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Abstract
As the counterpart of electric and magnetic dipoles, the toroidal dipole is of paramount importance in the flourishing fields of metamaterials and Nanophotonics, and has therefore been getting much attention recently. Here, we demonstrate that a toroidal dipole can be formed in (dielectric core)@(plasmonic shell) nanostructures as the core refractive index is increased. For nanostructures with relatively large core refractive indices, polarized charge oscillation can be induced in the core by the oscillation of free electrons in the shell. These two oscillations create a toroidal dipole in the nanostructure. The formation of a toroidal dipole induces a scattering dip and a new absorption peak spectrally close to the created scattering dip. We also show that the toroidal dipole-induced absorption and scattering dip become weaker as the imaginary part of the core refractive index is increased. To the best of our knowledge, this is the first observation that the absorption becomes weaker with the increase of the imaginary part of the refractive index. Moreover, because almost all of the electric and magnetic fields are concentrated within the nanostructure, the toroidal dipole-induced scattering dip and absorption peak show resonance wavelength that is independent of the refractive index of the surrounding medium. Such an invariable property can be used as optical references or marks.
© 2017 Optical Society of America
1. Introduction
The toroidal dipole is produced by currents flowing on the surface of a torus along its meridians. It was proposed in 1957 by Zel’dovich, who suggested that such an excitation is produced by static currents, to explain the parity violation of the weak interaction [1]. Since then, toroidal dipoles and higher toroidal multipoles attract growing interest because of their unusual electromagnetic properties. For example, it has been shown that the strength of their interaction with electromagnetic field is not dependent on the strength of the field, but on the time derivative of electric field strength [2]. It has also been predicted that the nonstationary charge-current configurations involving toroidal multipoles could produce oscillating and propagating vector potential in the absence of electromagnetic fields [3,4]. The interactions between electrical currents producing toroidal multipoles have been shown to lead to the violation of the reciprocity theorem [5]. Moreover, materials with domains of toroidal polarization are expected to have different optical properties along opposite directions [6].
Although the existence of the static toroidal dipole has been predicted and its importance has been discussed for a number of solid-state systems, such as ferroelectric and ferromagnetic nano-microstructures, multiferroics, macromolecules, molecular magnets, etc [7–9], the dynamic toroidal dipoles are less known. On the one hand, toroidal multipoles are not included in the traditional multipole expansion in classical electrodynamics [10,11]. On the other hand, the far-fields of toroidal multipoles and the electric counterparts are indistinguishable [12], which results in the situation that in various scattering theories concerning particle scattering, toroidal multipoles are not considered as separate entities [13]. Toroidal multipoles are therefore often omitted in classical electrodynamics. Until recently, the toroidal dipoles were experimentally observed in microwave in metamaterials consisting of specially organized split ring resonators [14]. The importance of toroidal multipoles in classical electrodynamics has been realized and the toroidal multipoles within photonic nano-systems have attracted surging attention [12,15–23]. However, toroidal dipole can neither be obtained in a metallic torus [24], nor be excited magnetically in a torus-like metamaterial by straightforwardly rotating a magnetic-dipole structure [14]. Such a treatment just produces a linear superposition of individual magnetic dipoles, leading to a nonvanishing net magnetic dipole moment. To date, most structures that can support toroidal dipole are a type of toroid-shaped composite structures [15–17,23]. Only a few structures, for example Si disk, dielectric spheres and core@shell structures, are found to have ability to support toroidal dipoles in single structures [19–22]. The quest of simple structures that can support toroidal dipoles in single structures is therefore paramount importance to the investigations of the fundamental properties and applications of toroidal dipoles.
Here, we theoretically show the existence of toroidal dipoles in (dielectric core)@(plasmonic shell) nanostructures and the toroidal dipole-induced absorption and scattering dip. The plasmonic shells can support the electric dipole and high-order modes but cannot support the toroidal dipole. However, as the refractive index of the core is increased, the toroidal dipole can be formed in the core@shell nanostructures. The formation of toroidal dipole in the nanostructures results from the increased polarizability of core. The appearance of toroidal dipole creates a clear scattering dip and an intense absorption close to the scattering dip. We find that the toroidal dipole-induced scattering dip and absorption are reduced as the imaginary part of the core refractive index is increased. To the best of our knowledge, this is the first observation that the absorption of a nanostructure decreases with the increase of imaginary part of refractive index.
