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Abstract
Hyperbolic metamaterials (HMMs) exhibit rich optical nonlinear responses for the epsilon-near-zero (ENZ) and anisotropy. In this study, we extract the time-dependent change in the effective permittivity of an Ag nanorod array under femtosecond pulses pumping around its ENZ wavelength. The transmittance and transient absorption spectra measured by s- and p-polarizations are used in the extraction process. We experimentally confirm the existence of an ultrafast recovery process with a relaxation time of 0.24 ps in the transient absorption spectra. The calculation based on the extracted nonlinear effective permittivity indicates that the ultrafast signal originates from the superposition of two slower recovery processes, with relaxation times of 0.74 ps and 1.19 ps, respectively. The results indicate that when the responses of two nonlinear processes have different signs and recovery speeds, their superposition may cause faster signal recovery in the combined process than in the two individual processes.
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1. Introduction
The optical nonlinear characteristics of noble metal nanostructures have attracted wide-scale attention in recent years owing to their unprecedented potential for light control [1–4]. The surface plasmon polarizations of the nanostructures feature an active path for the modulation of the optical characteristics of samples [5,6]. Plasmonic materials reveal significant and ultrafast nonlinear optical responses after intense excitation by femtosecond pulses, and have many applications in nanophotonics, such as all-optical switches and optical chips [7–10]. Hyperbolic metamaterials are metamaterials with hyperbolic dispersion [11]. It reveals many interesting nonlinear optical characteristics owing to its unique dispersion and high anisotropy [12,13]. The ENZ wavelength may shift when excited by femtosecond pulses. Utilizing this phenomenon, HMMs can realize interesting applications, such as ultrafast polarization conversion [14].
The nonlinear signals of noble metals are governed mainly by the recovery of hot electrons. The recovery processes of hot electrons in noble metals after the pumping of femtosecond pulses mainly consist of two phases: electron–electron (e–e) scattering predominates in the first few hundred femtoseconds and can only be probed in well-designed structures or under specific detection conditions [15,16], and in the latter phase, electron–phonon (e–p) scattering occurs. These processes often occur within a few picoseconds [15,17,18]. The durations of the two processes mainly depend on the state of electrons in metals that cannot be affected by the detected conditions as such. However, some recent reports have found some unusual ultrafast recovery signals under certain specific probe conditions [19,20]. These phenomena are often attributed to ultrafast surface plasmon resonance (SPR). SPR is directly related to the effect of the permittivity of samples. In addition, different types of electromagnetic resonances may exist in metal nanostructures. Their nonlinear responses may superimpose and generate unusual nonlinear optical signals [21,22]. The effective permittivity can be used to model these resonances and trace the origin of nonlinear signals. Therefore, extracting the dynamic change in the effective permittivity of metal nanostructures facilitates analyzing and controlling these phenomena.
The two methods proposed in recent years enable the extraction of the dynamic change in permittivity of HMMs. First, some reports successfully extracted uninterrupted accurate χ(3) spectra of plasmonic materials from their transient absorption spectra and linear permittivity. These materials include Au, Ag, and TiN nanoparticles [23–25]. This method successfully solved the interference of the strong absorption and dispersion of SPR on the experimental results. However, this method has not been extended to anisotropic materials or used to extract time-dependent changes in permittivity. Therefore, this technique cannot be directly used for extracting the nonlinear parameters of HMMs. Besides that, this method needs the linear effective permittivity of the sample. Therefore, a technology to extract the linear effective permittivity of HMMs are needed. Second, some studies modeled the linear effective permittivity of HMMs using B-spline parameters. Thereafter, they were extracted from the ellipsometry parameters. This method can accurately extract the linear permittivity of HMMs [26,27]. In our previous report, the complex permittivity of Ag nanorod arrays (a type of type-I HMM) was successfully extracted by this method [28].
In this study, we combined and extended these two methods to extract the time-dependent change in the permittivity of HMMs under a pump of femtosecond pulses near the ENZ wavelength for the first time. We modeled the in-plane and out-of-plane linear effective permittivity of Ag nanorod arrays by B-spline-free perimeters and subsequently extracted them from the transmission spectra for s- and p- polarization. Thereafter, we performed the transient absorption spectrum in both s- and p- polarization and extracted the time-dependent changes in the in-plane and out-of-plane permittivity during the recovery process. We found that an ultrafast signal occurred in the transient absorption spectrum. Subsequently, we demonstrated the use of the extracted nonlinear permittivity to calculate the separate effects of resonances in Ag nanorod arrays and trace the source of the ultrafast signal. It is revealed that the recovery speed of nonlinear signals may be different from the recovery speed of the nonlinear effective permittivity. In addition, nonlinear responses generated by two different resonances may combine into one ultrafast signal when the two nonlinear optical responses have different signs and recovery speeds.
