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Abstract
Recently, using various intelligent approaches to achieve the efficient inverse design of photonic nanostructures with predefined and appropriate functionalities has attracted considerable attention. We propose a method to design subwavelength metal-dielectric nanoantennas and optimize the scattering directionality using a Bayesian optimization approach. The nanoantennas consisted of three gold disks separated by two dielectric layers. The geometrical parameters were optimized in an intelligent and fully automatic process. We showed that with the aid of the machine learning method, strong forward scattering or backward scattering at a specific wavelength could be efficiently achieved. We further showed that unidirectional scattering in opposite directions at two separate wavelengths can be designed. Moreover, it is possible to exchange the forward and backward directionality at two target wavelengths. The multipole decomposition approach was applied to analyze the multipole moments of the scattering field up to the third order. In the optimized unidirectional nanoantennas the electric and magnetic dipole moments satisfied the Kerker or anti-Kerker conditions at the wavelengths of interest. Our results demonstrated the possibility of automatically designing nanoantennas for specific applications via a machine learning scheme.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Given its profound implications and vast applicability, scattering of light by nanoparticles has gained considerable attention in various fields [1–7]. It has been used in various applications, including the directional coupling of light [8,9], nonlinear second harmonic scattering [10,11], and directional excitation of quantum emitters [12]. In most scenarios, such as in optical nanoantennas [13,14], ultrasensitive sensing [15] and photovoltaic devices [16], control over the scattering directionality of nanoparticles is of great significance.
Generally, the unidirectional scattering can be achieved in two distinct ways. The first approach, known as the detuned electric dipoles approach [17], uses two or more nanoparticles, supporting electric dipolar responses with deliberately tuned phases in each element to strengthen the scattering in a desired direction while reducing or totally suppressing it in the other directions. One particular example of such an approach is the Yagi-Uda antenna, which has been demonstrated to provide strong directivity in optical domain [18–20]. However, for this type of antenna, it is difficult to realize miniaturization and integration since the separation of the constitutent nanoparticles is about a quarter of a wavelength [21].
The second approach for creating unidirectional scattering relies on the excitation of electric and magnetic multipoles, e.g., electric dipole (ED) and magnetic dipole (MD) resonance, simultaneously. This approach presents an additional degree of freedom for tailoring the light emission characteristics in terms of frequency, efficiency, phase, and direction. Through tuning the parameters of the structure or/and constituent materials, the ED and MD can be engineered to coincide spectrally and be of the same strength (at the desired wavelength). An asymmetric field radiation with zero backward scattering or zero forward scattering can be obtained, which are often referred to as the Kerker and anti-Kerker conditions, respectively [22]. Nevertheless, an intrinsic MD is rarely shown in natural materials at the optical wavelength [23]. Even if it is present, it is often too weak to be effectively utilized. Fortunately, a stronger optical magnetic response can be achieved using some judiciously designed metal nanostructures or with high-index dielectric inclusions. This kind of antenna has been realized utilizing silicon nanodisks [24–28], core-shell particles [29,30], and dimer nanoantennas [31–33] in the optical regime and textured metallic disks supporting localized spoof plasmons at much lower frequencies [34].
In this work, we describe a scheme based on geometric parameter optimization to realize superior unidirectional scattering at one or multiple wavelengths. As an example, we studied the unidirectional scattering of a subwavelength multilayer metal-dielectric nanoantenna, which combines the advantages of both all-dielectric and plasmonic nanoantenna. The antenna structure is shown in Fig. 1. We show that by combining a computational electromagnetic solver and the Bayesian optimization method [35], a strong forward or backward scattering can be automatically achieved, with reasonable time requirement and cost. We also show that, similar to previous work [36], the unidirectional scattering in opposite directions at different wavelengths can be acquired rapidly. Furthermore, the enhanced forward and backward scattering at different wavelengths can be exchanged using different structures.
Fig. 1. Flowchart for the design optimization of the unidirectional scattering nanoantenna. The dashed box labeled COMSOL shows the schematic of the nanoantenna consisting of three gold nanodisks with diameters $d$ and thicknesses $t$, and they are separated by two different layers with thicknesses ${t_1}$ and ${t_2}$. The FS (forward scattering) is along the wave vector of the incident light ${\bf k}$, and it is opposite for the BS (backward scattering).
