Open Access
Add to BibSonomy
Abstract
We report the giant enhanced optical harmonic generation in all-dielectric silicon nitride (SiN) based resonant waveguide gratings (RWGs) of quasi-bound states in the continuum (BICs) of ultra-high Q factor and localized field. The BICs are realized by tuning the excitation of the guided modes modulated by geometry parameters of four-part grating layer. At a feasible structure of quasi-BIC for nanofabrication, the SHG and THG are enhanced by 103 and 106, compared with those from the RWGs of traditional two-part grating layer, respectively, and even up to 108 and 1010 compared with those from the planar SiN film, respectively. The resonance wavelength of quasi-BICs can be effectively tuned by the angle of incidence, while almost not affect the enhancement of SHG and THG response. Our results show that the efficiency harmonic generation from all-nonlinear-dielectric RWGs of quasi-BICs has potential applications for the integrated nonlinear photonic devices.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Nonlinear frequency conversion in complementary metal-oxide-semiconductor (CMOS) compatible materials has drawn much attention due to the applications in integrated photonic devices [1–4]. The generated frequencies via the harmonic generation or wave mixing process offer the new light sources used for the on-chip optical signal processing, communication, imaging, and so on. Silicon nitride (SiN) has been widely reported as a promising CMOS-compatible material of large second- and third-order nonlinear susceptibility for on-chip nonlinear optical devices [2–9]. In addition, SiN has superior linear optical properties, such as transparent at visible wavelength, tunable refractive index, compared with traditional silicon. In recent years, the enhanced nonlinear frequency conversion in SiN nanostructures has been intensively studied in ring resonators [10–12], photonic crystals [13,14], resonant waveguide gratings (RWGs) [7,15], and so on. Among these resonant structures, the RWGs have its particular properties due to the grating diffraction behaviors. Firstly, the angular separation of the fundamental light beam and the generated light beams of new frequency provide the pure frequency light sources without using filters [16]. Secondly, different from the in-plane ring resonators, photonic crystals and the other cavities, the fundamental and generated light travels out of the RWGs structure, which gives the high efficiency light of new frequency generated in the on-chip devices but traveling in the free space for the further usages, not like the generated new light in the ring resonators or photonic crystal slabs that is confined in the nanostructures or scattering out with low efficiency. Thirdly, the diffracted light travelling out the RWGs structure also offers the possibility to design the layer-by-layer integrated photonic devices and to extend the functions of devices. Further, the RWGs can be fabricated in a large scale using high-throughput and low-cost laser interference lithography or nanoimprint lithography [17]. The three-order of magnitude enhancement from the resonance nanostructures was achieved [7]. Different from the traditional RWGs structure of a regular grating layer, very recently, the novel RWGs structure consisting of a four-part periodic grating layer and a waveguide layer possessing the bound states in the continuum (BICs) was reported [18]. Quasi-BICs of ultrahigh Q factors and enhanced local field were obtained by changing the geometry parameters.
The BICs refer as perfectly confined states without any radiation of an infinity Q factor [19]. The symmetry-protected systems are the simplest place to realize the BICs since the coupling of resonances to the radiation modes are forbidden. The symmetry breaking via the change of asymmetry structure or angle of incidence will produce the so-called quasi-BICs [20–24]. The line width of resonant modes and thus the Q-factors can be tuned and controlled. Based on the ultra-high Q factor and ultra-narrow resonance width BICs and quasi-BICs were widely used in optical and photonic applications, such as lasers [25], filters [26], sensors [27], and nonlinear optics [28–30]. In this paper, we study the nonlinear optical harmonic generation in the SiN-based RWGs of quasi-BICs under transverse magnetic (TM) polarization. The second- and third-harmonic generation (SHG and THG) in the feasible structures are dramatically enhanced by several orders of magnitude compared with those from the traditional RWGs and the planar SiN film of the same thickness, respectively. Thus, the extremely large conversion efficiency is obtained. Especially the resonance wavelength can be effectively tuned by the angle of incidence, while the enhanced SHG and THG response are kept. Importantly, the dielectric material is not limited by SiN, but the other CMOS-compatible nonlinear dielectrics, such as semiconductors and lithium niobite, and so on. Our results show that high efficiency harmonic generation from the all-nonlinear-dielectric RWGs of quasi-BICs is of great significance for the integrated nonlinear photonic devices.
