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Bound states in the continuum in photonic structures

Bound states in the continuum provide a remarkable example of how a simple problem solved about a century ago in quantum mechanics can drive the research on a whole spectrum of resonant phenomena in wave physics. Due to their huge radiative lifetime, bound states in the continuum have found multiple applications in various areas of physics devoted to wave processes, including hydrodynamics, atomic physics, and acoustics. In this review paper, we present a comprehensive description of bound states in the continuum and related effects, focusing mainly on photonic dielectric structures. We review the history of this area, basic physical mechanisms in the formation of bound states in the continuum, and specific examples of structures supporting such states. We also discuss their possible applications in optics, photonics, and radiophysics.


Introduction
One of the basic problems in quantum mechanics is the energy eigenvalue problem of a particle in a spherical quantum well. Below the barrier (E < 0), the spectrum is discrete and the wavefunctions ψ(r) are bounded, i.e. R 3 ψ(r)dr < ∞. Above the barrier (E > 0), the spectrum is continuous and the wavefunctions cannot be normalized in the classical sense [see Fig. 1(a)]. These solutions can be presented as propagating modes of the free space surrounding the quantum well. However, E. Wigner and J. von Neumann revealed in 1929 that this classification can be broken for specific potentials that asymptotically tend to zero away from the quantum well. 1 They showed that for some potentials, there are bound states embedded in the continuum of propagating modes [see . Today, such modes are known as bound states in the continuum (BICs). It seems that this term was introduced by L. Fonda in 1960. 2 We should mention that the work by E. Wigner and J. von Neumann contains an algebraic error that was mentioned and corrected by Stillenger and Herrick in Ref. 3 Later, the theory of BICs was extended for the description of various atomic, molecular, and quantum mechanical systems, [3][4][5][6][7][8][9][10] but the potential proposed by E. Wigner and J. von Neumann is rather specific and it has been never implemented. Some ideas on how to construct potentials supporting BIC in semiconductor superlattices were developed in Refs. 11,12 However, they also were not implemented experimentally. We should mention that the experiment in 13 has nothing in common with the observation of BICs in semiconductor superlattices. The Authors of this work simply observed a defect state in the bandgap spatially localized by electron Bragg reflectors. BICs are not a unique feature of quantum mechanical systems. In contrast, they are specific solutions of wave equations in general and they can exist in acoustics, hydrodynamics, aerodynamics, [14][15][16][17][18][19][20] and optics. 21,22 In acoustics and hydrodynamics, BICs have been known for a long time as trapped modes. 19,[23][24][25] In recent years, BICs are actively studied in the areas of optics and photonics, as these states open up tremendous opportunities for implementing compact high-Q resonators and metasurfaces required for biosensing and enhancement of nonlinear optical effects and light-matter interaction.
In the last few years, several reviews of the state-of-the-art achievements in BICs were published Figure 1: (a) Spherical quantum well: the spectrum is continuous for E > 0 and discrete for E < 0; (b) A specific potential results in the formation of bound states in the continuum [26][27][28][29][30] . In this review, we focus on BICs in electrodynamicsystems in various spectral ranges, including visible, infrared, terahertz, and microwaveones. In this review, we present history, modern achievements, and different physical models explaining the nature of this beautiful phenomenon.
In the first section, the history of BICs in optics is described, and a selection of pioneering scientific papers is presented, including those that were published before the introduction of the term 'optical bound states in the continuum'. The second section is devoted to the description of various mechanisms of BICs formation in dielectric structures. Their quantum-mechanical nature, the importance and role of periodic potential for BICs implementation, and topological properties are discussed; an explanation of BICs properties in terms of multipole analysis is also provided. In the third section, examples of photonic structures with BICs are presented, and features of BICs are described in structures of various dimensionality: from single nanoparticles to periodic metasurfaces. In this section, we also briefly talk about quasi-BICs, which are formed from BICs due to the violation of symmetry of a photonic structure. In the fourth section, the applications of BICs are discussed for the detection of biological objects, laser generation, twisted light beams, and optical harmonics.

Historical reference
In modern literature, it is widely assumed that BICs in optical systems were predicted in 2008 in two works published by Marinica, Borisov, and Shabanov, 21 and by Bulgakov and Sadreev,22 and that the first experimental study of BICs in optics was performed in 2011 in the work, 31  One of these structures is shown in Fig. 2(a). This is a two-layer dielectric grating periodic along the x-axis, and it has translational symmetry along the y-axis. The Authors showed that at certain distances between the gratings, the resonances in the reflection spectrum at oblique incidence become infinitely narrow, i.e. disappear from the spectra. They also mentioned that this state corresponds to the light perfectly propagating along the structure with no radiation losses. Bulgakov and Sadreev also considered a single-mode waveguide formed by two identical photonic crystals.
They showed that light can be perfectly trapped at the defects of the photonic crystal despite the fact that the frequency of the trapped mode lies in the transmission band of the waveguide. Plotnik and colleagues performed a beautiful experiment on observation of the symmetryprotected BICs in a photonic structure that is an array of parallel dielectric single-mode waveguides fabricated of fused silica by direct laser writing [Fig. 2(c)]. 31 The near-field coupling between the waveguides results in the formation of a transmission band. Two additional waveguides fabricated above and below the array support the anti-symmetric mode with a frequency lying in the transmission band of the waveguide array. This anti-symmetric mode was excited from one side of the sample, and the intensity distribution was observed on the other side. The lower panel in Fig. 2(c) shows that the energy of the initial anti-symmetric mode does not leak to the waveguide array. In order to break the vertical symmetry, a gradient of the refractive index was created by heating the top side of the sample while cooling the bottom. Such heating results in the coupling of the excited anti-symmetric mode to the modes of the array. Thus, as the mode propagates through the sample, Figure 2: (a) Upper: schematic of a double grating structure in vacuum. Lower: specular reflection coefficient as a function of the wave vector k of the incident radiation and the distance h between the gratings for the fixed value of k x = 0.2G. Adapted from Marinica et al. 21 (b) The probability of transmission of a propagating photonic mode as a function of the frequency in the vicinity of the BIC for two sets of permittivity values. Inset: the mode profile. Adapted from Bulgakov et al. 22 (c) Upper: schematic of a one-dimensional array of 51 and two additional waveguides above and below the array. Lower: light intensity at the output plane of the structure. Adapted from Plotnik et al. 31 its energy is distributed between the waveguides forming the array. We would like to highlight that these three works revealed only the explicit connection between BICs in quantum mechanics and BICs optics for the first time. However, there are many earlier workswhere electromagnetic BICs were studied theoretically and experimentally but were not associated with quantum mechanics and the pioneering work by E. Wigner and J. von Neumann.