2. Theoretical background
Figure 1 depicts the framework and flow of the unidirectional scattering nanoantenna design, which has four basic ingredients: a candidate set selection, evaluator, EM solver, and Bayesian optimization. The nanoantenna we considered consisted of three gold disks with diameters $d$ and thicknesses $t$, and they were separated by two different dielectric layers with thicknesses ${t_1}$ and ${t_2}$. The corresponding refractive indices of the two different layers were denoted as ${n_1}$ and ${n_2}$, respectively, and the permittivity of gold was obtained from the report by Johnson and Christy [37]. All the full-wave electromagnetic calculations were performed using the commercial simulation package COMSOL Multiphysics, which is based on the finite element method (FEM) [38]. For the evaluator, the directivity ${G_{FB}} = 10{\log _{10}}({S_F}/{S_B})$ was chosen to quantitatively evaluate the performance of each configuration, where ${S_F}$ and ${S_B}$ are the radiated powers evaluated in the forward $(z ={+} \infty )$ and the backward $(z ={-} \infty )$ directions, respectively. After confirming the candidate set and evaluator, a bridge between the COMSOL and Bayesian optimization was built. Bayesian optimization, an experimental design algorithm based on machine learning [39,40], was utilized to achieve the optimal size of a structure that exhibited strong asymmetric scattering. Gaussian processes were adopted as surrogate models for the Bayesian optimization due to their flexibility and tractability.
The directivities of $N$ initial candidates are calculated (i.e., training examples) and a probabilistic regression model is learned from the training examples. For each iteration, a predictive distribution of the directivities is estimated for the remaining candidates, and a new candidate is selected based on the criterion of expected improvement [39]. The directivity for the new candidate is subsequently calculated and added to the training examples, which is consequently used to update the regression model for generating the next candidate suggestion. This iteration can be repeated until an optimum or pre-set criterion is achieved.
To understand the physical origin of the unidirectional scattering at the wavelength of interest, we employed a multipole decomposition method [41–46]. Generally, when light interacts with a nanoparticle, the scattered light for a far-field observer can be ascribed to radiation from different multipoles. The multipole decomposition starts from the polarization ${\bf P = }{\varepsilon _0}({\varepsilon _p} - {\varepsilon _d}){\bf E}$ induced by the incident light, where ${\varepsilon _0}$, ${\varepsilon _p}$, and ${\varepsilon _d}$ are the vacuum dielectric permittivity, the relative dielectric permittivity of the particle and the relative dielectric permittivity of the surrounding medium, respectively, and ${\bf E}$ is the electric field inside the particle. The time-harmonic fields of the incident plane waves are defined by $\exp ( - i\omega t)$, where $\omega$ is the angular frequency.
In the Cartesian coordinate system with the origin located at the mass center of the nanoparticle, the ordinary ED moment of the nanoparticle is expressed as follows [46]:
Supposed the other higher order terms are neglected in the considered nanoantennas, the total scattering cross-section ${\sigma _{sca}}$ from the above multipoles can be written as [41]
3. Results and discussion
Two optimization problems were tested: the first one was the BS or FS optimization at the wavelength of interest, and the second one was the optimization of FS and BS at different wavelengths and for their scattering directionality interchanged at fixed wavelengths. Considering the precision of the size and the properties of the nanoantenna, we set the $d$-parameter scope to $30,50,70,90,100$ nm, and the thicknesses of each layer $t,{t_i}(i = 1,2)$ were in the range $\Lambda = 10n$ nm $(n = 1,2 \ldots ,10)$. The refractive indices of the two layers ${n_j}(j = 1,2)$ were allowed to vary from 1 to 5. In addition, the nanoantenna was placed in a homogeneous surrounding medium with ${\varepsilon _d} = 1$.
