1. Introduction

Nonlinear dynamics in the microelectronic devices plays an important role in the modern applications due to the fact that it provides a majority of functions [1]. Generating the deterministic chaos is one of the most interesting concepts which has been widely realized in the nonlinear microelectronic [2]. Chua’s circuit [3] is the classic example of such scenario, and the chaotic behavior is well studied in optics for multimode lasers [4,5]. Chaotic regimes could exhibit very complicated dynamic features, therefore they have potential applications in secure data processing and true random numbers generation [6–8]. Recently, the concept of lumped optical nanoelements [9–18] has attracted much attention, whose aim is to engineer different nanophotonic devices with the same functions as these in microelectronics at nanometer scale. It can extend the work frequency domains of the microelectronics-based functional devices from GHz into infrared (IR) and visible ranges. Most previous studies of this topic were predominantly focused on the linear functionalities and recently researchers start to explore the nonlinear parts of this field. In some pioneering works, one theoretically and experimentally demonstrated modal symmetry controlled second-harmonic generation (SHG) by propagating plasmons in an optical nanocircuit [19]. A grating-coupled nanobelt-on-Au-nanoplate platform to realize the efficient SHG by orthogonal dual resonances of the hybrid plasmonic cavities was developed [20]. The plasmonic half-subtractor and demultiplexer circuits based on transmission-lines was experimentally demonstrated [21]. And also, the nonlinear chaotic behavior was recently demonstrated and quantitatively characterized in plasmonic resonators system [22,23] at nanometer scale, it demonstrated that nonlinear dimer nanoantenna could act as nonlinear nanophotonic circuitry.

Driven from the demand for large-scale integration of photonic nanocircuits, continuous efforts have been made to further shrink the size of on-chip photonic nanoelements. With the rapid development of nanofabrication technologies, the size of the plasmonic nanostructure can be reduced to few nanometers. When the size of nanoelements gets smaller than 10 nm [24–26], the quantum interaction among the electrons is no longer negligible. Besides the density-functional theory (DFT) [27,28] and Feibelman $d$ parameters framework [29,30], taking into account the spatial dispersion or nonlocality of the dielectric response in the framework of classical electrodynamics [31,32] can offer an efficient and relatively accurate description of the optical properties of the plasmonic nanostructures. The support of the resonance above the plasma frequency which is known as the longitudinal mode is one of the major characteristics of nonlocality [33–36]. The confinement of nonlocal field can significantly alter the linear and nonlinear responses of materials involving optical bistability [37,38], harmonic generation [39,40], near-field thermal rectification [41], Cherenkov radiation [42], and even Casimir force [43]. Therefore, the nonlinear dynamics as well as the chaos in the plasmonic nanostructures are expected to be influenced by nonlocality.

In this work, we propose a novel plasmonic dimer consisting of core-shell nanoparticles with plasmonic core and Kerr-type nonlinear shell, and take into account the nonlocality of the plasmonic core to investigate the nonlinear dynamics of the dimer. The Kerr-type nonlinearity in dimer enhanced by the localized surface plasmon resonance could provide a positive feedback at moderate optical intensity, so that this allows one to achieve desired nonlinear dynamics by tuning the external driving field. We explore the chaotic dynamics regime qualitatively based on the modulation instability and discuss the influence of the nonlocality and aspect ratio on the nonlinear dynamical behavior as well as the chaos regime of the nonlinear core-shell nanoparticle dimer.