2. Calculation details
The scattering, absorption and extinction spectra of the nanostructures were calculated using Mie theory and its extension to core@multishell nanostructures on the basis of the recursion algorithm [13,25]. The electric and magnetic near-field distributions were calculated using a commercial software FDTD solutions v8.6 (Lumerical Solutions). The refractive index of the surrounding medium of the nanostructures was taken as 1.0, except in the refractive index sensitivity study. For the hollow shell, the refractive index of the interior was set as that of the surrounding medium. The dielectric function of gold and silver were taken from previously measured values [26,27].
3. Results and discussion
In Fig. 1, we schematically show the resonances in (dielectric core)@(plasmonic shell) nanostructures with different core refractive indices. When the core refractive index is the same as the surrounding medium refractive index of 1.0, the core@shell nanostructure can be seen as plasmonic shell. The scattering and absorption of plasmonic shells are dependent on their geometries [28]. When plasmonic nanoshells have large size and relatively thick shell, the scattering is dominated by the electric dipole but the absorption is contributed jointly by the electric dipole and high order electric modes [28], such as the electric quadrupole and octupole. The absorption and scattering peaks of large and thick nanoshells are therefore well separated from each other. The near zero absorption can be obtained at the scattering peak [Fig. 1]. In contrast, when the core@shell nanostructure has relatively large core refractive index, the polarized charge oscillation can be created in the core by the electric dipole oscillation in the shell. The currents created by two oscillations form a torus shape, creating a magnetic ring, i.e. a toroidal dipole [Fig. 1]. The formation of toroidal dipole creates a scattering dip and a new absorption peak [Fig. 1]. The scattering dip results from the destructive interference between the toroidal and electric dipole, which is also called as anapole. The new absorption peak originates from the changes of current distribution induced by the interference. The attractive interaction between the polarized charges and the free electrons results in the movement of the free electron oscillation from the nanostructure surface toward the inner shell. The movement of the free electron oscillation makes most currents locate within the Au shell, which gives rise to the absorption of the shell owing to the Ohmic loss, producing a new peak on the absorption spectrum [Fig. 1]. Here, we point out that although dielectric spheres can support intrinsic resonances when they have relatively large size [29,30], the toroidal dipole cannot be created by the interaction between these intrinsic resonance modes and plasmonic dipole in the shell.
Fig. 1 Schematic of the electromagnetic resonances of (dielectric core)@(plasmonic shell) nanostructures with different core refractive indices. The red circle arrows schematically show the electric currents produced by the free electron oscillation in the shell. The blue arrow schematically displays the electric current created by the polarization charge oscillation in the core. When the core has a relatively high refractive index, toroidal dipole is created by the currents induced by free electric and polarized charge oscillations. The formation of toroidal dipole gives rise to a scattering dip and a new absorption peak.
To confirm our assumption, we carried out theoretical simulations on a (dielectric core)@(plasmonic shell) nanostructure with core refractive index of 3.0. We find that, for a dielectric nanosphere with radius of 50 nm and refractive index of 3.0, the scattering increases toward short wavelength and does not show resonance peak in the visible and near infrared regions, and the absorption is zero owing to the zero imaginary part of refractive index [Fig. 2(a)]. A 50-nm thick Au nanoshell with outer radius of 100 nm exhibit a broad scattering band with peak at 618 nm and absorption peak at 504 nm [Fig. 2(b)]. The well spectrally separation of the scattering and absorption peaks arise from the different dependences of scattering and absorption on the polarizability of nanostructures [31]. The well separation of the absorption peak from the scattering peak produces a bump at the shorter wavelength than the main peak on extinction spectrum. One should note that the scattering peak of the Au shell is two orders of magnitude stronger than the scattering of the dielectric nanosphere. When the dielectric nanosphere and the Au nanoshell form a (dielectric core)@(Au shell) nanostructure, the scattering and absorption spectra become clearly different from either those of core, shell or their mathematical addition [Fig. 2(c)]. The scattering spectrum displays a clear scattering dip located at 646 nm. In addition to the original absorption peak, a new sharp peak appears in the absorption spectrum with resonance wavelength close to that of the scattering dip. Because the depth of the scattering dip is larger than the intensity of the new absorption peak, a dip is still produced on the extinction spectrum of the core@shell nanostructure. The wavelengths of the low and high energy resonance peaks and the dip are 673, 575 and 640 nm, respectively.