In the first example, we fixed the wavelength at $\lambda = 670$ nm and checked the performance of the Bayesian optimization. 5 rounds of the optimization process were conducted with five different initial candidate sets. As shown in Fig. 2(a), the same global minimum or maximum could be obtained within 900 iterations (the total number of the candidate set was approximately 2.6 million) for all the optimizations processes. The optimal parameters are given in Table 1. Nanoantennas-I and II represent the structures of the BS and FS, respectively. The directivities ${G_{FB}}$ of the two nanoantennas were plotted in Fig. 2(b). For nanoantenna-I, the directivity was maintained below −10 dB for wavelengths from 625 to 725 nm and above 15 dB for nanoantenna-II with the same wavelength range. We found that the two nanoantennas exhibited a reasonably good bandwidth of unidirectional scattering. Figures 2(c) and 2(d) show the far-field radiation pattern of nanoantennas-I and II at $\lambda = 670$ nm, respectively. A nearly complete cancellation of the BS or FS was evident.
Fig. 2. FS and BS structure optimization for wavelength $\lambda = 670$ nm. (a) 5 optimization runs with different initial choices of the candidate sets for the case of forward and backward unidirectional scattering, respectively. (b) Far-field directivity ${G_{FB}}$ of the two nanoantennas: nanoantenna-I corresponds to the structure for BS and II corresponds to the structure for FS. Three-dimensional (3D) radiation patterns for (c) nanoantenna-I and (d) nanoantenna-II at $\lambda = 670$ nm.
Table 1. The optimal parameters of the nanoantenna-I and II (unit: nm)
To gain further insight into the physical mechanism of the unidirectional scattering, the multipole decompositions of the scattering cross-sections for these two nanoantennas were calculated. The results were shown in Fig. 3. It was noticed that the total scattering cross-section “Total Scat (COMSOL)” agreed with the multipoles summation “Sum Scat”, and all resonances agreed with a particular resonant multipole moment. Moreover, the EQ contribute negligibly to the scattering in the wavelength band considered. For the case of BS (see Fig. 3(a)), two MD resonances were excited at wavelengths $\lambda = 490$ and 725 nm, and one ED resonance was located at $\lambda = 535$ nm. From the classical electromagnetic theory, a pair of ED and MD with perpendicular orientations and equal contributions in the scattering can lead to the Kerker or anti-Kerker effects if the dipoles are in phase or out of phase, respectively [22,34]. An intersecting point of the ED and MD curves is approximately at $\lambda = 670$ nm, which corresponded to nearly zero-forward scattering (the anti-Kerker effect) (Fig. 2(c)). For the case of FS (Fig. 3(b) and Fig. 2(d)), it is evident that the Kerker effect was achieved at $\lambda = 670$ nm due to the equal contributions of the ED and MD. Furthermore, the intersection points of the ED, MD, and MQ curves shown in Fig. 3 are of great interest, and the corresponding far-field radiation patterns are shown in Fig. 4. For nanoantenna-I, the interactions between the ED, MD, and MQ moments led to predominant side scattering, as demonstrated in Fig. 4(a). However, a strong suppression of the side scattering was realized for $\lambda = 512$ nm (Fig. 4(b)). At $\lambda = 794$ nm, a strong suppression of the backward scattering occurred (Fig. 4(c)). For nanoantenna-II, Figs. 4(d)–4(e) demonstrated the side scattering suppression and anti-Kerker effect at $\lambda = 412$ and 582 nm, respectively.
Fig. 3. Scattering cross-sections and the multipole decompositions of the scattering spectra, including ED, MD, EQ and MQ, for nanoantenna-(a) I and (b) II. “Total Scat (COMSOL)” is the result calculated by the FEM method and “Sum Scat” is the sum of ED, MD, EQ and MQ.
Fig. 4. Radiation patterns for different wavelengths. The wavelengths correspond to the black arrows in Figs. 3(a) and 3(b). The wave vector of the incident light ${\bf k}$ was along the $+ z$ direction.
