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Zhong, H.; He, T.; Meng, Y.; Xiao, Q. Photonic Bound States in the Continuum in Nanostructures. Encyclopedia. Available online: https://encyclopedia.pub/entry/52720 (accessed on 15 August 2024).

Zhong H, He T, Meng Y, Xiao Q. Photonic Bound States in the Continuum in Nanostructures. Encyclopedia. Available at: https://encyclopedia.pub/entry/52720. Accessed August 15, 2024.

Zhong, Hongkun, Tiantian He, Yuan Meng, Qirong Xiao. "Photonic Bound States in the Continuum in Nanostructures" *Encyclopedia*, https://encyclopedia.pub/entry/52720 (accessed August 15, 2024).

Zhong, H., He, T., Meng, Y., & Xiao, Q. (2023, December 14). Photonic Bound States in the Continuum in Nanostructures. In *Encyclopedia*. https://encyclopedia.pub/entry/52720

Zhong, Hongkun, et al. "Photonic Bound States in the Continuum in Nanostructures." *Encyclopedia*. Web. 14 December, 2023.

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Bound states in the continuum (BIC) have garnered considerable attention recently for their unique capacity to confine electromagnetic waves within an open or non-Hermitian system. Utilizing a variety of light confinement mechanisms, nanostructures can achieve ultra-high quality factors and intense field localization with BIC, offering advantages such as long-living resonance modes, adaptable light control, and enhanced light-matter interactions, paving the way for innovative developments in photonics.

bound states in the continuum
nanostructures
2D materials

Bound states in the continuum (BIC) in photonic structures have emerged as a pivotal concept with vast applications in optics and photonics ^{[1]}^{[2]}^{[3]}. By judiciously designing the potential structure, BIC was first constructed mathematically in 1929 by von Neumann and Wigner as resonant states existing within the radiation continuum that does not radiate ^{[4]}. Since the BIC concept was further refined by Stillinger and Herrick ^{[5]}, the wisdom has been widely accepted and applied to various wave phenomena, such as acoustic waves ^{[6]}^{[7]}^{[8]}^{[9]}, water waves ^{[10]}^{[11]}^{[12]}, and electromagnetic waves ^{[13]}^{[14]}^{[15]}. After the first observation of BIC in optical systems ^{[13]}^{[16]}, this nascent topic has witnessed rapid development and expansion in photonics ^{[17]}^{[18]}^{[19]}^{[20]}.

The high-quality factor, strongly localized field, non-radiative property, unique light confinement mechanism, and intriguing topological nature of BIC have infused new vigor into areas such as resonator design ^{[21]}^{[22]}^{[23]}^{[24]}, low-loss optical transmission ^{[18]}, efficient nonlinear generation ^{[25]}^{[26]}, and advanced light–matter interactions ^{[27]}^{[28]}^{[29]} by applying BIC within diverse nanostructures ^{[30]}^{[31]}^{[32]}^{[33]}. The adjacent photonic structures depict the enhanced properties derived from the incorporation of BIC. In gratings, BIC can significantly enhance light–matter interactions ^{[34]}, leading to more efficient diffraction and light manipulation ^{[35]}. Photonic crystals with periodic structures can leverage BIC to achieve complete bandgap properties, further fine-tuning light propagation and confinement ^{[36]}. Waveguides also benefit from BIC to design long-lived resonances, optimizing light transmission with minimal loss ^{[18]}. Meanwhile, metasurfaces, known for their ability to manipulate electromagnetic waves on subwavelength scales, can pair with BIC to realize sharp resonances ^{[37]} and improved control over wavefronts ^{[38]}. By harnessing BIC’s superior attributes across these nanostructures, researchers have unlocked new horizons in photonic applications and innovations ^{[1]}^{[39]}. Furthermore, the synergy of BIC with 2D materials also enhances the performance of versatile optoelectronic devices such as dynamic switching ability ^{[40]}, harmonic generation efficiency ^{[34]}, and photoluminescence (PL) intensity ^{[41]} and provides opportunities for observing novel physical phenomena like polariton-induced nonlinearity ^{[26]} and collective behavior of Bose–Eistein condensate ^{[32]}.

A burgeoning body of literature has emerged in recent years to discuss the fundamentals of BIC ^{[1]}^{[3]}^{[39]}, reflecting the profound interest in this intriguing optical phenomenon. Photonic BIC refers to an electromagnetic mode that is confined to a finite region of space without radiating, similar to a traditional bound state, but exists within the continuum spectrum of energy or frequencies that only permits extended states with inevitable radiation loss in the traditional situation ^{[4]}^{[5]}.