To the best of our knowledge, the history of BICs in optics began in 1976 with the work of Kazarinov, Sokolova, and Suris 32 who considered a corrugated waveguide playing the role of a distributed-feedback resonator of a semiconductor laser. The Authors mentioned that if the eigenmode in the center of the Brillouin zone is formed by two counter-propagating waves which are π out of phase, then the radiation losses are canceled. Independently from this work, Vincent and Neviere considered theoretically the band structure of a dielectric corrugated waveguide [see  33 They showed that some resonances at the Γ-point of k-space are completely decoupled from the radiation continuum due to the symmetry mismatch between the mode and the external field, which leads to an infinite radiative lifetime of the mode. The authors of these works did not associate the observed resonances with BICs and they did not recognize the connection with earlier Figure 3: (a) Upper: Schematic representation of a corrugated waveguide. Lower: Dispersion curves for the normalized frequency. The curves correspond to the two distinct branches: the anti-symmetric one (A) and the symmetric one (B). Adapted from the work by Vincent and Neviere 34 (b) Upper: schematic of a multilayer slab waveguide with two-dimensional periodic texture. Lower: Imaginary part of frequency near the anti-crossing between TE-like and TM-like modes. The imaginary part of frequency turns to zero for the higher-energy band-edge states near the anti-crossing, indicating a truly bound state. Adapted from Paddon et al. 35 .
works in quantum mechanics, but the physics of non-radiating states were studied in detail.
Nonradiative states were also found in a two-dimensional periodic array of coupled dielectric spheres by M. Inoue and colleagues. 36 They showed that at the center of the Brillouin zone, optically inactive states exist that cannot be excited by a normally incident wave, regardless of its polarization. At the same time, they can be excited at oblique incidence, and in this case, extremely narrow peaks appear in the reflection/transmission spectrum, see Fig. 4(a). The existence of nonradiating modes in photonic crystal structures was also discussed by K. Sakoda 37,38 and later by P. Paddon and J. Young. 35 The latter have developed a method based on Green's functions that allows analyzing a complex photonic band structure, i.e. the spectral positions of resonances and their radiative lifetimes [see Fig. 3(b)]. This is probably the first work where BICs with a non-zero Bloch wavenumber, the so-called parametric (tunable) BICs or accidental BICs, were theoretically predicted. The authors explained that the coupling between the TE and TM modes leads to the anti-crossing of their dispersion curves and the formation of a mode with zero imaginary part of its eigenfrequency, i.e., the formation of BIC. In fact, the described mechanism is identical to that analyzed in the work of H. Friedrich and D. Wintgen for quantum mechanical systems. 7 Therefore, tunable BICs are also called Friedrich-Wintgen BICs. In 2003, S. Shipman and S. Venakides also observed BICs numerically in the transmission spectra through an array of parallel dielectric cylinders [ Fig. 4(b)]. In the same year, they developed a theory explaining the formation of bound states and anomalies in the transmission spectrum corresponding to these states. 39 Figure 4(b) (bottom panel) shows the numerical transmission spectrum for various Bloch wave numbers. It is explicitly shown that at normal incidence, a collapse of the Fano resonance occurs. The inset in the middle part of the figure shows the electric field distribution of the bound state. In fact, it is anti-symmetric with respect to the plane of symmetry of the unit cell; however, the authors plotted the distribution amplitude, which is an even function. In 2003, Bonnet et al 40 discovered ultra-narrowband resonant reflection from a one-dimensional corrugated waveguide at oblique incidence, i.e. at non-zero Bloch wave number, and analyzed the effect of a finite beam size on the reflection spectrum.
It seems that the BICs were observed in experiment for the first time in the work by Henry et al in 1985. 42 The Authors considered distributed-feedback resonator with a second-order grating.
They showed that the losses of the lasing mode mainly occur at the ends of the structure, while at its center they nearly vanish due to the destructive interference of the scattered radiation from the counter-propagating waves forming the lasing mode, as it was predicted in the work 32 . Later on, in 1986, 43 Avrutskii et al. analyzed the reflectance spectra from the corrugated ZnO waveguide deposited on a glass substrate in the visible range. It was found that a peak in the second stop band disappears from the spectrum at the normal incidence, manifesting the bound state formed by counter-propagating waveguide modes.
Another experiment was done by Robertson and coauthors in 1992. 44 They analyzed the transmission spectra of a two-dimensional dielectric structure consisting of alumina-ceramic cylinders arranged in a square array. The experiment was performed in the GHz frequency range (10-150 GHz). The Authors mentioned that one of the bands was not observed in the experiment because of the symmetry mismatch between the eigenmode and the exciting field. An experiment on the observation of BIC in the visible range was made in 45 by Pacradouni and colleagues. They measured the reflection spectra with angular resolution from the perforated AlGaAs membrane, confirming the narrowing lines in the vicinity of both at-Γ BIC (at normal incidence) and off-Γ BIC (at oblique incidence). However, we should note that this work contains a minor inaccuracy. Specifically, the existence of BICs with non-zero Bloch wave vector requires up-down mirror symmetry 46 or a very fine adjustment of the geometric and material parameters of the system, 47 which were not performed, according to the article. Thus, Pacradouni and colleagues observed only the increase of the Q factor, but not a BIC. For observation of off-Γ BIC, i.e. with a nonzero Bloch vector, in photonic crystal membranes, the structure is usually suspended 48 or covered by an optical liquid whose refractive index matches that of the substrate, 49 which provides the up-down mirror symmetry. The at-Γ BIC was also observed experimentally in a polaritonic system in 1998, by Fujita and colleagues. 50 The Authors analyzed the angular-resolved transmission spectra of a distributed-feedback microcavity consisting of quartz grating substrate covered by an organic-inorganic perovskite-type semiconductor. They discovered that at normal incidence the resonances disappear. These experimental results were comprehensively described by Yablonskii and colleagues in. 51 In this section, we tried to review the key works on BICs in electromagnetic systems. However, we should note that the existence of non-radiating states in photonic structures was also discussed in many other works (see, for example, Refs. 43,[52][53][54]

BICs in dielectric photonic structures
From quantum mechanics to photonics Let us set a link between quantum mechanics and optics by the example of a 1D quantum well with a translational symmetry along the z-direction. For such a system, we can define the domain of the continuum spectrum asẼ = E −  certain direction, BICs will be localized only in the orthogonal directions. The analogy between quantum and optical systems is very illustrative, but it is not complete. The vector structure of electromagnetic fields (polarization) makes the electromagnetic systems more diverse. We should also note that the Helmholtz equation written in the form of the stationary Schrödinger equation (Fig. 5) is a so-called generalized eigenvalue problem, since the required frequency ω is included both in the eigenvalue (the right side of the equation) and in the potential. However, if we consider the wave vector k z as an eigenvalue, then we obtain a classical eigenvalue problem.