Fig. 2 Electromagnetic resonances of a dielectric nanosphere, a Au nanoshell and the (dielectric core)@(Au shell) nanostructure. (a) Optical cross-sections of a dielectric nanosphere with radius of 50 nm and refractive index of 3.0. (b) Optical cross-sections of a Au nanoshell with inner and outer radii of 50 and 100 nm, respectively. (c) Optical cross-sections of the nanostructures formed with the dielectric nanosphere shown in (a) and the Au nanoshell shown in (b). (d) Near-field electric field distribution of the core@shell nanostructure at the two resonance peaks and the dip. (e) Near-field magnetic field distribution of core@shell nanostructure at the two resonance peaks and the dip. The electric and magnetic field are monitored in the plane across the center of the nanostructure. The electric and magnetic field are normalized to the incident field. The dashed circles in (d) and (e) stand for the interface between the core and the shell.
Since the resonance frequency of the dielectric core is far from that of the Au shell and the resonance intensity of the dielectric core is also much weaker than that of the Au shell, the induced scattering dip and the new absorption peak in the core@shell nanostructures should not come from the direct interference of the resonances in the dielectric nanosphere and the Au shell when separately excited by plane waves. In order to find the underlying mechanism, we carried out near field distribution analysis for the core@shell nanostructure at the two resonance peaks and dip wavelengths. The electric and magnetic near field distributions in different planes are shown in Figs. 2(d) and 2(e), respectively. At the high-energy resonance peak, the enhanced electric field mainly locates at the surface of the nanostructures, while the magnetic field enhancement is negligible. Such electric and magnetic field distributions indicate that this resonance corresponds to the electric dipole resonance of Au shell. For the dip resonance, the electric field is greatly enhanced in the core while is slightly enhanced on the surface of the shell. The magnetic field is mainly concentrated in the nanostructure and forms a ring in the plane normal to the incident electric field [Fig. 2(e)], which is the typical characteristic of toroidal dipole. According to the spatial distribution of the toroidal dipole, a part of currents forming toroidal dipole locates into the Au shell, which gives rise to the new absorption peak. Therefore, the scattering dip in the scattering spectrum and the new peak in the absorption spectrum result from the formation of toroidal dipole. At the low energy resonance peak, relatively strong electric field enhancements are obtained both in the core and at the nanostructure surface, with the intensity at the surface stronger than that in the core. One should note that the electric field distribution at the nanostructure surface has the electric dipole characteristic. The magnetic field at the low energy resonance peak mainly locates in the nanostructure with slightly leaked magnetic field at the surface of the nanostructure. The magnetic field in the nanostructure also forms a ring in the plane normal to the incident electric field, which is the characteristic of toroidal dipole. In contrast the magnetic field ring at the dip resonance, the magnetic field intensity at the low energy resonance peak is very nonuniform in the ring, which increases along the incident direction of the excitation light [Figs. 2(c) and 2(e)]. The low energy resonance peak is therefore jointly contributed from toroidal and electric dipoles. Such a phenomenon has been observed in dielectric nanowires and disks [19,21].