2. Theoretical framework

The dynamical system under study consists of two identical core-shell spherical nanoparticles made of nonlocal plasmonic core and Kerr-type nonlinear shell with inner radius a, outer radius b, center-to-center spacing d and embedded in host medium with the permittivity ${\varepsilon _h}$ as shown in Fig. 1. The sub-wavelength core-shell nanoparticles in dimer is described within the point-dipole approximation, and we assume that the dimer is driven by a normally incident external plane wave with the frequency $\omega $ close to the plasmonic resonant frequency ${\omega _0}$ of an individual core-shell nanoparticle, whose electric field polarization is parallel to the dimer axis. Hence, the longitudinal dipoles of the nanoparticles are excited on this configuration. While the permittivity for the Kerr-type dielectric shell takes the form ${\tilde{\varepsilon }_s} = {\varepsilon _s} + {\chi _s}{|{{E_s}} |^2}$, where ${\varepsilon _s}$ is the linear part of the permittivity taken as ${\varepsilon _s} = 2.2$, the nonlinear susceptibility of the Kerr-type shell is ${\chi _s} = 4.4 \times {10^{ - 22}}{\textrm{m}^2} \cdot {\textrm{V}^{ - 2}}$, and ${|{{E_s}} |^2}$ indicates the local field intensity inside the shell [38,44]. For the sake of simplicity, we adopt the conventional linear hydrodynamic model to describe the permittivity of the nonlocal plasmonic core since the situation is out of range of considering the electron spill-out effects. Therefore, a spatially dispersive permittivity ${\varepsilon _c} = {\varepsilon _g} - \omega _p^2/[{\omega ({\omega + i\gamma } )- {\beta^2}{k^2}} ]$ [38] is introduced with ${\varepsilon _g} = 3.7$, ${\omega _p}\textrm{ = }8.9\textrm{ eV}$, $\gamma \textrm{ = }0.021\textrm{ eV}$, and $\beta \textrm{ = (3/5}{\textrm{)}^{1/2}}{\upsilon _\textrm{F}}$ where ${\upsilon _\textrm{F}}$ is the Fermi velocity. In the following discussion, we fix $b = 10\textrm{ nm}$ and $d = 30\textrm{ nm}$ to make the point-dipole approximation valid, and define the aspect ratio of the core-shell nanoparticle as $\eta = a/b$.

 

Fig. 1. Schematics of a core-shell nanoparticle dimer illuminated by a plane wave with electric field parallel to the dimer axis.

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The dynamics of the slowly varying amplitudes of the excited dipole moments for the core-shell nanoparticles in dimer could be written in dimensionless units as follows:

In Eq. (1), the normalized expressions of dipole moments ${p_{1,2}}$ and incident driving field ${E_0}$ is given by

The general stationary states of this structure are described by the solutions of the following nonlinear equations,

The above relation indicates two kinds of the steady states, one is symmetric states with ${P_1} = {P_2} = {P_0}$ and asymmetric one with ${P_1} \ne {P_2}$. The stationary solutions are depicted in Fig. 2(a). For the symmetric case, this set has a bistable solution for $\Omega < - \textrm{Re} [G] - \sqrt 3 |{\Gamma - {\mathop{\rm Im}\nolimits} [G]} |$, in which the particle polarizations become a three-valued function of ${|E |^2}$.

3. Results and discussions

In order to analyze the stability of equilibria within the proposed system, we perform the linear stability analysis and find four eigenvalues, which are referred to as the instability growth rates (${\lambda _i}$ and $i = 1,2,3,4$), of the Jacobian matrix of Eq. (1). Note that the real part of the eigenvalue determines the stability of equilibria, and at least one eigenvalue once has a positive real part of value, the corresponding state is unstable. We calculate all of them numerically both in the symmetric and asymmetric cases, and specify the stable and unstable steady states in Fig. 2(a). It should be remarked that the temporal evolution of the polarization of the dimer would show several complex dynamics [22,23] in the unstable steady state regime, including a chaotic regime especially where no stable state exists. Figure 2(a) indicates a narrow band of no stable state regime for the domain of ${|E |^2}$ from $1.19 \times {10^{ - 5}}$ to $1.29 \times {10^{ - 5}}$ for both symmetric and asymmetric cases.

 

Fig. 2. Stationary state solutions and nonlinear dynamics in the core-shell nanoparticle dimer. (a) shows the stationary steady states for $|{{P_1}} |$ and $|{{P_2}} |$ in the function of the incident field intensity with $\eta = 0.9$ and $\Omega ={-} 0.06$. Continuous and dotted curves denote the stable and unstable branches for symmetric states (black line) and asymmetric states (blue and red lines correspond to $|{{P_1}} |$ and $|{{P_2}} |$). (b) shows the bifurcation diagrams of the nonlinear dynamics obtained for the same parameters as in (a). (c) is the local enlarged drawing of (b), and the gray solid lines indicate the saturation intensity levels for the temporal dynamical responses illustrated in Fig. 3,4. (d), (e) and (f) are the same as (a), (b) and (c) but for $\eta = 0.9$ and $\Omega ={-} 0.03$, and Panel (g), (h) and (i) are for $\eta = 0.7$ and $\Omega ={-} 0.03$ similarly. Panel (j), (k) and (l) are the same as (a), (b) and (c) but under local description.