One of the general principles that can explain the appearance of BICs in most electromagnetic systems is the destructive interference of two interacting leaky waves. This mechanism was originally proposed by Friedrich and Wintgen, 7 and it is schematically shown in Fig. 6(a). Let an open system (resonator) has two leaky modes |ψ s (s = 1, 2) with close or even equal eigenfrequencies Ω s = ω s + iγ s (s = 1, 2). Then, let's introduce some perturbationV that makes these states coupled. In the framework of the perturbation theory, the eigenmodes in the system, with account for the interaction, can be presented as a linear superposition of the initial states |ψ = C 1 |ψ 1 + C 2 |ψ 2 . If we can continuously tune the interaction potentialV , then at some specific conditions, the radiative losses can be suppressed completely (substantially). In this case, a genuine BIC (or a quasi-BIC) appears.
More formally, an open system within the two-mode approximation can be described in the framework of the temporal coupled-mode theory, where a = [a 1 (t), a 1 (t)] T are the complex amplitudes of states |ψ 1 and |ψ 1 . The complex amplitudes evolve in time as: Here, κ is responsible for the internal coupling, √ γ 1 γ 2 accounts for the coupling through the radiation continuum, and φ is the phase shift between the modes. The condition of the BIC appearance in the two-mode approximation can be written as 55 These conditions can be fulfilled through the tuning of the parameters of two coupled resonances. Figure 6(b) shows schematically how the radiative losses of the resonant states |ψ 1 and |ψ 2 depend on the coupling strength κ. It is worth mentioning that exactly at the point where the BIC appears, the radiative losses for the second mode are equal to γ 1 + γ 2 . This is an analogue of the Dicke superradiance for two emitters. 56,57 In the framework of this simple model, the initial resonant states should radiate to the same radiative continuum (scattering channel). Only in this case they can interfere destructively and This is the so-called empty lattice approximation, which is well-known in solid state physics 59 .
Therefore, the second bandgap is closed, and the eigenstates in the Γ-point are degenerate. They . This mechanism of BIC formation is quite clear, and it was discussed in many works, for example, in Refs. 32,33,60,61 BIC and diffraction orders A BIC can be formed from a leaky resonance if the scattering amplitudes to all the channels turn to zero, i.e. in the case of decoupling from all the open scattering channels. This can be achieved by varying parameters of the system. In this case, the BIC formation is possible only if the number of the adjusted parameters is larger than the number of the scattering channels. For finite-size structures, the number of scattering channels is infinite, and the existence of BICs in such systems is prohibited by the non-existence theorem. 26 Until recently, it was believed that the only exception is the structures surrounded by a completely opaque shell providing the decoupling of the internal resonances from the outside radiation continuum, which in quantum mechanics corresponds to an infinite potential barrier, in acoustics to hard-wall boundaries, and in optics to perfectly conducting walls or epsilon-near-zero barriers. 62,63 One more exception from the 'non-existence' theorem was found in, 64 where the authors revealed that finite-size solid acoustic resonators can support genuine BICs completely localized inside the resonator. Despite possible exceptions from the 'nonexistence' theorem, the BICs are usually formedin structures with a finite number of scattering channels. A typical example of such a system is a resonator coupled to one or several waveguide modes 20,65,66 or infinite photonic structures periodic in one or two directions. 21,46,49,67,68 Now, let us consider in more detail the mechanism of the BIC formation in periodic structures, by the example of a dielectric grating with a period L shown in Fig. 8(a). The electric field, being a Bloch function, can be written as Here, k b is the Bloch wavenumber, k y is the wavenumber component along the y-axis (the direction of the translational symmetry), and n is the index of the photonic band. Periodic function u n,k b (x, z) can be expanded into the Fourier series Here, s is an integer. Each term in this series corresponds to a diffraction channel, which could be open or closed. Outside the structure, the expansion coefficients describe a plane wave as If K s is real, then the diffraction channel is open and c n,s is the complex amplitude of the wave outgoing via the s-th diffraction channel. If K s is imaginary, then the diffraction channel is closed and c n,s is the complex amplitude of the near field.
For the state in the center of Brillouin zone (k b = 0) one can write Therefore, for BIC in the Γ-point, the z-averaged field should be equal to zero. If the unit cell of the grating is symmetric with respect to the z → −z transformation, then the eigenstates in the Γ-point can be even or odd functions of z. For odd functions, their z-averaged value is zero. Therefore, all such states are BICs. These BICs are called symmetry-protected, which means that the cancellation of radiation is protected by the symmetry of the structure. It is noteworthy that symmetryprotected BICs emerge in both low-contract and high-contrast photonic structures. 61,69,70 As opposed to symmetry-protected BIC, there are the so-called Friedrich-Wintgen BICs, also known as accidental or tunable BICs, for which c 0 (x) nullifies not because of the symmetry reasons but due to the tuning of parameters of the system. 46,49 In the general case, the Fourier amplitude c 0 is a complex vector function of the grating parameteres. Thus, for BIC formation, both its real and imaginary parts should be zero. However, one can show that for BIC, the components of c 0 can be chosen real everywhere in the k-space if the structure possesses time-reversal symmetry ε * (r) = ε(r), inversion symmetry ε(−r) = ε(r), and up-down mirror symmetry. 46,49 However, a symmetry-protected BIC does not require up-down mirror symmetry, and therefore it can be implemented in structures with substrate. This symmetry is necessary for observation of a tunable (off-Γ) BIC. Thus, as it was mentioned above, in experiments the samples are usually immersed in an optical liquid that is index-matched to the substrate, or suspended structures are used. 49,71 In order to create a BIC, it is not necessary to work in a region with only one open diffraction channel. If several diffraction channels are open, one needs to tune several parameters of the system to achieve a BIC. It is hard to implement in practice, but there are theoretical works showing it is possible. 72,73 BICs and the multipole expansion The appearance of BICs in periodic photonic structures can be explained in terms of the multipole expansion. The multipole approach is based on the formalism of the vector spherical harmonics (multipoles) representing a complete basis set of orthogonal vector functions that are the solutions of the vector Helmholtz equation. 74 Usually, these functions are used in the Mie theory as independent scattering channels. They give the connection between the directivity diagram of the outgoing radiation and the polarization currents induced inside a scatterer by the incident wave. 75 A number of beautiful scattering phenomena in the area of electromagnetic waves were explained within the multipole approach including the anapole effect, 76,77 the Kerker effect, [78][79][80] and superscattering. [81][82][83][84] The multipole approach can also enable a deeper understanding of the physics of BICs in periodic structures. 85,86 Following the book by Bohren and Huffman, 87 we define the vector spherical harmonics M p m and N p m as Here, {r, θ, ϕ} are spherical coordinates, P m is an associated Legendre polynomial, z Here, Y j is the Fourier transform of W j , k b is the Bloch wavenumber, K s is the reciprocal lattice vector, D j is the amplitude corresponding to Y j . The coefficient of the expansion of the far field D j is directly connected with the expansion coefficient of the polarization inside the unit cell. 86 Equation (14) shows that the far field of the structure is defined by multipole content of the unit i.e. to the Γ-point. The only allowed direction of the diffraction is normal to the metasurface, however, dipoles do not radiate in the direction of their axis. Therefore, this mode does not radiate, being a symmetry-protected BIC. 88 It is shown in Ref. 86  To find all the multipoles allowed for a certain mode, one can address to the group theory.
