the lower one exhibits an anti-symmetric feature. The band diagram obtained by solving the eigenvalue problem is shown in Fig. 3(d). The calculated $Q$ factors of band 1 is presented in Fig. 3(e). The divergence of the $Q$ factor occurs at the same place as the resonance linewidth vanishes in Fig. 3(a). The field profile of the BIC point, marked by B in Figs. 3(d) and 3(e), is plotted in Fig. 3(f). The field profile corresponding to the ${\it\Gamma}$ point is also plotted for comparison. Unlike the field of the ${\it\Gamma}$ point which extends to infinity, the electric field of point B is localized in the vicinity of the plasmonic bars, manifesting the characteristic of a BIC.
cpl-37-6-067802-fig3.png
Fig. 3. (a) The simulated transmission spectrum of the dimerized grating. (b) The transmission corresponding to the incident angle $\theta = 0$. (c) The near-field patterns of the electric field for the two resonant peaks indicated by the blue and red dots marked in (b). (d) Band diagrams of the system. The dashed line indicates the light line. (e) The $Q$ factors corresponding to band 1 indicated by the green curve in (d). (f) The field profiles ($E_{x}$) corresponding to points A and B marked in (d) and (e). Here the parameters are set as $L = 80$ nm, $g = 50$ nm, $a = 360$ nm.
The off-${\it\Gamma}$ (accidental) BIC on this dimerized grating can be tuned flexibly by modifying the structural parameter. For instance, the off-${\it\Gamma}$ BIC can be shifted to the ${\it\Gamma}$ point by modifying the gap as $g = 76$ nm and keeping other parameters fixed, or by changing period as $a = 426$ nm and leaving other parameters unchanged. The band diagrams and the corresponding $Q$-factors of the two cases are shown in Figs. 4(a) and 4(b) and Figs. 4(d) and 4(e), respectively. The divergence behaviors of the BICs of the two case are calculated and plotted in Figs. 4(c) and 4(g), respectively. For the off-${\it\Gamma}$ BICs, the $Q$ factor diverges with $Q \sim 1/(q -q_{\rm BIC})^{2}$, where $q_{\rm BIC}$ is the wavenumber of the BIC. When the BIC point is tuned to the ${\it\Gamma}$ point, $Q \sim 1/q^{4}$, which is consistent with the previous reports.[44,45] For each off-${\it\Gamma}$ BIC corresponding to $q_{\rm BIC}$, there is another off-${\it\Gamma}$ BIC corresponding to $-q_{\rm BIC}$, which diverges with $Q \sim 1/(q + q_{\rm BIC})^{2}$. The tuning of the parameters can merge the two isolated BIC into one, known as the merging BIC. The merging BIC diverges with $Q \sim 1/q^{4}$. In fact, it is an effective method to achieve resonances with ultrahigh $Q$ by merging multiple BICs.[46]
cpl-37-6-067802-fig4.png
Fig. 4. Tuning of the BIC by modifying the geometry parameters. (a) Band diagrams of the system with the gap $g = 76$ nm. (b) The $Q$ factors corresponding to band 1 in (a). (c) Divergence behavior in the vicinity of the BIC points, with the black dots for the isolated off-${\it\Gamma}$ BIC before tuning and the blue dots for the merging BIC, respectively. The corresponding lines indicate the power function fitting. (d) Band diagrams of the system with the periodicity $a = 426$ nm. (e) The $Q$ factors corresponding to band 1 in (d). (f) Divergence behavior in the vicinity of the BIC point.
