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Broadband electromagnetic dipole scattering by coupled multiple nanospheres
Superlattices and Microstructures xxx (2017) 1e11
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Broadband electromagnetic dipole scattering by coupled multiple nanospheres Xufeng Jing a, *, Qiufeng Ye a, Zhi Hong b, Dongshuo Zhu c, Guohua Shi d a
Institute of Optoelectronic Technology, China Jiliang University, Hangzhou 310018, China Centre for THz Research, China Jiliang University, Hangzhou 310018, China c Institute of Art and Design, Renmin University of China, China d Jiangsu Key Laboratory of Medical Optics, Suzhou Institute of Biomedical Engineering and Technology, Chinese Academy of Sciences, Suzhou 215163, China b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 12 June 2017 Accepted 20 July 2017 Available online xxx
With the development of nanotechnology, the ability to manipulate light at the nanoscale is critical to future optical functional devices. The use of high refractive index dielectric single silicon nanoparticle can achieve electromagnetic dipole resonant properties. Compared with single nanosphere, the use of dimer and trimer introduces an additional dimension (gap size) for improving the performance of dielectric optical devices through the coupling between closely connected silicon nanospheres. When changing the gap size between the nanospheres, the interaction between the particles can be from weak to strong. Compared with single nanospheres, dimerized or trimeric nanospheres exhibit more pronounced broadband scattering properties. In addition, by introducing more complex interaction, the trimericed silicon nanospheres exhibit a more significant increase in bandwidth than expected. In addition, the presence of the substrate will also contribute to the increase in the bandwidth of the nanospheres. The broadband response in dielectric nanostructures can be effectively applied to broadband applications such as dielectric nanoantennas or solar cells. © 2017 Published by Elsevier Ltd.
1. Introduction With the development of nanotechnology, more and more optical functional devices are miniaturized in preparation and design, so the ability to manipulate light at nanoscale will be critical for future optical functional devices. The ability to precisely control and design the scattering of electromagnetic waves in subwavelength nanostructures is of great significance not only in the emerging fields of nanometer photonics, such as nanoantennas, nanosensors, and photovoltaics, but even in biomedical applications [1e9]. Metallic nanoparticles, which capture, focus and manipulate nanoscale light through their surface plasmon resonances, are considered to be one of the promising solutions [7,9,10]. Thus, different types of nanoantennas such as nanoapertures antenna array [11], nanorod, bowtie [12], Yagi-Uda nanoantenna [13,14] and cross-optical antenna [15] have been studied. However, metallic plasmonic nanoparticles often suffer from intrinsic losses at optical frequencies. This drawback affects their possible practical applications in nanophotonics devices in optical region. Fortunately, the use of high refractive index dielectric nanoparticles to achieve similar resonance properties compared with metal
* Corresponding author. E-mail address: [email protected] (X. Jing). http://dx.doi.org/10.1016/j.spmi.2017.07.048 0749-6036/© 2017 Published by Elsevier Ltd.
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X. Jing et al. / Superlattices and Microstructures xxx (2017) 1e11
particles provides an alternative method [2,16]. Thus, low-loss optical functional devices such as all-dielectric or hybrid metal-dielectric nanoantennas can be realized [17e20]. It is known that the metal nanoparticles just exhibit the electric resonance, yet the dielectric nanoparticles with high refractive index can show simultaneously electric resonance and magnetic resonance in single particle [21e25]. The metallic nanoparticles can realize the magnetic optical resonance performance in complex metallic structure, such as metal-split resonators [26] or multilayer metal [27], by circular currents, the high refractive index dielectric nanoparticles are advantageous in terms of cost reduction and manufacturing simplicity. At present, most studies referring to high refractive index dielectric nanoparticles focused on single nanoparticle or periodic arrays of single particles [28e32]. Although the interaction between multiple high refractive index dielectric nanoparticles was investigated, they pay close attention to electric and magnetic field enhancement [33,34]. Here, we theoretically investigated the scattering behavior of resonances in single nanosphere, dimeric and trimeric silicon nanospheres. The coupled effect between nanosphere is also demonstrated. In addition, the influence of substrate on the scattering properties of the nanosphere structure was studied.