Symmetry-protected BIC. A state qualifies as a symmetry-protected BIC when it cannot couple to leakage modes due to a symmetry mismatch. Consequently, this state manifests as a bound state, not radiating even within the continuum band. The systems consist of horizontally aligned waveguide arrays with two supplementary vertical waveguides. With careful design, the horizontally arranged array was optimized to support a guide band with modes that exhibit symmetry in the y-direction. Conversely, the two vertically aligned arrays were tailored to support modes with y-directional anti-symmetry, and their frequencies are precisely located within the guide band. Within the guide band, modes from the vertically arranged array should ideally transmit within the horizontal array. However, in practice, a highly localized electromagnetic field was observed in the vertically aligned arrays.

Interference-based BIC. An interference-based BIC arises due to the destructive interference of individual radiation channels in resonant structures by judiciously tunning parameters. A unique characteristic of interference-based BIC is its emergence when the electromagnetic modes lack symmetry ^{[17]}, as opposed to the previous situation. To achieve interference cancellation, the number of tunning parameters typically needs to surpass the radiation channels. However, as the parameter count rises, tuning becomes increasingly complex, making this method more effective with fewer radiation channels ^{[42]}^{[43]}^{[44]}^{[45]}.
### 2.2. Multipole Expansion

### 2.3. Topological Origin

## 3. BIC in Various Photonic Structures

### 3.1. Gratings

The multipole expansion method offers a systematic approach to represent intricate wave fields by decomposing them into localized source terms, including monopoles, dipoles, and higher-order multipoles ^{[46]}. This decomposition not only simplifies complex fields but also provides a comprehensive insight into wave phenomena. Historically, the method has been instrumental in areas such as the Mie scattering theory ^{[47]}^{[48]}^{[49]}, antenna theory ^{[50]}^{[51]}, and resonance design ^{[52]}^{[53]}.

Analyzing the multipole components of BIC states offers a fresh perspective on the origins of BIC ^{[54]}. The rise of symmetry-protected BIC is linked to the non-transverse radiation characteristics of the multipole components, and the accidental off-Γ BIC stems from the destructive interference effect between various multipole orders Valued for its capacity to highlight dominant multipole modes, the multipole expansion method has been extensively used in BIC design for metasurfaces and photonic crystals ^{[37]}^{[55]}^{[56]}, which aids in comprehending the far-field radiation behavior of BIC modes ^{[22]}^{[57]}.

Inspired by condensed matter physics, topological photonics aims to design and harness optical structures that possess unique properties derived from their topological nature. This field has witnessed significant advancements in recent years ^{[58]}^{[59]}^{[60]}. Due to their robustness against fabrication defects and noise, a range of photonic devices leveraging topological principles has been developed, including topological microcavity polariton lasers ^{[61]}^{[62]} and signal transporters facilitated by the terahertz quantum valley Hall effect ^{[63]}. Additionally, the topological method offers a fresh lens to comprehend the BIC mechanism. Pioneering works have revealed that BIC in photonic crystals aligns with vortex centers, carrying quantized topological charges in the polarization far-field space ^{[64]}. Subsequent research has expanded these findings to periodic arrays of dielectric spheres ^{[65]} and one-dimensional (1D) gratings system ^{[57]}.

The inherent topological properties of BIC in momentum space present a unique approach to generating high-purity, efficient vortex beams by leveraging its quantized topological charges and suppressed-side-radiation characteristics ^{[22]}^{[66]}^{[67]}^{[68]}^{[69]}. Furthermore, dynamic switching capabilities have been achieved using a subwavelength-thin phase-change halide perovskite BIC metasurface, which allows for the alteration of emission patterns between polarization vortices with opposing topological charges at distinct wavelengths ^{[40]}^{[70]}.

Optical gratings are periodic structures that can manipulate the direction and wavelengths of incident light waves in specific ways. With their ability to disperse incident light, optical gratings provide a flexible platform for engineering BIC ^{[71]}^{[72]}.

With carefully designed groove spacing ^{[73]}^{[74]}^{[75]}, geometry ^{[19]}^{[76]} and dielectric constant ^{[77]}^{[78]}, one can easily tailor the resonance conditions of BIC to achieve customized control over specific wavelengths or frequency ranges, which bears immense potential in applications such as various environmental sensors ^{[73]}^{[79]}^{[80]}, narrowband filters ^{[35]}^{[81]}, and microlasers ^{[82]}^{[83]}. Stemming from the destructive interference between two modes, the perovskite microlaser sustains an interference-based BIC at a normal incident angle with a quality factor approaching 10^{10} and an observed lasing action in 548.5 nm when the pump density reaches 49 μJ/cm^{2} ^{[74]}. Beyond its considerable quality factor, the union of BIC with periodic grating also leads to the high confinement of electromagnetic fields within a specific localized area. This enhanced field confinement leads to stronger light–matter interactions ^{[84]} exhibiting a typical Fano resonant line shape ^{[73]} and also dramatically enhances optical harmonic generation with a boost of several orders of magnitude ^{[34]}^{[85]}.

