1. Introduction

Core shell nanoparticles have attracted a lot of attentions recent year due to the strong plasmonic coupling with tunable resonances by controlling the shell thickness and the size of the nanoparticles [1–3]. The electromagnetic fields interacting with a core shell particle have been studied in 1951 [4] using the Mie theory [5]. An intuitive model analogous to the molecular orbital theory was developed to explain the plasmon resonances for particles smaller than 1/10 of the wavelength of the illuminating light [6,7]. It has been demonstrated that the resonances of the core shell nanoparticles can be determined by the plasmon hybridization of the induced charge distributions in the inner and outer sides of the shells. As the dimension of nanoshell increases, the different plasmon modes will couple to each other owing to the retardation effect [2,8,9] and the quasistatic approximation will no longer be able to predict the resonances. Core shell nanoparticles, which may exist in various geometries and can be fabricated from many different materials, have found extensive applications in surface-enhanced Raman spectroscopy [10], fluorescence enhancement [11], photothermal cancer therapy [12,13], solar cell [14], nanoantenna [15], and photocatalyst [16]. However, few of previous theoretical and experimental studies of core-shell nanoparticles paid attention to the energy flows or manipulated energy flow in nanoshells with the retardation effect taken into consideration. This motivated the present investigation of optical vortices inside core-shell nanoparticles.

The light energy flows for small particle and high-order resonances were descripted in references [17,18]. Some peculiarities of light scattering, which are different from the electric dipole scattering (Rayleigh scattering), were found near plasmon resonant frequencies [19]. This is because the radiative damping related to inverse transformation of localized resonant plasmons into scattered light prevails over the usual dissipations. Under this circumstance, the interferences between the scattered light and the incident light can produce the complicated patterns of the energy flows such as the optical vortex. The optical whirlpool is a special case of the optical vortices. This kind of coupled optical vortices also relates to the absorption cross section [20]. Furthermore, the coupled optical vortex chains can be created by plasmonic nanodisc arrays [21]. The bifurcations of energy flow around the nanowire have been demonstrated and can be controlled by the size and relative permittivity of the nanowire [22]. Recently, a new concept called vortex nanogear transmissions (VNTs) was designed by a microsphere-coupled gold dimer for multi-functional phase-operated photonics machinery [23,24]. This design takes researchers one step closer to the information processing on chip scale using the optic vortex generated by the photonic nanostructures.

The electromagnetic power flow is determined by the local time-averaged Poynting vector S = 1/2Re $[E\times H^{*}]$, where E is the electric field and H is the magnetic field. The phase of the Poynting vector, ${\varphi }_s$, is defined as $\sin {\varphi }_s=S_z\left(x,z\right)/\left|\stackrel{⃑}{S}\right|,$ and $\cos {\varphi }_s=S_x\left(x,z\right)/\left|\stackrel{⃑}{S}\right|,$ where $\left|\stackrel{⃑}{S}\right|$ is the modulus of $\stackrel{⃑}{S}$. The topological features of the Poynting vector are related to the phase singularities or the stationary maxima, minima and saddle nodes. The saddle nodes occur at a relative minimum along one axial direction and where the crossing axis is a relative maximum. A singularity always accompanies the circulation of power flow which is called the optical vortex. In this paper, the optical vortices generated by the nanospheres are studied. The nanospheres are composed by a dielectric core with gold-shell cladding structures. The relation between cross sections and optical vortices was found by sweeping the extinction, scattering, and absorption cross sections of the single nanoshell of different radius ratios. Optical vortices were found to be near the low scattering and relatively strong absorption cross section. This explains why the zero scattering of nanoshell happens and make the optical invisibility cloak [25]. A pair of optical vortices inside a nanoshell evolves into a dimer form by a pair of nanoshells instead of a single nanoshell. Using the dielectric nanospheres to affect the energy flows is discussed in this paper.

