Nanoparticle Oligomers
Observation of Fano Resonances in All-Dielectric Nanoparticle Oligomers Katie E. Chong, Ben Hopkins, Isabelle Staude,* Andrey E. Miroshnichenko, Jason Dominguez, Manuel Decker, Dragomir N. Neshev, Igal Brener, and Yuri S. Kivshar
It is well-known that oligomers made of metallic nanoparticles are able to support sharp Fano resonances originating from the interference of two plasmonic resonant modes with different spectral width. While such plasmonic oligomers suffer from high dissipative losses, a new route for achieving Fano resonances in nanoparticle oligomers has opened up after the recent experimental observations of electric and magnetic resonances in low-loss dielectric nanoparticles. Here, light scattering by all-dielectric oligomers composed of silicon nanoparticles is studied experimentally for the first time. Pronounced Fano resonances are observed for a variety of lithographically-fabricated heptamer nanostructures consisting of a central particle of varying size, encircled by six nanoparticles of constant size. Based on a full collective mode analysis, the origin of the observed Fano resonances is revealed as a result of interference of the optically-induced magnetic dipole mode of the central particle with the collective mode of the nanoparticle structure. This allows for effective tuning of the Fano resonance to a desired spectral position by a controlled size variation of the central particle. Such optically-induced magnetic Fano resonances in all-dielectric oligomers offer new opportunities for sensing and nonlinear applications.
1. Introduction Inspired by their analogy to complex chemical molecules,[1] metallic nanoparticle clusters supporting collective plasmon resonances have become an active topic of current research.[1,2]
K. E. Chong, B. Hopkins, Dr. I Staude, Dr. A. E. Miroshnichenko, Dr. M. Decker, Dr. D. N. Neshev, Prof. Y. S. Kivshar Nonlinear Physics Centre Research School of Physics and Engineering The Australian National University Canberra, ACT 0200, Australia E-mail: [email protected] J. Dominguez, Dr. I. Brener Center for Integrated Nanotechnologies Sandia National Laboratories Albuquerque, New Mexico 87185, USA DOI: 10.1002/smll.201303612 small 2014, DOI: 10.1002/smll.201303612
The optical behavior of such systems is governed by the formation of delocalized collective modes originating from the coupling between localized surface plasmon resonances of the constituent elements. The study of these collective plasmon modes opens up new opportunities for investigating interactions between plasmonic nanostructures as well as for tailoring the properties of their resonances toward specific applications. Of particular interest are Fano resonances, which usually arise from the interference of a narrow resonance with a broader background.[3,4] Fano resonances have attracted a flurry of research interest due to their asymmetric lineshape, sharp spectral features, and sensitivity to structural and environmental parameters,[3–6] characteristics which make them ideal candidates for a variety of nanophotonics applications such as highly-sensitive all-optical sensors.[7–9] Fano resonances in the visible and near-infrared spectral range have been successfully observed experimentally for a multitude of plasmonic nanostructures and metamaterial resonators,[4,10] including many different types of metallic
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nanoparticle oligomers.[11–17] In these plasmonic oligomers, Fano resonances are highly sensitive to changes of the dielectric environment in the gaps between the metal nanoparticles, the size of these gaps and also to the symmetry of the oligomer, thereby enabling very efficient refractive index, structural, and symmetry tuning.[18–21] However, up to now, experimental studies on optical Fano resonances in oligomer nanoclusters have been exclusively focused on plasmonic systems. Recent experimental observations of strong resonant response from subwavelength all-dielectric particles from the visible to the mid-infrared spectral range[22–27] now open up a completely new route for creating Fano resonances in nanoparticle clusters, employing the Mie-type resonances of low-loss all-dielectric nanoparticles instead of the plasmonic resonances supported by metallic nanostructures. Generally, a variety of subwavelength nanophotonic structures fabricated from high-refractive-index dielectrics are expected to support Fano resonances similar to their nanoplasmonic counterparts. For example, an initial experimental observation has been reported recently for all-dielectric metamaterials.[28] Fano resonances in all-dielectric oligomers, however, have not been demonstrated experimentally so far. They are of particular interest, as high-refractive-index all-dielectric nanoparticles like spheres, disks or cubes, which are highly symmetric, do not only support electric, but also strong magnetic resonances. This is in contrast to plasmonic particles, where a non-negligible magnetic response can only occur for more sophisticated geometries, like split-ring resonators,[29] that support a ring current. As such, in comparison to plasmonic oligomers, the possibility of exciting both electric and magnetic resonances in the individual constituent elements of the all-dielectric oligomer structure opens up a whole new dimension for mutual inter-element coupling and, consequently, for the formation of collective modes. Indeed, it was recently predicted in theory that light scattering by silicon oligomers should exhibit well-pronounced Fano resonances originating from the predominant excitation of opticallyinduced magnetic dipole modes in the individual elements of the oligomer structure.