2. Theoretical model When the feature size of the high refractive index dielectric nanoparticles is significantly smaller than the incident wavelength, the resulting optical resonance can be described by the effective electric dipole and magnetic dipole as [35]:
! ! ! ! p ¼ ε0 aE E 0 ; m ¼ cH H 0
(1)
! ! where p and m denote the induced electric and magnetic dipoles, respectively, and aE and cH are the electric and magnetic ! ! polarizabilities, respectively, and E 0 and H 0 are the incident fields. Considering the interaction between multiple nanoparticles, a coupled electric and magnetic dipole methods proposed by Mulholland et al. [35] can be introduced. The electric and magnetic fields at the ith particle caused by the jth particle can be represented by
8 1=2 ! ! ! > ! ! ! m0 . > > n ¼ a a E þ b a $ n d c Hj E n E > E E ji i ij j ij j ji ji ij H < ε 0
1=2 > > ! ! ! ! ! ! ε > > aE ! n ji E j : H i ¼ aij cH H j þ bij cH H j $ n ji n ji þ dij 0
(2)
m0
! where n ji denote the direction vector from the jth particle to ith particle. The shorthand coefficients aij , bij and dij are expressed as:
8 > > > > > > > > > > > > < > > > > > > > > > > > > :
aij ¼ bij ¼
1 eikrij 4p rij
1 eikrij 4p rij
dij ¼
k2
1
rij
k2 þ
1 eikrij 4p rij
þ 2 3
ik rij
!
3ik rij !
!
þ 2
rij
k2 þ
ik rij
(3)
where rij denotes the distance between ith and jth particles and k ¼ 2p=l. Thus, the total induced electric and magnetic dipoles can be derived as [35]:
2 3 X ! 8! ! ! ! ! ! ! aij p j þ bij p j $ n ji n ji dij c n ji m j 5 p ¼ aE 4ε0 E 0 þ > > < i jsi 3 2 > X ! > ! ! ! ! : ! ! ! m i ¼ cH 4 H 0 þ aij m j þ bij m j $ n ji n ji dij c n ji p j 5
(4)
jsi
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where c ¼ 1=ε0 m0 represents the speed of light in vacuum. When aij , bij , dij /0 with rij /∞, the electric and magnetic dipoles can be regarded as equivalent to that of a single nanoparticle. When rij /0, the scattering characteristics changes dramatically. According to Eq. (4), the interactions, the electric-to-electric interaction, the electric-to-magnetic interaction, and the magnetic-to-magnetic interaction can be found. The coupled nanoparticle can achieve versatile functions by these induced interactions. Please cite this article in press as: X. Jing et al., Broadband electromagnetic dipole scattering by coupled multiple nanospheres, Superlattices and Microstructures (2017), http://dx.doi.org/10.1016/j.spmi.2017.07.048
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3. Simulation results and discussion In this work, we study the scattering properties of single, dimeric and trimeric silicon nanospheres, and the influence of the presence of substrates on their scattering properties. Silicon nanospheres as the base unit was studied with a constant radius R ¼ 75 nm, and we chose silica (SiO2) as the substrate material. The optical constants of silicon are taken from the optical handbook [36]. Fig. 1(a) is a schematic representation of the scattering process when light is incident on dimeric silicon nanospheres. The dimeric silicon nanospheres are irradiated by a light source (yellow wave), which propagates from the top to the bottom (-z) and has a polarization along the center line of the dimeric silicon nanospheres (x direction), as indicated by the gray arrows. In the following time-domain finite difference simulations (FDTD) simulation analysis [37], we use commercial software (Lumerical, FDTD, Solutions) for simulation analysis. In the simulation, we use the full field scattered field source (TFSF), which is a special planar wave source. Scattered light (green wave) is collected by a monitor box outside the full field scattered field and in this way, a scattering spectrum of nanostructure is obtained. Fig. 2(b) shows a threedimensional view and a cross-sectional view (x-y plane) of the single, dimeric and trimeric dielectric nanospheres, respectively. The radius of the dielectric nanospheres is always kept at R ¼ 75 nm, while the gap (g) in the dimeric and trimeric nanostructures can vary.
3.1. Scattering characteristics of single silicon nanosphere without substrate First, we investigated the scattering properties of single in a uniform air environment without substrate. Our main concern is the apparent electric resonance or magnetic resonance, as shown in the dashed line in Fig. 2(a). Note that there are two resonant modes: a magnetic resonance mode near l ¼ 609 nm, and an electric resonance mode near l ¼ 502 nm.
Fig. 1. (a) Schematic diagram of scattering process of dimeric nanospheres. The yellow wave and the green wave are incident light and scattered light, respectively. Polarization directions and wave vectors are represented by gray arrows. (b) Three-dimensional and cross-sectional views of single, dimeric and trimeric silicon nanospheres. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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Fig. 2. (a) The scattering cross-section of a single silicon nanosphere, where the electric resonance and the magnetic resonance are represented by a dashed line. (b) Electric field intensity (jEj) distributions and magnetic field (Hy) distributions in the xez plane of the magnetic resonance at l ¼ 609 nm and electric resonance at l ¼ 502 nm.
