Transition from Optical Bound States in the Continuum to Leaky

Letter pubs.acs.org/journal/apchd5

Transition from Optical Bound States in the Continuum to Leaky Resonances: Role of Substrate and Roughness Zarina F. Sadrieva,*,† Ivan S. Sinev,† Kirill L. Koshelev,† Anton Samusev,† Ivan V. Iorsh,† Osamu Takayama,‡ Radu Malureanu,‡ Andrey A. Bogdanov,† and Andrei V. Lavrinenko†,‡ †

Department of Nanophotonics and Metamaterials, ITMO University, St. Petersburg 197101, Russia DTU Fotonik, Technical University of Denmark, Oersteds Plads 343, DK-2800 Kgs. Lyngby, Denmark

‡

ACS Photonics 2017.4:723-727. Downloaded from pubs.acs.org by WESTERN SYDNEY UNIV on 01/14/19. For personal use only.

S Supporting Information *

ABSTRACT: Optical bound states in the continuum (BIC) are localized states with energy lying above the light line and having infinite lifetime. Any losses taking place in real systems result in transformation of the bound states into resonant states with finite lifetime. In this Letter, we analyze properties of BIC in CMOS-compatible one-dimensional photonic structure based on silicon-on-insulator wafer at telecommunication wavelengths, where the absorption of silicon is negligible. We reveal that a high-index substrate could destroy both off-Γ BIC and in-plane symmetry protected at-Γ BIC turning them into resonant states due to leakage into the diffraction channels opening in the substrate. We show how two concurrent loss mechanisms, scattering due to surface roughness and leakage into substrate, contribute to the suppression of the resonance lifetime and specify the condition when one of the mechanisms becomes dominant. The obtained results provide useful guidelines for practical implementations of structures supporting optical bound states in the continuum. KEYWORDS: bound states in the continuum, high Q-factor, leakage losses, roughness, silicon-on-insulator

W

In theory, BICs have infinite high quality factor, however, for real systems, it is limited because of material absorption, technological imperfections, roughness, finite lateral size of samples, and leakage into the substrate. In this Letter we analyze, for the first time to the best of our knowledge, transformation of BIC into a resonant state due to surface roughness and high-index substrate beneath the structure opening the diffraction channels. We implement a one-dimensional photonic structure based on silicon-oninsulator (SOI) wafer and reveal the critical role of the substrate on BIC. We demonstrate its transformation to a resonant state,31 having a finite radiation lifetime, with the change of the thickness of the silica spacer separating the photonic structure and the substrate of SOI wafer. Specifically, we study the optical modes excited in such structures numerically by calculating the dispersion and angular-dependent Q-factor using finite element method (FEM). We also simulate the reflectance spectrum with finite-difference timedomain method (FDTD) and compare the results from both calculations. We corroborate the numerical calculations with the experimental reflectance spectra, obtained from the angleresolved reflectivity measurements. The design of the photonic structure under study is shown in Figure 1a. It consists of rectangular bars made of crystalline silicon surrounded by fused silica. Since the array is periodic in the x-direction, the eigenstates of the structure are charac-

hile localized states with energies under the light line of the surrounding space are ubiquitous,1−3 optical bound states in the continuum (BICs) are a remarkable exception for this general rule. The BICs are spatially bounded (localized in one, two, or even three dimensions4,5) eigenstates of an optical system with quality factor that could, in principle, approach infinity despite lying within the light cone of the surrounding space, that is, in the continuum spectrum of radiative modes.6−10 The concept of BIC was first proposed by von Neumann and Wigner in 1929 for electron placed in a specific potential.11 More recently, another examples of electronic BICs have been predicted theoretically in atomic and molecular systems12,13 and in artificial systems.14,15 In optics, the term bound state in the continuum first appeared around 2008,16,17 although this phenomenon has been studied earlier.18−21 Experimental observation of optical BIC followed only in 2011.22 The inherent features of BIC, namely, an infinitely high quality factor (Q-factor) and robustness against changing of system parameters, make it very promising for many applications ranging from on-chip photonics and optical communications23−25 to biological sensing26 and photovoltaics. The BICs may be exploited, for example, for quantum information processing27,28 and tunable channel dropping.29 It has been already shown30 that, in nonlinear materials due to the Kerr effect, a robust BIC arises in a self-adaptive way without necessity to tune the material parameters. Also, BICs are very promising for study interaction of light with matter due to their high quality factor.28 © 2017 American Chemical Society

