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Second harmonic generation enhancement and directional emission from topological corner state based on the quantum spin Hall effect

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Abstract

Topological corner state has attracted much research interests since it does not obey the conventional bulk-edge correspondence and enables tightly confined light within small volumes. In this work, we demonstrate an enhanced second harmonic generation (SHG) from a topological corner state and its directional emission. To this end, we design an all-dielectric topological photonic crystal based on optical quantum spin Hall effect. In this framework, pseudospin states of photons, topological phase, and topological corner state are subsequently constructed by engineering the structures. It is shown that a high Q-factor of 3.66×1011 can be obtained at the corner state, showing strong confinement of light at the corner. Consequently, SHG is significantly boosted and manifests directional out-of-plane emission. More importantly, the enhanced SHG has robustness against a broad class of defects. These demonstrated properties offer practical advantages for integrated optical circuits.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Topological states, which are originated from condensed matter physics, have been widely extended in photonics since they may offer novel approaches in light engineering with intriguing topologically protected characteristics, such as immunity to backscattering from perturbation and suppressed scattering losses [13]. Recently, higher-order topological insulators (HOTIs) have been developed to demonstrate a breakdown of the conventional bulk-edge correspondence [4]. It has been shown that HOTIs may support topological states of two or more dimensions lower than them [5]. Particularly, a two-dimensional HOTI could present a zero-dimensional topological corner state, which has an extremely high Q-factor [6] and consequently a great ability to trap light in a small volume at their corner [7]. Together with the topological robustness against a broad class of defects [8], topological corner states have shown great potentials applications in various fields such as nanolasing and photoluminescence [9,10].

Concurrently, combining topological photonics with nonlinear optical effects has unlocked many fascinating phenomena and functionalities which have been absent in condensed matter physics [1113]. The emerging field of nonlinear topological photonics could be divided into two major aspects: the effects of topological states on nonlinearity and the converse. On one hand, nonlinearity may drive a topological trivial system into a nontrivial one, resulting in the topologically protected modes [14,15]. On the other hand, topological states offer a direct approach to investigate nonlinear optics due to their topological characteristics of free from scattering, low radiation loss, and highly localization of light [16,17]. Nonlinear responses in topological photonics opened up an avenue to develop advanced functionalities, such as nonlinear nonreciprocity and frequency conversion [18,19]. It has been observed that a strong enhancement of nonlinear photon generation could exist at topological edge state [17], showing topological robustness against various perturbations and controllable unidirectional excitation. Very recently, all dielectric topological photonics hosting double valley-Hall kink modes were designed to achieve tunable and bidirectional phase-matched second harmonic generation (SHG) [20]. Obviously, these results demonstrated the advantages of topological photonics over metamaterials in nonlinear photon generation owing to the unique features of topological modes [2125]. Nevertheless, there still remain challenges to put forward the nonlinear photon generation with higher performances from the aspect of application.

In this paper, we design an all-dielectric topological photonic crystal to achieve topological corner states with an extremely high Q-factors, thereby obtaining enhanced SHG and its directional emission. The designed topological photonic crystal is two-dimensional and based on the optical quantum spin Hall effect. We numerically investigate concentration of fundamental light at the corner and the corresponding value of Q-factor. Then, the SHG from the corner state and the emission direction are studied. Furthermore, the influences of corner size and structure perturbation on the SHG are discussed. These results may provide new opportunities for the practical application of topology-enabled nonlinear photon generation in integrated optical systems.

2. Structure and theory

The designed two-dimensional photonic crystal, which supports a zero-dimensional corner state, consists of two kinds of hexagonal array of all dielectric cylinders in air, as shown in Figs. 1(a) and 1(b). These two different photonic crystals are indicated by the blue and yellow regions. The lattice constant of the hexagonal lattice photonic crystals is a=1µm, which is the center-to-center distance between two neighboring lattices. The radius of each cylinder is r = a/11, and the distance from the center of each cylinder to the center of the lattice is L, as shown in Fig. 1(b). All simulations were carried out by using the commercial software COMSOL Multiphysics. The investigations on eigenmodes were calculated in the frequency domain with a unit lattice and periodic boundary conditions. Herein, we consider the transverse magnetic (TM) mode, containing out-of-plane electric field and in-plane magnetic field. While, the excitation of topological states and nonlinear emission were simulated in the time domain, where our designed structure is covered by perfectly matched layers with scattering boundaries.

 figure: Fig. 1.

Fig. 1. (a) Designed photonic crystal structure to obtain corner state. a is the lattice constant. (b) Two lattices of PCs consist of six cylinders made of dielectric material, which are arranged in the air. L is the distance from center of each cylinder to center of the lattice, and r is the radius of cylinders. The blue and yellow colors represent different distances L. Band structures of PCs with (c) a/L=4.25, (d) a/L=3, and (e) a/L=2.5. (Inset: Brillouin zone of triangular lattice; d and p represent dipole and quadrupole modes, respectively; ‘±’ represents the parity of first three bands at Γ and M point.). The Ez field distributions (TM mode) of dipoles and quadrupoles under topologically (f) trivial and (g) nontrivial conditions.

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We start by investigating the evolution of photonic band structures of the designed system when L changes. Figures 1(c), 1(d), and 1(e) plot the results of the cases with a/L=4.25, a/L=3, and a/L=2.5, respectively. When the a/L=3, double degeneracy forms the double Dirac cones at the Γ point due to the band folding process, in which the Dirac cones at K and K´ points in the first Brillouin zone fold together to form a fourfold degenerated point. It could be attributed to the equivalence between the designed system (a/L=3) and the honeycomb lattice of cylinders. When the lattice is either contracted (a/L=4.25) or expanded (a/L=2.5), as schematically shown in Fig. 1(b), the lattice symmetry will be broken, and the fourfold degenerated point is opened and a band gap is formed. In principle, a band inversion takes place upon varying the value of L in the designed system. The ${E_z}$ fields at the Γ point are calculated for the cases in Figs. 1(c) and 1(e), as shown in Figs. 1(f) and 1(g), respectively. It can be seen that the unit structures carry orbitals of px (py) and dxy (${d_{{x^2} - {y^2}}}$), which have similar symmetries as electronic orbitals of conventional atoms in solid. Based on the ${C_6}$ symmetry group of the designed system, two kinds of pseudospin states are formed and given by [24]

$${p_{\pm}} = {{({{p_{x}} \pm i{p_{y}}} )} / {\sqrt 2 }};{d_{\pm} } = {{({{d_{{x^2} - {y^2}}} \pm i{d_{xy}}} )} / {\sqrt 2 }},$$
where the sign denotes up and down pseudospins, which are equivalent to positive and negative orbital angular momenta of the ${E_z}$ field wave function. For the contracted case, the photonic band below and above the gap is occupied by ${p_ \pm }$ and ${d_ \pm }$ states, respectively. For the extended case, the ${E_z}$ field at the high and low frequency side of the band gap exhibit characters of ${p_ \pm }$ and ${d_ \pm }$, respectively, showing opposite behavior to the contracted case which could be attributed to spin-orbit coupling.