Indeed, the eigenmodes are transformed by irreducible representations of the structure's symmetry Out of the Γ point, the irreducible representation changes. Three main multipoles contributing to the symmetry-protected BIC in the considered example are shown in Fig. 9(c) (left panel). As we mentioned previously, for the accidental BIC, the sum of the vector spherical harmonics (Eq. (14)) is nullified in the direction of the open diffraction channel. The multipole content of the accidental BIC is shown in the third panel in Fig. 9(c).
One can show that in the square lattice (D 4h , D 4v ) all singlet states in the Γ-point with the frequency ω/c < 2πc/d are symmetry-protected BICs, while the bright modes are doubly degenerate. 86 However, in the triangular lattice (D 6h , D 6v ), there are two two-dimensional representations. One of them does not contain the multipoles with m = 1. Therefore, such structures can host doubly-degenerate symmetry-protected BICs. 92 Recently, Overvig and co-authors presented a quite detailed catalogue of the selection rules for symmetry-protected BICs in two-dimensional photonic crystal structures. 93 If the metasurface consists of meta-atoms characterized by a single multipole (dipole, quadrupole, octupole, etc), then the position of the accidental BIC in the k-space is determined by the direction of the nodal line of the multipole. Of course, the multipole approach is more natural for the metasurfaces consisting of resonant meta-atoms, but strictly speaking, it can be applied fairly for the photonic crystal slabs in the case of low-contrast materials or when the filling factor is low.
As an example, Fig. 9(d) shows the field distribution for two modes of the same symmetry in a metasurface and a photonic crystal slab with D 4h symmetry.

BICs and topological charges
Usually, BICs are robust to the change of some parameters of the system -they do not disappear, in the reciprocal space.
According to the Bloch theorem, the electric field of a mode in photonic crystal slab [see Fig. 10(a)] can be written (similar to the case of 1D grating considered earlier) can be written as . The topological charge q characterizing the BIC can be introduced as follows Here φ(k) = arg [c x (k) + ic y (k)], and C is a simple path in k-space that goes around the BIC in the counter-clockwise direction. The polarization vector has to come back to itself after the closed loop, so the overall angle change must be an integer multiple of 2π, and, thus, q must be an integer. For the considered structure, the symmetry-protected BIC carries a charge of q = +1, see For example, a BIC can be destroyed through the annihilation when two or several topological charges with total zero charge collide. A BIC with an integer topological charge can decay into several BICs with integer topological charges or into circularly polarized states with half-integer charges. 46,95,98,99 Some examples of the topological charges' migration and decay will be considered in the section 'Dielectric Gratings'. The topological charge carried by a BIC can be used for the generation of optical vortex beams. [100][101][102] In Ref., 96 the Authors developed quite a general approach for defining topological charge of BIC in a wide class of systems and then applied the developed formalism to linear periodic chains of coupled resonators [ Fig. 11(a)]. The Authors introduced a complex function W (ω, k z ) (quasimode coupling strength) characterizing the coupling efficiency between the scattering channel and the resonant states of the system. Here, k z is the wavenumber of the incident wave and ω is its frequency. Function W (ω, k z ) can be interpreted as a projection of the incident wave onto the eigenmode, or which is the same, the projection of the resonant state onto the wave outgoing through the scattering channel. Obviously, W (ω, k z ) = 0 for BIC, because the BIC appears in ω − k z space exactly at the crossing of lines corresponding to Re{W (ω, k z )} = f (ω, k z ) = 0 and Im{W (ω, k z )} = g(ω, k z ) = 0. The topological charge q can be introduced as can be equal to ±1. By variation of the waveguide layer thickness in region B, the Authors showed that the BICs can move in w − θ space and even annihilate, if two BICs with opposite charges collide. We should note that instead of varying w at a fixed frequency ω, one can vary ω at a fixed ridge width w. In this case, the results will be completely the same. To conclude, we should add that the topological charge can be introduced in many ways, but not all of the introduced charges will have a clear physical meaning, and thus, they cannot be directly measured in the experiment.

Losses and Q factor of quasi-BICs
Losses limit the total Q-factor Q tot of BIC. In general, Q tot can be decomposed into partial contributions: Here, the radiative losses account for surface roughness (Q surf ), structural disorder (Q str ), diffraction losses due to finite size of the sample (Q size ), and diffraction into high-index substrate (Q subs ).
The non-radiative losses Q abs include all types of absorption (fundamental absorption, free-carrier absorption, multiphoton absorption, etc). The loss mechanisms in a periodic photonic structure are shown schemtically in Fig. 12(a). The statement that BICs have infinite Q factor is quite common in the literature, but it is not completely correct. Only the radiative Q factor diverges for BICs (Q rad → ∞), as they are completely decoupled from the radiative continuum, but the total Q factor can remain finite. For example, BICs with finite Q tot can be observed in plasmonic structures, where the losses are essential. [107][108][109] Strictly speaking, BICs are a mathematical idealization, and unavoidable radiative losses turn them into quasi-BICs, which manifest themselves in the scattering spectra as sharp Fano resonances. 110 Quasi-BICs are important for applications, as they are still strongly localized and provide a giant enhancement of the incident field, and they can be excited by an external incident wave.
