Fano Interference of Electromagnetic Modes in Subwavelength

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Fano Interference of Electromagnetic Modes in Subwavelength Dielectric Nanocrosses Zhong-Jian Yang J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.6b07902 • Publication Date (Web): 08 Sep 2016 Downloaded from http://pubs.acs.org on September 12, 2016

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The Journal of Physical Chemistry

Fano Interference of Electromagnetic Modes in Subwavelength Dielectric Nanocrosses Zhong-Jian Yang* Hunan Key Laboratory of Super Microstructure and Ultrafast Process, School of Physics and Electronics, Central South University, Changsha, Hunan 410083, P.R. China

Abstract: The electromagnetic mode couplings in subwavelength high-index dielectric nanocrosses are theoretically investigated. The nanocross is constructed by two orthogonal rectangular silicon nanoblocks. A pronounced dip appears on the scattering spectrum of the nanocross in the visible region, which is found to be caused by the Fano interference between the broadband electric dipole mode on one nanoblock and the narrowband magnetic dipole and quadrupole modes on the other one. Near field distributions are calculated to analyze the destructive and constructive couplings between the electric fields associated with the electric and magnetic modes in the Fano interference. By spatially separating the two blocks and varying the geometry parameters of each block, the Fano interferences can be further confirmed and the tuning of Fano resonance lineshape is also demonstrated.

Introduction Metallic nanostructures have been intensively investigated over recent decades as they support surface plasmon resonances which can concentrate and manipulate light at deep subwavelength scale.1 However, the high metal loss has impeded many of their applications. Recently, dielectric nanostructures with high refractive index have drawn much attention as they exhibit similar strong optical responses2 but their material losses are low. Furthermore, they also support strong magnetic resonant modes besides electric modes.3,4 These properties make dielectric structures an attractive alternative to plasmonic structures for nanophotonic functionalities. The use of dielectric structures for many applications such as matematerials,5 solar cells,6,7 optical antannas,8-13 photodetectors,14,15 nonlinear optics16,17 and metasurfaces18-20 have been reported. 1

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The optical properties of many individual dielectric resonators have been studied. These resonators include spherical particles,3,9,21-26 cylinders or disks,27-33 rectangular nanowires,34,35 nanoblocks.36 Nanospheres are probably the most extensively studied structures among these objects because of the analytical treatment with the Mie theory and the progress in their fabrication.37 Both electric and magnetic dipole resonances can be excited easily in nanospheres. Rectangular shaped three-dimensional nanoparticles (nanoblocks, NBs) have begun to attract attention very recently.36 The optical properties could be different from that of cylinders due to the less symmetric morphology. The length dependent scattering responses by two basic electromagnetic modes on nanoblocks have been demonstrated. The couplings between dielectric structures have also drawn a lot of interest. The phenomena of hybridization of electric and magnetic modes in some nanosphere and square dimers have been studied by analogy with that in plasmonic dimers.38,39 But the hybridization scheme could be much richer in the dielectric nanoparticles due to the magnetic resonances.38 The coupling between two silicon cylinders can give rise to both magnetic and electric near field hotspots which has been demonstrated experimentally.40 It was also found that the interaction between electromagnetic modes can induce Fano resonance in some coupled dielectric structures.41-46 Fano resonance usually involves the interaction between broadband and narrowband spectrum responses. These spectral features in photonics structures can be obtained by constructing “bright” and “dark” modes, which have been extensively studied in plasmonic systems.47-49 The Fano interference between broadband and narrowband optical modes usually results in a sharp dip on scattering (absorption or extinction) response spectrum (or sharp peak on transmission spectrum). Inspired by the plasmonic Fano resonant structures some similar coupled dielectric resonators have been proposed to exploit Fano resonances.41,42,45 For example, a silicon nanorods and rings coupled structure has been investigated,41 where nanorods support bright electric dipole modes while silicon rings support dark magnetic modes. Their couplings result in a sharp peak on the transmission spectrum. Although the optical properties in many dielectric structures can be well explained by analogy to the relevant studies in their plasmonic counterparts such as hybridization and Fano resonance phenomena,38,41,42,45 there are still some other optical behaviors which are quite different from that in metal resonators. It has been shown that even a simple subwavelength dielectric resonator could support many 2


