1. Introduction

Optical bistability (OB) is a process in which there are two stable optical output states for a single light input. The phenomenon has been extensively studied since the seminal work of Gibbs [1] owing potential applications in all-optical devices, such as switches, logic gates, transistors, and memory. Various resonant optical systems consisting of a material with an intensity-dependent refractive index, i.e., Kerr effect, have been theoretically predicted and experimentally demonstrated as simple and elegant ways to realize OB [1]. Optical resonators can localize and enhance the optical fields to boost the weak Kerr effect of materials, and thus their optical properties can be changed with high sensitivity by low power input light. Traditionally, the resonant optical systems are Fabry-Perot (F-P) cavities [2–4], photonic crystal cavities [5–7], quantum well structures [8–10], waveguide-ring resonators [11–13], and surface plasmon (SP) resonance nanostructures [14–16]. Especially, plasmonic nanostructures have received much attention because the light can be confined into subwavelength volumes, well below the diffraction limit, and thus the local optical field can be dramatically enhanced and devices can be miniaturized [17]. OB has also been widely demonstrated in different metal nanostructures such as SPP crystals, metal-dielectric multilayers, and metal gratings [14–20].

The gap between coupled metallic nanostructures is critical for most key photonic applications because the local field is extremely confined in the gap regime for the gap-plasmon mode and conventionally the local field enhancement and confinement can significantly increase with the decrease of the gap. With the development of growth and nanofabrication technology, the gap can be well controlled by dielectric layer grown down to sub-ten nanometers by atomic layer deposition or laser molecular beam epitaxy (MBE) techniques [21] and even to subnanometer levels using molecules [22,23] or two-dimensional materials [24,25]. It is evident that as the nanoscale feature size of plasmonic systems is reduced below 20 nm, the classical description has considerable discrepancy with the experimental observations [26]. The spatial nonlocal effect of conducting materials is not negligible. The effect of classical nonlocality in the metallic structures can limit the field enhancement [22,27–29]. When the gap of coupled metallic nanostructures is less than 0.5 nm, the quantum tunneling effect is considered and the local field enhancement is dramatically reduced compared with the classical local effect [30–33]. Besides the linear optical response, nonlinear optical properties, such as harmonic generation, have been studied in gap-plasmon nanostructures. The surface charges that permeate the volume beneath the surface of the metal allow the fields to access the intrinsic third-order nonlinearity of metals and dramatically enhance the efficiency of third-order responses, such as third harmonic generation [34,35].

In this paper, we theoretically investigate how the nonlocal effects of metallic nanostructures influence the OB behavior in plasmonic systems. We choose a typical nanostructure of gap-plasmon modes that is a metasurface consisting of a metallic grating coupled with metallic film spaced with Kerr materials to demonstrate the premise. Especially, we will not only study the traditional OB of linear optical response, but also study the third-harmonic generation (THG) affected by both the linear OB and nonlocal effect in the nanostructures. Compared with the classical electromagnetic theoretical frame, the blue-shifted linear optical reflection and enlarged input light intensity for OB behavior are observed in the presence of classical nonlocal effects. The results are significant for the design of plasmonic all-optical devices based on OB.

2. Numerical model

A schematic of the planar metasurface consisting of a metallic grating coupled with metallic film spaced with Kerr materials is shown in Fig. 1. The period of a unit cell, and the width and height of the metallic grating are P, La, and Ha, respectively. P, La, and Ha are varied to ensure the gap-plasmon resonance at near-infrared wavelengths. The free-electron model of a metal is considered in this wavelength regime. The metallic film thickness is chosen to be 200 nm, which is thick enough to ensure no light is transmitted. The thickness of the Kerr material between the metallic grating and metallic film is g, which varies in the range of 0.5 to 20 nm in this study. The effect of quantum tunneling on the local electric field is not considered here though 0.5 nm is considered as the threshold tunnel distance [31,36,37]. The process of quantum conductivity for metal-insulator-metal nanostructures can be found in Ref. [38]. The relative dielectric permittivity of nonlinear dielectric materials $\varepsilon _r^d$ dependence on the local electric field |E_loc| can be expressed as $\varepsilon _r^d = \varepsilon _L^d + {\chi ^{(3)}}{|{{E_{loc}}} |^2}$, where $\varepsilon _L^d$ is the linear relative permittivity and χ⁽³⁾ the third-order nonlinear susceptibility of Kerr material, respectively. It is noting that the |E_loc| is the exact non-uniform local field in the gap, not the average local field or maximum field, as demonstrated in [16].

