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Abstract
In this paper we investigate the Fano resonances of a ring-disc nanostructure that consists of two nanodiscs and two concentric nanorings. The dark modes of both nanorings can couple to the bright mode of the nanodiscs, leading to separate Fano resonances from the outer and the inner nanoring. The concentric arrangement of the two nanorings allows for a coupling between the dark modes of the outer and the inner nanoring, thus creating an additional interaction that influences the Fano resonances of the dual-ring nanostructure. This interaction is investigated by comparing the Fano resonances of the complete dual-ring structure with the isolated Fano resonances of the individual single-ring structures. The effect of the coupling between dark modes on the Fano resonances is verified using a model of coupled harmonic oscillators that describe the Fano resonances of this system in a classical analogy. Lastly we compare the sensitivity of a single-ring nanostructure with that of a dual-ring nanostructure to investigate the effects of a coupling between dark modes on the sensing performance.
Published by Optica Publishing Group under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.
1. Introduction
Plasmonic resonances occur through an interaction between electromagnetic radiation and metallic nanostructures. These resonances can be described as a collective oscillation of the conducting electrons inside the metals as a response to the electromagnetic wave [1]. The plasmon modes can be divided into broad bright modes that can be directly excited by linearly polarized light and narrow dark modes that cannot be directly excited by light [2,3].
Bright modes and dark modes can couple with each other as well, leading to an interference that is known as the Fano resonance [4–6]. A Fano resonance leads to a sharp, asymmetric lineshape that is highly tunable and sensitive to changes of the refractive index. These properties make Fano resonances desirable in many fields of application, such as sensing, non-linear optics, optical switching and optical modulators [7–10], leading to a widespread interest in nanostructures that can support Fano resonances.
Investigations of Fano resonances are usually performed with numerical simulations or experimental fabrication and measurement of such a nanostructure. Common parameters that are investigated are for example the scattering cross section or the reflectivity of the nanostructure. These parameters are used to identify the spectral positions of the Fano resonances, which are then investigated by simulating the electric near-field at these positions.
There are already a number of different device geometries that were investigated this way, such as a nanoscale dolmen structure [11,12], an oligomer of nanodiscs [13,14] or a split-nanoring cluster [15]. These structures exhibit both bright modes and dark modes that couple to each other, forming a Fano resonance.
A ring-disc nanostructure is another geometry that is frequently used to investigate Fano resonances [16–21]. These nanostructures consist of at least a single nanodisc and a larger nanoring. The multipolar modes of the nanoring are harnessed as dark modes, the bright mode is provided by the dipole-like mode of the nanodisc. Placing a nanodisc and a nanoring close together results in a coupling between the bright dipolar mode of the nanodisc and the dark multipolar modes of the nanoring, thus creating Fano resonances.
While the coupling between bright and dark modes is already well investigated, there is as of yet very little discussion about the coupling of separate dark modes, with the publications being mainly in the field of electron-induced transparency [22–24]. A single nanoobject can have multiple dark modes, but these dark modes cannot interact with each other since they originate from the same nanoobject and are spectrally separated [25,26].
To facilitate an interaction between dark modes, a nanostructure must therefore have at least two separate sources for dark modes that are in close vicinity to each other [27,28]. This paper investigates the interaction between dark modes by simulating a ring-disc nanostructure with two nanorings that are placed in a concentric arrangement. Both the inner and the outer nanoring have dark modes that can couple to the bright mode of a nanodisc, creating Fano resonances. The dark modes of the outer and the inner nanoring can couple with each other as well, leading to additional interactions that influence the Fano resonances.
2. Theoretical methods
Figure 1(a) shows the geometry of the nanostructure, which consists of two concentric nanorings and two nanodiscs. Both nanodiscs have a radius of 70 nm each and a distance of 20 nm to the outer nanoring. The outer nanoring has an outer radius of 210 nm and a thickness of 40 nm. The inner nanoring has an outer radius of 150 nm and a thickness of 40 nm as well; this leads to a gap of 20 nm between the outer and the inner nanoring.
