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Curr. Opt. Photon. 2022; 6(5): 497-505

Published online October 25, 2022 https://doi.org/10.3807/COPP.2022.6.5.497

## Effective Coupling of a Topological Corner-state Nanocavity to Various Plasmon Nanoantennas

Na Ma, Ping Jiang , You Tao Zeng, Xiao Zhen Qiao, Xian Feng Xu

College of Science, China University of Petroleum (East China), Qingdao 266580, China

Corresponding author: *pjiang@upc.edu.cn, ORCID 0000-0002-1128-8818

Received: February 25, 2022; Revised: July 11, 2022; Accepted: July 20, 2022

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Topological photonic nanocavities are considered to possess outstanding optical performance, and provide new platforms for realizing strong interaction between light and matter, due to their robustness to impurities and defects. Here hybrid plasmonic topological photonic nanocavities are proposed, by embedding various plasmon nanoantennas such as gold nanospheres, cylinders, and rectangles in a topological photonic crystal corner-state nanocavity. The maximum quality factor Q and minimum effective mode volume Veff of these hybrid nanocavities can reach the order of 104 and 10−4 (λ/n)3 respectively, and the high figures of merit Q/Veff for all of these hybrid nanocavites are stable and on the order of 105 (λ/n)−3. The relative positions of the plasmon nanoantennas will influence the coupling strength between the plasmon structures and the topological nanocavity. The hybrid nanocavity with gold nanospheres possesses much higher Q, but relatively large Veff. The presence of a gold rectangular structure can confine more electromagnetic energy within a smaller space, since its Veff is smallest, although Q is lowest among these structures. This work provides an outstanding platform for cavity quantum electrodynamics and has a wide range of applications in topological quantum light sources, such as single-photon sources and nanolasers.

Keywords: Cavity quantum electrodynamics, Plasmon nanoantennas, Purcell enhancement, Topological photonic nanocavity, Topological protection

OCIS codes: (230.5298) Photonic crystals; (350.4238) Nanophotonics and photonic crystals; (350.5400) Plasmas

Cavity quantum electrodynamics (CQED) has wide range of applications for the realization of quantum information processing (QIP) [1, 2]. In the weak-coupling region, the spontaneous emission rate of quantum emitters can be dramatically improved based on the Purcell effect, which can be used to optimize the performance of a single-photon source [35]. A wide range of nanocavities has been used to establish solid-state CQED systems, including the whispering-gallery cavity [6, 7] and the photonic crystal (PhC) cavity [810]. To enhance the coupling efficiency, these nanocavities have to be optimized to have high a quality factor Q and small mode volume V [11]. A hybrid photonic plasmonic nanocavity based on a PhC cavity is an effective solution [12, 13] that combines the advantages of a high Q value for the photonic nanocavity and an ultrasmall mode volume of the plasmonic nanostructure. In experimental study, though, the performance of a normal photonic nanostructure is always strongly affected by the defects and disorder that are introduced by environmental perturbations as well as fabrication imperfections.

Recently, topological protection has been used in photonic devices, providing a new way to develop photonic devices with robustness to defects and disorders [1416]. The topological concept also offers a different approach for designing PhC nanocavities [1720]. An effective approach for constructing a topological nanocavity is the so-called photonic crystal corner-state nanocavity, utilizing a topological corner state that appears as a higher-order topological state in a two-dimensional (2D) system [21, 22]. However, if improving Purcell enhancement only depends on edge states, the emission rate is not high enough [23]. By embedding a plasmonic antenna in the topological photonic nanocavity, the performance of the cavity will be dramatically improved. Zhang et al. [24] proposed a topological hybrid nanocavity by coupling a topological corner-state nanocavity to a gold bowtie plasmonic nanoantenna. The strong local field near the plasmonic antenna leads to ultrasmall mode volume, to ensure large Purcell enhancement. Meanwhile, nonscattering edge states protect all scattering photons propagatingalong some specific direction. We can imagine that the Purcell effect should be sensitive to the shape, size, and relative position of the nanoantenna in the topological photonic nanocavity. Besides the bowtie nanoantenna, other plasmonic nanoantennas will also have different effects on the topological hybrid nanocavities, which has not yet been confirmed, though. In this paper, various kinds of plasmonic nanoantennas, such as Au nanospheres and cylindrical and rectangular nanoantennas, are embedded in PhC corner-state nanocavities to construct hybrid plasmonic topological photonic nanocavities. The performances of all of these hybrid nanocavities are simulated using the 3D finite-difference time-domain method. The maximum quality factor Q and the minimum effective mode volume Veff can reach the order of 104 and 10−4 (λ/n)3 respectively. The high figure of merit Q/Veff is always stable, on the order of 105 (λ/n)−3. This kind of hybrid plasmonic topological photonic nanocavity will provide a nonscattering, ultralarge Purcell enhancement with robustness, which has potential applications in topological quantum light sources such as single-photon sources and nanolasers.

