1. Introduction

As a typical optical nonlinear phenomenon, the optical bistability (OB) refers to an optical effect where a system displays two possible outputs for a single input field, which provides an applicable way for the manipulation of light by light in modern science and engineering [1,2]. For instance, it plays an important role in the design of modern photonic functionalities in the field of optical communication and optical computer [3], including optical memory elements [4], optical transistors [5], ultrafast optical switches [6], optical multivibrators [7] and so on.

Since governed by photon-photon interactions, the optical nonlinearities are superlinearly dependent on the applied field and inherently weak in general. However, research suggests that nonlinear optical response can be boosted dramatically through local field effects induced by the surface plasmon resonance (SPR) [8], which is quite common in metallic plasmonic nanostructures [9–11]. Incidentally, the well-known graphene is also a good candidate in the plasmonic community due to the large enhancement of optical nonlinearity [12] and even the realization of optical bistability [13–15]. Without involving resonators or external feedback structures, such nanocomposites can exhibit extremely fast operations.

Because of the existence of the nonlinearity, it is quite intractable to deal with the problem of OB in such plasmonic nanocomposite materials. In the past decades, a number of theoretical works concentrated on the microstructures, which are exactly solvable, like parallel slabs with nonlinear plasmonics, low-density nanocomposites with nonlinear plasmonic inclusions [16], and two-level system [17]. In particular, Bergman et al. investigated the OB of nanocomposites containing plasmonic nanospheres in the nonlinear host based on the variational principle [18]. Moreover, with the self-consistent mean-field approximation, a general framework was proposed to study the OB in nonlinear and plasmonic nanocomposites [19,20]. A semi-analytical method for studying nonlinear light scattering from multilayer cylindrical nanostructures was developed [21]. With the advancement of computers, numerical simulations of nonlinear nanostructures based on the finite-difference time-domain method are possible [22]. Incidentally, the experimental realizations of OB in spherical CdS coated with plasmonic silver [23] and in bucked dome microcavities [24] were also reported.

On the other hand, stemming from symmetry breaking of local basic elements, the radial dielectric anisotropy can be widely found in both natural and artificial materials with different length-scales, including plasma membranes [25], liquid crystal [26], artificial cloak [27], and even bulk of immigrant cells [28]. The intuitive property of the dielectric radial anisotropy is that the permittivity is a tensor with nonidentical diagonal elements, and one should note that such dielectric anisotropy is controllable with designed nanostructures [29]. Historically, based on the directional dependence, the radially anisotropic nanocomposites containing plasmonic nanoparticles were extensively studied in the tunability of light scattering [30], second and third harmonic generations [31], and even the optical bistable behavior [32,33]. It is particularly encouraging to apply the radial anisotropy to nonlinear plasmonic devices.

In this work, we would like to investigate the optical bistable behavior in plasmonic coated nanospheres containing nonlinear plasmonic shell and dielectric core with radial anisotropy. In our analysis, the self-consistent mean-field approximation is applied in the calculation of the relationship between the spatial average local field in the nonlinear shell and the applied external field. The model and the method can give rise to a modulational way for realizing two optical bistabilities and even tristability through the adjustment of the core-shell radii ratio and the radial anisotropic parameter. Based on the spectral representation analysis, we shall show that the mechanism for the presence of optical tristability in our system is due to the interaction between two dipole surface plasmon resonant modes, which is quite different from that proposed in the hybrid optomechanical system due to the introduction of the optical parametric amplifier [34]. The modulation of such optical nonlinearity may lead to some potential applications including low-intensity optical memories, switches, and sensitive tunable sensors.

The paper is organized as follows. In section 2, we describe our model and derive the theoretical formulae. In section 3, with further numerical calculation, we demonstrate the existence of single OB, double OB, and optical tristability (OT) in plasmonic coated nanospheres with radial anisotropy, and we also discuss the origin of nonlinear optical behavior qualitatively by casting the problem into spectral representation theory. Possible observation of optical bistable and tristable behavior was proposed in the semi-infinite nanocomposite materials. And we end this paper with a conclusion and expectation in section 4.