It has been demonstrated the band inversion process experiences the topological phase transition, which could be characterized by a dipole moment P [26]. For the extended case, the dipole moments of all the three bands below the gap are 1/2, 0, and 1/2, respectively, indicating the nontrivial phase of the system. In contrast, the dipole moments are zero for the first three bands in the contracted case, showing trivial state of the system. To judge the existence of topological corner state, a quadrupole moment Q [26] is required to represent the topological corner charge, which is well related with the dipole moments P. The values of Q are calculated to be 0 and 1/2 in the trivial and nontrivial cases, respectively. The nonzero topological corner charge ensures the emergence of topological corner state. One may refer to the literature for detailed calculations [2729].

To obtain a topological corner state, the topologically trivial and nontrivial lattices are arranged as shown in Fig. 1(a). The first and third quadrants are nontrivial, in contrast, the second and fourth quadrants are trivial. Along the x-axis, the two types of lattice couple with each other through the armchair-type interface [30], while along the y-axis, the broken zigzag-type interface is arranged and the unit cells are cut in half along the interface. It should be mentioned that the whole photonic crystals contain about 400 lattices (20×20 lattices), which are surrounded by perfectly matched layers with scattering boundaries.

To demonstrate the existence of the topological corner state, the eigenmodes of the designed system in Fig. 1(a) are analyzed near the frequency of the first double Dirac cone. The calculated eigenfrequency spectrum in Fig. 2(a) indicates the existences of bulk, edge and corner states in the designed system. To validate, the normalized electric field intensity at these states is simulated. Figure 2(b) shows diffusion of the bulk mode across the whole photonic crystal lattice. Within the bulk gap, the topological protected edge state appears, showing electric field confinement along the interfaces between the trivial and nontrivial regions, as shown in Fig. 2(c). Beyond the bulk-edge correspondence, the corner state arises at the center of the designed system where the interfaces intersect, presenting strong confinement of electric field inside a small volume, as shown in Fig. 2(d). This corner state confined in a small volume may function as a nanocavity with higher Q-factor compared with the bulk and edge states [31,32]. Figure 2(e) illustrates the dependence of Q-factor on mode number. The value of Q-factor could reach a maximum of 1.65×108, further confirming the great ability of restricting light in a small area.

 figure: Fig. 2.

Fig. 2. (a) Calculated eigenfrequency spectrum of the designed photonic crystal system. Blue dots, green dots and red star denote bulk, edge and corner states, respectively. Electric field intensities at the (b) bulk, (c) edge and (d) corner states. The inset in (d) enlarges the area near the corner marked by white dashed lines. (e) Dependence of Q-factor on mode number.

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3. Results and discussion

The above results obviously show energy confinement at corner state, thereby together with intrinsic topological robustness [33], corner state may significantly improve the characteristic of nonlinear emission from nanostructures. To demonstrate, SHG from corner, edge and bulk states are calculated considering the second-order susceptibility of AlGaAs to be χ(2)=100 pm/V [34]. Theoretically, the emission of SHG is determined by the nonlinear polarization oscillating at second harmonic frequency $P_\textrm{z}^{{\mathrm{\Omega }_\textrm{2}}} = {\chi _2}E_{\textrm{1z}}^\textrm{2}$, where E1z is the electric field at fundamental frequency. Therefore, the SHG can be enhanced when electric field is highly localized at the corner state. The time-domain responses of the designed system are simulated with a sinusoidal point source, which can be described as E1z=E0sin1×t). E0 is set to be 3e9 V/m and Ω1=2πf1, where f1 is the fundamental frequency of excitation source. f1=154.99THz, 162.99THz, and 186.79THz corresponds to bulk, edge and corner states, respectively. For simplicity, we do not consider the circular polarized sources. Nevertheless, it should be noted that the excitation of corner state is spin-dependent since the designed system is based on quantum spin Hall effect [5]. After Fourier transformations, the frequency spectra obtained from the designed system at bulk, edge and corner states are plotted in Fig. 3(a). The blue solid line shows that second harmonic signal strongly emits from the corner state. The green and red solid lines show the obtained frequency spectra from the bulk and edge states, presenting no obvious second harmonic signal. Comparisons between these frequency spectra indicate the enhancement of SHG from topological corner state. It should be noted that the position of excitation source is optimized, and the emission signal are detected at the positions where the electric field has maximum value in each case. In these three cases, a same sinusoidal current is utilized as excitation source. The detected electric fields have different intensities at their fundamental frequency, which could be attributed to the different scattering behaviors of light at bulk, edge and corner states. Figures 3(b)–3(d) plot the SHG electric field distributions at three representative pump frequencies: in Fig. 3(b), the pump corresponds to corner state and dazzling SHG emission from a small volume near the corner can be viewed; in Figs. 3(c) and 3(d), the pumps are tuned to the bulk and edge frequencies, therefore the SHG comes from the bulk and edge regions, respectively. The obtained SHG distribution at the topological corner state is 2 and 3 orders of magnitude higher than those coming from edge and bulk states, respectively.

 figure: Fig. 3.

Fig. 3. (a) Detected emission frequency spectra from the designed system at bulk, edge and corner states. Simulated SHG electric field distributions at (b) corner, (c) bulk, and (d) edge states. The white and yellow circles indicate the positions of excitation and detection, respectively.

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As demonstrated previously, the topological corner state can be treated as a nanocavity with a high Q-factor [31]. To improve the SHG emission, the Q-factor of the corner state is optimized by tuning the size of nanocavity. Figure 4(a) schematically shows the designed system before and after increasing the distance D between trivial and nontrivial region in our designed system. The black line with filled dots and blue line with hollow dots in Fig. 4(b) represent the simulated Q-factor and eigen-frequency of the corner state for the case with different values of D. When the value of D continuously increases, the Q-factor first rises and then descends, while, the eigen-frequency shifts to lower energy. When D=0.2µm, the Q-factor of corner state may reach the highest value of 3.66×1011, which could be attributed to balance between radiation loss and transverse loss [32]. Figure 4(c) plots the emission spectra in frequency domain from corner states with different values of D. It could be seen that the intensity peak of second harmonic emission goes up and down when D increases, in the meantime, the emitted frequency red shifts.

 figure: Fig. 4.