Although a single nanodisc is sufficient to excite Fano resonances, a second disc is placed on the opposite side of the nanorings to achieve symmetry. The symmetrical geometry of the nanostructure suppresses even modes [29], reducing the total amount of Fano resonances that occur in this structure. The entire nanostructure is made of gold, as gold shows plasmonic properties in the visible spectrum. Moreover, gold nanostructures are stable and do not oxidize in air, unlike e.g. silver or aluminum, which makes it an ideal material for plasmonic nanostructures. The nanostructure has a height of 50 nm and is placed on a substrate that consists of a semi-infinite glass layer. A polarized plane wave is used as a radiation source. The polarization of the electric field component $\vec {E}$ as well as the propagation direction $\vec {k}$ are shown as a blue and a purple arrow in Fig. 1.
Fig. 1. a) Schematic representation of the complete nanostructure. b) Partial nanostructure consisting of two nanodiscs and the outer nanoring. c) Partial nanostructure consisting of two nanodiscs and the inner nanoring.
The investigation of the properties of this nanostructure is pursued using the finite difference time domain method (FDTD) [30–32]. We employed the commercial software Lumerical to calculate the scattering cross section and the electric near-field of the nanostructures. The electric near-field enhancement of this nanostructure is monitored in a 2D slice of the nanostructure at a height of z=25 nm, exactly in the middle of the nanostructure.
The simulations used a uniform Yee-grid with a cell size of 1 nm x 1 nm x 1 nm. The dielectric function of gold is taken from the experimental values of Johnson and Christie [33], the values for glass are taken from Palik [34]. The top medium that surrounds the nanostructure is assumed to be air and is described with a refractive index of 1.
3. Results and discussion
3.1 Simulation results
To understand whether a coupling between the dark modes of the two nanorings occurs, a frame of reference of the "isolated" Fano resonances of each nanoring has to be established. This can be done by investigating the Fano resonances of a single-ring structure. The composite nanostructure is now separated into two single-ring structures. These systems only consist of two nanodiscs and either the outer or the inner nanoring (see Fig. 1(b) and (c)).
The Fano resonances of these structures are now investigated by calculating their scattering cross sections. The bright and dark modes that constitute the Fano resonances can be investigated by themselves as well. The bright mode of this structure is provided by the two nanodiscs, its position and shape is shown by calculating the scattering cross-section of the two nanodiscs by themselves (shown as the dashed line in Figs. 2(a), 3(a), and 4).
Fig. 2. a) Scattering cross section of the outer-ring structure. The scattering cross section of the two discs by themselves is shown for reference. A decapolar (D) and a hexapolar (H) Fano resonance cause two dips and two scattering peaks in the lineshape of the cross section. b) Electric near-field showing the decapolar dark mode at a wavelength of 614 nm. c) Electric near-field showing the hexapolar dark mode at a wavelength of 812 nm.
Fig. 3. a) Scattering cross section of the inner-ring structure. The scattering cross section of the two discs by themselves is shown for reference. A hexapolar (H) Fano resonance causes a dip and a scattering peak in the lineshape of the cross section. b) Electric near-field showing the hexapolar dark mode at a wavelength of 647 nm.
Fig. 4. Scattering cross section of the dual-ring structure. The scattering cross section of the two discs by themselves is shown for reference. A decapolar (D) and a hexapolar (H) Fano resonance cause dips and scattering peaks in the lineshape.
Dark modes on the other hand do not appear in the scattering cross section of an isolated nanoring, as they cannot be directly excited by a plane wave at normal incidence [26,35]. However, their position and shape can be determined by placing oscillating dipoles close to an isolated nanoring. These dipoles can excite the dark modes, leading to an increased power emission of the dipoles at their resonance positions [36,37]. Monitoring the power that is emitted by these dipoles gives direct information about the dark modes (see Supplement 1 Fig. S2-S4).
Fano resonances themselves appear in the scattering cross section of single-ring structures as additional scattering peaks and dips, with their resonance position being located on the slope between the dip and the peak (see Supplement 1 Fig. S1).
This paper highlights each Fano resonance at their approximate resonance position and labels each of them by their corresponding dark mode, which is a multipolar mode of the nanoring. The dark modes can be determined by simulating the electric near-field at the approximate resonance position where the near-field enhancement of the dark modes is most pronounced. At this position one can determine the multipolar dark mode by simply counting the number of high near-field spots that are visible along the nanoring.
Figure 2 compares the scattering cross section of the outer-ring system with the scattering cross section of the discs themselves. The scattering cross section of the nanodiscs has only a single broad scattering peak, showing that the nanodiscs only have a single dipolar bright mode in this spectral range.
