1. Introduction

Nonlinear plasmonics is a young but fast growing research field [1], it encompasses both a nonlinear response of an active medium-such as metal-and that of a surrounding dielectric medium. Nonlinear response of plasmonic systems has been observed both in metal films and in metallic nanostructures. It plays an important role in modern photonic functionalities, including control over the frequency spectrum of laser light, generation of ultrashort pluses, all-optical signal processing and ultrafast switching [2]. Optical bistability (OB) is a typical feature of nonlinear optical effects [3,4]. OB could provide the optical structure the function to control two distinguish stale states with the history of the input light. Due to its ability of controlling light with light, OB has a lot of potential applications in optical communications and computing [5,6] as a remarkable all-optical signal processing. To show this bistable nonlinear response, a light intensity dependent refractive index is required in a typical OB configuration. The general medium to realize OB is called Kerr nonlinear material [7], whose refractive index is proportional to the square of field intensity. To achieve significant strong optical bistability at small excitation power, high Kerr nonlinearity is required. However, conventional Kerr-type materials generally have very weak nonlinear response. Therefore, the certain mechanism of near field enhanced process is adopted to boost the nonlinear response inside the Kerr medium. An increased effective nonlinear optical response can be achieved by plasmonic effects. Such effects arise from coherent oscillations of conduction electrons near the surface of noble-metal structures [8,9], and can result in dramatic local electromagnetic field enhancement [8,10], hence boost the Kerr nonlinearity [1].

Metal have long been compelling candidates for plasmonic materials which can support surface plasmon that confine and enhance the light in deeply subwavelength volume. However, this enhanced field by surface plasmonic resonances would be limited when the electron-electron interaction in the metal cannot be neglected [11]. It happens once the dimension of a single plasmonic structure [12–14] or the gap between plasmonic elements [15–17] is small. So that the conventional local description of the electronic response of metallic nanostructure fails, as one should take into account the finite spatial profile of the plasmon-induced screening charge (nonlocality). When the expected field enhancement is modified by the above nonlocal effects in the extreme coupling regime, the nonlinear processes should also be significant affected. In fact, nonlocalty have showed the reduced field enhancements [18–23] in plasmonic nanostructure, and the influence of surface electrons interaction on nonlinear current in the gap of two dimer [24] was reported. Besides that, studies of the nonlocal effects on third harmonic generation in gap nanostructure [25–27] were investigated as well. For artificial nonlocal metamaterial, nonlocality enhanced ultrafast nonlinear optical response was found [28] recently. In this paper, we shall perform the investigation of the nonlocal effects on the bistable near field and bistable transmittance of a two-dimensinal composite slab composed of nonlocal core-Kerr dielectric shell cylindrical core-shell inclusion embedded in a host medium. We show the nonlocality of the plasmonic meal alters the optical bistable curves in both near field and transmittance of the proposed composite slab. It demonstrates that the nonlocal effects could provide more parameter space for the existence of optical bistability in near field and transmittance.

2. Model and theory

As shown in Fig. 1, we first consider the two-dimensional composite consisting of coated nano-cylinders in the linear case. The outer radius of the core-shell inclusions is assumed as$b$, and the inner radius $a=\eta b$ where $\eta$ denotes the aspect ratio. The inclusions are embedded in the host medium ${\epsilon }_{\text{h}}$with a volume factor$f$, and the core material is described by a spatial dispersive dielectric function$\epsilon (k,\omega )$. We assume the total size of the inclusions is much less than the incident wavelength, hence the retardation effect is neglected here. Within the quasistatic approximation, the total electric potential outside the inclusions is given by,

 

Fig. 1 2D composite compose of core-shell inclusions embedded in the host medium.

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As to the nonlocal core, its electrostatic potentials could be derived by introducing the semi-classical infinite barrier model (SCIB) [29,30]. The electric field and displacement vector satisfy the electrostatic equations,

With a similar process to the case of nonlocal spherical shell [31], we achieve the Poisson-type equation for the displacement potentials in the core region as,

Taking the Fourier transform of Eq. (5), we yield

Up to now, we get the general expressions of the electric potentials and displacement potentials throughout the whole space in the composite, taking into account the spatial dispersive dielectric response of the nonlocal core. The unknown coefficients can be determined by the boundary conditions [31,32] as follows,

Solving the above equations, we yield

Note that if the spatial dispersion of the nonlocal core is neglected, $f(a,a)$will reduce to$\epsilon (0,\omega )$according to the integral in Eq. (12). Substituting $f(a,a)$ for the local dielectric function of the core$\epsilon (0,\omega )$, Eq. (13) will immediately become the results in the local case.

