In 1908, German physicist Gustav Mie gave the solution of Maxwell’s equations while he was analyzing the scattering problem of wavelength-comparable particles (e.g. gold nanoparticle) in a homogeneous surrounding (e.g. water) under the excitation of electromagnetic plane wave, which afterward was well known as the Mie solution ,  A century later, the current research of ubiquitous scattering phenomena in artificially engineered micro/nanostructures has been actively developing from the start point of Mie scattering. Photonic crystals , plasmonics , plasmonic metamaterials , and dielectric metamaterials  continuously challenge our common wisdom, expand the boundary of science and technology, and facilitate the advancement of interdisciplinary research. Among a wide range of artificial structures in photonics, we focus on high-index dielectric metastructures in this review. In contrast to photonic crystals, dielectric metastructures are not restricted in periodic arranged structures; the characteristic Mie resonant peaks can even be observed from one single structure. Moreover, out-of-plane excitation (normal incidence or oblique incidence) is conventionally used in dielectric metastructures, instead of in-plane excitation or propagation in photonic crystals. In contrast to plasmonic metastructures, low loss and optical magnetism (wavelength-sized structures in the visible range) in single dielectric structures are unique advantages of dielectric metastructures. For the choice of materials of building blocks in dielectric metastructures, several common high-index and low-loss materials are frequently used in the visible range, like Si, Ge, TiO2. Similar to photonic crystals, a unit cell of dielectric metastructures (e.g. a particle) is typically wavelength sized (d~λ/n, where d is the typical size of a dielectric particle, λ is the wavelength in free space, and n is the refractive index of dielectric), resulting in the breakdown of the quasi-static approximation. To accurately describe and thoroughly explain the scattering phenomenon of wavelength-sized homogeneous sphere, Mie solution is well-suited to analyze the scattered fields in near-field and far-field regions , , and the total scattering cross-section (Csca) can be expanded into an infinite series of spherical multipolar terms (Mie expansion)
where ɑm and bm are Mie coefficients of the mth-order electric and magnetic multipolar modes, respectively. k is the wavenumber. Electric dipole (ED, ~ɑ12), magnetic dipole (MD, ~b12), electric quadrupole (EQ, ~ɑ22), and magnetic quadrupole (MQ, ~b22) are the first four multipole terms in the Mie expansion . In mathematics, multipole terms in the Mie expansion form a complete and orthogonal basis for the decomposition of the scattered fields. For a 100-nm-radius Si nanosphere in free space, the scattering spectra and the electric fields (E-fields) distributions of dipole and quadrupole modes at multipolar scattering peaks (P1–P4) are shown in Figure 1A. One can simply distinguish the contribution of ED, MD, EQ, and MQ from the Mie expansion of scattering spectra (color lines in Figure 1A) and from the orientation of E-fields (black arrow in the insets of Figure 1A). Note that ɑm and bm are the functions of the particle size and the refractive index difference between the particle and the surroundings. High-index particles embedded in low-index medium are favorable to improve the Mie scattering efficiency. At the resonance of each multipolar mode, the electromagnetic fields are strongly confined inside the particle and the remarkable enhancement is comparable to the plasmonic counterpart. In contrast, a broadened ED scattering peak (Pplasmon) is obtained from a 100-nm-radius Au nanosphere (Figure 1B), due to the strongly damped plasmon mode. The expansion results indicate the lack of MD (blue line in Figure 1B) and high-order magnetic multipole modes in subwavelength-sized metal nanoparticles at the visible and near-infrared regions. Enlightened by the simplest case of nanosphere, we should clarify that the Mie solution, which gives an exact solution for the case of spherical particles and a first-order approximation for the case of nonspherical particles , is merely a special case of standard multipole expansion in electromagnetics . For readers who are interested in numerically addressing the scattering problem of arbitrary particles or particles array, the analytical framework is available elsewhere . Additionally, originating from toroidal current configuration (currents flowing along the merdians of a fictitious torus), toroidal multipoles cannot be directly extracted from the standard electric and magnetic multipole expansion; the specific formulism and detailed interpretation are available elsewhere , .
Recently, the artificial manipulation of multipolar and multimodal properties in dielectric metastructures have been extensively reported in zero- (0D; e.g. single nanoparticle , , , , ), one- (1D; e.g. single nanowire and nanorods , , , , , two- (2D; e.g. meta-array , , , and even three- (3D; e.g. nanohelices , , metamaterials ) dimensional metastructures. Here, we mainly focus on 0D (single or few particles) and 2D (array) metastructures, due to their compact and simplified geometrical configuration. We unambiguously give the concept of multipole engineering, which indicates the manipulation of multipolar interaction under linearly polarized planar wave, such as near-field coupling and multipolar interference effect appearing in single or few metastructures (Figure 1C). Beyond multipole engineering, the concept of multimode engineering indicates the manipulation of interaction (e.g. weak and strong coupling) between multipolar terms and exotic optical modes or states, like cavity mode (bound mode), lattice (diffraction) mode, circular polarization, plasmon mode, exciton mode, and others (Figure 1C). In this review, we aim at the recent progress and provide an insightful overview on the aforementioned areas, highlighting the crucial role of multipole and multimode engineering in Mie resonance-based metastructures.
Based on the above concept of multipole and multimode engineering, this review is organized as follows: single and cluster metastructure (meta-atom and meta-molecule)-mediated multipolar interference phenomena, which are responsible for the generation of several novel dipole modes and their multipole family, are investigated in Section 2. Fano resonance and bound states in the continuum are also discussed as the nontrivial multimode coupling phenomena. In Section 3, we extend our discussions to planar meta-film, including periodic meta-lattice and aperiodic meta-assembly, revealing the importance of the collective coupling effect and surface lattice modes. In Section 4, we overview the multimode engineering in chiral photonics, including achiral and chiral metastructures. In Section 5, we discuss the recent advancement of hybrid metastructures, highlighting the resonant coupling phenomena in dielectric-plasmonic and Mie-excitonic systems. In Section 6, we summarize this review and present the outlook of this fast-developing research area.