Fig. 4. (a) Schematic of the proposed system before and after increasing the size of nanocavity. D is the distance between trivial and nontrivial regions. (b) Dependences of the Q-factor and eigen-frequency of the corner state on the value of D, and (c) the corresponding emission spectra from the corner state.

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Besides the enhancement of SHG emission, topological corner state has another unique advantage of robustness against a broad class of perturbations around the corner. In other word, the SHG emission from corner state could exist with high performance if the perturbations do not break the topological characteristic of the designed photonic crystals. Now we introduce several typical perturbations and observe their robustness, and their corresponding SHG distribution patterns are plotted in Figs. 5(a)–5(c). In perturbation I, we introduced lattice defect by removing a unit lattice, which is denoted by the white circle in Fig. 5(a); in perturbation II, we introduced a bend interface between the second and third quadrants, as shown by the white dashed lines in Fig. 5(b); in perturbation III, the lattice defect and the bend interface exist simultaneously. Note that, the white dashed lines represent the interfaces between trivial and nontrivial regions, the white circles in Figs. 5(a) and 5(c) marks the position of the removed unit lattice, and the bend angles in Figs. 5(b) and 5(c) are α=π/3. From the above results, the SHG from topological corner states can be excited effectively. The emission frequency spectra are detected and plotted in Fig. 5(d), showing the topologically protected SHG emission. Further, Fig. 5(e) summarizes the frequency and full-width half-maximum (FWHM) of SHG emission. In comparison with the case without perturbation, the frequency shifts of emitted SHG induced by the perturbations are within 1.28 THz. In addition, the FWHM at emitted SHG frequency remains smaller than 0.005THz which is negligible compared to the central frequencies. The robustness of topological SHG from the corner state against the perturbations further demonstrates their great potential in practical applications, such as nanolasers, frequency conversion, and quantum information in nanooptics [3537].

 figure: Fig. 5.

Fig. 5. Topological robustness of the SHG emission from corner state. SHG distribution patterns for the cases with (a) lattice defect (Perturbation I), (b) bend interface (Perturbation II), and (c) bend interface with lattice defect (Perturbation III). A unit lattice inside the white circle is removed. The white dashed lines represent the interfaces between trivial and nontrivial regions. (d) The emitted frequency spectra from corners with the perturbations in (a-c). (e) Full-width half-maximum and SHG frequency for the cases with and without perturbations.

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The SHG enhancement and emission performances discussed above are originated from the physical nature of topological corner state, where the band inversion between p band and d band happens at the Γ point. It has been demonstrated that this topological nature may result in zero in-plane wavevector, thereby providing an orthogonal channel between the topological system and free space [38,39]. For a comprehensive analysis, it is important to evaluate the angular intensity distributions in far-field free-space, since they may provide additional information [40]. Therefore, the angle-resolved far field pattern of the SHG from corner state is calculated by Fourier transformation from the near field distribution in Fig. 3(b) and plotted in Fig. 6(a). It can be seen that the SHG directionally and perpendicularly emits out of plane. Figures 6(b) and 6(c) show the emissions in x-z and y-z planes with divergence angles of 5° and 3°, respectively. More importantly, the topological protection of SHG emission extends to the directional feature, as demonstrated in Figs. 6(d)–6(f). With perturbations I, II, III, the divergence angles are always smaller than 5° in both x-z and y-z planes.

 figure: Fig. 6.

Fig. 6. Directional emission of SHG from the topological corner state. (a) Angle-resolved far field pattern of SHG emission. Far field patterns in (b) x-z and y-z planes which are extracted from (a). Far field patterns in both x-z and y-z planes from the corners with perturbations (d) I, (e) II, and (f) III discussed in Fig. 5.

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4. Conclusion

In this work, we demonstrated the enhancement and directional emission of SHG from a topological corner state. For this purpose, an all-dielectric topological photonic crystal is designed to achieve optical quantum spin Hall effect. The pseudospin states of photons, topological phase, and topological corner state were subsequently constructed by engineering the structures. It has been demonstrated that the value of Q-factor can be as high as $3.66 \times {10^{11}}$ at the corner state, resulting in strong confinement of light at the corner. Consequently, SHG was significantly boosted and directionally emitted out-of-plane. More importantly, the enhanced and directional SHG emission showed robustness against a broad class of defects. These demonstrated properties offer practical advantages for integrated optical circuits.

Funding

National Natural Science Foundation of China (11804073, 61775050); Fundamental Research Funds for the Central Universities (JD2020JGPY0009, PA2019GDZC0098).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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References

  • View by:

