## Abstract

The topological magnetoelectric effect is associated with the photonic spin-orbit interaction. However, due to the proportionate fine structure constant of the topological term, the topological magnetoelectric effect is usually weak. In this paper, we demonstrate that the axion term enables manipulation of the spin Hall shift of light around Fano resonance. And, the excited surface plasmon near the nanoparticle's interface could enhance the topological magnetoelectric effect for several orders. Numerical simulation of near field and far-field scattering confirms our theoretical results. Our work may pave the way to exploit the topological magnetoelectric effect in practical applications, such as optical sensing and nanoprobing.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Recently, the topological magnetoelectric (TME) effect has attracted intensive interest in optical physics. TME effect is predicted by effective topological field theory [1,2], which modifies the Maxwell-Lagrangian of classical electromagnetism by adding an additional term $(\Theta \alpha /4{\pi ^2}){\bf E} \cdot {\bf B}$, where $\Theta \textrm{ = (2N + 1)}\pi $ is the axion angle in the effective Landau-Ginzburg theory, N is an integer indicating the highest fully filled Landau level and $\alpha \textrm{ = }{{{e^\textrm{2}}} / {\hbar c}}$ is the fine structure constant [3]. The axion angle $\Theta $ describes the topological magneto-electric polarizability. This parameter in TME effect shows that magnetization can be induced by an electric field, and electric polarization can be induced by a magnetic field [4]. Materials with TME effect have been predicted and demonstrated in various systems, such as Bi_{2}Te_{3}, Bi_{2}Se_{3}, Sb_{2}Te_{3} and SnTe [5–7]. Recently, the studies on TME effect have revealed many fascinating phenomena. For instance, a magnetic monopole charge can be induced by an electric charge owing to the TME effect [8]. And quantized Faraday and Kerr rotation are experimentally demonstrated in three-dimensional Bi_{2}Se_{3} films with TME effect, which could be used to directly measure the fine-structure constant [3]. Moreover, topological surface states are experimentally observed in topological metals, facilitating robust photon transport [9,10].

The quantized Hall effect is usually accompanied by the topological magnetoelectric effect. A quantized Hall current can be induced on the topological surface [11,12]. And this Hall current may give rise to additional light scattering from the surface. Meanwhile, if the light possesses both spin angular momentum (SAM) and orbital angular momentum (OAM), the topological magnetoelectric effect may produce nontrivial response on the spin-orbit interaction (SOI), which is the origin of spin Hall shift of light [13–16]. The spin Hall effect of electron is a phenomenon that moving electron varies its spatial trajectory due to the coupling of the electron’s spin and orbital motion. Analogous to Hall effects in electronic systems, spin Hall shift of light is a transversal shift of scattered light in the observed location to original location due to the spin-orbit interaction of light. Compared with the spin Hall effect in photonic and electronic systems, the polarization of light corresponds to the electron spin, and the refractive index gradient in inhomogeneous medium corresponds to the applied electric field to the electron [13]. Generally, SOI is relatively weak, while it can be enhanced in plasmonic system [17]. Due to the inherent sensitivity, Fano resonance is a good candidate for observing such small interactions. Fano resonance charactered by asymmetric profiles arises from the interference of discrete state with a continuum state, which is discovered firstly in quantum systems [18,19], and then in classical optics [20]. In Mie theory, Fano resonances could appear when a broad mode spectrally overlaps with a narrow discrete mode [21].

In this paper, we propose a plasmonic nanoparticle with topological surface to manipulate the spin Hall shift. The collective plasmonic resonance can enhance both the spin-orbit interaction and the topological magneto-electric effect. Hence, the topological parameter $\Theta $ can effectively tune the spin Hall shift at the plasmonic resonances. By properly selecting geometric parameters, the sharp electric quadrupole modes can interact with the broadband electric dipolar modes, giving rise to the Fano resonance [20,22,23]. The inherent high sensitivity renders us to probe spin-orbit interaction with far-field optical technique [24–26] and nano-mechanical measurement [27].

