We study the effect of structural disorder on the transition from the bound states in the continuum (BICs) to quasi-BICs by the example of the periodic photonic structure composed of two layers of parallel dielectric rods. We uncover the specificity in the robustness of the symmetry-protected and accidental BICs against various types of structural disorder. We analyze how the spatial mode localization induced by the structural disorder results in an effective reduction of the system length and limits the Q factor of quasi-BICs. Our results are essential for the practical implementation of BICs especially in natural and self-assembled photonic structures, where the structural disorder plays a crucial role.
Bound states in the continuum (BICs) are the specific solutions of the wave equation which are spatially localized despite the fact that their frequencies lay in the radiation continuum. Initially, BICs were proposed in quantum mechanics by von Neumann and Wigner . In the last two decades, BICs have been observed in many photonic structures such as photonic crystals, metasurfaces [2–4], chains of coupled resonators [5–8], waveguides [9–13], and even single subwavelength resonators supporting quasi-BIC in a form of supercavity modes [14–16]. A divergent radiative quality (Q) factor, strong spatial localization, and drastic enhancement of the incident field make BICs very prospective for many applications including lasers [17–25], optical filters [26–28], biological and chemical sensors [29–33], and nonlinear photonics [34–43].
A genuine BIC with an infinitely large radiative Q factor is a mathematical idealization. In real samples, the radiative Q factor becomes finite due to roughness, diffraction on the edges and into the substrate, and other imperfections inevitably appearing during the fabrication. Thus, in practice, a genuine BIC turns into a quasi-BIC (q-BIC) manifesting itself in the scattering spectra as a high-Q resonance [44–47]. The material absorption also results in a decrease in the total Q factor of q-BIC and makes it hardly recognizable in the experimental scattering spectra. The identification of the dominant loss mechanism is the crucial point allowing enhancing the performance of the photonic devices based on BICs.
Most of the systems supporting q-BIC are periodic (metasurfaces, grating, chains etc.). The structural disorder in such systems drastically affects their optical properties resulting in nontrivial Fano resonance evolution, light localization, and coherent back-scattering etc. [48–52]. The disorder effects are most essential in self-assembled and natural photonic structures [53–55]. Despite the important role of the structural disorder in periodic photonic structures, it is weakly studied in application to BICs and their transition to q-BICs. In particular, the coupled-wave theory (CWT) framework was developed to study the Q factor of symmetry-protected BIC in dielectric gratings with randomly perturbed filling factor and the element position . The radiation losses of symmetry-protected BICs were shown to decrease quadratically with the fluctuation amplitude. It was shown in Ref.  that disorder can results in the formation of BIC band in multichain and multilayer systems. The robustness of symmetry-protected BICs was partly analyzed in Ref.  in the framework of the Fano–Anderson model applied to the system of parallel coupled waveguide.
Here we provide a rigorous analysis of the Q factor of both symmetry-protected and accidental BICs in a two-layered periodic array of infinitely long dielectric rods accounting for the structural disorder in the position of the rods keeping them parallel, i.e. preserving the translation symmetry. In contrast to the Ref. , we uncover two concurrent radiation loss mechanisms due to a finite number of periods and scattering due to the structural disorder. We find the disorder amplitude up to which the vertical and horizontal fluctuations in the rod position contribute to the radiation losses independently. We demonstrate that accidental BIC is more resistant to the fluctuation of the position along the direction of periodicity rather than to the perturbation of distance between two layers of rods. Contrary, the symmetry-protected BIC shows greater sensitivity to the fluctuation of the distance between the layers. Also, we reveal the disorder-induced spatial localization of the q-BICs and prove that it limits the Q factor of q-BICs if the array of rods is longer than the localization length.
2 Results and discussions
2.1 Infinite array without structural disorder
We start the analysis from the consideration of the infinite periodic structure without structural disorder shown in Figure 1(a). It consists of two identical periodic arrays of infinitely long dielectric rods. The radius of the rods r = 0.5 cm and the period a = 3 cm. Thus, the unit cell of the structure contains two rods spaced apart by distant h. For further analysis we will use permittivity of the rods ɛ = 2.1 that corresponds to low-loss polymers, for example, Teflon in the frequency range 0.1–10 GHz [59, 60]. The advantage of such low-refractive-index materials is that the electromagnetic field is mostly localized outside the rods, i.e. in air, where the losses are negligible. Thus, the material absorption is substantially suppressed substantially and we will not account for it in the further analysis. The considered two-layer system is convenient as it provides a simple way to precisely tune the accidental and symmetry-protected BICs [61–67].