2. Representation of the sample
An Ag nanorod array surrounded by Al2O3 was selected for analysis in this study. The Ag nanorod arrays in our experiment were fabricated by sputtering deposition in a co-sputtering system (Jsputter8000, ULVAC Inc.). A radio frequency (RF) system was used for the plasma-assisted deposition. The sputter power of the Ag target was 10 W, and that of the Al2O3 target was 120 W. The power of the RF substrate bias was 45 W. The detailed fabrication process and theory are reported in [29]. The TEM image of the nanorod array is shown in Fig. 1 (a). A schematic of the nanorod array is shown in Fig. 1 (b). The diameter, distance and length of the nanorods are approximately 4 nm,7 nm and 150 nm, respectively. The volume fraction of silver is approximate 25.6% in the sample. The noble metal nanorod array exhibits two absorption peaks. The transmittance spectra for the s-polarization and p-polarization light of the sample in our experiment were measured using an ellipsometer at 25°, 30°, and 35°. For clarity, only the transmittance measured at 30° is illustrated in Fig. 2 (a). The positions of the two absorption peaks are approximately 370 nm and 515 nm, respectively.
Fig. 2. (a) Transmittance spectrum for s- and p-polarization incident light. The experimental result is shown by the solid line and the result calculated based on the extracted effective permittivity is shown by dots. (b) Extracted out-of-plane effective permittivity of nanorod arrays. (c) Extracted in-plane effective permittivity of nanorod arrays.
To quantitatively analyze the linear and nonlinear optical characteristics of the plasmonic nanorod arrays, samples were mostly treated as homogeneous, uniaxial films in the previous reports [30,31]. Noble metal nanorod arrays are type-I HMMs. The effective permittivity tensor of a type-I HMM can be expressed as follows:
3. Extraction of the dynamic change of effective permittivity
To obtain the dynamic change in the optical properties of the sample and extract its nonlinear effective permittivity, the change in transmittance near the ENZ position was measured by the transient absorption spectrum. The femtosecond pulses were pumped by a mode-locked Yb: KGW-based fiber laser (PHAROS, Light Conversion) and generated by an optical parametric amplifier (OPA, ORPHEUS, Light Conversion). The sample was pumped by s-polarization femtosecond pulses with a full width at half maximum of 190 fs. The wavelength was 400 nm, and the intensity was 14.2 GW/cm2. The probe pulses were a supercontinuum, and the wavelength range of 460 nm – 710 nm including the ENZ wavelength was detected in the experiment. The incident angle of the probe pulses was fixed at 30°, and there was a small angle between the pump and probe pulses. The experimental light path was the same as that in our previous work [21]. In the experiment, the permittivity of the metal in the sample changes while pumping the pump pulses, and recovers to the quasi-steady-state in several picoseconds after the pumping of the pump pulses. The pump and recovery processes lead to a dynamic change in the transmittance of the probe pulses. To distinguish the effect of ɛ⊥ and ɛ||, both the experimental results obtained with p-polarization and s-polarization probe pulses were measured, and are shown in Fig. 3 (a), (b). The changes in transmittance were represented by ΔOD = -log10(Tnol/Tlin), where the sign ‘nol’ represents the pumped state and the sign ‘lin’ represents the origin state. ΔOD probed with s-polarization pulses decreased with the wavelength, and remained larger than zero during the following recovery process. ΔOD probed with p-polarization changed from negative to positive with increasing wavelength. The minimum value of ΔOD appears at 502 nm that is slightly shorter than the wavelength of the linear absorption peak. Both results recover to the quasi-steady-state when the delay time is approximately 5 ps.
Fig. 3. ΔOD measured in experiment ((a), (b)) and ΔOD calculated with the extracted Δɛ ((c), (d)). (a), (c) are the result probed with s-polarization light and (b), (d) are the result probed with p-polarization light.