Beyond simple gratings, there have emerged compound structures broadening the boundaries of applications. Take asymmetry dual-gratings as an example: this kind of grating consists of two parallel gratings with different adjacent gaps separated by a fixed distance, adding new design dimensions to be exploited for controlling quality factor and operating wavelength ^{[86]}^{[87]}^{[88]}^{[89]}. The composite integration of waveguides and gratings is another common approach to harness the unique characters of both platforms. The incident light can couple with guide mode in waveguide ^{[90]}^{[91]}^{[92]}, owning to the tangential wave vector provided by discrete periodic gratings ^{[93]}^{[94]}^{[95]}, namely, the Guide Mode Resonance (GMR) phenomenon. By synergizing the GMR condition with the BIC concept, the Goos-Hänchen shift ^{[96]}^{[97]} and spin Hall effect ^{[98]} can be greatly enhanced due to an ultra-high-quality factor provided by BIC modes.
### 3.2. Photonic Crystals

Photonic crystals (PhCs), also known as photonic bandgap materials, have been a captivating area of research in photonics for decades ^{[99]}^{[100]}. These nanostructured materials possess unique optical properties that arise from their ability to control the propagation of electromagnetic waves through periodic modulation of the refractive index ^{[101]}. While under the extensive exploration of BIC, this exotic optical concept breaks the traditional wisdom that provides an alternative way to confine light and achieve high-quality resonance in PhCs in addition to the photonic bandgap design method ^{[16]}. By capitalizing on the unique properties of both BIC and PhCs, a new generation of compact, efficient, and versatile photonic devices has been prototyped ranging from low threshold lasers ^{[21]} to second harmonic generation ^{[25]}, and other intensively active research field ^{[102]}.

Composed of a series of periodic dielectrics, 1D PhCs are relatively convenient for designing BIC due to their simple structure ready for exploring conditions for supporting BIC in 1D PhC systems ^{[103]}^{[104]}^{[105]}. Applying BIC in 1D PhCs has enabled enhanced light–matter interactions ^{[28]}, but other application-oriented research remains relatively scarce and limited by their simplistic structure. For 2D PhC slab systems, BIC can be achieved by judiciously tailoring the lattice geometry parameter ^{[106]} and slab thickness ^{[107]}, exhibiting more design flexibility compared with 1D periodic structure to functionalize a broader range of applications without significantly increasing design complexity. For instance, by harvesting the merits of strong optical field localization, BIC within heterostructure cavity PhC slabs offers an avenue for highly efficient nonlinear frequency conversion ^{[34]}, which proves to be more pragmatic than directly designing a photonic bandgap at the second-harmonic frequency. The side-radiation suppression property of BIC ^{[20]} also enabled some high-performance lasers with lower thresholds to work at room temperature ^{[21]}^{[22]}^{[24]}. Laser beam quality can be further improved by leveraging the topological nature of BIC. As vortex centers exist in far-field polarization fields ^{[64]}, BIC continuously shifts but does not disappear when changing geometrical parameters. Such robustness allows for merging a cluster of BIC singularities into a single point in momentum space, leading to an enhanced quality factor and better lasing directionality ^{[108]}^{[109]}^{[110]}, namely, the merging BIC techniques.
### 3.3. Waveguides

Photonic waveguides, with their capacity to control light at microscopic levels, act as the primary channels in a circuit. They guarantee the precise direction of photons to the appropriate components at the necessary moments ^{[70]}^{[111]}^{[112]}. Incorporating the principles of BIC into photonic waveguide designs has yielded numerous benefits. Notably, utilizing BIC broadens the material choices available. Traditionally, light confinement in waveguides has been largely achieved with total internal reflection. This method often mandates the use of low-refractive index substrates, limiting the choice of materials for waveguide fabrication ^{[113]}^{[114]}^{[115]}. The inherent characteristic of BIC to sustain localized without decay within a radiative spectrum indicates the potential to use higher-refractive index substrates without leakage, thus preventing unwanted transmission loss ^{[18]}^{[116]}^{[117]}^{[118]}. By utilizing BIC, an organic polymer waveguide on a diamond substrate was demonstrated with an ultra-low propagation loss. Using structural parameter optimization, BIC was realized at specific waveguide widths, exhibiting near-zero coupling strength. This indicates that the bound mode was entirely decoupled from the continuum mode in the waveguide, resulting in the calculated ultra-low propagation loss ^{[18]}. Beyond diamond substrates, BIC also facilitates reduced losses in organic polymer waveguides positioned on lithium niobate substrates ^{[119]}^{[120]}, which presents a promising avenue for crafting versatile on-chip integrated photonic devices, including photodetectors ^{[121]}, modulators ^{[122]}, and other essential components ^{[123]}. Building on this foundation, an on-chip four-channel TM mode (de)multiplexer with data transmission at 40 Gps/channel has been demonstrated for high-dimensional communication ^{[124]}. The large nonlinear coefficients and wide transparency window of LiNbO_{3} also suggest that it is an ideal platform for achieving efficient second-harmonic generation ^{[125]}^{[126]}^{[127]}.
### 3.4. Metasurfaces