 

Fig. 1. (a) Schematic of the geometry consisting of a metallic nanowire grating coupled to metallic substrate via a gap of Kerr dielectric materials. In the cross section, P represents the grating period, g is the gap distance, and La and Ha are the width and height of the metallic nanowires, respectively. (b) Reflectance spectra from a metasurface with 0.5-nm gap for local and nonlocal models of different parameters of Ag.

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Light with transverse-magnetic (TM) polarization shines on the nanostructure normally. The linear optical reflection from the metasurface is obtained by solving the wave equation with a time-harmonic propagation exp(-iωt), $\nabla \times \nabla \times {\textbf{E}_\omega } - k_\omega ^2{\textbf{E}_\omega } = {\mu _0}{\omega ^2}{\textbf{P}_\omega }$, where ω is the angular frequency of the light, E_ω is the electric field at fundamental frequency, ${k_\omega } = {\omega / c}$ is the wave vector of the light with c being light velocity in vacuum, µ₀ is the vacuum permeability, and P_ω is the polarization of materials at ω. In the local model for the metal, the P_ω(r, ω) = D(r, ω)-ε₀E(r, ω) with the electric displacement field D(r, ω) = ε₀ε(r, ω)E(r, ω), where the ε(r, ω) is the permittivity of metal, r is the position vector, and ε₀ is the permittivity of vacuum. The permittivity ε(r,ω) can be written as $\varepsilon (\textbf{r},\omega ) = {\varepsilon _\infty } - \frac{{\omega _p^2}}{{{\omega ^2} + i\omega \gamma }}$ using the local free-electron Drude model in the near-infrared wavelength regime that we are concerned with, where ${\varepsilon _\infty }$ is the dielectric constant due to bound charges, $\gamma $ is the damping frequency, and the plasma frequency ${\omega _p} = \sqrt {{{n{e^2}} / {{\varepsilon _0}m_e^\ast }}} $ with n being the electron density, e the electron charge, and $m_e^\ast $ the effective electron mass. Thus in the metal domain, P_ω is expressed as ${\textbf{P}_\omega } = {\varepsilon _0}({\varepsilon _\infty } - 1 - \frac{{\omega _p^2}}{{{\omega ^2} + i\omega \gamma }}){\textbf{E}_\omega }$. When the spatial nonlocality of a metal is not negligible, the wave-vector dependent dielectric constant of metal must be considered, and the permittivity of metal is expressed using the hydrodynamic Drude model as $\varepsilon ({\textbf{k}_\omega },\omega ) = {\varepsilon _\infty } - \frac{{\omega _p^2}}{{{\omega ^2} + i\omega \gamma - {\beta ^2}\textbf{k}_\omega ^2}}$, where $\beta = \sqrt {{3 / 5}} {v_f}$ with Fermi velocity v_f. The polarization of the metal P_ω(r, ω) is then expressed using ${\beta ^2}\nabla (\nabla \cdot {\textbf{P}_\omega }) + ({\omega ^2} + i\omega \gamma ){\textbf{P}_\omega } ={-} \frac{{{n_0}{e^2}}}{{m_e^\ast }}{\textbf{E}_\omega }$ [27,34]. In the gap dielectric domain, the polarization P_ω is always expressed as ${\textbf{P}_\omega } = {\varepsilon _0}(\varepsilon _r^d - 1){\textbf{E}_\omega }$.