Usually, the efficient performance of photonic structures requires a critical coupling of the eigenmode to the incident field, which is achieved at the condition Q abs = Q rad . [111][112][113][114][115] The absorption Figure 12: (a) Mechanisms of losses in a periodic photonic structure. They include surface roughness, radiative loss due to structural disorder, diffraction losses due to finite size of the sample, and diffraction to high-index substrate. (b) A symmetry-protected BIC in a grating etched on the silicon-on-insulator wafer. Diffraction that is normal to the surface plane is prohibited by the symmetry of the BIC, while diffraction into the substrate is allowed.
rate in semiconductors, for example, can be controlled in the visible, infrared, and terahertz ranges via free electrons induced by an external optical pulse. [116][117][118] The radiative losses of quasi-BICs in periodic photonic structures can be controlled by the size of the sample, 67,119,120 the angle of incidence 49 or asymmetry of the unit cell. 110 It seems that the latter is the most precise and powerful technique, and thus, more suitable for applications requiring normal incidence excitation. The optical properties of the asymmetric photonic structures with quasi-BICs and their applications are discussed in the following Sections. Another mechanism of the dynamic all-optical control over the radiative losses of BICs was proposed in Ref. 121 It was shown that the Kerr-type nonlinearity can result in radiative losses, which appear due to the coupling between the symmetry-protected BIC and a bright mode of the system. Thus, the radiative losses of BICs can be controlled by the intensity of the incident light. A similar mechanism of the radiative loss control was proposed for the implementation of optical memory based on BICs. 122,123 One of the loss mechanisms is the diffraction into the substrate with a high refractive index.
This mechanism is crucial, for example, for photonic structures fabricated from a silicon-oninsulator wafer. 124 In this case, the refractive index of the substrate is higher than the effective refractive index of the BIC, thus, there is a diffraction into the substrate [see Fig. 12(b)]. It is important that despite the presence of the substrate with a high refractive index, the zero-order diffraction into the substrate is closed for symmetry-protected BICs, as the substrate does not break Recently, a symmetry-protected BIC in a photonic crystal slab with total Q factor of about 10 6 was demonstrated experimentally. 133 The radiation from the edges of the structure was suppressed by surrounding the sample with a photonic crystal with another lattice constant. Thus, the frequency of BIC was in the bandgap of the surrounding photonic crystal.
Another source of radiative losses is the radiation induced by fabrication imperfections or structural disorder. Such losses are a common problem for high-Q photonic structures. [134][135][136] Structural fluctuations are much more difficult to control than, for example, the sample size. Therefore, the radiation losses due to structural fluctuation and fabrication imperfections usually are the main loss mechanism limiting the Q factor of BIC. It is well-known that the structural disorder in periodic photonic structures drastically affects their optical properties, resulting in non-trivial Fano resonance evolution, light localization, coherent back-scattering, etc . [137][138][139][140][141] The disorder effects are most essential in self-assembled and natural photonic structures. [142][143][144] According to the general theory, the Q factor of BIC due to the structural fluctuations behaves as Q −1 str ∝ σ 2 , where σ is the disorder amplitude. 145 In Ref., 146  The Anderson localization effect in structures with BICs is discussed in Ref. 148 In this work, the Authors considered the structure similar to that considered by Plotnik et al. [see Fig. 2(c)] 31 and analyzed in detail the stability of the BIC against disorder. They showed that due to the non-trivial interplay between the BIC and disorder-induced localized states, the entangled biphoton survives after averaging over all the disorder configurations.
The Authors of Ref. 48 proposed the idea how to increase the resistance of BICs to structural disorder. They experimentally demonstrated a BIC with the total Q factor about 5 × 10 5 . The main idea is to merge several BICs in the k-space in the vicinity of the Γ-point. Later, merged BICs, or super-BICs, in photonic crystal slabs were used to achieve record-low threshold lasing. 120 As we mentioned above, the radiation of BIC due to structural disorder occurs via the fractional-order Bloch waves near the Γ-point. Thus, in the proposed design, the rescattering takes place mainly between the extremely high-Q states, making the BIC immune against the structural disorder.

BICs in photonic structures of various dimensions Gratings
In this section, we overview the BICs in 1D dielectric gratings -photonic structures of finite thickness periodic in one direction and having a translational symmetry in other direction. The study of such structures has a long history, which began in 1887 from Lord Rayleigh's works. 149 The waveguide gratings were intensively studied since the middle of the 20-th century, and they found numerous applications in distributed-feedback lasers, laser mirrors, bandpass filters, wavelength demultiplexers, polarizers, chemical and bio-sensors, and other optic and optoelectronic devices.
An interested reader may consult numerous reviews. [150][151][152][153][154] A pronounced twist in the physics of optical gratings was related to the development of nanotechnology, namely, photonic crystals, metasurfaces, flat optics, and high-contrast gratings. 155 Due to simple geometry, the BICs in dielectric gratings can be described easily using the Fourier modal method (FMM, also called rigorous coupled-wave analysis, RCWA) 156 or the true modal method (TMM) 157 designed to be particularly efficient for gratings, resonant state expansion, 158,159 guided mode expansion, 160 multiple scattering theory (Korringa-Kohn-Rostoker method), 161 and others. FMM and TMM methods supplemented with the S-matrix technique provide a powerful tool for analysis of complex multilayer and photonic crystal structures. 53,162 Along with the symmetry-protected BICs which we discussed previously, the gratings allow implementing another type of BICs, usually called Fabry-Perot BIC. It is well-known that in the vicinity of the optical resonances of non-absorbing gratings, the transmission becomes zero [see  1) and (2), assuming that ω 1 = ω 2 and γ 1 = γ 2 . The Fabry-Perot BICs can appear at both normal k b = 0 and oblique k b = 0 incidence.
Actually, there is no need to use a double-layer structure to obtain a Fabry-Perot BIC. It can also appear in gratings with varying thickness. This is well-described by modal methods developed in Refs. [163][164][165][166][167] In these approaches, Bloch modes are used that are propagating eigenmodes of the infinitely thick grating, similar to plane waves in free space. 163 A plane interface between a homogeneous medium and a photonic crystal couples the modes with each other (upon reflection) and with plane waves of the surrounding space or substrate (upon transmission), as illustrated in Lyndin, but without any reference to BICs. 170 A further characterization of BICs in 1D gratings can be done in terms of S-matrix poles. 53,[171][172][173] The collapse of the Fano resonance for BICs means merging the zeros and poles of the S-matrix. 174 This condition can used to find particular parameters at which the Fabry-Perot BIC appears. 97 Blanchard and co-authors 175  As we discussed previously, BICs in periodic structures can be considered as a polarization vortex with a certain topological charge, and this vortex can demonstrate non-trivial dynamics in k-space with the variation of the system's parameters, including annihilation, merging, and decay.