[30] The conventional understanding of the Fano resonance is that it arises from the interference of broad and narrow spectral lines, which then results in an asymmetric scattering profile. It turns out that this concept can be further extended to the situation of interference between several modes of comparable spectral width.[31] Importantly, the asymmetric lineshape of the Fano resonance is still clearly observed since the destructive and constructive interference take place in a narrow spectral range exhibiting strongly resonant response. Furthermore, it is worth noting that the fields of the modes are mainly concentrated inside the dielectric particles, and not mostly at their surfaces as for plasmonic particles. As an important consequence, this novel type of Fano resonances is expected to be less sensitive to unintentional structural variations like surface roughness, which makes it very promising for a range of future applications in nanophotonics. Moreover, owing to their higher resistance to heat as compared to optical metal nanostructures and to the possibility of free carrier generation in the material, Fano resonances of silicon oligomers might offer
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a range of new opportunities not attainable with plasmonic nanoclusters, for example harsh-environment sensing or nonlinear applications. In this paper, for the first time to our knowledge, we experimentally verify the concept of magnetic Fano resonances in all-dielectric oligomers. We observe the characteristic suppression of the extinction associated with the Fano resonances in the measured linear-optical spectra of fabricated oligomer nanostructures. In support of our experimental results, we further discuss the underlying physics of the observed Fano resonances by employing a novel theoretical approach[31] based on collective-mode decomposition of the localized field excitation of the nanoparticle structures. This analysis confirms that the observed Fano resonances originate from the predominant excitation of opticallyinduced magnetic dipole modes, and furthermore provides a useful insight into their tuning behavior.
2. Results For the experiment we create a variety of different oligomer structures consisting of silicon nanodisks on a silicon oxide substrate. Details of the fabrication process can be found in the Experimental Section. We focus on heptamer structures, namely oligomers composed of seven individual nanoparticles. A schematic of the fabricated heptamer structures and scanning electron micrographs (SEM) of a typical sample are shown in Figure 1(a) and Figure 1(b), respectively, with their dimensions specified. Each heptamer consists of a hexamer (six particles arranged in a ring-like fashion) and a central nanodisk. Top view SEM images of each type of the fabricated nanoclusters are displayed in Figure 2(a). All heptamer structures are arranged in two-dimensional arrays with a lattice constant of 2840 nm [see Figure 1(b)]. This value has been chosen in order to ensure that the separation between neighboring structures is large enough for dominant coupling to remain within each oligomer structure, while the in-plane density of oligomer structures is still sufficiently high to create pronounced resonances in the far-field spectra. Linear-optical transmittance spectra of four silicon nanodisk heptamer arrays with varying central nanodisk diameter d2 are measured with unpolarized incident light (see Experimental Section for details on the measurement process) and the results are shown in Figure 2(b). For a subsequent comparison with theoretical data, the experimental transmission measurements are presented in terms of extinction crosssection σext using its relation to transmittance T given by the first-order approximation to the Beer-Lambert Law:[32]
σ ext ∝ − ln(T ) ≈ (1 − T ).
(1)
In order to gain a theoretical understanding of the optical response of these heptamer systems, we compare the experimental extinction spectra to those of individual heptamers in free space calculated using the discrete dipole approach. For plasmonic heptamer systems, such an analysis is typically based on observing the electric dipole moments or charge distributions of each constituent particle.[12,19,33–35]
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small 2014, DOI: 10.1002/smll.201303612
Observation of Fano Resonances in All-Dielectric Nanoparticle Oligomers
Figure 2. (a) Scanning electron micrographs of typical fabricated heptamer oligomers. Diameters d2 of the central nanodisk are 320, 350, 380 and 400 nm. (b,c) Extinction spectra for a variety of silicon nanodisk heptamer structures featuring a systematic variation of the central nanodisk diameter d2. Shown are (b) experimentally measured results for arrayed structures with unpolarized incident light, and (c) theoretically calculated results for single heptamers for arbitrarilypolarized light obtained using the coupled-dipole approximation. A Fano resonance is created in the heptamers (see gray-shaded region and colored arrows), and it moves across the spectrum as the diameter of the central particle is varied.
Figure 1. (a) Sketch of the geometry of the heptamer structures composed of silicon nanodisks fabricated on a silicon oxide substrate. The nanodisks of the surrounding hexamer have a constant diameter d1 = 460 nm, while the diameter d2 of the central nanodisk is systematically varied through 320, 350, 380 and 400 nm. All nanodisks have a height h of 260 nm and the radius r of the hexamer ring is 568 nm. (b) Scanning electron micrograph of a typical fabricated sample. The insets show magnified (top) and oblique (bottom) views of a single heptamer structure.
