Nanophotonics 2020; 9(5): 1115–1137 Review Tianji Liu, Rongyang Xu, Peng Yu, Zhiming Wang and Junichi Takahara* Multipole and multimode engineering in Mie resonance-based metastructures https://doi.org/10.1515/nanoph-2019-0505 this area are discussed in-depth, in order to lead the mul- Received December 7, 2019; revised January 30, 2020; accepted tifunctional metastructures for novel applications in the January 30, 2020 future. Abstract: Benefited from the well-known Mie resonance, Keywords: Mie resonance; dielectric metamaterials; a plethora of physical phenomena and applications are dipole family; ­chirality; strong coupling. attracting attention in current research on dielectric- based nanophotonics. High-index dielectric metastruc- tures are favorable to enhance light-matter interaction in nanoscale with advantages such as low loss, optical 1 Introduction magnetism, and multipolar responses, which are supe- rior to their plasmonic counterpart. In this review, we In 1908, German physicist Gustav Mie gave the solution of highlight the important role played by Mie resonance- Maxwell’s equations while he was analyzing the scatter- based multipolar and multimodal interaction in nano- ing problem of wavelength-comparable particles (e.g. gold photonics, introducing the concept of “multipole and nanoparticle) in a homogeneous surrounding (e.g. water) multimode engineering” in artificially engineered under the excitation of electromagnetic plane wave, which dielectric-based metastructures and providing an over- afterward was well known as the Mie solution [1, 2] A view of the recent progress of this fast-developing area. century later, the current research of ubiquitous scattering The scope of multipole and multimode engineering is phenomena in artificially engineered micro/nanostruc- restricted not only in multipolar interferences of meta- tures has been actively developing from the start point of atom and meta-molecule but also in the nontrivial inter- Mie scattering. Photonic crystals [3], plasmonics [4], plas- modal coupling (Fano resonance and bound states in the monic metamaterials [5], and dielectric metamaterials [6] continuum), in the collective mode and the surface lattice continuously challenge our common wisdom, expand mode appearing via periodic meta-lattices­ and aperiodic the boundary of science and technology, and facilitate meta-assembly, in chiral enhancement via chiral and the advancement of interdisciplinary research. Among a achiral dielectric metastructures, and in Mie resonance- wide range of artificial structures in photonics, we focus mediated hybrid structures (Mie-plasmon and Mie-exci- on high-index dielectric metastructures in this review. In ton). Detailed examples and the underlying physics of contrast to photonic crystals, dielectric metastructures are not restricted in periodic arranged structures; the charac- teristic Mie resonant peaks can even be observed from one single structure. Moreover, out-of-plane excitation *Corresponding author: Junichi Takahara, Photonics Center, Graduate (normal incidence or oblique incidence) is conventionally School of Engineering, Osaka University, 2-1 Yamadaoka, Suita, used in dielectric metastructures, instead of in-plane exci- Osaka 565-0871, Japan, e-mail: [email protected]. tation or propagation in photonic crystals. In contrast to https://orcid.org/0000-0002-3683-626X plasmonic metastructures, low loss and optical magnet- Tianji Liu: Institute of Fundamental and Frontier Sciences, University of Electronic Science and Technology of China, Chengdu ism (wavelength-sized structures in the visible range) in 610054, China; and Graduate School of Engineering, Osaka single dielectric structures are unique advantages of die- University, 2-1 Yamadaoka, Suita, Osaka 565-0871, Japan. lectric metastructures. For the choice of materials of build- https://orcid.org/0000-0002-5862-2449 ing blocks in dielectric metastructures, several common Rongyang Xu: Graduate School of Engineering, Osaka University, high-index and low-loss materials are frequently used 2-1 Yamadaoka, Suita, Osaka 565-0871, Japan Peng Yu and Zhiming Wang: Institute of Fundamental and Frontier in the visible range, like Si, Ge, TiO2. Similar to photonic Sciences, University of Electronic Science and Technology of China, crystals, a unit cell of dielectric metastructures (e.g. a Chengdu 610054, China particle) is typically wavelength sized (d ~ λ/n, where d is Open Access. © 2020 Junichi Takahara et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 Public License. 