The addition of a nanoring has caused the appearance of two scattering peaks (along with their dips) to the left and to the right of the dipolar disc mode. These dips and peaks were caused by two different Fano resonances that occur in this system.
The Fano resonances are now labeled by their corresponding dark modes as described above. The electric near-field shows a decapolar and a hexapolar dark mode, the two Fano resonances are therefore called decapolar resonance and hexapolar resonance. The decapolar dark mode is less visible than the hexapolar dark mode both in the lineshape of the cross section and in the electric near-field. The lower visibility of the decapolar mode is due to the higher damping of plasmons in gold at lower wavelengths due to interband transitions [1,38].
No quadrupolar or octupolar mode were found, this is due to the aforementioned suppression of even modes through the symmetry of the nanostructure. The mode structure seen in the scattering cross section of this structure is confirmed in the calculated absorption cross section, transmission, and reflectivity spectra (not shown).
We now repeat the examination of the Fano resonances for the inner-ring structure in Fig. 3. The scattering cross section of the inner-ring structure shows an additional peak (and thus a new dip) when compared to the cross section of the discs by themselves, which means the inner-ring structure exhibits a Fano resonance as well.
Simulating the electric near-field at the dip position shows that this is a hexapolar resonance. The position of that resonance occurs at lower wavelengths compared to the hexapolar resonance of the outer-ring structure. This blueshift of this dark mode is due to the smaller size of the inner nanoring. Lowering the size of a nanoobject increases the restoring forces that affect the surface charges that are caused by a plasmon, increasing the resonance energy of that plasmon and blueshifting the resonance [1,39].
Now that the Fano resonances of the isolated single-ring structures have been investigated, the entire dual-ring nanostructure can be considered. The scattering cross section of the dual-ring structure seen in Fig. 4 has little resemblance to either the cross sections of the single-ring structures, or to the cross section of the discs by themselves. There are three different scattering peaks that are caused by Fano resonances, these are found at 630 nm, 730 nm, and 920 nm. Two of these peaks are clearly visible, but the rightmost scattering peak at 920 nm has a lower intensity and is barely visible.
The central scattering peak has the highest intensity and results from the scattering peak of the bright dipolar mode of the nanodiscs. That peak has been redshifted through the coupling to the dark modes of the inner and outer nanoring. The two side-peaks are caused by the Fano resonances that occur in this nanostructure, but these Fano resonances are affected by the additional coupling between the dark modes of the inner and the outer nanoring.
The left side-peak at $\lambda \approx$ 620 nm is at the same position as the decapolar scattering peak that occurs in the outer-ring system (see Fig. 2), which indicates that this peak is caused by the decapolar resonance of the outer ring. Adding a second nanoring has made that peak more pronounced. This is due to the presence of an additional hexapolar mode at the inner nanoring which interacts with the other modes. The rightmost scattering peak can be assigned similarly to the hexapolar scattering peak of the outer-ring system. The addition of a second nanoring has redshifted that scattering peak and lowered its intensity, making it barely visible.
In order to better understand the coupling mechanism that occurs between the dark modes of the inner and the outer nanoring, we now simulate the electric near-field at the approximate positions of both Fano resonances. These are found at the slopes between the dip and the scattering peak and labeled in Fig. 4 with a D and H. The near-fields of these modes are shown at the position of their highest near-field intensity in Fig. 5.
As expected, the simulated near-fields show a decapolar dark mode at $\lambda \approx$ 640 nm and a hexapolar dark mode at $\lambda \approx$ 890 nm, confirming our previous assumptions. The electric near-field of these dark modes is confined between the two nanorings, increasing the near-field enhancement compared to the respective dark modes of the single-ring systems. The electric field intensity of the hexapolar mode is roughly constant for each near-field spot, but the field enhancement of the decapolar mode depends on the position of the near-field spot. The four "central" near-field spots have a higher field enhancement than the spots that are adjacent to the nanodiscs, while the spots between them are barely visible.