2. Optical vortexes in a core-shell nanoparticle

To study the optical energy flowing around the core-shell nanospheres illuminated by light of different wavelengths, a single dielectric-core gold-shell nanosphere with inner radius r₁ and outer radius r₂ was considered as shown in Fig. 1(a). The extinction, scattering, and absorption efficiencies are calculated for different radius ratios r₁/r₂. To make it easy to understand the physical properties of the optical vortices in a core-shell nanosphere, the outer radius r₂ was set to be 140 nm and the relative permittivity of the core material was set to be 4.0 in our study. The comparison of analytical and numerical solutions for r₁/r₂ = 0.5 are shown in Fig. 1(b). In this figure the analytical solutions are shown as the solid lines and the numerical results of the finite-difference time-domain method [26] are shown as the dashed lines. The incident plane waves with x-polarization and wavelengths ranging from 300 nm to 1300 nm were considered. The wave propagated vertically in the -z direction. Perfectly matched layers were assumed for the boundaries in our computation domain. Johnson and Christy’s experimental data [27] were chosen to fit the optical properties of gold. The effects of the relative permittivity of the core material and the dimensions of the nanosphere on optical vortices are discussed below.

 

Fig. 1 (a) Schematic drawing of a single dielectric-core and gold-shell nanosphere, where r₁ and r₂ denote the inner radius and outer radii. The incident wave with x-polarization propagates in the negative z direction. (b) Extinction, scattering, and absorption efficiencies of analytical (Q_ex-A, Q_sc-A, and Q_ab-A), and numerical solutions (Q_ex-N, Q_sc-N, and Q_ab-N).

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The extinction, scattering, and absorption cross section spectra of the core-shell nanoparticle with fixed outer radius r₂ = 140 nm and a fixed relative permittivity of the core material 4.0 are shown in Figs. 2(a)-2(c) for r₁/r₂ from 0.5 to 1.0. The extinction cross section in Fig. 2(a) equals to the summation of the scattering cross section in Fig. 2(b) and the absorption cross section in Fig. 2(c). The results shown in Fig. 2 indicate that there are two modes at least. For a fixed radius ratio, one mode is in the long wavelength region and the other mode is in the short wavelength region. The maxima of the extinction, scattering, and absorption cross sections are red-shifted as the radius ratio increases. For the mode with a long wavelength, it has the largest scattering cross section at radius ratio about 0.85 in Fig. 2(b), and its absorption cross section increases with radius ratio gets larger which means the thinner gold shell in Fig. 2(c). For the mode with a short wavelength, the scattering cross section becomes smaller as the radius ratio increases as shown in Fig. 2(b), but the absorption cross section does not change much for different radius ratios in Fig. 2(c). It should be bear in mind that the two bands of scattering cross section spectra in Fig. 2(b) and the two bands of absorption cross section spectra in Fig. 2(c) are not at the same wavelength. The case with the radius ratio equal to 1.0 is a single dielectric nanosphere without gold coating shell. Scattering only happens in the short wavelength region in this case.

 

Fig. 2 (a) Extinction, (b) scattering, and (c) absorption cross section spectra of a single dielectric-core gold-shell nanoparticle for radius ratio r₁/r₂ from 0.5 to 1.0 with the outer radius equal to 140 nm and the relative permittivity of the dielectric core equal to 4.0.