2. Theoretical development

2.1 Linear electrostatic solution of coated spheres with radial anisotropy

Let us first consider linear composites in which nondilute coated spheres with a shell of radius b and the permittivity ${\varepsilon _s}$, and the anisotropic core of radius a, are randomly embedded in the host medium with permittivity ${\varepsilon _s}$, as shown in Fig. 1. The volume fraction of the coated spheres and the host are f and $1 - f$ respectively. Since the coated particles are randomly oriented, the whole system is still isotropic. Without loss of generality, we restrict our work to the quasi-static approximation.

 

Fig. 1. Schematic diagram of the nondilute composite in which the anisotropic spherical inclusions are coated by nonlinear metal shell.

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The permittivity tensor of the anisotropic core with radial anisotropy is diagonal with a value ${\varepsilon _r}$ in the radial direction and ${\varepsilon _t}$ in the tangential directions in spherical-polar coordinates [25,35], i.e.,

The radial anisotropy can be easily established from a problem of spherically stratified medium [29,35,36]. With the static Maxwell equations $\nabla \cdot \textbf{D} = 0$ and $\nabla \times \textbf{E} = 0$, we arrive at the “tensorial Laplace equation” describing the electrostatic potential $\nabla \cdot \left( {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over \varepsilon } \cdot \nabla \phi } \right) = 0$, which can be cast into the spherical coordinates as,

In the case of the spherical symmetry, the general form of potential $\phi $ can be chosen as,

Substituting Eq. (3) into Eq. (2), one can easily obtain the general solution of $\Theta (\theta )= {P_n}({\cos \theta } )$ and the ordinary differential equation for the radial function ${R_n}(r )$,

Here, we would like to mention that the dipole interactions among the coated inclusions are taken into account by introducing the Lorentz local field, With the boundary conditions, one can obtain the electric potentials inside the three regions (the spherical core, shell and host medium) respectively, which are [31]

2.2 Optical bistable behavior in the general nonlinear case

In this subsection, the whole system is assumed to be composed of coated spheres with the nonlinear shell, whose permittivity is field-dependent with

As the magnitude of the applied field increases, one can expect that the nonlinear term may be comparable to or even larger than the linear one. In this regard, optical bistability may arise, and the field-dependent effective response should be estimated. Due to the existence of both nonlinearity and radial anisotropy, the local field in the core (or shell) will be non-uniform. For solving this nonlinear local field, we resort to the mean-field approximation [15,19,20]. In this connection, the field-dependent permittivity of the shell can be approximated by

Hence, for the nonlinear system, the electric potentials in Eq. (5) can be rewritten by substituting ${\varepsilon _s}$ with ${\tilde{\varepsilon }_s}$. And the average of local field squared inside the nonlinear shell can then be written as,

Due to ${\tilde{\varepsilon }_s} \propto {\left\langle {{{|\textbf{E} |}^2}} \right\rangle _s}$, we can find that Eq. (9) is a fifth-order equation for ${\left\langle {{{|\textbf{E} |}^2}} \right\rangle _s}$, which can be self-consistently solved for a given applied field ${\textbf{E}_0}$ with a set of parameters of the system. Hence, the optical bistability or even tristability should be expected in the present system.

2.3. Effective permittivity and reflectivity of the nonlinear composite system

In this subsection, we derive the effective field-dependent permittivity and the reflectivity from the nonlinear composite system.

The effective field-dependent permittivity ${\tilde{\varepsilon }_\textrm{e}}$ can be determined by ${\tilde{\varepsilon }_e} = \left\langle \textbf{D} \right\rangle /{\textbf{E}_0}$. And the spatial average of the electric displacement $\left\langle \textbf{D} \right\rangle$ is,

As a consequence, the effective field-dependent permittivity is derived to be

Hence, at normal incidence, the reflectivity of the nonlinear composite system can be easily obtained by,

3. Numerical results and discussion

In this section, we perform numerical calculations on the nonlinear optical behavior in plasmonic coated nanospheres with radial anisotropy. The linear optical parameters of the plasmonic shell and the host are selected as ${\varepsilon _s} ={-} 7.1 + 0.22i$ (the permittivity of Ag at the visible frequency) and ${\varepsilon _m} = 1.772$ respectively. Also, we consider the nonlinear susceptibility ${\chi _s} \approx {10^{ - 11}}{(m/V)^2}$ [37]. For simplicity, we set $f = 0.1$ to deal with the nondilute situation. We aim at the effect of the interfacial parameter $\eta $ and the radial anisotropy ratio $\gamma = {\varepsilon _t}/{\varepsilon _r}$ on the tunability of optical nonlinear behavior.