The origin of the BICs in the gratings mentioned above can be explained from the viewpoints of mode symmetry and multipole interference. The BIC formed in the simple grating is a symmetry protected BIC. The field profile corresponding to the ${\it\Gamma}$ point, as shown in Fig. 2, exhibits an even symmetry under the mirror reflection in the $x$ direction. While the only radiation channel corresponding to the ${\it\Gamma}$ point in the subwavelength regime is the plane wave propagating in the $z$ direction, with the corresponding electric and magnetic field vectors being odd under the mirror reflection. The two symmetries are incompatible with each other and hence a BIC forms. Similarly, the BICs supported in bands 3 and 4 in the dimerized grating are also originated from the symmetry incompatibility since the corresponding field profiles possess an even mirror symmetry (not shown here). As for bands 1 and 2, though the profiles corresponding to the ${\it\Gamma}$ points exhibit an anti-symmetric and symmetric characteristic, none of the profiles are even under the mirror reflection and hence no BICs form here. The microscopic mechanism for the accidental BIC in the systems can be understood from the viewpoint of multipole interference.[18–20] As is known, the radiating property of a periodic system depends on the radiation property of each unit cell and the interference between the unit cells. The latter corresponds to the diffraction channels, and for subwavelength regime, only the 0$^{\rm th}$-order diffraction exists. If there is a direction in which the radiation of the unit cell drops to zero and this direction overlaps with the radiation channel of the system, no radiation will happen, and thus a BIC will form. On the other hand, as an electromagnetic radiation source, each unit cell can be decomposed into multipoles. Then the zero-radiation direction of the unit cell can be regarded as the result of destructive interference between the multipoles. Therefore, the BICs can be interpreted in the viewpoint of multipole interference. For the simple grating, as the lateral size of each single bar is much smaller than the wavelength in vacuum, the multipoles supported on a single bar is dominated by the electric dipole. There is no significant interference between multipoles and thus no off-${\it\Gamma}$ BIC forms. For the dimerized grating, the currents in the two bars in a unit cell give rise to a plenty of multipoles with significant moments. By solving the eigenproblem of the system, the currents in the grating are obtained, then the multipole moments corresponding to a unit cell are calculated according to the following definitions.[47–49] Electric dipole moment: $$ {\boldsymbol p}= -\frac{1}{i\omega }\int {{\boldsymbol j}{d}^{3}} r, $$ magnetic dipole moment: $$ {\boldsymbol m}= \frac{1}{2c}\int {({\boldsymbol r}\times {\boldsymbol j})} d^{3}r, $$ toroidal dipole moment: $$ \boldsymbol{T}\mathbf{ = }\frac{1}{10c}\int {[({\boldsymbol r}\cdot {\boldsymbol j}){\boldsymbol r}-2r^{2}{\boldsymbol j}]d^{3}r}, $$ electric quadrupole moment: $$ Q_{{\rm (e)}\alpha \beta } = -\frac{1}{2i\omega }\int {\left[r_{\alpha } j_{\beta } +r_{\beta } j_{\alpha } -\frac{2}{3}({\boldsymbol r}\cdot {\boldsymbol j})\delta_{\alpha \beta } \right]d^{3}r}, $$ magnetic quadrupole moment: $$ Q_{{\rm (m)}\alpha \beta } =\frac{1}{3c}\int {[({\boldsymbol r}\times {\boldsymbol j})_{\alpha } r_{\beta } +({\boldsymbol r}\times {\boldsymbol j})_{\beta } r_{\alpha } ]d^{3}r,} $$ electric octupole moment:% $$\begin{align} O_{{\rm (e)}\alpha,\beta,\gamma} =\,&\frac{1}{6i\omega }\Big\{ \int \Big[ j_{\alpha } \Big({\frac{r_{\beta } r_{\gamma } }{3}-\frac{1}{5}r^{2}\delta_{\beta,\gamma } } \Big)\\ &+r_{\alpha } \Big({\frac{j_{\beta } r_{\gamma } }{3}+\frac{j_{\gamma } r_{\beta } }{3}-\frac{2}{5}\Big({{\boldsymbol r}\cdot {\boldsymbol j}}\Big)\delta_{\beta,\gamma } } \Big) \Big] d^{3}r \\ &+\int \Big[ j_{\beta } \Big(\frac{r_{\alpha } r_{\gamma } }{3}-\frac{1}{5}r^{2}\delta_{\alpha,\gamma } \Big)\\ &+r_{\beta } \Big({\frac{j_{\alpha } r_{\gamma } }{3}+\frac{j_{\gamma } r_{\alpha } }{3}-\frac{2}{5}\Big({{\boldsymbol r}\cdot {\boldsymbol j}} \Big)\delta_{\alpha,\gamma } } \Big) \Big] d^{3}r \\ &+\int \Big[ j_{\gamma } \Big({\frac{r_{\beta } r_{\alpha } }{3}-\frac{1}{5}r^{2}\delta_{\beta,\alpha } } \Big)\\ &+r_{\gamma } \Big({\frac{j_{\beta } r_{\alpha } }{3}+\frac{j_{\alpha } r_{\beta } }{3}-\frac{2}{5}\Big({{\boldsymbol r}\cdot {\boldsymbol j}} \Big)\delta_{\beta,\alpha } } \Big) \Big] d^{3}r \Big\}. \end{align} $$ Here ${\boldsymbol j}$ is the current density, $c$ is the speed of light, ${\boldsymbol r}$ is the position vector, and the subscripts $\alpha$ and $\beta$ represent the coordinates $x$, $y$, $z$. As the lateral size of the grating bars is much smaller than the incident wavelength, other higher-order terms are neglected. Accordingly, the far-field distribution from the unit cell is the sum of the multipole radiation fields:[41] $$\begin{align} \boldsymbol{E}(\boldsymbol{r})\!\approx &\frac{k^{2}e^{ikr}}{4\pi \varepsilon_{0}r}\big(\hat{\boldsymbol{r}}\!\times\! [({\boldsymbol p}\!+\!ik{\boldsymbol T})\!+\!{\boldsymbol m}\!\times \!\hat{\boldsymbol{r}}\!+\!ik\hat{\boldsymbol{r}}\!\times\! [\hat{\boldsymbol{r}}\!\times\! {\boldsymbol Q}_{\rm (e)} \hat{\boldsymbol{r}}]\\ &+\frac{ik}{2}\hat{\boldsymbol{r}}\times {\boldsymbol Q}_{\rm (m)} \hat{\boldsymbol{r}}+k^{2}\hat{\boldsymbol{r}}\times [\hat{\boldsymbol{r}}\times \boldsymbol{O}_{\rm (e)}(\boldsymbol{\hat{r}\hat{r}})] \big), \end{align} $$ where $\hat{\boldsymbol{r}}$ is the unit vector directed along $r$, ${\boldsymbol Q}_{\rm (e)} \hat{\boldsymbol{r}}$, ${\boldsymbol Q}_{\rm (m)} \hat{\boldsymbol{r}}$ and $\boldsymbol{O}_{\rm (e)}(\boldsymbol{\hat{r}\hat{r}})$ correspond to the dyadic products. For our system which is uniform in $y$ direction, only the field distributions in $x$–$z$ plane are considered. Figure 5(a) presents the far-field radiation profiles ($|E|^{2}$) of the multipole components contributing to the total field at the BIC point shown in Fig. 3. The total radiation profile is shown in Fig. 5(b), where the zero-radiation directions are indicated by the arrows with red crosses. When the multipoles up to the electric octupole are considered, the calculated zero-radiation direction corresponds to $\sin\theta =0.191$. For the grating considered here, the only radiation channel corresponding to $q$ is the zeroth-order diffraction with diffraction angle $\sin^{-1}(q/k_{0})$, where $k_{0}$ is the wavenumber of the plane wave in free space. For the BIC point shown in Fig. 3, $q_{\rm BIC}/k_{0} =0.193$, which is very close to the value of 0.191. It can be seen that the zero-radiation direction of the unit cell overlaps with the radiation channel of the grating and hence a BIC forms here.
cpl-37-6-067802-fig5.png
Fig. 5. (a) The profiles ($|E|^{2}$) of the dominant multipole components in the dimerized grating. Here the electric dipole ($p_{z}$), magnetic dipole ($m_{y}$), toroidal dipole ($T_{z}$), electric quadrupole ($Q_{{\rm (e)} xz}$, $Q_{{\rm (e)} zx}$), magnetic quadrupole ($Q_{{\rm (m)} xy}$, $Q_{{\rm (m)} yx}$), and electric octupole ($O_{{\rm (e)} xxz}$, $O_{{\rm (e)} xzx}$, $O_{{\rm (e)} zxx}$, $O_{{\rm (e)} zzz}$) are considered. (b) The total radiation profile resulted from the superposition of the multipoles. The arrows with red crosses indicate the zero-radiation direction.