3. Bistability in near field

To include the nonlinear effects, we introduce the field-intensity dependent dielectric function to describe the shell material which has the following expression

Figure 2 shows the bistable relation between the average local field $E_s$ and the external incident field $E_0$ when the nonlocal nature of the core materials is taken into account. Gold [15,25] and silver [14,35] are typical materials which admit nonlocal effects on their dielectric response. Here, we introduce the hydrodynamic model [15,31,36] of silver to describe the core medium: $\epsilon (k,\omega )={\epsilon }_{\text{g}}-{\omega }_{\text{p}}^2/[\omega (\omega +i\Gamma )-{\beta }^2{k}^2]$, where ${\epsilon }_{\text{g}}=3.7$ is the background permittivity relating to the interband transition, ${\omega }_{\text{p}}=1.348\times {10}^{16}{\text{s}}^{-1}$ and $\Gamma =3.179\times {10}^{13}{\text{s}}^{-1}$ are the plasma frequency and damping constant. $\beta$ indicates the pressure term of electron gas which is proportional to the Fermi velocity. As to the nonlinear shell, despite some particular poly, such as polyDCHD-HS [37] or polyMEH-PPV [38], which gains much large third-order nonlinear coefficients up to $\text{~}{10}^{\text{-}17}{\text{m}}^2/{\text{V}}^2$, the typical Kerr nonlinearity of dielectric medium is not particularly large. For instance, experimentally measured third-order nonlinear coefficients of Al₂O₃ and Fused silica are in the level of $\text{~}{10}^{\text{-}22}{\text{m}}^2/{\text{V}}^2$ [39]. Without loss of generality, the linear permittivity of the shell here is${\epsilon }_{\text{s}}=2.2$, and the third order nonlinear constant is chosen as ${\chi }_{\text{s}}=4.4\times {10}^{-20}{\text{m}}^2/{\text{V}}^2$. For the sake of simplicity, the host medium is assumed to have a permittivity${\epsilon }_{\text{h}}=1$ . To make comparisons, the curves in local quasistatic approximation are plotted in Fig. 2 as well.

 

Fig. 2 The average local field $E_s$ as a function of the incident field $E_0$ for various aspect ratios of the core-shell inclusions: (a) $\eta =0.1$, (b) $\eta =0.5$ and (c) $\eta =0.7$under nonlocal (red) and local (black) descriptions respectively. The incident wavelength is $\lambda =340\text{nm}$ and the outer radius of the inclusion is$b=10\text{nm}$.

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It is found that difference between nonlocal and local cases becomes obvious when the aspect ratio is small ($\eta =0.1$). Moreover, Fig. 2(a) reveals that local case doesn’t even exhibit optical bistability. This phenomenon can be expected because the spatial dispersion (or nonlocality) of the core medium tends to dominate when the size is small, and spatial dispersion will dramatically alter the dielectric response of the nonlocal core, so that bistable curves in nonlocal case will show large deviation from these in local when they share the same incident wavelength and aspect ratio. Figures 2(b) and 2(c) illustrate that generally nonlocality will dramatically increase the switching-up threshold fields and slightly reduce the switching-down thresholds. Besides that, with the capability of field concentration in the core-shell inclusions, one could use low external field power to achieve high near field intensity in the shell region$E_{\text{s}}$. In addition, the upper branch field $E_{\text{s}}$ we can achieve at the switching-up point in nonlocal case is also higher than that in local one.

To exhibit optical bistability, parameters in Eq. (17) shall satisfy some specific conditions [40,41]. Otherwise, we could not achieve bistable curves as the local case in Fig. 2(a). Once the material of each component is determined, the incident wavelength $\lambda$ and aspect ratio $\eta$ would play important roles in the appearance of OB and OB’s switching threshold fields. For instance, when aspect ratio is 0.5, Fig. 2(b) shows lower switching threshold field $E_0$ compared to the others. In what follows, we’d like to give rigorous calculation based on Eq. (17) to further demonstrate the dependence of OB on incident wavelength and aspect ratio. In fact, Eq. (17) is a seven-order equation for${〈{\left|E\right|}^2〉}_{\text{s}}$ if is expanded. By differentiating Eq. (17) with respect to ${〈{\left|E\right|}^2〉}_{\text{s}}$and setting $\partial ({\left|E_0\right|}^2)/\partial {〈{\left|E\right|}^2〉}_{\text{s}}=0$, we obtain a six-order equation for ${〈{\left|E\right|}^2〉}_{\text{s}}$ by solving which the OB condition can be determined as,