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To study the optical energy flows of different structures, we took the results of radius ratios r₁/r₂ = 0.5, 0.7, and 0.9 in Fig. 2 and studied their electric field intensities and optical energy flows for the wavelengths in Figs. 3-5. The extinction, scattering, and absorption cross section spectra for the radius ratio r₁/r₂ = 0.5 are shown in Fig. 3(a). It can be seen that there are two peaks at wavelengths of 764 nm and 542 nm in the extinction cross section spectra (denoted by the black arrow in this figure). These two peaks are contributed by multi-modes due to the geometry of the nanosell locating in the phase retardation region. There is a dip at wavelength of 560 nm in the scattering cross section spectra as denoted by red arrow in Fig. 3(a). Although this dip is not very low, it may correspond to a specific physical mechanism comparing to the cases in the other wavelengths. Thus, we plot the Poynting vector lines for the cases of three wavelengths: 764 nm, 560 nm, and 542 nm in Figs. 3(b)-3(d). In this plots, the color intensities indicate the amplitudes of Poynting vectors and the stream lines indicate the directions of the energy flows. The corresponding electric fields of these three cases are shown in Figs. 3(e)-3(g), where the color intensities indicate the amplitudes of electric fields and the stream lines indicate the directions of the electric fields. It is can be seen in Fig. 3(b) that the energy flow pattern is quite similar to the Rayleigh scattering which has been pointed in Ref. 22. This is confirmed by the electric field plot in Fig. 3(e). The plot shows two strong electric charges on the outer surface of the gold shell in the lower part of the nanosphere. The amplitude of the energy flows in Fig. 3(b) are high on the outer surface but low on the inner surface of the gold shell and within the dielectric core material. This explains why the strong scattering but weak absorption happens at wavelength of 764 nm in Fig. 3(a). As shown in Fig. 3(c), some energy flows penetrate into the shell from the upper side of the nanosphere and reach the bottom of the core material, and some energy flows enter and go through the shell from the lower side of the nanosphere. These energy flows form the circulation located at the interfaces of the lower sides of the core and the shell. Comparing to the energy flow pattern of the extinction peak at the wavelength of 542 nm in Fig. 3(d), the circulations at (x = −55 nm, z = −35) and (x = 55 nm, z = −35) in Fig. 3(c) are very visible. This phenomenon correlated to the electric field distributions in Figs. 3(f) and 3(g), which show stronger electric field intensities inside the core in Fig. 3(f). The electric charges in Figs. 3(f) and 3(g) are not symmetric as a classic quadrupole mode due to the effect of the higher order modes. The absorption of the case in Fig. 3(g) is larger than the case in Fig. 3(f) because the gold material is lossier at shorter wavelengths. The corresponding phases of Poynting vectors in Figs. 3(b)-3(d) are shown in Figs. 3(h)-3(j). It should be noticed that the phase + π and -π are the phase although they show as deep red and deep blue colors in the figures. The plots of Poynting vectors can be used to understand the change of the energy flow. The quick change of phase of Poynting vectors means that the energy flow vibrates in those regions.

 

Fig. 3 (a) Extinction (black line), scattering (blue line), and absorption (red line) spectra of a single dielectric-core gold-shell nanoparticle with r₁/r₂ = 0.5. The energy flow of the coreshell at wavelength of (b) 764 nm, (c) 560 nm, and (d) 542 nm, corresponding to the peak of the extinction cross section, the dip of the scattering cross section, and the other extinction peak. The color bar indicates the amplitudes of the Poynting vectors in logarithmic scale. The corresponding electric field distribution of (b)- (d) are (e)-(g). The color bar indicates the amplitudes of the electric fields in linear scale. The corresponding phases of Poynting vectors of (b)-(d) are (h)-(j).

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Fig. 4 (a) Extinction (black line), scattering (red line), and absorption (blue line) spectra of a single dielectric-core gold-shell nanoparticle with r₁/r₂ = 0.7. The energy flow of the coreshell at wavelength of (b) 658 nm, (c) 650 nm, and (d)586 nm, corresponding to the peak of the absorption cross section, the dip of the scattering cross section and the other absorption peak. The color bar indicates the amplitudes of the Poynting vectors in logarithmic scale. The corresponding electric field distributions of (b)-(d) are (e)-(g). The color bar indicates the amplitudes of the electric fields in linear scale. The corresponding phases of Poynting vectors of (b)-(d) are (h)-(j).

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Fig. 5 (a) Extinction (black line), scattering (red line), and absorption (blue line) spectra of a single dielectric-core gold-shell nanoparticle with r₁/r₂ = 0.9. The energy of the coreshell at wavelength of (b) 850 nm, (c) 930 nm and (d) 998 nm, corresponding to the dip of the scattering cross section, the cross point of the absorption and scattering cross section and the peak of the absorption cross section. The amplitude of the Poynting vector in logarithmic scale. The corresponding electric field distribution of (b)-(d) are (e)-(g). The color bar indicates the amplitudes of the electric fields in linear scale. The corresponding phases of Poynting vectors of (b)-(d) are (h)-(j).