According to the Bloch theorem, the electric field of eigenmodes can be written as . Here, k z is the wavenumber corresponding to the direction of the translation symmetry, k x is the Bloch wavenumber, s is the index of the band and we will omit for simplicity, is the periodic function of x. The BICs in such system appear only if either k z = 0 or k x = 0. Further, we will limit our analysis to the case of TE-polarized modes [E = (0, 0, E z )] with k z = 0. For TE modes with k z = 0, we can write . The periodic Bloch amplitude can be expanded into the Fourier series as follows:
where n is the index of the diffraction channel. At the frequencies above the light line (ω/c > |k x |), the mode leaks from the structure to the radiation continuum via the open diffraction channels. BIC appears when the leakage into all open diffraction channels is forbidden, i.e. the complex Fourier coefficients – the amplitudes of the outgoing waves – are zero. In the subwavelength regime λ < a, only the zeroth diffraction channel is open. Thus, the amplitude of the outgoing leaky wave is defined by the zeroth Fourier coefficient . By the definition, it is equal to the field averaged over the period, i.e. . For the structures having the time-reversal and π-rotational symmetries, denoted as , the coefficient becomes pure real for BICs [68–70]. At the Γ-point (the center of the Brillouin zone), is either odd or even function of x since the photonic structure is -invariant [71, 72]. Obviously, for an odd function the zero-order Fourier coefficient vanishes. In this case, the symmetry-protected BIC occurs – the coupling to the radiation continuum disappears due to the point-symmetry of the structure. In the case of the even mode, the spatial average may vanish not only due to the symmetry reasons but at specific values of the structure parameters such as period, the radius of rods, permittivity, interlayer distance resulting in the appearance of accidental BIC also called tunable BIC.
Figure 1(b) and (c) show the spectra of TE-modes and the electric field distribution for possible BICs appeared in the Γ-point. Mode 3 is odd with respect to transformation and, thus, it has a symmetry-protected BIC labeled as sp-BIC. Modes 1 and 2 are even with respect to transformation and, thus, they can have only accidental BICs. The dependence of the Q factor for all three modes in the Γ-point versus the distance between the layers h is shown in Figure 1(d). One can see from this figure that the BICs in the Γ-point of modes 1 and 2 appear only at the specific distances between the layers, while the Q factor of the sp-BIC is insensitive to h. The spectral position of the Γ-point for all three modes versus h is shown in Figure 6 in Appendix A. Modes 1 and 2 have different symmetries with respect to the y → −y transformation. Mode 1 is even (symmetric) and mode 2 is odd (anti-symmetric). Thus, we denote the corresponding BICs as s-BIC and a-BIC. These BICs can be considered as Fabry–Perot resonances formed due to the total reflection from the array of the rods [5, 73, 74].
2.2 Structural disorder and Q factor
In this section, we consider the structure of finite size, consisting of N periods, accounting for the structural disorder. In this case, the total Q factor is contributed by two parts
as we neglect absorption and the effect of surface roughness. Here, Q dis is responsible for the radiation due to the structural disorder; Q ord is responsible for the radiation due to the finite size of the structure and it is a function of N. Since the finite structure is not true periodic, the diffraction directions are smeared out resulting in additional diffraction losses. Therefore, a genuine BIC transforms into a resonant state (quasi-BIC) with finite Q factor [7, 75]. Recently, the Q factor of a symmetry-protected quasi-BIC was shown to grow as , see Ref. , while the Q factor of an accidental quasi-BIC emerging at the center of the Brillouin zone increases as , see Refs. [76, 77]. The dependence of Q ord on N calculated numerically for the structure under consideration is shown in Table 1.
|Number of periods, N||100||150||200|
As a next step, we account for the contribution of the structural disorder Q dis to the total quality factor Q tot. We will consider a model of the uncorrelated disorder introducing a fluctuation to the position of the rod keeping all the rods parallel. To gain deeper insight into how the structural disorder affects symmetry-protected and accidental BICs, we first consider the fluctuations in the rod positions along the x and y axes independently and then account for these shifts simultaneously. In both cases, the shift of the rod is given by δt = R ⋅ σ ⋅ a. Here, a is the period of the structure, σ is a fluctuation amplitude (disorder amplitude); R is a uniformly distributed random number in the range [−1; 1].
To analyze Q dis we use the finite element method calculating the complex eigenfrequencies and total Q factors of the disordered array of rods with the finite number of periods N. We collect the data from ensembles of 100 structures for each value of amplitude σ at the given number of periods N. Then we average the calculated Q factors over the ensemble as , where m is the number of ensembles.
