Open Access
Add to BibSonomy
Abstract
We theoretically propose a broadband transverse unidirectional scattering scheme based on the interaction between a tightly focused azimuthally polarized beam (APB) and a silicon hollow nanostructure. When the nanostructure is located at a specific position in the focal plane of the APB, the transverse scattering fields can be decomposed into contributions from transverse components of the electric dipoles, longitudinal components of magnetic dipoles and magnetic quadrupole components. In order to satisfy the transverse Kerker conditions for these multipoles within a wide infrared spectrum, we design a novel nanostructure with hollow parallelepiped shape. Through numerical simulations and theoretical calculations, this scheme exhibits efficient transverse unidirectional scattering effects in the wavelength range of 1440 nm to 1820 nm (380 nm). In addition, by adjusting the position of the nanostructure on the x-axis, efficient nanoscale displacement sensing with large measuring ranges can be achieved. After analyses, the results prove that our research may have potential applications in the field of high-precision on-chip displacement sensors.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Interaction between nanostructures and beams enables manipulation of light fields at sub-wavelength scales, opening up a range of applications including photo-detection [1,2], imaging [3], and optical sensing [4]. Among them, nano-antennas are gaining increasing attention due to their superior near-field and far-field energy conversion capabilities [5], particularly in research involving directional far-field radiation [6–9]. For the realization of directional far-field radiation, metal nano-antennas with various configurations, such as core/shell structures [10], polymers [11], and asymmetric structures [12], have been extensively exploited in recent years. However, due to the unavoidable limitations from Ohmic losses and complicated geometrical configurations, schemes of metal nano-antennas have an adverse influence on performance of far-field radiation, hence limiting their practical application. All-dielectric materials with high refractive index, such as silicon, have low Ohmic losses in both visible and infrared bands [13], abundant electromagnetic modes [14,15], and high on-chip integration [16], which can provide effective solutions for this problem. It is possible to excite electric and magnetic modes with a single silicon nano-antenna yet no sophisticated structural design [17–20], thus achieving directional far-field radiation with high freedom. Particularly, transverse unidirectional scattering can be induced by altering the amplitude and phase difference between the electric and magnetic multipole moments to satisfy the transverse Kerker condition [21–24]. Moreover, illuminating silicon nanostructures with cylindrical vector beams (CVBs) rather than planar beams opens up entirely new applications for transverse unidirectional scattering, providing practical solutions for nanoscale displacement measurements [25–30].
CVBs contain radially (RPBs) and azimuthally (APBs) polarized beams and are widely used in fields of light-matter interaction due to special amplitude and phase distributions of their electromagnetic fields [31–37]. When CVB is tightly focused, depending on the type of polarization, a strong longitudinal electric or magnetic field is generated [38,39]; and the transverse scattering power of a nano-antenna moving in the focal plane with this beam is extremely sensitive to the distance between the nano-antenna and the focal point. As a result of the amplification of nanoscale displacements in near-field to scattering powers in far-field, highly accurate displacement measurements are achievable [25–30]. It has been experimentally demonstrated that silicon nanosphere is capable of displacement measurements with great resolution at the Åm level under CVBs illumination [27]. However, in addition to resolution, broadband operating frequencies and larger measuring range are also necessary for certain practical applications. Currently, transverse unidirectional scattering based on dipoles of silicon nanosphere under CVBs illumination can only operate at a single visible wavelength and all within a measuring range of 100 nm [25–28]. At the same time, broadband transverse unidirectional scattering has only been reported under RPBs illumination, with bandwidth only around 100 nm [30,40]; and investigation under APBs illumination remains an undiscovered field.
In order to achieve broadband transverse unidirectional scattering with wider bandwidth and larger displacement measurement range, we innovatively propose a scheme based on the interaction of tightly focused APB with silicon nanostructure featuring a hollow parallelepiped shape by numerical simulations. Due to the enormous longitudinal magnetic field at the center of APB and larger characteristic size of the nanostructure, it is discovered that the contribution of magnetic quadrupole moments to far-field radiation is no longer negligible. Therefore, a novel nanostructure is created so that dipoles and magnetic quadrupole excited by APB conform to transverse Kerker conditions within the broadband of the incident wave. Eventually, the scheme accomplishes broadband transverse unidirectional scattering in the near-infrared band from 1440 nm to 1820nm (380 nm), with a large range nanoscale displacement measuring.