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In order to understand the magnetic resonance mode and the electric resonance mode more clearly, the distribution of the electric field (jEj) and the magnetic field (Hy) in the x-z plane are given as shown in Fig. 2(b). Fig. 2(b) shows the magnetic resonance mode at l ¼ 609 nm and the field distribution of the electric resonance mode at l ¼ 502 nm. From the electric field distribution (left column), we observed a displacement current loop associated with a circular electric field at l ¼ 609 nm and a linear electric field distribution at l ¼ 502 nm. From the magnetic field distribution (right column), we note that the magnetic field distribution indicates the presence of an antinode at the center at l ¼ 609 nm, which is consistent with a magnetic dipole, while at l ¼ 502 nm, the distribution indicates the presence of a node at the center, consistent with an electric dipole. All these observations show that the resonance modes at l ¼ 609 nm originate from the magnetic resonance, and the resonance modes at l ¼ 502 nm originate from the electric resonance. 3.2. Broadband scattering of dimeric silicon nanospheres In the following section, we study the effect of the interaction of very close nanospheres on the scattering properties by changing the gap size (g) between two silicon nanospheres. Fig. 3 shows the scattering spectra of dimeric silicon nanospheres with the gap size (g) varying from 200 nm to 20 nm. In the case of a large gap of g ¼ 200 nm (green curve), a scattering spectrum similar to that of a single silicon nanosphere is obtained, which corresponds to a weak interaction between the silicon nanospheres. It can be observed that with the reduction of the gap between the dimer nanospheres, the interaction between each other becomes stronger and stronger, and electric and magnetic resonance are closer to each other to form a mixed resonance mode. At the same time, it shows a clear broadband response, which may be used in optical antenna and solar energy applications. In addition, the electric and magnetic field intensity distribution along the longitudinal section and the cross section were studied, respectively. Here, we chose the scattering spectrum of dimeric silicon nanospheres with the g ¼ 20 nm gap (magenta curve) as shown in Fig. 4(a). From the scattering spectra, we pick out two spectral positions marked by two dashed lines: the magnetic resonance mode at l ¼ 600 nm and the electric resonance mode at l ¼ 509 nm. Fig. 4(b)e(e) show the field intensity distributions along the longitudinal and cross-sectional directions, respectively. From Fig. 4(b), we note that the electric field roughly forms a displacement current loop and the magnetic field shows an antinode in each nanosphere (indicated by the black stars) locating close to the dimer center at l ¼ 600 nm. In the cross-sectional view shown in Fig. 4(c), the electric field shows a node in the center, and the magnetic field shows an antinode in the center, showing the known magnetic properties as shown in Fig. 4(b). At l ¼ 509 nm (as shown in Fig. 4(d) and (e)), it can be observed that the electric field is roughly linearly distributed and the magnetic field exhibits a node of electric characteristics, which is more pronounced in the cross sectional view (Fig. 4(e)). It should be noted that the field distribution in the dimeric nanospheres shows asymmetry relative to a single nanosphere. The electric displacement current loop corresponding to the magnetic resonance is not equivalent to the opposite direction along the x and z directions, and the linear electric distribution corresponding to the electric resonance is no longer simply along the x direction. These changes can be attributed to the interaction of different directions introduced in Eq. (4). From the above analysis and scattering diagram, it can be concluded that: As the gap size of the dimeric silicon nanospheres decreases, the interaction between the two spheres becomes stronger, and the magnetic resonance at l ¼ 609 nm and the electric resonance at l ¼ 502 nm in the single nanosphere move closer to each other and form a mixed resonance mode. 3.3. Broadband scattering of trimeric silicon nanospheres Fig. 5(a) shows the scattering spectrum of single nanosphere (upper left, red curve), dimeric nanospheres with g ¼ 20 nm (upper right, magenta curve) and trimeric nanospheres with g ¼ 40 nm (lower left, green curve) and g ¼ 20 nm (lower right, pink curve). As can be seen from the diagram, in the case of trimer nanospheres with g ¼ 40 nm and g ¼ 20 nm, it corresponds to strong interaction, and the interaction in g ¼ 20 nm is stronger than that in g ¼ 40 nm. The well-separated resonant mode due to the strong interaction between the nanospheres leads to a significant decrease in scattering cross section near of l ¼ 600 nm, which may be disadvantageous in broadband applications. Referring to the full width at half maximum (FWHM), we use the bandwidth of the half of the maximum scattering peak to represent the response bandwidth. Fig. 5(b) shows the total bandwidth histogram of single nanosphere (red column), dimer with g ¼ 20 nm (magenta column), and trimer with g ¼ 40 nm (green column) and g ¼ 20 nm (pink column). Compared with single nanosphere, the bandwidth increases by 10% for the dimer with gap g ¼ 20 nm (magenta column) and by 35% in the trimer with gap g ¼ 40 nm (green column). And the bandwidth of the trimer nanospheres with gap g ¼ 20 nm (pink column) can even increase by 55%. This bandwidth-scatter response indicates that solar cells and silicon nanodevices will have great potential for application. 