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It was demonstrated that the chaotic dynamics in this kind of system generally appears along with the no stable state regime [22]. Besides that, the partial unstable state regime, where the steady states are all unstable in asymmetric case but could be stable in symmetric one, might give rise to chaotic dynamics as well. Figure 2(a) illustrates a broader band of partial unstable regime than no stable steady state regime for the domain of ${|E |^2}$ from $0.79 \times {10^{ - 5}}$ to $1.19 \times {10^{ - 5}}$, and we find chaotic dynamics could exist in this regime as demonstrated in the following. Moreover, the band width of both no stable state regime and partial unstable state regime could be tuned by varying the aspect ratio of nonlocal core-shell particle as shown in Fig. 5 and Fig. 2(d)-(i).

To further reveal the dynamical behavior of the nonlocal core-shell nanoparticle dimer, we make separate simulations of Eq. (1) varying the background saturated field intensity as well as the peak intensity of external signal pulse. With the similar method [22], all absolute values of the nanoparticle dipole moments are recorded after fixed time interval, and we plot them in the function of saturated incident field intensity in Fig. 2(b). We set the recording time as ${\tau _{\textrm{rec}}} = 3 \times {10^4}$, and the time interval ${\tau _{\textrm{int}}} = 72$ is adjusted to ensure covering all features of dynamical process. The continuous branches indicate the stable steady states, and the results are entirely consistent with that of the linear stability analysis shown in Fig. 2(a). Here, a chaotic regime with symmetry breaking exists in the no stable state regime, as well as in the partial unstable regime. And it indicates a broader band of chaotic regime in the partial unstable regime than in the no stable state regime. Moreover, the regime with self-oscillations is discovered and exists separately from the chaotic attractor. Such behavior is quite different from that for the graphene-wrapped particle dimer [22] that the regime of self-oscillation and chaotic dynamics are near each other in the case of normal incident angle. It is of great interest to consider the transition to this regime. One will see the Hopf bifurcation at $5.25 \times {10^{ - 5}}$ associated with a stable limiting circle from a stable state via losing its stability, when decreasing the incident field intensity from the high level for the stable asymmetric branch. Then at $2.80 \times {10^{ - 5}}$ we observe numerically that it jumps into the stable steady state again and keeps the stability until the intensity is reduced to $1.29 \times {10^{ - 5}}$, and after that the chaotic dynamics occurs.

To further demonstrate the temporal chaotic dynamics, we depict the time evolutions of particles’ polarizations in Fig. 3 and 4. Figure 3(a) illustrates the transitions to the regime of chaotic dynamics resulting from the spontaneous switching from quasi-stable steady states. The background driven field intensity starts to grow from zero at $\tau = 0$, reaching a saturation level of ${|E |^2} = 1.25 \times {10^{ - 5}}$ which is located in the no stable state regime as discussed in the linear stability analysis shown in Fig. 2. As can be seen from Fig. 3(a) that strong damped oscillations are formed at early time evolution, and then it comes to the quasi-stable steady state for a while. The quasi-stable steady state could be identified by the linear stability analysis, characterized by 1 unstable and 3 stable dimensions. Repulsion from quasi-stable steady state becomes significant only when the state of system approaches its vicinity along all other stable dimensions. After that, it leaves the quasi-stable state spontaneously without any external signal pulse, and finally results in a transition to the chaotic dynamics regime with symmetric breaking where one symmetric stationary solution splits into two asymmetric curves with complex oscillations presenting a high degree of incoherence.

 

Fig. 3. Temporal evolutions of the dipole moments with (a) spontaneous switching for ${|E |^2} = 1.25 \times {10^{ - 5}}$ and (b) hard excitation switching for ${|E |^2} = 0.70 \times {10^{ - 5}}$. Bottom panels (c) and (d) show the corresponding external driving field intensity respectively.

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Fig. 4. Temporal evolutions of the dipole moments for ${|E |^2} = 1.00 \times {10^{ - 5}}$ with (a) hard excitation switching and (b) spontaneous switching. Bottom panels (c) and (d) show the corresponding external driving field intensity respectively. The insets are the local enlarged drawings of (c) and (d).