Figure 2 exhibits decreasing of the Q factor for both symmetry-protected and accidental BICs (sp-BIC and s-BIC) with the growing amplitude of disorder σ. As it was mentioned above, the in-Γ accidental BICs (a-BIC and s-BIC) require particular values of the distance between layers h at which the round-trip phase shift accumulates to a value being multiple of 2π. Therefore, the accidental BICs are more sensitive to the fluctuations in rod position along the y axis rather than along the x axis [see Figure 2(a)]. Thus, when we account the shift of rods along both coordinates, we observe that the random shifts along the y-axis make the main contribution to the decreasing of the Q factor for the accidental BICs. In contrast, the symmetry-protected BIC (sp-BIC) exist at any distance between the layers (Figure 1). Therefore, the y-axis disorder does not affect much its Q factor. However, the random shifts of rods along the x-axis induce an additional polarization (dipole moment) along the x-axis resulting in radiative losses. Thus, the symmetry-protected BIC is most sensitive to the lateral shifts of meta-atoms inside the unit cell [Figure 2(b)].
Next, we calculate the dependence of the total Q factor (Q tot) of the accidental s-BIC on the disorder amplitude σ for three structures of different length (N = 100, 150, 200) accounting for random shifts of the rods only along the x-axis [Figure 3(a)]. One can see from this figure that the Q factor becomes sensitive to structural disorder only when the disorder amplitude reaches a threshold σ ∼ 0.01. For smaller σ, the Q factor is completely defined by the radiation losses due to the finite size of the structure. In contrast, for σ > 0.01 the main contribution to the radiation losses is made by the scattering events on the structural disorder resulting in quadratic decaying of the Q factor with σ. This quadratic dependence is observed for both types of the structural disorder (along the x and y axes) (see Figure 7 in Appendix A.1). The similar dependence of the Q dis factor was shown recently for eigenmodes of the periodic dielectric grating composed of supercells with fluctuations in grooves positions and filling factor .
Next, we examine whether the losses due to disorder along the x and y axes can be decomposed into two independent terms corresponding to the disorder along each axis separately
Usually, it is true only when the radiative losses caused due to both of these shifts are small. For a large amplitude of the disorder, the total losses might not be expanded into a sum of simple terms from each disorder obtained independently. We expect that some correlation between shifts along x and y appearing since the shape of a sole rod is conserved may impact on . To analyze the validity of this decomposition, we consider an auxiliary quantity characterizing the deviation from it
If the composition (3) is correct then Δ has to be zero. Figure 4 demonstrates the dependence of and Δ on an amplitude of disorder σ for the symmetry-protected (panel a) and accidental s-BIC (panel b). As it can be seen from the figure, decomposition (3) breaks for the accidental s-BIC for σ > 0.7. This can be caused by correlation between shifts for a sole rod which impact becomes comparable with separate contributions with σ increasing. In addition to correlation, large disorder amplitudes can result in the spatial localization of the mode at a scale less than the length of the structure, thus, reducing its effective length. For the symmetry-protected BIC, there are some deviations of Δ for large σ but they are substantially smaller than ones for the accidental s-BIC. We suppose that a symmetry of field distribution plays a crucial role causing stability of sp-BICs rather than accidental BICs.
2.3 Effect of spatial localization
As we discussed previously, the Q factor of all types of BICs depends on the size of the sample. From other hand it is known that disorder can result in spatial localization of wavefunctions (eigenmodes) that can be considered as effective reduction of the sample size [50, 78], , , . In this section, we analyze how the structural disorder results in spatial localization of BICs and how it affects their Q factors.
As we discussed in the previous section, there is a deviation from Eq. (3) for large disorder amplitudes for the accidental BIC (s-BIC). We suppose that this deviation can be related to the wave localization inherited to one-dimension systems with structural disorder [50, 78], which leads to an effective reduction of the structure length. We examine the amplitude of the electric field distribution along the structures of different length.
Figure 5 shows the envelop curves (mode profiles) of the accidental BIC (s-BIC) for the structure of various length (N = 100, 150, 200) for the fixed disorder amplitude σ = 0.085. The shaded areas of different colors correspond to different realization from the ensemble. The gray curves correspond to the envelope function of the accidental s-BIC in structures without disorder. To estimate the localization length, we superpose the envelope function of the accidental s-BIC in structure with N = 60 (dashed curve). One can see that for disorder amplitude σ = 0.085 the wavefunction is localized on around 60 periods for all x-, y- or xy-oriented disorder. From one hand, the structural disorder localizes the wavefunction inside the structure, thus, the losses due radiation from the ends of the chain are negligible. Therefore, this loss mechanism is essential as for regular structure. From other hand, the structural disorder reduces the effective length of the structure limiting the Q factor. The localization can be neglected for the most practical shorter structures; it has to be taken into account for larger, self-assembled, and natural structures, though.