  1. Y. Yang, Z. Jia, Y. Wu, R. C. Xiao, Z. H. Hang, H. Jiang, and X. C. Xie, “Gapped topological kink states and topological corner states in honeycomb lattice,” Sci. Bull. 65(7), 531–537 (2020).
    [Crossref]
  2. J. Noh, W. A. Benalcazar, S. Huang, M. J. Collins, K. P. Chen, T. L. Hughes, and M. C. Rechtsman, “Topological protection of photonic mid-gap defect modes,” Nat. Photonics 12(7), 408–415 (2018).
    [Crossref]
  3. C. W. Chen, R. Chaunsali, J. Christensen, G. Theocharis, and J. Yang, “Corner states in second-order mechanical topological insulator,” Commun. Mater. 2(1), 62 (2021).
    [Crossref]
  4. M. R. López, Z. Zhang, D. Torrent, and J. Christensen, “Multiple scattering theory of non-Hermitian sonic second-order topological insulators,” Commun. Phys. 2(1), 132 (2019).
    [Crossref]
  5. B. Xie, G. Su, H. F. Wang, F. Liu, L. Hu, S. Y. Yu, P. Zhan, M. H. Lu, Z. Wang, and Y. F. Chen, “Higher-order quantum spin Hall effect in a photonic crystal,” Nat. Commun. 11(1), 3768 (2020).
    [Crossref]
  6. X. Xie, W. Zhang, X. He, S. Wu, J. Dang, K. Peng, F. Song, L. Yang, H. Ni, Z. Niu, C. Wang, K. Jin, X. Zhang, and X. Xu, “Cavity quantum electrodynamics with second-order topological corner state,” Laser Photonics Rev. 14(8), 1900425 (2020).
    [Crossref]
  7. A. Shi, B. Yan, R. Ge, J. Xie, Y. Peng, H. Li, W. E. I. Sha, and J. Liu, “Coupled cavity-waveguide based on topological corner state and edge state,” Opt. Lett. 46(5), 1089–1092 (2021).
    [Crossref]
  8. G. Siroki, P. A. Huidobro, and V. Giannini, “Topological photonics: From crystals to particles,” Phys. Rev. B 96(4), 041408 (2017).
    [Crossref]
  9. C. Han, M. Kang, and H. Jeon, “Lasing at multidimensional topological states in a two-dimensional photonic crystal structure,” ACS Photonics 7(8), 2027–2036 (2020).
    [Crossref]
  10. A. S. Berestennikov, A. Vakulenko, S. Kiriushechkina, M. Li, Y. Li, L. E. Zelenkov, A. P. Pushkarev, M. A. Gorlach, A. L. Rogach, S. V. Makarov, and A. B. Khanikaev, “Enhanced photoluminescence of halide perovskite nanocrystals mediated by a higher-order topological metasurface,” J. Phys. Chem. C 125(18), 9884–9890 (2021).
    [Crossref]
  11. S. Kruk, W. Gao, D. Y. Choi, T. Zentgraf, S. Zhang, and Y. Kivshar, “Nanoscale topological corner states in nonlinear optics,” Nano Lett. 21(11), 4592–4597 (2021).
    [Crossref]
  12. D. Smirnova, S. Kruk, D. Leykam, E. Melik-Gaykazyan, D. Y. Choi, and Y. Kivshar, “Third-harmonic generation in photonic topological metasurfaces,” Phys. Rev. Lett. 123(10), 103901 (2019).
    [Crossref]
  13. Z. Lan, J. W. You, and N. C. Panoiu, “Nonlinear one-way edge-mode interactions for frequency mixing in topological photonic crystals,” Phys. Rev. B 101(15), 155422 (2020).
    [Crossref]
  14. L. J. Maczewsky, M. Heinrich, M. Kremer, S. K. Ivanov, M. Ehrhardt, F. Martinez, Y. V. Kartashov, V. V. Konotop, L. Torner, D. Bauer, and A. Szameit, “Nonlinearity-induced photonic topological insulator,” Science 370(6517), 701–704 (2020).
    [Crossref]
  15. D. A. Dobrykh, A. V. Yulin, A. P. Slobozhanyuk, A. N. Poddubny, and Y. S. Kivshar, “Nonlinear control of electromagnetic topological edge states,” Phys. Rev. Lett. 121(16), 163901 (2018).
    [Crossref]
  16. F. Zangeneh-Nejad and R. Fleury, “Nonlinear second-order topological insulators,” Phys. Rev. Lett. 123(5), 053902 (2019).
    [Crossref]
  17. S. Kruk, A. Poddubny, D. Smirnova, L. Wang, A. Slobozhanyuk, A. Shorokhov, I. Kravchenko, B. Luther-Davies, and Y. Kivshar, “Nonlinear light generation in topological nanostructures,” Nat. Nanotechnol. 14(2), 126–130 (2019).
    [Crossref]
  18. W. Zhang, J. Tang, Y. Ming, C. Zhang, and Y. Lu, “Optical-field topological phase transition in nonlinear frequency conversion,” Opt. Express 28(3), 2818–2827 (2020).
    [Crossref]
  19. X. Huang, C. Lu, C. Liang, H. Tao, and Y. C. Liu, “Loss-induced nonreciprocity,” Light: Sci. Appl. 10(1), 30 (2021).
    [Crossref]
  20. Z. Lan, J. W. You, Q. Ren, E. I. Wei, and N. C. Panoiu, “Second-harmonic generation via double topological valley-Hall kink modes in all-dielectric photonic crystals,” Phys. Rev. A 103(4), L041502 (2021).
    [Crossref]
  21. M. Scalora, M. A. Vincenti, D. de Ceglia, V. Roppo, M. Centini, N. Akozbek, and M. J. Bloemer, “Second- and third-harmonic generation in metal-based structures,” Phys. Rev. A 82(4), 043828 (2010).
    [Crossref]
  22. C. Ciracì, E. Poutrina, M. Scalora, and D. R. Smith, “Origin of second-harmonic generation enhancement in optical split-ring resonators,” Phys. Rev. B 85(20), 201403 (2012).
    [Crossref]
  23. C. Ciraci, E. Poutrina, M. Scalora, and D. R. Smith, “Second-harmonic generation in metallic nanoparticles: Clarification of the role of the surface,” Phys. Rev. B 86(11), 115451 (2012).
    [Crossref]
  24. L. H. Wu and X. Hu, “Scheme for achieving a topological photonic crystal by using dielectric material,” Phys. Rev. Lett. 114(22), 223901 (2015).
    [Crossref]
  25. Y. Yang, Y. F. Xu, T. Xu, H. X. Wang, J. H. Jiang, X. Hu, and Z. H. Hang, “Visualization of a unidirectional electromagnetic waveguide using topological photonic crystals made of dielectric materials,” Phys. Rev. Lett. 120(21), 217401 (2018).
    [Crossref]
  26. Z. Zhang, B. Hu, F. Liu, Y. Cheng, X. Liu, and J. Christensen, “Pseudospin induced topological corner state at intersecting sonic lattices,” Phys. Rev. B 101(22), 220102 (2020).
    [Crossref]
  27. F. Liu, H. Y. Deng, and K. Wakabayashi, “Helical topological edge states in a quadrupole phase,” Phys. Rev. Lett. 122(8), 086804 (2019).
    [Crossref]
  28. W. A. Benalcazar, T. Li, and T. L. Hughes, “Quantization of fractional corner charge in C n-symmetric higher-order topological crystalline insulators,” Phys. Rev. B 99(24), 245151 (2019).
    [Crossref]
  29. Z. Zhang, H. Long, C. Liu, C. Shao, Y. Cheng, X. Liu, and J. Christensen, “Deep-Subwavelength Holey Acoustic Second-Order Topological Insulators,” Adv. Mater. 31(49), 1904682 (2019).
    [Crossref]
  30. S. Wang, L. Talirz, C. A. Pignedoli, X. Feng, K. Müllen, R. Fasel, and P. Ruffieux, “Giant edge state splitting at atomically precise graphene zigzag edges,” Nat. Commun. 7(1), 11507 (2016).
    [Crossref]
  31. Y. Ota, F. Liu, R. Katsumi, K. Watanabe, K. Wakabayashi, Y. Arakawa, and S. Iwamoto, “Photonic crystal nanocavity based on a topological corner state,” Optica 6(6), 786–789 (2019).
    [Crossref]
  32. W. Zhang, X. Xie, H. Hao, J. Dang, S. Xiao, S. Shi, H. Ni, Z. Niu, C. Wang, K. Jin, X. Zhang, and X. Xu, “Low-threshold topological nanolasers based on the second-order corner state,” Light: Sci. Appl. 9(1), 109 (2020).
    [Crossref]
  33. X. D. Chen, W. M. Deng, F. L. Shi, F. L. Zhao, M. Chen, and J. W. Dong, “Direct observation of corner states in second-order topological photonic crystals slab,” Phys. Rev. Lett. 122(23), 233902 (2019).
    [Crossref]
  34. L. Xu, M. Rahmani, D. Smirnova, K. Zangeneh Kamali, G. Zhang, D. Neshev, and A. E. Miroshnichenko, “Highly-efficient longitudinal second-harmonic generation from doubly-resonant AlGaAs nanoantennas,” Photonics 5(3), 29 (2018).
    [Crossref]
  35. D. Smirnova, D. Leykam, Y. Chong, and Y. Kivshar, “Nonlinear topological photonics,” Appl. Phys. Rev. 7(2), 021306 (2020).
    [Crossref]
  36. A. S. Kupriianov, Y. Xu, A. Sayanskiy, V. Dmitriev, Y. S. Kivshar, and V. R. Tuz, “Metasurface engineering through bound states in the continuum,” Phys. Rev. Appl. 12(1), 014024 (2019).
    [Crossref]
  37. Q. Ren, F. Feng, X. Yao, Q. Xu, M. Xin, Z. Lan, J. You, X. Xiao, and W. E. I. Sha, “Multiplexing-oriented plasmon-MoS2 hybrid metasurfaces driven by nonlinear quasi bound in the continuum,” Opt. Express 29(4), 5384–5396 (2021).
    [Crossref]
  38. Z. K. Shao, H. Z. Chen, S. Wang, X. R. Mao, Z. Q. Yang, S. L. Wang, X. X. Wang, X. Hu, and R. M. Ma, “A high-performance topological bulk laser based on band-inversion-induced reflection,” Nat. Nanotechnol. 15(1), 67–72 (2020).
    [Crossref]
  39. S. Tong and C. Ren, “Directional acoustic emission via topological insulators based on cavity-channel networks,” Appl. Phys. Lett. 117(9), 093504 (2020).
    [Crossref]
  40. J. Yang, J. P. Hugonin, and P. Lalanne, “Near-to-far field transformations for radiative and guided waves,” ACS Photonics 3(3), 395–402 (2016).
    [Crossref]