All these effects were demonstrated in the Ref. 178  The value of topological charge determines the asymptotic dependence of the radiative Q factor on the Bloch wavenumber. Due to the C 2 symmetry, one-dimensional periodic structures possess BICs with topological charges q = 0 or q = ±1. 46 The Q factor of an isolated BIC with |q| = 1 48,178,183 For example, the Q factor of the symmetry-protected BIC decreases as k −2 . Approaching the point of the annihilation, the dependence changes to Q ∼ 1/(k − k BIC ) 2 (k + k BIC ) 2 for both accidental in-Γ and symmetry-protected BICs. The inverse fourth-power relation indicates that the quality factor can be very large, even when |k − k BIC | is not small. 183 However, if the system parameters are detuned from this regime, the Q factor of the symmetry-protected BICs decreases as Q ∼ 1/k 2 again. These asymptotic dependencies are extremely important, as they determine the radiative losses of quasi-BICs in finite-size structures Refs. 69,96,99 Owing to the axial symmetry and periodicity of the structure along the z-direction, the solution can be written in the following form where ϕ is an azimuthal angle, m is the azimuthal quantum number, k z is the Bloch wavenumber, and u m,kz is a periodic function of z. In the special case of m = 0, the solutions of Maxwell's equations in cylindrical coordinates can always be divided into T E and T M polarizations. 185 It is worth mentioning that in contrast to uncorrugated waveguides, in our case, the solutions do not split into two independent polarizations for k z = 0, as k z is the quasi-wavenumber which is determined  186 They also demonstrated that BICs with m = 0 and k z = 0 can be created by fine tuning of the chain parameters. As we mentioned previously, BICs in the periodic chain can be characterized by the topological charge, similar to BICs in 2D periodic structures. However, it is still unclear how to measure this charge experimentally.
BICs in a periodic chain were observed experimentally for the first time in the GHz frequency range, in an 1D array of coupled ceramic disks [see Fig. 15(a)]. 67 To observe the symmetryprotected BIC, two identical loop antennas were used as magnetic dipoles: they were placed coaxially with the chain and connected to ports of a vector network analyzer. The measured and calculated transmission spectra are shown in Fig. 15(a). The last peak in the series corresponds to a quasi-BIC, which turns into a genuine BIC in the infinite chain. It was shown that the total Q factor of the quasi-BIC grows quadratically with the increase of the number of periods and then saturates at the level Q = 4000 due to the material absorption. An accidental BIC (m = 0, k b = 0) was observed experimentally in a similar system in Ref. 129 The linear growth of Q factor with the number of disks was demonstrated for the observed accidental off-Γ BIC. On the contrary, the radiative Q factor of the accidental in-Γ BIC follows the scaling law as Q ∼ N 3 with N being the number of scatterers in the array. 125 The first experimental observation of a BIC in an axially symmetric periodic structure in the optical range was presented in 2019 . 184 The Authors fabricated a silicon nanowire with a periodic grating at the surface as shown in Fig. 15(b). The surface grating forms a structure similar to a periodic array of disks placed coaxially on a core. Since the nanowire was illuminated by a plane wave at normal incidence, quasi-BICs with different m were excited. The Authors analyzed both theoretically and experimentally how the appearance of a BIC depends on illumination and geometry of cylindrical nanowire. The BIC-induced light confinement in such structures can be used to substantially enhance absorption, which can be applied in photodetector or photovoltaic devices incorporating a p-i-n diode design and based on nanowires with superlattices. 187 Nanowires with a periodic superlattice, and arrays of disks have advantages over the array of spheres due to a larger number of degrees of freedom. By independent adjustment of the period, height, and radius of the disks, one can provide precise mode engineering, and obtain several BICs with different orbital angular momenta and Bloch vectors. Such linear chains supporting BICs can be used as a compact source of optical beams with angular momentum.

2D periodicity. Photonic crystal slabs
In this section, we review the main properties of BICs in photonic crystal slabs (waveguides) -2D dielectric photonic structures that have a band gap for the waves propagating in the waveguide plane and that use index guiding to confine light in the third dimension. Since 1990s, such structures are considered easier for fabrication than photonic crystals with full three-dimensional band gaps, while retaining many of the latter's desirable properties. 188,189 Compared to 1D structures, photonic crystal slabs provide a broader variety of designs for unit cells, different types of their arrangement and, consequently, more degrees of freedom for flexible control over their optical properties. 160,190 Due to this fact, the variety of BICs in such structures is more extensive than for one-dimensional gratings or chains, including the possibility of BICs with high-order (≥ 2) topological charges 46 and robustness to disorder. 48 The first systematic study of BICs in photonic crystal slabs was performed in 2013 49 for a Si 3 N 4 slab patterned with a periodic arrangement of circular holes as shown in Fig. 16, left panel. The slab was grown on a SiO 2 /Si substrate and immersed in a liquid, index-matched to silica, which was important to keep the up-down reflection symmetry required for observation of accidental BICs. The band structure for TM-like modes along the Γ-X direction is shown in Fig. 16, central panel. The structure supports a symmetry-protected BIC at the normal incidence and an accidental BIC at a non-zero Bloch vector corresponding to an oblique angle of incidence around 35 degrees, as can be seen in Fig. 16, right panel. The red crosses show the values of the radiative Q factor calculated from the total Q factor, which was extracted from the measured reflectivity spectra by assuming the value of non-radiative Q factor equal to 10 4 . The achieved value of radiative Q factor for the accidental BIC reaches 10 6 ; however, the value of the total Q factor remains limited to 10 4 because of parasitic and non-radiative losses. This work was the first direct evidence of true accidental BICs measured experimentally, to the best of our knowledge.
The physics and origins of accidental BICs were analytically described in Ref. 191 within a coupled-wave theory (CWT) developed specially for description of response for 2D photonic crys-tal slabs . 192 Within this approach, the electromagnetic fields of eigenmodes of a photonic system are expanded into guided modes and radiative modes of a uniform waveguide, which are used as basis functions. Due to the interaction with the lattice, the guided modes also have radiative losses.
Thus, both the guided and leaky modes can contribute to the same radiation channels (diffraction channels) due to the interaction with the periodic potential of the structure. In the work, 191 it was shown that accidental BICs are formed due to the destructive interference of all the basis functions: both the guided and leaky ones. Two reasons of destructive interference were indicated.