In conclusion, we have uncovered how the uncorrelated structural disorder affects the symmetry-protected and accidental in-Γ BICs in the one-dimensional periodic structure composed of two layers of dielectric rods. Regardless of the direction along which the position of rods are perturbed, the Q factor of both symmetry-protected and accidental BICs decays quadratically with the amplitude of the disorder σ. Importantly, the symmetry-protected BIC is more resistant to the fluctuation of the distance between layers (along y) rather than the fluctuation of position along x. And vice versa, the accidental BIC is more robust against the fluctuation along the direction of periodicity (along x). We have shown that the fluctuations of position along the direction of periodicity and distance between the layers contribute to the losses independently for low disorder amplitudes. For the large values of disorder amplitude σ or long structures, the localization of electromagnetic energy suppresses the total Q factor due to the effective reduction of the system length. We anticipate our findings provide useful guidelines for practical implementation of resonators supporting bound states in the continuum in self-assembled and natural structures, where the role of structural disorder is the most relevant.
Funding source: Russian Science Foundation
Award Identifier / Grant number: (20-72-00152) (21-19-00677)
Funding source: BASIS foundation and the Council on Grants of the President of the Russian Federation
Award Identifier / Grant number: (MK-2224.2020.2)
Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.
Research funding: Simulations of eigenmodes of disordered structures were supported by the Russian Science Foundation (20-72-00152). Analysis of disorder strength on BIC and their robustness and effect of spatial localization was supported by Russian Science Foundation (21-19-00677). A.B. acknowledges the BASIS foundation and the Council on Grants of the President of the Russian Federation (MK-2224.2020.2).
Conflict of interest statement: The authors declare no conflicts of interest regarding this article.
Appendix A: BIC’s frequency in a periodic structure without disorder
Figure 6 shows how the frequencies of sp-, a-, and s-BICs depend on the distance between layers h of the infinite periodic structure. While the frequency of the sp-BIC increases slowly, the frequencies of the mode 1 and mode 2, at which the accidental BICs occur, exhibit complicate behavior.
A.1 Radiative losses due to structural disorder
Figure 7 shows that, for both symmetry-protected and accidental BICs, the losses due to various types of structural disorder exhibit quadratic dependence on the disorder amplitude.
 C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljačić, “Bound states in the continuum,” Nat. Rev. Mater., vol. 1, p. 16048, 2016. https://doi.org/10.1038/natrevmats.2016.48.Search in Google Scholar
 S. Han, L. Cong, Y. K. Srivastava, et al.., “All-dielectric active terahertz photonics driven by bound states in the continuum,” Adv. Mater., vol. 31, p. 1901921, 2019. https://doi.org/10.1002/adma.201901921.Search in Google Scholar PubMed
 E. N. Bulgakov and A. F. Sadreev, “Bound states in the continuum with high orbital angular momentum in a dielectric rod with periodically modulated permittivity,” Phys. Rev. A, vol. 96, p. 013841, 2017. https://doi.org/10.1103/physreva.96.013841.Search in Google Scholar
 E. N. Bulgakov and A. F. Sadreev, “Light trapping above the light cone in a one-dimensional array of dielectric spheres,” Phys. Rev. A, vol. 92, p. 023816, 2015. https://doi.org/10.1103/physreva.92.023816.Search in Google Scholar
 Z. Sadrieva, M. Belyakov, M. Balezin, et al.., “Experimental observation of a symmetry-protected bound state in the continuum in a chain of dielectric disks,” Phys. Rev. A, vol. 99, p. 053804, 2019. https://doi.org/10.1103/physreva.99.053804.Search in Google Scholar
 M. Sidorenko, O. Sergaeva, Z. Sadrieva, et al.., “Observation of an accidental bound state in the continuum in a chain of dielectric disks,” Phys. Rev. Appl., vol. 15, p. 034041, 2021. https://doi.org/10.1103/physrevapplied.15.034041.Search in Google Scholar