4. Influence of substrate on scattering properties of structures In the actual activity, the nanostructures can not be separated from the substrate and used alone, and the presence of the substrate will have a certain impact on the scattering properties of the nanostructures, so then we simply study the effect of the substrate on the scattering properties of each structure. We also use R ¼ 75 nm silicon nanospheres as the basic unit for Please cite this article in press as: X. Jing et al., Broadband electromagnetic dipole scattering by coupled multiple nanospheres, Superlattices and Microstructures (2017), http://dx.doi.org/10.1016/j.spmi.2017.07.048
Fig. 3. Scattering spectra of dimeric nanospheres with different gap sizes. Two nanospheres are identical (R ¼ 75 nm). The illustrations show the configuration of the corresponding dimeric nanospheres. (a)e(d) The scattering spectra of dimeric nanospheres, g ¼ 200 nm (green curve), g ¼ 100 nm (blue curve), g ¼ 50 nm (cyan curve) and g ¼ 20 nm (magenta curve), respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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Fig. 4. (a) Scattering spectra of dimeric nanospheres with gap size g ¼ 20 nm. In the figure, two spectral positions marked with dashed lines are used for field distribution analysis. (b)e(e) The intensity distributions of electric field (jEj) and magnetic field (jHyj) of the dimer with gap g ¼ 20 nm at l ¼ 600 nm, 509 nm, respectively, along the longitudinal section (xez plane, y ¼ 0 nm) and cross-section (xey plane, z ¼ 0 nm).
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Fig. 5. (a) The scattering spectrum of single nanosphere (upper left, red curve), dimeric nanospheres with g ¼ 20 nm (upper right, magenta curve) and trimeric nanospheres with g ¼ 40 nm (lower left, green curve) and g ¼ 20 nm (lower right, pink curve). The illustrations show the corresponding single or dimeric or trimeric nanospheres configuration. (b) Summarized bandwidth column diagram of single nanosphere (red column), dimer with g ¼ 20 nm (magenta column), and trimer with g ¼ 40 nm (green column) and g ¼ 20 nm (pink column). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 6. Schematic diagram of a single silicon nanosphere, dimeric and trimeric silicon nanospheres on a substrate. The dashed line represents the semi-infinite substrate surface in contact with the nanostructures.
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Fig. 7. (a)e(c) Comparison of the scattering spectra of (a) single nanosphere, (b) dimeric and (c) trimeric nanospheres with different gap sizes. The red curve and green curve indicates the scattering spectrum of each structure without and with substrate. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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comparative analysis, semi-infinite substrates using common silica materials. In the study, we adhere the silicon nanospheres to semi-infinite silica substrate 10 nm, as shown in Fig. 6. Fig. 7 shows the scattering spectra of single nanospheres, dimeric and trimeric nanospheres with or without a substrate. It can be seen from the figure that in the presence of the substrate, the scattering spectra of either single nanosphere, or dimeric and trimeric nanospheres exhibit varying degrees of bandwidth increase. In addition, the resonant peak of the trimer is obviously increased. All of these changes may be due to the fact that the presence of the substrate provides an additional radius for the nanosphere, providing additional delay for the excitation of the resonant mode so that the resonant mode extends into the substrate. 5. Conclusion According to the electric resonance and magnetic resonance in a single silicon nanosphere, we have theoretically proved the broadband scattering properties of dimeric and trimeric silicon nanospheres and the influence of the existence of the substrate on their scattering properties. With the decrease of the gap size between the dimeric silicon nanospheres, the electric resonance and the magnetic resonance of the single nanosphere move closer to each other and form a mixed resonance mode. The use of trimeric silicon nanospheres to introduce a stronger interaction can result in a further increase in the scattering response bandwidth. Referring to the FWHM, we use the bandwidth span of the half maxima scattering peak to characterize the response bandwidth. Compared with single silicon nanospheres, dimeric and trimeric silicon nanospheres exhibit a significant increase in continuous bandwidth. By adjusting the gap between the trimeric silicon nanospheres, the increase in continuous bandwidth can reach more than 50%. And the presence of the substrate will also increase the scattering bandwidth of the nanosphere structure to a certain extent. The theoretical finding may be potentially used in the construction of dielectric nanosensors as well as in broadband applications for solar cells. Acknowledgement The authors acknowledge the support from Natural Science Foundation of Zhejiang Province (LY17F050009), National Natural Science Foundation of China (NSFC) (No.61405182, No.61377108, No.61675226, and No.51401197), the National Instrumentation Program (2016YFF0102000), the Jiangsu Province Science Fund for Distinguished Young Scholars (BK20060010), the Frontier Science research project of the Chinese Academy of Sciences (QYZDB-SSW-JSC03), and the Strategic Priority Research Program of the Chinese Academy of Sciences (XDB02060000). References [1] L. Novotny, B. 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Broadband electromagnetic dipole scattering by coupled multiple nanospheres Xufeng Jing a, *, Qiufeng Ye a, Zhi Hong b, Dongshuo Zhu c, Guohua Shi d a