2 Multimode engineering in meta-atoms and meta-molecules
Recently, high-index dielectric single nanoparticle and cluster have attracted much attention in the photonic area  due to their clear physics and potential for multimodal engineering in the design of novel functional nanophotonic structure. Like atom and molecule operated as a basic constituent unit of matter, such an individual nanoparticle and single cluster are termed as “meta-atom” and “meta-molecule” in metastructures, respectively, highlighting their role as a structural unit in the interpretation of light-matter interaction (see Figure 1C). Although the concept was initially proposed in plasmonics and metamaterials , we limit the discussion to high-index dielectric.
We firstly introduce the concept of multipole family frequently used in dielectric-based metastructures. The normal multipole family includes ED, MD, toroidal dipole (TD), and their high-order multipoles derived by multipole expansion or decomposition of the scattering fields; such an idea of decomposition deepens the understanding of complicated scattering fields generated by metastructures and has already been demonstrated via many applications according to the different engineering demands . With the fast development of dielectric metastructures, multipole interference plays an important role in the engineering of the multipole family; recently, several types of extended multipoles have been proposed and demonstrated in dielectric-based metastructures by the superposition of normal multipole modes with different phase symmetry and vector structure, like Huygens dipole (HD), Janus dipole (JD), circular dipole (CD), anapole mode (AM), and their high-order multipoles. Such an engineered multipole family is dubbed as the “extended multipole family” and will be highlighted in this section, in order to distinguish their performance with that of the normal multipole family. The concept of extended multipole family is helpful for a better explanation of these gradually complicated physical phenomena in such a fast-developing area. We emphasize again that toroidal multipole mode and AM are associated with toroidal current configuration, which are not compatible to the framework of standard multipolar expansion (i.e. electric and magnetic multipoles). Besides, multipole family-based interparticle and intermodal coupling enables nontrivial but intriguing phenomena, such as Fano resonance and bound states in the continuum. Despite the complicated physical origin of the above normal and extended multipole family, a carefully designed Mie resonator (dielectric meta-atom or meta-molecule) provides a platform to improve their performance and enables them to be experimentally accessible in nanophotonics.
HD is composed of in-phase orthogonal ED moment (p) and MD moment (m) with equal intensities of induced dipole fields (i.e. |p|=|m|/c, c is speed of light in vacuum), releasing a unidirectional dipolar emission via satisfying the well-known Kerker condition (Figure 2A) , . Usually, a simple subwavelength-sized plasmonic nanoparticle cannot be regarded as a HD source, due to the insufficient MD response in the visible range. Instead, a high-index Mie resonator is frequently used for unidirectional emission. HD is a powerful example to show the crucial phase symmetry in dipole interference (Figure 2A, B). Note that we mainly concern dipole and quadrupole interactions and skip a detailed discussion of other high-order multipoles in this section. Fundamentally, the electric fields induced by ED and MD (orthogonal arrangement) present even parity and odd parity, respectively, leading to constructive interference at one side but destructive interference at the other (red arrows in Figure 2B). Note that the realization of unidirectional emission is not restricted in HD, but other combinations like dipole-quadrupole (ED+EQ, MD+MQ) and quadrupole-quadrupole (EQ+MQ), which are attributed to the generalized Kerker effects in nanophotonics (Figure 2B) . One special case is the superposition of dipole combinations (in-phase ED+MD) and antiphase quadrupole combinations (in-phase EQ+MQ), retaining only transverse scattering and suppressing both forward and backward scattering (Figure 2C) .
JD is composed of π/2 out-of-phase orthogonal ED mode and MD mode (|p|/c=|im|) (Figure 2A), resulting in interesting face-dependent coupling or emission in adjacent nanophotonic structures (Figure 2A) . Distinct with the well-known spin-momentum locking effect of CD (Figure 2D), JD does not show left-right directionality in adjacent dielectric slab. In contrast to HD or other generalized Kerker effect-based dipole combinations (Figure 2E), JD does not present the directionality in free space or far-field and the face dependence can be found only in adjacent evanescent-wave supporting waveguides, originating from the topology of complex vectorial electromagnetic fields in intermodal coupling (Figure 2F, G). Such face dependence is switchable by reversing the relative phase between ED and MD (i.e. from π/2 to –π/2; Figure 2F, G). Additionally, the introduction of helicity in JD (like CD) may form a novel type of dipole source termed “spinning JD,” which effectively expands the scope of near-field directionality and combines with the spin-momentum locking phenomenon . Practically, JD can be implemented by high-index Mie resonator (meta-atom)  or metal-dielectric-metal (plasmonic) resonator , which is promising for directional control in nanophotonic circuitry and quantum optics .
CD is composed of two π/2 out-of-phase orthogonal EDs, which is equivalent to point dipole with circular polarization in most functionalities (Figure 2A, D). Distinguished by rotational direction of dipole orientation, left-handed and right-handed CDs are crucial for the circular dichroism (CDC) , the spin-orbit interaction , and the spin-Hall effect of light . Detailed discussions can be found in Section 4. HD, JD, and CD are inherently associated with near-field and far-field directionality, leading to the manipulation of the light flow via single meta-atom mediated light-matter interaction in adjacent nanostructures.
Dynamic TD can be approximately interpreted as an azimuthally distributed MD (green circle in Figure 3A), produced by a displacement current (blue circle in Figure 3A) flowing along the meridians at the surface of imaginary torus (coil-like). Although the radiation patterns of TD mode in far-field are identical to ED mode (red grid in Figure 3A), the intrinsic difference between them is found in the near-field properties (i.e. the distinct distribution of E-fields). Therefore, several reports have argued that the TD (T) and its multipole family compose the third elementary dipole family (besides ED and MD), in order to emphasize its distinct parity-time symmetry (P-odd and T-odd) of dipole moment and charge-current configurations , . However, the nonexistence and inexplicit meaning of TD and toroidal multipoles terms are clarified from complete spherical Mie expansion; therefore, TD can only be extracted from toroidal current expansion inside nanoparticle . Different types of meta-molecules (meta-oligomer) are feasible to enhance TD in an all-dielectric configuration, such as trimer, tetramer, pentamer, and hexamer. In these meta-molecules, the rotational arrangement of nanoparticles is crucial to obtain high-quality factor (Q-factor) and enhanced TD mode (low transmission) ,  (Figure 3B, C); interestingly, the fine tuning in the partial radius of Si nanodisk tetramers enables the further enhancement of the TD mode in meta-molecules, while the slightly breaking rotational symmetry results in the formation of trapped mode (dark mode-like) (Figure 3C, D) .