The THG nonlinear process can be described under the undepleted pump approximation as $\nabla \times \nabla \times {\textbf{E}_{3\omega }} - k_{3\omega }^2{\textbf{E}_{3\omega }} = {\mu _0}{(3\omega )^2}[\textbf{P}_{3\omega }^{(1)} + \textbf{P}_{3\omega }^{(3)}]$, where E_3ω is the electric field at third-harmonic frequency, ${k_{3\omega }} = {{3\omega } / c}$ is the wave vector of the THG light, $\textbf{P}_{3\omega }^{(1)}$ and $\textbf{P}_{3\omega }^{(3)}$ are the linear polarization and THG polarization in the materials at 3ω, respectively. $\textbf{P}_{3\omega }^{(1)}$ can be expressed as $\textbf{P}_{3\omega }^{(1)}\textrm{ = }{\varepsilon _0}\chi _{3\omega }^{(1)}{\textbf{E}_{3\omega }}$ and $\textbf{P}_{3\omega }^{(3)}$ is written as, ${\textbf{P}_{3\omega }}\textrm{ = }{\varepsilon _0}\chi _{3\omega }^{(3)}:{\textbf{E}_\omega }{\textbf{E}_\omega }{\textbf{E}_\omega }$, where $\chi _{3\omega }^{(1)}$ and $\chi _{3\omega }^{(3)}$ are the linear and third-order susceptibility of materials at THG frequency. The linear dielectric permittivity of materials ${\varepsilon _r}(3\omega )$ at 3ω thus written as ${\varepsilon _r}(3\omega )\textrm{ = }1\textrm{ + }\chi _{3\omega }^{(1)}$. In this letter, we will not consider the dispersion of dielectric materials, i.e. the linear dielectric constant of materials $\varepsilon _L^d$ is the same at ω and 3ω frequencies. When study the effect of linear optical bistability on THG, the intensity-dependent permittivity of materials ${\varepsilon _r}(3\omega )$ at 3ω frequency domain is inherited from $\varepsilon _r^d$ at fundamental ω frequency domain. The expressions of permittivity of Ag at classical local Drude and hydrodynamic Drude model are the same as those at the fundamental frequency, just replacing the ω with 3ω, respectively.

The aforementioned coupled equations can be numerically solved by using finite element method (Comsol Multiphysics) to study the linear and nonlinear optical properties of our proposed metasurface. A hard-wall boundary of metal surface, i.e., $\textbf{n} \cdot {\textbf{P}_\omega } = 0$ with n the unit vector normal to the surface, is employed because the spill-out effect of electrons from metal is negligible. The corners of metallic gratings are rounded to a radius of 5 nm to avoid field divergence (Fig. 1(a)). In this letter, the linear permittivity $\varepsilon _L^d$ of dielectrics is assumed to be 2.2, the Kerr susceptibility χ⁽³⁾ of purely real part is assumed to be 8.4×10⁻¹⁸ m²/V² [39], which is a typical value for polymers [40,41], glasses [42,43] and dielectric doped with metallic nanoparticles [44]. The dielectrics are considered as isotropic materials of THG susceptibility $\chi _{3\omega }^{(3)}$ also of 8.4×10⁻¹⁸ m²/V². It is noting that the exact value of $\chi _{3\omega }^{(3)}$ is not important here, because such value only affect the magnitude of THG but will not influence the trend of THG during the optical bistability of reflectance in the nanostructures. The optical nonlinearity of metal is neglected since the local field is dominantly concentrated in the gap dielectric domain at gap-plasmon mode, which is confirmed by the comparison of obtained optical bistability behaviors including/excluding the nonlinearity of Ag. The parameters for Ag are taken from the literatures with ε_∞=3.7, ω_p=1.38×10¹⁶ rad/s, γ = 2.73×10¹³ rad/s, and v_f= 1.39×10⁶ m/s [28,45].