### Ⅱ. TWO CONFIGURATIONS OF HYBRID TOPOLOGICAL NANOCAVITIES WITH AU NANOSPHERES

We firstly consider the bare topological corner-state nanocavity, which is composed of two topologically distinct PhC structures with square-shaped air holes. These two structures possess an identical square lattice with lattice constant a = 270 nm and hole length d = 88 nm. The difference is that one is formed by shrinking four air holes to the center of the unit cell, resulting in a trivial phase of θZak = (0, 0), and the other is formed by expanding four air holes far away from the center, leading to a topological phase of θZak = (π, π), respectively marked by yellow and red dotted frames in Fig. 1. When combining these two topologically distinct PhC structures in a suitable manner, a topological corner-state nanocavity is formed and the most electromagnetic energy is tightly concentrated inside the nanocavity, with a smaller mode volume compared to other topological nanocavities. Moreover, we choose GaAs with n = 3.46 as the substrate material, because its high refractive index is conducive to confining more electromagnetic energy within the PhC slab, due to total internal reflection. The slab’s thickness is set to T = 150 nm. Then the topological corner-state nanocavity is optimized by shifting the topological PhC structure to be far away from the center of the cavity, to improve the intrinsic optical properties. A dipole source is placed in the center of the PhC slab to excite the field mode of the nanocavity. By performing 3D finite-difference time-domain calculations, a high quality factor Q = 14882.7, a small effective mode volume Veff = 0.61 (λ/n)3, and the corresponding figure of merit Q/Veff = 2.426445 × 104 (λ/n)−3 are obtained. It is seen from Fig. 1(a) that most of the light is localized to a small area around the center of the bare topological corner-state nanocavity.

Figure 1.Structural schematics and electric field intensity maps. (a) Calculated electric field intensity distribution |E| for the bare topological corner state nanocavity. (b, c) Schematics of the hybrid plasmonictopological photonic nanocavity with Au nanospheres, marked as Type A and Type B. The yellow and red dotted frames represent trivial and topological regions respectively. (d, e) Calculated electric field intensity distribution |E| for the above two configurations, along the cross section of the center of the nanospheres. The radius and gap of the two nanospheres are 20 nm and 5 nm respectively. All insets correspond to enlarged views of the two gold nanospheres. The electric field strength is normalized to the maximum field intensity in the same cross section of the bare cavity.

Although the corner state cavity has an excellent ability to concentrate much of the light field, there is still a large amount of light scattered into the upper- and right-side waveguides, as well as the interior of the PhCs, resulting in a relatively large mode volume, as shown in Fig. 1(a). To obtain a high-performance optical nanocavity with ultralarge Q/Veff, a pair of gold nanospheres is placed on the topological corner-state nanocavity to form a hybrid plasmonic topological photonic nanocavity. The localization of the electric field is greatly enhanced and the mode volume of the hybrid nanocavity is significantly reduced, owing to the presence of the metallic antenna. The optical properties of the electric field profiles, quality factor Q, effective mode volume Veff, and figure of merit Q/Veff of the designed hybrid nanocavity are investigated.

Different positions of the Au nanospheres relative to the cavity potentially exert different influence on the characteristics of the hybrid nanocavity. Here, we mainly consider two different structures, as shown in Figs. 1(b) and 1(c). The direction of the two nanospheres is aligned parallel or perpendicular to the direction of maximum electric field strength, denoted as Type A and Type B respectively. The electric field intensity distribution |E| of each of the two hybrid nanocavities is calculated, as shown in Figs. 1(d) and 1(e). It is obviously seen from these results that there is a big difference between the electric field profiles of the two structures. For the hybrid nanocavity marked as Type A, although only a small part of the electric field is still scattered into the surrounding PhC, more electric field is confined inside the center of the hybrid nanocavity. To be specific, it is observed from the insets that most of the localized electric field is concentrated in the gap between the two nanospheres, and thus a pronounced hot spot is formed, which makes the electric field strength more than 2.8 times as large as that of the bare cavity. Some weak electric field profiles exist near the edges on both sides, away from the gap between the nanospheres. On the contrary, for Type B only a small part of the electric field is confined near the edges of the nanospheres and the electric field in the gap between the nanospheres is much weaker. Even so, the electric field strength is about 1.2 times as large as that of the bare cavity.

To further study the influence of different positions of the Au nanospheres on the hybrid nanocavities, we quantitatively characterize the optical properties of the hybrid nanocavities by calculating the quality factor Q, effective mode volume Veff, and figure of merit Q/Veff. The quality factor Q is defined as Q=ωcΔω, and the mode volume is defined as V=V ε(r) E(r)2d3rmax(ε(r)E(r)2), which is usually expressed in units of the cubic wavelength (λ/n)3, termed the effective mode volume Veff. Thus the figure of merit Q/Veff is directly defined as the ratio of the quality factor Q to the effective mode volume Veff. As shown in Fig. 2, although these three quantities for the above two configurations all decrease as the radius of the nanospheres increases, there are a number of differences between the two structures. It is shown in Fig. 2(a) that the quality factor Q of the two structures is almost equal before the radius of the nanospheres is 15 nm, and after that Q for Type A is always lower than that for Type B. Besides, Veff for Type A is much smaller than that for Type B when the radius of the nanospheres is 25 nm or 40 nm, as shown in Fig. 2(b). The strong fluctuation of Type A is due to the stronger coupling between the Au nanospheres and the topological nanocavity. A possible reason for the sudden drop in the effective mode volume at 25 nm and 40 nm is the larger redshift of the resonance wavelength with the offset Δλ = 0.25 nm, resulting in a relatively large reduction of effective mode volume. Moreover, the bigger the radius of the nanospheres, the smaller Veff of the hybrid cavity, which indicates that bigger Au nanospheres are conducive to confining the light field within a smaller volume. Additionally, the values of the figure of merit Q/Veff for the two structures are equal before the radius of the nanospheres is 15 nm, and the figure of merit for Type A is lower than that for Type B after the radius of the nanospheres is 30 nm, as shown in Fig. 2(c). Consequently, these results also demonstrate that the smaller Au nanospheres with radius less than 15 nm can couple to the topological corner-state nanocavity in the two cases because these performance parameters are almost equivalent. However, for the bigger Au nanospheres the coupling effect is more significant when the nanospheres are aligned parallel to the direction of maximum field strength, which leads to relatively larger loss, making Q and Q/Veff much lower. Intriguingly, more electromagnetic field is confined within a smaller space, resulting in a smaller mode volume, owing to the strong coupling between the Au nanospheres and the nanocavity.