Received: November 2, 2016 Published: March 24, 2017 723

DOI: 10.1021/acsphotonics.6b00860 ACS Photonics 2017, 4, 723−727

ACS Photonics

Letter

Figure 1. (a) Schematic image of photonic structure under consideration. Parameters of the structure: w = 150 nm; a = 1000 nm, t = 350 nm. Thickness of the bottom silica layer H is equal to 1 μm. (b, c) SEM images of the fabricated structure in side and top view, respectively. (d) Calculated dispersion curves for the first three TE modes of the photonic structure.

terized by the frequency f, the wave vector along the bars ky and by the Bloch quasi-wave vector kx restricted to the first Brillouin zone. It has been shown35,36 that such one-dimensional photonic structures support BICs either if the wave vector ky or quasi-wave vector kx is zero, that is, an excitation wave goes along the x- or y-direction. For the proposed design, BIC occurs when the ky component of wave vector is zero. Hence, in this case, the spectrum of the structure consists of pure TE and TM modes. Here we focus on TE modes only (electric field along the bars). The case of TM mode and BICs at normal and oblique incidence is analyzed in Supporting Information. We consider two cases: (i) the first one is referred to as “ideal”, because it describes a periodic array of silicon bars and infinite silica medium; (ii) the second one is termed “practical” and differs from the first case by adding a silicon substrate and surrounding air. The geometries of the first and second cases are shown in Figure 2a,c. Let us first consider an “ideal” structure without silicon substrate (see Figure 2a). The calculated dispersion curves of the three lower TE modes are shown in Figure 1d. The modes were numbered in order of occurrence in the spectrum, so the numbers do not indicate the characteristics of electric field. Since the eigenvalue solver returns complex eigenfrequencies f, we straightforwardly calculate the radiative Q-factor as Q=

Re(f ) 2Im(f )

Figure 2. Design of the photonic structure in “ideal” (a) and “practical” (c) cases. (b) Calculated distribution of Ey component for TE1 mode in the “ideal” case. (d) Calculated distribution of Ey component for TE1 mode in the “practical” case when H = 1 μm. The TE-polarized excitation wave has normal incidence for both cases. Calculated Qfactor (e) and dispersion curves (f) for both numerical models. Refractive indices of silica and silicon in the considered wavelength range have very weak dispersion,32−34 so, we put the constants nSiO2 = 1.44 and nSi = 3.48.

substrate), all diffraction channels, except zeroth one, are closed since the frequency of at-Γ BIC fΓ = 206 THz is below the diffraction limit: fΓ < c/(nSiO2a) ≈ 207.4 THz, where c is the vacuum speed of light. In the “practical” case, the photonic structure is placed on the silica spacer separating the structure and the silicon substrate. It might seem that the substrate should not destroy the at-Γ BIC since it is protected by in-plane symmetry, which is not broken by the substrate.35 However, it is true only for the subwavelength regime when the zeroth diffraction channel is forbidden due to the in-plane symmetry and higher diffraction channels are closed. In our case, the evanescent fields of the at-Γ BIC transform in the silicon substrate to propagating (diffractive) waves and the BIC becomes a resonant state. So, for the frequency fΓ = 206 THz, we have four waves diffracting at the angles αs = arcsin[snSic/ ( fΓa)], where s = ±1, ±2. The diffraction angles αs are defined with respect to the z-axis. Figure 2d shows how the energy stored in BIC leaks into the silicon substrate via open diffraction channels forming the characteristic interference pattern. Hence, a high-index substrate could destroy even a symmetry protected at-Γ BIC turning it into a resonant mode with a finite Q-factor. A similar effect is observed for off-Γ BICs (see Supporting Information). As the leaky losses of the BIC appear because of its near fields penetrating into the silicon substrate, the Q-factor exhibits an exponential dependence on the thickness of the oxide layer H (see blue curve in Figure 3). Infinite increasing of oxide layer thickness corresponds to absence of the substrate, that is, to the “ideal” case with BIC at the Γ-point. It is important that in spite of the fact the Q-factor rapidly drops with H (see Figure 2e), namely, it decreases almost 400-fold from 8 × 105 to 2 × 103 with the increment of the spacer