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Figure 4(a) shows that, as the radius ratio r₁/r₂ increases to 0.7 with a fixed outer radius of 140 nm, the dip of scattering cross section is red-shifted to 650 nm (red arrow) as compared to the case for radius ratio r₁/r₂ = 0.5 in Fig. 3(a). In the vicinity of the dip which occurs at scattering cross section equal to 650 nm, there is a peak in the absorption cross section at the wavelength of 658 nm (black arrow). The corresponding energy flows and electric field are shown in Figs. 4(b) and 4(e), respectively. The energy flows which penetrate into the core from the upper side rotate inwardly and form a saddle point at the bottom of the core at x = 0 nm, z = −85 nm, with the energy flows penetrating from the bottom side of the nanoshell in Fig. 4(b). Just like the Rayleigh scattering, the other saddle point can be found under the nanoparticle at x = 0 nm, z = −170 nm. Between these two saddle points, two whirlpools at x = 99 nm, z = −99 nm and x = −99 nm, z = −99 nm appear on the opposite sides of the shell. The existence of the two optical vortices at about x = 72 nm, z = −5 nm and x = −72 nm, z = −5 nm in the core depends on the two singularities at about x = −103 nm, z = −5 nm and x = 103 nm, z = −5 nm in the metal shell. These circulations of energy flow cause the peak of the absorption in Fig. 4(a) and are related to the dip of the scattering cross section which represents the formation transparency phenomenon induced by the cancellation of the opposite polarization vectors and hybrid multi-pole mode [28]. From its electric field distribution in Fig. 4(e), the charge distribution is in the dipole mode at the inner surface and quadrupole mode at the outer surface of gold shell, and the strong electric fields concentrate at the center of core rather than on the surface of the shell. At the dip of the scattering cross section which occurs at the wavelength of 650 nm (red arrow) in Fig. 4(a), a new pair of saddle points at x = 0 nm, z = 77 nm and x = 0 nm, z = 142 nm and a new pair of whirlpools at about x = −47 nm, z = 133 nm and x = 47 nm, z = 133 nm appear at the upper side of the nanoparticle in Fig. 4(c). Its corresponding electric field distribution is shown in Fig. 4(f) which looks similar to Fig. 4(e), but the electric field distribution is more symmetric in Fig. 4(f). The symmetric distribution of the quadrupole mode at the outer surface interferes with the dipole mode at the inner surface, decreasing the scattering light. The dipole mode at the inner surface and the quadrupole mode at the outer surface dominate the charge distribution of the absorption peak in the region of the wavelengths ranging from 628 nm to 670 nm in Fig. 4(a). Comparing to Figs. 4(c) and 4(d), the saddle points and the pair of whirlpools move downward if the wavelength decreases. The two singularities at x = −103 nm, z = −5 nm and x = 103 nm, z = −5 nm in the middle of the shell in Fig. 4(c) also move downward and absorb the whirlpools at the bottom of the shell when the wavelength decreases to 586 nm, as shown in Fig. 4(d), where is the other absorption peak at shorter wavelength in Fig. 4(a). It is found that the charge distributions at wavelength of 586 nm in Fig. 4(g) are the quadrupole modes at the inner and outer shell surfaces, which are different charge distributions for the longer wavelength mode in Fig. 4(e). The optical vortices can be found in the wavelength region of these two absorption peaks in Fig. 4 (a). For this case of r₁/r₂ = 0.7, the optical vortices can’t be found if the wavelength is longer than 670 nm or shorter than 560 nm. The corresponding phases of Poynting vectors in Figs. 4(b)-4(d) are shown in Figs. 4(h)-4(j). From these phase plots, the singular points can be easily determined.