Here E i0 ( H i0 ) , αˆ iE (αˆ iH ) and pi (mi) are the electric (magnetic) applied field, tensor dipole polarizability and dipole moment, respectively, for the ith particle. The coupling between particles is described using dyadic Greens functions, Gˆ ij , between the locations of the ith and jth particles and k, 0 and µ0 are the free space wavenumber, permittivity and permeability respectively. In such a dipole system the extinction cross section can then be written analytically as
σ ext = However, since high-refractive-index all-dielectric particles do not only support electric but also magnetic modes,[22–27,36] this common approach is not applicable in our case. Instead, in order to adequately describe the optical properties of the realized silicon oligomers, both electric and magnetic dipole moments have to be considered.[30,37,38] Therefore, to calculate the extinction spectra of the silicon nanodisk heptamers, we employ the more general coupled electric and magnetic dipole approach[39] where each nanodisk is represented as an electric and a magnetic dipole. In this approach the electric and magnetic dipole responses of all particles are related to the externally-applied electric and magnetic field through ⎛ ⎞ μ0 p i = αˆ iE ∈0 E i0 + αˆ iE ∈0 k 2 ⎜ ∑ i ≠ j Gˆ ij p j − ∇ × Gˆ ij m j ⎟ ∈0 ⎝ ⎠
(2)
⎛ ⎞ 1 m i = αˆ iH H i0 + αˆ iH k 2 ⎜ ∑ i ≠ j Gˆ ij m j + ∇ × Gˆ ij p j ⎟ . ∈0 μ 0 ⎝ ⎠
(3)
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k Im ∈0 | E 0 | 2
{∑ E i
0† i pi
}
+ H i0 † mi .
(4)
The results of this approach applied to our heptamer system are summarized in Figure 2(c).
3. Discussion Experimental and theoretical extinction spectra show a good qualitative agreement. In both sets of results we observe two distinct extinction peaks for each spectrum. The peak at longer wavelengths is associated with the magnetic resonance of the collective structure. The second peak, which is observed at shorter wavelengths, corresponds to the magnetic resonance of the central particle. In accordance with earlier work on individual silicon nanodisks,[26] this peak experiences a red-shift as the central particle diameter is increased. The Fano resonance occurs as a region of resonantly suppressed extinction in-between
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of a silicon oxide layer and a silicon handle wafer in the experiment.
3.1. Coupled-Mode Analysis
Figure 3. (a) Simulated total extinction spectrum of the silicon disk heptamer with a 400-nm-diameter central particle. The contributions of electric and magnetic eigenmodes to the extinction cross-section can be completely separated and are shown as well. (b) Decomposition of the heptamer’s magnetic response into three eigenmodes and their raw sum, the latter not exhibiting Fano-like characteristics. (c) Distribution of the magnetic dipoles in each eigenmode for horizontally-polarized light. It can be seen that an asymmetric lineshape Fano resonance is induced in the extinction spectra around the spectral position where the resonance curves for the eigenmodes i. and ii. intersect.
the two peaks. This region is highlighted by the grayshaded area and the colored arrows in Figure 2. When the diameter of the central particle is increased and its associated resonance peak shifts closer towards the resonance peak of the collective structure, the Fano signature is spectrally shifted correspondingly. Interestingly, the interaction between the two resonances leads to a small, but clearly noticeable, variation in the height of the collective resonance peak. By increasing the central disk diameter d2 and shifting the peaks closer together, the overlap of the two peaks is increased and therefore the observed reduction (rather than growth) in height of the collective peak is indicative of destructive interference between these two resonances. This is observed both in theoretical and experimental spectra. Due to rotational symmetry, the extinction and other cross-sections of the heptamer nanostructures are independent of polarization.[38,40] It is worth noting that sharp asymmetric Fano resonances can be observed for unpolarized incident light, which is experimentally demonstrated in this paper. It also demonstrates the importance of the rotational symmetry of our system which provides the freedom of polarization choices for theoretical analysis. Differences in the exact resonance positions in numerical calculations compared to experimental spectra are primarily due to approximating the nanodisks as pairs of electric and magnetic dipoles in the theoretical model. Further deviations will arise from sample imperfections like roughness and not perfectly straight sidewalls of the silicon nanodisks, in addition to the presence of a substrate consisting
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Based on the qualitative agreement of the experimental results with the theoretical coupled-dipole approximation, in the following we analyze the observed extinction spectra in terms of the eigenmodes[41] of the heptamer structure.[31] In general, given that each individual dielectric nanoparticle supports both electric and magnetic dipole-like resonances, one would naively expect the collective eigenmodes of the heptamer to arise from hybridization of the entire set of supported modes. However, for planar nonchiral structures the contributions of electric and magnetic eigenmodes to the extinction cross-section can be completely separated.