Ning Ji, Tao Shui^,1, Yi-Lou Liu, Wang-Rui Zhang, Xiu-Mei Chen, Wen-Xing Yang^,1School of Physics and Optoelectronic Engineering, Yangtze University, Jingzhou, 434023, China

First author contact: ¹Authors to whom any correspondence should be addressed.
Received:2021-05-16Revised:2021-08-12Accepted:2021-08-16Online:2021-10-01


Abstract
Quantum coherence can be enhanced by placing metal nanoparticles (MNPs) in optical microcavities. Combining localized-surface plasmon resonances (LSPRs), nonlinear interaction between the LSPR and microcavity arrays of a MNP-microcavity complex offer a unique playground to observe novel optical phenomena and develop novel concepts for quantum manipulation. Here we theoretically demonstrate that optical solitons are achievable with a one-dimensional array which consists of a chain of periodically spaced identical MNP-microcavity complex systems. These differ from the solitons which stem from the MNPs with nonlinear Kerr-like response; the optical soliton here originates from LSPR-microcavity interaction. Using experimentally achievable parameters, we identify the conditions under which the nonlinearity induced by LSPR-microcavity interaction allows us to compensate for the dispersion caused by photon hopping of adjacent microcavities. More interestingly, the dynamics of solitons can be modulated by varying the radius of the MNP. The presented results illustrate the potential to utilize the MNP-microcavity complex for light manipulation, as well as to guide the design of photon switch and on-chip photon architecture.
Keywords： nanoparticle-microcavity;Hamiltonian structures;soliton and rational solutions


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Ning Ji, Tao Shui, Yi-Lou Liu, Wang-Rui Zhang, Xiu-Mei Chen, Wen-Xing Yang. Optical soliton in a one-dimensional array of a metal nanoparticle-microcavity complex. Communications in Theoretical Physics, 2021, 73(11): 115105- doi:10.1088/1572-9494/ac1d9e

1. Introduction

The surface plasmonic origins from noble metal nanostructures have become valuable tools in a considerable amount of applications ranging from precision sensing to quantum information processing due to their ability to localize light into a nanometer size regime [1, 2]. It is worth noting that the dissipation of the plasmonic modes which originates from the radiation and Ohmic absorption is a challenge for applications in photon nanocircuitry and precision sensors [3]. Some attempts [4–8] have been made to suppress dissipation, such as elongating metal nanoparticles (MNPs) to cancel the absorption of multiple modes, and forming arrays to obtain better coherence for a certain plasmonic mode. Despite these attempts, the nonradiative decay of plasmonic modes in MNPs is still an obstacle for coherent interactions [9]. Fortunately, reports indicate that the hybrid plasmonic photon modes in the microcavity where the localized-surface plasmon resonances (LSPRs) occur can modify the cavity mode, generating local hot spots and significantly enhancing coherent interactions [10–14]. Recently, based on the LSPR-microcavity interaction, precision sensing for single nanoparticles by monitoring the nanoparticle-dependent transmission spectra of the high-quality microcavity has been reported [15–21]. However, another remarkable property, nonlinearity originating from the interaction between the plasmonic mode and the microcavity mode, remains obscure.

In this work, we find that a self-localization phenomenon refers to an optical soliton that is achievable in a one-dimensional array consisting of a chain of periodically spaced identical MNP-microcavity complex systems. Since the word soliton was coined in 1965, the field of solitons in general and the field of optical solitons, in particular, have grown enormously [22]. In contrast to the solitons which stem from the MNPs with a nonlinear Kerr-like response [23–26], the optical soliton originates from a different mechanism—namely the interaction between the plasmonic mode and microcavity mode. The photon hopping between adjacent optical microcavities will lead to dispersion, while the interaction between MNPs and cavities brings in nonlinearity. Using experimentally achievable parameters, we show that the dispersion induced by the photon hopping can be balanced by the nonlinearity that originates from the LSPR-microcavity interaction. As a result, the dynamics of the cavity mode can be well described by using the standard nonlinear Schrödinger equation (NLSE). Interestingly enough, the dynamics of solitons can be modulated by varying the radius of the MNP. In addition, we analyzed the modulation instability in an array of a MNP-microcavity complex. The present work also shows that the hybrid MNP-microcavity array supports the long-lived localized mode, which may provide a potential solution for precision measurement and a better understanding of extreme rogue wave events.

