Photonics Insights

**Introduction**

At the invitation of Photonics Insights' founding editor, Prof. Zi Jian and Prof. Lei Shi from Fudan University, Prof. Chao Peng from Peking University, and Prof. Dezhuan Han from Chongqing University jointly authored a review article titled "Optical bound states in the continuum in periodic structures: mechanisms, effects, and applications". The article was published in the first issue of Photonics Insights in 2024 and was selected as the "On the Cover". (Jiajun Wang, Peishen Li, Xingqi Zhao, et al. Optical bound states in the continuum in periodic structures: mechanisms, effects, and applications[J]. Photonics Insights, 2024, 3(1): R01)

This review systematically summarizes the recent developments of optical bound states in the continuum (BICs) in periodic structures, including the physical mechanisms of BICs, explored effects enabled by BICs, and applications of BICs, and provides a perspective on future developments for BICs.

**1. Physical Mechanisms**

The journey into the realm of BICs began with early theoretical insights into scattering resonances in quantum mechanics in 1929 by von Neumann and Wigner. Due to the ability to confine waves in open systems, BICs aroused great research interest and were further extended to other wave systems, such as the familiar acoustic waves, water waves, and electro-magnetic waves. Among them, BICs are actively studied in optical systems in recent years. With ultrahigh quality (Q) factors, optical BICs have powerful abilities to trap light in optical structures from the continuum of propagation waves in free space. Especially in periodic structures with well-defined wave vectors, many hidden topological characteristics were discovered in optical BICs. Both high Q factors and topological vortex configurations in momentum space facilitated by BICs bring new degrees of freedom to modulate light, leading to many novel discoveries in light–matter interactions and spin–orbit interactions of light, and BIC applications in lasing and sensing have also been well explored with many advantages.

Elucidating the physical mechanisms of optical BICs in periodic structures is a key topic in the related research field. The physical mechanisms behind BICs have been elucidated from different physical perspectives, covering Friedrich-Wintgen (FW) mechanism of BICs, plane-wave and Bloch-wave perspectives, multipolar perspectives, and the intriguing topological perspective, which enable more effective applications of BICs.

**FW mechanism of BICs:**

According to the FW BICs principle, the formation of BIC in periodic optical structures can be elucidated through coupling and interference. BICs are then classified into original FW BICs, symmetry-protected BICs, and accidental BICs based on different FW origins. Although these types of BICs appear to have different features, they can be uniformly described using the FW BIC mechanism. They can all be characterized as the coupling of different modes of guided resonances or Fabry-Perot modes, and can be analytically explained through the temporal coupled-mode theory. At a specific value of the tuning parameter, the width of one resonance vanishes and hence it becomes a BIC.

Fig. 1 Representative examples of different types of BICs in periodic photonic systems.

**Plane-wave and Bloch-wave perspective of BICs:**

To explore other properties of BICs, there are also other alternative bases to analyze the interference of modes, such as plane waves or Bloch waves. By treating the modes in photonic crystal slabs as the combination of a series of plane waves, the origin of BICs can be derived from an analytical iterative method. Besides, the Bloch waves, as another kind of extended wave, contain the complete information from the in-plane periodicity and the formation of BICs can also be interpreted by utilizing this basis. These choices are beneficial in facilitating a more rigorous derivation based on the first principles and more efficient computational methods to study BICs.

**Multipolar perspective of BICs:**

Multipole expansion is extensively used to analyze the properties of optical systems, and it can also be employed to elucidate optical BICs. For periodic structures, the overall optical properties can be regarded as the interferences of radiations from all the local sources: unit cells. For each unit cell, its radiation pattern can be decomposed by a series of multipoles, each with certain directions (poles) where radiation vanishes. These multipole components are composed of a dominant component and other minor components. The dominant component will determine the properties of the modes, and its singularity index is closely associated with the topological charge of the BIC. Through this method, insight can be gained into the underlying mechanisms that give rise to the non-radiative behavior of BICs, as shown in Fig. 2. Furthermore, the mechanism of multipole expansion not only describes BICs in photonic quasicrystals and similar systems but also analyzes the generation of other polarization singularities.