We obtained ɛ⊥ and ɛ|| in the extraction process for the linear permittivity. Thereafter, the Levenberg–Marquardt algorithm was used to determine the change in ɛ⊥ and ɛ|| from the experimental ΔOD. We marked the change in the complex permittivity as Δɛ|| = ɛnol ||-ɛlin || and Δɛ⊥ = ɛnol ⊥-ɛlin ⊥. Both ɛnol || and ɛnol ⊥ are delay time- and wavelength-dependent. The first step was to extract Δɛ||. The Levenberg–Marquardt algorithm was used to determine the optimal Δɛ|| to satisfy ΔOD measured by s-polarization probe pulses. Linear ɛ|| values were used as the original values. The result was a set of complex Δɛ|| dependent on the delay times and wavelengths. The permittivity can satisfy the change in the ΔOD spectrum probed by the s-polarization pulses. The following step was used to extract Δɛ⊥ based on the extracted ɛnol ||. The extracted ɛnol || was used as the in-plane permittivity of the film, and the Levenberg–Marquardt algorithm was used to determine the optimal Δɛnol ⊥ to satisfy ΔOD measured by p-polarization probe pulses. After the two steps, ΔOD measured at other delay times was used to extract Δɛ|| and Δɛ⊥ at other delay times. Consequently, the extracted effective permittivity was a complex tensor. The effective permittivity was dependent on both the wavelength and delay time. The ΔOD calculated using the extracted nonlinear permittivity is shown in Fig. 3 (c), (d) for comparison. The extracted complexes, Δɛ⊥ and Δɛ||, are displayed in Fig. 4.
Fig. 4. Change of the effective permittivity. (a) Change in the real part of the in-plane permittivity. (b) Change in the imaginary part of in-plane permittivity. (c) Change in the real part of the out-of-plane permittivity. (d) Change in the imaginary part of the out-of-plane permittivity.
The transient absorption spectrum measured using s-polarization pulses (shown in Fig. 3 (a)) is mainly affected by the change in the absorption characteristics of Ag. Consequently, a larger ΔOD measured using s-polarization light indicates a larger imaginary part of Δɛ||. Δɛ|| in Fig. 4 (b) remains larger than zero and continues to decrease with the wavelength. The absorption peak near 515 nm arises mainly from the ENZ resonance [32]. The effect permittivity of the sample can be expressed by an effective medium theory [30,32]:
4. Ultrafast process in transient absorption spectra
The transient absorption spectrum (shown in Fig. 3) exhibited an unusual ultrafast recovery process at 543 nm. As shown in Fig. 5 (a), (b), the relaxation time measured in the transient spectrum shows an obvious sensitivity to the polarization of the probe light. The kinetics measured using p-polarization probe light recovery were much faster than those measured using s-polarization probe light. Figure 5 (c) shows the change in the effective permittivity at 543 nm. To compare the recovery speed of the effective permittivity and transmittance, we fit the curves with exponential functions, as shown in Eq. (4).
where A represents the variables that include the change in the effective permittivity or transmittance. B is a constant that represents the intensity of the nonlinear process. t and t0 are the delay and starting times of the recovery process, respectively. τ is the relaxation time of the recovery process. C is a constant related to the quasi-steady state.Fig. 5. ΔT and Δɛ at 543 nm. (a). Change in transmittance measured using s-polarization probe light. Both the results obtained from the experiment and calculated by the nonlinear permittivity are shown. (b). Change in transmittance measured using p-polarization probe light. The legends in (a) are also applicable to (b). (c). Change of Re(ɛ||), Im(ɛ||), Re(ɛ⊥), Im(ɛ⊥). The normalized change of Re(ɛ||), Im(ɛ||), Re(ɛ⊥) and Im(ɛ⊥) are also shown in the insert of (c).