AM can be regarded as a nonradiating source (trapped mode) via destructive interference between the ED mode and the TD mode due to their identical far-field radiation patterns and out-of-phase configurations (Figure 4A) , , , . Although such engineered anapole reveals vanishing far-field radiations (does not hold for all excitation directions ), one must note that AM is not an eigenmode of an open cavity (like multipole in Mie expansion) , . In other words, the continuity of the boundary condition requires the existence of external excitation fields, implying the interaction of damping channels from different complex eigenmodes and holding the nonradiating property only in steady state. But the intriguing zero-scattering property and well-confined stored electromagnetic energy in AM are still favorable for a plethora of applications in near-field and far-field, such as Purcell enhancement , anapole nanolaser , optical cloaking , and active nanophotonic devices . As shown in Figure 4B–D , the first and the second orders of AM (AM1 and AM2) are supported in a single slotted Si nanodisk (see two dips of blue dotted line in Figure 4B), leading to multiresonance-based electromagnetic energy intensity (~|E2|) enhancement exceeding 103 times. Although AM1 shows more suppressed scattering behavior than AM2 (Figure 4C), AM2 enables larger electromagnetic energy intensity generated by the combination of poloidal current mode and Fabry-Pérot (FP) mode (standing wave; Figure 4D); in contrast, AM1 is primarily a poloidal current mode (insets of Figure 4C). Moreover, the slotted geometry is accessible for the integration with quantum emitter or fluorescent substance. Realization of active control of AM is also important in nanophotonics; to this end, structured phase-change alloy Ge2Sb2Te5 (GST) microdisks are demonstrated for multipolar Mie resonances and large mode shifting (Δλ/λ~15%) between ED and AM (Figure 4E) . The dramatic change of optical constant during the phase transition in GST leads to an ED-AM switching, functionalizing a thermal-optical scattering mode convertor (bright mode-dark mode) in the infrared range.
Fano resonance is generally described as a physical phenomenon that a discrete localized state (weakly damping oscillator) destructively interferes with a continuum of states (strongly damping oscillator), accompanying a π-phase shift at the resonance of discrete state (Lorentz oscillator model) and producing an asymmetrical spectral line shape , . Note that such line shape is controlled by Fano parameter (q) and external coupling from excitation (e.g. q=0 and q→∞ correspond to uncoupled discrete state and continuum state, respectively); q is directly associated with the phase shift (δ) between the discrete state and a continuum by q=cotδ (Figure 5A). In Mie resonant nanostructures, dipole-active mode and dipole-forbidden mode are frequently employed as bright mode (strongly damping) and dark mode (weakly damping) under the plane wave excitation, respectively. As demonstrated by Si heptamer (Figure 5B), magnetic resonance induced a phase difference between the center nanoparticle (blue line) and the outer ring (hexamer, black line), forming an individual dark mode and collective bright mode, respectively , . The appearance of Fano-dip in scattering spectra manifests the small q value and out-of-phase bright mode (δ~π/2). Interestingly, less sensitive dependence on interparticle separations is shown in Si heptamer than their plasmonic counterpart. The occurrence of Fano resonance is not limited in meta-molecules, but even in a single Si nanorod (Figure 5C) , . Two types of resonant modes, including Mie modes and FP modes, coexist in such nanorod with a large aspect ratio. Asymmetrically sharp Fano peak appears at λ=670 nm due to the well-confined HE311 FP mode overlapping a broad background of Mie modes. For applications, refractive index sensing ,  and electromagnetically induced transparency (a special type of Fano resonance characterized by q=0 and almost the same resonant frequency in bright mode and dark mode)  are attractive in near-field coupling control due to their unique properties.
As another type of nonradiating state in photonics, bound states in the continuum (BIC) is revisiting and attractive to obtain the infinite Q-value (divergent lifetime) and energy trapping state in dielectric metastructures . Implied by the name, BIC can be interpreted as a bound state coexisting with a radiation continuum (extended wave) at a certain frequency in an open system, but the bound state is still spatially confined without any leakage of radiation . The authentic BIC phenomenon exists only in infinitely extended structures at least in one dimension. On the contrary, BIC phenomenon is forbidden in a single particle system with nonzero finite permittivity and permeability, arising from the continuity condition at the boundary of particle . For a finite number of particles (e.g. 1D chain), quasi-BIC with a finite but ultrahigh Q-factor appears as an approximation of authentic BIC in infinite structures . Although photonic BIC and quasi-BIC have already been demonstrated in many extended structures, like photonic crystal slabs , , , metasurfaces , , and 2D nanowire array , , we also notice an emerging type of quasi-BIC analogue or the so-called “supercavity mode” mediated by a single Mie resonator (Figure 6A) , , . According to the Friedrich-Wintgen scenario , , while two resonant modes, Mie-like mode and FP-like mode, are excited in a single high-index Mie nanocylinder and strongly coupled to the same radiation channel (radiation continuum), while the aspect ratio of Mie resonator (r/l) is tuned continuously, one can find a totally suppressed radiation channel via destructive interference between Mie mode and FP mode. Therefore, quasi-BIC can be regarded as a singularity of the Fano parameter (q→∞, radiation continuum is uncoupled from external excitation), resulting in the occurrence of an ultrahigh Q-value mode (the so-called supercavity mode) in the scattering spectrum. Interestingly, the anticrossing behavior (large red circle in Figure 6B) indicates the strong coupling between Mie-like mode (TE1,1,0, mode A in Figure 6B, C) and FP-like mode (TM1,1,1, mode B in Figure 6B, C), resulting from their same azimuthal mode symmetry and the inversion symmetry of Mie resonator. For quasi-BIC of TM1,1,1 (Figure 6C), magnetic quadrupole (MQ in Figure 6D) contributes dominantly to the total far-field radiation pattern and the radiation channel. Multipole expansion enables clear explanation and sufficient details of phase and vectorial distribution in each multipole term , , . Additionally, a fast-changing far-field pattern with respect to the aspect ratio of Mie resonator corresponds to the contributions from other resonant modes. Note that such quasi-BIC phenomenon is not only limited in 3D single nanocylinder but has also be reported in single microcavity , few nanodisks , and nanowires . Very recently, a general comparison for evaluation of Q-factor of single subwavelength dielectric nanocavity has been reported elsewhere .