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electric and magnetic modes which are close to each other on spectrum.30,35 In the case of simple shape metal structures, they do not show such property generally. The muti-mode resonances including magnetic responses in dielectric structures could cause electric−electric, electric−magnetic and magnetic−magnetic modes couplings. The complex couplings make the understanding of optical responses such as scattering and absorption spectra not so intuitive. For example, magnetic and electric modes in silicon sphere dimers could undergo destructive couplings,28 while a favorable overlap between magnetic and electric modes has been reported in nanoblocks.36 The touching (or overlapping) of two metal structures usually makes a totally new resonator where the resonance cannot be explained from the point view of the couplings between the modes on the untouched elements anymore.50 This is due to the fact that surface plasmon modes are associated with oscillations of free electrons. In contrast, the optical modes in dielectric nanostructures are leaky geometrical resonances. Thus, touching of dielectric cavities usually only causes stronger mode couplings compared to nontouching cases.28 This feature allows more spatial tunability for mode interactions in coupled dielectric resonators. Based on the facts discussed above, more studies are still needed to fully understand the interaction properties between electromagnetic modes in coupled subwavelength dielectric structures. Here, we theoretically investigate the couplings of basic electric and magnetic modes in individual nanocrosses. A nanocross is formed by two orthogonal rectangular silicon NBs. On one NB, there is a broadband electric dipole mode response. On the other one, there are magnetic dipole and quadrupole mode resonances which are spectrally narrow and close to each other. These electric and magnetic modes are spectrally overlapped and they undergo Fano interference in nanocross, which results in a pronounced dip on the scattering spectrum. Near fields distributions are calculated to analyze the destructive and constructive couplings between the electric fields associated with the electric and magnetic modes in the Fano interference. Geometry parameters are also varied to further confirm the Fano resonance and understand the Fano lineshape tunability property in this system.

Methods The simulations were carried out by using commercial finite-difference time-domain 3



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(FDTD) software (Lumerical FDTD). The mesh size near the simulated structures is 2*2*2 nm3. For the case of separated NB1 and NB2 with distance d = 10 nm, the mesh size along z-axis in the gap is 1 nm. The excitation source is the Total-field scattered-field (TFSF) plane wave. The polarization is along x-axis. Perfectly matched layer (PML) boundary conditions were used in the simulations. The surrounding index for simulations is n = 1. The dielectric function of silicon is taken from Palik’s book.51 The absorption cross section for a structure is obtained by normalizing the power flowing into the structure by the plane wave intensity. In calculations, we set a power monitor box surrounding the structure (6 two-dimensional monitors in total), and then get the transmission of each monitor. The transmission function in the software returns the amount of power transmitted through a power monitor or a profile monitor, normalized to the source power. By adding the 6 transmissions and multiplied by the source area one can get the absorption cross section. The scattering cross section can be obtained in similar way, where the only difference is that the monitors are placed in only scattered light region (for absorption the monitors are in the total field region). The extinction cross section equals the sum of absorption and scattering cross sections.

Results and Discussion We first investigate the optical properties of individual NBs. The index of surrounding medium is taken to be 1. Silicon is taken as the high-index material. The optical spectra of the first NB (NB1) with excitation planewave polarization along the long axis of the block are shown in Figure 1a. The length, width, and thickness are L1 = 360, W1 = 30 and T1 = 90 nm, respectively. There are two main responses, an electric dipole mode with broad spectral response and a magnetic dipole mode (around λ = 460 nm) with relatively small line width. The main features of the optical responses are similar to that reported in Ref.36. These electric and magnetic dipole mode responses can also be further confirmed by calculations based on the mutipole decomposition approach (see Figure S1 in Supporting Information).52 The second NB (NB2) is excited with the planewave polarization along the short axis of the block (Figure 1b). The length, width, and thickness are L2 = 200, W2 = 90 4