2. Numerical structure and method
A unit of the RWGs structure composed of a four-part periodic grating layer and a waveguide layer is shown in Fig. 1(a). For the grating layer, the first and third parts are dielectric of width da, and the second and fourth parts are air of width db and dc, respectively. Thus the periodic of grating layer Λ is Λ = 2*da+db+dc. Here we set da=0.2Λ, db=d-Δd and dc=d+Δd with d=0.3Λ. We define an adjustable geometric parameter δ=Δd/d∈[0,1] to reflect the difference between the second and fourth parts of the grating. The waveguide layer is also made of SiN. The thickness of grating layer and waveguide layer is denoted as hw and hg, respectively. The fused silica is chosen as the substrate. The incident plane is the x-o-z plane. Without loss of generality, we only consider the TM polarized light of an angle of incidence θ.
Fig. 1. (a) Schematic diagram of the section of RWG structure as a simulation domain. da is the width of SiN, db and dc are the width of air. hg and hw are the height of grating layer and waveguide layer, respectively. The incidence light of TM-polarization and angle of incidence θ is excited on port 1. k, E and H are the wavevector, electric- and magnetic-field of the incident light, respectively. The periodic boundary condition is used for the left- and right-boundary. (b) Guided mode resonance of the grating waveguide structure. 0 and −1 represents the 0th and −1st order of diffraction. β is the propagating constant of the guided mode. (c) Dispersion relation of the TM0 guided mode in the waveguide layer (black solid line), and kx=kx,i (i=-1,-2) under different angle of incidence, 1° (red dashed lines), 5° (green dashed lines), 10° (blue dashed lines), and 15° (cyan dashed lines), respectively.
Under the undepleted pump approximation, the fundamental, SHG and THG fields in the frequency domain (time-harmonic factor exp(-iωt)) can be expressed as [31]
The above coupling equations can be numerically solved using finite element method (Comsol Multiphysics) to obtain the linear optical reflectance, transmitted SHG and THG from the nanostructures. The simulation setup is similar as that used in the Ref. [32]. We select one period of grating as the simulation domain, as shown in Fig. 1(a). The top boundary of the air domain is set as the port 1 for the light incidence, and the bottom boundary of the fused silica substrate is set as port 2. The reflectance of light is equal to the square of the S-parameter S11. A TM-polarized fundamental light of an angle θ shins on the periodic structure from port 1 as shown in Fig. 1(a). The wave vectors along the x and z directions are kx=k0sinθ and kz=k0cosθ, respectively. The electric field components of light are Ex=E0cosθ and Ey=E0sinθ, respectively, where E0 is the fundamental electric field. The left and right boundaries are set to be Bloch-Floquet periodic boundary conditions, and the Floquet wavevector kF is defined as kFx=kx and kFy=0 at the fundamental frequency. The solved fundamental electric field is used to express the components of P(2)(2ω) and P(3)(3ω), respectively. The Eqs. (2) and (3) are then solved using P(2)(2ω) and P(3)(3ω) as a source, respectively. The electric and magnetic fields at SHG and THG frequency can thus be obtained. The transmitted power of SHG and THG is obtained by integrating Poynting flow on the bottom port 2 using $\textrm{P(}2\omega \textrm{)} = \int {{{Re {\{ }{\textbf E}(2\omega ) \times {\textbf H}(2\omega ){\} }} / 2}} \cdot {\textbf n}ds$ and $\textrm{P(}3\omega \textrm{)} = {{\int {Re \{ {\textbf E}(3\omega ) \times {\textbf H}(3\omega )} \} } / 2} \cdot {\textbf n}ds$, respectively, where H(2ω) and H(3ω) are magnetic fields at SHG and THG frequency, respectively, and n represents the normal direction of bottom boundary. ds = dl·1[m] with dl the element of line along port 2 boundary and assuming the length along y of an unit meter.