Institute of Optoelectronic Technology, China Jiliang University, Hangzhou 310018, China Centre for THz Research, China Jiliang University, Hangzhou 310018, China c Institute of Art and Design, Renmin University of China, China d Jiangsu Key Laboratory of Medical Optics, Suzhou Institute of Biomedical Engineering and Technology, Chinese Academy of Sciences, Suzhou 215163, China b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 12 June 2017 Accepted 20 July 2017 Available online xxx
With the development of nanotechnology, the ability to manipulate light at the nanoscale is critical to future optical functional devices. The use of high refractive index dielectric single silicon nanoparticle can achieve electromagnetic dipole resonant properties. Compared with single nanosphere, the use of dimer and trimer introduces an additional dimension (gap size) for improving the performance of dielectric optical devices through the coupling between closely connected silicon nanospheres. When changing the gap size between the nanospheres, the interaction between the particles can be from weak to strong. Compared with single nanospheres, dimerized or trimeric nanospheres exhibit more pronounced broadband scattering properties. In addition, by introducing more complex interaction, the trimericed silicon nanospheres exhibit a more significant increase in bandwidth than expected. In addition, the presence of the substrate will also contribute to the increase in the bandwidth of the nanospheres. The broadband response in dielectric nanostructures can be effectively applied to broadband applications such as dielectric nanoantennas or solar cells. © 2017 Published by Elsevier Ltd.
1. Introduction With the development of nanotechnology, more and more optical functional devices are miniaturized in preparation and design, so the ability to manipulate light at nanoscale will be critical for future optical functional devices. The ability to precisely control and design the scattering of electromagnetic waves in subwavelength nanostructures is of great significance not only in the emerging fields of nanometer photonics, such as nanoantennas, nanosensors, and photovoltaics, but even in biomedical applications [1e9]. Metallic nanoparticles, which capture, focus and manipulate nanoscale light through their surface plasmon resonances, are considered to be one of the promising solutions [7,9,10]. Thus, different types of nanoantennas such as nanoapertures antenna array [11], nanorod, bowtie [12], Yagi-Uda nanoantenna [13,14] and cross-optical antenna [15] have been studied. However, metallic plasmonic nanoparticles often suffer from intrinsic losses at optical frequencies. This drawback affects their possible practical applications in nanophotonics devices in optical region. Fortunately, the use of high refractive index dielectric nanoparticles to achieve similar resonance properties compared with metal
* Corresponding author. E-mail address: [email protected] (X. Jing). http://dx.doi.org/10.1016/j.spmi.2017.07.048 0749-6036/© 2017 Published by Elsevier Ltd.
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particles provides an alternative method [2,16]. Thus, low-loss optical functional devices such as all-dielectric or hybrid metal-dielectric nanoantennas can be realized [17e20]. It is known that the metal nanoparticles just exhibit the electric resonance, yet the dielectric nanoparticles with high refractive index can show simultaneously electric resonance and magnetic resonance in single particle [21e25]. The metallic nanoparticles can realize the magnetic optical resonance performance in complex metallic structure, such as metal-split resonators [26] or multilayer metal [27], by circular currents, the high refractive index dielectric nanoparticles are advantageous in terms of cost reduction and manufacturing simplicity. At present, most studies referring to high refractive index dielectric nanoparticles focused on single nanoparticle or periodic arrays of single particles [28e32]. Although the interaction between multiple high refractive index dielectric nanoparticles was investigated, they pay close attention to electric and magnetic field enhancement [33,34]. Here, we theoretically investigated the scattering behavior of resonances in single nanosphere, dimeric and trimeric silicon nanospheres. The coupled effect between nanosphere is also demonstrated. In addition, the influence of substrate on the scattering properties of the nanosphere structure was studied.