2. Geometry and theoretical model
The proposed scheme is depicted in Fig. 1(a), in which the silicon nanostructure featuring a hollow parallelepiped shape is illuminated by tightly focused APB at the focal plane. The numerical aperture of the objective lens utilized to focus the APB is 0.3, and the dielectric permittivity of silicon is obtained from Palik’s handbook [41]. The side lengths of the nanostructure and the hollow are L1 = 620 nm and L2 = 320 nm, respectively, with a height of H = 210 nm. The size of the nanostructure is optimized to generate transverse unidirectional scattering in the infrared spectrum, and the hollow structure of it delivers broadband characteristics of transverse unidirectional scattering [30,40,42]. Some works have demonstrated that the presence or absence of a silica substrate has little influence on the spectrum features for a given all-dielectric nanostructure [43]; we also discussed the existence of a silica substrate in Supplement 1 (section 4), and proved that our scheme is effective in both situations. Hence, the nanostructure is considered to be in a homogeneous air medium with εd = 1. In addition, full-wave numerical calculations are performed utilizing COMSOL Multiphysics.
Fig. 1. Silicon nanostructure interacts with tightly focused azimuthally polarized beam (APB). (a) The APB focused by an objective lens with NA = 0.3 propagates along the z axis. The nanostructure has the following dimensions: H = 210 nm, L1 = 620 nm, and L2 = 320 nm (hollow). (b)-(e) Theoretical field intensity distribution of tightly focused APB in the focal plane.
In accordance with the vector diffraction integral theorem [38], the tightly focused APB is created in COMSOL Multiphysics, and results of electromagnetic field distributions in focal plane are depicted in Fig. 1(b)-(e). It is clear from the figures that the transverse electric field Et (|Et|2=|Ex|2+|Ey|2) and magnetic field Ht (|Ht|2=|Hx|2+|Hy|2) are azimuthally and radially polarized, respectively. Their field intensity distribution is null at the center of APB, increasing with the distance from the center and then decreasing, displaying a donut-like pattern. The absence of a longitudinal electric field component in APB is demonstrated, as evidenced by the fact that |Ez|2 = 0. Additionally, the longitudinal magnetic field Hz is found to be higher in the vicinity of the center and decrease with increasing distance from the center.
The multipole decomposition method is an analytical tool used to investigate the composition of the electromagnetic multipole moments in nanostructures, which can adequately delineate the scattering characteristics of the nanostructure [44–48]. Most previous researches adopted multipole decomposition method under the long-wave approximation [30,40]. However, this likely results in an imprecise computation of the multipoles and imposes matching requirement for corresponding size of nanostructures and incident wavelength, thus significantly restricting the design flexibility of the nanostructures. The multipole decomposition method beyond the long-wave approximation we adopted is efficient for all wavelengths and size dimensions, consequently facilitating a more accurate assessment of the scattering characteristics of nanostructures [48]. The exact expressions can be found in the first section of Supplement 1. Here, we investigated the interaction between APB and the nanostructure placed at x = 300 nm away from the center of APB (i.e. (x, y) = (300, 0) nm in the focal plane), as shown in Fig. 2(a). The multipole decomposition results reveal that the far-field scattering is composed of total electric dipole (TED), magnetic dipole (MD), electric quadrupole (EQ), and magnetic quadrupole (MQ). Interference between electric dipole (ED) and toroidal dipole (TD) forms TED mode, with the same far-field radiation pattern, making it impossible to distinguish between them [49]. It is evident from the results in Fig. 2(a) that contributions of TED and MD dominate in scattering intensity when the wavelength is greater than 1300 nm. However, we recognize that it is insufficient to evaluate only the scattering cross-section at a single position of the nanostructure, as was done in previous researches [30,40], since this may lead to inaccuracies. Multipole scattering cross-sections reflect the squared relationship between multipole moments, illustrated by Fig. 2(a). When quadrupole moments are relatively tiny in comparison to the dipole moments, quadrupole moments may be disregarded in scattering cross-sections, despite the fact that the quadrupole moments may still have an apparent impact on far-field radiation. Hence, we have creatively considered the radiation electric field of the nanostructure situated in various locations and determined that magnetic quadrupoles are not negligible.