The behaviour seen in Fig. 5 already indicates that these electric near-fields are influenced by a coupling between the dark modes of the outer and inner ring. Each dark mode causes the accumulation of alternating positive and negative charges along the surface of the nanoring, these charges cause the characteristic electric near-field of each dark mode [26,40]. The concentric arrangement of the two nanorings now allows for an interaction between the charges of the outer and the inner nanoring (and vice-versa) at the gap between the two nanorings. The increased electric near-field occurs when two charges with opposite signs face each other. This arrangement also leads to Coulomb attraction and a bonding between the modes.
The electric near-field in Fig. 5(b) can thus be neatly explained as a coupling between the hexapolar modes of the inner and the outer nanoring. Since both modes have the same symmetry, there will be a large spatial overlap between them. Each charge of the outer ring faces an opposite charge at the inner ring at this wavelength, leading to a near-field enhancement at each spot and to a bonding between the modes. However, the charges of the inner nanoring have to have the opposite sign compared to the corresponding charge of the outer ring to enable this coupling, meaning that the dark mode of the inner ring oscillates with a phase shift of 180$^{\circ }$ compared to that of the outer nanoring. Simulating the surface charges of the two nanorings at a wavelength of 904 nm (see Supplement 1 Fig. S5) confirms this behaviour.
Understanding the behaviour seen in Fig. 5(a) is more difficult however. The electric near-fields between the two nanorings are influenced by the coupling between the decapolar mode of the outer ring and the hexapolar mode of the inner nanoring. Here there are only four pairs of charges with an opposite sign, these can be found at the top and the bottom of the two nanorings where they cause the four strong near-field spots. There are also two pairs of surface charges where the sign of the charge is equal (found at the near-field spots adjacent to the nanodiscs). The resulting repulsive interaction then flips the surface charge of the inner nanoring.
The remaining four near-field spots are only weakly visible, these spots are caused by the surface charges of the outer nanoring that have no counterpart at the inner nanoring. They do not contribute to the coupling between the dark modes, but they do attract opposing charges in the inner nanoring through electric forces.
A simulation of the surface charges of the nanorings at a wavelength of 641 nm visualizes this behaviour and is shown in the Supplement 1 as well (see Fig. S6).
Based on the interactions between the surface charges of the nanorings, we can now estimate the coupling strength between the dark modes. The coupling strength between the decapolar mode and the hexapolar mode should be lower than the coupling strength between two hexapolar modes, as two identical dark modes overlap much better than two different ones. However, the bonding between the decapolar and the hexapolar mode does not require a phase shift of 180$^{\circ }$ between the two dark modes, as these modes appear to oscillate in phase.
The examination of the dual-ring structure has shown that its scattering behaviour changes clearly compared to the two single-ring systems. Investigating the electric near-field of this structure indicates that the dark modes interact and couple with each other through the interaction between their surface charges, leading to the dark modes having a stronger electric near-field that is confined between the nanorings.
3.2 Verification with the coupled oscillator model
In order to confirm the effect of this additional coupling mechanism between dark modes, we describe the nanostructures and their bright and dark modes as a system of coupled mechanical oscillators. The coupled-oscillator model was proposed by Joe et al. [5] to describe Fano resonances, but similar models were also used to describe strong coupling or electron-induced transparency [41,42]. Each oscillator in this model represents either a bright mode or a dark mode, with the coupling between bright and dark modes leading to a Fano resonance. The advantage of this approach is that a coupling between dark modes can be added and be deliberately enabled or disabled, which allows for a direct investigation of the effects of this coupling.
The investigation with the coupled-oscillator model will follow the same steps as the investigation of the nanostructure with simulations. At first the outer-ring and the inner-ring nanostructure will be described with the coupled-oscillator model, afterwards the two structures will be again combined into the dual-ring nanostructure.
The first step is now to describe the outer-ring nanostructure with the coupled-oscillator model. Our simulation of the outer-ring structure has shown that this system consists of one bright mode and two dark modes. This means that we need a system of three harmonic oscillators to describe this system. Oscillator 1 serves as the bright mode and is driven by an external oscillating force, representing the excitation by the incident light. Oscillators 2 and 3 represent the two dark modes of the outer-ring system, with oscillator 2 standing for the decapolar dark mode and oscillator 3 for the hexapolar dark mode (see Fig. 2). Dark modes cannot be excited by the incident light, which is why they are not driven by an oscillating force. However, there is a coupling between oscillators 1 and 2 and a second coupling between oscillators 1 and 3, which causes the Fano resonances. There is no coupling between oscillators 2 and 3 since they represent dark modes from the same nanoring, which prevents an interaction between them.