3 Multipole and multimode engineering in meta-lattices and meta-assembly
To realize multipole engineering in planar optics, an array or assembly of nanostructures is frequently used. In this section, we physically focus on multimode engineering-based collective coupling and resonance in 2D periodic or aperiodic metastructures. As a well-known example, the phase-gradient metasurface is typically defined as an interface evaluated by effective permittivity and permeability, leading to discontinuity in a certain direction and enforcing an abrupt change in both the amplitude and phase of the impinging wavefront. Several types of phase-gradient dielectric metasurface, like high-contrast array, Pancharatnam-Berry phase-based metasurface, and dispersion-engineered metasurface, are usually explained by phase jumps, impedance matching, and generalized refraction law, which present less or ambiguous associations with multipole and multimode engineering exhibited in Mie resonant metastructures, and readers may find comprehensive reviews elsewhere , , , , , . To distinguish from the above metasurface and clearly indicate the role of multimode engineering in periodic and aperiodic configurations, here, we use “meta-lattices” and “meta-assembly” to describe such structures highlighting the multipolar and multimodal property via collective coupling and resonances. Although single meta-atom and meta-molecule provide an insightful interpretation of fundamental light-matter interactions in nanoscale, low scattering efficiency of a single structure is incapable in nanostructure-array-based practical applications. More seriously, the open cavity-like Mie resonator (unavoidable radiation loss) is restricted in obtaining high-quality factor and sufficient spectral resolution at resonance. Therefore, Mie resonant meta-lattices and meta-assembly exhibit not only the powerful potential of applications but also some novel physical phenomena that disappear in a single metastructure (such as coupled dipole resonance, surface lattice resonance).
3.1 Collective coupling effect in Mie resonant meta-lattices
In Mie resonant meta-lattices with subwavelength-sized period, the collective coupling effect is ubiquitous and sometimes manifested as period dependence while diffraction modes are lacking. Many have papers reported period dependence in dielectric nanoarray in 1D and 2D. The introduction of period is not a simple expansion and superposition of single meta-atom or meta-molecule; instead, weak near-field coupling may change the collective electromagnetic response like effective permittivity and permeability. Effective medium theory enables to quantitatively analyze such collective effect; however, scattering matrix of collective mode is nontrivial to obtain from the arbitrary geometry of resonators array in metafilm. Several numerical methods, like finite elements method and finite-difference time-domain, are employed to evaluate the spectral responses of reflection, transmission, and absorption. As shown in Figure 7A, Lorentzian oscillator-like Si nanocubes induce the abrupt phase change of reflected wave at the resonant wavelength (the so-called backscattering property in Mie scattering) , ; two main peaks in visible wavelengths originate from ED and MD resonances. Strong period dependence existed in the array of Si nanocubes, indicating the role of interparticle coupling in meta-lattices (Figure 7A, bottom). Such period dependence is feasible to manipulate structural color generation by using dielectric meta-lattices, resulting in controllable color saturation (peak intensity) but maintaining color hue (peak wavelength). Additionally, Si oxidation enables further manipulation of structural color generation (Figure 7B, top) . Si nanocube meta-lattices are oxidized by thermal annealing at a high-temperature oxygen atmosphere, encapsulated by a SiO2 cover layer. The high permittivity contrast ensures drastic change in backscattering spectra, such as blue-shift color hue and gradually decreased color saturation, which can be observed with respect to annealing time; the results demonstrated remarkable color tunability and the functionality of invisible ink (Figure 7B, bottom). Besides oxidation-based spectral tunability, flexible substrate supporting hexagonal Si nanocylinder meta-lattices can be utilized as reconfigurable matastructures for sensing applications . Polyethylene (PE) terephthalate-based flexible substrate displays sensitive response to externally applied force; a mechanical sensor is demonstrated by measuring the peak shift at MD resonance in transmission spectra, due to the deformation of meta-lattice-induced interparticle coupling (separation between hexagonal lattices is changed) (Figure 7C). Multiphysical field-mediated external modulation or tunability is not limited in oxidation and mechanical effect, but also by thermal-optical effect , electric-optical effect , and phase-change materials , to name a few.
In Section 2, single meta-atom-mediated HD is introduced as the combination of in-phase orthogonal ED and MD, which is featured by phase symmetry-based unidirectional emission. Similarly, dielectric perfect reflector , ,  and transmitted Huygens surface (HS) , ,  are proposed as phase-controlled metadevices to steer reflection/transmission, which are usually designed by periodic dielectric meta-lattices. The existence of ED and MD provides the possibility to control electric and magnetic resonances, respectively . For magnetic perfect reflector (left in Figure 7D), near-zero phase change from incident excitation can be observed by the E-field profile of reflection, accompanying with the remarkable E-field enhancement at the interface of structure. On the contrary, for electric perfect reflector (right in Figure 7D), π-phase change and cancelled-out E-field are observed at the interface. Note that the magnetic perfect reflector is not limited in MD, but also in EQ and other high-order multipolar resonances, due to the same phase symmetry (parity) . Magnetic dielectric perfect reflector is promising in near-field applications like Raman spectroscopy and fluorescence enhancement. One must also notice the importance of impedance matching in near-unity reflection/transmission meta-lattices, formulating into effective electric and magnetic polarizabilities in metastructures, which is related to interparticle coupling of meta-lattices , and the optical constant of substrate/superstrate . For full 2π phase coverage modulation, transmitted HS (all-dielectric) can be regarded as the combination of orthogonally arranged ED and MD resonances. While ED and MD resonances are frequency-overlapped, a full 2π phase coverage is obtained. As one practical structure of HS, a Si nanodisk array ensures polarization-insensitive high transmittance at telecommunication wavelength , , . At optical frequency, such all-dielectric HS provides a platform for low-loss, highly efficient, and full phase coverage modulation, which is promising for imaging applications like metalens , , , ,  and holography , , . As shown in Figure 7E, a thermal optic modulation-based varifocal Huygens metalens is demonstrated. The whole device is composed of the all-dielectric HS (top, metalens part) and the 700-μm-thick polydimethylsiloxane (PDMS) slab loaded by gold heater (bottom, thermal-optic part). For metalens part, the diameter (D)-variable Si nanodisk array (160–260 nm in Figure 7F) is carefully designed for high focusing efficiency and guarantees 2π phase coverage at a specific wavelength (fulfilling Kerker condition for a certain diameter; Figure 7F). For the thermal-optic part, a voltage-controlled gold heater that generates Ohmic heat is responsible for thermal-optic modulation of PDMS (temperature-dependent refractive index), imparting the varifocal functionality to Huygens metalens (Figure 7G). Aside from varifocal functionality, near-unity numerical aperture , a narrow field, and a wide field of view , , ,  are also demonstrated in Huygens metalens.