In summary, we have investigated the BICs supported on a dimerized plasmonic grating. Both the symmetry protected BIC and the accidental BIC can be supported on the dimerized grating. The accidental BIC can be tuned easily by adjusting the geometry parameters such as the lattice periodicity or the gap between the two bars in a unit cell. The BICs on the simple grating and the dimerized grating are interpreted from the viewpoints of mode symmetry and multipole interference. The dimerized unit cell provides plentiful multipoles and the interference between them can give rise to zero-radiation directions. The accidental BIC results from the overlapping of the radiation channel of the system with the zero-radiation direction of the unit cell. This scheme applied in this work may provide useful guidance for BIC related researches and applications. References Interfering resonances and bound states in the continuumFormation of bound states in the continuum for a quantum dot with variable widthBound States in the Continuum Realized in the One-Dimensional Two-Particle Hubbard Model with an ImpuritySurface Bound States in the ContinuumEmbedded trapped modes in water waves and acousticsBound States in the Continuum in PhotonicsBound states in the continuum in photonic waveguides inspired by defectsExperimental Observation of Optical Bound States in the ContinuumBound states in the continuumToroidal dipole bound states in the continuumStrong terahertz magneto-optical phenomena based on quasi-bound states in the continuum and Fano resonancesAnalytical Perspective for Bound States in the Continuum in Photonic Crystal SlabsTopological Nature of Optical Bound States in the ContinuumTopological Bound States in the Continuum in Arrays of Dielectric SpheresObservation of bound states in the continuum in the dimerized chainFrom topologically protected coherent perfect reflection to bound states in the continuumSingularities and Poincaré Indices of Electromagnetic MultipolesMultipolar origin of bound states in the continuumExperimental observation of a polarization vortex at an optical bound state in the continuumObservation of trapped light within the radiation continuumTransition from Optical Bound States in the Continuum to Leaky Resonances: Role of Substrate and RoughnessObservation of Polarization Vortices in Momentum SpaceUltrasensitive optical absorption in graphene based on bound states in the continuumCritical field enhancement of asymptotic optical bound states in the continuumLight enhancement by quasi-bound states in the continuum in dielectric arraysTerahertz sensing of 7 nm dielectric film with bound states in the continuum metasurfacesLasing action from photonic bound states in continuumSymmetry-protected mode coupling near normal incidence for narrow-band transmission filtering in a dielectric gratingResonant transmission near nonrobust periodic slab modesBloch bound states in the radiation continuum in a periodic array of dielectric rodsDiffraction of electromagnetic waves by plane metallic gratingsDiffraction of electromagnetic waves by periodic arrays of rectangular cylindersEmbedded scattering eigenstates using resonant metasurfacesFormation of Bound States in the Continuum in Hybrid Plasmonic-Photonic SystemsSolitons in PolyacetyleneDirect measurement of the Zak phase in topological Bloch bandsSurface Impedance and Bulk Band Geometric Phases in One-Dimensional SystemsSelective enhancement of topologically induced interface states in a dielectric resonator chainMeasurement of the Zak phase of photonic bands through the interface states of a metasurface/photonic crystalLasing in topological edge states of a one-dimensional latticeDimerized high contrast gratingsTopological Subspace-Induced Bound State in the ContinuumStrong resonances on periodic arrays of cylinders and optical bistability with weak incident wavesBound states in the continuum on periodic structures surrounded by strong resonancesTopologically enabled ultrahigh-Q guided resonances robust to out-of-plane scatteringToroidal dipolar excitation and macroscopic electromagnetic properties of metamaterialsOptical theorem and multipole scattering of light by arbitrarily shaped nanoparticlesA minimal discrete model for toroidal moments and its experimental realization