[30,42] Since the oligomer structures exhibit rotational symmetry, their extinction cross-section is polarization independent as mentioned previously. Thus, we can fix the polarization of the incident light to be horizontal without loss of generality. Figure 3(a) illustrates this for the example of a heptamer structure with a central particle diameter d2 = 400 nm. Shown are the contributions of the electric and magnetic eigenmodes of the heptamer to the extinction spectrum, as well as the total calculated extinction spectrum of the heptamer. Clearly, destructive Fano interference is only observed for the magnetic eigenmode contribution.[30,31] In order to further discuss the origin of the observed Fano resonance, it is therefore sufficient to limit our analysis to the magnetic eigenmodes of the heptamer only. Figure 3(b) shows a decomposition of the heptamer’s magnetic response into the three eigenmodes. For comparison, Figure 3(b) also includes the raw sum i+ii+iii of the mode amplitudes, emulating the artificial case of absent mode interaction and showing that a qualitatively different lineshape results in this situation. By comparing Figure 3(a) and (b), one can clearly see that the Fano lineshape in extinction seen in Figure 3(a) occurs at the intersection of the resonance curves of the two magnetic eigenmodes denoted by i and ii in Figure 3(b), indicating that the Fano lineshape arises from the interference of these two resonant eigenmodes. In Figure 3(c) we show the dipole distributions of these eigenmodes for a horizontally-polarized incident field ( E 0 polarized in the x-direction) to provide an intuitive explanation of the observed changes of the heptamer optical response upon variation of the central nanodisk diameter d2. The resonant mode denoted by i. is heavily localized at the central particle, making it very sensitive to a variation of d2. The mode denoted by ii., on the other hand, extends over the entire heptamer structure with only a small fraction of the mode concentrated at the central particle. Mode ii. is therefore not significantly influenced by a variation of d2. This is further illustrated in Figure 4, which shows the heptamers’ decomposed magnetic responses for the four different diameters realized in the experiment. Clearly, only the resonance peak of mode i. experiences a significant shift with variation of the diameter, altogether
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small 2014, DOI: 10.1002/smll.201303612
Observation of Fano Resonances in All-Dielectric Nanoparticle Oligomers
spectra are in good qualitative agreement with theoretical results obtained by using the coupled electric and magnetic dipole approximation. By analyzing the major collective modes of the oligomer structures associated with its dominant resonances, our numerical calculations further unveil the main mechanism of wave interference realized in the heptamer structures. This confirms that the observed Fano resonances indeed originate from interference between the optically-induced magnetic resonances of the central particles and those of the collective structures.
5. Experimental Section Fabrication: The heptamer samples are fabricated via electronbeam lithography (EBL) on backside polished silicon-on-insulator wafers (SOITEC, 260 nm top silicon thickness, 2 µm buried oxide thickness) using the negative-tone electron-beam resist NEB-31A. The EBL process is followed by a directive reactive-ion etching process using the obtained electron-beam resist pattern as an etch mask. As the final step, the etched sample is placed into oxygen plasma and piranha solution to remove any electron-beam resist mask residue. Measurement: Linear-optical transmittance spectra of the silicon nanodisk heptamer arrays are measured for unpolarized incident light using a custom-built white light spectroscopy setup with an optical spectrum analyzer. The sample is placed between a pair of 20× Mitutoyo Plan Apo NIR infinity-corrected objectives with numerical aperture NA = 0.4 which focus the incident light onto the sample and collect the light transmitted through it. Transmittance through the sample is referenced to the transmittance through the unstructured wafer. The range of incident angles is reduced to ±3° by an aperture to achieve near-normal incidence.
Figure 4. Decomposition of the heptamer’s magnetic response into three eigenmodes (i. red, ii. cyan, and iii. green) for a systematic variation of the central nanodisk diameter d2. Curves corresponding to different diameters are vertically displaced by 10 units for clarity. It can be seen that only the resonant peak of mode i. experiences a noticeable shift with the diameter of the central nanodisk, resulting in a deformation of the Fano resonance profile of the heptamer oligomer [see Figure 2].
resulting in the observed easily traceable tuning characteristics of the Fano resonance.
4. Conclusion We have experimentally demonstrated that magnetic Fano resonances can be observed in high-refractive-index alldielectric nanoparticle oligomers. We have fabricated and optically characterized arrays of heptamer structures composed of silicon nanodisks on a silicon oxide substrate. Their measured linear-optical spectra reveal a clear manifestation of Fano resonances. Furthermore, we have observed that the Fano resonance can be spectrally tuned by a systematic size variation of the central nanoparticle. Our experimental small 2014, DOI: 10.1002/smll.201303612
Acknowledgements This work was performed, in part, at the Center for Integrated Nanotechnologies, an Office of Science User Facility operated for the U.S. Department of Energy (DOE) Office of Science. Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-AC04– 94AL85000. The authors also acknowledge a support from the Australian Research Council.