 C.-L. Zou, J.-M. Cui, F.-W. Sun, et al.., “Guiding light through optical bound states in the continuum for ultrahigh-q microresonators,” Laser Photon. Rev., vol. 9, p. 114, 2015. https://doi.org/10.1002/lpor.201400178.Search in Google Scholar
 Z. Yu, X. Xi, J. Ma, H. K. Tsang, C.-L. Zou, and X. Sun, “Photonic integrated circuits with bound states in the continuum,” Optica, vol. 6, p. 1342, 2019. https://doi.org/10.1364/optica.6.001342.Search in Google Scholar
 E. N. Bulgakov and A. F. Sadreev, “Bound states in the continuum in photonic waveguides inspired by defects,” Phys. Rev. B, vol. 78, p. 075105, 2008. https://doi.org/10.1103/physrevb.78.075105.Search in Google Scholar
 E. A. Bezus, D. A. Bykov, and L. L. Doskolovich, “Bound states in the continuum and high-q resonances supported by a dielectric ridge on a slab waveguide,” Photon. Res., vol. 6, p. 1084, 2018. https://doi.org/10.1364/prj.6.001084.Search in Google Scholar
 D. A. Bykov, E. A. Bezus, and L. L. Doskolovich, “Bound states in the continuum and strong phase resonances in integrated Gires–Tournois interferometer,” Nanophotonics, vol. 9, p. 83, 2020. https://doi.org/10.1515/nanoph-2019-0316.Search in Google Scholar
 M. V. Rybin, K. L. Koshelev, Z. F. Sadrieva, et al.., “High-q supercavity modes in subwavelength dielectric resonators,” Phys. Rev. Lett., vol. 119, p. 243901, 2017. https://doi.org/10.1103/physrevlett.119.243901.Search in Google Scholar
 L. Carletti, K. Koshelev, C. De Angelis, and Y. Kivshar, “Giant nonlinear response at the nanoscale driven by bound states in the continuum,” Phys. Rev. Lett., vol. 121, p. 033903, 2018. https://doi.org/10.1103/PhysRevLett.121.033903.Search in Google Scholar PubMed
 M. Odit, K. Koshelev, S. Gladyshev, K. Ladutenko, Y. Kivshar, and A. Bogdanov, “Observation of supercavity modes in subwavelength dielectric resonators,” Adv. Mater., vol. 33, p. 2003804, 2021. https://doi.org/10.1002/adma.202003804.Search in Google Scholar PubMed
 A. Kodigala, T. Lepetit, Q. Gu, B. Bahari, Y. Fainman, and B. Kanté, “Lasing action from photonic bound states in continuum,” Nature, vol. 541, p. 196, 2017. https://doi.org/10.1038/nature20799.Search in Google Scholar PubMed
 B. Wang, W. Liu, M. Zhao, et al.., “Generating optical vortex beams by momentum-space polarization vortices centred at bound states in the continuum,” Nat. Photonics, vol. 14, pp. 1–6, 2020. https://doi.org/10.1038/s41566-020-0658-1.Search in Google Scholar
 C. Cui, C. Zhou, S. Yuan, et al.., “Multiple fano resonances in symmetry-breaking silicon metasurface for manipulating light emission,” ACS Photonics, vol. 5, p. 4074, 2018. https://doi.org/10.1021/acsphotonics.8b00754.Search in Google Scholar
 B. Midya and V. V. Konotop, “Coherent-perfect-absorber and laser for bound states in a continuum,” Opt. Lett., vol. 43, p. 607, 2018. https://doi.org/10.1364/ol.43.000607.Search in Google Scholar PubMed
 Q. Song, J. Hu, S. Dai, et al.., “Coexistence of a new type of bound state in the continuum and a lasing threshold mode induced by pt symmetry,” Sci. Adv., vol. 6, p. eabc1160, 2020. https://doi.org/10.1126/sciadv.abc1160.Search in Google Scholar PubMed PubMed Central
 S. Mohamed, J. Wang, H. Rekola, et al.., Topological charge engineering in lasing bound states in continuum, 2020, arXiv preprint arXiv:2012.15642.Search in Google Scholar
 M.-S. Hwang, H.-C. Lee, K.-H. Kim, et al.., “Ultralow-threshold laser using super-bound states in the continuum,” Nat. Commun., vol. 12, p. 1, 2021. https://doi.org/10.1038/s41467-021-24502-0.Search in Google Scholar PubMed PubMed Central
 M. Wu, S. T. Ha, S. Shendre, et al.., “Room-temperature lasing in colloidal nanoplatelets via mie-resonant bound states in the continuum,” Nano Lett., vol. 20, p. 6005, 2020. https://doi.org/10.1021/acs.nanolett.0c01975.Search in Google Scholar PubMed
 S. T. Ha, Y. H. Fu, N. K. Emani, et al.., “Directional lasing in resonant semiconductor nanoantenna arrays,” Nat. Nanotechnol., vol. 13, p. 1042, 2018. https://doi.org/10.1038/s41565-018-0245-5.Search in Google Scholar PubMed