2021 (7)

C. W. Chen, R. Chaunsali, J. Christensen, G. Theocharis, and J. Yang, “Corner states in second-order mechanical topological insulator,” Commun. Mater. 2(1), 62 (2021).
[Crossref]

A. Shi, B. Yan, R. Ge, J. Xie, Y. Peng, H. Li, W. E. I. Sha, and J. Liu, “Coupled cavity-waveguide based on topological corner state and edge state,” Opt. Lett. 46(5), 1089–1092 (2021).
[Crossref]

A. S. Berestennikov, A. Vakulenko, S. Kiriushechkina, M. Li, Y. Li, L. E. Zelenkov, A. P. Pushkarev, M. A. Gorlach, A. L. Rogach, S. V. Makarov, and A. B. Khanikaev, “Enhanced photoluminescence of halide perovskite nanocrystals mediated by a higher-order topological metasurface,” J. Phys. Chem. C 125(18), 9884–9890 (2021).
[Crossref]

S. Kruk, W. Gao, D. Y. Choi, T. Zentgraf, S. Zhang, and Y. Kivshar, “Nanoscale topological corner states in nonlinear optics,” Nano Lett. 21(11), 4592–4597 (2021).
[Crossref]

X. Huang, C. Lu, C. Liang, H. Tao, and Y. C. Liu, “Loss-induced nonreciprocity,” Light: Sci. Appl. 10(1), 30 (2021).
[Crossref]

Z. Lan, J. W. You, Q. Ren, E. I. Wei, and N. C. Panoiu, “Second-harmonic generation via double topological valley-Hall kink modes in all-dielectric photonic crystals,” Phys. Rev. A 103(4), L041502 (2021).
[Crossref]

Q. Ren, F. Feng, X. Yao, Q. Xu, M. Xin, Z. Lan, J. You, X. Xiao, and W. E. I. Sha, “Multiplexing-oriented plasmon-MoS2 hybrid metasurfaces driven by nonlinear quasi bound in the continuum,” Opt. Express 29(4), 5384–5396 (2021).
[Crossref]

2020 (12)

Z. K. Shao, H. Z. Chen, S. Wang, X. R. Mao, Z. Q. Yang, S. L. Wang, X. X. Wang, X. Hu, and R. M. Ma, “A high-performance topological bulk laser based on band-inversion-induced reflection,” Nat. Nanotechnol. 15(1), 67–72 (2020).
[Crossref]

S. Tong and C. Ren, “Directional acoustic emission via topological insulators based on cavity-channel networks,” Appl. Phys. Lett. 117(9), 093504 (2020).
[Crossref]

W. Zhang, X. Xie, H. Hao, J. Dang, S. Xiao, S. Shi, H. Ni, Z. Niu, C. Wang, K. Jin, X. Zhang, and X. Xu, “Low-threshold topological nanolasers based on the second-order corner state,” Light: Sci. Appl. 9(1), 109 (2020).
[Crossref]

Z. Zhang, B. Hu, F. Liu, Y. Cheng, X. Liu, and J. Christensen, “Pseudospin induced topological corner state at intersecting sonic lattices,” Phys. Rev. B 101(22), 220102 (2020).
[Crossref]

Z. Lan, J. W. You, and N. C. Panoiu, “Nonlinear one-way edge-mode interactions for frequency mixing in topological photonic crystals,” Phys. Rev. B 101(15), 155422 (2020).
[Crossref]

L. J. Maczewsky, M. Heinrich, M. Kremer, S. K. Ivanov, M. Ehrhardt, F. Martinez, Y. V. Kartashov, V. V. Konotop, L. Torner, D. Bauer, and A. Szameit, “Nonlinearity-induced photonic topological insulator,” Science 370(6517), 701–704 (2020).
[Crossref]

C. Han, M. Kang, and H. Jeon, “Lasing at multidimensional topological states in a two-dimensional photonic crystal structure,” ACS Photonics 7(8), 2027–2036 (2020).
[Crossref]

W. Zhang, J. Tang, Y. Ming, C. Zhang, and Y. Lu, “Optical-field topological phase transition in nonlinear frequency conversion,” Opt. Express 28(3), 2818–2827 (2020).
[Crossref]

B. Xie, G. Su, H. F. Wang, F. Liu, L. Hu, S. Y. Yu, P. Zhan, M. H. Lu, Z. Wang, and Y. F. Chen, “Higher-order quantum spin Hall effect in a photonic crystal,” Nat. Commun. 11(1), 3768 (2020).
[Crossref]