3.2 Surface lattice resonance in Mie resonant meta-lattices
In Section 3.1, we introduced the collective coupling in subwavelength-sized meta-lattices, which displays weak interparticle coupling in periodic structures and recognizable dipole or multipole resonances originating from a single meta-atom or meta-molecule. While the period of meta-lattices is comparable with wavelength, coupled resonances in meta-lattices may induce a diffraction mode (Rayleigh anomaly) in the surrounding ,  and guided mode in adjacent waveguide , inducing an intermodal coupling between diffraction modes and multipole modes (supported by meta-atom or meta-molecule). Interestingly, the coupling between the ED, MD, and lattice modes enables two new modes termed as ED-LR and MD-LR, respectively, which can be separately controlled by different dimensions of periodicity in lattices (Px and Py). As an expansion of HS, ED-LR is spectrally overlapped with MD resonance, resulting in the occurrence of resonant lattice Kerker effect , , . The surface lattice mode can also facilitate obtaining a high Q-factor in metastructures , , . As shown in Figure 8A, a Si cylinder meta-lattice (a one unit cell with 100 nm height and variable diameters [dd] is shown in the inset of Figure 8A) is embedded into fused silica (extended structures, not waveguide) . In contrast to Fano resonance-based surface lattice resonance in plasmonic structures, the collective resonant modes  here are formed by strong coupling between single particle dipole modes (ED, MD) and surface lattice modes (Rayleigh anomaly), revealing a large detuning in mode coupling (Figure 8B). Both sharp peaks in the range of 600–650 nm correspond ED-LR and MD-LR, respectively. The red shift of peak positions is observed with respect to the increasing dd (numbers are labeled in Figure 8B). The sharp peaks are favorable for field enhancement and high Q-factor applications. In waveguide configuration (Figure 8C), a finite-thick high-index layer is utilized as core layer (brown layer, and the top cladding is air) in slab waveguide, in order to support quasi-guided modes. ED (MD) mode is efficiently coupled to transverse electric (TE) mode transverse magnetic (TM) mode, due to the symmetry of mode profiles. Again, the strong coupling between single particle dipole modes and quasi-guided modes (fundamental modes and high-order modes) leads to a large detuning and ultranarrow lattice-based resonant peaks (Figure 8D). Two pairs of peaks in the range of 600–700 nm are TE0(TM0)-LR and TE1(TM1)-LR modes (high-order modes), respectively. The quasi-guided modes provide strongly confined mode profiles, leading to a further field enhancement and higher Q-factor with respect to ED-LR and MD-LR. Instead of dipole mode, such mode coupling can also occur between Mie resonance-based AM and lattice modes; an ultrahigh Q-factor (~106) is numerally demonstrated in Si hollow nanocuboids meta-lattices .
3.3 Light-matter interactions in meta-assembly
In contrast to ordered photonic arrays, disordered photonic structures are more generalized photonic structures in meta-assembly. Understanding the evolution and differences between ordered and disordered meta-structures is crucial in nanotechnology. As shown in Figure 8E, size-disordered meta-assembly is composed of Si nanospherical arrays with unequal radii ; here, σR is the radius deviations in assembly and the radius of Si nanospheres (R) is randomly changed by a uniform distribution: R0–σR<R<R0+σR, where R0 is the radius of ordered Si nanospheres. In both ED coupling (with respect to hy; Figure 8F) and MD coupling (with respect to hx; Figure 8G), MD modes are suppressed while sR is increased, indicating the radius-sensitive MD modes in size-disordered meta-assembly. In contrast, ED modes are more stable with respect to σR; counterintuitively, lattice mode coupling-like ED-LR modes can also be observed in such disordered structures. Other disordered meta-assembly-like position disorder (in two-dimension) and quasi-random structures  are also intriguing in controlling collective resonances and inducing phase transitions .
Randomized micro/nanoparticle assemblies are feasible for large-area and cost-effective applications. Multiscattering-induced broadband absorption is favorable for passive cooling  and photothermal conversion , . As the requirement of the passive cooling, expectedly, near-unity thermal emission should only appear within 8–13 μm, in order to cover the atmospheric transmitted window and use ultracold universe as a heat sink. Meanwhile, daytime passive cooling requires the suppression of absorption in visible wavelengths. To this end, randomized SiC and SiO-overcoated SiO2 (to further broaden absorption band) nanospheres are mixed and doped into PE thin film (Figure 8H) . The emission spectrum is effectively expanded by SiC and SiO2 nanocomposite and lies within atmospheric transmittance window, due to their strong surface phonon resonances (Figure 8I; SiC: 10.5–13 μm, SiO2+SiO: 8–13 μm). The cooling performance can be evaluated from the power difference between emission and absorption.