In our simulation, the linear refractive index of SiN is obainted from the experimental data measured by ellipsometery, and the refractive index of fused silica is used as in the Ref. [5]. The determined tensor components of second-order susceptibility are $\chi _{zzz}^{(2)} = 2.47$ pm/V, $\chi _{xxz}^{(2)} = 0.49$ pm/V, and $\chi _{zxx}^{(2)} = 0.47$ pm/V, and the tensor component of third-order susceptibility is $\chi _{iiii}^{(3)} = 3 \times {10^{ - 20}}$m2/V2 [5,7]. The geometry parameters of the RWGs are chosen as Λ=683 nm, hg=50 nm, and hg=300 nm.
3. Results and discussion
As demonstrated in the Ref. [18], the BICs in the compound structure of four-part periodic grating and waveguide originate from the change of diffraction due to the change of period. As the parameter δ=1, it is corresponding to a traditional RWGs structure with a grating layer of period Λ and a fill factor 0.4. As the δ=0, it is also a traditional RWGs structure but the period of grating layer becomes Λ/2. Since the reciprocal lattice G induced by the grating doubles when the period of the grating changes from Λ to Λ/2, i.e. G=2π/Λ becomes G’=4π/Λ, the phase matching condition of the guided mode resonance (GMR) that is satisfied under grating period Λ for δ≠0 is not valid any more for grating of period Λ/2 [18]. So excitable odd-order guided resonance mode in the structure δ≠0 cannot be excited in the structure δ=0. Such discrete dark modes are BICs of an infinite Q factor [33]. The perturbation of the structure when δ changes from zero to nonzero excites the guided mode of strong confined localized resonance modes called quasi-BICs.
The dispersion relation of the TM0 guided mode in the waveguide layer can be calculated by,
The reflectance spectra as the geometric parameter δ changing from 0 to 1.0 at the angle of incidence θ=1° are shown in Fig. 2(a). When the δ=0.1, the nearly perfect reflection at the wavelength λ = 1063.5 nm is obtained, which corresponds to the resonance waveguide mode. The value is some larger than that determined from the relation of dispersion (Fig. 1(c)) since the effective refractive index nc in the cladding layer should be considered. The resonance peaks have slight redshift and become broader as the increase of δ, because the refractive index distribution of the grating layer varies with δ. The magnetic field |Hy/H0| distributions at the corresponding resonance modes when δ=0.1 and 1.0 are shown in the insert of Fig. 2(a). The waveguide modes are clearly seen, and the maximum enhancement at δ=0.1 arrives up to 341, which is much larger than that in the traditional RWG structure when δ=1 (around 40), and larger than that in the traditional RWG structures we ever designed [15]. The magnetic field distributions under the corresponding resonance states at the other angles of incidence θ=5°, 10°, 15° are similar as those at θ=1°. At the same RWG structure, the resonance wavelength has a blueshift with the increase of angle of incidence. At the same structure of δ=0.1, the resonance wavelength is at around 1025.6 nm, 979.2 nm and 934.2 nm when θ=5°, 10° and 15°, respectively. The Fig. 2(b) gives a clearer vision of such trend in the structure of δ=0.4 when changing the angle of incidence from 1° to 15°. The dependence of the resonance wavelength on the angle of incidence is summarized in Fig. 2(c). The broad range of resonance wavelength is obtained from 1072.78 nm at 0° to 934.23 nm at 15°, which provides the flexible tunability of the generation of harmonic wavelengths for the new light sources in one structure. The continuity of the change of resonance wavelength with the angle of incidence is verified in the Fig. 2(d) for the angle of incidence in the range of 0° to 1°.
Fig. 2. (a)The reflectance spectra of RWGs of different parameter δ at the different angles of incidence θ=1°. The inset shows the magnetic field |Hy/H0| distribution at the resonance modes in the structure of δ=0.1 and 1.0, respectively. (b) The dependence of reflectance of RWGs of δ=0.4 on the angle of incidence. (c) The relation of resonance wavelength with the angle of incidence at the RWG structure of δ=0.4. (d) The detail of reflectance spectra changing with the angle of incidence in the range of 0° to 1° in the RWG structure of δ=0.4.