In Fig. 2, we show the relationship between the incident field ${E_0}$ and the average local field $\left\langle {{{|\textbf{E} |}^2}} \right\rangle _s^{1/2}$ in the nonlinear shell. The cases are chosen as $\gamma = 1/4$, $1$ (the isotropic case) and $5/2$ with different $\eta $ for comparison. For the isotropic case ($\gamma = 1$), we can classify the curves into two groups, one with only single OB behavior ($\eta = 0.3$,$0.5$), and the other with double OB behavior which has no overlap between the two OB regions ($\eta = 0.02$,$0.06$,$0.1$). While for the case of $\gamma = 5/2$, the groups can be classified into three cases, the single OB behavior ($\eta = 0.1$,$0.3$,$0.5$), the double OB behavior with no overlap ($\eta = 0.06$) and the double OB with overlap ($\eta = 0.02$) which is called the optical tristability (OT) [19,38]. These all-optical switching properties, combined with the tunable interfacial and anisotropic parameters may directly lead to the design of nonlinear optical sensors and nanocircuit components.

 

Fig. 2. The average local electric field in the nonlinear shell $\left\langle {{{|\textbf{E} |}^2}} \right\rangle _s^{1/2}$ as a function of the external incident field ${E_0}$ for various interfacial parameters $\eta $ with different anisotropic ratios (a) $\gamma = 1/4$, (b) $\gamma = 1$, (c) $\gamma = 5/2$, respectively. The steady and unsteady states are not distinguished from each other as the Fig. 3 shown.

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To distinguish these three nonlinear optical properties in detail, we show their typical representatives in Fig. 3. The typical single OB behavior is shown in Fig. 3(a) whose changing process is apparently similar to the magnetic hysteresis loop. As increasing ${E_0}$, the system reaches the first abrupt point, and $\left\langle {{{|\textbf{E} |}^2}} \right\rangle _s^{1/2}$ jumps discontinuously to the upper branch with a higher value. Further increasing ${E_0}$ leads to a monotonic increase in $\left\langle {{{|\textbf{E} |}^2}} \right\rangle _s^{1/2}$. On the contrary, with the decrease of ${E_0}$, $\left\langle {{{|\textbf{E} |}^2}} \right\rangle _s^{1/2}$ is declined to the second abrupt point monotonically, and then triggers dumping to the lower value. The abrupt points are denoted as the switch-up threshold ${E_{1,U}}$ and switch-down threshold ${E_{1,L}}$, respectively. Different from the single OB behavior, the double OB or OT behavior undergoes four abrupt changes of $\left\langle {{{|\textbf{E} |}^2}} \right\rangle _s^{1/2}$ within the whole loops. As a consequence, one observes four corresponding thresholds, which are denoted as ${E_{1,U}}$, ${E_{2,U}}$, ${E_{1,L}}$ and ${E_{2,L}}$ respectively as shown in Figs. 3(b) and (c). It is quite obvious to describe the major difference between double OB and OT. That is, when ${E_{1,U}} < {E_{2,L}}$, the two OB regions have no overlap, which refers to the double OB behavior; while for ${E_{1,U}} > {E_{2,L}}$, the overlapping region of the two OB just indicates the OT behavior.

 

Fig. 3. Three typical optical nonlinear properties are shown. They are, respectively, (a) the single OB with $\gamma = 1$, $\eta = 0.3$; (b) the double OB with $\gamma = 1/4$, $\eta = 0.06$; and (c) the OT with $\gamma = 5/2$, $\eta = 0.02$. The negative slope (dotted lines) represents the unsteady state. The arrows indicate the process of the whole evolution.