Fig. 2. Analyses of the components of multipoles. (a) Normalized scattering intensity and the contribution of multipoles of the nanostructure positioned at x = 300 nm, where solid lines represent total electric dipoles (TED), magnetic dipoles (MD), electric quadrupoles (EQ), and magnetic quadrupoles (MQ), as well as their sums. The dashed line represents the total value calculated from full-wave simulations. (b) y-component of electric dipole moment (${p_y}$), z-component of magnetic dipole moment (${m_z}/c$), and xz-component of magnetic quadrupole moment ($ik/6c\ast {M_{xz}}$) versus x positions of nanostructure at incident wavelength λ = 1450 nm.
The relationship between multipole moments and radiation electric field is expressed by the following equation [48,50].
It can be inferred that the radiative field along the x direction is proportional to the three components ${p_y}$, ${\textrm{m}_\textrm{z}}\textrm{/c}$ and $ik/6c\ast {M_{xz}}$, with the magnitudes being indicated in Fig. 2(b). As the x position of the nanostructure increases, the contribution of the magnetic quadrupole grows more and more noticeably. Hence, when striving for higher accuracy in displacement measurement, both dipoles and magnetic quadrupole need to be taken into consideration for the successful implementation of unidirectional scattering along the x direction. Figure 3(a)-(d) depict the far-field scattering patterns of ${p_y}$, ${m_z}/c$ and $ik/6c\ast {M_{xz}}$ in the x-z plane, as well as the unidirectional scattering patterns caused by their interference. The marks on either side of the x-axis denote the phase symmetry of far-field radiation patterns, which is a simple expression of the interference between two or more multipole moments [51]. ${p_y}$ and $ik/6c\ast {M_{xz}}$ are observed to exhibit even symmetry, whereas ${m_z}/c$ displays odd symmetry. When the phase difference between even symmetry multipoles and odd symmetry multipoles is π (0), coherent subtraction occurs in the -x (+x) direction, resulting in a zero-radiated electric field, which in turn produces a unidirectional scattering in the + x (-x) direction. Consequently ${p_y}$, ${m_z}/c$ and $ik/6c\ast {M_{xz}}$ must concurrently accord with certain amplitude-matching and phase-matching (in-phase or anti-phase). The consequence of the amplitude varying with the x position of the nanostructure has been presented in Fig. 2(b) and the phase differences between them shall be discussed below.
Fig. 3. Patterns of multipole radiation and associated phase difference. Far-field scattering patterns of (a) electric dipole moment (${p_y}$); (b) magnetic dipole moment (${m_z}/c$); (c) magnetic quadrupole moment ($ik/6c\ast {M_{xz}}$), and (d) the interference between them results in unidirectional scattering along the x-axis. (e) Phase differences between ${p_y}$, ${m_z}/c$ and $ik/6c\ast {M_{xz}}$ vary with incident wavelength.
The phase differences between ${p_y}$, ${m_z}/c$ and $ik/6c*{M_{xz}}$ remain constant while the x location varies [27,30], as supported by third section of Supplement 1 (Fig. S5). As illustrated in Fig. 3(e), phase differences are simply related to the incident wavelength with dash lines denoting the markers deviating from 0 or π by 10°. Within the 1440 nm to 1820nm spectrum region, the phase difference between ${p_y}$, and ${m_z}/c$ is close to π (i.e., anti-phase), the phase difference between ${p_y}$ and $ik/6c\ast {M_{xz}}$ is close to 0 (i.e., in-phase), and the phase difference between ${m_z}/c$ and $ik/6c\ast {M_{xz}}$ is close to π (i.e., anti-phase), with all deviations within 10°. Obviously, amplitude-matching can be achieved through adjustment of lateral x position of the nanostructure, while phase-matching can be realized by selecting an appropriate incident wavelength. Therefore, nanostructure positioned in a particular location interacts with APB in the spectrum of 1440 nm-1820nm, theoretically leading to E-x = 0 and permitting the realization of broadband unidirectional scattering along the + x direction. The bandwidth of the wavelength is 380 nm (∼0.233λ), a value far larger than that in any previous reports [30,40]. In the process of reducing x from the amplitude-matching position to zero, the electric field intensity along x direction undergoes a regular and sensitive change. Consequently, measuring the electric field intensity along the x-axis enables displacement sensing in x direction.