## 2. Theoretical formulations

In this section, the scattering behavior of a topological plasmonic nanoparticle with respect to the circularly-polarized wave is investigated. Without loss of generality, we consider a symmetric system, in which a topological nanosphere illuminated by a left-handed circularly polarized (LCP) light. The radius of the nanosphere is a. Here, we treat circularly-polarized wave as a superposition of *x*-polarized wave and *y*-polarized wave with a quadrature phase shift $\pi /2$, and suppress the time-dependence factor ${e^{ - i\omega t}}$. The LCP light takes the form ${\bf E}_{\textrm{inc}}^{} = {E_0}{e^{ikz}}(\hat{x}\textrm{ - }i\hat{y})$, where ${E_0}$ is the amplitude of the incidence.

According to Mie theory [28], the incident LCP fields can be expanded in vector spherical harmonics.

The constitutive relations for topological nanoparticle are described by ${\bf D}\textrm{ = }\varepsilon {\bf E}\textrm{ - }\bar{\alpha }{\bf B}\textrm{, }{\bf H}\textrm{ = }{{\bf B} / \mathrm{\mu}}$$\textrm{ + }\bar{\alpha }{\bf E}$, where $\bar{\alpha }\textrm{ = }{{\Theta \alpha } / \pi }$ is proportional to the axion angle. Compared with the topologically trivial case, it has been verified that no change takes place in the corresponding equations if the modified constitutive relations are applied in the Maxwell equations.

At the boundary between the nanoparticle and surrounding medium ($\textrm{r = a}$), we impose the boundary conditions.

*x*refers to the size parameter $x = ka$, and ${\Omega = }\bar{\alpha }\sqrt {{{\varepsilon }_\textrm{b}}\textrm{/}{\mathrm{\mu}_\textrm{b}}} .$

It’s convenient for us to define two new scattering coefficients ${A_n}$, ${B_n}$, where

to simplify the expressions in Eq. (3). Consequently, we can express our scattered fields in a nicer way.As is shown in Fig. 1, the Spin Hall (SH) shift is defined as the transverse shift in the perceived far-field location of the source [14], ${\Delta _{\rm{SH}}}\textrm{ = }\mathop {\lim }\limits_{r \to \infty } r{{{{\bf S}_\mathrm{\phi}}\hat{\phi }} / {|{{{\bf S}_r}} |}}$ where ${{\bf S}_\phi }$ and ${{\bf S}_\textrm{r}}$ are the azimuthal and radial component of the scattered Poynting vector ${\bf S} = {{{\bf E}_{\rm{sca}}^{} \times {\bf H}_{\rm{sca}}^\ast } / {\bf 2}}$.

For a topological nanosphere, the SH shift of the scattered light can obtain the following explicit form by applying equations (5).

_{1}and S

_{2}are the modified amplitude scattering matrix elements, ${\textrm{S}_\textrm{1}} = \sum\nolimits_{n = 1}^{n = \infty } {({({\textrm{2n + 1}} )/\textrm{n}({\textrm{n + 1}} )} )({{A_n}{\mathrm{\pi}_n} + {B_n}{\tau_n}} )} ,\;{\textrm{S}_\textrm{2}} = \sum\nolimits_{n = 1}^{n = \infty } {({({\textrm{2n + 1}} )\textrm{/n}({\textrm{n + 1}} )} )({{A_n}{\mathrm{\tau}_n} + {B_n}{\mathrm{\pi}_n}} )} .$ And the angle-dependent functions ${\mathrm{\pi} _{\rm n}}(\cos \theta )$ and ${\tau _{\rm n}}(\cos \theta )$ are defined as, ${\mathrm{\pi} _{\rm n}}(\cos \theta ) = \rm{P}_{\rm n}^1(\cos \theta )/$$\sin \theta $, ${\tau _{\rm n}}(cos\theta ) = d{\rm P}_{\rm n}^1(\cos \theta )/d\theta .$ Note that in the modified scattering coefficients and corresponding amplitude scattering matrix elements, the cross-polarized topological terms $a_n^T$ and $b_n^T$ play nontrivial role in tuning the near-field scattering fields and far-field spin Hall shift. These additional cross-polarized terms are produced by the TME effect.