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and T2 = 90 nm, respectively. There are two main peaks, where the right one corresponds to a magnetic dipole mode (the magnetic field direction is along the long axis of the block, this specific mode is denoted by MD) and the left one is an electric dipole mode (see Figure S1 for multipole decomposition calculations and magnetic field for the MD). In fact, there are some other modes for the same NB2, and one of them is a magnetic quadrupole mode. This mode can be excited by introducing retardation effect with the planewave excitation wave vector along the long axis of the block (Figure 1c). There are two main resonances on the spectra shown in Figure 1c, the right one is also a magnetic dipole mode but the magnetic dipole moment is along the short axis of the block, which is different from the MD in Figure 1b. The left one is a magnetic quadrupole mode which can be identified by the magnetic field distribution as shown in Figure 1c (this specific mode is denoted by MQ) and the resonance is close to the MD mode in Figure 1b. Although this MQ mode can be hardly excited with the configuration in Figure 1b, it plays an important role in the mode couplings in nanocross which will be the focus of this work. It is noted that there is also significant electric field enhancement for a magnetic mode.36,53 This is an important feature which can help us to understand the mode couplings in our dielectric system.

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Figure 1. The optical responses of individual NBs. (a) The absorption, scattering and extinction cross sections of the NB1. The inset shows the schematic of the NB1 and plane wave excitation configuration. The length (L1), width (W1), and thickness (T1) are 360, 30 and 90 nm, respectively. (b) The absorption, scattering and extinction cross sections of the NB2. The inset shows the schematic of the NB2 and plane wave excitation configuration. The length (L2), width (W2), and thickness (T2) are 200, 90 and 90 nm, respectively. (c) The same plot for the NB2 but the plane wave is propagating along the long axis of the block. The top inset shows the || at λ = 505 nm, where the arrows denote the magnetic field directions; the bottom one shows the excitation configuration.

Now we consider the nanocross as shown in Figure 2a. It consists of the two blocks discussed above (NB1 and NB2) and they overlap spatially with their long axis being orthogonal to each other. The index of surrounding medium is also 1. The coordinate system is shown in Figure 2a and its origin is chosen at the center of the nanocross. The polarization and wave vector of the excitation plane wave are along the x-axis and z-axis directions, respectively, and this will be the default excitation configuration in the following calculations unless specified. Figure 2b shows the scattering spectrum of the nanocross, where a pronounced dip appears on the broad spectrum and this dip is close to the magnetic mode of the individual NB2. Another 6


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feature is also seen that the scattering intensity is enhanced compared to individual nanoblocks with a wide range in the longer wavelength region. Here, the extinction spectrum of the nanocross is also calculated, and its lineshape is similar to that of the scattering (Figure 2c). Near field investigations show that these two features, namely the dip and enhanced intensity on the scattering spectrum, are caused by Fano interference of electric and magnetic modes in this system.

Figure 2. Optical spectra with Fano resonance. (a) The schematic of the nanocross with x-polarized plane wave excitation. The origin of coordinate system is placed at the nanocross center. (b) Scattering spectra for the nanocross and individual NBs with the same source excitation configuration. (c) The extinction, scattering and absorption cross section spectra of the nanocross.

We first focus on the near field couplings near the dip at λ = 505 nm. The scattering response of the individual NB1 near the dip position is mainly due to the electric dipole mode (Figures 1a and S1). This indicates that the interactions here could involve the electric dipole mode on NB1 and the magnetic modes on NB2. Figure 3a shows the electric field enhancement |E| on the y = 0 plane, where the arrows denote the main feature of the calculated E-field directions (see Figure S2 for the detail about the calculated electric field vector distributions). The two arrows on two sides show the E-field directions on NB1, which is similar to that of the electric dipole mode on the individual NB1 (Figure 3b). The three arrows in the middle part 7