As the radius ratio increases to 0.9 from 0.7 with a fixed outer radius of 140 nm, the peak of the absorption cross section at longer wavelength (black arrow) and the dip of the scattering cross section (red arrow) start to separate in Fig. 5(a). For the case of the dip of the scattering cross section with radius ratio r₁/r₂ = 0.9, which is at wavelength of 850 nm in Fig. 5(a), the saddle points appear at the top and the bottom of the core material for the energy flow shown in Fig. 5(b). This phenomenon is similar to the energy flow pattern for the case of the dip of the scattering cross section with radius ratio r₁/r₂ = 0.7 in Fig. 4(c). The corresponding electric field distribution of Fig. 5(b) is shown in Fig. 5(e), which has a charge distribution different from that shown in Fig. 4(f). By increasing the wavelength from 850 nm, the two optical vortices in the core move from the core material to the interface between the core and the shell. As the wavelength approaches to 930 nm, which is the crossing point of the curves of the absorption and scattering cross sections (the blue arrow in Fig. 5(a)), the two optical vortices are on the interface of the core and the shell as shown in Fig. 5(c). If the wavelength becomes longer, these two optical vortices disappear as shown in Fig. 5(d), which is indicated by the black arrow at the peak of the absorption cross section (i.e., wavelength of 998 nm, in Fig. 5(a)). The corresponding electric field distributions of the cases in Figs. 5(c) and 5(d) are shown in Figs. 5(f) and 5(g) respectively. It can be seen that the latter case has symmetric dipole mode at the inner and outer shell surfaces. This symmetric dipole mode in Fig. 5(g) shows that the electrons at the inner surface of the core shell oscillates symmetrically with the electrons at the outer surface because the shell is very thin in this case. This symmetric dipole mode, which occurs at longer wavelengths thus corresponding to lower energy states, can be explained by the hybridization model [6]. This symmetric dipole mode which induces strong electric fields at the surface of the shell often be used in SERS applications [10].

This peak of absorption in longer wavelength shown in Fig. 5(a) is broader than that of the radius ratio r₁/r₂ = 0.7 in Fig. 4(a). This absorption is resulted from the charge interaction between the outer and the inner surfaces of the shell. It is why the magnitude of the peak of absorption in Fig. 5(a) is larger than that in Fig. 4(a) because the gold shell is thinner in Fig. 5. This can also explain why there is no such absorption peak in Fig. 3(a) because the gold shell is too thick in Fig. 3 to allow the charge interaction between the outer and inner surfaces of the shell. Therefore, the absorption peak in longer wavelengths for r₁/r₂ = 0.9 in Fig. 5 is contributed by the strong interaction between the inner and the outer surface of the shell instead of the circulation of energy flows inside the core material. The optical vortices inside the core exist in the wavelength region around 800 nm to 930 nm which is close to the dip of scattering cross section with the strong absorption cross section. If the wavelength is longer than that at the dip of the scattering cross section at 850 nm, the optical vortices will look like Fig. 4(b) in the middle of the core. If the wavelength is shorter than that at the dip of the scattering cross section at 850 nm, the optical vortices will look like Fig. 4(d) in the bottom of the core. To see the change of the energy flow, the corresponding phases of Poynting vectors in Figs. 5(b)-5(d) are Fig. 5(h)-5(j).

3. Core-shell nanoparticles with different core permittivities

The core shell nanoparticles studied in the preceding section were made from the same material. In this section, the vortices in different dielectric core materials are investigated. Figure 6 shows the extinction, scattering and absorption spectra for the core shell nanoparticles with the relative permittivity of the core, ε_r, changing from 1 to 9 for a fixed inner radius of 98 nm and an outer radius of 140 nm (i.e., the radius ratio r₁/r₂ = 0.7). It can be seen that the dips of the scattering cross section are red shifted when the relative permittivity of the core increases. Two split lines in the absorption cross section spectra appear when the relative permittivity of the core is larger than 2.89. As discussed in Sec. 2, it can be seen that the line of longer wavelengths in the absorption spectra represents the optical vortex similar to Fig. 4(b) and the other line represents the optical vortex similar to Fig. 4(d). It indicates that the optical vortex can’t be found in the core when the relative permittivity of core material is smaller than 2.89.

 

Fig. 6 (a) Extinction, (b) scattering, and (c) absorption cross section spectra of a core-shell nanoparticle for r₂ = 140 nm, r₁/r₂ = 7/10 and the relative permittivity ranging from 1 to 9.