## 3. Discussions

Topological particles can be obtained in experiment by various methods. For example, topological particles have been obtained by laser ablation technique at room temperature [27]. In addition, designed metamaterials have been demonstrated a photonic analogue of a topological insulator [30]. In this paper, we consider a solid plasmonic sphere with topological surface around Fano resonance. The particle’s permittivity ɛ is described by the Drude formula, and the related parameters are chosen at the typical Fano resonance with ${\mathrm{\omega} _p} = 5.25 \times {10^{15}}\;\textrm{rad }{\textrm{s}^{\textrm{ - 1}}}$ and $\gamma = 5.25 \times {10^{13}}\;\textrm{rad }{\textrm{s}^{\textrm{ - 1}}}$ (weak dissipation) [22] and size parameter $q = {{\mathrm{\omega} a} / c} \approx 0.64$ ($\mathrm{\omega}$ is the angular frequency of incident light, *a* is the radius of the particle and *c* is the speed of light in vacuum). As is shown in Fig. 2, higher-order modes (quadrupole n = 2) exhibit sharp resonance around incident wavelength $\lambda = 590\;nm$, giving rise to Fano resonance by interfering with the board dipole mode (n = 1).

For a topological insulator sphere, usually, the TME effect cannot be manifested in the scattering coefficients ${a_n}$ [29]. Because the topological term is proportional to the fine structure constant $\alpha$, which is of the order of 0.01. The TME effect on the scattering coefficients can be safely neglect. However, for a topological plasmonic sphere, we find ${a_1}$ and ${a_2}$ undergo obvious red-shift with the increase of axion angle $\Theta $ around dipole and quadrupole resonances, as is shown in Fig. 2(a). Besides the red-shift, the cross-polarized scattering coefficient $a_1^T$ and $a_2^T$ increase with the increase of the $\Theta $, as is shown in Fig. 2(b).

As for the far-field scattering, ${a_n}$ and $a_n^T$ contribute to the bulk scattering ${Q_{\textrm{bulk}}} = ({2 / {{q^2}}})\sum\nolimits_{n = 1}^\infty {(2n + 1)({{|{{a_n}} |}^2} + {{|{{b_n}} |}^2})}$ and surface scattering ${Q_{\textrm{surf}}} = ({2 / {{q^2}}})\sum\nolimits_{n = 1}^\infty {(2n + 1)({{|{a_n^T} |}^2} + {{|{b_n^T} |}^2})}$, respectively. The sum of ${Q_{\textrm{bulk}}}$ and ${Q_{\textrm{surf}}}$ is the total scattering efficiency ${Q_{\textrm{sca}}}$. The TEM effect will induce surface Hall currents at the surface of the sphere [31], and hence give rise to the extra surface scattering. As is shown in Fig. 2(d), for a conventional plasmonic sphere ($\Theta \textrm{ = }0$), there is no surface scattering. With the increase of $\Theta $ the surface scattering also increases accordingly. Importantly, around dipole and quadrupole resonances, the magnitude of $a_n^T$ and caused ${Q_{\textrm{surf}}}$ by TME effect are much larger than that of a topological insulator sphere. For topological plasmonic sphere, $a_n^T$ is enhanced by about two orders of magnitude, and ${Q_{\textrm{surf}}}$ is enhanced by about four orders of magnitude. These giant enhancements may due to the strong localized surface plasmon (LSP) near the interface of the sphere around the resonances.

We plot the electric field enhancements in the vicinity of the particle around Fano resonance ($\lambda = 590\;nm$), where the sharp electric quadrupole modes overlap with the board electric dipole modes. As is shown in Fig. 3, the electric fields show a typical electric quadrupole pattern, and the magnitude of the electric field near the surface is enhanced by as much as 40 times. It is well known that collective plasmons can effectively enhance light-matter interaction in various plasmonic systems [17]. In this work, we find a similar effect in the topological plasmonic system, where the TME effect is enhanced by surface plasmon. According to the modified constituent equations or the additional term $\Theta {\bf E} \cdot {\bf B}$ in the electrodynamic Lagrangian, [1] the localized surface plasmon will enhance the topological magnetoelectric effect in proportion. Therefore, the system can be topologically tuned by changing the axion angle $\Theta $ around plasmonic resonances. In Fig. 3, we plot the local spin angular momentum (SAM) density ($\textrm{SAM} = {(4\mathrm{\omega} )^{ - 1}}Im[{\varepsilon _0}{{\bf E}^\ast } \times {\bf E} + {\mu _0}{{\bf H}^\ast } \times {\bf H}]$) and its corresponding spin flow direction (black arrows) at the surface of the particle. The TME effect changes both the magnitude and directions of the SAM. As is shown in Fig. 3(b), the SAM is reduced, indicating that more SAM are transformed to local orbital angular momentum (OAM). It means the the spin-orbit interaction of light is enhanced as well.