Based on the above results, we now discuss the second optimization problem: nanoantenna design for FS and BS at different wavelengths and their directionality interchange for a fixed wavelength. In this example, the evaluator of the Bayesian optimization should be redefined as:
We randomly selected the operating wavelengths, e.g., $\lambda = 650$ and 750 nm, and the candidate parameter region was the same as in the previous case. The optimal parameters of the structure, referred as nanoantenna-III, are listed in Table 2, and the corresponding directivity is shown in Fig. 5(a). It is seen that the directivity ${G_{FB}}$ reached approximately −24 dB at $\lambda = 650$ nm and increased to nearly 90 dB at $\lambda = 750$ nm. In contrast to the previous case, we speculate that the strong FS and BS occurred separately at the two wavelengths, which was confirmed by the scattering patterns (insets in Fig. 5(a)). Figure 5(b) shows the scattering cross-section spectra and the corresponding multipole decomposition. Clearly, compared to the ED and MD moments, the EQ and MQ moments were vanishingly small in the whole wavelength region, and could be neglected. For the ED and MD moments, there were two crossover points near $\lambda = 650$ and 750 nm, respectively, which implied that the Kerker and anti-Kerker effects were realized as shown in the inset of Fig. 5(a). Furthermore, we show the two-dimension (2D) radiation patterns for the resonance peaks in the total scattering cross-section (insets in Fig. 5(b)). Clearly, there is an extremely weak interaction between the ED and MD moments at the resonance peaks.Fig. 5. (a) Far-field directivity ${G_{FB}}$ for nanoantenna-III. The insets show the 3D radiation patterns for wavelength $\lambda = 650$ and 750 nm, respectively. (b) Corresponding scattering cross-section spectra and the multipole decompositions. The insets shows the 2D radiation pattern in the $xz$ plane for the peak wavelength. The black curve in the inset corresponds to the peak at $\lambda = 475$ nm and the red curve corresponds to the high peak at $\lambda = 700$ nm.
Table 2. The optimal parameters of the nanoantenna-III (unit: nm)
Finally, by allowing the Bayesian optimization to incorporate inequality constraints [47], it is possible to switch the directionality of the unidirectional scattering for the same wavelength. To verify this, we designed unidirectional scattering at wavelengths $\lambda = 600$ and 725 nm with the constraint that one of the directivities was less than zero. Table 3 lists the optimal results of the structures labeled as nanoantenna-IV and V, which were subjected to the constraints $G_{FB}^\lambda < 0$ ($\lambda = 600$ nm) and $G_{FB}^\lambda < 0\,$ ($\lambda = 725$ nm), respectively. The directivity spectra of nanoantennas-IV and V, shown in Fig. 6(a), show that the direction of the scattering reversed several times. Moreover, the directivity for nanoantenna-IV (V) reached a minimum (maximum) value of about −9 (47) dB at $\lambda = 600$ nm and a maximum (minimum) value of 38 (−16) dB at $\lambda = 725$ nm. This unambiguously demonstrated the interchange between FS and BS at the same wavelength of interest (insets in Figs. 6(b)–6(c)).
Fig. 6. (a) Far-field directivity ${G_{FB}}$ of nanoantenna-IV and V. Corresponding scattering cross-section spectra and the multipole decompositions for nanoantenna (b) IV and (c) V. The insets in (b) and (c) show the tion pattern in the $yz$ plane for $\lambda = 600$ and 725 nm. In the inset of (c) the black curve corresponds to $\lambda = 600$ nm and the red curve corresponds to $\lambda = 725$ nm.
Table 3. The optimal parameters of the nanoantenna-IV and V (unit: nm)
To analyze the directionality reversal in more detail, the scattering cross-section and the corresponding multipole decompositions for nanoantennas-IV and V are shown in Figs. 6(b) and 6(c), respectively. At the considered wavelengths, we did not observe any single EQ resonance. For nanoantenna-IV, there was an overlapping of the ED and MQ moments in the short wavelength region, whereas in the long wavelength region, the ED and MD resonances overlapped. The scattering contribution of MQ became completely invisible with the increase in wavelength. Figures 7(a)–7(d) show the far-field radiation patterns for points “a”, “b”, “c” and “d” marked in Fig. 6(b). Evidently, two unidirectional scattering in opposite directions could be realized due to the interaction between the ED, MD and MQ moments [i.e., for points “a” and “b”, Figs. 7(a) and 7(b)]. Figures 7(c) and 7(d) show the realization of the Kerker effect at wavelengths and 670 nm. For nanoantenna-V, a strong anti-Kerker effect was realized at a wavelength nm (Fig. 7(e)).