3. Results and discussion

We first focus the study on the most extreme coupling configuration of the metasurface of the gap g = 0.5 nm. The linear optical response is calculated under the low incident electric field E₀ = 1 V/m (corresponding to an intensity of around 13 W/cm²) considering the linear dielectric material. The reflectance spectra of the metasurface with P = 100 nm, La = 45 nm, and Ha = 40 nm is shown in Fig. 1(b). The resonance wavelength λ_r of the plasmonic gap mode under the local Drude model of Ag is around 1625 nm. A dramatic blue-shift of resonance wavelength is obtained around 1405 nm when considering the classical nonlocal effect of the parameter β. The electric field distribution at the resonance plasmonic gap mode considering the local and nonlocal effects is shown in Fig. 2(a) and (b), respectively. The electric fields both concentrate in the gap. The maximum enhancement factor is around 546 and 443 for local Drude and nonlocal models of Ag, respectively. The enhancement factor considering nonlocal effect is less than that in the local model, owing to the screening effect of electron-electron interaction [26,27]. The electric field along the center line (x-axis at y = 0.25 nm) of the gap and along the vertical line (y-axis) perpendicular to the Ag substrate at x = 20 nm (near the corner of Ag grating) is shown in Fig. 2(c) and (d), respectively.

 

Fig. 2. Electric field distribution at the resonance wavelengths of a metasurface with 0.5-nm gap under (a) local model and (b) nonlocal model of Ag. (c) Electric field along the center line of the gap and (d) along the vertical line y when x = 20 nm for the local and nonlocal models of different nonlocal parameters.

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The different parameters of nonlocal effect which are related with the thickness of free-electron shell are also considered [34]. The reflectance when the nonlocal parameters are 0.5β and 2β are also shown in Fig. 2. The larger nonlocal parameter will introduce larger blue-shift of resonance wavelength compared with the local model. The electric field along the center line of the gap and the vertical line at x = 20 nm is also shown in Fig. 2(c) and (d). The larger nonlocal parameter dramatically reduces the local field in the dielectric gap, because the Thomas–Fermi screening length becomes thicker with the increase of nonlinear parameter [34].

We then study the linear reflectance from the metasurface when the nonlinear refraction of dielectric is not neglected under the intense light input. Figure 3(a) and (b) show the change of the reflectance spectra under different laser intensities under local and nonlocal models of nonlinear parameter β, respectively. The resonance wavelength changes from 1625 nm at 13 W/cm², i.e., the dielectric nonlinearity can be neglected, to 1704nm at 140 kW/cm² under the local model, while the resonance wavelength changes from 1405 nm with the linear dielectric to 1475 nm at 300 kW/cm² in the presence of nonlocal effects. The reduced electric field in the gap when considering the nonlocal effect leads to larger input laser intensity being required to obtain the same shift of the resonance wavelength under the traditional local Drude model of Ag.

 

Fig. 3. Reflectance spectra from metasurface with 0.5-nm gap under different light intensities for (a) local model and (b) nonlocal model of parameter β, respectively. Optical bistability of reflectance (c) and THG (d) from the metasurface with 0.5-nm gap for local and nonlocal models of different nonlocal parameters. The THG field distribution of unit V/m at low intensity state “1” (e) and high intensity state “2” (f). The states “1” and “2” are labeled in (d) for THG in metasurface considering nonlocal model of parameter β.

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The optical bistable behavior of linear reflectance in the metasurface is calculated as the incident light intensity increases and decreases. Each step is performed by using the solution obtained at the previous step [16,46]. The working wavelength is chosen as the wavelengths for which the linear reflectance is around 85%, which are 1710nm, 1580 nm, 1478 nm, and 1321 nm for local and nonlocal models of parameters 0.5β, β, and 2β, respectively. The hysteresis loops of the reflectance as a function of input intensity, which exhibit two stable branches, are shown in Fig. 3(c). The threshold intensity I_th from the higher reflectance state to the lower reflectance state is around 140 kW/cm² under the local Drude model of Ag, while I_th increases to 200 kW/cm², 300 kW/cm², and 525 kW/cm² in the presence of nonlocal parameters 0.5β, β, and 2β, respectively. The increased threshold intensity is ascribed to the decreased enhancement factor of electric field in the dielectric layer considering the nonlocal effect, especially the OB behavior depends on its historical evolution, i.e., the current state is influenced by the previous state.

