1. Introduction

Nanoparticles (NPs) have the ability to manipulate the light scattering and optical field at the far- and near-fields, which is elementary to nanoscale photonic devices [1,2]. Thus, it is of significant importance to gain a deep insight into the different resonant phenomena and understand their connection for the successful design of photonic devices [3–5]. In recent years, Fano resonances have attracted enormous attention in nanophotonics due to their sharp features in the spectra [6,7]. As a universal phenomenon exhibiting a characteristic asymmetric spectral lineshape, it has been observed in a wide range of systems, such as quantum field [8], topological structure [9], metallic [10–13] and semiconductor nanostructures [14], and microwave antennas [15]. This attractive feature shows numerous potential applications including lasing [16], optical switch [17], and nanoantenna with high directivities [18]. It is well known that the Fano lineshape resonance is excited by the interference between a broad continuum state (CS) and a narrow discrete state (DS). The coexistence of CS and DS modes in a certain spectral range results in the resonant phenomenon with the conventional asymmetric Fano lineshape, and the sharp phase change guarantees the near vertical spectral transit from peak to dip. Depending on the situation in plasmonic systems, DS can be introduced by various methods like the grating effect, a dark plasmonic mode, or symmetry-breaking effect like geometric asymmetry [3–7]. Hence, a large research effort has been devoted to design a complex plasmonic nanostructure for subtly tuning the Fano resonance. An individual NP can also support Fano resonance as well like metallic nanorod [19], and core shell nanostructures [20]. Actually, this is a classical topic in optics like the interference phenomenon from the Mie scattering of dielectric objects [21,22]. When the size of NP is much smaller than the light wavelength, so-called quasistatic or Rayleigh approximation, the scattering performance shows uniformity in the forward and backward directions due to the induced dipole moment [1,2]. When the size of NP breaks the Rayleigh approximation, retardation effect has to be included to analyse the light scattering of NP [5]. Due to the phase retardation effect inside the large NP, higher modes like quadrupole and octupole appear. The constructive and destructive interferences between the narrow higher modes as the DS and the broad dipolar mode as the CS thereafter result in the fascinating Fano resonance and the asymmetric forward scattering (FS) and backward scattering (BS). Meanwhile, the introduction of dielectric substrate beneath the NP, a general case in the nanophotonic experiment, will lead to a symmetry-breaking effect of the dielectric environment surrounding NPs [23]. Such effect has a significant impact on the scattering properties compared to the one embedded in a homogeneous medium. Therefore, it is of great importance to understand the influence of supporting substrates for the design of various NP-based devices like solar cell and directional lighting [24,25]. Recently, evidence has been reported on its influence on the plasmonic resonances of NPs: under the external optical excitation, the dielectric substrate is polarized and then induces a non-uniform electric field across the metallic NP, which can excite the out-of-plane (perpendicular to the dielectric substrate) multipolar plasmonic modes [26–30]. Many of these reports base their understanding on the hybridization theory: the dielectric substrate mediates the interactions among the different plasmonic modes of metallic NP and gives rise to their hybridization into a bonding mode and an anti-bonding mode.

In our previous reports, we revealed the out-of-plane quadrupolar mode Q around 250 nm of Al nanodisk (ND) on the quartz substrate in air, which is very sensitive to the oxidation and top surface roughness [31–35]. To be precise, the light is incident from z direction with x polarization, and the surface of substrate is xy plane, thus here the out-of-plane means xz plane, and the in-plane means xy plane. Obviously, due to its nature of out-of-plane multipolar mode, it is also readily influenced by the height of ND. Surprisingly, this effect of height on the optical properties of ND has been rarely reported in the nano-optic field up to now, and it is widely neglected in designing the nanostructures [36–38]. Here, the Fano resonances in the FS and BS spectra of single Al ND in the symmetric (environment of refractive index (RI) 1 and 1.46) and asymmetric (on a quartz substrate in air) environment are investigated by finite difference time domain (FDTD) simulations. By tuning the height of Al ND, mode Q arises even in a uniform environment due to the retardation effect. A detailed spectral analysis is done by using a two-pole Fano model to fit the Fano lineshape scattering spectra and quantitatively analyse the Fano interference between the quadrupolar and dipolar modes. Furthermore, the transmission spectra of periodic Al ND arrays with different heights and pitches from the simulation and experiment are compared to confirm the influence. The appearances of high modes for the Al ND in the asymmetric/symmetric environment confirm the retardation effect in different dielectric environments and inside the ND with different heights.