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Let us now analyze the impact of the interfacial parameter $\eta $ and the anisotropic parameter $\gamma $ on the single OB, double OB and OT in detail. To this end, we plot the threshold values of ${E_{i,L}}$ and ${E_{i,U}}$ as a function of $\eta $ for different $\gamma $ in Figs. 4(a)–4(c). It is evident that both ${E_{1,L}}$ and ${E_{1,U}}$ are decreased with increasing $\eta $ in the first OB with lower ${E_0}$. However, with the increase of $\eta $, ${E_{1,U}}$ is decreased more rapidly than ${E_{1,L}}$, and hence the first OB region gets narrower and vanishes ultimately. This leads to one critical interfacial parameter ${\eta _{c1}}$ above which the first OB vanishes. Moreover, ${\eta _{c1}}$ is found to be decreased with the increase of $\gamma $ by the comparison among Figs. 4(a)–4(c). As for the second OB with higher ${E_0}$, the phenomenon is quite different. With increasing $\eta $, ${E_{2,L}}$ is decreased rapidly first and keeps almost invariant with a saturation value, whereas ${E_{2,U}}$ is increased monotonously. As a result, the second OB region gets wider for larger $\eta $. Here, we would like to mention that the second OB only arises for $\eta > {\eta _{\textrm{c}2}} \sim {10^{ - 3}}$ [or see the insert of Fig. 4(a)], which indicates the existence of finite core-size. Therefore, the second OB (with higher ${E_0}$) may result from the local plasmon resonance at the interface between the anisotropic core and nonlinear plasmonic shell. This also supports the fact that the upper and lower threshold fields for the second OB are much more dependent on the anisotropic parameter $\gamma $ than those for the first OB.

 

Fig. 4. The switch-up and switch-down threshold fields as a function of $\eta $ for (a) $\gamma = 1/4$, (b) $\gamma = 1$, (c) $\gamma = 5/2$; and as a function of $\gamma $ for (d) $\eta = 0.01$. The colored regions indicate the existence of OB and OT with grey and yellow respectively. The blue dots and green dots represent the critical points ${\eta _{c1}}$ and ${\eta _{c2}}$ respectively. The insert in (a): with log-plot of x-axis in the small ${\eta _{}}$.

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Due to the variation tendency of two types of OB in Figs. 4(a)–4(c), there is double OB for ${\eta _{c2}} < \eta < {\eta _{c1}}$, and single OB mainly exists for $\eta > {\eta _{c1}}$. Moreover, it is intuitively shown that for the cases of $\gamma = 1/4$ (the tangential permittivity is smaller than the radial one) and $\gamma = 1$ (the tangential permittivity equals the radial one, i.e., the isotropic case), there is no OT, which indicates there is a weak coupling between the two local plasmon resonant modes. According to what we described above, the necessary condition to have OT is ${E_{1,U}} > {E_{2,L}}$. In Fig. 4(d), we can see that ${E_{2,L}}$ is significantly reduced with the increase of $\gamma $, the emergence of OT behavior is quite possible for large $\gamma $. As a consequence, the anisotropic parameter $\gamma $ is highly related to the modulation of OT behavior in such coated plasmonic nanospheres with radial anisotropy. This example emphasizes the feasibility of the nonlinear coated nanospheres as an optical nano-memory with large storage due to the optical tristability.

To explain the modulation qualitatively on the optical nonlinear phenomena by $\eta $ and $\gamma $, we try to cast the problem into the spectral representation [19]. Let’s set $\tilde{s} \equiv {\varepsilon _m}/({{\varepsilon_m} - {{\tilde{\varepsilon }}_s}} )$, $s \equiv {\varepsilon _m}/({{\varepsilon_m} - {\varepsilon_s}} )$ and $x \equiv {\nu _1}{\varepsilon _r}/{\varepsilon _m}$, then Eq. (9) admits,