Although the SH shift cannot be tuned by a dipole sphere, which is independent of the optical properties of a dipole; we have enhanced SH shifts of nanoparticles by overlapping their electric and magnetic dipolar modes, and electric dipolar and electric quadrupolar modes [32,33]. Here we show that the TME effect can be used to tune the SH shift around Fano resonance. In Fig. 4, we demonstrate the SOI-associated spin Hall shifts of the particle and their corresponding far-field scattering patterns. The incident wavelength is chosen as $\lambda \textrm{ = }590{\kern 1pt} \textrm{nm}$ around Fano resonance (see the inset in Fig. 3(a)). The axion angle $\Theta $ can switch the particle behind and before the Fano resonance, which originates from the interaction between the electric dipole mode and electric quadrupole mode (see Fig. 2(c)). One the other hand, the localized surface plasmons produce radiative modes that constructively or destructively interfere with each other. This strong interference, on the one hand, produces complex near-field distributions around the particles; on the other hand, the interference possesses high sensitivity to the particle’s optical properties and boundary conditions. The change of axion angle $\Theta $ can tune the SH shift (see Fig. 4(a)) and swift the far-field scattering (see Fig. 4(b)). The flip of forward scattering and backward scattering is also a typical phenomenon in Fano resonances. Interestingly, the changing of far-field SH shifts can be explained from the variation of near field spin flow. As analyzed in Fig. 3, the SH shifts are increased (blue dot line) due to the enhancement of SOI. The change of direction of spin flow on the particle’s surface may shed light on the changing of far-field SH shift. As is shown in Fig. 3(b), the net spin flow circulate from the backside to the front of the particle, which possibly enhances the SOI and then increase the SH shift on the backward (scattering angle $\theta \approx {160^\textrm{o}}$).

Finally, in order to verify our theoretical topological model, we use the finite element method (COMSOL Multiphysics V.5.4) to obtain the far-field scattering efficiency ${Q_{\textrm{sca}}}\textrm{ = }{Q_{\textrm{bulk}}} + {Q_{\textrm{surf}}}$ around the electric dipolar and electric quadrupolar resonances. The topological constitutive relations are set in the software by manually modifying its built-in variables for Maxwell’s equations. As shown in Fig. 5, we find the simulation results are in good agreement with the theoretical solutions.

## 4. Conclusion

To conclude, we have proposed to exploit topological magnetoelectric effect to tune the spin Hall shift of scattered light around Fano resonance. The topological magnetoelectric effect could render extra flexibility to light scattering and corresponding spin Hall shift of light. This flexibility may find application in measuring magnetoelectric response and establishing precise values of fundamental constants, identifying variations of structure parameters of nanostructure, and designing photonic analogue of a topological insulator with metamaterials [30,34,35]. In addition, the system is sensitive to the small variation of topological parameters around Fano resonance. The surface plasmon will greatly enhance the topological magnetoelectric effect. Hence, a small change in axion term can lead to dramatic variation in the spin Hall shift and far-field scattering pattern. The inherent sensitivity can offer new applications in many fields, including optical sensing dynamic tuning light-emission and plasmonic detection [36–39].

## Funding

National Natural Science Foundation of China (11504252, 11774252); China Postdoctoral Science Foundation (2018M630596); Qinglan Project of Jiangsu Province of China; “333” project (BRA2015353).

## Disclosures

The authors declare no conflicts of interest.

## References

**1. **X.-L. Qi, T. L. Hughes, and S.-C. Zhang, “Topological field theory of time-reversal invariant insulators,” Phys. Rev. B **78**(19), 195424 (2008). [CrossRef]

**2. **M.-C. Chang and M.-F. Yang, “Optical signature of topological insulators,” Phys. Rev. B **80**(11), 113304 (2009). [CrossRef]

**3. **L. Wu, M. Salehi, N. Koirala, J. Moon, S. Oh, and N. P. Armitage, “Quantized Faraday and Kerr rotation and axion electrodynamics of a 3D topological insulator,” Science **354**(6316), 1124–1127 (2016). [CrossRef]

**4. **A. Karch, “Electric-magnetic duality and topological insulators,” Phys. Rev. Lett. **103**(17), 171601 (2009). [CrossRef]