2. Methods

FDTD Simulation: FDTD simulations were done with a commercial software (Lumerical FDTD solutions). For the simulation of a standalone Al ND, Total Field/Scattered Field (TFSF) source with polarization along x direction in the wavelength range from 200 to 700 nm was used with perfectly matched layers (PML) at x, y and z directions. Al ND was illuminated at normal incidence (z direction) from the top side in the symmetric/asymmetric environment, as illustrated in Figs. 1(a)-(c). RI (n) of 1 for air and 1.46 for quartz were used to create a symmetric environment for the Al ND in air (denoted as Sym., Fig. 1(a)) and in quartz (denoted as Sym.(n), Fig. 1(c)) and an asymmetric environment (Al ND on a quartz substrate in air, denoted as Asym., Fig. 1(b)). In contrast to the simplified model of Al ND in our previous researches [31–35], Al ND of diameter (D) 100 nm with increasing height (H) from 20 to 100 nm was modelled by a fully closed uniform 4 nm Al₂O₃ layer to eliminate the possible asymmetrical influence on the FS and BS, as shown in Fig. 1(a). The scattering efficiency was obtained by the scattering cross section normalized by the geometrical cross section πr² (r = 50 nm). The nearfield electric field distributions (${(\vec{E}/{\vec{E}_0})^2}$, where $\vec{E}$ and ${\vec{E}_0}$ stand for the local and incident electric fields respectively) were recorded in the xz plane where y = 0 nm, and the charge density (ρ) distribution was calculated using the electric field from simulation and Gauss’s law $\rho = {\varepsilon _0}\nabla \bullet \vec{E}$ where ε₀ stands for the permittivity of the vacuum. For the periodic arrays, the plane wave source was employed with PML at z direction and periodic boundary conditions at x and y directions. The period was varied at both x and y directions from 200 to 320 nm at a step of 20 nm. Dielectric functions of different materials were all taken from the database of the software (Palik data for Al and Al₂O₃), and calculations were done with 1 nm minimal mesh size in the modelling region.

 

Fig. 1. Schematics illustrating the simulation setup to record FS and BS spectra of single Al ND as a function of height (H) in the symmetric ((a) n = 1 denoted as Sym., (c) n = 1.46 denoted as Sym.(n)) and asymmetric ((b), denoted as Asym.) environment. The inset indicates xz plane, and wave vector (k) along z direction with polarization (E) along x direction. The simulated FS (bottom panel) and BS (up panel) spectra of single Al ND with varying H in corresponding environments are shown respectively in (d) Sym., (e) Asym., and (f) Sym.(n) environment. The shaded gray lines represent the fitted spectra using two-pole Fano model, while the parameters are tabulated in Appendix Table 1. In Appendix Fig. 7, the same data as in (d)-(f) are plotted in the vertically-shifted way for better comparing the intensities between FS and BS for each H.

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Experiment: Al ND arrays with different pitches from 200 to 320 nm in a step of 20 nm were fabricated on the quartz substrates by electron beam lithography. Al films of thickness 20 and 100 nm were evaporated on the nanostructured resist on the quartz substrate in a chamber of 5×10⁻⁵ Torr. The oxidation effect and long-term stability of Al ND were reported in our previous research [31–35]. The transmission spectra were measured on an ultraviolet–visible–near-infrared spectrophotometer (V770, JASCO) with an unpolarised halogen lamp, while the reference was taken on the area without NDs. To create a uniform dielectric environment of n = 1.46, the immersion oil was used with another quartz coverslip. The wave vectors of incident light in the experiment are in accordance with Figs. 1(b)-(c).