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Two different core relative permittivity values, ε_r = 2.25 and ε_r = 6.25, were selected to vindicate our deduction. In Fig. 7(a), the blue and red lines represent the cross section of ε_r = 2.25 and ε_r = 6.25, respectively. The dashed and solid lines represent the scattering and absorption cross section, respectively. The peak of the absorption of ε_r = 6.25 is at 796 nm, and the corresponding power flows, electric fields and phase of the Poynting vectors are shown in Figs. 7(b)-7(d). The circulation in Fig. 7(b) is similar to Fig. 4(b) in which the relative permittivity of core is 4. The peak of the absorption of ε_r = 2.25 is at 550 nm, and the corresponding power flows, electric fields and phase of the Poynting vectors are shown in Figs. 7(e)-7(g). As shown in Fig. 7(e), the optical vortices are not found in the core material even at the peak of the absorption of ε_r = 2.25, i.e., wavelength of 550 nm.

 

Fig. 7 (a) Scattering (dashed lines), and absorption (sold lines) spectra of dielectric-core gold-shell nanoparticles with relative permittivity ε_r = 2.25 (blue lines) and ε_r = 6.25 (red lines). The energy of the coreshell at wavelength of (b) 796 nm and (e) 550 nm, corresponding to the peak of the absorption cross section of ε_r = 6.25 and ε_r = 2.25, respectively. The corresponding electric field distribution of (b) and (e) are (c) and (f). The corresponding phases of Poynting vectors of (b) and (e) are (d) and (g).

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4. Optical vortex manipulation by a core shell dimer

In order to investigate the variation of optical vortices in a dimer of two core shell nanoparticles, we used the same conditions for the single nanoparticle in Fig. 1 for the pair of core shell nanoparticles. The radius ratio r₁/r₂ varied from 0.5 to 1 and the distance between the two nanoparticles was 70 nm. The extinction, scattering and absorption spectra of a dimer of core shell nanoparticle were calculated and presented in Figs. 8(a)-8(c), respectively. It should be kept in mind that Fig. 2 is the extinction, scattering and absorption spectra of a single core shell nanoparticle. Comparing Fig. 8 to Fig. 2, it reveals that the dipole mode for longer wavelength in the cross section spectra is red-shifted. According to the plasmon hybridization model, the dipole mode red shifts when a dimer instead of the individual nanoshell [7] is used.

 

Fig. 8 (a) Extinction, (b) scattering, and (c) absorption cross section spectra of a pair of dielectric-core gold-shell nanoparticle for radius ratio r₁/r₂ from 0.5 to 1.0 with the outer radius equal to 140 nm and the relative permittivity of the dielectric core equal to 4.0. The distance between two nanoparticles is 70 nm.

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Figure 9(a) shows the comparisons of the scattering and absorption cross sections between a single core shell and a pair of the core shell nanoparticles for the same radius ratio r₁/r₂ = 0.7. The dip of the scattering cross section and the absorption peak of the dimer both slightly blue shift. Except for the redshift of the dipole resonance, the cross section of a single nanoshell and a dimer nanoshell are alike. At the absorption peak of 658 nm, the interaction of these two nanoshell breaks the bifurcation [22] as shown in Fig. 9(b). One of the expected optical vortices inside the core and its relevant singularity in the shell both disappear and the distribution of the energy flow around a nanoshell is no longer symmetric. This is because the induced charges in the two core shell nanoparticles attract each other which results in the electric field vector shown in Fig. 9(c). Therefore these two optical vortices rotate inwardly. A similar phenomenon has been observed in two nanowires [29]. The phase distribution of the Poynting vectors of a dimer nanoshell in Fig. 9(b) is shown in Fig. 9(d). Because the optical energy rotates circularly around the center of the optical vortex, causing the optical vortex to generate a singularity in its center point. The phase of the Poynting vector in the center of the optical vortex inside the core is undefined. To understand the evolution of the pair of optical vortices inside the core, the nanoshell with r₁/r₂ = 0.8 and r₁/r₂ = 0.9 are selected for investigation, and the results are presented in Figs. 9(e) and 9(f). The former frequency is at 736 nm, which is the peak of the absorption cross section, and the latter frequency is at 850 nm as Fig. 5(b). At 850 nm, the pair of the optical vortices inside the core are visible but one of them is oblique. In Fig. 9(e), one of the circulations inside the core shifts to the shell and couples to the lower whirlpool. In Fig. 9(b), one of the circulations inside the core either moves away from this xz-plane or annihilates by coupling other singular points near the surface of the shell. Therefore, a pair of optical vortices with opposite polarization left in the cores of a pair of nanoshell individually.