X. Xie, W. Zhang, X. He, S. Wu, J. Dang, K. Peng, F. Song, L. Yang, H. Ni, Z. Niu, C. Wang, K. Jin, X. Zhang, and X. Xu, “Cavity quantum electrodynamics with second-order topological corner state,” Laser Photonics Rev. 14(8), 1900425 (2020).
[Crossref]

Y. Yang, Z. Jia, Y. Wu, R. C. Xiao, Z. H. Hang, H. Jiang, and X. C. Xie, “Gapped topological kink states and topological corner states in honeycomb lattice,” Sci. Bull. 65(7), 531–537 (2020).
[Crossref]

D. Smirnova, D. Leykam, Y. Chong, and Y. Kivshar, “Nonlinear topological photonics,” Appl. Phys. Rev. 7(2), 021306 (2020).
[Crossref]

2019 (10)

A. S. Kupriianov, Y. Xu, A. Sayanskiy, V. Dmitriev, Y. S. Kivshar, and V. R. Tuz, “Metasurface engineering through bound states in the continuum,” Phys. Rev. Appl. 12(1), 014024 (2019).
[Crossref]

Y. Ota, F. Liu, R. Katsumi, K. Watanabe, K. Wakabayashi, Y. Arakawa, and S. Iwamoto, “Photonic crystal nanocavity based on a topological corner state,” Optica 6(6), 786–789 (2019).
[Crossref]

M. R. López, Z. Zhang, D. Torrent, and J. Christensen, “Multiple scattering theory of non-Hermitian sonic second-order topological insulators,” Commun. Phys. 2(1), 132 (2019).
[Crossref]

F. Zangeneh-Nejad and R. Fleury, “Nonlinear second-order topological insulators,” Phys. Rev. Lett. 123(5), 053902 (2019).
[Crossref]

S. Kruk, A. Poddubny, D. Smirnova, L. Wang, A. Slobozhanyuk, A. Shorokhov, I. Kravchenko, B. Luther-Davies, and Y. Kivshar, “Nonlinear light generation in topological nanostructures,” Nat. Nanotechnol. 14(2), 126–130 (2019).
[Crossref]

D. Smirnova, S. Kruk, D. Leykam, E. Melik-Gaykazyan, D. Y. Choi, and Y. Kivshar, “Third-harmonic generation in photonic topological metasurfaces,” Phys. Rev. Lett. 123(10), 103901 (2019).
[Crossref]

F. Liu, H. Y. Deng, and K. Wakabayashi, “Helical topological edge states in a quadrupole phase,” Phys. Rev. Lett. 122(8), 086804 (2019).
[Crossref]

W. A. Benalcazar, T. Li, and T. L. Hughes, “Quantization of fractional corner charge in C n-symmetric higher-order topological crystalline insulators,” Phys. Rev. B 99(24), 245151 (2019).
[Crossref]

Z. Zhang, H. Long, C. Liu, C. Shao, Y. Cheng, X. Liu, and J. Christensen, “Deep-Subwavelength Holey Acoustic Second-Order Topological Insulators,” Adv. Mater. 31(49), 1904682 (2019).
[Crossref]

X. D. Chen, W. M. Deng, F. L. Shi, F. L. Zhao, M. Chen, and J. W. Dong, “Direct observation of corner states in second-order topological photonic crystals slab,” Phys. Rev. Lett. 122(23), 233902 (2019).
[Crossref]

2018 (4)

L. Xu, M. Rahmani, D. Smirnova, K. Zangeneh Kamali, G. Zhang, D. Neshev, and A. E. Miroshnichenko, “Highly-efficient longitudinal second-harmonic generation from doubly-resonant AlGaAs nanoantennas,” Photonics 5(3), 29 (2018).
[Crossref]

Y. Yang, Y. F. Xu, T. Xu, H. X. Wang, J. H. Jiang, X. Hu, and Z. H. Hang, “Visualization of a unidirectional electromagnetic waveguide using topological photonic crystals made of dielectric materials,” Phys. Rev. Lett. 120(21), 217401 (2018).
[Crossref]

D. A. Dobrykh, A. V. Yulin, A. P. Slobozhanyuk, A. N. Poddubny, and Y. S. Kivshar, “Nonlinear control of electromagnetic topological edge states,” Phys. Rev. Lett. 121(16), 163901 (2018).
[Crossref]

J. Noh, W. A. Benalcazar, S. Huang, M. J. Collins, K. P. Chen, T. L. Hughes, and M. C. Rechtsman, “Topological protection of photonic mid-gap defect modes,” Nat. Photonics 12(7), 408–415 (2018).
[Crossref]

2017 (1)

G. Siroki, P. A. Huidobro, and V. Giannini, “Topological photonics: From crystals to particles,” Phys. Rev. B 96(4), 041408 (2017).
[Crossref]

2016 (2)

J. Yang, J. P. Hugonin, and P. Lalanne, “Near-to-far field transformations for radiative and guided waves,” ACS Photonics 3(3), 395–402 (2016).
[Crossref]

S. Wang, L. Talirz, C. A. Pignedoli, X. Feng, K. Müllen, R. Fasel, and P. Ruffieux, “Giant edge state splitting at atomically precise graphene zigzag edges,” Nat. Commun. 7(1), 11507 (2016).
[Crossref]

2015 (1)

L. H. Wu and X. Hu, “Scheme for achieving a topological photonic crystal by using dielectric material,” Phys. Rev. Lett. 114(22), 223901 (2015).
[Crossref]

2012 (2)

C. Ciracì, E. Poutrina, M. Scalora, and D. R. Smith, “Origin of second-harmonic generation enhancement in optical split-ring resonators,” Phys. Rev. B 85(20), 201403 (2012).
[Crossref]

C. Ciraci, E. Poutrina, M. Scalora, and D. R. Smith, “Second-harmonic generation in metallic nanoparticles: Clarification of the role of the surface,” Phys. Rev. B 86(11), 115451 (2012).
[Crossref]

2010 (1)

M. Scalora, M. A. Vincenti, D. de Ceglia, V. Roppo, M. Centini, N. Akozbek, and M. J. Bloemer, “Second- and third-harmonic generation in metal-based structures,” Phys. Rev. A 82(4), 043828 (2010).
[Crossref]

Akozbek, N.

M. Scalora, M. A. Vincenti, D. de Ceglia, V. Roppo, M. Centini, N. Akozbek, and M. J. Bloemer, “Second- and third-harmonic generation in metal-based structures,” Phys. Rev. A 82(4), 043828 (2010).
[Crossref]

Arakawa, Y.