4 Multimode engineering in achiral and chiral metastructures
4.1 Achiral dielectric metastructures
Generally, chiral geometry is the structure lacking reflection and inversion symmetry. A chiral object and its mirror image are usually called “left-handed” and “right-handed” enantiomers, due to a very familiar example of our own hands. In biomolecules, S-enantiomer and R-enantiomer may show different chemical properties (e.g. in some drugs, one enantiomer is active, the other is inactive or adverse), which can be effectively distinguished by CDC spectroscopy. However, only a very small dissymmetry factor (g), which is a parameter to evaluate the strength of chiral effect, can be observed in molecules solution (typical value: 10–3–10–2) and most of molecules show resonant peaks in ultraviolet (UV) wavelength. Thanks to field enhancement and confinement by nanostructures, plasmonic and dielectric are suitable for visible-light chiral enhancement and promising in chiral sensing and separation, chirality sensitive photochemistry, and the growth of chiral crystal. To analyze chiral response in a local area, the optical density of chirality,
For a Si sphere, as shown in Figure 9A , observable CDC enhancement (C/CCPL) is obtained from each Mie resonant multipolar peak, especially at high-order magnetic multipolar peaks (magnetic octupole [MO], magnetic 16-pole [MS], magnetic 32-pole [MT] in the CDC enhancement, Figure 9B), due to the strongly enhanced and localized E and H hotspots matching the vibrational band of typical chiral molecules nearby . Additionally, the Si sphere can be used as an optical antenna to tailor the far-field emission from chiral quantum emitters (point dipole source composed of ED and MD modes); in contrast, using the plasmonic counterpart can only obtain almost achiral scattered fields due to the excitation of dominant ED moment . A Si cube-dimer  is practical to further enhance chiral hotspots via spectrally and parallelly overlapped E and H fields at the subwavelength-sized gap, which can be interpreted as the superposition of hotspot by orthogonal linear polarizations of incident light. Interestingly, in Si cylinder-dimer, the net chiral density and flux in far-field can be observed even in linear polarized excitation (Figure 9C) . Because linearly polarized light is equivalently decomposed to LCP and RCP, but undergo different dissipation induced by dimer axis for LCP and RCP, the result is an observable chiral signal-like chiral density in near-field and the third Stokes parameter (S3) in far-field. For near-UV enhancement, TiO2-dimer is suitable for enhanced chiral sensing, due to the optical constant of material .
Achiral dielectric metasurfaces are widely used for the control of transmitted chiral spectra, due to the advantages of low loss, low heat generation, and high Q-factors . Also, ordered array and collective response are indispensable for large-area chiral sensing and separation , , , . Inspired by Si-dimers, holey Si nanodisk metasurface  effectively generates parallel and π/2 out-of-phase E and H fields, producing a superchiral field at the center of the a holey nanodisk (Figure 9D). According to the duality principle in classical electromagnetics, the direction of E and H fields of the ED mode is parallel with the corresponding H and E fields of MD mode, which guarantees the parallelly overlapped E and H fields as the aforementioned design rules for chiral nanostructures. Other design rules, like the spectrally overlapped and the π/2 out-of-phase E and H fields, are equivalent to the well-known Kerker condition. Note that the π/2 out-of-phase E and H fields are naturally fulfilled from incident plane wave (according to Maxwell’s equations), implying the requirement of in-phase ED and MD modes in holey nanodisk metasurface. High transmission efficiency is also guaranteed in such all-dielectric metastructures.
4.2 Chiral dielectric metastructures
Artificially engineered chiral metastructures enable to further improve the CDC and g-factor of chiral response, which are promising for chiral optical devices such as polarization-resolved imaging and detection , reconfigurable chiral devices , and enhanced chiral sensing . The lossless dielectric metastructures show multipolar excitation  and high efficiency chiral signal transmission ,  with respect to their plasmonic counterpart. Planar chiral metastructures are widely applied for their relatively simple fabrication process and high compatibility in nanophotonics. As shown in Figure 9E, TiO2-based gammadion array metastructures are deposited on SiO2 substrate . Near-unity CDC is shown in transmitted spectrum at 540 nm for wavelength-sized metastructures (periodicity: 500 nm), which can be attributed to zero-order diffraction mode (transmission) under normally incident CPL. To clarify multimodal contribution, the results of multipole expansion extracted from far-field radiations are shown in Figure 9F, G. For the case of LCP, two remarkable dips appear around 540 nm, generated from toroidal quadrupole mode (TQ, purple) and MO mode (MO, yellow) (Figure 9F). In contrast, the radiation peaks of TQ and MO are observed around 540 nm (Figure 9G). The results indicate the near-unity CDC obtained by broken symmetry in gammadion metastructures and the suppressed dipole modes (or low-order modes) by wavelength-sized periodicity. Besides gammadion shape, other asymmetrical planar metastructures ,  also support the strong chiral signal and accompanying more physical effects like Fano resonance .
The dielectric chiral oligomer is another important type of dielectric chiral metastructure, which can be composed by the assembly of achiral nanoparticles , ,  or chiral nanohelices . Dielectric chiral oligomers are mainly based on scattering CDC in extinction spectra, instead of absorption-based plasmonic counterpart, avoiding the photothermal effect induced photo-destruction . As shown in Figure 9H, a Si-tetramer is helically positioned with the equal interparticle spacing . From the extinction spectra of Si-tetramer under z-direction incident RCP, ED mode (σE) and MD mode (σM) appear in the visible range (Figure 9I). The corresponding contributions from ED and MD are calculated from multipole expansion of CDC spectra (Figure 9J); peak 1 is a pure ED mode, while peaks 2–4 are mainly induced by MD mode. One should note that such enhanced chiral fields are usually well confined within Si particles (Mie resonators), which implies the insensitivity of interparticle spacing in dielectric oligomers, in contrast to their plasmonic counterpart .
5 Multimode engineering in hybrid metastructures
On one hand, high-index dielectric metastructures provide a powerful platform for obtaining multipolar responses of Mie resonances and enhanced electromagnetic fields in metastructures. On the other hand, energy transfer between Mie resonant mode and adjacent optical emitters or other types of resonant modes (quasiparticles) is confirmed via weak and strong mode coupling behaviors in hybrid metastructures like Mie excitons  and dielectric-plasmonic structures , , . In this section, our discussions are concentrated on the light-matter interaction in Mie resonance-mediated hybrid metastructures.