3. Results and discussion
In this article, we explore the broadband unidirectional scattering performance utilizing four incident wavelengths in the 1440 nm-820 nm region, namely 1450 nm, 1550 nm, 1650 nm, and 1750nm, and the related simulation results are depicted in Fig. 4. According to Eq. (4), E-x = 0 can be attained when $- {p_y} - {m_z}/c + ik/6c\ast {M_{xz}} = 0$. We calculate the x position that can achieve amplitude-matching under the consideration of dipole moments and magnetic quadrupole moment ($- {p_y} - {m_z}/c + ik/6c\ast {M_{xz}}$), and dipole moments only ($- {p_y} - {m_z}/c$), and the results are displayed in Fig. 4(a), (d), (j), and (g). Obviously, the x position calculated involving only dipole moments is always less than the x position calculated involving both dipole moments and magnetic quadrupole moment. To further analyse these two methods, the nanostructure is placed in the respective positions obtained from these calculations, and the far-field scattering patterns are calculated and shown in the right side of Fig. 4. The far-field scattering patterns computed with Eq. (2) in the x-z plane (shown by the black solid line) are consistent with the far-field scattering patterns obtained from full-wave simulations in the x-z plane (represented by the red dashed line), validating the simulation's accuracy. In addition, the full-wave simulation vividly displays 3D far-field scattering patterns (in the middle). In the analysis of far-field scattering patterns, it has been demonstrated that the only consideration of dipole moments is insufficient; therefore, both simulation and theoretical calculation indicate that E-x ≠ 0, as depicted in Fig. 4(b), (e), (h), and (k). After taking into account the magnetic quadrupole moment, the x position for amplitude-matching is precisely identified, allowing the nanostructure to accomplish the completely unidirectional scattering depicted in Fig. 4(c), (f), (i). and (l).
Fig. 4. Unidirectional scattering along the + x axis at various wavelengths. (a), (d), (g), and (j) are theoretical calculations of the x position when unidirectional scattering occurs for 1450 nm, 1550 nm, 1650 nm, and λ = 1750nm, with red representing the absence of magnetic quadrupole moment and black representing its presence. (b), (e), (h), and (k) are scattering patterns for computed x locations at various wavelengths without taking magnetic quadrupole moment into account, where the dashed and solid lines represent the full-wave simulation results and theoretical calculations in the xz-plane, respectively, with the 3D radiation patterns in the middle. (c), (f), (i), and (l) illustrate the results of calculations involving magnetic quadrupole moment.
To quantitatively represent the broadband feature of unidirectional scattering, we establish a ratio Dx of intensity in the + x direction to that in the -x direction [40]:
According to the definition of Dx, when Dx = 1 (-1), the electric field scattering fully in the + x (-x) direction, and when Dx = 0, the intensity of scattering is the same in the + x and -x directions. Hence, the closer |Dx| is to 1, the greater the performance of unidirectional scattering. The results of calculating Dx at various x positions under 1450 nm, 1550 nm, 1650 nm, and 1750nm are depicted in Fig. 5. As the x position increases, it can be observed that Dx always tends to increase substantially before decreasing slowly. Furthermore, Dx is consistently greater than 0.9 for x positions between 525 nm and 600 nm, which is larger than previous research [40]. This indicates that when the nanostructure is positioned between 525 nm and 600 nm, effective unidirectional scattering occurs across the whole wavelength spectrum. In conclusion, the unidirectional scattering of the nanostructure has great broadband features.
To further investigate the dependence of scattering fields along the x-axis on the position of the nanostructure in the broadband, numerical calculations were used to validate the scattering characteristics of nanostructures at various positions. Figure 6 depicts far-field radiation patterns in the x-z plane of nanostructure for various transverse displacements (i.e., x = 0 nm, 100 nm, 200 nm, and 300 nm). The dashed lines represent the results of full-wave simulations, whereas the solid lines represent the theoretical calculation results. Using Fig. 6(d) as an illustration (λ = 1750nm), it can be seen that when the nanostructure is located at the origin, the radiation pattern exhibits the same variation trend in ± x directions, demonstrating a high level of symmetry; as the transverse displacement of the nanostructure increases, the intensity of radiation electric field in the -x direction decreases, and the intensity in the + x direction increases, with the shape of the scattering field patterns being highly correlated with x positions. Consequently, within the whole bandwidth, the position of the nanostructure may be inferred by measuring the intensity of radiation electric fields in ± x directions, providing a feasible resolution for sensing nanoscale displacement.