In Fig. 5, we plot two poles ${s_ - }$ and ${s_ + }$ as a function of $\eta $. In addition, with the given permittivities, the real part of s is in the vicinity of $0.2$. For $\gamma = 1/4$ (or $\gamma = 1$), we have ${s_ - } < {\mathop{\rm Re}\nolimits} (s )< {s_ + }$ with the large geometric parameter such as $\eta > 0.24$ (or $\eta > 0.15$). In this region, with increasing the applied incident field gradually, ${\mathop{\rm Re}\nolimits} ({\tilde{s}} )\textrm{ = }{\mathop{\rm Re}\nolimits} \left[ {{\varepsilon_m}/\left( {{\varepsilon_m} - {\varepsilon_s} - {\chi_s}{{\left\langle {{{|\textbf{E} |}^2}} \right\rangle }_s}} \right)} \right]$ increases monotonically for the given $\eta $, and hence it gets far away from ${s_ - }$, but gets close to ${s_ + }$. As a consequence, the contribution from the terms of ${({\tilde{s} - {s_ + }} )^{ - 1}}$ in Eq. (13) to ${\left\langle {{{|\textbf{E} |}^2}} \right\rangle _s}$ becomes much more important than that from the terms of ${({\tilde{s} - {s_ - }} )^{ - 1}}$, which can be omitted. In this sense, Eq. (13) is reduced to a cubic equation of ${\left\langle {{{|\textbf{E} |}^2}} \right\rangle _s}$, which signifies a single OB behavior.

 

Fig. 5. Dependence of the poles ${s_ - }$ and ${s_ + }$ on the geometric parameter $\eta $ with various anisotropic ratio $\gamma $. The separation distance between ${s_ - }$ and ${s_ + }$ increases with $\eta $, but decreases with $\gamma $, especially in the region of small $\eta $. The black dotted line is the real part of s, which indicates the linear situation. The corresponding values of $\eta $ for the intersections are about $0.24$, $0.15$, $0.08$ with $\gamma = 1/4$, $1$, $5/2$ respectively.

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On the other hand, in the region of $\eta < 0.24$ (or $\eta < 0.15$) for $\gamma = 1/4$ (or $\gamma = 1$), both poles ${s_ - }$ and ${s_ + }$ are larger than ${\mathop{\rm Re}\nolimits} (s )$. In this connection, with increasing ${E_0}$, ${\mathop{\rm Re}\nolimits} ({\tilde{s}} )$ gets close to ${s_ - }$ at first, and the terms of ${({\tilde{s} - {s_ - }} )^{ - 1}}$ in Eq. (13) becomes the resonant part, while the terms of ${({\tilde{s} - {s_ + }} )^{ - 1}}$ is much less than the former ones, resulting in the first OB. With the further increase of ${E_0}$, ${\mathop{\rm Re}\nolimits} ({\tilde{s}} )$ approaches to ${s_ + }$, and thus the second OB is observed similarly. Since ${s_ - }$ and ${s_ + }$ are well separated from each other, the two OB behaviors can hardly have the overlapping region of ${E_0}$. In other words, no OT takes place.

However, for large anisotropic parameters such as $\gamma \textrm{ = }5/2$, it is easy to find that ${s_ - }$ and ${s_ + }$ are getting close, especially for small $\eta $. In this situation, since the poles ${s_ - }$ and ${s_ + }$ approach to each other (but not the same), the terms containing ${({\tilde{s} - {s_ - }} )^{ - 1}}$ and the terms containing ${({\tilde{s} - {s_ + }} )^{ - 1}}$ almost have the equal contribution to ${\left\langle {{{|\textbf{E} |}^2}} \right\rangle _s}$. Then, Eq. (13) becomes a fifth-order polynomial equation of ${\left\langle {{{|\textbf{E} |}^2}} \right\rangle _s}$, which possibly produces five real roots for a given applied field ${E_0}$, thus implying the OT behavior.

Actually, we predict the OT in such plasmonic nanospheres due to the fact that two dipole symmetric and asymmetric resonant modes exist for metallic plasmonics in the shell [32]. This mechanism is quite different from the previous OT predicted in nonlinear composite media containing nonspherical particles, for which one electric dipole resonant mode along one direction interacts with the other electric dipole one along the other direction [19]. Here, the two electric dipole resonant modes are stemmed from the two metal-dielectric interfaces of the core-shell structure. So the coupling of dipole modes is quite sensitive to the distance between them, which means a proper $\eta $ for realizing OT. Also, as the first OB vanishes with the increase of $\eta $, the tunability of $\gamma $ is the essential part in our system. In addition, in nonlinear graphene-wrapped nanospheres, the OTs are associated with the magnetic dipolar mode and the magnetic quadrupolar mode [15].