 J. M. Foley, S. M. Young, and J. D. Phillips, “Symmetry-protected mode coupling near normal incidence for narrow-band transmission filtering in a dielectric grating,” Phys. Rev. B, vol. 89, p. 165111, 2014. https://doi.org/10.1103/physrevb.89.165111.Search in Google Scholar
 J. M. Foley and J. D. Phillips, “Normal incidence narrowband transmission filtering capabilities using symmetry-protected modes of a subwavelength, dielectric grating,” Opt. Lett., vol. 40, p. 2637, 2015. https://doi.org/10.1364/ol.40.002637.Search in Google Scholar
 X. Cui, H. Tian, Y. Du, G. Shi, and Z. Zhou, “Normal incidence filters using symmetry-protected modes in dielectric subwavelength gratings,” Sci. Rep., vol. 6, p. 36066, 2016. https://doi.org/10.1038/srep36066.Search in Google Scholar PubMed PubMed Central
 S. Romano, G. Zito, S. Managò, et al.., “Surface-enhanced Raman and fluorescence spectroscopy with an all-dielectric metasurface,” J. Phys. Chem. C, vol. 122, p. 19738, 2018. https://doi.org/10.1021/acs.jpcc.8b03190.Search in Google Scholar
 S. Romano, G. Zito, S. Torino, et al.., “Label-free sensing of ultralow-weight molecules with all-dielectric metasurfaces supporting bound states in the continuum,” Photon. Res., vol. 6, p. 726, 2018. https://doi.org/10.1364/prj.6.000726.Search in Google Scholar
 Y. K. Srivastava, R. T. Ako, M. Gupta, M. Bhaskaran, S. Sriram, and R. Singh, “Terahertz sensing of 7 nm dielectric film with bound states in the continuum metasurfaces,” Appl. Phys. Lett., vol. 115, p. 151105, 2019. https://doi.org/10.1063/1.5110383.Search in Google Scholar
 X. Chen, W. Fan, and H. Yan, “Toroidal dipole bound states in the continuum metasurfaces for terahertz nanofilm sensing,” Opt. Express, vol. 28, p. 17102, 2020. https://doi.org/10.1364/oe.394416.Search in Google Scholar PubMed
 D. N. Maksimov, V. S. Gerasimov, S. Romano, and S. P. Polyutov, “Refractive index sensing with optical bound states in the continuum,” Opt. Express, vol. 28, p. 38907, 2020. https://doi.org/10.1364/oe.411749.Search in Google Scholar
 K. Koshelev, Y. Tang, K. Li, D.-Y. Choi, G. Li, and Y. Kivshar, “Nonlinear metasurfaces governed by bound states in the continuum,” ACS Photonics, vol. 6, p. 1639, 2019. https://doi.org/10.1021/acsphotonics.9b00700.Search in Google Scholar
 S. Krasikov, A. Bogdanov, and I. Iorsh, “Nonlinear bound states in the continuum of a one-dimensional photonic crystal slab,” Phys. Rev. B, vol. 97, p. 224309, 2018. https://doi.org/10.1103/physrevb.97.224309.Search in Google Scholar
 L. Carletti, K. Koshelev, C. De Angelis, and Y. Kivshar, “Giant nonlinear response at the nanoscale driven by bound states in the continuum,” Phys. Rev. Lett., vol. 121, p. 033903, 2018. https://doi.org/10.1103/PhysRevLett.121.033903.Search in Google Scholar PubMed
 E. N. Bulgakov and D. N. Maksimov, “Nonlinear response from optical bound states in the continuum,” Sci. Rep., vol. 9, p. 1, 2019. https://doi.org/10.1038/s41598-019-43672-y.Search in Google Scholar PubMed PubMed Central
 V. Kravtsov, E. Khestanova, F. A. Benimetskiy, et al.., “Nonlinear polaritons in a monolayer semiconductor coupled to optical bound states in the continuum,” Light Sci. Appl., vol. 9, p. 1, 2020. https://doi.org/10.1038/s41377-020-0286-z.Search in Google Scholar PubMed PubMed Central
 L. Carletti, S. S. Kruk, A. A. Bogdanov, C. De Angelis, and Y. Kivshar, “High-harmonic generation at the nanoscale boosted by bound states in the continuum,” Phys. Rev. Res., vol. 1, p. 023016, 2019. https://doi.org/10.1103/physrevresearch.1.023016.Search in Google Scholar
 A. P. Anthur, H. Zhang, R. Paniagua-Dominguez, et al.., “Continuous wave second harmonic generation enabled by quasi-bound-states in the continuum on gallium phosphide metasurfaces,” Nano Lett., vol. 20, p. 8745, 2020. https://doi.org/10.1021/acs.nanolett.0c03601.Search in Google Scholar PubMed
 D. N. Maksimov, A. A. Bogdanov, and E. N. Bulgakov, “Optical bistability with bound states in the continuum in dielectric gratings,” Phys. Rev. A, vol. 102, p. 033511, 2020. https://doi.org/10.1103/physreva.102.033511.Search in Google Scholar