We now have a system of three equations of motion:
The left-hand side of these equations is a typical equation of motion of a damped oscillator, with a damping constant $\gamma _i$ and a resonance frequency $\omega _i$ for each oscillator. On the right-hand side of the equation are the accelerations that affect each oscillator. These are the external oscillator that drives oscillator 1 as well as the coupling between the oscillators. The strength of the coupling is described with the two coupling constants $\nu _{12}$ and $\nu _{13}$.
To analyze the behaviour of this system we have to select specific values for these constants. First we determine the resonance frequency $\omega _i$ and the damping constant $\gamma _i$ for oscillator 1. These parameters can be determined from the scattering cross section of the isolated nanodiscs by fitting a Breit–Wigner function to the scattering peak. This function is the absolute square of the complex amplitude of a driven harmonic oscillator and is defined here as:
Our previous scattering spectra were all plotted against the wavelength, but the wavelength can be easily converted to the radial frequency using the following equation.
This relation is also valid to find the resonance frequency of the plasmons, as they oscillate with the same frequency as their exciting light [1].
The parameters of oscillator 1 are now determined by fitting the Breit-Wigner function to the scattering cross section of the two isolated nanodiscs. This results in the following fit parameters.
The parameters of oscillators 2 and 3 are determined by placing oscillating dipoles near the nanoring and measuring the increased power emission (see Supplement 1 Fig. S2-S4). Plotting the emitted power against the oscillation frequency of that dipole results in a curve that has peaks for each dark mode. Fitting a Breit-Wigner function to these peaks results in the following parameters:
The amplitude of the driving oscillator $a_1$ does not influence the resulting lineshape of this model, only its amplitude. It is set in the order of $10^{30}\ \frac {1}{s^2}$ to prevent numerical errors during calculations.
The coupling strength between the bright modes and the two dark modes cannot be quantitatively derived from previous simulation results. A higher coupling strength results in stronger Fano resonances, leading to more pronounced peaks and dips in the lineshape (see Supplement 1 Fig. S7 and S8). It is therefore possible to estimate the coupling strength from the simulated scattering cross sections, which results in:
The only parameter that is not specified at this point is the angular frequency of the driving force $\omega _0$. This parameter is calculated from the driving wavelength $\lambda _0$ using Eq. (3) as well.
After defining the equations of motion as well as their parameters, we can now describe the behaviour of this system for different wavelengths $\lambda _0$, with $\lambda _0$ serving as the excitation wavelength of the incident light in this model. The actual motion of the coupled oscillators can be calculated with the scipy package for python. A forward Euler approach [32] is used to describe the motions of the system for more than 100 oscillations, at this point all oscillators have reached their steady state amplitude |$c_i\ (\lambda _0)$|. Only the steady-state amplitude of oscillator 1 ( |$c_1\ (\lambda _0)$| ) is actually relevant for the comparison to the simulations. This is due to the fact that only the bright mode can be directly excited by light, therefore reciprocally only the bright mode can actually emit radiation into the far-field [43]. Here we use the absolute square of the steady-state amplitude |$c_1\ (\lambda _0)$|$^2$ as a comparison to the cross section, as $c_1$ can be treated analogous to an oscillating dipole moment.
Figure 6(a) shows the absolute square of oscillator 1 at different excitation wavelengths. Comparing the results of the coupled-oscillator model of this figure to the simulated scattering cross section seen in Fig. 6(b), one can see that the spectral positions and the lineshape of the Fano resonances coincide relatively well with each other.
Fig. 6. a) Absolute square of the steady-state amplitude of the coupled oscillator model that describes the outer-ring system. b) Scattering cross section of the outer-ring structure, shown for comparison.
The same approach is now repeated for the inner-ring system. The first step is again to set up an equation of motion. The previous simulation of the inner-ring system has shown only a single recognizable Fano resonance that has to be considered in addition to the previous bright mode. Thus, we need a system of two coupled oscillators, one bright mode and one dark mode.