Figure 2.Calculated (a) quality factor Q, (b) effective mode volume Veff, and (c) figure of merit Q/Veff, varying with the radius of Au nanospheres. The subscripts A (black square) and B (red circle) represent the two kinds of designed hybrid nanocavities featuring Au nanospheres.

We next investigate the influence of the gap width between the two gold nanospheres on the optical properties of the hybrid nanocavity. When gradually increasing the gap width, Q, Veff, and Q/Veff for Type B are always stable, which suggests that the coupling between the Au nanospheres and the nanocavity is much weaker, and is consistent with the above opinion obtained from the electric field profile. But for Type A, these parameters experience an obvious change. It is shown in Fig. 3(a) that Q for Type A gradually decreases with increasing gap, and is always lower than that for Type B, but Q for Type A is still on the order of 104. In Fig. 3(b) Veff for Type A is stable around 0.127 (λ/n)3 and greater than VeffB before the gap is 50 nm. After that, VeffA declines and becomes smaller than VeffB, which indicates that the Au nanospheres with bigger gap have an outstanding ability to localize electromagnetic field. The sudden drop of the effective mode volume at a gap of 60 nm is due to the larger redshift of the resonance wavelength with the offset Δλ = 0.25 nm, leading to a relatively large reduction in the effective mode volume. As shown in Fig. 3(c), Q/Veff reaches minimum and maximum values when the gap is 50 nm and 60 nm respectively. Q/Veff is on the order of 105 (λ/n)−3, which is an order of magnitude larger than that for the bare cavity. Therefore, both configurations can reach a relatively large quality factor Q and figure of merit Q/Veff, but Type A is more beneficial for improving the emission property of quantum emitters due to higher electric field strength. These conclusions provide a novel method to tailor the light-matter interaction.

Figure 3.Calculated (a) quality factor Q, (b) effective mode volume Veff, and (c) figure of merit Q/Veff, varying with the gap width of the Au nanospheres. The subscripts A (black square) and B (red circle) represent the two kinds of designed hybrid nanocavities featuring Au nanospheres.

### Ⅲ. COUPLING BETWEEN THE CORNER-STATE CAVITY AND OTHER GOLD NANOANTENNAS

Apart from the Au nanospheres, other Au nanoantennas can also remarkably decrease the mode volume to improve the localization of the light field. In the following, different gold structures such as cylinders and rectangles are considered and the optical properties of diverse hybrid nanocavities are intensively investigated, as shown in Figs. 4(a) and 4(b). The electronic field strength profiles of various structures are calculated and mapped in Figs. 4(c) and 4(d). Most of the electromagnetic energy is confined inside the gap between these two Au nanoparticles to form a pronounced hot spot. Moreover, the electric field strength of the topological hybrid nanocavities with cylinder antennas is enhanced by a factor of 4.13, compared to the bare cavity. The presence of the rectangular antenna structure may weaken the electronic field strength to some extent, owing to the intense metal loss that is induced. As a consequence, using plasmonic nanoantennas with metal nanospheres and cylinders is easier for strengthening the light-matter interaction via enhanced electric field distribution.

Figure 4.(a, b) Structural diagrams and (c, d) electric field intensity distributions of the hybrid plasmonictopological corner-state nanocavities with cylindrical and rectangular nanoantenna structures respectively. All insets correspond to enlarged views of the gold structures. The electric field strength is normalized to the maximum field intensity in the same cross section of the bare cavity.

The quality factor Q, effective mode volume Veff, figure of merit Q/Veff, and total transmittance are calculated by the 3D finite-difference time-domain method to compare the differences between these gold antenna structures. Here the total transmittance is defined as the sum of the transmittances of the upper and left waveguides. The length for the sphere presented in Fig. 5 corresponds to the radius of the nanosphere in Fig. 2. As shown in Fig. 5(a), for all structures Q declines as the length of the gold structure increases. For the hybrid nanocavities with Au nanospheres and cylinders, Q is still above 5,000, but for the hybrid nanocavities with the rectangular nanoantenna, Q is always much smaller. The reason for this result is that more intense plasmons are induced at the interface between the Au rectangular antenna and the topological nanocavity, resulting in severe metal loss. In particular, Q drops dramatically to around 200 before a length of 25 nm. These results indicate that a larger Au rectangular structure produces more plasmons and stronger coupling to the corner-state nanocavity, which directly leads to more serious metal loss to deteriorate the quality factor of the hybrid cavity. Veff for the hybrid nanocavity with Au cylinders is stable around 0.12 (λ/n)3, like that for the nanosphere structure before the length of 30 nm, and then drops to the order of 10−2 (λ/n)3. The presence of the Au rectangular structure makes Veff of the hybrid nanocavity decrease from 10−2 (λ/n)3 to 10−3 (λ/n)3 before a length of 30 nm, and then Veff stabilizes on the order of 10−3 (λ/n)3. It is obvious from the simulation results in Fig. 5(b) that the bigger Au rectangular structures are much better at confining electromagnetic energy within a smaller volume. Although Q and Veff have a distinct difference between these structures, Q/Veff for all is still on the order of 105 (λ/n)−3, as shown in Fig. 5(c). When the length of the rectangular structure is 30 nm, Q/Veff reaches a maximum value of 2.52 × 105 (λ/n)−3. Additionally, the total transmittance values of the hybrid nanocavities with Au nanospheres and cylinders both increase with length, while that of the rectangular structures declines from 21% to 1.5% as the length increases, as shown in Fig. 5(d). This indicates that the rectangular antenna possesses a better ability to enhance the localization of the light field in the hybrid nanocavity, compared to other antennas. Less electromagnetic energy is leaked into the upper- and right-side waveguides, resulting in lower total transmittance. The larger rectangular antennas also exert a stronger influence on the performance of the hybrid nanocavity, which is basically consistent with the conclusion obtained from Fig. 5(b).