The THG in the metasurface as the incident light intensity increases and decreases is simultaneously calculated with the simulation of OB behavior of linear reflectance, as shown in Fig. 3(d). The fundamental wavelengths at different models are the same as those for linear reflectance, respectively. Similar as the linear reflectance, the hysteresis loops of the THG power as a function of input intensity are observed. The OB behavior of THG directly originates in the OB of linear reflectance at which the fundamental local fields have two stable distributions at higher reflectance state and lower reflectance state, respectively. The threshold intensities from the lower THG power state to the higher THG state are almost in consistence with those in linear reflectance OB. It is worth noting that, different from the traditional relation of THG power P_3ω and fundamental intensity I_in as ${P_{3\omega }} \propto I_{in}^3$, such relation is broken because the different fundamental intensity will change the resonance condition of metasurface (Fig. 3(a) and (b)) along with the local intensity in the gap, and the local intensity is also related with the previous state. At the relative lower fundamental intensity (<100 kW/cm²) when the resonance peaks have not changed much, the P_3ω under the local model is larger than those under the nonlocal model, and P_3ω is of the larger value at the smaller nonlocal parameters due to the quenching local electric field under the nonlocal model of metal. The THG fields |E_3ω| at the two stable states “1” and “2” (labeled in Fig. 3(d)) under the same input intensity 270 kW/cm² for the metasurface considering Ag of nonlinear parameter β are shown in the Fig. 3(e) and (f), respectively. The different distribution and magnitude of electric field can be obviously observed.

The nonlocal effect on linear and nonlinear optical response of the thickness of the gap is further investigated. In this section, we limit the nonlinear parameter of traditional β. The reflectance spectra from various structures of different gaps considering the local and nonlocal models when E₀= 1 V/m are shown in Fig. 4. The period P of metasurfaces in Fig. 4(a) is 150 nm, (b) to (d) is 500 nm, and (e) is 700 nm, with La = 55 nm and Ha = 50 nm in Fig. 4(a) and (b), La = 100 nm and Ha = 20 nm in Fig. 4(c), and La = 180 nm and Ha = 20 nm for Fig. 4(d) and La = 200 nm and Ha = 20 nm for Fig. 4(e). The resonance is mainly from the coupling of Ag nanowire and Ag film which is determined by the geometry of Ag nanowire and the gap, so the period is not sensitive to the period, as shown in the inset of Fig. 4(a) for the nanostructure of gap 1 nm of period 500 nm. The blue-shifts of resonance wavelength in the presence of nonlocal effects are observed in all structures. As the gap increases, the blue-shift becomes smaller due to the weaker coupling between Ag nanowire grating and Ag substrate, as shown in Fig. 4(f). The enhancement of electric field also decreases with the increase of the gap. The electric field distributions at local and nonlocal models when the gap is 10 nm are shown in Fig. 5(a) and (b) at their respective resonance wavelengths. The maximum enhancement factor at the bottom corner of the Ag nanowire is around 318 and 247 for local and nonlocal models of Ag, respectively. The electric field along the center line of the gap and along the line perpendicular to the gap at x = 85 nm is shown in Fig. 5(c) and (d), respectively. The difference of electric field is quite small at such gap dimensions.

 

Fig. 4. Reflectance spectra from metasurfaces with gap of (a) 1 nm, (b) 2 nm, (c) 5 nm, (d) 10 nm, and (e) 20 nm. (f) Difference of resonance wavelength for local and nonlocal models vs. the gap. The inset of (a) shows the reflectance spectra from metasurface with gap 1 nm but of a different period.