Bauer, D.

L. J. Maczewsky, M. Heinrich, M. Kremer, S. K. Ivanov, M. Ehrhardt, F. Martinez, Y. V. Kartashov, V. V. Konotop, L. Torner, D. Bauer, and A. Szameit, “Nonlinearity-induced photonic topological insulator,” Science 370(6517), 701–704 (2020).
[Crossref]

Benalcazar, W. A.

W. A. Benalcazar, T. Li, and T. L. Hughes, “Quantization of fractional corner charge in C n-symmetric higher-order topological crystalline insulators,” Phys. Rev. B 99(24), 245151 (2019).
[Crossref]

J. Noh, W. A. Benalcazar, S. Huang, M. J. Collins, K. P. Chen, T. L. Hughes, and M. C. Rechtsman, “Topological protection of photonic mid-gap defect modes,” Nat. Photonics 12(7), 408–415 (2018).
[Crossref]

Berestennikov, A. S.

A. S. Berestennikov, A. Vakulenko, S. Kiriushechkina, M. Li, Y. Li, L. E. Zelenkov, A. P. Pushkarev, M. A. Gorlach, A. L. Rogach, S. V. Makarov, and A. B. Khanikaev, “Enhanced photoluminescence of halide perovskite nanocrystals mediated by a higher-order topological metasurface,” J. Phys. Chem. C 125(18), 9884–9890 (2021).
[Crossref]

Bloemer, M. J.

M. Scalora, M. A. Vincenti, D. de Ceglia, V. Roppo, M. Centini, N. Akozbek, and M. J. Bloemer, “Second- and third-harmonic generation in metal-based structures,” Phys. Rev. A 82(4), 043828 (2010).
[Crossref]

Centini, M.

M. Scalora, M. A. Vincenti, D. de Ceglia, V. Roppo, M. Centini, N. Akozbek, and M. J. Bloemer, “Second- and third-harmonic generation in metal-based structures,” Phys. Rev. A 82(4), 043828 (2010).
[Crossref]

Chaunsali, R.

C. W. Chen, R. Chaunsali, J. Christensen, G. Theocharis, and J. Yang, “Corner states in second-order mechanical topological insulator,” Commun. Mater. 2(1), 62 (2021).
[Crossref]

Chen, C. W.

C. W. Chen, R. Chaunsali, J. Christensen, G. Theocharis, and J. Yang, “Corner states in second-order mechanical topological insulator,” Commun. Mater. 2(1), 62 (2021).
[Crossref]

Chen, H. Z.

Z. K. Shao, H. Z. Chen, S. Wang, X. R. Mao, Z. Q. Yang, S. L. Wang, X. X. Wang, X. Hu, and R. M. Ma, “A high-performance topological bulk laser based on band-inversion-induced reflection,” Nat. Nanotechnol. 15(1), 67–72 (2020).
[Crossref]

Chen, K. P.

J. Noh, W. A. Benalcazar, S. Huang, M. J. Collins, K. P. Chen, T. L. Hughes, and M. C. Rechtsman, “Topological protection of photonic mid-gap defect modes,” Nat. Photonics 12(7), 408–415 (2018).
[Crossref]

Chen, M.

X. D. Chen, W. M. Deng, F. L. Shi, F. L. Zhao, M. Chen, and J. W. Dong, “Direct observation of corner states in second-order topological photonic crystals slab,” Phys. Rev. Lett. 122(23), 233902 (2019).
[Crossref]

Chen, X. D.

X. D. Chen, W. M. Deng, F. L. Shi, F. L. Zhao, M. Chen, and J. W. Dong, “Direct observation of corner states in second-order topological photonic crystals slab,” Phys. Rev. Lett. 122(23), 233902 (2019).
[Crossref]

Chen, Y. F.

B. Xie, G. Su, H. F. Wang, F. Liu, L. Hu, S. Y. Yu, P. Zhan, M. H. Lu, Z. Wang, and Y. F. Chen, “Higher-order quantum spin Hall effect in a photonic crystal,” Nat. Commun. 11(1), 3768 (2020).
[Crossref]

Cheng, Y.

Z. Zhang, B. Hu, F. Liu, Y. Cheng, X. Liu, and J. Christensen, “Pseudospin induced topological corner state at intersecting sonic lattices,” Phys. Rev. B 101(22), 220102 (2020).
[Crossref]

Z. Zhang, H. Long, C. Liu, C. Shao, Y. Cheng, X. Liu, and J. Christensen, “Deep-Subwavelength Holey Acoustic Second-Order Topological Insulators,” Adv. Mater. 31(49), 1904682 (2019).
[Crossref]

Choi, D. Y.

S. Kruk, W. Gao, D. Y. Choi, T. Zentgraf, S. Zhang, and Y. Kivshar, “Nanoscale topological corner states in nonlinear optics,” Nano Lett. 21(11), 4592–4597 (2021).
[Crossref]

D. Smirnova, S. Kruk, D. Leykam, E. Melik-Gaykazyan, D. Y. Choi, and Y. Kivshar, “Third-harmonic generation in photonic topological metasurfaces,” Phys. Rev. Lett. 123(10), 103901 (2019).
[Crossref]

Chong, Y.

D. Smirnova, D. Leykam, Y. Chong, and Y. Kivshar, “Nonlinear topological photonics,” Appl. Phys. Rev. 7(2), 021306 (2020).
[Crossref]

Christensen, J.

C. W. Chen, R. Chaunsali, J. Christensen, G. Theocharis, and J. Yang, “Corner states in second-order mechanical topological insulator,” Commun. Mater. 2(1), 62 (2021).
[Crossref]

Z. Zhang, B. Hu, F. Liu, Y. Cheng, X. Liu, and J. Christensen, “Pseudospin induced topological corner state at intersecting sonic lattices,” Phys. Rev. B 101(22), 220102 (2020).
[Crossref]

Z. Zhang, H. Long, C. Liu, C. Shao, Y. Cheng, X. Liu, and J. Christensen, “Deep-Subwavelength Holey Acoustic Second-Order Topological Insulators,” Adv. Mater. 31(49), 1904682 (2019).
[Crossref]

M. R. López, Z. Zhang, D. Torrent, and J. Christensen, “Multiple scattering theory of non-Hermitian sonic second-order topological insulators,” Commun. Phys. 2(1), 132 (2019).
[Crossref]

Ciraci, C.

C. Ciraci, E. Poutrina, M. Scalora, and D. R. Smith, “Second-harmonic generation in metallic nanoparticles: Clarification of the role of the surface,” Phys. Rev. B 86(11), 115451 (2012).
[Crossref]

Ciracì, C.