Figure 5.Variation laws of various optical properties. (a) Quality factor Q, (b) effective mode volume Veff, (c) figure of merit Q/Veff, and (d) total transmittance as functions of length, for various gold antenna structures.

The gap between these gold structures plays a crucial role in tuning the optical properties of the hybrid nanocavity. When the gap increases, it is seen in Fig. 6(a) that the quality factor Q for the hybrid cavities with Au nanospheres and rectangles gradually declines, while that for Au cylinders increases. Among these hybrid nanocavities, the presence of the Au rectangles keeps Q at its lowest values, below 600, due to intense metal loss. The Au nanospheres and cylinders keep Q much higher, above 104. It is seen from Fig. 6(b) that the Au nanospheres and cylinders yield effective mode volume Veff on the order of 10−1 (λ/n)3 and 10−2 (λ/n)3 respectively. Veff for the hybrid nanocavity with Au rectangular nanoantenna gradually decreases from 10−3 (λ/n)3 to 10−4 (λ/n)3, due to the continuous redshift of the resonance wavelength for the hybrid nanocavity. Therefore, the Au rectangular structure is more helpful for reducing the localization volume of the light field. It is worth mentioning that the figure of merit Q/Veff for all structures is always stable and on the order of 105 (λ/n)−3, as displayed in Fig. 6(c). The values of total transmittance for the hybrid nanocavities with Au nanospheres and cylinders are stable around 25% and 20% respectively. The total transmittance for the other structures drops dramatically to 2% when the gap increases, as shown in Fig. 6(d). Among these hybrid structures, the rectangular antennas cause Q to decrease, evidently due to greater induced metal loss, but simultaneously they greatly reduce Veff, compared to the other gold antennas. A possible reason for this is that the larger overlap area between the gold antennas and the dielectric nanocavity produces more intense plasmons, which directly leads to the obvious decrease in mode volume. Therefore, we could choose the most desirable hybrid plasmonic topological photonic nanocavity according to the optical performance of the different nano antennas.

Figure 6.Variation laws of various optical properties. (a) Quality factor Q, (b) effective mode volume Veff, (c) figure of merit Q/Veff, and (d) total transmittance as functions of gap width, for different gold antenna structures.

In conclusion, a topological corner-state nanocavity with ultrahigh quality factor Q of 104 is designed, and this topological nanocavity can couple to various gold antenna structures to enhance the light-matter interaction. Au nanospheres parallel to the maximum field strength couple effectively to the topological cavity, owing to a greater enhancement of the electric field, although the quality factor Q is relatively low, due to serious metal loss. Besides, other gold structures such as cylinders and rectangles can also obviously improve the optical properties of the hybrid topological cavity. The mapped electric field profiles of various hybrid nanocavities exhibit pronounced hot spots in the gap between these two Au nanoparticles, and the maximum electric field intensity for these structures is enhanced to different degrees. Especially, the presence of the Au rectangular structure greatly reduces the effective mode volume Veff, so that more electromagnetic field is concentrated within a much smaller space. The minimum effective mode volume Veff even reaches 10−4 (λ/n)3. Moreover, the figure of merit Q/Veff for all of these structures stays on the order of 105 (λ/n)−3. Therefore, hybrid plasmonic topological nanocavities with robustness against structural imperfections and unidirectional transmission can be realized via combining various gold nanostructures. This ultrahigh Purcell enhancement is beneficial to improving the optical properties of nanolasers and provides potential applications for effective and stable single-photon sources.

The authors declare no conflict of interest.

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

Central University Basic Research Fund (22CX03018A); Graduate Innovation Engineering Project (YCX2021145).

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### Article

#### Article

Curr. Opt. Photon. 2022; 6(5): 497-505

Published online October 25, 2022 https://doi.org/10.3807/COPP.2022.6.5.497

## Effective Coupling of a Topological Corner-state Nanocavity to Various Plasmon Nanoantennas

Na Ma, Ping Jiang , You Tao Zeng, Xiao Zhen Qiao, Xian Feng Xu

College of Science, China University of Petroleum (East China), Qingdao 266580, China

Correspondence to:*pjiang@upc.edu.cn, ORCID 0000-0002-1128-8818

Received: February 25, 2022; Revised: July 11, 2022; Accepted: July 20, 2022

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

Topological photonic nanocavities are considered to possess outstanding optical performance, and provide new platforms for realizing strong interaction between light and matter, due to their robustness to impurities and defects. Here hybrid plasmonic topological photonic nanocavities are proposed, by embedding various plasmon nanoantennas such as gold nanospheres, cylinders, and rectangles in a topological photonic crystal corner-state nanocavity. The maximum quality factor Q and minimum effective mode volume Veff of these hybrid nanocavities can reach the order of 104 and 10−4 (λ/n)3 respectively, and the high figures of merit Q/Veff for all of these hybrid nanocavites are stable and on the order of 105 (λ/n)−3. The relative positions of the plasmon nanoantennas will influence the coupling strength between the plasmon structures and the topological nanocavity. The hybrid nanocavity with gold nanospheres possesses much higher Q, but relatively large Veff. The presence of a gold rectangular structure can confine more electromagnetic energy within a smaller space, since its Veff is smallest, although Q is lowest among these structures. This work provides an outstanding platform for cavity quantum electrodynamics and has a wide range of applications in topological quantum light sources, such as single-photon sources and nanolasers.