2. Theoretical model When the feature size of the high refractive index dielectric nanoparticles is significantly smaller than the incident wavelength, the resulting optical resonance can be described by the effective electric dipole and magnetic dipole as [35]:
! ! ! ! p ¼ ε0 aE E 0 ; m ¼ cH H 0
(1)
! ! where p and m denote the induced electric and magnetic dipoles, respectively, and aE and cH are the electric and magnetic ! ! polarizabilities, respectively, and E 0 and H 0 are the incident fields. Considering the interaction between multiple nanoparticles, a coupled electric and magnetic dipole methods proposed by Mulholland et al. [35] can be introduced. The electric and magnetic fields at the ith particle caused by the jth particle can be represented by
8 1=2 ! ! ! > ! ! ! m0 . > > n ¼ a a E þ b a $ n d c Hj E n E > E E ji i ij j ij j ji ji ij H < ε 0
1=2 > > ! ! ! ! ! ! ε > > aE ! n ji E j : H i ¼ aij cH H j þ bij cH H j $ n ji n ji þ dij 0
(2)
m0
! where n ji denote the direction vector from the jth particle to ith particle. The shorthand coefficients aij , bij and dij are expressed as:
8 > > > > > > > > > > > > < > > > > > > > > > > > > :
aij ¼ bij ¼
1 eikrij 4p rij
1 eikrij 4p rij
dij ¼
k2
1
rij
k2 þ
1 eikrij 4p rij
þ 2 3
ik rij
!
3ik rij !
!
þ 2
rij
k2 þ
ik rij
(3)
where rij denotes the distance between ith and jth particles and k ¼ 2p=l. Thus, the total induced electric and magnetic dipoles can be derived as [35]:
2 3 X ! 8! ! ! ! ! ! ! aij p j þ bij p j $ n ji n ji dij c n ji m j 5 p ¼ aE 4ε0 E 0 þ > > < i jsi 3 2 > X ! > ! ! ! ! : ! ! ! m i ¼ cH 4 H 0 þ aij m j þ bij m j $ n ji n ji dij c n ji p j 5
(4)
jsi
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where c ¼ 1=ε0 m0 represents the speed of light in vacuum. When aij , bij , dij /0 with rij /∞, the electric and magnetic dipoles can be regarded as equivalent to that of a single nanoparticle. When rij /0, the scattering characteristics changes dramatically. According to Eq. (4), the interactions, the electric-to-electric interaction, the electric-to-magnetic interaction, and the magnetic-to-magnetic interaction can be found. The coupled nanoparticle can achieve versatile functions by these induced interactions. Please cite this article in press as: X. Jing et al., Broadband electromagnetic dipole scattering by coupled multiple nanospheres, Superlattices and Microstructures (2017), http://dx.doi.org/10.1016/j.spmi.2017.07.048
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3. Simulation results and discussion In this work, we study the scattering properties of single, dimeric and trimeric silicon nanospheres, and the influence of the presence of substrates on their scattering properties. Silicon nanospheres as the base unit was studied with a constant radius R ¼ 75 nm, and we chose silica (SiO2) as the substrate material. The optical constants of silicon are taken from the optical handbook [36]. Fig. 1(a) is a schematic representation of the scattering process when light is incident on dimeric silicon nanospheres. The dimeric silicon nanospheres are irradiated by a light source (yellow wave), which propagates from the top to the bottom (-z) and has a polarization along the center line of the dimeric silicon nanospheres (x direction), as indicated by the gray arrows. In the following time-domain finite difference simulations (FDTD) simulation analysis [37], we use commercial software (Lumerical, FDTD, Solutions) for simulation analysis. In the simulation, we use the full field scattered field source (TFSF), which is a special planar wave source. Scattered light (green wave) is collected by a monitor box outside the full field scattered field and in this way, a scattering spectrum of nanostructure is obtained. Fig. 2(b) shows a threedimensional view and a cross-sectional view (x-y plane) of the single, dimeric and trimeric dielectric nanospheres, respectively. The radius of the dielectric nanospheres is always kept at R ¼ 75 nm, while the gap (g) in the dimeric and trimeric nanostructures can vary.
3.1. Scattering characteristics of single silicon nanosphere without substrate First, we investigated the scattering properties of single in a uniform air environment without substrate. Our main concern is the apparent electric resonance or magnetic resonance, as shown in the dashed line in Fig. 2(a). Note that there are two resonant modes: a magnetic resonance mode near l ¼ 609 nm, and an electric resonance mode near l ¼ 502 nm.