We further study the evolution of the optical response of the core@shell nanostructure with the increase of core refractive index to find when the toroidal dipole can be created. When the refractive index of core is smaller than ~2.0, there is only a broad band in the scattering spectrum. When the core refractive index is larger than ~2.0, a dip appears on the scattering spectrum [Fig. 3(a)]. The dip wavelength exhibits near linear red shifts with the increase of core refractive index. The linear fitting shows that the variation of dip wavelength with core refractive index follows the equation of λdip = 139ncore + 230, where λdip and ncore are the dip wavelength and core refractive index, respectively. The dip arises from the interaction of toroidal dipole and electric dipole, which gives rise to nonradiating anapole modes [19]. Moreover, we find that the appearance of dip induces negligible shift and broadening of the entire scattering [Fig. 3(a)]. In another words, the increase of core refractive index only produces a dip on the scattering band but does not bring shift or broadening of the scattering band. Figure 3b displays the evolution of the absorption of the core@shell nanostructure with the increase of core refractive index. Only a broad absorption band appears in the absorption spectrum when the refractive index of core is smaller than ~2.0. As the core refractive index is further increased to larger than ~2.0, a new sharp absorption peak is produced besides the original broad absorption band. The new absorption peak displays a near linear red shift with the increase of core refractive index [Fig. 3(b)]. Because the depth of the dip in the scattering is larger than the intensity of the new absorption peak, a dip is produced in the extinction spectrum when the core refractive index is larger than ~2.0 [Fig. 3(c)].
Fig. 3 Evolution of electromagnetic resonances of the (dielectric core)@(Au shell) nanostructure with core radius of 50 nm and shell thickness of 50 nm with respect to the refractive index of core. (a) Scattering cross-section. (b) Absorption cross-section. (c) Extinction cross-section. (d) Scattering cross-section at the dip position (black circle) and the ratio of the scattering cross-section at the dip position to the scattering cross-section at the same wavelength but with core refractive index of 1 (blue square). (e) Absorption cross-section at the new induced absorption peak. (f) Extinction cross-section at the dip position (black circle) and the ratio of the extinction cross-section at the dip position to the extinction cross-section at the same wavelength but with core refractive index of 1 (blue square).
Figure 3(d) shows the evolution of the scattering intensity at the dip wavelength with the increase of core refractive index. The scattering intensity shows a monotonous decrease as the core refractive index is increased. We also studied the evolution of the ratio between dip scattering and the original scattering, which is defined as the scattering of the core@shell nanostructure with a core refractive index of 1.0 at the same wavelength. Although the scattering intensity at the dip decreases monotonously, the ratio of the dip scattering to the original scattering decreases with the increase of core refractive index at first, and then increases. Such an evolution results from the evolution of original scattering intensity. Owing to the red-shift of the dip resonance with the increase of core refractive index, the corresponding original scattering first increases and then decreases, making the ratio first decrease and then increase. When the core refractive index is about 3.0, the dip has the wavelength close to the original scattering peak and thus the ratio reaches its minimum, indicating the scattering is efficiently suppressed. The intensity of the new absorption peak increases with the increase of core refractive index [Fig. 3(e)]. Because the depth of the dip is large than the intensity of the new absorption peak, the evolution of dip extinction is similar to that of dip scattering [Fig. 3(f)].
Since the magnetic field distributions can give the direct evidence for the formation toroidal dipole, we then carried out magnetic field distributions of nanostructures with different core refractive indices. The core refractive indices of 1, 1.5, 2.0, 2.5 and 3.0 are chosen in the study. Figure 4a shows the excitation condition. The magnetic field is monitored in the plane normal to the incident electric field and across the center of the nanostructures [Figs. 4(b)−4(f)]. For the core with refractive index of 1.0 and 1.5, the magnetic fields are extremely weak and most of magnetic field distributes on the surface of the nanostructures [Figs. 4(b) and 4(c)]. For the nanostructure with core refractive index of 2.0, a very small dip and new absorption peak form on the scattering and absorption spectra, respectively. The magnetic field distributions at the low energy resonance peak and the small dip are shown in Figs. 4(d) and 4(g), respectively. At the low energy resonance peak, most of magnetic field still distributes at the surface of the nanostructure, with small part locating into the nanostructure [Fig. 4(d)]. The magnetic field at the dip wavelength is slightly enhanced within the nanostructure and forms a very weak magnetic ring [Fig. 4(g)]. The formation of magnetic ring indicates the appearance of toroidal dipole. The nanostructures with core refractive indices of 2.5 and 3.0 produce a clear dip and a new absorption peak on the scattering and absorption spectra, respectively. For these two nanostructures, the magnetic field is greatly enhanced and forms a ring shape within the nanostructures at the low resonance peak and the dip wavelengths [Figs. 4(e), 4(f), 4(h) and 4(i)]. The enhancement at the dip wavelength is stronger than that at the low energy resonance peak. Another difference is that the magnetic field at the low energy wavelength is also slightly enhanced at the surface of the nanostructures, while it is barely enhanced at the dip wavelength. It should be noted that the slight magnetic field enhancement at the surface of all nanostructures comes from the surface current of the electric dipole oscillation. Therefore, the low energy resonance peak of the whole nanostructures contains the contribution from electric dipole. We can therefore conclude that the appearances of the scattering dip and the new absorption peak are related to the formation of toroidal dipole.