3. Results and discussion

Figure 1 shows the calculated FS and BS spectra of an isolated Al ND with varying H in air (Fig. 1(d)), on quartz substrate in air (Fig. 1(e)), and in the environment of n = 1.46 (Fig. 1(f)). When Al ND is located in a symmetric environment of n = 1 (Figs. 1(a) and (d)), the BS and FS spectra show a broad dipolar resonance (mode D) when H varies from 20 to 40 nm. Meanwhile, the overlap of BS and FS spectra proves the uniform scattering over the space (as shown in Appendix Fig. 7(a)). However, the dipolar resonance undergoes a blueshift from λ = 386 to 345 nm due to the coupling effect between the higher modes and the dipolar mode, which is more evident when H increases over 50 nm. As a result of the enhanced phase retardation effect with increasing H, an out-of-plane quadrupolar mode (mode Q) beyond the in-plane dipolar mode (mode D) starts to emerge around λ = 250 nm as shown in Fig. 1(d). The origination and proof of these two modes can be referred to our previous report [33]. Although the higher modes are not strong enough to appear in the scattering spectrum when H is below 50 nm, the FS and BS spectra display a slightly asymmetric resonance when H increases from 30 to 50 nm. When H is over 50 nm, FS spectra with higher intensities start to separate with BS spectra at the shorter wavelength side of mode D peak (below 300 nm), as better shown in Appendix Fig. 7(a). In the FS spectra, mode Q arises about λ = 250 nm when H = 50 nm and it redshifts to about 278 nm when H increases to 100 nm. In detail, an equal scattering intensity with mode D is rapidly reached by mode Q at H = 70 nm in the FS spectra. With even larger H, mode Q is strong enough to submerge the peak of mode D in the FS spectra, and in this case the Fano dip becomes unapparent. In the BS spectra, mode Q is only distinct from λ = 246 nm (H = 80 nm) to 252 nm (H = 100 nm). As for the longer wavelength side of the mode D peak, the FS and BS spectra keep overlapping with each other. When Al ND is located in a uniform dielectric environment with higher RI n = 1.46 (Figs. 1(c) and (f)), similar evolutions on the FS and BS spectra are observed as in Fig. 1(d). However, redshift and broadening occur for both mode Q and D due to the stronger retardation effect in the dielectric environment with higher RI [1,2]. Meanwhile, the higher modes beyond quadrupole appear below λ = 300 nm, as shown in Figs. 1(f) and 7(c).

When the Al ND was placed in the asymmetric environment, i.e., on the quartz substrate in air (Figs. 1(b) and (e)), FS spectra already show a difference with BS spectra on the scattering intensity even when H = 20 nm in spite of the similarity on the spectral lineshapes (Fig. 7(b)). Meanwhile, the BS spectra always show a lower intensity than the FS spectra regardless of the height of Al ND. In general, plasmonic NP placed on the dielectric interface prefers to scatter more light into the higher RI side because of its greater optical density of state [24]. The existence of quartz substrate reinforces the phase accumulation at the interface of ND and substrate, thus FS and BS spectra exhibit the difference even H is as low as 20 nm. In the BS spectra of Fig. 1(e), a small bump (mode Q) appears at around λ = 235 nm for H = 30 nm as shown by the corresponding inset of Fig. 7(b), then two distinct peaks can be observed with increasing H. Finally, it develops into a prominent asymmetric peak at around λ = 280 nm in BS spectrum for H = 100 nm. For the FS spectra, mode Q is only strong enough from λ = 262 nm (H = 50 nm) to 310 nm (H = 100 nm), and two peaks in the FS spectra show a comparable intensity when H = 80 nm. When H increases over 80 nm, the mode D is so weak that it emerges in the longer wavelength side of mode Q in the FS spectrum.

In all the three dielectric environments, the higher mode Q with a narrow width plays as the DS, and the dipolar mode D plays as the CS, thus resulting in the Fano resonance in the scattering spectra. The interference effects between the two modes can be observed and proven by the differential FS and BS spectra. Additional peak around λ = 200 nm arises partially due to the oxide layer with the higher RI value as shown by the spectra with large height (H ≥ 80 nm) in Figs. 1(d)-(f), which is also demonstrated in our previous report [33]. As seen in Figs. 1(d)-(f), the broadening in linewidth for both modes with increasing H arises from the radiation damping that is proportional to the volume of the ND and dissipation, and the retardation effect that is stronger for large NP and in higher RI environments [1,2,39,40].