Fig. 7. Radiation patterns for points “a”, “b”, “c”, “d” and “e” marked in Figs. 6(b) and 6(c). (a) $\lambda = 511$ nm, (b) $\lambda = 536$ nm, (c)$\lambda = 563$ nm, (d)$\lambda = 670$ nm, (e)$\lambda = 543$ nm. The wave vector of the incident light ${\bf k}$ was along the $+ z$ direction.
4. Conclusions
In conclusion, combining a well-established electromagnetic solver and a custom Bayesian optimization implementation, we can quickly design nanoantennas with unidirectional scattering at one wavelength and unidirectional scattering in the opposite direction at different wavelengths. Furthermore, we used the multipole decomposition approach to analyze the underlying mechanism for the unidirectional scattering. Based on this machine learning method we could find particular designs that satisfied the Kerker or anti-Kerker conditions at specific predefined wavelengths. We also shown that the forward and backward scattering interchanges could be realized due to the interaction between the ED, MQ and MD moments. We expect that our scheme can provide an alternative method for efficient design of any unidirectional nanoantennas.
Funding
Science and Technology Planning Project of Shenzhen Municipality (GRCK20170822164729349, JCYJ20170811154119292, JCYJ20180306172003963); Natural Science Foundation of Guangdong Province (2015A030313748); China Postdoctoral Science Foundation (2018M630356).
Disclosures
The authors declare no conflict of interest.
References
1. S. Nie and S. R. Emory, “Probing single molecules and single nanoparticles by surface-enhanced Raman scattering,” Science 275(5303), 1102–1106 (1997). [CrossRef]
2. G. M. Akselrod, C. Argyropoulos, T. B. Hoang, C. Ciracì, C. Fang, J. Huang, D. R. Smith, and M. H. Mikkelsen, “enhancement in plasmonic nanoantennas,” Nat. Photonics 8(11), 835–840 (2014). [CrossRef]
3. P. Y. Chen, C. Argyropoulos, G. D’Aguanno, and A. Alù, “Enhanced second-harmonic generation by metasurface nanomixer and nanocavity,” ACS Photonics 2(8), 1000–1006 (2015). [CrossRef]
4. D. Sikdar, I. D. Rukhlenko, W. Cheng, and M. Premaratne, “Optimized gold nanoshell ensembles for biomedical applications,” Nanoscale Res. Lett. 8(1), 142 (2013). [CrossRef]
5. P. J. Schuck, D. P. Fromm, A. Sundaramurthy, G. S. Kino, and W. E. Moerner, “Improving the mismatch between light and nanoscale objects with gold bowtie nanoantennas,” Phys. Rev. Lett. 94(1), 017402 (2005). [CrossRef]
6. Z. Ruan and S. Fan, “Superscattering of light from subwavelength nanostructures,” Phys. Rev. Lett. 105(1), 013901 (2010). [CrossRef]
7. A. E. Miroshnichenko, B. Luk’yanchuk, S. A. Maier, and Y. S. Kivshar, “Optically induced interaction of magnetic moments in hybrid metamaterials,” ACS Nano 6(1), 837–842 (2012). [CrossRef]
8. F. J. Rodríguez-Fortuño, G. Marino, P. Ginzburg, D. O’Connor, A. Martínez, G. A. Wurtz, and A. V. Zayats, “Near-field interference for the unidirectional excitation of electromagnetic guided modes,” Science 340(6130), 328–330 (2013). [CrossRef]
9. Z. Xi, Y. Lu, W. Yu, P. Wang, and H. Ming, “Unidirectional surface plasmon launcher: rotating dipole mimicked by optical antennas,” J. Optics 16(10), 105002 (2014). [CrossRef]
10. X. J. Ni, N. K. Emani, A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Broadband light bending with plasmonic nanoantennas,” Science 335(6067), 427 (2012). [CrossRef]