Fig. 1. (a) Schematic diagram of scattering process of dimeric nanospheres. The yellow wave and the green wave are incident light and scattered light, respectively. Polarization directions and wave vectors are represented by gray arrows. (b) Three-dimensional and cross-sectional views of single, dimeric and trimeric silicon nanospheres. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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Fig. 2. (a) The scattering cross-section of a single silicon nanosphere, where the electric resonance and the magnetic resonance are represented by a dashed line. (b) Electric field intensity (jEj) distributions and magnetic field (Hy) distributions in the xez plane of the magnetic resonance at l ¼ 609 nm and electric resonance at l ¼ 502 nm.
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In order to understand the magnetic resonance mode and the electric resonance mode more clearly, the distribution of the electric field (jEj) and the magnetic field (Hy) in the x-z plane are given as shown in Fig. 2(b). Fig. 2(b) shows the magnetic resonance mode at l ¼ 609 nm and the field distribution of the electric resonance mode at l ¼ 502 nm. From the electric field distribution (left column), we observed a displacement current loop associated with a circular electric field at l ¼ 609 nm and a linear electric field distribution at l ¼ 502 nm. From the magnetic field distribution (right column), we note that the magnetic field distribution indicates the presence of an antinode at the center at l ¼ 609 nm, which is consistent with a magnetic dipole, while at l ¼ 502 nm, the distribution indicates the presence of a node at the center, consistent with an electric dipole. All these observations show that the resonance modes at l ¼ 609 nm originate from the magnetic resonance, and the resonance modes at l ¼ 502 nm originate from the electric resonance. 3.2. Broadband scattering of dimeric silicon nanospheres In the following section, we study the effect of the interaction of very close nanospheres on the scattering properties by changing the gap size (g) between two silicon nanospheres. Fig. 3 shows the scattering spectra of dimeric silicon nanospheres with the gap size (g) varying from 200 nm to 20 nm. In the case of a large gap of g ¼ 200 nm (green curve), a scattering spectrum similar to that of a single silicon nanosphere is obtained, which corresponds to a weak interaction between the silicon nanospheres. It can be observed that with the reduction of the gap between the dimer nanospheres, the interaction between each other becomes stronger and stronger, and electric and magnetic resonance are closer to each other to form a mixed resonance mode. At the same time, it shows a clear broadband response, which may be used in optical antenna and solar energy applications. In addition, the electric and magnetic field intensity distribution along the longitudinal section and the cross section were studied, respectively. Here, we chose the scattering spectrum of dimeric silicon nanospheres with the g ¼ 20 nm gap (magenta curve) as shown in Fig. 4(a). From the scattering spectra, we pick out two spectral positions marked by two dashed lines: the magnetic resonance mode at l ¼ 600 nm and the electric resonance mode at l ¼ 509 nm. Fig. 4(b)e(e) show the field intensity distributions along the longitudinal and cross-sectional directions, respectively. From Fig. 4(b), we note that the electric field roughly forms a displacement current loop and the magnetic field shows an antinode in each nanosphere (indicated by the black stars) locating close to the dimer center at l ¼ 600 nm. In the cross-sectional view shown in Fig. 4(c), the electric field shows a node in the center, and the magnetic field shows an antinode in the center, showing the known magnetic properties as shown in Fig. 4(b). At l ¼ 509 nm (as shown in Fig. 4(d) and (e)), it can be observed that the electric field is roughly linearly distributed and the magnetic field exhibits a node of electric characteristics, which is more pronounced in the cross sectional view (Fig. 4(e)). It should be noted that the field distribution in the dimeric nanospheres shows asymmetry relative to a single nanosphere. The electric displacement current loop corresponding to the magnetic resonance is not equivalent to the opposite direction along the x and z directions, and the linear electric distribution corresponding to the electric resonance is no longer simply along the x direction. These changes can be attributed to the interaction of different directions introduced in Eq. (4). From the above analysis and scattering diagram, it can be concluded that: As the gap size of the dimeric silicon nanospheres decreases, the interaction between the two spheres becomes stronger, and the magnetic resonance at l ¼ 609 nm and the electric resonance at l ¼ 502 nm in the single nanosphere move closer to each other and form a mixed resonance mode. 3.3. Broadband scattering of trimeric silicon nanospheres Fig. 5(a) shows the scattering spectrum of single nanosphere (upper left, red curve), dimeric nanospheres with g ¼ 20 nm (upper right, magenta curve) and trimeric nanospheres with g ¼ 40 nm (lower left, green curve) and g ¼ 20 nm (lower right, pink curve). As can be seen from the diagram, in the case of trimer nanospheres with g ¼ 40 nm and g ¼ 20 nm, it corresponds to strong interaction, and the interaction in g ¼ 20 nm is stronger than that in g ¼ 40 nm. The well-separated resonant mode due to the strong