**5. **M. Z. Hasan and C. L. Kane, “Colloquium: Topological insulators,” Rev. Mod. Phys. **82**(4), 3045–3067 (2010). [CrossRef]

**6. **J. Yu, K. Zhu, X. Zeng, L. Chen, Y. Chen, Y. Liu, C. Yin, S. Cheng, Y. Lai, J. Huang, K. He, and Q. Xue, “Helicity-dependent photocurrent of the top and bottom Dirac surface states of epitaxial thin films of three-dimensional topological insulators Sb_{2}Te_{3},” Phys. Rev. B **100**(23), 235108 (2019). [CrossRef]

**7. **J. Yu, W. Wu, Y. Wang, K. Zhu, X. Zeng, Y. Chen, Y. Liu, C. Yin, S. Cheng, Y. Lai, K. He, and Q. Xue, “Giant photoinduced anomalous Hall effect of the topological surface states in three dimensional topological insulators Bi_{2}Te_{3},” Appl. Phys. Lett. **116**(14), 141603 (2020). [CrossRef]

**8. **X. L. Qi, R. Li, J. Zang, and S. C. Zhang, “Inducing a magnetic monopole with topological surface states,” Science **323**(5918), 1184–1187 (2009). [CrossRef]

**9. **S. Y. Xu, C. Liu, S. K. Kushwaha, R. Sankar, J. W. Krizan, I. Belopolski, M. Neupane, G. Bian, N. Alidoust, T. R. Chang, H. T. Jeng, C. Y. Huang, W. F. Tsai, H. Lin, P. P. Shibayev, F. C. Chou, R. J. Cava, and M. Z. Hasan, “Observation of Fermi arc surface states in a topological metal,” Science **347**(6219), 294–298 (2015). [CrossRef]

**10. **Z. Wang, D. Gresch, A. A. Soluyanov, W. Xie, S. Kushwaha, X. Dai, M. Troyer, R. J. Cava, and B. A. Bernevig, “MoTe_{2}: A type-II Weyl topological metal,” Phys. Rev. Lett. **117**(5), 056805 (2016). [CrossRef]

**11. **L. Ge, T. Zhan, D. Han, X. Liu, and J. Zi, “Unusual electromagnetic scattering by cylinders of topological insulator,” Opt. Express **22**(25), 30833–30842 (2014). [CrossRef]

**12. **A. Lakhtakia and T. G. Mackay, “Electromagnetic scattering by homogeneous, isotropic, dielectric-magnetic sphere with topologically insulating surface states,” J. Opt. Soc. Am. B **33**(4), 603–609 (2016). [CrossRef]

**13. **D. Haefner, S. Sukhov, and A. Dogariu, “Spin Hall effect of light in spherical geometry,” Phys. Rev. Lett. **102**(12), 123903 (2009). [CrossRef]

**14. **J. Soni, S. Mansha, S. Dutta Gupta, A. Banerjee, and N. Ghosh, “Giant Goos-Hänchen shift in scattering: the role of interfering localized plasmon modes,” Opt. Lett. **39**(14), 4100–4103 (2014). [CrossRef]

**15. **J. Soni, S. Ghosh, S. Mansha, A. Kumar, S. Dutta Gupta, A. Banerjee, and N. Ghosh, “Enhancing spin-orbit interaction of light by plasmonic nanostructures,” Opt. Lett. **38**(10), 1748–1750 (2013). [CrossRef]

**16. **W. Li, J. Liu, Y. Gao, K. Zhou, and S. Liu, “Photonic spin Hall effect on an ellipsoidal Rayleigh particle in scattering far-field,” Opt. Express **27**(20), 28194–28203 (2019). [CrossRef]

**17. **L.-K. Shi and J. C. W. Song, “Plasmon geometric phase and plasmon Hall shift,” Phys. Rev. X **8**(2), 021020 (2018). [CrossRef]

**18. **U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev. **124**(6), 1866–1878 (1961). [CrossRef]

**19. **A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, “Fano resonances in nanoscale structures,” Rev. Mod. Phys. **82**(3), 2257–2298 (2010). [CrossRef]

**20. **M. F. Limonov, M. V. Rybin, A. N. Poddubny, and Y. S. Kivshar, “Fano resonances in photonics,” Nat. Photonics **11**(9), 543–554 (2017). [CrossRef]