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Received: November 21, 2013 Revised: January 12, 2014 Published online:
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Observation of Fano Resonances in All-Dielectric Nanoparticle Oligomers Katie E. Chong, Ben Hopkins, Isabelle Staude,* Andrey E. Miroshnichenko, Jason Dominguez, Manuel Decker, Dragomir N. Neshev, Igal Brener, and Yuri S. Kivshar
It is well-known that oligomers made of metallic nanoparticles are able to support sharp Fano resonances originating from the interference of two plasmonic resonant modes with different spectral width. While such plasmonic oligomers suffer from high dissipative losses, a new route for achieving Fano resonances in nanoparticle oligomers has opened up after the recent experimental observations of electric and magnetic resonances in low-loss dielectric nanoparticles. Here, light scattering by all-dielectric oligomers composed of silicon nanoparticles is studied experimentally for the first time. Pronounced Fano resonances are observed for a variety of lithographically-fabricated heptamer nanostructures consisting of a central particle of varying size, encircled by six nanoparticles of constant size. Based on a full collective mode analysis, the origin of the observed Fano resonances is revealed as a result of interference of the optically-induced magnetic dipole mode of the central particle with the collective mode of the nanoparticle structure. This allows for effective tuning of the Fano resonance to a desired spectral position by a controlled size variation of the central particle. Such optically-induced magnetic Fano resonances in all-dielectric oligomers offer new opportunities for sensing and nonlinear applications.
1. Introduction Inspired by their analogy to complex chemical molecules,[1] metallic nanoparticle clusters supporting collective plasmon resonances have become an active topic of current research.[1,2]
K. E. Chong, B. Hopkins, Dr. I Staude, Dr. A. E. Miroshnichenko, Dr. M. Decker, Dr. D. N. Neshev, Prof. Y. S. Kivshar Nonlinear Physics Centre Research School of Physics and Engineering The Australian National University Canberra, ACT 0200, Australia E-mail: [email protected] J. Dominguez, Dr. I. Brener Center for Integrated Nanotechnologies Sandia National Laboratories Albuquerque, New Mexico 87185, USA DOI: 10.1002/smll.201303612 small 2014, DOI: 10.1002/smll.201303612
The optical behavior of such systems is governed by the formation of delocalized collective modes originating from the coupling between localized surface plasmon resonances of the constituent elements. The study of these collective plasmon modes opens up new opportunities for investigating interactions between plasmonic nanostructures as well as for tailoring the properties of their resonances toward specific applications. Of particular interest are Fano resonances, which usually arise from the interference of a narrow resonance with a broader background.[3,4] Fano resonances have attracted a flurry of research interest due to their asymmetric lineshape, sharp spectral features, and sensitivity to structural and environmental parameters,[3–6] characteristics which make them ideal candidates for a variety of nanophotonics applications such as highly-sensitive all-optical sensors.[7–9] Fano resonances in the visible and near-infrared spectral range have been successfully observed experimentally for a multitude of plasmonic nanostructures and metamaterial resonators,[4,10] including many different types of metallic
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nanoparticle oligomers.[11–17] In these plasmonic oligomers, Fano resonances are highly sensitive to changes of the dielectric environment in the gaps between the metal nanoparticles, the size of these gaps and also to the symmetry of the oligomer, thereby enabling very efficient refractive index, structural, and symmetry tuning.[18–21] However, up to now, experimental studies on optical Fano resonances in oligomer nanoclusters have been exclusively focused on plasmonic systems. Recent experimental observations of strong resonant response from subwavelength all-dielectric particles from the visible to the mid-infrared spectral range[22–27] now open up a completely new route for creating Fano resonances in nanoparticle clusters, employing the Mie-type resonances of low-loss all-dielectric nanoparticles instead of the plasmonic resonances supported by metallic nanostructures. Generally, a variety of subwavelength nanophotonic structures fabricated from high-refractive-index dielectrics are expected to support Fano resonances similar to their nanoplasmonic counterparts. For example, an initial experimental observation has been reported recently for all-dielectric metamaterials.[28] Fano resonances in all-dielectric oligomers, however, have not been demonstrated experimentally so far. They are of particular interest, as high-refractive-index all-dielectric nanoparticles like spheres, disks or cubes, which are highly symmetric, do not only support electric, but also strong magnetic resonances. This is in contrast to plasmonic particles, where a non-negligible magnetic response can only occur for more sophisticated geometries, like split-ring resonators,[29] that support a ring current. As such, in comparison to plasmonic oligomers, the possibility of exciting both electric and magnetic resonances in the individual constituent elements of the all-dielectric oligomer structure opens up a whole new dimension for mutual inter-element coupling and, consequently, for the formation of collective modes. Indeed, it was recently predicted in theory that light scattering by silicon oligomers should exhibit well-pronounced Fano resonances originating from the predominant excitation of opticallyinduced magnetic dipole modes in the individual elements of the oligomer structure.