The Q factors, defined as Q=λpeak/Δλ with Δλ the full width at half maximum (FWHM) of the resonance peak, are calculated for different values of δ. The typical Q factors at θ=1° in the different nanostructures are shown in Fig. 3. When δ=1 for the traditional RWGs of the guided mode, the Q factor is around 1.55×103. As δ gradually decreases to near zero, the Q factor increases rapidly. For example, the Q factor reaches up to 1.32×105 at δ=0.1 and 5.32×105 at δ=0.05. When δ=0, the resonance peak vanishes completely of Δλ=0 and the Q factor becomes infinite. Though the first-part and third-part of the SiN grating has the same width without breaking the in-plane symmetry, the structure still can be considered as asymmetric metasurfaces considering the second- and four-part of air when δ≠0. The linear relation Q factors with δ−2 is valid [34,35], as shown in the inset of Fig. 3.
Fig. 3. Dependence of Q factor on δ. The dash line is a guide for the eye. The inset shows the linear relationship between Q factor and δ−2, and the dash line is a linear fitting.
The nonlinear response can be significantly enhanced by the strong localized fields at resonance modes. We first investigate SHG from our RWG structures under the TMin-TMout configuration. The SHG from the planar SiN film of thickness hw+hg=350 nm was first calculated, and the maximum SHG response at the angle of incidence 60° was recorded as the reference. The enhancement factor is determined by comparing with the SHG power from the RWGs with that from the planar SiN film. The SHG enhancement factor in RWGs at the angle of incidence θ=1° of different parameter δ are shown in Fig. 4(a). The enhancement factor arrives 108 when δ=0.1 due to the strong localized field in the waveguide layer. The SHG power increases significantly with decrease of δ due to the increased local field enhancement. For example, the SHG power in δ=0.1 structure is three orders of magnitude larger than in structure of δ=1, which is the traditional RWG structure. The magnetic field Hy(2ω) of the unit A/m at the resonance mode in the RWGs with δ=0.1 when the fundamental field E0=1×106 V/m is shown in Fig. 4(b). The SHG wavelength should be tunable as indicated by the reflectance spectra at the different angle of incidence, as shown in Figs. 3(b) and (d). Figure 4(c) gives the SHG spectra at the different angles of incidence from the RWG structure of δ=0.4. It is clear that the peaks of SHG spectra correspond to the resonance modes at different angles of incidence. The enhancement of SHG response is almost independent of the angle of incidence, but has a slight increase with the increase of angle of incidence. The behavior can be ascribed to the property of χ(2) in SiN, which has the domain component $\chi _{zzz}^{(2)}$, and thus the larger electric field along z direction at the larger angle of incidence gives rise to more efficient SHG response. The detail of the change of resonance wavelength with the angle of incidence in the range of 0° to 1° is shown in the Fig. 4(d).
Fig. 4. (a) SHG enhancement factor in the RWGs of different parameter δ at angle of incidence θ=1°. (b) Hy(2ω) distribution of SHG at θ=1° under the fundamental field E0=1×106 V/m. The unit s A/m. (c) SHG spectra in the RWG structure of δ=0.4 at different angles of incidence. (d) The SHG spectra changing with the angle of incidence in the range of 0° to 1° in the RWG structure of δ=0.4. The colorbar shows the logarithm of SHG enhancement factor.
We then study THG from our RWG structures also under the TMin-TMout configuration. Figure 5(a) shows the THG enhancement factor from the RWGs at the angle of incidence θ=1° of different parameter δ. The maximum THG response from the planar SiN film of thickness 350 nm at normal incidence is calculated and used as the reference. The enhancement factor reaches up to 1010 when δ=0.1, and the THG power in δ=0.1 structure is six orders of magnitude larger than that in the structure of δ=1 as a traditional RWG structure. The magnetic field Hy(3ω) of the unit A/m at the resonance mode in the RWGs of δ=0.1 is shown in Fig. 5(b). The fundamental field E0=1×106 V/m is also used. The THG spectra at the different angles of incidence from the RWG structure of δ=0.4 is shown in Fig. 5(b). The peaks of THG spectra locate at the resonance reflectance wavelength at different angles of incidence are also observed. The 6 to 7 orders of magnitude of enhancement were observed in the broad range of resonance wavelength. Different from the behavior of SHG, the enhancement factor of THG is some decreasing with the increase of angle of incidence, that is because the extinction coefficient κ is larger at shorter THG wavelength.