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Fig. 5. Regime of no stable steady states for asymmetric cases where $\textrm{Min}({\lambda _i}) > 0$(for each pair of ${P_1}$ and ${P_2}$) for nonlocal case (left panel) and local case (right panel) respectively. The color bar denotes the value of $\textrm{Max}({\lambda _i})$ and domain with lower value indicates the modulation instability.

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The situation is different in the partial unstable regime due to the existence of a stable symmetric steady state. The system will come into the symmetric equilibria when the background driven field intensity growing from zero, reaching to a saturation level of ${|E |^2} = 1.00 \times {10^{ - 5}}$, and it indicates a switching from the symmetric equilibria to the chaos via a quasi-stable steady state with the help of a signal pulse as shown in Fig. 4(a). The duration time of the quasi-stable steady state is $\Delta \tau \approx 5000$, which is ${\approx} 150$ times longer than the external signal pulse. This processing gives us the opportunity to transform a sharp pulse into much longer one as in a monostable multivibrator, and provides a delay to produce the chaotic signal. It should be mentioned that besides the hard excitation with an external signal pulse, the quasi-stable steady state in this case could be driven by the soft excitation as well, that is, with the help of the rapidly growing background driven field intensity to the saturation level. This process is similar to the excitation of stable symmetric steady state shown in Fig. 3(a). The triggering between these two initial states can be induced by small variations in the rapid growing rate of background driven field intensity at the beginning as shown in the insets of Fig. 4. Figure 4(b) illustrates the soft excitation of the quasi-stable steady state with a higher rapid growing rate of ${|E |^2}$, and the system eventually goes to the chaotic dynamics. The quasi-stable states in these two cases shows modulation instability, and will have the chance to be directly induced by this kind of initial condition while ignoring the stable state in the linear analysis.

Further decreasing the background driven field intensity to a lower saturation level ${|E |^2} = 0.70 \times {10^{ - 5}}$, we will find a quasi-stable steady state located between two stable branches. It provides us the possibility to trigger the polarizations between them via a quasi-stable state, whose functionality is referred as the tristable multivibrator [22]. Figure 3(b) illustrates the quasi-stable equilibrium co-existing with two stable steady states and one chaos. And it demonstrates that one could perform the transition from a quasi-stable state to the asymmetric stable steady state via a chaotic process. Moreover, it could be demonstrated (though not shown here) that the duration time of keeping the quasi-stable steady state, as well as the chaotic dynamical process can be modulated by the position and peak value of the input signal pulse. This functionality can be used to convert one short pulse into the comparatively long chaotic dynamical processing with a fixed duration time defined by the property of the dimer.

Note that the core-shell particle has an additional degree of freedom to tune the plasmonic properties via the aspect ratios, hence the dynamical behavior of the dimer would be different corresponding to various aspect ratios. As we discussed above, it is concluded that the chaotic regime of dipole dynamical process is generally accompanied by the no stable steady state, as well as partial unstable state regime with modulation instability. Since only symmetry-breaking chaos exists in the present system, it is easy to find this kind of chaotic dynamics in regime where no asymmetric stable steady states exist, i.e., both the partial unstable state regime and no stable state regime we mentioned before. To qualitatively give a brief analysis on the influence of the aspect ratio on potential chaotic dynamics of the system, we plot the no stable steady state regime for the asymmetric states in the control parameter $({\Omega , {{|E |}^2}} )$ plane based on the linear stability analysis as shown in Fig. 5. Note that no stable steady state means each pair of ${P_{1,2}}$ in a same point $({\Omega , {{|E |}^2}} )$ has at least a positive value of instability growth rate, and the domains in Fig. 5 with color indicate the maximum value of all positive instability growth rates for one point $({\Omega , {{|E |}^2}} )$. As we can see, there exist two branches of no stable steady state regime in the parameter space. It is easy to understand in the view of linear stability analysis that only if each pair of ${P_{1,2}}$ are unstable and with low positive instability growth rate, the asymmetric stats can exist in the form of self-oscillations or chaos. So that only the branch with low positive value corresponding to modulation instability might lead to the chaotic dynamics.