First, the contributions from equivalent directions in the reciprocal space can result in spontaneous appearance of an additional symmetry and, thus, to the formation of an accidental BIC. An example of such a BIC is presented in the work. 49 Second, it was shown that accidental BICs do not necessarily occur exactly at a high-symmetry point in the reciprocal space. The reason is, in contrast to symmetry-protected BICs, for accidental BICs, the guided basis modes contribute to the open radiation channels of the photonic structure with different weights and, therefore, after the radiation is cancelled at the symmetry point, some residual radiation can still remain. In other words, the point corresponding to a BIC in the reciprocal space does not always have a high symmetry, but is located near such a point. This is typical for the Friedrich-Wintgen mechanism. 7,193 Furthermore, it was shown analytically that the locations of accidental BICs can be shifted by changing various parameters, such as the cladding permittivity or geometrical sizes of the waveguide. It is worth mentioning that in other works, analytical considerations demonstrate that the energy of symmetry-protected BICs is confined dominantly in the closed diffraction channels of the ± orders.
The moderate value of the total Q factor of 10 4 in Ref. 49 is related to the out-of-plane losses originating in realistic samples due to finite size of samples and fabrication imperfections. Recently, a new approach was suggested, which allows suppressing the out-of-plane losses, using the topological nature of BICs, namely, by merging several BICs at the Γ point in the k-space. 48,194 In the paper, the Authors considered a 2D dielectric photonic crystal membrane suspended in air. The fundamental TE-band of the membrane supported one symmetry-protected BIC at the Γ point and eight accidental BICs located symmetrically around the Γ point. The tunability of accidental BICs allows moving them away from an off-Γ position towards the center of the Brillouin zone. The experimentally measured value of 5 × 10 5 was demonstrated for the total Q factor in the merging regime. For a single isolated BIC with a ±1 charge, the Q factor of standard symmetry-protected BICs decays quadratically (Q ∼ 1/k 2 ) with respect to the distance k from the Γ point. However, it was shown that the scaling changes to Q ∼ 1/k 6 in the configuration in which all nine BICs merge. Later, it was shown that for finite-size samples, the highest Q factor can be achieved not for the complete merging condition, but in the pre-merging regime, when the accidental BICs lie on the circle of a small radius in the k-space. 120 One of the remarkable properties of 2D periodic photonic structures is that a BIC can exist even in chiral samples without the in-plane inversion symmetry. 195 In general, a BIC in the subdiffractive regime can be achieved via cancellation of radiation into both polarization channels.
In Ref., 195 the Authors showed that by using a cross-shaped periodic photonic structure without a second-order symmetry axis, it is possible to cancel radiation amplitudes for both polarizations: one polarization channel is suppressed due to the vertical mirror plane, and another one parametrically, tuning the geometry of the unit cell. This mechanism of BIC formation is very similar to the one considered by Bulgakov and Sadreev for obtaining partially symmetry-protected BICs in a chain of dielectric spheres. 186

Individual subwavelength resonators
For individual subwavelength resonators, genuine nonradiative states require extreme values of permittivity, tending toward infinity or zero 62,198,199 or imitating periodic boundary conditions with metallic waveguides. 65,200,201 In realistic individual resonators, there is always an infinite number of radiation channels, which limits the Q factor substantially. However, the concept of quasi-BICs allows reaching almost nonradiative states for individual dielectric resonators. Using the parametertuning approach originally developed by Friedrich and Wintgen 7 and later used for some extended geometries, 202 one can create high-Q quasi-BICs in compact geometries at subwavelength scales. The concept of the mentioned supercavity mode is shown in Fig. 17 indices is also possible, but it leads to lower Q factors due to lower symmetry of the field profile.
The first experimental observation of quasi-BICs was carried out quite recently in microwave 196 and near-IR 197,206 ranges. Figure 17(b) shows the measured spectral map of the reflection coefficient 1 − |S 11 | for a single ceramic disk with permittivity of 44.8 from Ref. 196 The excitation was performed in the near field with a loop antenna, which selectively excites the modes with zero azimuthal index. The spectra reveal four avoided resonance crossings corresponding to formation of quasi-BICs. The radiative Q factor of quasi-BICs reaches 200000, but due to the absorption in the ceramic material, the maximal measured unloaded total Q factor is 12500. The observation of supercavity modes in near-IR was carried out for isolated AlGaAs nanodisks on engineered threelayer substrate with a reflective layer of ITO in the middle. 197,206 The SEM image of individual nanodisks is shown in the upper panel of Fig. 17(c). The measured spectra and extracted Q factor for a range of the disk diameters is shown in the lower panel of Fig. 17(c). The maximum measured Q factor is approximately 180 for a disk with 930 nm diameter. The spectra reveal that the lineshape is Fano-like and changes its asymmetry in the vicinity of the quasi-BIC, as was predicted theoretically for a single disk in vacuum. To match the mode symmetry and excitation, a tightly focused azimuthally polarized Gaussian beam was used.
Later, several generalizations were suggested for quasi-BICs in textcolorblacknon-periodic finite resonators. For example, two coaxial disks were suggested as a structure supporting quasi-BICs with much higher values of Q factor than that of a single disk. 207 Additionally, Bragg reflectors were suggested to further increase the Q factor of an individual quasi-BIC disk. 208 The multipolar classification of eigenmodes in single resonators of different shapes was suggested later, based on the symmetry group theory. 209 These results lead to prediction of quasi-BICs in dielectric spheroids, 210 triangular prisms, 209 bar-shaped microcavities 211 and resonators of arbitrary shapes. 212

Asymmetric metasurfaces and quasi-BICs
Recently, it was demonstrated that broken-symmetry dielectric metasurfaces demonstrate sharp peaks in the normal incidence transmission spectra, which are associated with quasi-BICs excitation. 110 It was shown that the in-plane asymmetry induces an imbalance of the interference between counter-propagating leaky waves, resulting in the formation of a quasi-BIC.This effect was observed in a variety of metasurfaces in different frequency ranges and was connected to electromagnetic field-induced transparency, 213 tunable high-Q resonances, 214 trapped-mode resonances, 215 broken-symmetry Fano metasurfaces 216,217 and dark modes. 218 Figure 18(a) shows the scattering of light by an asymmetric metasurface. The radiative Q factor of quasi-BICs in asymmetric metasurfaces follows the typical inverse quadratic dependence on the meta-atom asymmetry parameter α, 110 as shown in Fig. 18(b), left panel. In the regime of transmission (or reflection), the quasi-BICs manifest themselves as sharp asymmetric Fano resonances whose width and depth decreases with the decrease of α, as shown in Fig. 18(b), right panel. The total Q factor of the quasi-BIC mode is limited by other losses (see Section 'Losses and Q factor of quasi-BICs'). Due to parasitic effect of other losses, the maximal field enhancement can be   281 and low-threshold lasing. 88 We will consider several applications in more detail. Passive photonic structures supporting BICs recently were used to increase the efficiency of biological objects' detection. In the work, 223 Tittl et al. implemented a nanophotonic sensor of biomolecules in the mid-IR range, based on the reflection from a dielectric matrix of metasurfaces supporting BICs. In Fig. 19(a) used for angle multiplexing of spectra, which also allowed a convenient distinguishing of absorption spectra of various biomolecules. Figure 19(b) shows a metasurface multiplexing scheme and reflection spectra from the structure. In another series of works by Romano et al., 131,244,282 the sensitivity of sensors based on photonic crystal waveguides with BICs was studied to changes in the refractive index of the external medium, and high sensitivity was experimentally shown to be achieved due to BICs. In a recent work by Jahani et al, 248 BIC-based silicon metasurfaces with a complex unit cell were developed for the detection of extracellular vesicles of cancer cells.