(1)

One can see from Figure 2b,e that the Q-factor of the TE1 mode tends to infinity at Γ-point of k-space (normal incidence), and light becomes perfectly confined in the slab. At this point, leaky resonance turns into a localized eigenmode that does not decay (see Figure 2b). It follows from Figure 2f that BIC for the “ideal” structure appears at a frequency of about 206 THz (1454 nm). For the “practical” case, the substrate weakly affects the position of the eigenmode, but results in leaky losses (see Figure 2d). The relative difference between the positions of modes in the “ideal” and “practical” cases is of an order of 0.005% or less and reaches the maximal value at the Γ-point, as shown in Figure 2f. The crucial role of the substrate on leaky losses of the waveguide modes was recently shown.37 For the SOI wafer, the leakage occurs if the bottom oxide layer is not sufficiently thick to fully optically insulate the photonic structure from the silicon substrate. In the “ideal” case (without 724

DOI: 10.1021/acsphotonics.6b00860 ACS Photonics 2017, 4, 723−727

ACS Photonics

Letter

Figure 3. Dependence of the Q-factor of resonant state (i.e., at normal incidence) on both roughness and distance to substrate H. The inset shows the silicon bar in side view and region dR with modified refractive index which is introduced to model losses from surface roughness. Refractive index of silicon n and width dR of roughness region are varied.

thickness from 3 to 1 μm, the resonant state does remain quite confined and has a Q-factor high enough for a number of applications. Another inherent feature of experimental samples is surface roughness, which leads to parasitic loss via scattering. Roughness of interfaces is equivalent to surface current resulting in radiation losses. These losses can be accounted for by introduction of an effective imaginary part to the refractive index of a thin near-surface layers.3,38−40 We analyze the influence of both roughness and substrate on the Q-factor of BIC using FEM simulation. The calculated Q-factors give account for the leakage into the silicon substrate and the −1 −1 scattering loss due to the roughness: Q−1 tot = Qrad + Qrough. The scattering loss does not depend on the thickness of the silica spacer. As it was mentioned earlier, the Q-factor of the resonant state for the structure without roughness shows an unlimited exponential growth with the thickness of the silica spacer H (see solid blue line in Figure 3). Figure 3 shows that for each case when roughness is taken into account there is a particular value H* from which the Q-factor becomes nearly constant. At the saturation point (H = H*), leaky losses into the substrate become approximately equal to losses due to roughness (Qrad ≈ Qrough) and almost independent of H. For H < H*, the leaky losses make the main contribution into the total ones (Qtot ≈ Qrad) and vice versa, for H > H*, the scattering due to roughness has a major contribution into the total losses (Qtot ≈ Qrough). Therefore, further increase in thickness H does not affect the Q-factor. To confirm our theoretical results experimentally, we fabricated a grating of silicon nanobars from SOI substrate with 1 μm thick silicon oxide layer covered by a 330 nm thick silicon slab shown in Figure 1 (details of the fabrication are presented in Methods). The parameters of the grating (period a = 1 μm, bar width w = 150 nm, bar height t = 330 nm) were chosen so that the guided modes are excited in the nearinfrared spectral range, where the absorption of silicon is negligible.33 To characterize the sample, we perform angle-resolved reflectivity measurements (details of the measurement are presented in Methods). Figure 4a shows the measured map of the angle-resolved reflectance spectra of the fabricated structure