In the end, in order to observe such behavior for more practical purposes, we suggest investigating the measurable physical parameter such as the reflectivity of such nonlinear composite system. We plot the reflectivity with normal incidence in Fig. 6. It is found that the loops of reflectance curves are quite consistent with the variations of local field $\left\langle {{{|\textbf{E} |}^2}} \right\rangle _s^{1/2}$, which indicates the existence of bistable or tristable states in the composite system obviously.

 

Fig. 6. The reflectance R at normal incidence (the solid black line) and the average local field of nonlinear shell $\left\langle {{{|\textbf{E} |}^2}} \right\rangle _s^{1/2}$ vs. ${E_0}$ (the red dash line) for $\gamma = 5/2$ and for (a) $\eta = 0.02$, (b) $\eta = 0.06$, and (c) $\eta = 0.1$. The loops of R indicate that it is possible to show different reflectances for a given applied field ${E_0}$, which is consistent with the solution of $\left\langle {{{|\textbf{E} |}^2}} \right\rangle _s^{1/2}$.

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For real experiments, as one OB exists because of the nonlinear shell, for the tristability, the practical device should have not only a small core to ensure the existence of two bisabilities, but also a core with $\gamma > 1$ to ensure the overlapping of the two OB region. For example, as shown in Fig. 4(d), we can choose $\eta = 0.01$ and $\gamma = 2$. As the observation of intrinsic OB in similar core-shell structure was reported in the early time [23], the design of tunable effective anisotropy in the core is quite essential. From the previous work, we can find the structure of multilayers is quite an effective way for realizing the anisotropy [36]. Furthermore, within the framework of transformation optics, one can deal with the radial anisotropy from a geometric viewpoint [40]. Therefore, from the viewpoint of possible applications, our system can be used in an all-optical switching nanodevices, because the phenomena can be switched among different nonlinear optical states by changing the optical anisotropy and the geometric parameter.

4. Conclusions

In this paper, based on self-consistent mean-field approximation, we investigate the optical nonlinear properties of nondilute suspension composed of nonlinear coated nanospheres with anisotropic cores. We deal with the nondilute situation by introducing the Lorentz local field. Consequently, the established formulae give a fifth-order polynomial equation, which mathematically predicts the appearance of optical multistability. We demonstrate that the existence of single OB, double OB, and OT behaviors strongly depend on the anisotropy parameter $\gamma $ and the interfacial parameter $\eta $. We predict two critical interfacial parameters, between which ${\eta _{c2}} < \eta < {\eta _{c1}}$ the emergence of double OB is possible. In the meanwhile, the emergence of OT behavior is mainly ensured by large $\gamma $. With the adjustment of the anisotropy parameter $\gamma $ and the interfacial parameter $\eta $, one can realize the switchings among single OB, double OB, and OT, and hence achieve more functionality about optical switching in this proposed nano-devices.

Some comments are in order. For nonlinear plasmonic nanoparticles with dielectric anisotropy, one can take one step forward to study the nonlinear dynamic behavior [7,41,42], and the anisotropic parameter may serve as a powerful tool for steering the modulational instability and other nonlinear phenomena. It is a remarkable fact that such anisotropy can be designed and tuned through transformation optics [43] and so on. As for the plasmonic shell, we only introduce the local permittivity. Actually, at the nanoscale, the spatial dispersion or nonlocality should be taken into account. Along this line, the interaction between nonlocality and Kerr-nonlinearity in plasmonics may result in much abundant all-optical nonlinear behavior [44–46]. Furthermore, one can expect the effect of exciton-plasmon and exciton-phonon interactions on OB in coupled hybrid system via the adjusting the intensity or frequency of pump field [47,48]. Based on these, we can expect more tunable nonlinear optical nanodevices with the development of nanofabrication technology.