Keywords: Cavity quantum electrodynamics, Plasmon nanoantennas, Purcell enhancement, Topological photonic nanocavity, Topological protection

### I. INTRODUCTION

Cavity quantum electrodynamics (CQED) has wide range of applications for the realization of quantum information processing (QIP) [1, 2]. In the weak-coupling region, the spontaneous emission rate of quantum emitters can be dramatically improved based on the Purcell effect, which can be used to optimize the performance of a single-photon source [35]. A wide range of nanocavities has been used to establish solid-state CQED systems, including the whispering-gallery cavity [6, 7] and the photonic crystal (PhC) cavity [810]. To enhance the coupling efficiency, these nanocavities have to be optimized to have high a quality factor Q and small mode volume V [11]. A hybrid photonic plasmonic nanocavity based on a PhC cavity is an effective solution [12, 13] that combines the advantages of a high Q value for the photonic nanocavity and an ultrasmall mode volume of the plasmonic nanostructure. In experimental study, though, the performance of a normal photonic nanostructure is always strongly affected by the defects and disorder that are introduced by environmental perturbations as well as fabrication imperfections.

Recently, topological protection has been used in photonic devices, providing a new way to develop photonic devices with robustness to defects and disorders [1416]. The topological concept also offers a different approach for designing PhC nanocavities [1720]. An effective approach for constructing a topological nanocavity is the so-called photonic crystal corner-state nanocavity, utilizing a topological corner state that appears as a higher-order topological state in a two-dimensional (2D) system [21, 22]. However, if improving Purcell enhancement only depends on edge states, the emission rate is not high enough [23]. By embedding a plasmonic antenna in the topological photonic nanocavity, the performance of the cavity will be dramatically improved. Zhang et al. [24] proposed a topological hybrid nanocavity by coupling a topological corner-state nanocavity to a gold bowtie plasmonic nanoantenna. The strong local field near the plasmonic antenna leads to ultrasmall mode volume, to ensure large Purcell enhancement. Meanwhile, nonscattering edge states protect all scattering photons propagatingalong some specific direction. We can imagine that the Purcell effect should be sensitive to the shape, size, and relative position of the nanoantenna in the topological photonic nanocavity. Besides the bowtie nanoantenna, other plasmonic nanoantennas will also have different effects on the topological hybrid nanocavities, which has not yet been confirmed, though. In this paper, various kinds of plasmonic nanoantennas, such as Au nanospheres and cylindrical and rectangular nanoantennas, are embedded in PhC corner-state nanocavities to construct hybrid plasmonic topological photonic nanocavities. The performances of all of these hybrid nanocavities are simulated using the 3D finite-difference time-domain method. The maximum quality factor Q and the minimum effective mode volume Veff can reach the order of 104 and 10−4 (λ/n)3 respectively. The high figure of merit Q/Veff is always stable, on the order of 105 (λ/n)−3. This kind of hybrid plasmonic topological photonic nanocavity will provide a nonscattering, ultralarge Purcell enhancement with robustness, which has potential applications in topological quantum light sources such as single-photon sources and nanolasers.

### Ⅱ. TWO CONFIGURATIONS OF HYBRID TOPOLOGICAL NANOCAVITIES WITH AU NANOSPHERES

We firstly consider the bare topological corner-state nanocavity, which is composed of two topologically distinct PhC structures with square-shaped air holes. These two structures possess an identical square lattice with lattice constant a = 270 nm and hole length d = 88 nm. The difference is that one is formed by shrinking four air holes to the center of the unit cell, resulting in a trivial phase of θZak = (0, 0), and the other is formed by expanding four air holes far away from the center, leading to a topological phase of θZak = (π, π), respectively marked by yellow and red dotted frames in Fig. 1. When combining these two topologically distinct PhC structures in a suitable manner, a topological corner-state nanocavity is formed and the most electromagnetic energy is tightly concentrated inside the nanocavity, with a smaller mode volume compared to other topological nanocavities. Moreover, we choose GaAs with n = 3.46 as the substrate material, because its high refractive index is conducive to confining more electromagnetic energy within the PhC slab, due to total internal reflection. The slab’s thickness is set to T = 150 nm. Then the topological corner-state nanocavity is optimized by shifting the topological PhC structure to be far away from the center of the cavity, to improve the intrinsic optical properties. A dipole source is placed in the center of the PhC slab to excite the field mode of the nanocavity. By performing 3D finite-difference time-domain calculations, a high quality factor Q = 14882.7, a small effective mode volume Veff = 0.61 (λ/n)3, and the corresponding figure of merit Q/Veff = 2.426445 × 104 (λ/n)−3 are obtained. It is seen from Fig. 1(a) that most of the light is localized to a small area around the center of the bare topological corner-state nanocavity.

Figure 1. Structural schematics and electric field intensity maps. (a) Calculated electric field intensity distribution |E| for the bare topological corner state nanocavity. (b, c) Schematics of the hybrid plasmonictopological photonic nanocavity with Au nanospheres, marked as Type A and Type B. The yellow and red dotted frames represent trivial and topological regions respectively. (d, e) Calculated electric field intensity distribution |E| for the above two configurations, along the cross section of the center of the nanospheres. The radius and gap of the two nanospheres are 20 nm and 5 nm respectively. All insets correspond to enlarged views of the two gold nanospheres. The electric field strength is normalized to the maximum field intensity in the same cross section of the bare cavity.

