## 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 3.66×10^{11} 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 [1–3]. 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 [11–13]. 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 [21–25]. 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.

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 *p*_{x} (*p*_{y}) and *d*_{xy} (${d_{{x^2} - {y^2}}}$), 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]

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 [27–29].

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 1.65×10^{8}, further confirming the great ability of restricting light in a small area.

## 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 *E*_{1z} 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 *E*_{1z}*=E*_{0}sin*(Ω*_{1}*×t)*. *E*_{0} is set to be 3e^{9} V/m and *Ω*_{1}=2π*f*_{1}, where *f*_{1} is the fundamental frequency of excitation source. *f*_{1}=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.

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 3.66×10^{11}, 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.

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 [35–37].

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.

## 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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