The relaxation times of the recovery processes measured using s- and p- polarization probe light (shown in Fig. 5 (a) and (b)) were 1.11 ps and 0.24 ps when fitted using Eq. (4), respectively. In contrast, the relaxation times of Re(ɛ||), Im(ɛ||), Re(ɛ⊥), and Im(ɛ⊥) (shown in Fig. 5 (c)) were 1.10 ps, 1.19 ps, 2.12 ps, and 2.03 ps when fitted by exponential functions, respectively. These relaxation times are showed in Table 1. The minimum time-resolved scale in our experiment was 0.1 ps. Therefore, if two relaxation times differ within 0.1 picoseconds, the two relaxation times can be considered to be the same. Consequently, the change of Re(ɛ||) and Im(ɛ||) have the same relaxation time. And the change of Re(ɛ⊥) and Im(ɛ⊥) have the same relaxation time. The normalized permittivity changes are also illustrated in the insert of Fig. 5 (c) to show the difference. The different causes of the absorption characteristics result in different relaxation speeds of ɛ|| and ɛ⊥. In addition, the recovery speed of ΔT may shift with the polarization of probe pulses. The recovery speed of the transmittance measured using s-polarized light is close to the recovery speed of ɛ||. However, the change in transmittance measured using p-polarized light shows no similarity to any of the four components of the effective permittivity. It is obvious that all the four components of the effective permittivity cannot independently support the ultrafast processes of hundreds of femtoseconds, as shown in Fig. 5 (b). Here, we used the extracted nonlinear permittivity to analyze the origin of the ultrafast process.
Table 1. The relaxation time of the contribution of effective permittivity to ΔT at 543 nm
5. Analysis of the ultrafast process
To analyze the origin of the ultrafast process, we calculated the contributions of ENP resonance and ENZ resonance to the recovery process using the extracted nonlinear permittivity. Both the change of the ENP resonance and ENZ resonance origin from the change of permittivity of metal in the sample. The contributions of ENP resonance refers to the contribution of ɛ||. And the contributions of ENZ resonance refers to the contribution of ɛ⊥. When a metal nanostructure is pumped, the energy of photons is absorbed by electrons via intraband and interband transitions, leading to a non-thermal distribution of electrons in silver. The process is almost instantaneous. The thermalization process occurs among non-thermal electrons whereafter, resulting in a Fermi–Dirac distribution with a defined electron temperature. The process may take over several hundreds of femtoseconds. Then the energy of electrons will be transferred to the surrounding lattice, eventually reaching a quasi-static state. This process may consume several picoseconds. The changes in electronic states may affect both intraband and interband transitions of electrons, resulting in a fluctuation in both the real and imaginary parts of the permittivity of metals, thereby affecting the resonances supported by the metal nanostructure [34]. The increase in the imaginary parts of permittivity of metals is the most significant contribution to the change in transmittances of metal nanorod arrays [35]. Most nonlinear responses of metal nanostructures originate from the same kinetics of metal described above. However, the ENP resonance and ENZ resonance supported by the nanorod arrays may show different responses to the same change of permittivity of metal at the same wavelength. Their responses may also shift with the characteristics of pump and probe pulses, such as the pump power and the incident angle of the probe light, resulting in a novel nonlinear phenomenon [21]. Nonlinear permittivity can be used to quantitatively analyze the separate effects of these resonances. The transmittance recovery can be factorized into four parts that are related to the real and imaginary parts of ɛ⊥ and ɛ||, as shown in Eq. (5).
Fig. 6. Contributions of each component of the effective permittivity to the transient spectrum measured using s-polarization probe light. (a), (b), (d), and (e) are the contributions of the real part of ɛ||, the imaginary part of ɛ||, the real part of ɛ⊥ and the imaginary part of ɛ⊥, respectively. (c) is the total contribution of ɛ||, and (f) is the total contribution of ɛ⊥.
Fig. 7. Contributions of each component of the effective permittivity to the transient spectrum measured using p-polarization probe light. (a), (b), (d), and (e) are the contributions of the real part of ɛ||, the imaginary part of ɛ||, the real part of ɛ⊥ and the imaginary part of ɛ⊥, respectively. (c) is the total contribution of ɛ||, and (f) is the total contribution of ɛ⊥.
The s-polarization probe light is mainly affected by the ENP resonance. As shown in Fig. 6, the change in ɛ||, or the ENP resonance makes the main contribution to the transient spectrum measured using the s-polarization probe light. The contributions of ɛ|| mainly originate from its imaginary part. The s-polarization probe light cannot excite the ENZ resonance. Consequently, as shown in Fig. 6 (d)–(f), ɛ⊥ does not contribute as such to the experimental results. In contrast, the p-polarization probe light is affected by both the ENP resonance and the ENZ resonance. As shown in Fig. 7, the experimental results measured using p-polarization probe light exhibit richer nonlinear characteristics than the results measured using s-polarization probe light because of the participation of the ENZ resonance. ɛ⊥ is the main contributor to the transient spectrum. The real part of ɛ⊥ is a greater contributor than its imaginary part. The ENP resonance and the ENZ resonance may superimpose and produce some unusual phenomena [21].