5.1 Mie-plasmon modes in hybrid metastructures
In the application of Purcell enhancement, metal nanoparticle on a mirror (MNPoM) is a well-studied plasmonic metastructure , , , featured by the generation of gap-plasmon mode in a nm-sized gap layer. While the quantum emitters embedded nanogap shrinks from ~10 nm to ~1 nm, MNPoM undergoes an extremely sensitive change in coupling strength from weak coupling (Purcell enhancement) to strong coupling . More severely, quantum effects like the electron tunneling and nonlocal screening are dominant in a subnanometer gap . In contrast, low-loss Si nanoparticle on a mirror (SiNPoM) is promising in obtaining a gap-insensitively resonant peak position (Figure 10A ) while eliminating the nonradiative decay of metal nanoparticles . In such Purcell enhancement configuration, a nano-sized gap layer is composed of a Si quantum dot (QD) monolayer, which are used not only as a phosphor but also as an accurately controlled spacer. The Mie-plasmonic hybrid mode is strongly enhanced and well confined in the gap under the oblique excitation (Figure 10A). Physically, the hybrid mode arises from the coupling between a perpendicularly orientated ED mode (out of plane) and its image dipole mode supported by an Au mirror, which ensure the gap (spacer)-insensitive property of scattering peaks (color lines in Figure 10B). Distinguished with MNPoM, SiNPoM enables gap-controlled coupling strength but almost stable peak positions. Additionally, ED mode and MD mode are together responsible for broadband scattering (Figure 10B). Note that the different coupling performances in low-order and high-order hybrid modes may induce plasmon-like mode and dielectric mode, respectively . Such gap insensitivity and broadband emission are favorable for simplifying the design and fabrication of nanogap resonator.
Besides the assembly of nanogap resonator, the combination of plasmonic and dielectric nanocavity uncovers a rich physics and potential to tailor the multipolar scattering spectra of planar metastructures. In order to achieve ultrahigh-resolution meta-atom-based color printing, the key is to decrease the impact from interparticle coupling in densely arranged dielectric meta-atoms. As shown in visible reflection spectra (backscattering), multipolar property of scattering can be observed from the Si nanocylinder array and further broadened spectra with respect to the increased diameter of Si nanocylinder, resulting in the degradation in the performance of color printing like color mixing (Figure 10C) . In stark contrast, metal-masked Si Mie resonator is one feasible solution (Figure 10D); the addition of thin Cr layer (e.g. 30 nm) enables effective suppression of ED and MD peaks induced by Si nanocylinders. Importantly, such suppression is wavelength dependent and nonuniform for ED, MD, and high-order Mie peaks, generating brilliantly tunable reflection full colors with sufficient color saturation in ~μm2 area.
Optical nanoantennas facilitate local control of intensity, polarization, and directionality of light . Thanks to the excitation of localized plasmon, plasmonic antennas hold promise for obtaining the ultrasmall mode volume and high Purcell enhancement. However, unavoidable dissipation of metallic materials becomes a barrier for realizing highly efficient nanoantennas. Alternatively, dielectric-plasmonic hybrid nanoantennas may mitigate this situation , ; as shown in Figure 10E, high light collection and scattering efficiency are obtained by subwavelength-sized Au bowtie antennas, then the scattering light is coupled into Si Yagi-Uda antennas and the directional emission pattern is formed by asymmetrical shape of Si antennas (bottom inset in Figure 10E). Importantly, directionality is sensitive to the alignment between Au and Si antennas, which is useful for nanoscale positioning sensors. Aside from directional control, the research of hybrid nanoantennas is also active on nonlinear optics, especially for multiple harmonic generation , , due to the plasmon-enhanced conversion efficiency.
To investigate the strong mode coupling between plasmonic nanoantennas and Si nanodisc, a combination of Au nanorod and Si nanocavity is shown in Figure 10E; the entire metastructure is embedded into a homogenous dielectric medium (low index, n=1.45 is chosen). Plasmonic ED (PED) mode is dominantly supported in a subwavelength-sized Au nanorod, and Mie resonance-based ED (MED) and MD (MMD) modes are supported by wavelength-comparable (λ/dh, dh is the diameter of nanodisc) Si nanodisc (transmittance map in Figure 10E) . The hybrid metastructure is used as a unit cell of meta-lattices, and the transmittance is shown for such meta-lattices with respect to Au nanorod length. Dipolar modes excited from Au nanorod and Si nanodisc are labeled in the transmittance map; interestingly, different mode coupling behaviors are found in PED-MED and PED-MMD. For PED-MED coupling (point B in the bottom part of Figure 10E), a resonance crossing is observed indicating the weak coupling between two modes. For PED-MMD coupling, the featured resonance anticrossing is observed, signifying the entry of strong coupling regime and hybridization of plasmonic-photonic modes (antibonding hybrid mode). Analyzed from the mode profiles at anticrossing and crossing points, the strong mode coupling phenomenon enhances the field distributions at the gap between Au rod and Si disc at two anticrossing points, inducing the pair of antibonding and bonding modes. On the contrary, approximate isolated resonant behaviors in Au rod and Si disc can be observed at the crossing point (PED-MED), implying relatively weak coupling between them.