The ratio between FS and BS is calculated and plotted in Fig. 2. The asymmetric peaks in Figs. 2(a)-(c) demonstrate that at shorter wavelengths the scattering is dominated by FS (FS/BS ratio is mainly above one), and after reaching the maximum of FS/BS ratio the value approaches to be constant around 1 at the longer wavelengths. Obviously, it is a well-known phenomenon in the Mie theory that the retardation effect in the large sphere will greatly modify the scattering into forward direction [1,2]. It is noted that the wavelength of the maximum of FS/BS ratio occurs near the maximum of FS and the Fano dip of BS, thus the scattering directionality is greatly enhanced so that the forward direction is largely dominant. A prominent peak arises in the FS/BS ratio spectra and it is continuously redshifted and enhanced as a function of H in Figs. 2(a)-(c). At H = 100 nm, the maximum of FS/BS ratio shows about 10 at around λ = 280 nm for the Al ND in air (Fig. 2(a)), and it reaches about 17 at around λ = 336 nm for the Al ND on the quartz substrate in air (Fig. 2(b)). As for the Al ND in the environment of n = 1.46 (Fig. 2(c)), it displays around 10 at λ = 382 nm with another peak at λ = 200 nm. It is noted that in recent years the generalized Kerker effect has been under intensive investigation to obtain the directional scattering from either dielectric or metallic NPs [37,41]. Our result of enhanced FS/BS ratio demonstrates that the asymmetric environment and tall height of NP can be a simple routine to manipulate the directionality of scattering, thus it will be beneficial for preferred applications like plasmonic-enhanced photoluminescence of a phosphor [25].

 

Fig. 2. FS/BS ratios of single Al ND in the (a) Sym., (b) Asym., and (c) Sym.(n) environment calculated from the corresponding spectra of Figs. 1(d)-(f). Inset shows the zoomed spectra in the shorter wavelength range.

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To provide a deep insight into the Fano interference, the scattering spectra (S(E)) of single Al ND as plotted in Figs. 1(d)-(f) are further analysed by a phenomenological two-pole Fano interference model [10,11,33]

The peak information extracted from Figs. 1(d)-(f) and corresponding fitting parameters are plotted in Fig. 3. The energies of mode Q and D are shown separately in Figs. 3(a) and (b). For mode Q (Fig. 3(a)), it displays a continuous redshift with increasing H in all the situations. As for mode D (Fig. 3(b)), the peak position in the FS spectra (blue, orange and red solid lines with circular markers) displays a continuous shift to higher energy, while in the BS spectra (blue, orange and red dashed lines with circular markers) it firstly displays a blueshift and then a redshift from H = 50 nm. The fitted parameters of mode D shows the similar tendency, except the one from FS of Sym. (Cyan solid line with circular marker in Fig. 3(b)) undergoing a redshift from H = 70 nm and the one from BS of Asym. (green dashed line with circular marker in Fig. 3(b)) undergoing a continuous blueshift with increasing H. By comparing the mode energy in different dielectric environments in Figs. 3(a)-(b), a continuous redshift is observed as switching the environment from Sym. to Asym. and to Sym.(n). It is worth to note that the peaks of mode Q in the BS spectra (dashed line in Fig. 3(a)) show higher energies than the corresponding ones in the FS spectra (solid line in the same color in Fig. 3(a)). However, a different phenomenon is observed for peak D in Fig. 3(b). With lower heights, peak D is overlapped in the FS and BS spectra. From H = 40 nm, they start to separate with peak D of the FS located in the higher energy position, and this disparity is enhanced with increasing H. These result from the phase difference of the free charge oscillation at the top and bottom part of ND, thus the different interferences between the two modes are produced at the forward and backward directions.

 

Fig. 3. Information extracted from the FS and BS spectra in Fig. 1(d)-(f) and corresponding fitting parameters: peak positions of mode Q (a) and mode D (b), positions of Fano dip in the FS and BS spectra and maximum of FS/BS ratio (c), and calculated value of q (d). Corresponding legends are provided on the top.

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As depicted in Fig. 3(c), the maximum of FS/BS ratio firstly experiences a shift to the higher energy and afterwards to the lower energy from H = 40 nm. Moreover, the strongest directionality (the maximum of FS/BS ratio) occurs around the wavelength of the Fano dip in the BS spectra. Figure 3(d) displays the calculated asymmetry parameter q, showing an increment with increasing H in both FS and BS spectra. When the two scattering pathways for mode Q and D are of similar magnitudes, i.e. q = 1, Fano lineshapes are most pronounced and the interference of two modes are relatively stronger. For Sym., the value q approaches one on the FS spectra at H = 100 nm (cyan solid line in Fig. 3(d)). However, it approaches one on the BS spectra of Asym. at H = 80 nm and saturated for taller NDs (green dash line in Fig. 3(d)).