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denote the E-field directions on NB2, which show a feature of spatial superposition of the MD and MQ modes compared to each of the mode distribution on the individual NB2 (Figures 3c and 3d). The two arrows marked as AD1 and AD2 correspond to the E-field directions of the MD mode on NB2 and the arrow AQ is related to the E-field direction of the MQ mode on NB2. Here it should be pointed out that to demonstrate the E-field feature for the MQ mode in Figure 3d, the excitation configuration in Figure 1c was used. The magnetic field enhancement |H| on the x = 0 plane is also calculated for further verification of the two magnetic modes on NB2 as shown in Figure 3e. The M-field distribution which is related to the MQ mode can be easily verified by analogy with that in Figure 1c. As it is shown by the M-field direction arrows, there is also relatively small y-axis component of M-field because of the contribution from the MD mode (the dipole moment is along y-axis direction, see Figure S2 for the detail about the calculated magnetic field vector distributions). The magnetic near field vector distribution on z = 0 plane for the magnetic modes on NB2 is also calculated (Figure S2). And it is in consistent with the near field distribution about the MD and MQ modes in Figure 3e. It has been noted that the MQ mode can be hardly excited on individual NB2 under the excitation configuration used for nanocross, so it is induced by the near field coupling to the mode on NB1. As denoted by the arrows, the E-fields of the MD and MQ modes on NB2 show an overall destructive interference with that of the electric mode on NB1. The E-field associated with the electric dipole mode inside the NB1 disappears due to the near field energy transfer and destructive interference. This destructive coupling causes the dip on the scattering spectrum.

Figure 3. Destructive near field couplings near the Fano dip at λ = 505 nm. The electric field enhancement || distribution on the y = 0 plane for the nanocross, individual NB1 and individual NB2 are shown in (a), (b) and (c), respectively. The arrows show the electric field directions. (d) The same plot for the individual NB2 but the excitation configuration now is the same as that in Figure 1c. The color bar for || in (a)-(d) is the same. (e) The magnetic field enhancement || 8


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distribution of the nanocross on the x = 0 plane, the arrows show the magnetic field directions.

The coupling behaviors between the electric and magnetic modes for the enhanced scattering response on the nanocross can also be understood by analyzing the near field distributions (Figure 4). The wavelength position λ = 555 nm, which is around the right peak on the spectrum, is chosen to demonstrate the near field couplings. Figure 4a shows the electric field enhancement |E| on the y = 0 plane. The arrows are drawn in the same way as that in Figure 3a to show the main feature of the calculated E-field directions (see Figure S3 for the detail about the calculated electric field vector distributions and the magnetic field vectors discussed below). The magnetic field enhancement |H| distribution on the x = 0 plane is shown in Figure 4d. The two magnetic modes on the NB2 can also be verified at λ = 555 nm, but the MD dominant the response here (this can also be verified by the M-field vector distribution on the z = 0 plane, see Figure S3). It is seen that the E-fields of the MD and MQ modes on NB2 show an overall constructive interference with the electric dipole mode on NB1. This causes an overall enhanced |E| on nanocross compared to the individual NB1 (Figure 4b) and NB2 (Figure 4c). A remarkable enhanced |H| is also seen compared to that on the individual NB2 (Figure 4e). This constructive interference leads to the enhanced scattering intensity on the nanocross.

Figure 4. Constructive near field couplings at λ = 555 nm. The || distribution on the y = 0 plane for the nanocross, individual NB1 and individual NB2 are shown in (a), (b) and (c), respectively, and the arrows show the electric field directions. The color bar for || in (a)-(c) is the same. The || distribution on x = 0 plane for the nanocross and individual NB2 are shown in (d) and (e), respectively. The arrows show the magnetic field directions. The color bar for |H H| in (d)-(e) is the same.