Fig. 5. (a) THG enhancement factor in the RWGs of different parameter δ at angle of incidence θ=1°. (b) Hy(3ω) distribution of SHG at θ=1° under the fundamental field E0=1×106 V/m. The unit is A/m. (c) THG spectra in the RWG structure of δ=0.4 at different angles of incidence. (d) The THG spectra changing with the angle of incidence in the range of 0° to 1° in the RWG structure of δ=0.4. The colorbar shows the logarithm of THG enhancement factor.
The conversion efficiency of SHG and THG, ηSHG and ηTHG, are defined as ηSHG=PSHG/Pω, and ηTHG=PTHG/Pω, where PSHG, PTHG and Pω are the power of SHG, THG and fundamental light, respectively. The conversion efficiency of SHG and THG in the RWGs nanostructures at θ=1° is determined respectively, as shown in the Fig. 6. The SHG conversion efficiency around 10−7 can be achieved at the input intensity 1 GW/cm2, and 10−5 at the input intensity 100 GW/cm2 are expected. The laser intensity is easily obtained using a femtosecond laser. The large THG conversion efficiency up to 10−4 could be expected at the input intensity 1 GW/cm2. The assumption of undepleted pump light should be still valid here. For more intense input light and the higher conversion, the theory of coupled waves between fundamental and harmonic lights should be considered [31]. In the reality for the experimental design and measurements, more factors will be involved, such as the imperfections of fabrication, the nonlinear absorption and nonlinear refraction of the material under the intense laser, the damage threshold of the material, and so on. The high-quality materials of excellent optical properties and fabrication are required.
Fig. 6. Conversion efficiency of SHG and THG dependence on the fundamental input intensity. The slope 1 and 2 indicate the second- and third-order nonlinear process, respectively.
We finally point out that the parameter δ=0.1 in our designed structure means the Δd=20.5 nm. Such difference is feasible for the nanofabrication using state-of-art technology, such as electron-beam lithography or focused-ion beam. The designed grating structures can be fabricated using silicon, and then be used as the master for nanoimprint to fabricate the RWGs with low-cost ang high-throughput. The other structures of larger δ can also be prepared using the same process. The giant enhanced SHG and THG show great potential applications in nanophotonic devices.
4. Conclusions
We have demonstrated that the dramatically enhanced optical harmonic generation in all-dielectric SiN-based RWGs due to the strong localized field at quasi-BICs. At a reasonable parameter δ=0.1 for nanofabrication, the SHG and THG are enhanced by 103 and 106, compared with those from the traditional RWGs, and even up to 108 and 1010 compared with those from planar SiN film, respectively. Importantly, the resonance wavelength of quasi-BICs can be effectively tuned by the angle of incidence, while the enhancement of SHG and THG response is almost kept. The designed structures could be fabricated using nanoimprint combined with etching technology with low-cost ang high-throughput. Further, the dielectric material is not limited by SiN but the other CMOS-compatible nonlinear dielectrics, such as semiconductors and lithium niobite, and so on. Our results show that the all-nonlinear-dielectric RWGs of quasi-BICs are of potential applications for the integrated nonlinear photonic devices.
Funding
Natural Science Foundation of Shandong Province (ZR2019MF029, ZR2019MA024); National Natural Science Foundation of China (11404195, 12074225, 91950106).
Disclosures
The authors declare no conflicts of interest.