 

Fig. 9 (a) Scattering (dashed line) and absorption (solid line) cross section of a nanoshell (blue line) and a dimer of nanoshell (red line) with r₂ = 140 nm, (b) the distribution of the energy flows at the wavelength of 658 nm, (c) the corresponding phases of the Poynting vectors, and (d) the distributions of the electric fields. (e) the energy flow of the dimer at the wavelength of 736 nm with r₁/r₂ = 0.8. (f) the energy flow of the dimer at the wavelength of 850 nm with r₁/r₂ = 0.9.

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5. Optical vortex manipulation by the dielectric nanoparticles

The existence of the optical vortex is due to the unbalance distribution of the energy flows. This point has been highlighted in Ref [28]. Thus, the different surrounding media provide us with an additional degree of freedom for manipulating the energy flow in the core. A dielectric nanosphere was placed beside a nanoshell as shown in Fig. 10(a). The dielectric nanosphere has a relative permittivity of 4.0 as the core of the nanoshell. The outer radii are 140 nm for both the nanoshell and the dielectric nanospheres. The radius ratio r₁/r₂ of the nanoshell is 0.7. Comparing to the results in Figs. 10(b) and Fig. 4(b), it is found that the pair of the singularities at the left side disappear and the residual optical vortex inside the core rotates counterclockwise if a dielectric nanosphere occupies the right side of the nanoshell as shown in Fig. 10(b). The distance between two nanowires will affect the topological features inside the cores [28]. Two scenarios are selected in our study. One case has no dielectric sphere as Fig. 10(c) and its corresponding energy flow shown in Fig. 10(d). The other case has a dielectric sphere as Fig. 10(e) and its corresponding energy flow shown in Fig. 10(f). Two optical vortices reside in each core and rotate outwardly in Figs. 10(d) and 10(f). Adding a dielectric sphere in the middle of the dimer bends the power flows and shifts the locations of the singularities close to the centers of the cores.

 

Fig. 10 Schematic drawings of the structures of (a) a nanoshell and a nanosphere, (c) a pair of nanoshell, and (e) a pair of nanoshell a nanosphere. The outer radii of the nanoshell and the dielectric nanosphere are 140 nm, the relative permittivity of the dielectric materials are set as 4.0, and the radius ratio of the nanoshell is 0.7. The corresponding energy flow of (a), (c) and (e) at the wavelength of 658 nm are (b), (d), and (f).

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6. Conclusion

We studied the extinction, scattering, and absorption cross section properties of nanoparticles constructed by the dielectric core with the gold shell cladding in the phase retardation region. With proper selections of radius ratio and dielectric function of the core, we found that specific energy flow patterns inside the core of the nanoshell structure can be related to the cross section spectra. Due to the interference of muti-plasmonic modes of the outer and inner surface, the elimination of the scattering cross section and the increasing of the absorption imply the occurrence of the optical vortices, which only exist beyond the quasistatic limit.

According to the present analysis of the energy flow distribution and the electric field of core-shell nanoparticles, the singularities depend on the modes of the inner and outer surfaces. A pair of optical vortices, each centers in the cores of a pair of nanoshells can form a dimer by breaking the bifurcation of the singularity patterns. The effect of different surrounding media on the optical vortices demonstrated that the optical vortex inside the core can be manipulated. The effects of different geometries and permittivities of the core on the characteristics of optical vortices was also investigated for the nanoparticle design shown in this paper. More complex energy flow pattern can be expected to emerge by increasing the size of the nanoshell or the permittivity of the core.

The investigation paved the way for finding the optical vortex in the nanoparticles with dielectric core and gold shell based on limited parameters. The physical mechanisms described in this study are also useful for finding the optical vortices, which may be used for light harvesting or the development of quantum information technologies.