Here, we calculate the contribution of ɛ|| and ɛ⊥, or the ENP resonance and the ENZ resonance, to ΔT measured with s- and p- polarization probe pulses at a wavelength of 543 nm using Eq. (6). The results are shown in Fig. 8 (a), (b). As shown in Fig. 6, ɛ⊥ does not contribute as such when probed with s-polarized light. Therefore, as shown in Fig. 8 (a), ΔT measured with s-polarization pulses is almost equal to the contribution of ɛ||. The relaxation time of the contribution of ɛ|| to ΔT measured using s-polarization probe light is 1.19 ps. As shown in Fig. 6, the contribution of ɛ|| mainly originates from the imaginary part of ɛ||. Consequently, its relaxation time is approximately equal to the relaxation time of Im(ɛ||) (1.19 ps).
Fig. 8. Contributions of ɛ⊥ and ɛ|| of the effective permittivity at 543 nm. (a) is the contributions of ɛ⊥ and ɛ|| to the transmittance measured using s-polarization probe light. (b) is the contributions of ɛ⊥ and ɛ|| to the transmittance measured using p-polarization probe light. The normalized results are also shown. The legends in (a) are also applicable to (b). (c) is the contributions of Re(ɛ⊥), Im(ɛ⊥) and ɛ⊥ to the transmittance measured using p-polarized probe light. (d) is ∂ΔT/∂Re(ɛ⊥) and ∂ΔT/∂Im(ɛ⊥) when measured using p-polarized probe light.
As shown in Fig. 8 (b), the contribution of ɛ⊥ is greater than that of ɛ|| when probed with p-polarized light. The contribution of ɛ⊥ recovers faster than that of ɛ|| in the first 1 ps. The combined contribution of ɛ⊥ and ɛ|| remains almost unchanged after 1 ps. Consequently, the signal reaches a balance earlier than the electronic state, leading to an ultrafast recovery signal of ΔT with a relaxation time of 0.24 ps. There are three essential conditions for the combined process: 1. The two signals exhibit different signs. 2. The intensities of the two signals are comparable. 3. The recovery speeds of the two signals are different. The relaxation times of the contributions of ɛ⊥ and ɛ|| to ΔT measurements with p-polarization probe light are 0.74 ps and 1.15 ps, respectively. These relaxation times are also shown in Table 1. The relaxation time of the contribution of ɛ|| is approximately equal to the relaxation time of Im(ɛ||) (1.19 ps). However, the relaxation time of the contribution of ɛ⊥ (0.74 ps) is shorter than both the relaxation times of Re(ɛ⊥) (2.12 ps) and Im(ɛ⊥) (2.03 ps). This is because the change in Im(ɛ⊥) mainly leads to a change in absorption, and the change in Re(ɛ⊥) mainly leads to a shift in the absorption peak. Consequently, the relaxation times of the contribution of Im(ɛ⊥) and Re(ɛ⊥) are different. As shown in Fig. 8 (c), the signals of contribution of Im(ɛ⊥) and Re(ɛ⊥) are different. And the contribution of Im(ɛ⊥) recovers much faster at first. Consequently, as a superposition of the two contributions, the contribution of ɛ⊥ first drops rapidly until it does not change with time when the contribution of Im(ɛ⊥) and Re(ɛ⊥) reach a balance. The process makes the relaxation times of the contribution of ɛ⊥ different from both the contributions of Im(ɛ⊥) and Re(ɛ⊥).