OB occurs when the six-order equation admit two real, positive solutions in its six ones. Actually, there is no analytical solution for a six-order equation, so that we give the numerical results of it, as shown in Fig. 3. Within the present parameter space, only two kinds of real solutions exists, one associates with two positive solutions which corresponds to OB and the other is none. Figure 3 demonstrates that nonlocality will dramatically increase the bistable region in the parameter space for bistabel near field, especially in the low $\eta$ region. It explains why we could not achieve optical bistability in local case shown in Fig. 2(a). Figure 3 provide us a clear diagram showing where OB exists for both nonlocal and local cases and might be useful for the further design of OB devices. Besides that, two real solutions of Eq. (18) correspond to the switching-up and switching-down threshold fields of the bistable curves. Although not demonstrated here, we concluded that core-shell inclusions with parameters near the edge between “with OB” and “without OB” regions in Fig. 3 will have lower switching thresholds. It should be remarked that when the diameter of the nonlocal core is less than 2nm, the electron spill-out of the shell will dominate hence our approach is no longer valid, and a fully quantum techniques such as time dependent density functional theory (TDDFT) [42] should be required. Therefore, we restrain the down limit for the aspect ratio to 0.1.

 

Fig. 3 Numbers of real solutions of Eq. (18) in the functions of incident wavelength and aspect ratio in (a) nonlocal and (b) local conditions, respectively. Two real solutions means there existing optical bistability. The outer radii of the inclusions are fixed at $b=10\text{nm}$.

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4. Bistability in transmittance

The field intensity dependent dielectric for shell materials results in the optical bistability in the near field, and we show that nonlocality given by the ultra-nano-sized plasmonic core will dramatically alter this optical bistable behavior. In the following, we would like to further investigate the transmittance and reflectance of the subwavelength slab consisting of this kind of 2D composite. To model the dielectric response of the nonlinear composite, we employ 2D Clausius-Mossotti approximation, and the effective dielectric response of the composite system then can be written as [43]

 

Fig. 4 (a) Transmittance (T) and reflectance (R) spectra of the slab composed of liner composite material with thickness $d=100\text{nm}$at normal incidence ($\theta =0$). Solid and dashed lines denote the nonlocal and local cases respectively. The inserts in (a): (left) ${〈{\left|E\right|}^2〉}_{\text{s}}$as the function of incident wavelength; (right) the schematic diagram of the model. (b) R and T versus the incident angle $\theta$ at $\lambda =324\text{nm}$. The volume factor is$f=0.01$.

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To begin with, we first consider the transmittance (T) and reflectance (R) spectra in linear case. Figure 4(a) illustrate the T/R spectra for normal incidence and it is found nonlocality would lead to the blue shift of the transmittance dip (or reflectance peak). The transmittance dip results from the surface plasmon resonance at the interface between nonlocal core and the dielectric shell. This resonance, on one hand, will prevent the light from transmitting the composite slab so that there exist transmittance dip and reflectance peak at the resonant wavelength, on the other, will result in large damping hence $\text{T}+\text{R<1}$ in the resonant position compared with other off-resonant region. The insert in Fig. 4(a) shows the average field intensity as the function of incident wavelength in the shell region. It is found that the field intensity within the shell will be dramatically enhanced at the surface plasmon resonant wavelength, which can be adopted to enhance the nonlinear response in the shell materials. In Fig. 4(b) we plot T/R in the function of incident angle at the resonant wavelength ($\lambda =324\text{nm}$). With the increase of incident angle$\theta$, reflectance increases and transmittance decreases respectively.

Dramatic field enhancement inside the shell region at surface plasmon resonant wavelength provide a certain condition to boost the nonlinear effects on the Kerr medium. As a consequence, T/R will show nonlinear dependence on the incident field intensity which is plotted in Fig. 5(a). Similar to the optical bistable curves in near field shown in Fig. 2(a), with the specific values of $\lambda$ and $\eta$, only nonlocal case exhibit optical bistability in transmittance. Red solid line in Fig. 5(a) shows that transmittance at low incident field intensity $E_0$ keeps in a high level and jumps to a low level when $E_0$ reaches to $\text{~}1\times {10}^8\text{V/m}$. On the other hand, when decreasing$E_0$, it first goes low from a high level but finally jumps to a high transmittance once$E_0<5.9\times {10}^7\text{V/m}$. Within the bistable region, this nonlocal composite slab exhibit two different values of transmittance for one incident field intensity. However, if one neglect the nonlocal nature of the core medium, there is no such bistability exist. Nowadays, nano optical functional devices with ultra-small size are highly demanded. Nonlocal effects are important in the theoretical predication for electromagnetic properties of nano devices when the dimension is ultra-small. Actually, Fig. 5(a) merely provides us one example to show the difference between nonlocal and local cases. When more values for $\eta$ are considered, it is found that we could achieve broader region of bistable transmittance in nonlocal case than local case. Figure 5(b) illustrate the switching-up ($E_{\text{up}}$) and switching-down ($E_{\text{down}}$) threshold fields in the functions of the aspect ratio. The bistable region lays on the gap between switching-up and switching down curves. Like bistable near field in Fig. 3, nonlocality will dramatically increase bistable area for the bistable transmittance in the geometrical parameter space, and it offers an additional branch of bistable region in low $\eta$ part compared to local case. As to another branch in high $\eta$ part, nonlocality still provide a larger bistable region.