Although the corner state cavity has an excellent ability to concentrate much of the light field, there is still a large amount of light scattered into the upper- and right-side waveguides, as well as the interior of the PhCs, resulting in a relatively large mode volume, as shown in Fig. 1(a). To obtain a high-performance optical nanocavity with ultralarge Q/Veff, a pair of gold nanospheres is placed on the topological corner-state nanocavity to form a hybrid plasmonic topological photonic nanocavity. The localization of the electric field is greatly enhanced and the mode volume of the hybrid nanocavity is significantly reduced, owing to the presence of the metallic antenna. The optical properties of the electric field profiles, quality factor Q, effective mode volume Veff, and figure of merit Q/Veff of the designed hybrid nanocavity are investigated.

Different positions of the Au nanospheres relative to the cavity potentially exert different influence on the characteristics of the hybrid nanocavity. Here, we mainly consider two different structures, as shown in Figs. 1(b) and 1(c). The direction of the two nanospheres is aligned parallel or perpendicular to the direction of maximum electric field strength, denoted as Type A and Type B respectively. The electric field intensity distribution |E| of each of the two hybrid nanocavities is calculated, as shown in Figs. 1(d) and 1(e). It is obviously seen from these results that there is a big difference between the electric field profiles of the two structures. For the hybrid nanocavity marked as Type A, although only a small part of the electric field is still scattered into the surrounding PhC, more electric field is confined inside the center of the hybrid nanocavity. To be specific, it is observed from the insets that most of the localized electric field is concentrated in the gap between the two nanospheres, and thus a pronounced hot spot is formed, which makes the electric field strength more than 2.8 times as large as that of the bare cavity. Some weak electric field profiles exist near the edges on both sides, away from the gap between the nanospheres. On the contrary, for Type B only a small part of the electric field is confined near the edges of the nanospheres and the electric field in the gap between the nanospheres is much weaker. Even so, the electric field strength is about 1.2 times as large as that of the bare cavity.

To further study the influence of different positions of the Au nanospheres on the hybrid nanocavities, we quantitatively characterize the optical properties of the hybrid nanocavities by calculating the quality factor Q, effective mode volume Veff, and figure of merit Q/Veff. The quality factor Q is defined as $Q=ωcΔω$, and the mode volume is defined as $V=∫V ε(r) E(r)2d3rmax(ε(r)E(r)2)$, which is usually expressed in units of the cubic wavelength (λ/n)3, termed the effective mode volume Veff. Thus the figure of merit Q/Veff is directly defined as the ratio of the quality factor Q to the effective mode volume Veff. As shown in Fig. 2, although these three quantities for the above two configurations all decrease as the radius of the nanospheres increases, there are a number of differences between the two structures. It is shown in Fig. 2(a) that the quality factor Q of the two structures is almost equal before the radius of the nanospheres is 15 nm, and after that Q for Type A is always lower than that for Type B. Besides, Veff for Type A is much smaller than that for Type B when the radius of the nanospheres is 25 nm or 40 nm, as shown in Fig. 2(b). The strong fluctuation of Type A is due to the stronger coupling between the Au nanospheres and the topological nanocavity. A possible reason for the sudden drop in the effective mode volume at 25 nm and 40 nm is the larger redshift of the resonance wavelength with the offset Δλ = 0.25 nm, resulting in a relatively large reduction of effective mode volume. Moreover, the bigger the radius of the nanospheres, the smaller Veff of the hybrid cavity, which indicates that bigger Au nanospheres are conducive to confining the light field within a smaller volume. Additionally, the values of the figure of merit Q/Veff for the two structures are equal before the radius of the nanospheres is 15 nm, and the figure of merit for Type A is lower than that for Type B after the radius of the nanospheres is 30 nm, as shown in Fig. 2(c). Consequently, these results also demonstrate that the smaller Au nanospheres with radius less than 15 nm can couple to the topological corner-state nanocavity in the two cases because these performance parameters are almost equivalent. However, for the bigger Au nanospheres the coupling effect is more significant when the nanospheres are aligned parallel to the direction of maximum field strength, which leads to relatively larger loss, making Q and Q/Veff much lower. Intriguingly, more electromagnetic field is confined within a smaller space, resulting in a smaller mode volume, owing to the strong coupling between the Au nanospheres and the nanocavity.

Figure 2. Calculated (a) quality factor Q, (b) effective mode volume Veff, and (c) figure of merit Q/Veff, varying with the radius of Au nanospheres. The subscripts A (black square) and B (red circle) represent the two kinds of designed hybrid nanocavities featuring Au nanospheres.

We next investigate the influence of the gap width between the two gold nanospheres on the optical properties of the hybrid nanocavity. When gradually increasing the gap width, Q, Veff, and Q/Veff for Type B are always stable, which suggests that the coupling between the Au nanospheres and the nanocavity is much weaker, and is consistent with the above opinion obtained from the electric field profile. But for Type A, these parameters experience an obvious change. It is shown in Fig. 3(a) that Q for Type A gradually decreases with increasing gap, and is always lower than that for Type B, but Q for Type A is still on the order of 104. In Fig. 3(b) Veff for Type A is stable around 0.127 (λ/n)3 and greater than VeffB before the gap is 50 nm. After that, VeffA declines and becomes smaller than VeffB, which indicates that the Au nanospheres with bigger gap have an outstanding ability to localize electromagnetic field. The sudden drop of the effective mode volume at a gap of 60 nm is due to the larger redshift of the resonance wavelength with the offset Δλ = 0.25 nm, leading to a relatively large reduction in the effective mode volume. As shown in Fig. 3(c), Q/Veff reaches minimum and maximum values when the gap is 50 nm and 60 nm respectively. Q/Veff is on the order of 105 (λ/n)−3, which is an order of magnitude larger than that for the bare cavity. Therefore, both configurations can reach a relatively large quality factor Q and figure of merit Q/Veff, but Type A is more beneficial for improving the emission property of quantum emitters due to higher electric field strength. These conclusions provide a novel method to tailor the light-matter interaction.