Oscillator 1 is again the bright mode and oscillator 4 is the hexapolar dark mode of the inner-ring system. The previous parameters of oscillator 1 are unchanged. The resonance frequency and the damping constant of oscillator 4 can again be found by placing an oscillating dipole close to the nanoring and fitting the Breit-Wigner function to the resulting peak (see Supplement 1 Fig. S4), which results in the following parameters:
The coupling strength between the oscillators 1 and 4 has to be lower than the previous coupling strength between oscillators 1 and 2/3. This is due to the higher distance between the inner nanoring and the nanodiscs compared to the distance between the nanodiscs and the outer nanoring. The coupling strength is now estimated to be:
All new parameters have thus been assigned values as well. The new steady-state amplitude of oscillator 1 can now be investigated again by simulating the movement of the oscillators. Fig. 7 shows that the coupled-oscillator model of the inner-ring system can replicate the general position and lineshape of the Fano resonance of its corresponding structure as well.
Fig. 7. Absolute square of the steady-state amplitude of the coupled oscillator model that describes the inner-ring system. b) Scattering cross section of the inner-ring structure, shown for comparison.
The final step is now to describe the entire dual-ring nanostructure with the coupled-oscillator model. This can be done by combining the equations of motion of both the outer-ring system and the inner-ring system. This results in a system of four coupled oscillators. Oscillator 1 is again the bright mode that is driven by an external force, while oscillators 2 and 3 represent the decapolar and hexapolar dark modes of the outer nanoring, and oscillator 4 is the hexapolar dark mode of the inner nanoring.
As before there is a coupling between the bright mode and the three dark modes, but in addition we now assume that there may be a coupling between the dark modes of the inner and the outer nanoring. The dual-ring system can thus be described with the following equations of motion:
All previously used parameters of this equation remain unchanged. The new coupling between the dark modes is described with the coupling parameters $\nu _{24}$ for the coupling between the decapolar mode and the hexapolar mode and $\nu _{34}$ for the coupling between the two hexapolar modes. We can estimate these parameters based on the discussion of the coupling strength between dark modes for Fig. 5. Thus, we set the parameters to:
The negative sign of $\nu _{34}$ ensures that the two oscillators have a phase difference of 180$^{\circ }$ as the two hexapolar modes do (see Supplement 1 Fig. S5). On the other hand $\nu _{24}$ has a positive sign as there is no phase shift for the coupling between a hexapolar and a decapolar mode (see Supplement 1 Fig. S6). The lower absolute value of $\nu _{24}$ compared to $\nu _{34}$ denotes the lower coupling strength between the decapolar and the hexapolar mode compared to the coupling strength between two hexapolar modes.
The steady-state amplitude of oscillator 1 is now calculated again. For comparison, we also calculate the steady-state amplitude of a second system of oscillators without any coupling between the dark modes. This is done by setting $\nu _{24}$ and $\nu _{34}$ to zero.
The resulting lineshapes for the coupled and the uncoupled oscillators are compared in Fig. 8. The oscillator model without any dark mode coupling has a total of four visible peaks, while the oscillator model with coupling between dark modes has only three. It is clear that by adding a coupling between dark modes the small peak at $\lambda \approx$ 650 nm disappears. Additionally, the coupling between dark modes also makes the leftmost peak more pronounced, while the rightmost peak gets redshifted and diminished. The central peak is similarly redshifted by roughly 30 nm. All of these changes are in agreement with the simulated scattering cross section of the dual-ring structure (see Fig. 4), indicating a coupling of dark modes in that structure. Still, there are certain differences to the cross section of the dual-ring structure, like e.g. the lack of background scattering at higher wavelengths, but they are caused by the relative simplicity of the coupled-oscillator model. Nevertheless, the results from the coupled-oscillator model support that a coupling between the dark modes occurs in the dual-ring structure.
Fig. 8. a) Absolute square of the steady-state amplitude of the coupled oscillator model that describes the dual-ring system without coupling between the dark modes. b) Absolute square of the steady-state amplitude of the coupled oscillator model that describes the dual-ring system with coupling between the dark modes.