## 4. Summary

### 4.1. Chiral BIC

Consisting of two-dimensional arrays of subwavelength structures, metasurfaces offer unparalleled capabilities to manipulate light properties ^{[70]}^{[93]}^{[128]}^{[129]}^{[130]}^{[131]}, including its amplitude, phase, and polarization ^{[132]}. These distinctive traits have been harnessed for diverse applications such as imaging ^{[133]}, optical computing ^{[134]}, and optical anticounterfeit ^{[135]}. The fusion of BIC’s high Q characteristic with the adaptability of metasurface regulation further amplifies their potential ^{[136]}^{[137]}.

Sensing. By harnessing the high-quality factor property of BIC, metasurface-based sensors ^{[138]} can attain unparalleled sensitivity and accuracy. The breakthrough in gold split ring metasurfaces ^{[139]} with BIC realized by Srivastava et al. has paved the way for its application in sensing. To further enhance sensitivity, Chen et al. ^{[140]} utilized the toroidal dipole bound states in the continuum (TD-BIC) and achieved an impressive amplitude sensitivity of 0.32/RIU. Similarly, Cen et al. ^{[141]} demonstrated excellent performance in a refractive index sensor with a sensitivity of 465.74 GHz/RIU and a figure of merit of 32,984. In addition to a single sensing function, more flexible modulation and functions can be explored.

Imaging. Dynamic imaging and image processing have made rapid advancements with the integration of the BIC and metasurface, along with the utilization of the control of different materials. Yesilkoy et al. ^{[142]} demonstrated hyperspectral imaging with dielectric metasurfaces using a metal oxide semiconductor (CMOS) for the spectrometer in 2019. For reconfigurability, Ge_{2}Sb_{2}Te_{5} (GST) film ^{[143]} and graphene ^{[137]}^{[144]} were used to control dynamic imaging by tuning the resonance or voltage.

Nonlinear optics. The strong field confinement and enhanced light–matter interactions provided by metasurfaces open up avenues for investigating nonlinear effects at ultra-low light intensities when coupled with BIC. Metasurfaces are capable of hosting a perfect BIC mode in extremely symmetric geometries. However, it is undetectable. When disrupting geometric symmetry, the ideal BIC would be converted into a quasi-BIC mode, which can be detected

As a fundamental geometrical property, chirality refers to objects or systems where they are distinguishable from their mirror image. In the specific domain of photonics, chirality has garnered substantial interest for its capability of modulating the geometrical phase for arbitrary wavefront shaping ^{[145]}^{[146]}, efficiently manipulating the polarization state of light emission ^{[147]}^{[148]}^{[149]} and advancing light spin manipulation ^{[126]}^{[150]}. The high-quality factor, strong field enhancement, and far-field directionality properties of BIC align well with the requirements of the aforementioned applications. Therefore, chiral BIC has become one of the recent hotspots in BIC research.
### 4.2. BIC in Hybrid Structures with 2D Materials

With the advancement in BIC research, the focus has expanded from purely dielectric BIC to include BIC in hybrid systems. This expansion firstly facilitates the intricate interplay between metallic and dielectric materials ^{[151]}, giving rise to unprecedented optical characteristics. The fusion of plasmons with BIC mainly aims to leverage the high-quality factor property of BIC to address the inherent ohmic losses associated with metal-based systems ^{[152]}. By combining metals like gold with dielectrics like SiO_{2} ^{[153]}, plasmonic BIC can be achieved in such a hybrid system with more compressed modal volume ^{[154]} and stronger light–matter interactions ^{[155]}^{[156]}, which has been exploited for biomolecular sensing ^{[155]}^{[157]}.

Another particularly promising avenue is the integration of BIC with 2D materials. The novel functionalities offered by 2D materials were discussed in the prior sections. Characterized by large oscillator strengths in their excitons, monolayer TMDs present promising platforms for investigating strong light–matter interactions. By transferring monolayer TMDs onto dielectric metasurfaces ^{[158]} or photonic crystal slabs ^{[159]}, Rabi-splitting can be significantly enhanced due to the intense coupling between the excitons in 2D materials and the optical quasi-BIC in dielectrics. This coupling can be further optimized by adjusting the position of the monolayer and the thickness of the dielectrics, as highlighted in Ref. ^{[160]}.

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