Figure 3. Calculated (a) quality factor Q, (b) effective mode volume Veff, and (c) figure of merit Q/Veff, varying with the gap width of the Au nanospheres. The subscripts A (black square) and B (red circle) represent the two kinds of designed hybrid nanocavities featuring Au nanospheres.

### Ⅲ. COUPLING BETWEEN THE CORNER-STATE CAVITY AND OTHER GOLD NANOANTENNAS

Apart from the Au nanospheres, other Au nanoantennas can also remarkably decrease the mode volume to improve the localization of the light field. In the following, different gold structures such as cylinders and rectangles are considered and the optical properties of diverse hybrid nanocavities are intensively investigated, as shown in Figs. 4(a) and 4(b). The electronic field strength profiles of various structures are calculated and mapped in Figs. 4(c) and 4(d). Most of the electromagnetic energy is confined inside the gap between these two Au nanoparticles to form a pronounced hot spot. Moreover, the electric field strength of the topological hybrid nanocavities with cylinder antennas is enhanced by a factor of 4.13, compared to the bare cavity. The presence of the rectangular antenna structure may weaken the electronic field strength to some extent, owing to the intense metal loss that is induced. As a consequence, using plasmonic nanoantennas with metal nanospheres and cylinders is easier for strengthening the light-matter interaction via enhanced electric field distribution.

Figure 4. (a, b) Structural diagrams and (c, d) electric field intensity distributions of the hybrid plasmonictopological corner-state nanocavities with cylindrical and rectangular nanoantenna structures respectively. All insets correspond to enlarged views of the gold structures. The electric field strength is normalized to the maximum field intensity in the same cross section of the bare cavity.

The quality factor Q, effective mode volume Veff, figure of merit Q/Veff, and total transmittance are calculated by the 3D finite-difference time-domain method to compare the differences between these gold antenna structures. Here the total transmittance is defined as the sum of the transmittances of the upper and left waveguides. The length for the sphere presented in Fig. 5 corresponds to the radius of the nanosphere in Fig. 2. As shown in Fig. 5(a), for all structures Q declines as the length of the gold structure increases. For the hybrid nanocavities with Au nanospheres and cylinders, Q is still above 5,000, but for the hybrid nanocavities with the rectangular nanoantenna, Q is always much smaller. The reason for this result is that more intense plasmons are induced at the interface between the Au rectangular antenna and the topological nanocavity, resulting in severe metal loss. In particular, Q drops dramatically to around 200 before a length of 25 nm. These results indicate that a larger Au rectangular structure produces more plasmons and stronger coupling to the corner-state nanocavity, which directly leads to more serious metal loss to deteriorate the quality factor of the hybrid cavity. Veff for the hybrid nanocavity with Au cylinders is stable around 0.12 (λ/n)3, like that for the nanosphere structure before the length of 30 nm, and then drops to the order of 10−2 (λ/n)3. The presence of the Au rectangular structure makes Veff of the hybrid nanocavity decrease from 10−2 (λ/n)3 to 10−3 (λ/n)3 before a length of 30 nm, and then Veff stabilizes on the order of 10−3 (λ/n)3. It is obvious from the simulation results in Fig. 5(b) that the bigger Au rectangular structures are much better at confining electromagnetic energy within a smaller volume. Although Q and Veff have a distinct difference between these structures, Q/Veff for all is still on the order of 105 (λ/n)−3, as shown in Fig. 5(c). When the length of the rectangular structure is 30 nm, Q/Veff reaches a maximum value of 2.52 × 105 (λ/n)−3. Additionally, the total transmittance values of the hybrid nanocavities with Au nanospheres and cylinders both increase with length, while that of the rectangular structures declines from 21% to 1.5% as the length increases, as shown in Fig. 5(d). This indicates that the rectangular antenna possesses a better ability to enhance the localization of the light field in the hybrid nanocavity, compared to other antennas. Less electromagnetic energy is leaked into the upper- and right-side waveguides, resulting in lower total transmittance. The larger rectangular antennas also exert a stronger influence on the performance of the hybrid nanocavity, which is basically consistent with the conclusion obtained from Fig. 5(b).

Figure 5. Variation laws of various optical properties. (a) Quality factor Q, (b) effective mode volume Veff, (c) figure of merit Q/Veff, and (d) total transmittance as functions of length, for various gold antenna structures.