It is clearly illustrated in Fig. 5 and Fig. 2(d)-(i) that the modulation instability regime decreases with a smaller aspect ratio, and will disappear when $\eta $ continues to be reduced the value lower than $0.6$ (though not shown). Moreover, we could demonstrate that the potential chaos regime in the parameter space of ${|E |^2}$ is different for different $\Omega $, and generally smaller $|\Omega |$ indicates that the chaos will occur with a lower ${|E |^2}$.

All analysis and discussions above have considered the nonlocality, the nonlocal nature of the surface electron in the metal core will influence the dielectric response of the core-shell particle. Next, to give a detailed clarification of the influence of the nonlocal dielectric response of the plasmonic materials on the nonlinear dynamical process in the present dimer, we re-derive all the derivation in the theoretical part under conventional local description. Figure 6 illustrates the well-known nonlocal effects of the blue-shift of the plasmonic resonant frequency. It is clearly shown that the relationship between ${\omega _{0\_local}}$ and $\eta $ is monotonically increasing. However, the relationship between ${\omega _{0\_nonlocal}}$ and $\eta $ is not. The nonlocal effects will become dramatic when the size of the plasmonic core is reduced to an ultra-small scale, consequently ${\omega _{0\_nonlocal}}$ shows dramatical blue-shift with low $\eta $ in Fig. 6. With a similar process, we plot the no steady stable state regime of the dimer in the local case with the same geometric parameters in Fig. 5. It is found that the nonlocality will slightly reduce the area of the modulation instability regime in the control parameter $({\Omega , {{|E |}^2}} )$ plane which corresponds to chaos regime.

 

Fig. 6. Dipolar resonant frequency of the individual core-shell nanoparticle as the function of the aspect ratio with nonlocal (red solid) and local (black dashed) descriptions respectively.

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It is clearly shown that the asymmetric no steady stable state regime, as well as the chaotic regime is broader in the local case for a specific case with $\Omega ={-} 0.06$ by comparing Fig. 2(a)-(c) with Fig. 2(j)-(l). Moreover, the separated self-oscillations and chaos regimes in the nonlocal case are close to each other in the local case, and the stable steady states which was located between them disappear. So that the transition to this regime will be different from the nonlocal case, and the chaos comes directly after the self-oscillations. It should be mentioned that we use frequency detuning parameter $\Omega $ to describe the no steady stable regime in Fig. 2(a)-(c) and Fig. 2(j)-(l), if we introduce the specific driving frequency $\omega $ for comparison, the difference between nonlocal and local cases will be enlarged in view. Note that although ${\omega _{0\_nonlocal}}$ does not show obvious deviation from ${\omega _{0\_local}}$ in high $\eta $ case which regime we mainly focus on, the dynamical nonlinear process will be dramatically different especially for the chaos.

In the end, we should mention that it will be interesting to quantitatively characterize chaos in nonlinear dynamics with more robust tool –Lyapunov spectra which has been discussed in the previous work [23]. Here we qualitatively analyze the influence of the nonlocality and aspect ratio on the chaotic dynamics and chaos regime in the view of linear stability analysis. The further investigation with Lyapunov spectra will be more complicated task for the present model, and the conclusions we made are expected the same qualitatively.

4. Conclusion

In this work, we gave a general theoretical analysis of the nonlinear dynamics for a core-shell particle dimer by taking account the nonlocal nature of surface electrons in the plasmonic core at ultra-small nanoscale. This kind of simple system has been demonstrated to operate as tristable, astable multivibrators and chaos generator. And we have found a new kind of tristable multivibrator with functional switching from the symmetric to the asymmetric equilibria via a quasi-stable steady state and chaos under the normal incidence of external driving field. We also give a qualitative description on the relationship between the no steady stable state regime and chaos regime, and based on which explored the influence of the aspect ratio of core-shell particle on the chaos regime. We demonstrate that thinner nonlinear shell (lower aspect ratio) will reduce the chaos regime within the present model, and the core-shell structure could provide more transition modes due to the tunable aspect ratio. Compared to the local case, nonlocality of the plasmonic core will alter the plasmonic resonant frequency of the core-shell particle, thus change the no steady stable state regime. It is shown that nonlocality would reduce the potential chaos regime and influence the nonlinear dynamical behavior of the dimer. And this found would be very important in the design of on-chip nonlinear functional photonic nanoelements with ultra-small size.