interaction between the nanospheres leads to a significant decrease in scattering cross section near of l ¼ 600 nm, which may be disadvantageous in broadband applications. Referring to the full width at half maximum (FWHM), we use the bandwidth of the half of the maximum scattering peak to represent the response bandwidth. Fig. 5(b) shows the total bandwidth histogram of single nanosphere (red column), dimer with g ¼ 20 nm (magenta column), and trimer with g ¼ 40 nm (green column) and g ¼ 20 nm (pink column). Compared with single nanosphere, the bandwidth increases by 10% for the dimer with gap g ¼ 20 nm (magenta column) and by 35% in the trimer with gap g ¼ 40 nm (green column). And the bandwidth of the trimer nanospheres with gap g ¼ 20 nm (pink column) can even increase by 55%. This bandwidth-scatter response indicates that solar cells and silicon nanodevices will have great potential for application. 4. Influence of substrate on scattering properties of structures In the actual activity, the nanostructures can not be separated from the substrate and used alone, and the presence of the substrate will have a certain impact on the scattering properties of the nanostructures, so then we simply study the effect of the substrate on the scattering properties of each structure. We also use R ¼ 75 nm silicon nanospheres as the basic unit for Please cite this article in press as: X. Jing et al., Broadband electromagnetic dipole scattering by coupled multiple nanospheres, Superlattices and Microstructures (2017), http://dx.doi.org/10.1016/j.spmi.2017.07.048
Fig. 3. Scattering spectra of dimeric nanospheres with different gap sizes. Two nanospheres are identical (R ¼ 75 nm). The illustrations show the configuration of the corresponding dimeric nanospheres. (a)e(d) The scattering spectra of dimeric nanospheres, g ¼ 200 nm (green curve), g ¼ 100 nm (blue curve), g ¼ 50 nm (cyan curve) and g ¼ 20 nm (magenta curve), respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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Fig. 4. (a) Scattering spectra of dimeric nanospheres with gap size g ¼ 20 nm. In the figure, two spectral positions marked with dashed lines are used for field distribution analysis. (b)e(e) The intensity distributions of electric field (jEj) and magnetic field (jHyj) of the dimer with gap g ¼ 20 nm at l ¼ 600 nm, 509 nm, respectively, along the longitudinal section (xez plane, y ¼ 0 nm) and cross-section (xey plane, z ¼ 0 nm).
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Fig. 5. (a) The scattering spectrum of single nanosphere (upper left, red curve), dimeric nanospheres with g ¼ 20 nm (upper right, magenta curve) and trimeric nanospheres with g ¼ 40 nm (lower left, green curve) and g ¼ 20 nm (lower right, pink curve). The illustrations show the corresponding single or dimeric or trimeric nanospheres configuration. (b) Summarized bandwidth column diagram of single nanosphere (red column), dimer with g ¼ 20 nm (magenta column), and trimer with g ¼ 40 nm (green column) and g ¼ 20 nm (pink column). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 6. Schematic diagram of a single silicon nanosphere, dimeric and trimeric silicon nanospheres on a substrate. The dashed line represents the semi-infinite substrate surface in contact with the nanostructures.
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Fig. 7. (a)e(c) Comparison of the scattering spectra of (a) single nanosphere, (b) dimeric and (c) trimeric nanospheres with different gap sizes. The red curve and green curve indicates the scattering spectrum of each structure without and with substrate. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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comparative analysis, semi-infinite substrates using common silica materials. In the study, we adhere the silicon nanospheres to semi-infinite silica substrate 10 nm, as shown in Fig. 6. Fig. 7 shows the scattering spectra of single nanospheres, dimeric and trimeric nanospheres with or without a substrate. It can be seen from the figure that in the presence of the substrate, the scattering spectra of either single nanosphere, or dimeric and trimeric nanospheres exhibit varying degrees of bandwidth increase. In addition, the resonant peak of the trimer is obviously increased. All of these changes may be due to the fact that the presence of the substrate provides an additional radius for the nanosphere, providing additional delay for the excitation of the resonant mode so that the resonant mode extends into the substrate. 5. Conclusion According to the electric resonance and magnetic resonance in a single silicon nanosphere, we have theoretically proved the broadband scattering properties of dimeric and trimeric silicon nanospheres and the influence of the existence of the substrate on their scattering properties. With the decrease of the gap size between the dimeric silicon nanospheres, the electric resonance and the magnetic resonance of the single nanosphere move closer to each other and form a mixed resonance mode. The use of trimeric silicon nanospheres to introduce a stronger interaction can result in a further increase in the scattering response bandwidth. Referring to the FWHM, we use the bandwidth span of the half maxima scattering peak to characterize the response bandwidth. Compared with single silicon nanospheres, dimeric and trimeric silicon nanospheres exhibit a significant increase in continuous bandwidth. 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