Figure 4. (a) Measured reflectivity for TE polarization. The quality factor Q was calculated from experimental results for incident 4°. The inset shows reflection spectra for the angle of incidence from 2° to 4°. (b) Calculated reflectivity for TE polarization. The inset shows reflection spectra for incidence from 0.4° to 4°.

normalized to the reflectance spectra of the unprocessed SOI wafer. The observed modulation corresponds to the Fabry− Perot modes of the SOI wafer. The interference of the incident light with the leaky modes results in Fano-type resonances.19 The experimental spectra are in good agreement with the results of the FDTD simulation (Figure 4b). Comparison of Figures 1d and 4 shows that the two most pronounced lines correspond to TE1 and TE2 modes. As it is shown in the insets of Figure 4, for small angles of incidence, the Fano-type peak corresponding to the TE1 mode is less pronounced as compared to TE2 mode because of its higher Q-factor. The resonance lifetimes can be determined from reflection spectra by extracting the line widths of the observed Fano features.41 The results of the extraction for TE1 mode from both experimental and FDTD data are presented in Figure 5a along with the calculated Q-factor from the FEM simulations. Figure 5a shows that the Q-factor obtained from the FDTD calculation agrees well with the results of the FEM. The Qfactor extracted from the experimental reflectance spectra, however, reaches slightly lower values than the calculated Qfactor (Figure 5a). It should be noted that the resolution of the spectrometer allows to measure Q-factors up to 103. However, due to divergence of the excitation beam in the experiment, we measure the characteristics not of single states with specific Bloch wavenumbers but their convolution. It explains the difference between the experimental data and simulations. As it was mentioned above (see Figure 3), for photonic structures with thin bottom oxide layer, leakage into substrate is 725

DOI: 10.1021/acsphotonics.6b00860 ACS Photonics 2017, 4, 723−727

ACS Photonics

Letter

and could find a number of applications in optical communications, on-chip photonics, laser physics, and sensing.

■

METHODS Fabrication. The grating (Figure 1) is fabricated using electron beam lithography. The structure is then covered by 3 μm of silicon oxide via sputtering to create symmetric environment for guided modes of the grating. Finally, to suppress the unwanted Fabry−Perot interference from the thick silicon substrate, the rear face of the sample is processed via etching in SF6 plasma (“black silicon” process42). Optical Measurements. We perform angle-resolved reflectivity measurements using the setup schematically shown in Figure 6. We excite the sample with spectrally broadband supercontinuum source (NKT Photonics SuperK Extreme) mildly focused onto the sample surface via a set of parabolic mirrors. By using an adjustable iris diaphragm, we facilitate relatively low beam divergence (about 1°), while the size of the beam-spot on the sample is approximately 200 μm. The light reflected from the sample is filtered by a filter having a cut-on wavelength at 1200 nm and then coupled through a multimode fiber to an optical spectrum analyzer (Yokogawa AQ6375), allowing for 1 nm spectral resolution in the near-IR range. The sample and the collection accessories are mounted on separate rotation stages which allow for angle-resolved reflectance measurements with the angle of incidence from 2° and higher and a step of 0.5°. For a description of low-angle reflectance measurements, see Supporting Information.

Figure 5. (a) Quality factor Q vs incident angle extracted from experimental (shown with dots) and FDTD (dashed line) data and calculated with FEM for the “practical” case (solid line). During both calculations, substrate was taken into account. The Q-factor at the incident angle of 1° was obtained from reflectance measurements, described in Supporting Information. (b) Quality factor Q vs incident angle for “ideal structure” (without substrate). The data are obtained from eigenfrequency calculation (using FEM) and estimated from calculated reflectance (using FEM).

the main reason for the decreasing of the Q-factor at Γ-point (normal incidence). To confirm that the observed leaky resonance transforms back into BIC for symmetric environment, we repeat the same Fano fitting procedure for the simulated reflectance spectra of the “ideal” model and compare with the prediction of eigenfrequency (FEM) calculation in this case. One can see from Figure 6b that the BIC occurs at the normal incidence.