5.2 Mie-exciton modes in hybrid metastructures
Recently, the resonant mode coupling phenomena between excitonic emitters and Mie resonator are numerically or experimentally demonstrated by observation of a large Rabi-splitting phenomenon in molecular J aggregate ,  and transition-metal dichalcogenide (TMDC) materials , , ,  incorporated hybrid metastructures. J aggregate is well known for collective excitons featured by large oscillator strength and the ease of integration in organic-inorganic hybrid metastructures . The hybridization and coupling of plasmon mode and molecular exciton mode in J aggregates (the so-called plexcitons) have been extensively reported , , . In contrast to plasmonic system, the utilization of high-index Mie resonator not only decreases nonradiative decay but also generates unidirectional emission of hybrid modes, which is associated with the multipolar property of Mie scattering. One simple hybrid metastructure is Si-J aggregate core-shell nanoparticle . Si core can be directly analyzed by multipolar expansion, extracting the contributions from ED, MD, EQ, and MQ modes (SiEDR, SiMDR, SiEQR, Sand iMQR, respectively, in Figure 11A). The dielectric function of the J aggregate shell is formulated by using a one-oscillator Lorentzian model,
Here, ε∞ is the dielectric constant at the frequency infinity, assuming equal value as the permittivity of the surrounding (e.g. water). ω0, γ0, f are the resonant frequency of exciton, the damping coefficient, and the oscillator strength in the J-aggregate, respectively. While ω0 is carefully tuned to the same resonant frequency of SiMDR, two excitonic absorption peaks are observed in Figure 11B, due to the resonant property of the dielectric function εex(ω) and the specified value of f and γ0. Again, the absorption spectrum of J aggregate-shell can also be extracted into ED, MD, EQ, and MQ modes (JEDR, JMDR, JEQR, and JMQR in Figure 11B). The absorption peaks in J aggregate-shell correspond to scattering dips in hybrid metastructures (core shell). Interestingly, two scattering dips are mainly controlled by the resonant coupling of JEDR-SiMDR (HMDR in Figure 11C) and JEDR-SiEDR (HEDR in Figure 11C); moreover, two dips become deeper respect to the increasing f (Figure 11C). The strong coupling effect can be verified by the large Rabi-splitting (~100 meV) between high-energy branch and low-energy branch in the resonant peak map of SiMDR (Figure 11D). Note that the resonant peaks at both energy branches are unidirectional scattering modes, resulting from the Kerker condition in the strong mode coupling region of SiEDR, SiMDR, and excitonic modes.
As one promising candidate material in TMDCs, monolayer or multilayer WS2 exhibits a large transition dipole of excitons. Similar to Si-J aggregate hybrid structures, the combination of Si nanoparticle and monolayer WS2 enables the observation of mode splitting between Mie modes and exciton modes; moreover, the discovery of exotic optoelectronic properties of TMDCs is promising for applications of actively tunable devices , , . In the Si core and monolayer WS2 shell hybrid system, mode splitting phenomenon is obtained from resonant coupling between MD mode and excitonic mode (A-exciton) (left in Figure 11E) . The high index of monolayer WS2 enables the red-shift of MD mode in Si core, resulting in increased spectral overlap between two modes. However, mode splitting is lacking in absorption spectra, implying the weak coupling in this case. In comparison, Si nanoparticle on monolayer WS2 flake reveals Fano-like line shape in scattering spectra (right in Figure 11E), indicating the smaller coupling strength between MD mode in Si nanoparticle and exciton mode in WS2 flake . Interestingly, both configurations exhibit directional scattering pattern transferred from MD mode. In WS2/MoS2 heterostructures, interlayer excitons may appear in such misaligned energy band structure (Figure 11F) , which can be confirmed by small red shift or blue shift with respect to pure monolayer or few-layer WS2 and MoS2. Using Si nanoparticle deposition on WS2/MoS2 heterostructure and a designed SiO2/Si substrate, the hybrid system can be interpreted as a combination of the WS2/MoS2 heterostructures and the Mie-FP cavity (SiO2/Si substrate) configuration. Strong coupling phenomena are observed by anticrossing behavior (MD mode and interlayer exciton mode) in such a hybrid system (Figure 11G). Note the FP cavity can further enhance MD mode in Si nanoparticle. Additionally, multilayer WS2 can be regarded as an anisotropic high-index dielectric with excitonic resonance at a certain frequency . In the high aspect ratio geometry of WS2 nanodisk (excitonic Mie resonator), the dark mode-like AM can be excited and form a hybrid anapole-exciton polariton mode. Except the strong mode coupling phenomenon in Mie-TMDCs hybrid system, Si nanowire antenna enables the directional tailoring of radiation pattern in adjacent MoS2 monolayer , resulting from the resonant coupling between exciton mode in MoS2 and multipolar modes in Si nanowire.
6 Summary and outlook
The revisiting of century-old Mie scattering provides a powerful nanophotonic platform for many phenomena in metastructures. Multipole and multimode engineering manipulates mode interference in meta-atom or meta-molecule, collective resonance and lattice modes in meta-lattices and meta-assembly, spin angular momentum of light in achiral and chiral metastructures, and strong resonant coupling in hybrid systems, to name a few. High-index dielectric metastructures enable field enhancement and spatial confinement, a precondition of enhanced light-matter interaction in nanoscale. Low-loss dielectric materials definitely improve the efficiency of reflective/transmission or wavefront controlled metasurfaces. Optical magnetism like magnetic hotspot generation in the visible range is the unique advantage of dielectric metastructures for polarization and directionality associated applications.
Several emerging research subareas in dielectric metastructures are not discussed, due to the length and scope of this review. Here, we briefly mention achievements obtained in the following areas, which can be regarded as the outlook of dielectric metastructures. In nonlinear optics, nonlinear dielectric metasurfaces are feasible for efficient high-order harmonic generation and ultrafast control of optical switching , . In spin-orbit interaction of light, meta-atoms or meta-molecules may realize the generation of optical vortex and transverse spin-Hall shift , . In quantum optics, entangled photon states can be generated and manipulated in the platform of dielectric metastructures , . In topological photonics, topological edge states in 1D chiral chains, 2D meta-lattices, 3D Si bianisotropic metacrystals are recently demonstrated , , . We anticipate that the rapid development in these emerging areas will become an impetus of nanophotonics in near future.
This work was supported by JSPS KAKENHI grant number JP19H02630 and the JSPS Core-to-Core Program, A. Advanced Research Networks (Advanced Nanophotonics in the Emerging Fields of Nano-imaging, Spectroscopy, Nonlinear Optics, Plasmonics/Metamaterials and Devices). T.L. and P.Y. were supported by the Institute of Fundamental and Frontier Sciences, University of Electronic Science and Technology of China. P.Y. was also supported by the China Postdoctoral Science Foundation (2019M663467, Funder Id: http://dx.doi.org/10.13039/501100002858).
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