11. N. F. Yu, P. Genevet, M. A. Kats, F. Aieta, J. P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011). [CrossRef]
12. A. G. Curto, G. Volpe, T. H. Taminiau, M. P. Kreuzer, R. Quidant, and N. F. Van Hulst, “Unidirectional emission of a quantum dot coupled to a nanoantenna,” Science , 329(5994) 930–933 (2010). [CrossRef]
13. L. Novotny and N. van Hulst, “Antennas for light,” Nat. Photonics 5(2), 83–90 (2011). [CrossRef]
14. S. Kühn, U. Håkanson, L. Rogobete, and V. Sandoghdar, “Enhancement of single-molecule fluorescence using a gold nanoparticle as an optical nanoantenna,” Phys. Rev. Lett. 97(1), 017402 (2006). [CrossRef]
15. I. Ament, J. Prasad, A. Henkel, S. Schmachtel, and C. Sönnichsen, “Single unlabeled protein detection on individual plasmonic nanoparticles,” Nano Lett. 12(2), 1092–1095 (2012). [CrossRef]
16. H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9(3), 205–213 (2010). [CrossRef]
17. A. B. Evlyukhin, S. I. Bozhevolnyi, A. Pors, M. G. Nielsen, I. P. Radko, M. Willatzen, and O. Albrektsen, “Detuned electrical dipoles for plasmonic sensing,” Nano Lett. 10(11), 4571–4577 (2010). [CrossRef]
18. T. Kosako, Y. Kadoya, and H. F. Hofmann, “Directional control of light by a nano-optical Yagi-Uda antenna,” Nat. Photonics 4(5), 312–315 (2010). [CrossRef]
19. D. Dregely, R. Taubert, J. Dorfmüller, R. Vogelgesang, K. Kern, and H. Giessen, “3D optical Yagi–Uda nanoantenna array,” Nat. Commun. 2(1), 267 (2011). [CrossRef]
20. E. A. Jones and W. T. Joines, “Design of Yagi-Uda antennas using genetic algorithms,” IEEE Trans. Antennas Propag. 45(9), 1386–1392 (1997). [CrossRef]
21. I. S. Maksymov, I. Staude, A. E. Miroshnichenko, and Y. S. Kivshar, “Optical Yagi-Uda nanoantennas,” Nanophotonics 1, 65–81 (2012). [CrossRef]
22. M. Kerker, D. S. Wang, and C. L. Giles, “Electromagnetic scattering by magnetic spheres,” J. Opt. Soc. Am. 73(6), 765–767 (1983). [CrossRef]
23. N. Noginova, G. Zhu, M. Mavy, and A. Noginov, “Magnetic dipole based system for probing optical magnetism,” J. Appl. Phys. 103(7), 07E901 (2008). [CrossRef]
24. I. Staude, A. E. Miroshnichenko, M. Decker, N. T. Fofang, S. Liu, E. Gonzales, and Y. Kivshar, “Tailoring directional scattering through magnetic and electric resonances in subwavelength silicon nanodisks,” ACS Nano 7(9), 7824–7832 (2013). [CrossRef]
25. M. R. Shcherbakov, D. N. Neshev, B. Hopkins, A. S. Shorokhov, I. Staude, E. V. Melik-Gaykazyan, and A. A. Fedyanin, “Enhanced third-harmonic generation in silicon nanoparticles driven by magnetic response,” Nano Lett. 14(11), 6488–6492 (2014). [CrossRef]
26. P. D. Terekhov, K. V. Baryshnikova, A. S. Shalin, K. Alina, and A. B. Evlyukhin, “Resonant forward scattering of light by high-refractive-index dielectric nanoparticles with toroidal dipole contribution,” Opt. Lett. 42(4), 835–838 (2017). [CrossRef]
27. A. F. Cihan, A. G. Curto, S. Raza, P. G. Kik, and M. L. Brongersma, “Silicon mie resonators for highly directional light emission from monolayer mos 2,” Nat. Photonics 12(5), 284–290 (2018). [CrossRef]
28. E. A. Gurvitz, K. S. Ladutenko, P. A. Dergachev, A. B. Evlyukhin, A. E. Miroshnichenko, and A. S. Shalin, “All-dielectric nanophotonics: The high-order toroidal moments and anapole states in all-dielectric photonics,” Laser Photonics Rev. 13(5), 1970025 (2019). [CrossRef]