■

ASSOCIATED CONTENT

* Supporting Information S

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsphotonics.6b00860. Influence of surface roughness and leakage into the substrate on Q-factor of the off-Γ BIC; Details on lowangle reflectance measurements (PDF).

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected].

Figure 6. Scheme of setup for reflectance measurements.

ORCID

Zarina F. Sadrieva: 0000-0001-8299-3226 Andrey A. Bogdanov: 0000-0002-8215-0445

■

CONCLUSION In summary, we have theoretically and experimentally analyzed properties of the BIC in CMOS-compatible one-dimensional photonic structure based on silicon-on-insulator wafer at telecommunication wavelengths. We have shown how two concurrent loss mechanisms (leaky losses into the substrate and scattering due to roughenss) destroy the BIC. We have revealed that a high-index substrate could destroy even an at-Γ BIC protected by the in-plane symmetry and turn it into a resonant state. It happens owing to leakage into the diffraction channels opening in the substrate. We have shown that the leaky losses into the substrate have an exponential decrease with thickness of the silica spacer separating the photonic structure and the substrate. We have found a characteristic thickness of the spacer after which the total Q-factor becomes nearly constant since it is mainly contributed by the scattering losses. The obtained results provide useful guidelines for practical implementations of structures supporting optical bound states in the continuum

Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS The work has been supported by the Ministry of Education and Science of the Russian Federation (3.1668.2017), the President of Russian Federation (Grant MK-6462.2016.2), RFBR (16-3760064), and FP7-PEOPLE-2013-IRSES (GA 612564). K.L.K. acknowledges FASIE for the valuable support. The authors thank Maksym Plakhotnyuk for black silicon processing.

■

REFERENCES

(1) Joannopoulos, J. D.; Johnson, S.; Winn, J. N. J.; Meade, R. R. D. Photonic Crystals: Molding the Flow of Light; Princeton University Press, 2008; p 286. (2) Maier, S. A.; Atwater, H. A. Plasmonics: Localization and guiding of electromagnetic energy in metal/dielectric structures. J. Appl. Phys. 2005, 98, 011101.