[1]Von Neumann J and Wigner E 1929 Phys. Z. 30 465
[2] Friedrich H and Wintgen D 1985 Phys. Rev. A 32 3231
[3] Bulgakov E and Sadreev A 2011 Phys. Rev. B 83 235321
[4] Zhang J M, Braak, D and Kollar M 2012 Phys. Rev. Lett. 109 116405
[5] Molina M I, Miroshnichenko A E and Kivshar Y S 2012 Phys. Rev. Lett. 108 070401
[6] Linton C M and McIver P 2007 Wave Motion 45 16
[7] Marinica D C, Borisov A G and Shabanov S V 2008 Phys. Rev. Lett. 100 183902
[8] Bulgakov E N and Sadreev A F 2008 Phys. Rev. B 78 075105
[9] Plotnik Y, Peleg O, Dreisow F, Heinrich M, Nolte S, Szameit A and Segev M 2011 Phys. Rev. Lett. 107 183901
[10] Hsu C W, Zhen B, Stone A D, Joannopoulos J D and Soljačić M 2016 Nat. Rev. Mater. 1 16048
[11] He Y, Guo G, Feng T, Xu Y and Miroshnichenko A E 2018 Phys. Rev. B 98 161112
[12] Chen G Y, Zhang W X and Zhang X D 2019 Opt. Express 27 16449
[13] Yang Y, Peng C, Liang Y, Li Z and Noda S 2014 Phys. Rev. Lett. 113 037401
[14] Zhen B, Hsu C W, Lu L, Stone A D and Soljačić M 2014 Phys. Rev. Lett. 113 257401
[15] Bulgakov E N and Maksimov D N 2017 Phys. Rev. Lett. 118 267401
[16] Song Q J, Zhao M X, Liu L, Chai J W, He G H, Xiang H, Han D Z and Zi J 2019 Phys. Rev. A 100 023810
[17] Dai S, Liu L, Han D and Zi J 2018 Phys. Rev. B 98 081405
[18] Chen W, Chen Y and Liu W 2019 Phys. Rev. Lett. 122 153907
[19] Sadrieva Z, Frizyuk K, Petrov M, Kivshar Y and Bogdanov A 2019 Phys. Rev. B 100 115303
[20] Doeleman H M, Monticone F, Hollander W, Alù A and Koenderink A F 2018 Nat. Photon. 12 397
[21] Hsu C W, Zhen B, Lee J, Chua S L, Johnson S G, Joannopoulos J D and Soljačić M 2013 Nature 499 188
[22] Sadrieva Z F, Sinev I S, Koshelev K L, Samusev A, Iorsh I V, Takayama O, Malureanu R, Bogdanov A A and Lavrinenko A V 2017 ACS Photon. 4 723
[23] Zhang Y, Chen A, Liu W, Hsu C W, Wang B, Guan F, Liu X, Shi L, Lu L and Zi J 2018 Phys. Rev. Lett. 120 186103
[24] Zhang M and Zhang X 2015 Sci. Rep. 5 8266
[25] Yoon J W, Song S H and Magnusson R 2015 Sci. Rep. 5 18301
[26] Bulgakov E N and Maksimov D N 2017 Opt. Express 25 14134
[27] Srivastava Y K, Ako R T, Gupta M, Bhaskaran M, Sriram S and Singh R 2019 Appl. Phys. Lett. 115 151105
[28] Kodigala A, Lepetit T, Gu Q, Bahari B, Fainman Y and Kanté B 2017 Nature 541 196
[29] Foley J M, Young S M and Phillips J D 2014 Phys. Rev. B 89 165111
[30] Shipman S P and Venakides S 2005 Phys. Rev. E 71 026611
[31] Bulgakov E N and Sadreev A F 2014 Phys. Rev. A 90 053801
[32] Kalhor H A 1978 J. Opt. Soc. Am. 68 1202
[33] Zunoubi M R and Kalhor H A 2006 J. Opt. Soc. Am. A 23 306
[34] Krasnok A and Alú A 2018 J. Opt. 20 064002
[35] Azzam S I, Shalaev V M, Boltasseva A and Kildishev A V 2018 Phys. Rev. Lett. 121 253901
[36] Su W, Schrieffer J R and Heeger A J 1979 Phys. Rev. Lett. 42 1698
[37] Atala M, Aidelsburger M, Barreiro J T, Abanin D, Kitagawa T, Demler E, Bloch and I 2013 Nat. Phys. 9 795
[38] Xiao M, Zhang Z Q and Chan C T 2014 Phys. Rev. X 4 021017
[39] Poli C, Bellec M, Kuhl U, Mortessagne F and Schomerus H 2015 Nat. Commun. 6 6710
[40] Wang Q, Xiao M, Liu H, Zhu S, Chan and C T 2016 Phys. Rev. B 93 041415
[41] St-Jean P, Goblot V, Galopin E, Lemaître A, Ozawa T, Le Gratiet L, Sagnes I, Bloch J and Amo A 2017 Nat. Photon. 11 651
[42] Overvig A C, Shrestha S and Yu N 2018 Nanophotonics 7 1157
[43] Xiao Y X, Ma G, Zhang Z Q and Chan C T 2017 Phys. Rev. Lett. 118 166803
[44] Yuan L and Lu Y Y 2017 Phys. Rev. A 95 023834
[45] Yuan L and Lu Y Y 2018 Phys. Rev. A 97 043828
[46] Jin J, Yin X, Ni L, Soljačić M, Zhen B and Peng C 2019 Nature 574 501
[47] Savinov V, Fedotov V A and Zheludev N I 2014 Phys. Rev. B 89 205112
[48] Evlyukhin A B, Fischer T, Reinhardt C and Chichkov B N 2016 Phys. Rev. B 94 205434
[49] Xiang H, Ge L, Liu L, Jiang T, Zhang Z Q, Chan C T and Han D 2017 Phys. Rev. B 95 045403