Fig. 4 Magnetic field distribution of core@shell nanostructures with different core refractive indices. The nanostructures have a core radius of 50 nm and a Au shell thickness of 50 nm. (a) Schematic of the excitation condition. (b−f) Magnetic field distributions at the low energy resonance peak of nanostructures with core refractive indices of 1, 1.5, 2, 2.5 and 3, respectively. (g−i) Magnetic field distributions at the resonance dip of nanostructures with core refractive indices of 2, 2.5 and 3, respectively. All fields in (b−i) are monitored in x-y plane across the center of the nanostructures. The color bars in (c−i) are the same as that in (b). The dashed circles stand for the interface between the core and the shell.
Since many realisitic dielectric materials have nonzero imaginary part of refractive indices [27], we further studied the effect of the imaginary part of the core refractive index on the toroidal dipole of the nanostructure. The imaginary part of the core refractive index constant is used in the whole spectra range. Figure 5 shows the evolution of the optical response of the nanostructure as the imaginary part of the core refractive index is increased. With the increase of the imaginary part of the core refractive index, the depth of the dip on the scattering spectrum gradually reduces and disappears when the imaginary part is large than ~0.4 [Fig. 5(a)]. Further increase of the imaginary part above ~0.6, the scattering spectrum is very close to that of the nanostructure with core refractive index of 1.0. The new absorption peak exhibits gradual intensity decrease with the increase of the imaginary part of the core refractive index, and finally disappears as the imaginary part is larger than ~0.6 [Fig. 5(b)]. As far as we know, this is the first time for the observation that the absorption is reduced as the imaginary part of refractive index is increased. This phenomenum can be understood by the variation of electric field distribution in the Au shell. With the increase of the imaginary part of core refractive index, the oscillation of polarization charges in the core becomes weak and thereby the toroidal dipole becomes weak. Accompanied with the weakening of toroidal dipole, the electric field located in the Au shell is also weakened. As a result, the asborption is reduced as the imaginary part of core refractive index is increased. The evolution of the extinction spectrum with the imaginary part of the core refractive index is similar to that of the scattering spectrum. Such evolutions of scattering, absorption and extinction spectrum can be understood by the destruction of the polarization charge oscillation by the increase of imaginary part. As the imaginary part of core refractive index is increased, the polarizability of the core is reduced and the induced, the amount of the polarization charges becomes small and thus the current induced by the polarization charge oscillation is reduced. The disappearance of current produced by polarization charge oscillation results in the vanishement of the toroidal dipole. As a result, the dip on the scattering spectrum and the new peak on the absorption spectrum vanish.
Fig. 5 Evolution of electromagnetic resonances of the (dielectric core)@(Au shell) nanostructure with respect to the imaginary part of core refractive index. (a) Scattering cross-section. (b) Absorption cross-section. (c) Extinction cross-section. The nanostructure has a core radius of 50 nm and shell thickness of 50 nm. The real part of core refractive index is 3.0.
We further studied sensitivity of the scattering dip of the nanostructure to the refractive index of surrounding medium (nmedium). Although the nmedium increase leads to red-shifts and broadening of the lower energy scattering peak, the scattering dip almost keeps its resonance wavelength and linewidth when nmedium increases from 1.0 to ~1.35 [Fig. 6(a)]. When the nmedium gets larger than ~1.35, the depth of dip is reduced. Obviously, the toroidal dipole-induced scattering dip is different from the Fano dip in plasmonic oligomers. The latter is very sensitive to nmedium and shows great potential applications in sensing [32]. Such a difference can be understood by the fact that the electric and magnetic fields are mainly focused within the nanostructures for the toroidal dipole-induced scattering dip, while the electric fields are mainly distributed between the gap in Fano dip. The fields located within the nanostructures cannot be altered by the nmedium change, while the gap fields can be modified by the nmedium variation. Similarly, the toroidal dipole-induced absorption peak shows nearly constant resonance wavelength and linewidth when nmedium is smaller than ~1.35 [Fig. 6(b)]. The intensity of the new peak decreases as nmedium is increased. The narrow linewidth and invariable resonance wavelength to surrounding environment make the toroidal dipole-induced absorption and scattering dip be innate references for many applications in nanophotonics.