726

DOI: 10.1021/acsphotonics.6b00860 ACS Photonics 2017, 4, 723−727

ACS Photonics

Letter

filtering in a dielectric grating. Phys. Rev. B: Condens. Matter Mater. Phys. 2014, 89, 165111. (27) Zheng, H.; Baranger, H. U. Persistent quantum beats and longdistance entanglement from waveguide-mediated interactions. Phys. Rev. Lett. 2013, 110, 1−5. (28) van Loo, A. F.; Fedorov, A.; Lalumiere, K.; Sanders, B. C.; Blais, A.; Wallraff, A. Photon-Mediated Interactions Between Distant Artificial Atoms. Science 2013, 342, 1494−1496. (29) Bulgakov, E.; Pichugin, K.; Sadreev, A. Channel dropping via bound states in the continuum in a system of two nonlinear cavities between two linear waveguides. J. Phys.: Condens. Matter 2013, 25, 395304. (30) Bulgakov, E. N.; Sadreev, A. F. Robust bound state in the continuum in a nonlinear microcavity embedded in a photonic crystal waveguide. Opt. Lett. 2014, 39, 5212. (31) Hsu, C. W.; Zhen, B.; Stone, A. D.; Joannopoulos, J. D.; Soljačić, M. Bound states in the continuum. Nat. Rev. Mater. 2016, 1, 16048. (32) Li, H. H. Refractive index of silicon and germanium and its wavelength and temperature derivatives. J. Phys. Chem. Ref. Data 1980, 9, 561−658. (33) Green, M. A.; Keevers, M. J. Optical properties of intrinsic silicon at 300 K. Prog. Photovoltaics 1995, 3, 189−192. (34) Malitson, I. H. Interspecimen Comparison of the Refractive Index of Fused Silica. J. Opt. Soc. Am. 1965, 55, 1205. (35) Zhen, B.; Hsu, C. W.; Lu, L.; Stone, A. D.; Soljačić, M. Topological Nature of Optical Bound States in the Continuum. Phys. Rev. Lett. 2014, 113, 257401. (36) Sadrieva, Z. F.; Bogdanov, A. A. Bound state in the continuum in the one-dimensional photonic crystal slab. J. Phys.: Conf. Ser. 2016, 741, 12122. (37) Sarmiento-Merenguel, D. J.; Ortega-Moñux, A.; Fédéli, J.-M.; Gonzalo Wangü e mert-Pér ez, J.; Alonso-Ramos, C.; Durán Valdeiglesias, E.; Cheben, P.; Molina-Fernández, I.;́ Halir, R. Controlling leakage losses in subwavelength grating silicon metamaterial waveguides. Opt. Lett. 2016, 41, 3443−3446. (38) Azzam, R. M. A.; Bashara, N. M. Polarization Characteristics of Scattered Radiation from a Diffraction Grating by Ellipsometry with Application to Surface Roughness. Phys. Rev. B 1972, 5, 4721−4729. (39) Benisty, H.; Labilloy, D.; Weisbuch, C.; Smith, C. J. M.; Krauss, T. F.; Cassagne, D.; Béraud, A.; Jouanin, C. Radiation losses of waveguide-based two-dimensional photonic crystals: Positive role of the substrate. Appl. Phys. Lett. 2000, 76, 532−534. (40) Maradudin, A. A.; Mills, D. L. Scattering and absorption of electromagnetic radiation by a semi-infinite medium in the presence of surface roughness. Phys. Rev. B 1975, 11, 1392−1415. (41) Rybin, M. V.; Samusev, K. B.; Sinev, I. S.; Semouchkin, G.; Semouchkina, E.; Kivshar, Y. S.; Limonov, M. F. Mie scattering as a cascade of Fano resonances. Opt. Express 2013, 21, 30107−30113. (42) Jansen, H.; de Boer, M.; Legtenberg, R.; Elwenspoek, M. The black silicon method: a universal method for determining the parameter setting of a fluorine-based reactive ion etcher in deep silicon trench etching with profile control. J. Micromech. Microeng. 1995, 5, 115.