 K. Koshelev, S. Kruk, E. Melik-Gaykazyan, et al.., “Subwavelength dielectric resonators for nonlinear nanophotonics,” Science, vol. 367, p. 288, 2020. https://doi.org/10.1126/science.aaz3985.Search in Google Scholar PubMed
 I. V. Timofeev, D. N. Maksimov, and A. F. Sadreev, “Optical defect mode with tunable q factor in a one-dimensional anisotropic photonic crystal,” Phys. Rev. B, vol. 97, p. 024306, 2018. https://doi.org/10.1103/physrevb.97.024306.Search in Google Scholar
 A. Taghizadeh and I.-S. Chung, “Quasi bound states in the continuum with few unit cells of photonic crystal slab,” Appl. Phys. Lett., vol. 111, p. 031114, 2017. https://doi.org/10.1063/1.4990753.Search in Google Scholar
 Z. F. Sadrieva, I. S. Sinev, K. L. Koshelev, et al.., “Transition from optical bound states in the continuum to leaky resonances: role of substrate and roughness,” ACS Photonics, vol. 4, p. 723, 2017. https://doi.org/10.1021/acsphotonics.6b00860.Search in Google Scholar
 K. Koshelev, S. Lepeshov, M. Liu, A. Bogdanov, and Y. Kivshar, “Asymmetric metasurfaces with high-q resonances governed by bound states in the continuum,” Phys. Rev. Lett., vol. 121, p. 193903, 2018. https://doi.org/10.1103/physrevlett.121.193903.Search in Google Scholar
 P.-E. Wolf and G. Maret, “Weak localization and coherent backscattering of photons in disordered media,” Phys. Rev. Lett., vol. 55, p. 2696, 1985. https://doi.org/10.1103/physrevlett.55.2696.Search in Google Scholar
 A. N. Poddubny, M. V. Rybin, M. F. Limonov, and Y. S. Kivshar, “Fano interference governs wave transport in disordered systems,” Nat. Commun., vol. 3, p. 1, 2012. https://doi.org/10.1038/ncomms1924.Search in Google Scholar PubMed PubMed Central
 M. F. Limonov and M. Richard, Optical Properties of Photonic Structures: Interplay of Order and Disorder, Boca Raton, CRC Press, 2012.Search in Google Scholar
 C. Liu, M. V. Rybin, P. Mao, S. Zhang, and Y. Kivshar, “Disorder-immune photonics based on mie-resonant dielectric metamaterials,” Phys. Rev. Lett., vol. 123, p. 163901, 2019. https://doi.org/10.1103/physrevlett.123.163901.Search in Google Scholar
 J. F. Galisteo-López, M. Ibisate, R. Sapienza, L. S. Froufe-Pérez, Á. Blanco, and C. López, “Self-assembled photonic structures,” Adv. Mater., vol. 23, p. 30, 2011. https://doi.org/10.1002/adma.201000356.Search in Google Scholar PubMed
 V. Astratov, A. Adawi, S. Fricker, M. Skolnick, D. Whittaker, and P. Pusey, “Interplay of order and disorder in the optical properties of opal photonic crystals,” Phys. Rev. B, vol. 66, p. 165215, 2002. https://doi.org/10.1103/physrevb.66.165215.Search in Google Scholar
 S. Fan, P. R. Villeneuve, and J. Joannopoulos, “Theoretical investigation of fabrication-related disorder on the properties of photonic crystals,” J. Appl. Phys., vol. 78, p. 1415, 1995. https://doi.org/10.1063/1.360298.Search in Google Scholar
 L. Ni, J. Jin, C. Peng, and Z. Li, “Analytical and statistical investigation on structural fluctuations induced radiation in photonic crystal slabs,” Opt. Express, vol. 25, p. 5580, 2017. https://doi.org/10.1364/oe.25.005580.Search in Google Scholar PubMed
 Y.-X. Xiao, Z.-Q. Zhang, and C. Chan, “A band of bound states in the continuum induced by disorder,” Sci. Rep., vol. 8, p. 1, 2018. https://doi.org/10.1038/s41598-018-23576-z.Search in Google Scholar PubMed PubMed Central
 H. L. Chen, G. Wang, and R. K. Lee, “Nearly complete survival of an entangled biphoton through bound states in continuum in disordered photonic lattices,” Opt. Express, vol. 26, p. 33205, 2018. https://doi.org/10.1364/oe.26.033205.Search in Google Scholar
 Y.-S. Jin, G.-J. Kim, and S.-G. Jeon, “Terahertz dielectric properties of polymers,” J. Kor. Phys. Soc., vol. 49, p. 513, 2006.Search in Google Scholar
 R. Haas and P. Zimmermann, “22-Ghz measurements of dielectric constants and loss tangents of castable dielectrics at room and cryogenic temperatures,” IEEE T. Microw. Theory, vol. 24, p. 881, 1976. https://doi.org/10.1109/tmtt.1976.1128985.Search in Google Scholar