1116 T. Liu et al.: Multipole and multimode engineering in Mie resonance-based metastructures the typical size of a dielectric particle, λ is the wavelength array, the analytical framework is available elsewhere [11]. in free space, and n is the refractive index of dielectric), Additionally, originating from toroidal current configura- resulting in the breakdown of the quasi-static approxima- tion (currents flowing along the merdians of a fictitious tion. To accurately describe and thoroughly explain the torus), toroidal multipoles cannot be directly extracted scattering phenomenon of wavelength-sized homogene- from the standard electric and magnetic multipole expan- ous sphere, Mie solution is well-suited to analyze the scat- sion; the specific formulism and detailed interpretation tered fields in near-field and far-field regions [7, 8], and are available elsewhere [12, 13]. the total scattering cross-section (Csca) can be expanded Recently, the artificial manipulation of multipolar into an infinite series of spherical multipolar terms (Mie and multimodal properties in dielectric metastructures expansion) have been extensively reported in zero- (0D; e.g. single nanoparticle [14–18]), one- (1D; e.g. single nanowire and ∞ 2π 22 nanorods [19–23], two- (2D; e.g. meta-array [24–26], and Cm=+(2 1)(|ab||+ |), (1) sca 2 ∑m mm k even three- (3D; e.g. nanohelices [27, 28], metamaterials [29]) dimensional metastructures. Here, we mainly focus where ɑm and bm are Mie coefficients of the mth-order elec- on 0D (single or few particles) and 2D (array) metastruc- tric and magnetic multipolar modes, respectively. k is the tures, due to their compact and simplified geometrical 2 wavenumber. Electric dipole (ED, ~ɑ1 ), magnetic dipole configuration. We unambiguously give the concept of 2 2 (MD, ~b1 ), electric quadrupole (EQ, ~ɑ2 ), and magnetic multipole engineering, which indicates the manipulation 2 quadrupole (MQ, ~b2 ) are the first four multipole terms of multipolar interaction under linearly polarized planar in the Mie expansion [9]. In mathematics, multipole terms wave, such as near-field coupling and multipolar inter- in the Mie expansion form a complete and orthogonal ference effect appearing in single or few metastructures basis for the decomposition of the scattered fields. For a (Figure 1C). Beyond multipole engineering, the concept ­100-nm-radius Si nanosphere in free space, the scatter- of multimode engineering indicates the manipulation ing spectra and the electric fields (E-fields) distributions of interaction (e.g. weak and strong coupling) between of dipole and quadrupole modes at multipolar scattering multipolar terms and exotic optical modes or states, like peaks (P1–P4) are shown in Figure 1A. One can simply cavity mode (bound mode), lattice (diffraction) mode, ­distinguish the contribution of ED, MD, EQ, and MQ circular polarization, plasmon mode, exciton mode, and from the Mie expansion of scattering spectra (color lines others (Figure 1C). In this review, we aim at the recent pro- in Figure 1A) and from the orientation of E-fields (black gress and provide an insightful ­overview on the aforemen- arrow in the insets of Figure 1A). Note that ɑm and bm are tioned areas, ­highlighting the crucial role of multipole the functions of the particle size and the refractive index and multimode engineering in Mie resonance-based difference between the particle and the surroundings. metastructures. High-index particles embedded in low-index medium are Based on the above concept of multipole and mul- favorable to improve the Mie scattering efficiency. At the timode engineering, this review is organized as follows: resonance of each multipolar mode, the electromagnetic single and cluster metastructure (meta-atom and meta- fields are strongly confined inside the particle and the molecule)-mediated multipolar interference phenomena, remarkable enhancement is comparable to the plasmonic which are responsible for the generation of several novel counterpart. In contrast, a broadened ED scattering peak dipole modes and their multipole family, are investigated (Pplasmon) is obtained from a 100-nm-radius Au nanosphere in Section 2. Fano resonance and bound states in the con- (Figure 1B), due to the strongly damped plasmon mode. tinuum are also discussed as the nontrivial multimode The expansion results indicate the lack of MD (blue line coupling phenomena. In Section 3, we extend our discus- in Figure 1B) and high-order magnetic multipole modes sions to planar meta-film, including periodic meta-lattice in subwavelength-sized metal nanoparticles at the visible and aperiodic meta-assembly, revealing the importance of and near-infrared regions. Enlightened by the simplest the collective coupling effect and surface lattice modes. case of nanosphere, we should clarify that the Mie solu- In Section 4, we overview the multimode engineering in tion, which gives an exact solution for the case of spheri- chiral