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Fig. 5. Electric field distribution at the resonance wavelengths of metasurface with 10-nm gap under (a) local model and (b) nonlocal model of Ag. (c) Electric field along the center line of the gap and (d) along the vertical line y at x = 85 nm under the local and nonlocal models of nonlocal parameter β.

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The optical bistabilities for metasurfaces with different gaps are shown in Fig. 6. The working wavelengths of metasurfaces in Fig. 6(a) and (b) with gaps of 1 nm and 10 nm as the wavelengths for which the reflectance is 85%, i.e. around 1438 nm and 1332 nm for the nanostructure of gap 1 nm under local and nonlocal models of Ag, respectively, and 1590 nm and 1576.8 nm for the nanostructure of gap 10 nm under local and nonlocal models of Ag respectively. The corresponding wavelengths are used as the fundamental wavelengths of THG as shown in Fig. 6(d) and (e), respectively. The threshold intensity in the metasurfaces increases with the increase of gap due to the decrease of local field in the gap in principle, however, it is essentially dependent on the geometry of nanostructure and working wavelength. For example, the threshold of intensity for OB of reflectance in the nanostructure of gap 1 nm with period 500 nm will reduce to be around 35 kW/cm² and 50 kW/cm² at the working wavelengths of 1410 nm and 1304 nm for the local and nonlocal model of Ag, respectively. So the threshold intensity for the OB could be optimized for the experimental design. In general, the difference of I_th under the local and nonlocal models becomes smaller with the increase of the gap. The optical bistability of reflectance and THG in the metasurface with 20-nm gap (Fig. 6(c) and (f)) is calculated at the same working wavelength of 1440 nm for the local and nonlocal models. The difference of the threshold intensity is still large, i.e., I_th= 5.63 MW/cm² for the local model and I_th= 6.77 MW/cm² for the nonlocal model, although the reflectance spectra are almost overlapped (Fig. 4(e)). This is probably caused by dielectric permittivity of a nonlinear dielectric material being proportional to |E_loc|², so the difference of the local field could be amplified to change the resonance plasmon response, and further the OB behavior is influenced by the previous state of the historical evolution.

 

Fig. 6. Optical bistability of reflectance (a)-(c) and THG (d)-(f) from metasurfaces with gap of (a) (d) 1 nm, (b) (e) 10 nm, and (c) (f) 20 nm at local and nonlocal models. For (a) (d) and (b) (e), the working wavelength is chosen as the wavelengths for which the reflectance is 85%, and for (c) (f), the working wavelength is set as 1440 nm for both local and nonlocal models.

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4. Conclusions

The optical properties in metasurfaces consisting of Ag gratings coupled with Ag film spaced with a Kerr material are investigated considering the classical nonlocal effect. The blue-shift resonance wavelength and reduced enhancement of local field are obtained when the gap is less than 20 nm in the presence of the classical nonlocal effect compared with those of the local Drude model of Ag. When the Kerr nonlinearity of the dielectric material in the gap is not neglected under the intense incident light, hysteresis loops of the reflectance, along with the THG of two stable branches as a function of input intensity are obtained. The threshold intensity from the higher reflectance state to the lower reflectance state increases in the consideration of the nonlocal model compared with that obtained with the local model. The difference of linear optical properties, such as the resonance wavelength and electric field, decreases with the increase of the gap spacing. The difference between optical bistability predicted by the local and nonlocal models also decreases with as the gap increases, but is still not very close even when the gap is 20 nm, owing to the nonlinear refraction being related to |E_loc|² and the historical evolution of the optical bistability behavior. In addition, the influence of OB in consideration of nonlocal effect in plasmonic systems is not limited in our grating-dielectric-film coupled nanostructure, but in any ultra-compacted plasmonic nanostructures of small distance, e.g., less than 20 nm. And, the Kerr dielectric in the gap is not limited by only possessing purely real χ⁽³⁾, while the effect of two-photon absorption or saturable absorption on optical bistability behaviors will further be studied. The results are expected to have significance for the understanding and design the ultra-compacted plasmonic nonlinear photonic devices.