29. W. Liu, A. E. Miroshnichenko, D. N. Neshev, and Y. S. Kivshar, “Broadband unidirectional scattering by magneto-electric core–shell nanoparticles,” ACS Nano 6(6), 5489–5497 (2012). [CrossRef]
30. R. Li, X. Zhou, M. Panmai, J. Xiang, H. Liu, M. Ouyang, H. Fan, Q. Dai, and Z. Wei, “Broadband zero backward scattering by all-dielectric core-shell nanoparticles,” Opt. Express 26(22), 28891–28901 (2018). [CrossRef]
31. T. Pakizeh and M. Käll, “Unidirectional ultracompact optical nanoantennas,” Nano Lett. 9(6), 2343–2349 (2009). [CrossRef]
32. X. M. Zhang, J. J. Xiao, Q. Zhang, L. M. Li, and Y. Yao, “Plasmonic tm-like cavity modes and the hybridization in multilayer metal-dielectric nanoantenna,” Opt. Express 23(12), 16122–16132 (2015). [CrossRef]
33. Q. Zhang, J. J. Xiao, X. M. Zhang, D. Han, and L. Gao, “Core–shell-structured dielectric–metal circular nanodisk antenna: gap plasmon assisted magnetic toroid-like cavity modes,” ACS Photonics 2(1), 60–65 (2015). [CrossRef]
34. F. F. Qin, Q. Zhang, and J. J. Xiao, “Sub-wavelength unidirectional antenna realized by stacked spoof localized surface plasmon resonators,” Sci. Rep. 6(1), 29773 (2016). [CrossRef]
35. C. E. Rasmussen and C. K. I. Williams, “Gaussian Processes for Machine Learning,” MIT Press: Cambridge, MA (2006).
36. X. M. Zhang, J. J. Xiao, Q. Zhang, F. F. Qin, X. Cai, and F. Ye, “Dual-band unidirectional emission in a multilayered metal–dielectric nanoantenna,” ACS Omega 2(3), 774–783 (2017). [CrossRef]
37. P. B. Johnson and R. W. Christy, “Optical constants of the noblemetals,” Phys. Rev. B 6(12), 4370–4379 (1972). [CrossRef]
38. http://www.comsol.com (accessed Jan 10, 2017).
39. D. R. Jones, “A taxonomy of global optimization methods based on response surfaces,” J. Global Optim. 21(4), 345–383 (2001). [CrossRef]
40. B. Shahriari, K. Swersky, Z. Wang, R. P. Adams, and N. De Freitas, “Taking the human out of the loop: A review of bayesian optimization,” Proc. IEEE 104(1), 148–175 (2016). [CrossRef]
41. P. D. Terekhov, K. V. Baryshnikova, Y. A. Artemyev, A. Karabchevsky, A. S. Shalin, and A. B. Evlyukhin, “Multipolar response of nonspherical silicon nanoparticles in the visible and near-infrared spectral ranges,” Phys. Rev. B 96(3), 035443 (2017). [CrossRef]
42. E. E. Radescu and G. Vaman, “Exact calculation of the angular momentum loss, recoil force, and radiation intensity for an arbitrary source in terms of electric, magnetic, and toroid multipoles,” Phys,” Phys. Rev. E 65(4), 046609 (2002). [CrossRef]
43. A. B. Evlyukhin, C. Reinhardt, and B. N. Chichkov, “Multipole light scattering by nonspherical nanoparticles in the discrete dipole approximation,” Phys. Rev. B 84(23), 235429 (2011). [CrossRef]
44. A. Pors, S. K. Andersen, and S. I. Bozhevolnyi, “Unidirectional scattering by nanoparticles near substrates: generalized Kerker conditions,” Opt. Express 23(22), 28808–28828 (2015). [CrossRef]
45. Q. Zhang, J. J. Xiao, M. Li, D. Han, and L. Gao, “Coexistence of scattering enhancement and suppression by plasmonic cavity modes in loaded dimer gap-Antennas,” Sci. Rep. 5(1) 17234 (2015). [CrossRef]
46. R. Alaee, C. Rockstuhl, and I. Fernandez-Corbaton, “An electromagnetic multipole expansion beyond the long-wavelength approximation,” Opt. Commun. 407, 17–21 (2018). [CrossRef]
47. J. R. Gardner, M. J. Kusner, Z. E. Xu, K. Q. Weinberger, and J. P. Cunningham, “Bayesian Optimization with Inequality Constraints,” PMLR 32(2), 937–945 (2014).