Fig. 6. Scattering patterns at various x locations for nanostructure. Scattering patterns in the x-z plane for x = 0 nm, x = 100 nm, x = 200 nm and x = 300 nm at (a) 1450 nm, (b) 1550 nm, (c) 1650 nm and (d) 1750 nm, where the dotted lines represent results of full-wave simulations and the solid lines represent results of theoretical calculations.
We analyse the performance of the proposed scheme for measuring nanoscale displacements, namely the quantitative link between intensity of electric radiation field and x-position. Following previous researches [29,30], the scattering power in the + x and -x directions is defined to be the left and right powers, respectively, and the quantitative form of the right-to-left ratio (RLR) is shown below.
Fig. 7. Right-to-left ratios (RLR) for scattering power as functions of x positions for nanostructure. At (a) 1450 nm, (b) 1550 nm, (c) 1650 nm and (d) 1750 nm, the solid blue and black lines are calculated results for theory 1 (RLR1: calculated with dipoles and magnetic quadrupole) and theory 2 (RLR2: calculated only with dipoles) respectively, while the red dashed lines are results of full-wave simulations.
Moreover, we explored the universality of displacement measurements when nanostructure deviated from the focus plane. Figure 8 depicts the relationship between RLR and transverse displacement (x position) for the nanostructure with 0 nm, 50 nm, and 100 nm centers along the z-axis, respectively. The results of all three cases are identical, demonstrating that the nanostructure does not need to be precisely positioned on the focus plane, but having some freedom in the z-direction. Therefore, we present a method for measuring nanoscale displacement with broad spectral features, large measuring ranges, and universal capability in the z direction.
Fig. 8. Right-to-left ratios (RLR) of scattering powers as a function of x positions at various z positions. The black, red, and green lines in the (a) full-wave simulations and (b) theoretical calculations correspond to z = 0 nm, z = 50 nm, and z = 300 nm, respectively.
4. Conclusion
In conclusion, we have achieved broadband transverse unidirectional scattering at infrared wavelengths using tightly focused APB illuminating a silicon nanostructure with hollow parallelepiped shape. It was discovered through analysing the scattering cross-section and the far-field radiation at various positions of the nanostructure that the transverse scattering field can be decomposed into contributions of the electric dipole component (${p_y}$), the magnetic dipole component (${m_z}/c$) and the magnetic quadrupole component ($ik/6c\ast {M_{xz}}$). The shape of the nanostructure was delicately designed so that ${p_y}$, ${\textrm{m}_\textrm{z}}\textrm{/c}$ and $\mathrm{ik/6c\ast }{M_{xz}}$ interfere in the x-direction and satisfy the transverse Kerker conditions. As a result, this scheme enables efficient transvers unidirectional scattering in the 1440 nm -1820nm (380 nm) wavelength range. Meanwhile, our results also demonstrated that nanoscale displacement sensing along the x-direction can be achieved. As the incident wavelength varies between 1440 and 1820nm, the measuring range exceeds 100 nm. For the features of broad operating bandwidth, large measuring range and universal capability in the z direction, this scheme offers a new solution for nanoscale displacement sensing.