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To get more insight into the near field couplings around the Fano interference, we also investigated the system of separated NB1 and NB2 as shown in Figure 5a. Here, the only difference from the nanocross structure in Figure 2a is that the NB2 was moved away from NB1 in the z-axis direction. The distance d for this system is defined by the gap distance between the top surface of NB2 and the bottom surface of NB1 (NB2 is under NB1). In fact, in this system it is easier to identify the electric or magnetic modes on each NB since the two NBs are separated. The scattering of this structure also shows a dip on the spectrum, and the dip is also near the destructive interference position of the Fano resonant nanocross and it becomes more pronounced with decreasing the distance d. These spectra features indicate that there is also Fano interference here. The electric and magnetic near field distributions for the structure with d = 10 nm were also calculated (see Figures S4 and S5 for the detail discussion about the Fano resonance for this case) and their destructive and constructive coupling behavior is indeed quite similar to that in nanocross. So compared to the case of separated nanoblocks, the nanocross undergoes the same Fano resonance and the coupling strength is higher. Here, small differences are also noted, for destructive coupling the role of MD becomes relatively more important, and for constructive coupling the additional enhancement of |E| is more remarkable than |H|. It has been shown that the destructive interference in the nanocross involves both the MD and MQ modes on the NB2 and the MQ dominant in the coupling (Figures 2-4). The resonance wavelength positions of the MD and MQ modes on individual NB2 show different behaviors with varying the NB2 length L2. By increasing the L2, the MD mode is nearly the same, while the MQ mode shows a clear redshift (Figure S6). The resonance positions of MD and MQ modes for the L2 around 170 nm and 200 nm are close to each other (Figures 1b, 1c and S6). So when the L2 value is away from this range, the two modes will be more apart from each other spectrally. This can cause a separation between the two destructive couplings on the spectrum, and the coupling becomes relatively weak in the nanocross. As a result, the depth of the dip becomes smaller as shown in Figure 5b. With the point view of Fano resonance tunability in this system, the variation of the NB2 length L2 cannot produce Fano dips at other wavelengths with the depth similar to that near λ = 505 nm where MD and MQ is spectrally close to each other (Figure 5b). In fact, we can do this by varying the width W2 and thickness T2 of NB2. Here, T2 is always taken to be the same as W2 for simplicity. The NB2 with W2 = 80 10


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and 100 nm are investigated and their lengths are all L2 = 200 nm. Both the MD and MQ modes on individual NB2 show redshift with increasing the W2 and T2 (see Figure S6). And the two mode resonance positions are close to each other with each of the chosen geometry parameters for an individual NB2. The scattering responses for nanocrosses with these different NB2 are shown in Figure 5c. A redshift of Fano dip is seen with increasing the W2 and T2, and these Fano dips show similar depth.

Figure 5. Fano resonances for separated NB1 and NB2 and nanocrosses with different NB2 geometry parameters. (a) Scattering cross sections for separated NB1 and NB2 with distance d from 50 to 10 nm. The inset shows the structure configuration. The case for nanocross is also shown for comparison. (b) Scattering cross sections for nanocrosses with NB2 length L2 from 140 to 270 nm. Here, the width and thickness are fixed at W2 = T2 = 90 nm. (c) Scattering cross sections for nanocrosses with NB2 width (or thickness T2, T2 = W2 for each case) W2 = 80, 90 and 100 nm. The length L2 is fixed at L2 = 200 nm. The structure configuration of the nanocrosses in (b) and (c) is the same as that in Figure 2.

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The dependence of the Fano interference on the NB1 geometry parameters is also studied. The variation of NB1 geometry parameter will change its scattering magnitude and thus the Fano resonance lineshapes. Figure 6a shows the scattering spectra for nanocrosses with different NB1 lengths (L1 = 300, 360 and 420 nm), the thickness and width are fixed at 90 and 30 nm, respectively. The scattering cross section of an individual NB1 increases with the length L1 (Figure S7). As such, the scattering on two sides of the Fano dip increases with L1. Since the destructive coupling is so strong, the scattering value at dip is almost the same despite the variation of NB1 scattering. This makes the Fano dip of L1 = 420 nm be relatively more pronounced. It should be noted that with even larger L1 the value at the Fano dip will increase due to the increased mismatch between the scattering magnitudes of two blocks. See, for example, L1 = 550 nm in Figure S7. The scattering responses with different NB1 widths (W1 = 20, 30 and 60 nm) are also investigated (Figure 6b). Here, the length and thickness are fixed at 360 and 90 nm, respectively. The increasing of the width W1 of an individual NB1 will cause not only the increasing of its scattering magnitude but also the redshift of its magnetic and electric dipole mode resonances (Figure S7). The combine of these two effects lead to a stronger scattering on the right side of the dip with larger W1. In contrast, the left side of the dip does not necessarily show such a behavior due to the redshift of the scattering spectrum. As a result, the Fano resonance lineshape is less pronounced with a relatively large width (for example, W1 = 60 nm) compared to that of W1 = 30 nm.