[30] The conventional understanding of the Fano resonance is that it arises from the interference of broad and narrow spectral lines, which then results in an asymmetric scattering profile. It turns out that this concept can be further extended to the situation of interference between several modes of comparable spectral width.[31] Importantly, the asymmetric lineshape of the Fano resonance is still clearly observed since the destructive and constructive interference take place in a narrow spectral range exhibiting strongly resonant response. Furthermore, it is worth noting that the fields of the modes are mainly concentrated inside the dielectric particles, and not mostly at their surfaces as for plasmonic particles. As an important consequence, this novel type of Fano resonances is expected to be less sensitive to unintentional structural variations like surface roughness, which makes it very promising for a range of future applications in nanophotonics. Moreover, owing to their higher resistance to heat as compared to optical metal nanostructures and to the possibility of free carrier generation in the material, Fano resonances of silicon oligomers might offer
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a range of new opportunities not attainable with plasmonic nanoclusters, for example harsh-environment sensing or nonlinear applications. In this paper, for the first time to our knowledge, we experimentally verify the concept of magnetic Fano resonances in all-dielectric oligomers. We observe the characteristic suppression of the extinction associated with the Fano resonances in the measured linear-optical spectra of fabricated oligomer nanostructures. In support of our experimental results, we further discuss the underlying physics of the observed Fano resonances by employing a novel theoretical approach[31] based on collective-mode decomposition of the localized field excitation of the nanoparticle structures. This analysis confirms that the observed Fano resonances originate from the predominant excitation of opticallyinduced magnetic dipole modes, and furthermore provides a useful insight into their tuning behavior.
2. Results For the experiment we create a variety of different oligomer structures consisting of silicon nanodisks on a silicon oxide substrate. Details of the fabrication process can be found in the Experimental Section. We focus on heptamer structures, namely oligomers composed of seven individual nanoparticles. A schematic of the fabricated heptamer structures and scanning electron micrographs (SEM) of a typical sample are shown in Figure 1(a) and Figure 1(b), respectively, with their dimensions specified. Each heptamer consists of a hexamer (six particles arranged in a ring-like fashion) and a central nanodisk. Top view SEM images of each type of the fabricated nanoclusters are displayed in Figure 2(a). All heptamer structures are arranged in two-dimensional arrays with a lattice constant of 2840 nm [see Figure 1(b)]. This value has been chosen in order to ensure that the separation between neighboring structures is large enough for dominant coupling to remain within each oligomer structure, while the in-plane density of oligomer structures is still sufficiently high to create pronounced resonances in the far-field spectra. Linear-optical transmittance spectra of four silicon nanodisk heptamer arrays with varying central nanodisk diameter d2 are measured with unpolarized incident light (see Experimental Section for details on the measurement process) and the results are shown in Figure 2(b). For a subsequent comparison with theoretical data, the experimental transmission measurements are presented in terms of extinction crosssection σext using its relation to transmittance T given by the first-order approximation to the Beer-Lambert Law:[32]
σ ext ∝ − ln(T ) ≈ (1 − T ).
(1)
In order to gain a theoretical understanding of the optical response of these heptamer systems, we compare the experimental extinction spectra to those of individual heptamers in free space calculated using the discrete dipole approach. For plasmonic heptamer systems, such an analysis is typically based on observing the electric dipole moments or charge distributions of each constituent particle.[12,19,33–35]
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Observation of Fano Resonances in All-Dielectric Nanoparticle Oligomers
Figure 2. (a) Scanning electron micrographs of typical fabricated heptamer oligomers. Diameters d2 of the central nanodisk are 320, 350, 380 and 400 nm. (b,c) Extinction spectra for a variety of silicon nanodisk heptamer structures featuring a systematic variation of the central nanodisk diameter d2. Shown are (b) experimentally measured results for arrayed structures with unpolarized incident light, and (c) theoretically calculated results for single heptamers for arbitrarilypolarized light obtained using the coupled-dipole approximation. A Fano resonance is created in the heptamers (see gray-shaded region and colored arrows), and it moves across the spectrum as the diameter of the central particle is varied.
Figure 1. (a) Sketch of the geometry of the heptamer structures composed of silicon nanodisks fabricated on a silicon oxide substrate. The nanodisks of the surrounding hexamer have a constant diameter d1 = 460 nm, while the diameter d2 of the central nanodisk is systematically varied through 320, 350, 380 and 400 nm. All nanodisks have a height h of 260 nm and the radius r of the hexamer ring is 568 nm. (b) Scanning electron micrograph of a typical fabricated sample. The insets show magnified (top) and oblique (bottom) views of a single heptamer structure.