C. Ciracì, E. Poutrina, M. Scalora, and D. R. Smith, “Origin of second-harmonic generation enhancement in optical split-ring resonators,” Phys. Rev. B 85(20), 201403 (2012).
[Crossref]

Collins, M. J.

J. Noh, W. A. Benalcazar, S. Huang, M. J. Collins, K. P. Chen, T. L. Hughes, and M. C. Rechtsman, “Topological protection of photonic mid-gap defect modes,” Nat. Photonics 12(7), 408–415 (2018).
[Crossref]

Dang, J.

X. Xie, W. Zhang, X. He, S. Wu, J. Dang, K. Peng, F. Song, L. Yang, H. Ni, Z. Niu, C. Wang, K. Jin, X. Zhang, and X. Xu, “Cavity quantum electrodynamics with second-order topological corner state,” Laser Photonics Rev. 14(8), 1900425 (2020).
[Crossref]

W. Zhang, X. Xie, H. Hao, J. Dang, S. Xiao, S. Shi, H. Ni, Z. Niu, C. Wang, K. Jin, X. Zhang, and X. Xu, “Low-threshold topological nanolasers based on the second-order corner state,” Light: Sci. Appl. 9(1), 109 (2020).
[Crossref]

de Ceglia, D.

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Z. Zhang, B. Hu, F. Liu, Y. Cheng, X. Liu, and J. Christensen, “Pseudospin induced topological corner state at intersecting sonic lattices,” Phys. Rev. B 101(22), 220102 (2020).
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Z. Zhang, H. Long, C. Liu, C. Shao, Y. Cheng, X. Liu, and J. Christensen, “Deep-Subwavelength Holey Acoustic Second-Order Topological Insulators,” Adv. Mater. 31(49), 1904682 (2019).
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X. Huang, C. Lu, C. Liang, H. Tao, and Y. C. Liu, “Loss-induced nonreciprocity,” Light: Sci. Appl. 10(1), 30 (2021).
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A. S. Berestennikov, A. Vakulenko, S. Kiriushechkina, M. Li, Y. Li, L. E. Zelenkov, A. P. Pushkarev, M. A. Gorlach, A. L. Rogach, S. V. Makarov, and A. B. Khanikaev, “Enhanced photoluminescence of halide perovskite nanocrystals mediated by a higher-order topological metasurface,” J. Phys. Chem. C 125(18), 9884–9890 (2021).
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Z. K. Shao, H. Z. Chen, S. Wang, X. R. Mao, Z. Q. Yang, S. L. Wang, X. X. Wang, X. Hu, and R. M. Ma, “A high-performance topological bulk laser based on band-inversion-induced reflection,” Nat. Nanotechnol. 15(1), 67–72 (2020).
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L. J. Maczewsky, M. Heinrich, M. Kremer, S. K. Ivanov, M. Ehrhardt, F. Martinez, Y. V. Kartashov, V. V. Konotop, L. Torner, D. Bauer, and A. Szameit, “Nonlinearity-induced photonic topological insulator,” Science 370(6517), 701–704 (2020).
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S. Wang, L. Talirz, C. A. Pignedoli, X. Feng, K. Müllen, R. Fasel, and P. Ruffieux, “Giant edge state splitting at atomically precise graphene zigzag edges,” Nat. Commun. 7(1), 11507 (2016).
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W. Zhang, X. Xie, H. Hao, J. Dang, S. Xiao, S. Shi, H. Ni, Z. Niu, C. Wang, K. Jin, X. Zhang, and X. Xu, “Low-threshold topological nanolasers based on the second-order corner state,” Light: Sci. Appl. 9(1), 109 (2020).
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X. Xie, W. Zhang, X. He, S. Wu, J. Dang, K. Peng, F. Song, L. Yang, H. Ni, Z. Niu, C. Wang, K. Jin, X. Zhang, and X. Xu, “Cavity quantum electrodynamics with second-order topological corner state,” Laser Photonics Rev. 14(8), 1900425 (2020).
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Science (1)

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Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (6)

Fig. 1.
Fig. 1. (a) Designed photonic crystal structure to obtain corner state. a is the lattice constant. (b) Two lattices of PCs consist of six cylinders made of dielectric material, which are arranged in the air. L is the distance from center of each cylinder to center of the lattice, and r is the radius of cylinders. The blue and yellow colors represent different distances L. Band structures of PCs with (c) a/L=4.25, (d) a/L=3, and (e) a/L=2.5. (Inset: Brillouin zone of triangular lattice; d and p represent dipole and quadrupole modes, respectively; ‘±’ represents the parity of first three bands at Γ and M point.). The Ez field distributions (TM mode) of dipoles and quadrupoles under topologically (f) trivial and (g) nontrivial conditions.
Fig. 2.
Fig. 2. (a) Calculated eigenfrequency spectrum of the designed photonic crystal system. Blue dots, green dots and red star denote bulk, edge and corner states, respectively. Electric field intensities at the (b) bulk, (c) edge and (d) corner states. The inset in (d) enlarges the area near the corner marked by white dashed lines. (e) Dependence of Q-factor on mode number.
Fig. 3.
Fig. 3. (a) Detected emission frequency spectra from the designed system at bulk, edge and corner states. Simulated SHG electric field distributions at (b) corner, (c) bulk, and (d) edge states. The white and yellow circles indicate the positions of excitation and detection, respectively.
Fig. 4.
Fig. 4. (a) Schematic of the proposed system before and after increasing the size of nanocavity. D is the distance between trivial and nontrivial regions. (b) Dependences of the Q-factor and eigen-frequency of the corner state on the value of D, and (c) the corresponding emission spectra from the corner state.
Fig. 5.
Fig. 5. Topological robustness of the SHG emission from corner state. SHG distribution patterns for the cases with (a) lattice defect (Perturbation I), (b) bend interface (Perturbation II), and (c) bend interface with lattice defect (Perturbation III). A unit lattice inside the white circle is removed. The white dashed lines represent the interfaces between trivial and nontrivial regions. (d) The emitted frequency spectra from corners with the perturbations in (a-c). (e) Full-width half-maximum and SHG frequency for the cases with and without perturbations.
Fig. 6.
Fig. 6. Directional emission of SHG from the topological corner state. (a) Angle-resolved far field pattern of SHG emission. Far field patterns in (b) x-z and y-z planes which are extracted from (a). Far field patterns in both x-z and y-z planes from the corners with perturbations (d) I, (e) II, and (f) III discussed in Fig. 5.

Equations (1)

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p ± = ( p x ± i p y ) / 2 ; d ± = ( d x 2 y 2 ± i d x y ) / 2 ,

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