Funding
National Natural Science Foundation of China (12274462).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
References
1. C. Liu, J. Guo, L. Yu, J. Li, M. Zhang, H. li, Y. Shi, and D. Dai, “Silicon/2D-material photodetectors: from nearinfrared to mid-infrared,” Light: Sci. Appl. 10(1), 123–1184 (2021). [CrossRef]
2. L. Cao, J. Park, P. Fan, B. Clemens, and M. L. Brongersma, “Resonant Germanium Nanoantenna Photodetectors,” Nano Lett. 10(4), 1229–1233 (2010). [CrossRef]
3. M. Khorasaninejad, F. Aieta, P. Kanhaiya, M. A. Kats, P. Genevet, D. Rousso, and F. Capasso, “Achromatic Metasurface Lens at Telecommunication Wavelengths,” Nano Lett. 15(8), 5358–5362 (2015). [CrossRef]
4. M. A. Schmidt, D. Y. Lei, L. Wondraczek, V. Nazabal, and S. A. Maier, “Hybrid nanoparticle–microcavity-based plasmonic nanosensors with improved detection resolution and extended remote-sensing ability,” Nat. Commun. 3(1), 1108 (2012). [CrossRef]
5. P. M. Hlschlegel, H. J. Eisler, and O. J. F. Martin, “Resonant Optical Antennas,” Science 308(5728), 1607–1609 (2005). [CrossRef]
6. S. S. Kruk, L. Wang, B. Sain, Z. Dong, J. Yang, T. Zentgraf, and Y. Kivshar, “Asymmetric parametric generation of images with nonlinear dielectric metasurfaces,” Nat. Photonics 16(8), 561–565 (2022). [CrossRef]
7. P. D. Terekhov, A. B. Evlyukhin, A. S. Shalin, and A. Karabchevsky, “Polarization-dependent asymmetric light scattering by silicon nanopyramids and their multipoles resonances,” J. Appl. Phys. 125(17), 173108 (2019). [CrossRef]
8. A. G. Curto, G. Volpe, T. H. Taminiau, M. P. Kreuzer, R. Quidant, and N. F. V. Hulst, “Unidirectional Emission of a Quantum Dot Coupled to a Nanoantenna,” Science 329(5994), 930–933 (2010). [CrossRef]
9. T. Coenen, F. B. Arango, A. F. Koenderink, and A. Polman, “Directional emission from a single plasmonic scatterer,” Nat. Commun. 5(1), 3250 (2014). [CrossRef]
10. T. Feng, Y. Xu, W. Zhang, and A. E. Miroshnichenko, “Ideal Magnetic Dipole Scattering,” Phys. Rev. Lett. 118(17), 173901 (2017). [CrossRef]
11. G. Lu, Y. Wang, and R. Y. Chou, “Directional side scattering of light by a single plasmonic trimer,” Laser Photonics Rev. 9(5), 530–537 (2015). [CrossRef]
12. H. Zhang, W. Shang, H. Lu, F. Xiao, and J. Zhao, “Subwavelength asymmetric Au-VO2 nanodisk dimer for switchable directional scattering,” Chin. Phys. B 27(11), 117301 (2018). [CrossRef]
13. Y. H. Fu, A. I. Kuznetsov, A. E. Miroshnichenko, Y. F. Yu, and B. L. Yanchuk, “Directional visible light scattering by silicon nanoparticles,” Nat. Commun. 4(1), 1527 (2013). [CrossRef]
14. K. Yao and Y. Liu, “Controlling Electric and Magnetic Resonances for Ultracompact Nanoantennas with Tunable Directionality,” ACS Photonics 3(6), 953–963 (2016). [CrossRef]
15. D. Permyakov, I. Sinev, D. Markovich, P. Ginzburg, A. Samusev, P. Belov, V. Valuckas, A. I. Kuznetsov, B. S. Luk, A. E. Miroshnichenko, D. N. Neshev, and Y. S. Kivshar, “Probing magnetic and electric optical responses of silicon nanoparticles,” Appl. Phys. Lett. 106(17), 171110 (2015). [CrossRef]
16. H. Shu, L. Chang, Y. Tao, B. Shen, W. Xie, M. Jin, A. Netherton, Z. Tao, X. Zhang, R. Chen, B. Bai, J. Qin, S. Yu, X. Wang, and J. E. Bowers, “Microcomb-driven silicon photonic systems,” Nature 605(7910), 457–463 (2022). [CrossRef]
17. A. B. Evlyukhin, B. S. Luk’yanchuk, B. N. Chichkov, C. Reinhardt, and A. Seidel, “Optical response features of Si-nanoparticle arrays,” Phys. Rev. B 82(4), 045404 (2010). [CrossRef]