The gap between these gold structures plays a crucial role in tuning the optical properties of the hybrid nanocavity. When the gap increases, it is seen in Fig. 6(a) that the quality factor Q for the hybrid cavities with Au nanospheres and rectangles gradually declines, while that for Au cylinders increases. Among these hybrid nanocavities, the presence of the Au rectangles keeps Q at its lowest values, below 600, due to intense metal loss. The Au nanospheres and cylinders keep Q much higher, above 104. It is seen from Fig. 6(b) that the Au nanospheres and cylinders yield effective mode volume Veff on the order of 10−1 (λ/n)3 and 10−2 (λ/n)3 respectively. Veff for the hybrid nanocavity with Au rectangular nanoantenna gradually decreases from 10−3 (λ/n)3 to 10−4 (λ/n)3, due to the continuous redshift of the resonance wavelength for the hybrid nanocavity. Therefore, the Au rectangular structure is more helpful for reducing the localization volume of the light field. It is worth mentioning that the figure of merit Q/Veff for all structures is always stable and on the order of 105 (λ/n)−3, as displayed in Fig. 6(c). The values of total transmittance for the hybrid nanocavities with Au nanospheres and cylinders are stable around 25% and 20% respectively. The total transmittance for the other structures drops dramatically to 2% when the gap increases, as shown in Fig. 6(d). Among these hybrid structures, the rectangular antennas cause Q to decrease, evidently due to greater induced metal loss, but simultaneously they greatly reduce Veff, compared to the other gold antennas. A possible reason for this is that the larger overlap area between the gold antennas and the dielectric nanocavity produces more intense plasmons, which directly leads to the obvious decrease in mode volume. Therefore, we could choose the most desirable hybrid plasmonic topological photonic nanocavity according to the optical performance of the different nano antennas.

Figure 6. Variation laws of various optical properties. (a) Quality factor Q, (b) effective mode volume Veff, (c) figure of merit Q/Veff, and (d) total transmittance as functions of gap width, for different gold antenna structures.

### Ⅳ. CONCLUSION

In conclusion, a topological corner-state nanocavity with ultrahigh quality factor Q of 104 is designed, and this topological nanocavity can couple to various gold antenna structures to enhance the light-matter interaction. Au nanospheres parallel to the maximum field strength couple effectively to the topological cavity, owing to a greater enhancement of the electric field, although the quality factor Q is relatively low, due to serious metal loss. Besides, other gold structures such as cylinders and rectangles can also obviously improve the optical properties of the hybrid topological cavity. The mapped electric field profiles of various hybrid nanocavities exhibit pronounced hot spots in the gap between these two Au nanoparticles, and the maximum electric field intensity for these structures is enhanced to different degrees. Especially, the presence of the Au rectangular structure greatly reduces the effective mode volume Veff, so that more electromagnetic field is concentrated within a much smaller space. The minimum effective mode volume Veff even reaches 10−4 (λ/n)3. Moreover, the figure of merit Q/Veff for all of these structures stays on the order of 105 (λ/n)−3. Therefore, hybrid plasmonic topological nanocavities with robustness against structural imperfections and unidirectional transmission can be realized via combining various gold nanostructures. This ultrahigh Purcell enhancement is beneficial to improving the optical properties of nanolasers and provides potential applications for effective and stable single-photon sources.

### DISCLOSURES

The authors declare no conflict of interest.

### DATA AVAILABILITY

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

### FUNDING

Central University Basic Research Fund (22CX03018A); Graduate Innovation Engineering Project (YCX2021145).

### Fig 1.

Figure 1.Structural schematics and electric field intensity maps. (a) Calculated electric field intensity distribution |E| for the bare topological corner state nanocavity. (b, c) Schematics of the hybrid plasmonictopological photonic nanocavity with Au nanospheres, marked as Type A and Type B. The yellow and red dotted frames represent trivial and topological regions respectively. (d, e) Calculated electric field intensity distribution |E| for the above two configurations, along the cross section of the center of the nanospheres. The radius and gap of the two nanospheres are 20 nm and 5 nm respectively. All insets correspond to enlarged views of the two gold nanospheres. The electric field strength is normalized to the maximum field intensity in the same cross section of the bare cavity.
Current Optics and Photonics 2022; 6: 497-505https://doi.org/10.3807/COPP.2022.6.5.497

### Fig 2.

Figure 2.Calculated (a) quality factor Q, (b) effective mode volume Veff, and (c) figure of merit Q/Veff, varying with the radius of Au nanospheres. The subscripts A (black square) and B (red circle) represent the two kinds of designed hybrid nanocavities featuring Au nanospheres.
Current Optics and Photonics 2022; 6: 497-505https://doi.org/10.3807/COPP.2022.6.5.497

### Fig 3.

Figure 3.Calculated (a) quality factor Q, (b) effective mode volume Veff, and (c) figure of merit Q/Veff, varying with the gap width of the Au nanospheres. The subscripts A (black square) and B (red circle) represent the two kinds of designed hybrid nanocavities featuring Au nanospheres.
Current Optics and Photonics 2022; 6: 497-505https://doi.org/10.3807/COPP.2022.6.5.497

### Fig 4.

Figure 4.(a, b) Structural diagrams and (c, d) electric field intensity distributions of the hybrid plasmonictopological corner-state nanocavities with cylindrical and rectangular nanoantenna structures respectively. All insets correspond to enlarged views of the gold structures. The electric field strength is normalized to the maximum field intensity in the same cross section of the bare cavity.
Current Optics and Photonics 2022; 6: 497-505https://doi.org/10.3807/COPP.2022.6.5.497

### Fig 5.

Figure 5.Variation laws of various optical properties. (a) Quality factor Q, (b) effective mode volume Veff, (c) figure of merit Q/Veff, and (d) total transmittance as functions of length, for various gold antenna structures.
Current Optics and Photonics 2022; 6: 497-505https://doi.org/10.3807/COPP.2022.6.5.497

### Fig 6.

Figure 6.Variation laws of various optical properties. (a) Quality factor Q, (b) effective mode volume Veff, (c) figure of merit Q/Veff, and (d) total transmittance as functions of gap width, for different gold antenna structures.
Current Optics and Photonics 2022; 6: 497-505https://doi.org/10.3807/COPP.2022.6.5.497

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Wonshik Choi,
Editor-in-chief