Physicists have been exploring the possible existence of non-radiating sources, which can aid us in studying issues related to stable atomic and electronic structure models. Initially, while studying elementary particle physics, Yakov Zel’dovich introduced the academic term ‘anapole’, which means ‘without poles’ in Greek [1]. Recent research has demonstrated that in electrodynamics, anapole modes can be considered as destructive interference formed between similar radiation patterns of electric dipole moments and toroidal dipole moments. Due to the combination of these radiation patterns and destructive interference, there is no far-field radiation in the far-field region [2]. This non-radiating electromagnetic source, produced by the interference of electric dipole moments and toroidal dipole moments (known as ‘anapole modes’), has attracted great interest due to its non-radiating nature.

The bound states in the continuum (BIC) or embedded states were initially proposed by von Neumann and Wigner in 1929 in the context of quantum mechanics through the construction of artificial potentials [3]. Recent research indicates that it can be interpreted as a leaky mode within the continuous background spectrum with an infinite quality (Q) factor or zero radiation decay rate [4]. So far, it has been proven to be a universal phenomenon in waves [5] and has been realized in water waves [6], sound waves [[7], [8], [9]], and electromagnetic waves [10]. Moreover, four types of BICs have been demonstrated in theory and experiment, including symmetry-protected BIC [11], Fabry-Perot BIC [12], Friedrich-Wintgen BIC [13], and accidental BIC [10]. Among them, two resonances at the same position can lead to a BIC through interference of radiation. This type of BIC is called Friedrich–Wintgen BICs (coupled resonances). It generally occurs near frequency crossovers of uncoupled resonances [14].

The BIC is an emerging concept in nanophotonics. Recently, the construction of optical BIC has been successfully realized in photonic crystal [15], superstructural surface [16], and plasmonics [17]. Optical BICs are used in a wide range of applications, including lasers [18], sensors [19], fiber optic structures [20], and so on. As ideal BIC modes are non-radiating, they cannot be excited using propagating light to study their optical properties. Although a single or multiple nanostructures cannot support ideal BICs, research shows that Q-BIC modes can be constructed by carefully adjusting structural parameters [21]. A general method to explore Q-BIC modes is to construct a pair of high-Q and low-Q leaky modes with crossing or avoid-crossing features. The high-Q nature of the Q-BIC mode ensures extreme field confinement within the structure, thus offering broad prospects for enhancing light-matter interactions, such as lasers and enhanced nonlinear harmonic generation [22,23]. Rybin et al. [24] demonstrated that a single high-dielectric constant cylinder can support Q-BIC mode patterns, while Bogdanov et al. [25] conducted in-depth research on such Q-BICs from a fundamental physics perspective, and they found that Q-BICs are caused by Friedrich-Wintgen destructive interference. Recently, Huang et al. [26] developed a general method to find a series of high-Q Q-BICs in a single dielectric nanocavity, including rectangular lines, disks, and cuboids. Optical nanoantennas can effectively confine, localize resonances, and significantly enhance electromagnetic fields at sub-wavelength scales. BIC has been explored in optical nanoantennas, through nanoantenna arrays [[27], [28], [29]], but can also be in individual resonators [22,24]. Nanoantenna designs with broken symmetry provide small but sufficient coupling to form Q-BIC modes, thereby achieving far-field optical excitation, while ideal BIC modes do not couple to the far field. As nanoantennas support Q-BIC modes, controlling their geometric shape can achieve unidirectional scattering [30], sensitive high-spectral imaging [26], lasers, biosensing [29], nonlinear nanophotonics [22], and topological photonics [31].

Here, this work combines the high Q-factor of Q-BIC and the ability of plasmonic nanoantennas to confine light within a sub-wavelength range [32], designing a hybrid nanostructure based on plasmonic optical nanoantennas and all-dielectric nanodisks. It is characterized by the highly tunable Q-BIC modes excited in a simple hybrid structure. The hybrid system consists of a sub-wavelength high-refractive index dielectric disk and plasmonic antennas. The dielectric nanodisk supports an anapole mode, while the plasmonic antennas hold electric dipole resonances. By reasonably adjusting the phase difference between the two modes, the hybrid can exhibit Q-BIC behavior. Then we study hybrid structures with different configurations and hybrid structures with varying structural parameters. They further prove that the excitation of Q-BIC mode requires reasonable adjustment of the phase between the two modes. As the two modes involved in the construction of Q-BIC are on two separate structures, the Q-BIC mode of the hybrid structure here is highly tunable. Note that the modes involved in common reported studies with Q-BIC modes are provided by the same structure where the geometric requirement is stricter than our configuration. The proposed hybrid structure has a simple structure with high adjustability, which may find applications in fields such as nonlinear devices, biosensors, laser cavities, and optical communications [14].