18. A. I. Kuznetsov, A. E. Miroshnichenko, Y. H. Fu, J. Zhang, and B. L. Yanchuk, “Magnetic light,” Sci. Rep. 2(1), 492 (2012). [CrossRef]
19. B. S. Luk’yanchuk, N. V. Voshchinnikov, R. Paniagua-Dominguez, and A. I. Kuznetsov, “Optimum Forward Light Scattering by Spherical and Spheroidal Dielectric Nanoparticles with High Refractive Index,” ACS Photonics 2(7), 993–999 (2015). [CrossRef]
20. V. Valuckas, R. Paniagua-Domínguez, Y. H. Fu, B. Lukýanchuk, and A. I. Kuznetsov, “Direct observation of resonance scattering patterns in single silicon nanoparticles,” Appl. Phys. Lett. 110(9), 091108 (2017). [CrossRef]
21. H. K. Shamkhi, K. V. Baryshnikova, A. Sayanskiy, P. Kapitanova, P. D. Terekhov, P. Belov, A. Karabchevsky, A. B. Evlyukhin, Y. Kivshar, and A. S. Shalin, “Transverse scattering and generalized kerker effects in all-dielectric mie-resonant metaoptics,” Phys. Rev. Lett. 122(19), 193905 (2019). [CrossRef]
22. J. Li, N. Verellen, D. Vercruysse, T. Bearda, L. Lagae, and P. V. Dorpe, “All-Dielectric Antenna Wavelength Router with Bidirectional Scattering of Visible Light,” Nano Lett. 16(7), 4396–4403 (2016). [CrossRef]
23. J. Tian, Q. Li, Y. Yang, and M. Qiu, “Tailoring unidirectional angular radiation through multipolar interference in a single-element subwavelength all-dielectric stair-like nanoantenna,” Nanoscale 8(7), 4047–4053 (2016). [CrossRef]
24. Y. Yang, J. Liu, Y. Yu, D. Qiao, Y. Yong, and S.-M. Ralas, “Broadband unidirectional transverse light scattering in a V-shaped silicon nanoantenna,” Opt. Express 30(5), 7918–7927 (2022). [CrossRef]
25. A. Bag, M. Neugebauer, U. Mick, S. Christiansen, S. A. Schulz, and P. Banzer, “Towards fully integrated photonic displacement sensors,” Nat. Commun. 11(1), 2915 (2020). [CrossRef]
26. M. Neugebauer, P. Woźniak, A. Bag, G. Leuchs, and P. Banzer, “Polarization-controlled directional scattering for nanoscopic position sensing,” Nat. Commun. 7(1), 11286 (2016). [CrossRef]
27. A. Bag, M. Neugebauer, P. Woźniak, G. Leuchs, and P. Banzer, “Transverse Kerker Scattering for Angstrom Localization of Nanoparticles,” Phys. Rev. Lett. 121(19), 193902 (2018). [CrossRef]
28. H. Zhang, K. Gao, L. Han, S. Liu, T. Mei, F. Xiao, and J. Zhao, “Nanometric displacement sensor with a switchable measuring range using a cylindrical vector beam excited silicon nanoantenna,” Opt. Express 29(16), 25109–25117 (2021). [CrossRef]
29. W. Shang, F. Xiao, W. Zhu, L. Han, M. Premaratne, T. Mei, and J. Zhao, “Unidirectional scattering exploited transverse displacement sensor with tunable measuring range,” Opt. Express 27(4), 4944–4955 (2019). [CrossRef]
30. J. Wang, X. Wang, and M. Zeng, “Broadband transverse displacement sensing of silicon hollow nanodisk under focused radial polarization illumination in the near-infrared region,” Chin. Opt. Lett. 18(6), 063602 (2020). [CrossRef]
31. H. B. Sedeh, D. G. Pires, N. Chandra, J. Gao, D. Tsvetkov, P. Terekhov, I. Kravchenko, and N. Litchinitser, “Manipulation of Scattering Spectra with Topology of Light and Matter,” Laser Photonics Rev. 17(3), 2200472 (2023). [CrossRef]
32. Y. Chen, S. Chen, T. Shih, J. Wang, W. Yang, L. Qian, and Z. Yang, “Broadband unidirectional scattering in visible ranges and controllable hot-spot spatial transfer via a single nanoparticle,” Appl. Surf. Sci. 528, 146489 (2020). [CrossRef]