 R. F. Ndangali and S. V. Shabanov, “Electromagnetic bound states in the radiation continuum for periodic double arrays of subwavelength dielectric cylinders,” J. Math. Phys., vol. 51, p. 102901, 2010. https://doi.org/10.1063/1.3486358.Search in Google Scholar
 F. R. Ndangali and S. V. Shabanov, “The resonant nonlinear scattering theory with bound states in the radiation continuum and the second harmonic generation,” in Active Photonic Materials V, vol. 8808, San Diego, International Society for Optics and Photonics, 2013, p. 88081F.10.1117/12.2028277Search in Google Scholar
 E. Bulgakov, D. Maksimov, P. Semina, and S. Skorobogatov, “Propagating bound states in the continuum in dielectric gratings,” J. Opt. Soc. Am. B, vol. 35, p. 1218, 2018. https://doi.org/10.1364/josab.35.001218.Search in Google Scholar
 H. Hemmati and R. Magnusson, “Resonant dual-grating metamembranes supporting spectrally narrow bound states in the continuum,” Adv. Opt. Mater., vol. 7, p. 1900754, 2019. https://doi.org/10.1002/adom.201900754.Search in Google Scholar
 Y. Shuai, D. Zhao, Z. Tian, et al.., “Double-layer Fano resonance photonic crystal filters,” Opt. Express, vol. 21, p. 24582, 2013. https://doi.org/10.1364/oe.21.024582.Search in Google Scholar PubMed
 Y. Shuai, D. Zhao, A. S. Chadha, et al.., “Coupled double-layer Fano resonance photonic crystal filters with lattice-displacement,” Appl. Phys. Lett., vol. 103, p. 241106, 2013. https://doi.org/10.1063/1.4846475.Search in Google Scholar
 D. Marinica, A. Borisov, and S. Shabanov, “Bound states in the continuum in photonics,” Phys. Rev. Lett., vol. 100, p. 183902, 2008. https://doi.org/10.1103/physrevlett.100.183902.Search in Google Scholar
 C. W. Hsu, B. Zhen, J. Lee, et al.., “Observation of trapped light within the radiation continuum,” Nature, vol. 499, p. 188, 2013. https://doi.org/10.1038/nature12289.Search in Google Scholar PubMed
 B. Zhen, C. W. Hsu, L. Lu, A. D. Stone, and M. Soljačić, “Topological nature of optical bound states in the continuum,” Phys. Rev. Lett., vol. 113, p. 257401, 2014. https://doi.org/10.1103/physrevlett.113.257401.Search in Google Scholar
 E. N. Bulgakov and D. N. Maksimov, “Topological bound states in the continuum in arrays of dielectric spheres,” Phys. Rev. Lett., vol. 118, p. 267401, 2017. https://doi.org/10.1103/physrevlett.118.267401.Search in Google Scholar PubMed
 E. Ivchenko and G. Pikus, “Crystal symmetry,” in Superlattices and Other Heterostructures, Springer Series in Solid-State Sciences, vol. 110, Berlin, Heidelberg, Springer, 1995.10.1007/978-3-642-97589-9_2Search in Google Scholar
 V. Karagodsky, C. Chase, and C. J. Chang-Hasnain, “Matrix fabry–perot resonance mechanism in high-contrast gratings,” Opt. Lett., vol. 36, p. 1704, 2011. https://doi.org/10.1364/ol.36.001704.Search in Google Scholar PubMed
 V. Karagodsky and C. J. Chang-Hasnain, “Physics of near-wavelength high contrast gratings,” Opt. Express, vol. 20, p. 10888, 2012. https://doi.org/10.1364/oe.20.010888.Search in Google Scholar PubMed
 Z. F. Sadrieva, M. A. Belyakov, M. A. Balezin, et al.., “Experimental observation of a symmetry-protected bound state in the continuum in a chain of dielectric disks,” Phys. Rev. A, vol. 99, p. 053804, 2019. https://doi.org/10.1103/physreva.99.053804.Search in Google Scholar
 E. N. Bulgakov and D. N. Maksimov, “Light enhancement by quasi-bound states in the continuum in dielectric arrays,” Opt. Express, vol. 25, p. 14134, 2017. https://doi.org/10.1364/oe.25.014134.Search in Google Scholar
 E. N. Bulgakov and A. F. Sadreev, “High-q resonant modes in a finite array of dielectric particles,” Phys. Rev. A, vol. 99, p. 033851, 2019. https://doi.org/10.1103/physreva.99.033851.Search in Google Scholar
 I. Lifshits, S. Gredeskul, and L. Pastur, Introduction to the Theory of Disordered Systems, New York, Wiley, 1988.Search in Google Scholar
 J. Billy, V. Josse, Z. Zuo, et al.., “Direct observation of anderson localization of matter waves in a controlled disorder,” Nature, vol. 453, p. 891, 2008. https://doi.org/10.1038/nature07000.Search in Google Scholar PubMed
© 2021 Ekaterina E. Maslova et al., published by De Gruyter, Berlin/Boston
This work is licensed under the Creative Commons Attribution 4.0 International License.