Here E i0 ( H i0 ) , αˆ iE (αˆ iH ) and pi (mi) are the electric (magnetic) applied field, tensor dipole polarizability and dipole moment, respectively, for the ith particle. The coupling between particles is described using dyadic Greens functions, Gˆ ij , between the locations of the ith and jth particles and k, 0 and µ0 are the free space wavenumber, permittivity and permeability respectively. In such a dipole system the extinction cross section can then be written analytically as
σ ext = However, since high-refractive-index all-dielectric particles do not only support electric but also magnetic modes,[22–27,36] this common approach is not applicable in our case. Instead, in order to adequately describe the optical properties of the realized silicon oligomers, both electric and magnetic dipole moments have to be considered.[30,37,38] Therefore, to calculate the extinction spectra of the silicon nanodisk heptamers, we employ the more general coupled electric and magnetic dipole approach[39] where each nanodisk is represented as an electric and a magnetic dipole. In this approach the electric and magnetic dipole responses of all particles are related to the externally-applied electric and magnetic field through ⎛ ⎞ μ0 p i = αˆ iE ∈0 E i0 + αˆ iE ∈0 k 2 ⎜ ∑ i ≠ j Gˆ ij p j − ∇ × Gˆ ij m j ⎟ ∈0 ⎝ ⎠
(2)
⎛ ⎞ 1 m i = αˆ iH H i0 + αˆ iH k 2 ⎜ ∑ i ≠ j Gˆ ij m j + ∇ × Gˆ ij p j ⎟ . ∈0 μ 0 ⎝ ⎠
(3)
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k Im ∈0 | E 0 | 2
{∑ E i
0† i pi
}
+ H i0 † mi .
(4)
The results of this approach applied to our heptamer system are summarized in Figure 2(c).
3. Discussion Experimental and theoretical extinction spectra show a good qualitative agreement. In both sets of results we observe two distinct extinction peaks for each spectrum. The peak at longer wavelengths is associated with the magnetic resonance of the collective structure. The second peak, which is observed at shorter wavelengths, corresponds to the magnetic resonance of the central particle. In accordance with earlier work on individual silicon nanodisks,[26] this peak experiences a red-shift as the central particle diameter is increased. The Fano resonance occurs as a region of resonantly suppressed extinction in-between
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of a silicon oxide layer and a silicon handle wafer in the experiment.
3.1. Coupled-Mode Analysis
Figure 3. (a) Simulated total extinction spectrum of the silicon disk heptamer with a 400-nm-diameter central particle. The contributions of electric and magnetic eigenmodes to the extinction cross-section can be completely separated and are shown as well. (b) Decomposition of the heptamer’s magnetic response into three eigenmodes and their raw sum, the latter not exhibiting Fano-like characteristics. (c) Distribution of the magnetic dipoles in each eigenmode for horizontally-polarized light. It can be seen that an asymmetric lineshape Fano resonance is induced in the extinction spectra around the spectral position where the resonance curves for the eigenmodes i. and ii. intersect.
the two peaks. This region is highlighted by the grayshaded area and the colored arrows in Figure 2. When the diameter of the central particle is increased and its associated resonance peak shifts closer towards the resonance peak of the collective structure, the Fano signature is spectrally shifted correspondingly. Interestingly, the interaction between the two resonances leads to a small, but clearly noticeable, variation in the height of the collective resonance peak. By increasing the central disk diameter d2 and shifting the peaks closer together, the overlap of the two peaks is increased and therefore the observed reduction (rather than growth) in height of the collective peak is indicative of destructive interference between these two resonances. This is observed both in theoretical and experimental spectra. Due to rotational symmetry, the extinction and other cross-sections of the heptamer nanostructures are independent of polarization.[38,40] It is worth noting that sharp asymmetric Fano resonances can be observed for unpolarized incident light, which is experimentally demonstrated in this paper. It also demonstrates the importance of the rotational symmetry of our system which provides the freedom of polarization choices for theoretical analysis. Differences in the exact resonance positions in numerical calculations compared to experimental spectra are primarily due to approximating the nanodisks as pairs of electric and magnetic dipoles in the theoretical model. Further deviations will arise from sample imperfections like roughness and not perfectly straight sidewalls of the silicon nanodisks, in addition to the presence of a substrate consisting
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Based on the qualitative agreement of the experimental results with the theoretical coupled-dipole approximation, in the following we analyze the observed extinction spectra in terms of the eigenmodes[41] of the heptamer structure.[31] In general, given that each individual dielectric nanoparticle supports both electric and magnetic dipole-like resonances, one would naively expect the collective eigenmodes of the heptamer to arise from hybridization of the entire set of supported modes. However, for planar nonchiral structures the contributions of electric and magnetic eigenmodes to the extinction cross-section can be completely separated.