Photonic structures with BICs are widely used in active photonics, including for laser generation. Figure 20(a) shows the output power spectra depending on the wavelength and pump power for a 16 × 16 metasurface supporting BICs in the near-IR range from. 71 In this work, the Authors used a tunable accidental BIC with a wavelength of 1550 nm. The right panel of Fig. 20(a) shows the dependence of the output power on the pump power at resonance. As the pump power is increased to 60 µW, a distinct peak is observed in the emitted power spectra at the BIC wavelength.
The inset shows a SEM image of the structure. In a recent paper, 120 lasing from the so-called shows a SEM image of an InGaAsP photonic crystal waveguide and the measured far-field profile for a mode of the super-BIC laser. Laser generation in the structure is achieved through the use of quantum dots in the layer. As shown in Fig. 20(b), the spot in the Fourier space has a small angular divergence, which is a hallmark of strongly confined modes. The right panel of Fig. 20(b) shows the lasing threshold normalized to the pumping area (5.4 µm in size) as a function of the lattice parameter of the structure. The super-BIC state is realized at a lattice parameter of about 573 nm: in this regime, the lasing threshold is the lowest and has a record low value among all BIC lasers and lasers based on topologically protected states. 120 In the work of Huang et al., 100  BICs and quasi-BICs are studied most actively in nonlinear optics and photonics applications, mainly in order to enhance nonlinear generation and to observe the self-action of an exciting pulse.
In a series of works by K. Koshelev et al., 29,115 optical second-and third-harmonic generation was studied in nonlinear dielectric metasurfaces with asymmetric unit cell, supporting quasi-BICs with Q factor depending on the asymmetry. In particular, in the work, 115 the fabricated metasurface had a low Q factor of approximately 160 due to the presence of strong surface roughness. In this work, it was shown theoretically and experimentally that for such non-perfect photonic structures, the highest efficiency of harmonic generation is achieved not at the highest Q factor, but in the critical coupling regime, when the radiative Q factor is equal to the Q factor related to all the other Figure 21: (a) The spectrum of the signal from the 3rd to 11th harmonics from an asymmetric silicon metasurface with BICs, and the dependence of the harmonics' power on the pump pulse power. Based on G. Zograf et al. 259 (b) Statistics of the measured Q factors of the BICs for all the fabricated silicon metasurfaces with different sizes. The spectra of the second and third harmonics for a metasurface and an unstructured film. Based on J. Liu et al. 119 (c) Scheme of ultrafast selfaction of a pulse through a BIC. The spectrum of the third harmonic depending on the pump power. Carrier concentration and induced field amplitude inside the nanostructure at a pump intensity of 240 GW/cm 2 . Based on the work of I. Sinev and colleagues. 260 types of losses. It was also shown in the work that the critical coupling regime can be achieved by changing the asymmetry parameter of the meta-atom, which should be taken into account in the design of resonant nonlinear metasurfaces. G. Zograf et al. 259 studied the generation of higherorder odd harmonics (3,5,7,9,11) from an asymmetric silicon metasurface, shown schematically in the left panel of Fig. 21(a). The right panel of Fig. 21(a) shows the spectra of optical harmonics in the regime of excitation with 100 fs pulses and the dependence of the output power on the pump power for an optimized metasurface, using the critical coupling criterion. The dependence of the output power from pump powers in the range of 0.03 − 0.3 TW/cm 2 is defined by the same law, regardless of the harmonic number, which allows a conclusion that the structure operates in a nonperturbative regime. In this paper, the transition to the nonperturbative regime is explained through the mechanism of free-carrier generation in silicon due to multiphoton absorption. J.Liu et al. 119 studied asymmetric silicon metasurfaces with T-shaped meta-atoms, shown in Fig. 21(b). A record-high Q factor of a BIC of over 20000 was obtained in this work for a structure with 26×26 periods, achieved due to the high quality of nanolithography and special features of the structure design. In this work, the spectra of the optical second-and third-harmonic signals were measured, shown in the right panel of Fig. 21(b), and at the BIC wavelength, the harmonic signal increases by several times. I. Sinev et al. 260 studied an asymmetric silicon metasurface for third-harmonic generation at high pump power intensities of the order of 0.3 TW/cm 2 . It was demonstrated in the work that at a pump intensity higher than 240 GW/cm 2 , self-action of a pulse occurs due to the generation of free charge carriers in the multiphoton absorption mechanism, schematically shown in the left panel of Fig. 21(c). The calculated induced field amplitude and charge carrier concentration at a peak intensity of 240 GW/cm 2 reached 3 GW/m and 10 18 cm −3 , respectively, as shown in the right panel of Fig. 21(c).

Conclusion
In conclusion, bound states in the continuum have a long history in optics and radiophysics. Today, this area reaches a new level due to the fast development of the physics of metasurfaces, 2D materials, nonlinear nanophotonics, flat optics, and related directions. BICs can be observed in a wide variety of photonic structures, including metasurfaces, photonic crystal slabs, high-contrast gratings, corrugated planar waveguides and fibers, ridge waveguides, linear chains, and many others. A relatively recent discovery is a new class of BICs in Bragg resonators with an anisotropic defect layer, proposed and experimentally studied in the works. [283][284][285][286] Today, the physics of BICs is developing in acoustics. 20,[287][288][289] BICs are an illustrative example of how an idea suggested in one area of physics a century ago today affects many other fields and is already used in multiple practical applications.