33. P. Woźniak, P. Banzer, and G. Leuchs, “Selective switching of individual multipole resonances in single dielectric nanoparticles,” Laser Photonics Rev. 9(2), 231–240 (2015). [CrossRef]
34. S. Nechayev, J. S. Eismann, M. Neugebauer, P. Woźniak, A. Bag, G. Leuchs, and P. Banzer, “Huygens’ dipole for polarization-controlled nanoscale light routing,” Phys. Rev. A 99(4), 041801 (2019). [CrossRef]
35. M. Neugebauer, T. Bauer, P. Banzer, and G. Leuchs, “Polarization Tailored Light Driven Directional Optical Nanobeacon,” Nano Lett. 14(5), 2546–2551 (2014). [CrossRef]
36. F. Deng, H. Liu, M. Panmai, and S. Lan, “Sharp bending and power distribution of a focused radially polarized beam by using silicon nanoparticle dimers,” Opt. Express 26(16), 20051–20062 (2018). [CrossRef]
37. B. Wang, Y. Che, X. Zhong, W. Yan, T. Zhang, K. Chen, Y. I. Xu, X. Xu, and X. Li, “Cylindrical vector beam revealing multipolar nonlinear scattering for superlocalization of silicon nanostructures,” Photonics Res. 9(6), 950–957 (2021). [CrossRef]
38. K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7(2), 77–87 (2000). [CrossRef]
39. H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2(8), 501–505 (2008). [CrossRef]
40. F. Xue, H. Fan, Q. Dai, H. Liu, and S. Lan, “Broadband unidirectional scattering in the transverse direction and angular radiation realized by using a silicon hollow nanodisk under a radially polarized beam,” J. Phys. D: Appl. Phys. 55(9), 095111 (2022). [CrossRef]
41. E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, 1985).
42. T. Feng, W. Zhang, Z. Liang, and Y. Xu, “Unidirectional emission in an all-dielectric nanoantenna,” J. Phys.: Condens. Matter 30(12), 124002 (2018). [CrossRef]
43. V. R. Tuz, V. V. Khardikov, and Y. S. Kivshar, “All-Dielectric Resonant Metasurfaces with a Strong Toroidal Response,” ACS Photonics 5(5), 1871–1876 (2018). [CrossRef]
44. A. V. Kuznetsov, A. C. Valero, H. K. Shamkhi, P. Terekhov, X. Ni, V. Bobrovs, M. V. Rybin, and A. S. Shalin, “Special scattering regimes for conical all-dielectric nanoparticles,” Sci. Rep. 12(1), 21904 (2022). [CrossRef]
45. A. B. Evlyukhin, T. Fischer, C. Reinhardt, and B. N. Chichkov, “Optical theorem and multipole scattering of light by arbitrarily shaped nanoparticles,” Phys. Rev. B 94(20), 205434 (2016). [CrossRef]
46. K. V. Baryshnikova, P. D. Terekhov, Y. A. Artemyev, A. B. Evlyukhin, A. S. Shalin, and A. Karabchevsky, “Multipolar response of nonspherical silicon nanoparticles in the visible and near-infrared spectral ranges,” Phys. Rev. B 96(3), 035443 (2017). [CrossRef]
47. A. B. Evlyukhin and B. N. Chichkov, “Multipole decompositions for directional light scattering,” Phys. Rev. B 100(12), 125415 (2019). [CrossRef]
48. R. Alaee, C. Rockstuhl, and I. Fernandez-Corbaton, “An electromagnetic multipole expansion beyond the long-wavelength approximation,” Opt. Commun. 407, 17–21 (2018). [CrossRef]
49. A. E. Miroshnichenko, A. B. Evlyukhin, Y. F. Yu, R. M. Bakker, A. Chipouline, A. I. Kuznetsov, B. L. Yanchuk, B. N. Chichkov, and Y. S. Kivshar, “Nonradiating anapole modes in dielectric nanoparticles,” Nat. Commun. 6(1), 8069 (2015). [CrossRef]
50. J. D. Jackson, Classical Electrodynamics3rd ed. (Am. J. Phys, 1999).
51. W. Liu and Y. S. Kivshar, “Generalized Kerker effects in nanophotonics and meta-optics [Invited],” Opt. Express 26(10), 13085–13105 (2018). [CrossRef]