For rigorous analysis, we further divide the contribution of ɛ⊥ into the contributions of Re(ɛ⊥) and Im(ɛ⊥). The results are presented in Fig. 8 (c). The relaxation times of the contributions of Re(ɛ⊥) and Im(ɛ⊥) are 3.58 ps and 2.02 ps, respectively. These relaxation times are showed in Table 1 to facilitate comparison. A similar superposition process occurs again, thereby generating a faster signal. It is obvious that the relaxation speed of the contribution of Re(ɛ⊥) is slower than that of the contribution of Im(ɛ⊥). The relaxation time of the contribution of Im(ɛ⊥) is close to that of Im(ɛ⊥). However, the relaxation time of the contribution of Re(ɛ⊥) is longer than that of Re(ɛ⊥). The contributions of Re(ɛ⊥) and Im(ɛ⊥) were calculated using Eq. (6). The terms $\partial $Re(ɛ||)/$\partial $t and $\partial $Im(ɛ||)/$\partial $t represent the changes in Re(ɛ||) and Im(ɛ||) over the delay time, respectively. Consequently, if the terms $\partial $ΔT/$\partial $Re(ɛ⊥) and $\partial $ΔT/$\partial $Im(ɛ⊥) do not change with delay time, the relaxation times of the contributions of the two parts are the same as the change in effective permittivity when fitted by Eq. (4). To explain the deviation of absorption from the change in the effective permittivity, we calculated $\partial $ΔT/$\partial $Re(ɛ⊥) and $\partial $ΔT/$\partial $Im(ɛ⊥), as shown in Fig. 8 (d). The results indicate that $\partial $ΔT/$\partial $Im(ɛ⊥) does not change with the delay time. In contrast, $\partial $ΔT/$\partial $Re(ɛ⊥) is sensitive to the delay time. Its relaxation time is 2.24 ps when fitted by exponential equation. Consequently, the relaxation time of Re(ɛ⊥) is different from the relaxation time of its contribution to ΔT. As a result, two signals with different relaxation times may combine into one signal with a shorter relaxation time.
The relaxation times of ΔT and the permittivities at other wavelengths were also extracted and shown in Fig. 9 to further analyze the effect of superposition of ENZ and ENP resonance on the relaxation time. The extraction range was 480 nm – 640 nm to avoid the spectra ranges where ΔT is too small. As shown in Fig. 9, only ΔT measured with p-polarized probe pulses shows significant wavelength sensitivity among ΔT and the changes of permittivities. Its relaxation time decreases rapidly as the wavelength increases to 543 nm.
Fig. 9. (a) The relaxation times of ΔT measure with s- and p- polarized probe pulses at different wavelengths. (b) The relaxation times of the changes of permittivities at different wavelengths.
When the wavelength is shorter than 543 nm, the contribution of ENP resonance is larger than the contribution of ENZ resonance. ΔT has the same sign with the contribution of ENP resonance. The contribution of ENP resonance recovers faster than the contribution of ENZ resonance and leads to a faster recovery at the first 1 ps. The closer the two contributions are, the stronger the effect. When the wavelength increases to longer than 543 nm, the contribution of ENZ resonance is stronger than the contribution of ENP resonance. ΔT have the same sign with the contribution of ENZ resonance. The contribution of ENZ resonance recovers slower than the contribution of ENP resonance and lead to a slower recovery at the first 1 ps. Consequently, despite as shown in Fig. 9 (b), the permittivities almost unchanged with the wavelength, the relaxation time can be modulated.
6. Conclusion
In this study, we extracted the changes in the real and imaginary parts of both the in-plane and out-of-plane effective permittivity of Ag nanorod arrays near the ENZ wavelength when pumped by femtosecond pulses, to trace the origin of an ultrafast process. The effective permittivity satisfies the Kramers–Kronig relationship, and both components of the effective permittivity are time- and wavelength-dependent. The extraction method is also applicable to other hyperbolic materials. The extracted nonlinear effective permittivity can be used to analyze the independent contributions of each resonance appearing at the same wavelength. The sample in our experiment exhibits a polarization-sensitive ultrafast signal, whose relaxation time changes from 1.11 ps to 0.24 ps when the probe light changes from s- to p-polarization. Further analysis based on the extracted nonlinear effective permittivity indicates that the signal originates from the combination of the ENP resonance and ENZ resonance supported by the sample. The change in the relaxation speeds of the metal nanostructures is attributed to the superposition of multiple absorption characteristics. This phenomenon is triggered by the difference in the signs and relaxation speeds of the components. Some interesting nonlinear responses, such as ultrafast responses, can be extracted based on the superposition of multiple resonances. The nonlinear permittivity can be used to analyze and predict the combination phenomenon. This work is significant for the characterization of the nonlinearity of noble metal structures and may be useful for the design of nonlinear responses.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
Data availability
Data underlying the results presented in this paper are available from the corresponding author upon request.
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