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Figure 6. Fano resonances for nanocrosses with different NB1 geometry parameters. The structure configuration is the same as that in Figure 2. (a) Scattering cross sections for nanocrosses with NB1 length L1 from 300 to 420 nm. Here, the width and thickness are fixed at W1 = 30 nm and T1 = 90 nm, respectively. (b) Scattering cross sections for nanocrosses with NB1 width W1 from 10 to 60 nm. The length and thickness are fixed at L1 = 360 nm and T1 = 90 nm, respectively.

Based on the discussion about the geometry variations (Figures 5 and 6), we can see how to construct a nanocross with Fano resonances. The geometry parameters for the NB2 should be chosen based on the fact that its MD mode position is highly dependent on its thickness (its width can be chosen to be the same as the thickness for simplicity, W2 = T2), while the length (L2) hardly affects the MD mode position. Then the length L2 should be adjusted to make the MQ close to the MD mode. For NB1, the thickness is taken the same as that of NB2 to make the structure easier for fabrication. The value of width W1 should be chosen to make its electric dipole mode fully overlapped with the MQ and MD modes of NB2. The length L1 is taken to adjust its scattering magnitude to make the Fano dip more pronounced.

Conclusions In conclusion, we have theoretically investigated the electromagnetic modes couplings in individual subwavelength dielectric nanocrosses. A distinct dip can be found on the nanocross scattering spectrum. Electric and magnetic near field analyses 13



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show that this dip is caused by the Fano interferences between the electric dipole mode of NB1 and two magnetic modes (MD and MQ) of NB2. In this system, a magnetic mode (MD or MQ) is also accompanied by electric field enhancement which is closely related to the magnetic field of the mode. Near the dip, destructive coupling between the electric fields of the electric mode of NB1 and magnetic modes of NB2 can be clearly seen. For the broadband enhanced scattering, there is constructive coupling between the electric fields of these modes and the largely enhanced magnetic field is seen. By separating NB1 and NB2 spatially and varying NB1 or NB2 geometry parameters, the Fano interferences between the modes on NB1 and NB2 were further confirmed and the Fano resonance lineshape can be turned. Our study reveals that complex electromagnetic mode couplings can occur in even simple subwavelength dielectric nanostructures. It is also seen that as the mode couplings are complex, near field analysis which can identify the involved electromagnetic modes and directly show their destructive (and/or constructive) couplings is essential. The structure investigated here can be experimentally realized by using electron-beam lithography (EBL) and reactive-ion etching.34,39 The Si structures are usually patterned on quartz. Calculations show that the Fano resonance lineshape changes with nanocross placed on quartz (n ≈ 1.5), but the main feature, namely the pronounced Fano resonance dip, still remains (Figure S8). Substrate with higher refractive index further weakens the Fano resonance dip feature. The understanding of the complex mode couplings in basic dielectric nanostructures could help us to efficiently construct functional all dielectric metasurfaces and metamaterials.

Supporting Information Available: The mutipole decomposition calculations of individual NB1 and NB2; electric and magnetic near field vector distributions for destructive and constructive field couplings; discussion about the Fano resonance for the case of separated NB1 and NB2 with d = 10 nm; scattering spectra for individual NB1 and NB2 with different geometry parameters; scattering spectra for nanocross with L1 = 550 nm; scattering spectra of nanocross placed on substrates. This material is available free of charge via the Internet at http://pubs.acs.org.

Corresponding Author *Tel.: +86-731-88830323. E-mail: [email protected].

Acknowledgement: We acknowledge financial support from the Central South University. 14


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For Table of Contents Use Only:

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Figure 1. The optical responses of individual NBs. 118x233mm (300 x 300 DPI)



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Figure 2. Optical spectra with Fano resonance. 103x191mm (300 x 300 DPI)


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Figure 3. Destructive near field couplings near the Fano dip at λ = 505 nm. 43x22mm (300 x 300 DPI)



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Figure 4. Constructive near field couplings at λ = 555 nm. 60x44mm (300 x 300 DPI)


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Figure 5. Fano resonances for separated NB1 and NB2 and nanocrosses with different NB2 geometry parameters. 135x296mm (300 x 300 DPI)



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Figure 6. Fano resonances for nanocrosses with different NB1 geometry parameters. 89x140mm (300 x 300 DPI)


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TOC figure 35x16mm (300 x 300 DPI)


Fano Interference of Electromagnetic Modes in Subwavelength

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