 

Fig. 5 (a): Dependence of transmittance on external incident field in nonlocal (Red) and local (Black) cases. The incident wavelength $\lambda =328.5\text{nm}$ and the aspect ratio $\eta =0.5$. (b): Switching up and switching down threshold fields for the transmittance of the slab in the function of aspect ratio $\eta$ at $\lambda =328.5\text{nm}$. Blank space between the upper and lower lines shows the bistable region.

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In the end, the dependence of transmittance on the incident wavelength at different incident field intensity is investigated. Figure 6 shows that the bistable curves in transmittance spectra become obvious when the incident field intensity increases for both nonlocal and local cases. It leads to a wider gap between switching-up and switching-down threshold fields. In detail, the switching-up threshold fields are dramatically increased when $E_0$ increase. As to the switching-down case, however, it merely shows slightly red-shift. This $E_0$ dependent bistable transmittance in spectra would provide composite slab the function to vary its transmittance from a high value to a low one as a function of the input intensity or of the wavelength operation. Moreover, the variation of incident field intensity $E_0$would not lead to additional difference between nonlocal and local cases, i.e., the blue shift caused by nonlocality shows little dependence on $E_0$ in the transmittance spectrum. Since the size of the nonlocal core determine the degree of nonlocal effect, it can be expected that once we vary the aspect ratio $\eta$, the nonlocal and local bistable transmittance curves in Fig. 6 would show different deviations from each other.

 

Fig. 6 Dependence of transmittance on the incident wavelength at different incident field intensity (a)$E_0=5\times {10}^7\text{V}\cdot {\text{m}}^{-1}$, (b) $E_0=10\times {10}^7\text{V}\cdot {\text{m}}^{-1}$ and (c) $E_0=20\times {10}^7\text{V}\cdot {\text{m}}^{-1}$. Other parameters are the same as in Fig. (5).

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5. Conclusion

To conclude, in the quasistatic limit we developed an analytical theory to derive the near field inside a 2D nonlinear composite, where nonlocal core-Kerr dielectric shell inclusions are randomly embedded in the host medium. Using self-consistent mean field method, we establish the self-consistent nonlinear equation to describe the dependence of the field intensity in the nonlinear shell on the external incident field. Our numerical results show that once the nonlocal response of nonlocal core is taken into account, the bistable curves in near field will be affected. For instance, the switching-up threshold field $E_0$ in nonlocal case is much larger than that in local case with a same incident wavelength. Besides that, the upper branch field in the shell region we can achieve at the switching-up point in nonlocal case is found higher than that in local one. We give rigorous calculation to demonstrate in which parameter space (including incident wavelength and aspect ratio) the optical bistability of near field in shell exists, and show that within the present model, nonlocality could offers more parameter space than local case. Next, we employ 2D Clausius-Mossotti approximation to model the effective permittivity of the composite hence to study the field-dependent transmittance of the composite slab. In this connection, $E_0$controllable bistable transmittance is achieved, and similarly we found nonlocality would lead to much broader bistable region of the transmittance. In the geometrical parameter space, nonlocality can offer an additional branch of area for bistabiltiy in low aspect ratio region. Finally, we investigate the dependence of transmittance on the incident wavelength, and found bistable curve in the transmittance spectra.

Some comments should be remarked that, for the sake of simplicity, the composite is considered in the dilute limit and the electrostatic dipole interactions between each inclusion are not taken into account. Second, we use quasistatic approximation in the theoretical deviation. It was shown that some adoptions of quasistatic approximation might lead to unphysical results and spurious resonances would arise [44]. However, it was also proved that for small nonlocal spheres, the quasistatic approximation we adopted yields the similar results as the full-wave electromagnetic theory [19, 29]. Nevertheless, the quasistatic approximation could not extend to the case of large size. To overcome, the self-consistent mean field approximation in the framework of full wave Mie scattering theory [45] can be introduced, and we shall incorporate the a full wave self-consistent mean-field approximation with nonlocal electromagnetic theory [13, 36] to consider the contributions of higher order scattering terms beyond quasistatic approximation. Work along this line is in progress and should be reported elsewhere.