**21. **M. I. Tribelsky and B. S. Luk’yanchuk, “Anomalous light scattering by small particles,” Phys. Rev. Lett. **97**(26), 263902 (2006). [CrossRef]

**22. **B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. **9**(9), 707–715 (2010). [CrossRef]

**23. **I. Staude, T. Pertsch, and Y. S. Kivshar, “All-dielectric resonant meta-optics lightens up,” ACS Photonics **6**(4), 802–814 (2019). [CrossRef]

**24. **O. G. Rodriguez-Herrera, D. Lara, K. Y. Bliokh, E. A. Ostrovskaya, and C. Dainty, “Optical nanoprobing via spin-orbit interaction of light,” Phys. Rev. Lett. **104**(25), 253601 (2010). [CrossRef]

**25. **D. Rajesh, S. Nechayev, D. Cheskis, S. Sternklar, and Y. Gorodetski, “Probing spin-orbit interaction via Fano interference,” Appl. Phys. Lett. **113**(26), 261104 (2018). [CrossRef]

**26. **W. Liu, J. Zhang, B. Lei, H. Ma, W. Xie, and H. Hu, “Ultra-directional forward scattering by individual core-shell nanoparticles,” Opt. Express **22**(13), 16178–16187 (2014). [CrossRef]

**27. **W. H. Campos, J. M. Fonseca, V. E. de Carvalho, J. B. S. Mendes, M. S. Rocha, and W. A. Moura-Melo, “Topological insulator particles as optically induced oscillators: Toward dynamical force measurements and optical rheology,” ACS Photonics **5**(3), 741–745 (2018). [CrossRef]

**28. **C. F. Bohren and D. R. Huffman, * Absorption and scattering of light by small particles* (John Wiley & Sons, 1983).

**29. **L. Ge, D. Han, and J. Zi, “Electromagnetic scattering by spheres of topological insulators,” Opt. Commun. **354**, 225–230 (2015). [CrossRef]

**30. **A. B. Khanikaev, S. H. Mousavi, W. K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. **12**(3), 233–239 (2013). [CrossRef]

**31. **R. Schütky, C. Ertler, A. Trügler, and U. Hohenester, “Surface plasmons in doped topological insulators,” Phys. Rev. B **88**(19), 195311 (2013). [CrossRef]

**32. **R. Shi, D. L. Gao, H. Hu, Y. Q. Wang, and L. Gao, “Enhanced broadband spin Hall effects by core-shell nanoparticles,” Opt. Express **27**(4), 4808–4817 (2019). [CrossRef]

**33. **D. L. Gao, R. Shi, A. E. Miroshnichenko, and L. Gao, “Enhanced spin Hall effect of light in spheres with dual symmetry,” Laser Photonics Rev. **12**(11), 1800130 (2018). [CrossRef]

**34. **J. Maciejko, X. L. Qi, H. D. Drew, and S. C. Zhang, “Topological quantization in units of the fine structure constant,” Phys. Rev. Lett. **105**(16), 166803 (2010). [CrossRef]

**35. **X. Zhou, X. Ling, H. Luo, and S. Wen, “Identifying graphene layers via spin Hall effect of light,” Appl. Phys. Lett. **101**(25), 251602 (2012). [CrossRef]

**36. **X. Zhou, L. Sheng, and X. Ling, “Photonic spin Hall effect enabled refractive index sensor using weak measurements,” Sci. Rep. **8**(1), 1221 (2018). [CrossRef]

**37. **J. Bohn, T. Bucher, K. E. Chong, A. Komar, D. Y. Choi, D. N. Neshev, Y. S. Kivshar, T. Pertsch, and I. Staude, “Active tuning of spontaneous emission by Mie-resonant dielectric metasurfaces,” Nano Lett. **18**(6), 3461–3465 (2018). [CrossRef]

**38. **F. Zangeneh-Nejad and R. Fleury, “Topological Fano resonances,” Phys. Rev. Lett. **122**(1), 014301 (2019). [CrossRef]

**39. **G. Araneda, S. Walser, Y. Colombe, D. B. Higginbottom, J. Volz, R. Blatt, and A. Rauschenbeutel, “Wavelength-scale errors in optical localization due to spin-orbit coupling of light,” Nat. Phys. **15**(1), 17–21 (2019). [CrossRef]