The higher modes in the plasmonic ND are activated by the retardation of the incident light, and for the taller NDs the phase lagging between the top and bottom charge oscillation at normal incidence results in the out-of-plane higher modes shown in Fig. 1. To gain a deep insight into this performance, an integration of dipole moment is done by dividing the Al ND into four subsections from the center of Al ND, as illuminated in the schematic images of Figs. 4(a)-(c). From the xz plane charge distribution in the top (the green area in Figs. 4(a)-(c)) and bottom (the orange area in Figs. 4(a)-(c)) quarter, the top and bottom dipole moments along x direction P_x as a function of wavelength are calculated with the charge density ρ by

 

Fig. 4. Schematics illustrating the top (green) and bottom (orange) area to calculate P_x in the (a) Sym., (b) Asym., and (c) Sym.(n) environment. Normalized P_x to the absolute maximum is plotted for H = 20 nm (d, e, f), 60 nm (g, h, i), and 100 nm (j, k, l) in different columns in accordance with three dielectric environments in rows. Green and orange lines represent the normalized P_x for the top and bottom subdomains, respectively. Red regions (beyond zero) and blue regions (below zero) represent the dominant positive and negative charge distribution respectively. The normalized FS (black solid line) and BS (black dashed line) spectra are provided for convenience. The charge distributions at a specific wavelength are plotted in Appendix Fig. 8.

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By doing this integration, we are showing the dominant charge distribution in the top and bottom part of Al ND, thus we are ignoring the more complex modes beyond the out-of-plane quadrupole. The calculated top and bottom dipole moments dependent on the illumination wavelength are shown in Fig. 4 with the charge-distribution mapping at a specific wavelength supplied in Appendix Fig. 8. When they have the same signs, the two dipoles oscillate in phase, while the opposite signs produce the out-of-phase oscillations with a π phase difference. Owing to the out-of-phase alignment between the top and bottom dipoles, they cancel each other partially and results in a small net dipole moment, causing a subradiant mode with out-of-plane quadrupolar distribution with reduced scattering intensity. The phase differences mainly come from the phase difference Δφ between the propagating incident light as the driving force and the dipole oscillation, and the spatial separation between the top and bottom dipoles brings the additional phase difference 2πnd/λ, where n is the RI, d is the distance between two dipoles (∼H/2) [18]. In addition, the BS experiences a π shift comparing to incident light. In the forward direction, the constructive interference dominates, while in the backward direction destructive interference is predominant. For Al ND of H = 20 nm in Sym. and Sym.(n) (Figs. 4(d) and (f)), the green and orange lines overlap almost perfectly, meaning that the top and bottom dipoles show an in-phase oscillation over the whole range of wavelength. Both the FS and BS spectra feature a series of symmetric peaks, and the charge distribution at the resonant wavelength clearly shows an in-plane mode D in Fig. 8. At the shorter wavelength, a complicated charge distribution is observed in Fig. 8. However, when Al ND located in an asymmetric environment (Asym.) mode Q appears in the short wavelength around 200-220 nm (Fig. 4(e)). Mirror charges on the substrate interface nearby the ND appears in Fig. 8 due to the attraction from the charge of Al ND. When H is increased to 60 and 100 nm (Figs. 4(g)-(l)), multiple wavelength ranges display the domination of the mode Q. If we evaluate between NDs with the same height in Figs. 4(g)-(i) and (j)-(l), the spectral region of mode Q becomes enlarged as the environment switches subsequently from Sym. to Asym. and to Sym.(n). Without doubt, the interface between air and the substrate has an important role on the scattering performance of ND due to Fresnel reflections and mirror charges on the interface. However, an effective uniform environment with mediated RI between n = 1 and 1.46 can be applied to roughly simulate the asymmetric environment, especially for the redshifted resonance and appearance of high modes from Sym. to Asym. environment. Therefore, we reveal that the symmetry-breaking effect by the dielectric substrate beneath the metallic NP is mainly a retardation effect.