Fig. 2 Understandings of BICs from the perspective of multipolars.

**Topological perspective of BICs:**

Optical modes in periodic structures can have well-defined wave vectors k_{||}, and the optical BIC describes the optical mode of a certain k_{||}, which does not radiate along the direction without translational symmetry. In contrast, other optical modes surrounding this BIC with certain wave vectors are leaky and radiate light to free space with definite polarization states. The k_{||}-dependent polarization distribution forms a polarization field in momentum space. The optical BICs are discovered to be topological defects in momentum space, carrying topological charges defined by the winding major axes of polarization states surrounding BICs, as shown in Fig. 3. As the center of polarization vortex, the polarization state of BIC cannot be defined, corresponding to its nature as a bound state. The vortex configurations of optical BICs bring new polarization degrees of freedom, promoting research in topological photonics and singular optics. By combining graph theory with the topological charge of polarization singularities, they can further explore the non-local properties of BICs in momentum space, predicting the generation and evolution of BICs beyond high-symmetry directions in momentum space.

Fig. 3 Topological nature of BICs.

They also describe BICs with multiple radiation channels, such as those exceeding the diffraction limit, proposing an underexplored research area.

**2. Explored Effects Enabled by BICs**

BICs can be easily found in periodic photonic structures with certain rotational symmetry. While for better application, investigating modulation methods and evolution rules of BICs is crucial. With explored physical mechanisms of BICs, many approaches are proposed to modulate BICs in momentum space. Among them, changing the structural parameters is usually applied to modulate BICs. By tuning the periodicity, thickness or proportion of different constituents in a unit cell of the structures, diverse evolutions of BICs in momentum space have been realized, as shown in Fig. 4.

Fig. 4 Momentum-space evolution of BICs.

Symmetry, as the most prominent category of structural parameters, directly influences the polarization configurations and topological charges of BICs in periodic structures. When the structural symmetry degrees of freedom are further introduced, more evolution effects can be induced. As shown in Fig. 5, a more diverse range of polarization singularities and polarization configurations have been discovered under symmetry breaking. Furthermore, starting from polarization singularities with high-order topological charges, even richer modes of symmetry breaking and corresponding evolution of polarization singularities can be realized. Throughout these evolutionary processes, the topological charge of polarization singularities remains conserved.

Fig. 5 Momentum-space evolution of BICs by symmetry breaking.

In periodic structures, quasi-BICs have also garnered widespread attention, as shown in Fig. 6. Quasi-BICs are resonant states observed in optical systems that exhibit properties intermediate between traditional BICs and radiative modes. For quasi-BICs, imperfect destructive interference between different waves results in high Q factors while still allowing coupling with free space. Common approaches to obtain quasi-BICs involve breaking the in-plane rotational symmetry of the structure, transforming symmetry-protected BICs into quasi-BICs. This method also allows for modulation of the quality factor of quasi-BICs by controlling the degree of symmetry breaking. Quasi-BICs also exist in photonic quasicrystals and photonic moiré structures. Through further structural design and modulation, BICs can be endowed with new properties in multiple dimensions, constituting an important branch of BIC exploration. For instance, combining the confinement effect of BICs on the optical field with photonic bandgaps can create "Miniaturized BICs" with stronger field confinement effects. Additionally, based on quasi-BICs, polarization properties can be further controlled through symmetry modulation, resulting in chiral quasi-BICs.

Fig. 6 Various types of quasi-BICs.

When it comes to light field manipulations, the high Q factors of generalized BICs have sparked widespread research interest, giving rise to explorations of many exotic optical phenomena. Based on the high Q factors, optical BICs or quasi-BICs have been exploited to realize the zero-index propagation effect, negative refraction and giant enhancement Goos-Hänchen shift.

Surrounding BICs, the major axes of polarization state wind in the clockwise or anticlockwise direction. This winding topological nature of BICs has been explored with many topological effects, bringing new degrees of freedom to manipulate light. As shown in Fig. 7, researchers have achieved the generation of vortex beams and spin-Hall effect of light based on photonic crystal slabs. Moreover, the topological features of BICs enable full polarization control on the entire Poincaré sphere, and can also be associated with the topological features of optical force distribution in momentum space. The rich topological properties of BICs remain to be fully explored, and the light field control effects they bring forth hold immense potential for development.