References
1. L. Caspani, D. Duchesne, K. Dolgaleva, S. J. Wagner, M. Ferrera, L. Razzari, A. Pasquazi, M. Peccianti, D. J. Moss, J. S. Aitchison, and R. Morandotti, “Optical frequency conversion in integrated devices,” J. Opt. Soc. Am. B 28(12), A67–A82 (2011). [CrossRef]
2. A. L. Gaeta, M. Lipson, and T. J. Kippenberg, “Photonic-chip-based frequency combs,” Nat. Photonics 13(3), 158–169 (2019). [CrossRef]
3. D. T. H. Tan, K. J. A. Ooi, and D. K. T. Ng, “Nonlinear optics on silicon-rich nitride-a high nonlinear figure of merit CMOS platform,” Photonics Res. 6(5), B50–B66 (2018). [CrossRef]
4. D. J. Moss, R. Morandotti, A. L. Gaeta, and M. Lipson, “New CMOS-compatible platforms based on silicon nitride and Hydex for nonlinear optics,” Nat. Photonics 7(8), 597–607 (2013). [CrossRef]
5. T. Ning, H. Pietarinen, O. Hyvärinen, J. Simonen, G. Genty, and M. Kauranen, “Strong second-harmonic generation in silicon nitride films,” Appl. Phys. Lett. 100(16), 161902 (2012). [CrossRef]
6. K. Ikeda, R. E. Saperstein, N. Alic, and Y. Fainman, “Thermal and Kerr nonlinear properties of plasmadeposited silicon nitride/ silicon dioxide waveguides,” Opt. Express 16(17), 12987–12994 (2008). [CrossRef]
7. T. Ning, O. Hyvärinen, H. Pietarinen, T. Kaplas, M. Kauranen, and G. Genty, “Third-harmonic UV generation in silicon nitride nanostructures,” Opt. Express 21(2), 2012–2017 (2013). [CrossRef]
8. D. Grassani, M. H. R. Pfeiffer, T. J. Kippenberg, and C. S. Bres, “Second- and third-order nonlinear wavelength conversion in an all-optically poled Si3N4 waveguide,” Opt. Lett. 44(1), 106–109 (2019). [CrossRef]
9. A. Kitao, K. Imakita, I. Kawamura, and M. Fuji, “An investigation into second harmonic generation by Si-rich SiNx thin films deposited by RF sputtering over a wide range of Si concentrations,” J. Phys. D: Appl. Phys. 47(21), 215101 (2014). [CrossRef]
10. J. S. Levy, M. A. Foster, A. L. Gaeta, and M. Lipson, “Harmonic generation in silicon nitride ring resonators,” Opt. Express 19(12), 11415–11421 (2011). [CrossRef]
11. J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,” Nat. Photonics 4(1), 37–40 (2010). [CrossRef]
12. Z. C. Ye, K. Twayana, P. A. Andrekson, and V. Torres-Company, “High-Q Si3N4 microresonators based on a subtractive processing for Kerr nonlinear optics,” Opt. Express 27(24), 35719–35727 (2019). [CrossRef]
13. E. Sahin, K. J. A. Ooi, G. F. R. Chen, D. K. T. Ng, C. E. Png, and D. T. H. Tan, “Enhanced optical nonlinearities in CMOS-compatible ultra-silicon-rich nitride photonic crystal waveguides,” Appl. Phys. Lett. 111(12), 121104 (2017). [CrossRef]
14. N. Janrao and V. Janyani, “Nonlinear performance in silicon nitride slow light photonic crystal waveguides with elliptical holes,” Optik 125(13), 3081–3084 (2014). [CrossRef]
15. T. Ning, H. Pietarinen, O. Hyvärinen, R. Kumar, T. Kaplas, M. Kauranen, and G. Genty, “Efficient second- harmonic generation in silicon nitride resonant waveguide gratings,” Opt. Lett. 37(20), 4269–4271 (2012). [CrossRef]
16. F. Lagugné-Labarthet, F. Adamietz, V. Rodriguez, and C. Sourisseau, “Significant Enhancement of the Optical Second Harmonic Generation in a Poled Azopolymer Thin Grating,” J. Phys. Chem. B 110(28), 13689–13693 (2006). [CrossRef]
17. G. Quaranta, G. Basset, O. J. F. Martin, and B. Gallinet, “Recent Advances in Resonant Waveguide Gratings,” Laser Photonics Rev. 12(9), 1800017 (2018). [CrossRef]