For large height, retardation of incident electromagnetic field along the Al ND height results in the phase difference on the top and bottom charge oscillations, thus the higher modes including the mode Q with the out-of-plane (perpendicular to the dielectric substrate) distribution are excited beyond the in-plane mode D. Because of the subradiant nature of the quadrupolar modes Q, the radiative loss of hybridized mode Q is weak and results in the narrow width [2,40]. Therefore, the mode Q with a narrow width interferes with the mode D with a broad width, which gives rise to the Fano resonance on the scattering spectra. Even in a uniform dielectric environment (in air, Sym.), the Al ND with H = 60 nm (Fig. 4(g)) is sufficient to have a large phase retardation causing the discrepancy between the FS and BS. The interference effects in the FS and BS scattering spectra depend on the relative phases of the two modes into two directions. The phase lag of charge oscillation along the height of Al ND results in the non-uniform charge distribution along the height of Al ND, thus they radiate the asymmetric power in BS and FS. The enhanced retardation effect in the large ND make it worse, typically shown by Figs. 4(j)-(l). When the interference is predominantly constructive in the forward direction and almost exclusively destructive in the backward direction, the FS/BS ratio reaches around the maximum value [5]. The quadrupole charge distribution gradually modifies as the wavelength varies (Fig. 8). Beside the spatial separation, the phase difference also depends on the imaginary part of dielectric function of ND that the damping during the light propagation also causes phase lagging, therefore some ranges in the longer wavelength still shows the quadrupolar distribution in spite that it is out of resonance (Figs. 4(g)-(l) and Fig. 8).

The location of hot spots is a good indicator for the different modes [30,33], thus we did a similar integration on the electric field intensity (|E|²) around the Al ND, the integrated value EI is calculated according to

Comparing to the single Al ND, Al ND array is more realistic for the application and measurement in experiment. Therefore, periodic array of Al ND of H = 20 and 100 nm with a fixed diameter 100 nm were fabricated in different pitches to prove the retardation effect. The transmission spectra were measured respectively in Asym. and Sym.(n) environment, and the results are plotted in Fig. 6. ND array exhibits a sharp feature of Fano resonance because the scattering of one ND in the periodic array can be collected by the neighboring NDs, which brings the far-field coupling. The transmission spectra of periodic ND array at normal incidence are governed by the lattice mode equation

 

Fig. 5. Schematics illustrating the top (green) and bottom (orange) subdomain to calculate EI in the (a) Sym., (b) Asym., and (c) Sym.(n) environment. Normalized EI to the absolute maximum in the top (green line) and bottom (orange line) subdomains is plotted for H = 20 nm (d, e, f), 60 nm (g, h, i), and 100 nm (j, k, l) in different columns in accordance with the three dielectric environments in rows. The normalized FS (black solid line) and BS (black dashed line) spectra are provided for convenience. The electric-field intensity (|E|²) distribution at a specific wavelength can be checked in Appendix Fig. 9.

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Fig. 6. Transmission spectra of periodic Al ND arrays in the Asym. and Sym.(n) environments respectively with H = 20 nm (a, c) and 100 nm (b, d) as a function of array pitch (P) from 200 to 320 nm in simulation (top panel, Sim.) and experiment (bottom panel, Exp.). The gray lines represent the normalized FS spectra of the single Al ND from simulation that are inversed from peak to dip for the better comparison with the transmission spectra. Exemplified SEM images obtained at tilted angle 20° are shown on the two sides respectively for H = 20 nm (left) and 100 nm (right) and for P = 200, 260, and 320 nm. Scale bars show 250 nm.