Fig. 7 Manipulating light by utilizing the topological vortex configurations around BICs.

BICs have also attracted significant attention in the field of light-matter interactions. Light confinement with ultrahigh Q factors and small modal volume Vs can be achieved with BICs. The BIC structures then show great potential in optical nonlinearity enhancement under a large Purcell effect described by Q/V. Similarly, optical BICs can also be utilized to enhance the strong photon-exciton coupling instead of distributed Bragg reflectors due to their higher Q factor and near-infinite photon lifetime. Therefore, photonic cavity quantum electrodynamics systems can more easily achieve strong coupling and obtain large Rabi splitting, leading to strong implications for polaritons in quantum information processing. Polariton Bose-Einstein condensation (BEC) has been achieved based on optical BICs as well.

**3. Applications**

Given the BICs' innate ability to realize high-Q resonances, the most significant application of BICs is for lasing. The development of laser devices using periodic structures has a considerable history, such as early Photonic crystal surface-emitting lasers (PCSELs). The introduction of the BIC concept into periodic photonic structures has greatly advanced PCSELs, as shown in Fig 8. On the one hand, researches on the quality factors have led to BICs with higher Q factors, significantly reducing the threshold required for lasers. On the other hand, researches on the topological features of BICs can also be applied in lasers. Through tuning the topological charges, BICs can be utilized

to realize varied beam patterns, such as the vortex laser generated from BICs based on their polarization vortex configurations in momentum space. Quasi-BICs, due to their high Q factors and polarization modulation capabilities, have also attracted considerable attention in photonic crystal lasers. For example, chiral quasi-BICs obtained through symmetry modulation can achieve lasers with high circular polarization degrees.

Fig. 8 Applications of BIC in lasers.

Another prominent application of BICs in periodic photonic structures is highly sensitive sensing and detection. Structures incorporating BICs inherently exhibit high-Q resonance behavior, with the ideal scenario being an infinite Q factor. In a disturbed environment, the ideal BIC transforms into a quasi-BIC with an offset resonant wavelength while maintaining a fairly high Q factor. This unique property renders it highly applicable in various optical and photonic contexts. Moreover, the polarization distributions around BICs or quasi-BICs further enable their applications in chiral sensing. Chiral quasi-BICs designed through periodic structure engineering not only possess high quality factors but also exhibit a sharp response to the polarization of the optical field. Based on this, highly sensitive chiral detection has been achieved, holding extraordinary significance for the fields of medicine and pharmaceuticals.

**4. Summary and Outlook**

In this review, the authors summarize the burgeoning field of optical BICs in periodic structures. The physical mechanisms behind BICs were elucidated from different physical perspectives, covering FW BICs, plane-wave and Bloch-wave perspectives, multipolar perspectives, and the intriguing topological perspective. Diverse effects enabled by BICs were categorized and collected, encompassing evolution rules, modulation methods, quasi-BICs, and the exploitation of high Q factors for light propagation and intricate light-matter interactions. Applications of BICs in periodic structures are highlighted, with a focus on BIC lasing and sensing. The development of optical measurement systems utilized for BIC characterization were also reviewed.

The exploration of optical BICs in periodic structures has unveiled a wealth of intriguing phenomena and practical applications. Based on the demand for research in optical field modulation and light-matter interactions, there are still many more BICs with novel optical effects waiting to be discovered. Furthermore, the application scope of BICs is expected to expand beyond lasing and sensing. Exploring their potential in areas such as integrated optical circuits and quantum enhanced technologies will be of paramount importance.

In conclusion, the outlook for optical BICs in periodic structures is promising and multifaceted. As they embark on this journey of discovery, the synergy of theoretical insights, technological advancements, and interdisciplinary collaborations will propel optical BICs to the forefront of cutting-edge optics and photonics, shaping the landscape of future optical technologies.