3.3 Sensing performance
Now that a coupling between dark modes was evaluated, we investigate how the sensing performance of the dual-ring nanostructure compares to that of a single-ring nanostructure. This is done by calculating the scattering cross section of the nanostructures with three different surrounding media that have refractive indices of 1 (air), 1.33 (water) and 1.48 (PMMA). However, these changes do not affect the glass substrate, only the upper medium that surrounds the nanostructure. Increasing the refractive index of that medium causes a redshift of the Fano resonances and their respective scattering peaks [1]. These shifts of the scattering peaks can be used for sensing purposes. To investigate the effect of adding an additional nanoring, we compare the shift of a Fano peak of the dual-ring structure with the shift of similar Fano peaks of the outer ring-structure and the inner-ring structure. The investigated Fano peak of the dual-ring system appears at $\lambda \approx$ 750 nm, while the Fano peak of the outer-ring system appears at $\lambda \approx 670$ nm and that of the inner-ring system appears at $\lambda \approx 700$ nm.
The sensitivity of these peaks is now calculated by determining the position of each peak for each refractive index. The sensitivity $S$ is found by applying a linear fit to the peak positions. The slope of the linear fit is the sensitivity $S$ and has the unit nm/RIU (refractive index unit). Fig. 9 compares the sensitivity of the three scattering peaks. One can see that the fit of the dual-ring peak positions has a steeper slope than the fits of both the outer-ring peak positions and the inner-ring peak positions, which shows that the dual-ring scattering peak has a higher sensitivity than the corresponding single-ring peaks. The sensitivity of the dual-ring scattering peak is 305 nm/RIU while that of the outer-ring scattering peak is 196 nm/RIU, and that of the inner-ring scattering peak is 240 nm/RIU. Adding a second nanoring to the nanostructure has therefore increased the sensitivity by 56 % and 27 %, respectively.
Fig. 9. Scattering cross section of a) the dual-ring structure, b) the outer-ring structure, and c) the inner-ring structure for different refractive indices. The arrows highlight the peaks that are investigated. d) Calculation of the sensitivity of the scattering peaks with linear fits. The positions of the scattering peaks are shown as dots, the linear fit is shown as a line. The outer-ring peak is shown as black dots and line, the inner-ring structure is shown as red dots and line and the dual-ring peak is shown as blue dots and line.
Now that the sensitivity has been calculated, the figure of merit (FOM) of these peaks can be determined as well by dividing the sensitivity of each peak by its full width at half maximum (FWHM) [40]. This results in a FOM of 2.31 for the outer-ring structure, a FOM of 1.74 for the inner-ring structure and a FOM of 2.48 for the dual-ring structure. These values are comparable to those of other LSPR sensors, while much higher FOMs could be achieved with optimized geometries [44]. It also shows that the FOM of the dual-ring structure is improved over those of the two single-ring structures, especially the inner-ring system, confirming that adding a second nanoring improves the general sensing performance.
4. Conclusion
In this paper we investigated a composite nanostructure consisting of two nanodiscs and two concentric nanorings. Both nanorings are able to independently provide dark modes for Fano resonances. Using theoretical models like FDTD simulations and the coupled-oscillator model, we have shown that a coupling between dark modes of separate nanostructures occurs in addition to a coupling between bright and dark modes. This coupling was only shown for ring-disc nanostructures, but it is also possible to apply this coupling model to other nanostructures with plasmonic Fano resonances, simply by placing two separate sources of dark modes in close vicinity.
The coupling of dark modes with each other opens up a new way to manipulate the lineshape and the position of a Fano resonance by adding separate dark modes to the structure instead of changing the geometrical parameters of the nanostructure.
Another effect of the coupling between dark modes is the enhanced sensitivity of the Fano resonances, which is advantageous for chemical and biological sensing using plasmonic nanostructures. Moreover, the coupling between the dark modes of the nanorings has confined the enhanced electric near-field of these dark modes between the two nanorings while also increasing the strength of these near-fields. Enabling a coupling between dark modes can therefore be used to direct and to increase the near-field enhancement of a Fano resonance, which may be of interest for applications of plasmonic nano-antennas.
Funding
Ministerium für Wissenschaft, Forschung und Kunst Baden-Württemberg (Landesgraduiertenförderung); China Scholarship Council; Open Access Publication Fund of the University of Tübingen.
Acknowledgements
Maximilian Grimmer gratefully acknowledges financial support by the Landesgraduiertenförderung of the Ministry of Science, Research and the Arts Baden-Württemberg. Wei Tao gratefully acknowledges financial support by the Chinese Scholarship Council. The authors acknowledge support from the Open Access Publication Fund of the University of Tübingen. The authors also thank Lukas Lang for the useful discussions during the preparation of this paper.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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