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A more detailed description on the analysis of lattice modes and the transmission spectra of periodic Al ND array with a different height can be referred to our previous reports [25,31–34]. Transmission spectra of Al ND array with H = 20 nm in the Asym. environment (Fig. 6(a)) show the single broad dips, and the dip is narrowed with increasing pitch due to the lattice coupling. However, for the Al ND array with H = 100 nm a distinct difference can be found by comparing Fig. 6(b) with Fig. 6(a). Multiple dips and peaks appear in the transmission spectra (Fig. 6(b)) due to the retardation-induced higher modes, and characteristic Fano lineshapes appear owing to the grating effects where the wavelength can be described by fore-mentioned Eq. (4). If we create an uniform dielectric environment around the Al ND array (Sym.(n)), the remarkable characteristic lattice modes appear on the transmission spectra of Al ND array with H = 20 nm (Fig. 6(c)), meanwhile dips in the shorter wavelength range appears. These dips are enhanced when H increased to 100 nm (Fig. 6(d)), due to the coupling of lattice mode with the higher modes in the short wavelength side. The simulated results show a good similarity with the experimental transmission spectra in Figs. 6(a)-(d), while the discrepancies are ascribed to the experimental imperfections [31–35]. Beside more dips in the transmission spectra for taller Al ND (H = 100 nm), the main dips in the longer wavelength side are observed to be blueshifted, and this is true no matter the Al ND array is located in Asym. (Figures 6(a)-(b)) or Sym.(n) (Figs. 6(c)-(d)) environment. For example, the Al ND array of P = 200 nm in Asym. environment with H = 20 nm shows a main dip around 383 nm in the experiment (black line in bottom panel of Fig. 6(a)), while the one with H = 100 nm demonstrates the main dips around 345 nm (the one of Fig. 6(b)). When located in Sym.(n), they display as 424 nm (the one of Fig. 6(c)) and 386 nm (the one of Fig. 6(d)) respectively. This conflicts with our intuition that with increasing size of NP a redshifted resonance should be obtained. Indeed, with fixed height and increasing diameter of Al ND, we obtain the redshifted resonance [33]. However, with fixed diameter and increasing height of Al ND, we obtained the blueshifted resonance. This exactly demonstrates the retardation effect along the height of Al ND and coupling effect between higher modes and dipole mode with increasing height of Al ND, as we analysed in the previous part for the single Al ND.

4. Conclusions

To summarize, we have investigated the dramatic effects of height of Al ND on the near- and far-field properties in the symmetric/asymmetric dielectric environments. The illumination of an isolated Al ND with a sufficient height will excite a narrow out-of-plane quadrupolar mode owing to the phase retardation effect along the height of Al ND even in a uniform environment. The interference between this quadrupolar mode with the spectrally broad dipolar background continuum causes pronounced asymmetric Fano line profiles observed in the forward and backward scattering directions. The presence of dielectric substrate brings a non-uniform dielectric environment, and facilitates the appearance of Fano interference due to the enhanced phase accumulation and Fresnel reflection. Moreover, this Fano interference redirects the light into preferred forward direction at the shorter wavelength range. The evolutions of the out-of-plane quadrupolar mode and in-plane dipolar mode are revealed as a function of height in the symmetric/asymmetric dielectric environments. Furthermore, the simulated spectral change due to the retardation effect are verified experimentally by the Al ND periodic arrays. The counter-intuitive blueshifted main dips in the transmission spectra demonstrates the dominant role of retardation effect in the scattering performance of ND with large height. It is the ubiquitous situation in the nanophotonic research that the height of nanostructure is not optimised. Here, our investigation reveals its important role on the retardation effect to the appearance of out-of-plane high modes and the Fano interference, and it also affords a convenient opportunity to modulate the Fano resonance.

Appendix

Table 1. The fitting parameters from FS and BS spectra of single Al ND in the Sym. (Fig. 1(d)), Asym. (Fig. 1(e)), and Sym.(n) (Fig. 1(f)) environments.

View Table

 

Fig. 7. Simulated FS (solid line) and BS (dashed line) efficiencies of single Al ND in the (a) Sym., (b) Asym., and (c) Sym.(n) environments. The shaded gray lines represent the fitted spectra using two-pole Fano model.

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Fig. 8. Calculated charge distribution in xz plane (red and blue regions represent the positive and negative charges respectively) of single Al ND placed in different dielectric environments with H = 20 and 100 nm, while the corresponding wavelength is noted on the top of each mapping. The inset coordinate indicates xz plane.

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Fig. 9. Calculated electric-field intensity (|E|²) distributions of single Al ND placed in different dielectric environments with H = 20 and 100 nm. The black lines indicate the core-shell Al ND with a 4 nm oxide layer, while the substrate interface is also drawn in Asym. cases. Corresponding wavelength is noted on the top of each mapping, and the inset coordinate indicates xz plane.

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Funding

Ministry of Education, Culture, Sports, Science and Technology (17KK0133, 19H02434, 19K22058).

Acknowledgments

This work was partly supported by the Nanotechnology Hub, Kyoto University (JPMXP09F19NMC042).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are available.

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