(3) Dyakov, S. A.; Baldycheva, A.; Perova, T. S.; Li, G. V.; Astrova, E. V.; Gippius, N. a.; Tikhodeev, S. G. Surface states in the optical spectra of two-dimensional photonic crystals with various surface terminations. Phys. Rev. B: Condens. Matter Mater. Phys. 2012, 86, 115126. (4) Alù, A.; Engheta, N. Achieving transparency with plasmonic and metamaterial coatings. Phys. Rev. E - Stat. Nonlinear, Soft Matter Phys. 2005, 72, 1−9. (5) Monticone, F.; Alù, A. Embedded photonic eigenvalues in 3D nanostructures. Phys. Rev. Lett. 2014, 112, 1−5. (6) Hsu, C. W.; Zhen, B.; Lee, J.; Chua, S. L.; Johnson, S. G.; Joannopoulos, J. D.; Solijačić, M. Observation of trapped light within the radiation continuum. Nature 2013, 499, 188−91. (7) Longhi, S.; Della Valle, G. Floquet bound states in the continuum. Sci. Rep. 2013, 3, 2219. (8) Yang, Y.; Peng, C.; Liang, Y.; Li, Z.; Noda, S. Analytical Perspective for Bound States in the Continuum in Photonic Crystal Slabs. Phys. Rev. Lett. 2014, 113, 037401. (9) Lepetit, T.; Kanté, B. Controlling multipolar radiation with symmetries for electromagnetic bound states in the continuum. Phys. Rev. B: Condens. Matter Mater. Phys. 2014, 90, 241103. (10) Gansch, R.; Kalchmair, S.; Genevet, P.; Zederbauer, T.; Detz, H.; Andrews, A. M.; Schrenk, W.; Capasso, F.; Lončar, M.; Strasser, G. Measurement of bound states in the continuum by a detector embedded in a photonic crystal. Light: Sci. Appl. 2016, 5, e16147. (11) von Neumann, J.; Wigner, E. P. Uber merkwiirdige diskrete Eigenwerte. Phys. Z. 1929, 30, 465−467. (12) Stillinger, F. H.; Herrick, D. R.; Laboratories, B.; Hill, M. Bound states in the contimuun. Phys. Rev. A: At., Mol., Opt. Phys. 1975, 11, 446−454. (13) Cederbaum, L. S.; Friedman, R. S.; Ryaboy, V. M.; Moiseyev, N. Conical intersections and bound molecular states embedded in the continuum. Phys. Rev. Lett. 2003, 90, 13001. (14) Bulgakov, E. N.; Pichugin, K. N.; Sadreev, A. F.; Rotter, I. Bound states in the continuum in open Aharonov-Bohm rings. JETP Lett. 2006, 84, 430−435. (15) Sadreev, A. F.; Pilipchuk, A. S. Bound states in the continuum in zigzag quantum wire enforced by a finger gate. JETP Lett. 2015, 100, 585−590. (16) Marinica, D. C.; Borisov, A. G.; Shabanov, S. V. Bound States in the Continuum in Photonics. Phys. Rev. Lett. 2008, 100, 183902. (17) Bulgakov, E. N.; Sadreev, A. F. Bound states in the continuum in photonic waveguides inspired by defects. Phys. Rev. B: Condens. Matter Mater. Phys. 2008, 78, 075105. (18) Ochiai, T.; Sakoda, K. Dispersion relation and optical transmittance of a hexagonal photonic crystal slab. Phys. Rev. B: Condens. Matter Mater. Phys. 2001, 63, 125107. (19) Fan, S.; Joannopoulos, J. D. Analysis of guided resonances in photonic crystal slabs. Phys. Rev. B: Condens. Matter Mater. Phys. 2002, 65, 235112. (20) Tikhodeev, S. G.; Yablonskii, A. L.; Muljarov, E. A.; Gippius, N. A.; Ishihara, T. Quasiguided modes and optical properties of photonic crystal slabs. Phys. Rev. B: Condens. Matter Mater. Phys. 2002, 66, 45102. (21) Wiersig, J. Formation of Long-Lived, Scarlike Modes near Avoided Resonance Crossings in Optical Microcavities. Phys. Rev. Lett. 2006, 97, 253901. (22) Plotnik, Y.; Peleg, O.; Dreisow, F.; Heinrich, M.; Nolte, S.; Szameit, A.; Segev, M. Experimental Observation of Optical Bound States in the Continuum. Phys. Rev. Lett. 2011, 107, 183901. (23) Dreisow, F.; Szameit, a.; Heinrich, M.; Keil, R.; Nolte, S.; Tünnermann, a.; Longhi, S. Adiabatic transfer of light via a continuum in optical waveguides. Opt. Lett. 2009, 34, 2405−2407. (24) Kodigala, A.; Lepetit, T.; Gu, Q.; Bahari, B.; Fainman, Y.; Kanté, B. Lasing action from photonic bound states in continuum. Nature 2017, 541, 196−199. (25) Gentry, C. M.; Popović, M. A. Dark state lasers. Opt. Lett. 2014, 39, 4136. (26) Foley, J. M.; Young, S. M.; Phillips, J. D. Symmetry-protected mode coupling near normal incidence for narrow-band transmission 727

DOI: 10.1021/acsphotonics.6b00860 ACS Photonics 2017, 4, 723−727