[30,42] Since the oligomer structures exhibit rotational symmetry, their extinction cross-section is polarization independent as mentioned previously. Thus, we can fix the polarization of the incident light to be horizontal without loss of generality. Figure 3(a) illustrates this for the example of a heptamer structure with a central particle diameter d2 = 400 nm. Shown are the contributions of the electric and magnetic eigenmodes of the heptamer to the extinction spectrum, as well as the total calculated extinction spectrum of the heptamer. Clearly, destructive Fano interference is only observed for the magnetic eigenmode contribution.[30,31] In order to further discuss the origin of the observed Fano resonance, it is therefore sufficient to limit our analysis to the magnetic eigenmodes of the heptamer only. Figure 3(b) shows a decomposition of the heptamer’s magnetic response into the three eigenmodes. For comparison, Figure 3(b) also includes the raw sum i+ii+iii of the mode amplitudes, emulating the artificial case of absent mode interaction and showing that a qualitatively different lineshape results in this situation. By comparing Figure 3(a) and (b), one can clearly see that the Fano lineshape in extinction seen in Figure 3(a) occurs at the intersection of the resonance curves of the two magnetic eigenmodes denoted by i and ii in Figure 3(b), indicating that the Fano lineshape arises from the interference of these two resonant eigenmodes. In Figure 3(c) we show the dipole distributions of these eigenmodes for a horizontally-polarized incident field ( E 0 polarized in the x-direction) to provide an intuitive explanation of the observed changes of the heptamer optical response upon variation of the central nanodisk diameter d2. The resonant mode denoted by i. is heavily localized at the central particle, making it very sensitive to a variation of d2. The mode denoted by ii., on the other hand, extends over the entire heptamer structure with only a small fraction of the mode concentrated at the central particle. Mode ii. is therefore not significantly influenced by a variation of d2. This is further illustrated in Figure 4, which shows the heptamers’ decomposed magnetic responses for the four different diameters realized in the experiment. Clearly, only the resonance peak of mode i. experiences a significant shift with variation of the diameter, altogether
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Observation of Fano Resonances in All-Dielectric Nanoparticle Oligomers
spectra are in good qualitative agreement with theoretical results obtained by using the coupled electric and magnetic dipole approximation. By analyzing the major collective modes of the oligomer structures associated with its dominant resonances, our numerical calculations further unveil the main mechanism of wave interference realized in the heptamer structures. This confirms that the observed Fano resonances indeed originate from interference between the optically-induced magnetic resonances of the central particles and those of the collective structures.
5. Experimental Section Fabrication: The heptamer samples are fabricated via electronbeam lithography (EBL) on backside polished silicon-on-insulator wafers (SOITEC, 260 nm top silicon thickness, 2 µm buried oxide thickness) using the negative-tone electron-beam resist NEB-31A. The EBL process is followed by a directive reactive-ion etching process using the obtained electron-beam resist pattern as an etch mask. As the final step, the etched sample is placed into oxygen plasma and piranha solution to remove any electron-beam resist mask residue. Measurement: Linear-optical transmittance spectra of the silicon nanodisk heptamer arrays are measured for unpolarized incident light using a custom-built white light spectroscopy setup with an optical spectrum analyzer. The sample is placed between a pair of 20× Mitutoyo Plan Apo NIR infinity-corrected objectives with numerical aperture NA = 0.4 which focus the incident light onto the sample and collect the light transmitted through it. Transmittance through the sample is referenced to the transmittance through the unstructured wafer. The range of incident angles is reduced to ±3° by an aperture to achieve near-normal incidence.
Figure 4. Decomposition of the heptamer’s magnetic response into three eigenmodes (i. red, ii. cyan, and iii. green) for a systematic variation of the central nanodisk diameter d2. Curves corresponding to different diameters are vertically displaced by 10 units for clarity. It can be seen that only the resonant peak of mode i. experiences a noticeable shift with the diameter of the central nanodisk, resulting in a deformation of the Fano resonance profile of the heptamer oligomer [see Figure 2].
resulting in the observed easily traceable tuning characteristics of the Fano resonance.
4. Conclusion We have experimentally demonstrated that magnetic Fano resonances can be observed in high-refractive-index alldielectric nanoparticle oligomers. We have fabricated and optically characterized arrays of heptamer structures composed of silicon nanodisks on a silicon oxide substrate. Their measured linear-optical spectra reveal a clear manifestation of Fano resonances. Furthermore, we have observed that the Fano resonance can be spectrally tuned by a systematic size variation of the central nanoparticle. Our experimental small 2014, DOI: 10.1002/smll.201303612
Acknowledgements This work was performed, in part, at the Center for Integrated Nanotechnologies, an Office of Science User Facility operated for the U.S. Department of Energy (DOE) Office of Science. Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-AC04– 94AL85000. The authors also acknowledge a support from the Australian Research Council.
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Received: November 21, 2013 Revised: January 12, 2014 Published online:
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