18. F. Wu, J. Wu, Z. Guo, H. Jiang, Y. Sun, Y. Li, J. Ren, and H. Chen, “Giant Enhancement of the Goos-Hänchen Shift Assisted by Quasibound States in the Continuum,” Phys. Rev. A 12(1), 014028 (2019). [CrossRef]
19. C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljačić, “Bound states in the continuum,” Nat. Rev. Mater. 1(9), 16048 (2016). [CrossRef]
20. E. Pshenay-Severin, A. Chipouline, J. Petschulat, U. Hubner, A. Tunnermann, and T. Pertsch, “Optical properties of metamaterials based on asymmetric double-wire structures,” Opt. Express 19(7), 6269–6283 (2011). [CrossRef]
21. M. Song, H. Yu, C. Wang, N. Yao, M. Pu, J. Luo, Z. Zhang, and X. Luo, “Sharp Fano resonance induced by a single layer of nanorods with perturbed periodicity,” Opt. Express 23(3), 2895–2903 (2015). [CrossRef]
22. V. A. Fedotov, M. Rose, S. L. Prosvirnin, N. Papasimakis, and N. I. Zheludev, “Sharp Trapped-Mode Resonances in Planar Metamaterials with a Broken Structural Symmetry,” Phys. Rev. Lett. 99(14), 147401 (2007). [CrossRef]
23. A. B. Evlyukhin, S. I. Bozhevolnyi, A. Pors, M. G. Nielsen, I. P. Radko, M. Willatzen, and O. Albrektsen, “Detuned Electrical Dipoles for Plasmonic Sensing,” Nano Lett. 10(11), 4571–4577 (2010). [CrossRef]
24. S. Campione, S. Liu, L. I. Basilio, L. K. Warne, W. L. Langston, T. S. Luk, J. R. Wendt, J. L. Reno, G. A. Keeler, I. Brener, and M. B. Sinclair, “Broken symmetry dielectric resonators for high quality factor Fano metasurfaces,” ACS Photonics 3(12), 2362–2367 (2016). [CrossRef]
25. K. Hirose, Y. Liang, Y. Kurosaka, A. Watanabe, T. Sugiyama, and S. Noda, “Watt-class high-power, high-beamquality photonic-crystal lasers,” Nat. Photonics 8(5), 406–411 (2014). [CrossRef]
26. 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 89(16), 165111 (2014). [CrossRef]
27. B. Zhen, S. Chua, J. Lee, A. W. Rodriguez, X. Liang, S. G. Johnson, J. D. Joannopoulos, M. Soljačić, and O. Shapira, “Enabling enhanced emission and low threshold lasing of organic molecules using special Fano resonances of macroscopic photonic crystals,” Proc. Natl. Acad. Sci. U. S. A. 110(34), 13711–13716 (2013). [CrossRef]
28. S. Liu, M. B. Sinclair, S. Saravi, G. A. Keeler, Y. Yang, J. Reno, G. M. Peake, F. Setzpfandt, I. Staude, T. Pertsch, and I. Brene, “Resonantly Enhanced Second-Harmonic Generation Using III−V Semiconductor All-Dielectric Metasurfaces,” Nano Lett. 16(9), 5426–5432 (2016). [CrossRef]
29. S. D. Krasikov, A. A. Bogdanov, and I. V. Iorsh, “Nonlinear bound states in the continuum of a one-dimensional photonic crystal slab,” Phys. Rev. B 97(22), 224309 (2018). [CrossRef]
30. 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 6(7), 1639–1644 (2019). [CrossRef]
31. R. W. Boyd, Nonlinear Optics (Academic, 2003).
32. J. Lu, B. Y. Ding, Y. Y. Huo, and T. Y. Ning, “Geometric effect on second harmonic generation from gold grating,” Opt. Commun. 415, 146–150 (2018). [CrossRef]
33. W. Liu, Y. Li, H. Jiang, Z. Lai, and H. Chen, “Controlling the spectral width in compound waveguide grating structures,” Opt. Lett. 38(2), 163 (2013). [CrossRef]
34. 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 97(2), 024306 (2018). [CrossRef]
35. 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. 121(19), 193903 (2018). [CrossRef]