Fig. 6 Optical response of the (dielectric core)@(Au shell) nanostructures to the refractive index of surrounding medium. (a) Scattering spectra. (b) Absorption spectra. (c) Extinction spectra. The core has a radius of 50 nm and refractive index of 3.0. The shell has a thickness of 50 nm.
We finally studied the optical response of (dielectric core)@(Ag shell) nanostructures to find whether the toroidal dipole can be formed with the increase of core refractive index for different plasmonic shells [Fig. 7]. For a Ag shell with outer radius of 80 nm and shell thickness of 40 nm, the scattering spectrum, at the wavelength longer than 300 nm, is composed of a relatively broad electric dipole band and an electric high-order mode shoulder, the absorption spectrum exhibits a single peak contributed from electric high-order mode, and the extinction spectrum therefore shows two resonance peaks [Fig. 7]. As the core refractive index is increased, a scattering dip and a new peak appear on the scattering and absorption spectra, respectively [Figs. 7(a) and 7(b)], indicating that the toroidal dipole can also be supported by (dielectric core)@(Ag shell) nanostructures. Unlike the linear evolutions of dip and new peak resonance wavelength in (dielectric core)@(Au shell), the resonances of the scattering dip and the new absorption peak in (dielectric core)@(Ag shell) display two linear red-shift spectral ranges. The first range locates in the core refractive index smaller than ~2.2. In this range, the dip on the scattering spectrum is formed by the destructively interference between toroidal dipole and electric high-order mode. As a result, the dip locates at the spectral range of electric high-order mode in the scattering spectrum. When the core refractive index is larger than ~2.2, the evolution of toroidal dipole enters the second range. In this range, the destructively interference between toroidal dipole and electric dipole produces a clear dip at the spectral range of electric dipole resonance in the extinction and scattering spectra [Fig. 7]. We therefore believe that the two linear red-shift ranges of the scattering dip and the new absorption peak in (dielectric core)@(Ag shell) nanostructures originate from the different natures of the electric dipole and high-order mode. Considering that both (dielectric core)@(Au shell) and (dielectric core)@(Ag shell) nanostructures can support toroidal dipole, the toroidal dipole may be a universal property of (dielectric core)@(plasmonic shell) nanostructures.
Fig. 7 Optical cross-sections of (dielectric core)@(Ag shell) nanostructures with core diameter of 80 nm and shell thickness of 40 nm. (a) Evolution of scattering cross-section with the increase of core refractive index. (b) Evolution of absorption cross-section with the increase of core refractive index. (c) Evolution of extinction cross-section with the increase of core refractive index.
4. Conclusions
We demonstrate that toroidal dipole can be formed in (dielectric core)@(plasmonic shell) nanostructures as the core refractive index is increased. The increase of core refractive index endows the core with large polarizability and thus a great number of polarization charges can be induced by the plasmon resonance in the shell. The oscillations of polarization charges in the core and the free electrons in the shell give rise to the toroidal dipole in the nanostructure. The formation of toroidal dipole creates a scattering dip and induces a new absorption peak spectrally close to the scattering dip. We find that the toroidal dipole-induced scattering dip and absorption peak become weak as the imaginary part of core refractive index is increased. Moreover, owing to almost all electric and magnetic fields concentrated within the nanostructure, the toroidal dipole-induced scattering dip and the absorption peak show resonance wavelength independent of the refractive index of surrounding medium. As demonstrated in previous studies [33,34], the induced new absorption peak may generate hot-electrons, which can be utilized to enhance the efficiency of photocatalysts and solar cells.
Funding
National Natural Science Foundation of China (NSFC) (61505102, 61775129); Fundamental Research Funds for the Central Universities (GK201602004).
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