2. Model and dynamical equation

Let us consider a one-dimensional array with periodic spaced identical MNP-microcavity complex units, in which each of these units is made up of an optical microcavity and a MNP (see figure 1). For simplicity, we assume that all cavities have the same eigenfrequency ω_a and the photon hopping between the adjacent cavities has the same rate J. Without loss of generality, we focus on spherical particles and consider particles with a scale much smaller than the wavelength to apply quasistatic approximation, so that only dipolar LSPR with resonant frequency ω_c is coupled to the cavity mode [14] with coupling strength g₀. The dipole–dipole coupling between adjacent MNPs can be defined as G. With the rotating wave approximation, the Hamiltonian of the present array can be written as:(1)$\begin{eqnarray}\begin{array}{rcl}H & = & \sum _{j}\left[{\omega }_{a}{\hat{a}}_{j}^{\dagger }{\hat{a}}_{j}+{\omega }_{c}{\hat{c}}_{j}^{\dagger }{\hat{c}}_{j}-{g}_{0}{\hat{a}}_{j}^{\dagger }{\hat{a}}_{j}({\hat{c}}_{j}^{\dagger }+{\hat{c}}_{j})\right]\\ & & -\displaystyle \frac{1}{2}\sum _{j}J({\hat{a}}_{j}{\hat{a}}_{j+1}^{\dagger }+{\rm{H}}.{\rm{C}}.)\\ & & -\displaystyle \frac{1}{2}\sum _{j}G({\hat{c}}_{j}{\hat{c}}_{j+1}^{\dagger }+{\rm{H}}.{\rm{C}}.),\end{array}\end{eqnarray}$where a_j denotes the annihilation operator of the j-th cavity mode, and c_j denotes the annihilation operator of the j-th MNP. In equation (1), the first and second terms describe the free Hamiltonian of the cavities and MNPs, the third term describes the interaction between the MNP and cavity, the fourth term describes the interaction between the adjacent cavities, and the final term describes the interaction between the MNPs. The coupling strength g₀ between the MNP and the cavity can be expressed as ${g}_{0}=-2\pi {R}^{3}\sqrt{\tfrac{{\omega }_{c}{\omega }_{a}}{{\varepsilon }_{g}{V}_{a}{V}_{c}}}| f({r}_{0})| $, where ${\varepsilon }_{g}=2({\varepsilon }_{\infty }+{\varepsilon }_{b})$ with ${\varepsilon }_{\infty }$ and ϵ_b indicate the permittivity of the metal and the background, and f(r₀) is the normalized mode distribution function. Here we are interested in the mean response of the system to the plasmonic and cavity modes, so the operators can be reduced to their expected values. The dynamical evolution of the system can be described by the Heisenberg–Langevin equations (HLEs). Taking the damping of each cavity and MNP into consideration, the HLEs read as follows:(2)$\begin{eqnarray}\begin{array}{rcl}{\dot{a}}_{j} & = & -\left({\rm{i}}{\omega }_{a}+\kappa \right){a}_{j}\\ & & +{\rm{i}}{g}_{0}{a}_{j}\left({c}_{j}^{* }+{c}_{j}\right)+{\rm{i}}\displaystyle \frac{J}{2}\left({a}_{j+1}+{a}_{j-1}\right),\end{array}\end{eqnarray}$(3)$\begin{eqnarray}\begin{array}{rcl}{\dot{c}}_{j} & = & -\left({\rm{i}}{\omega }_{c}+{\rm{\Gamma }}\right){c}_{j}+{\rm{i}}{g}_{0}{\left|{a}_{j}\right|}^{2}\\ & & +{\rm{i}}\displaystyle \frac{G}{2}\left({c}_{j+1}+{c}_{j-1}\right),\end{array}\end{eqnarray}$where ${a}_{j}\equiv \left\langle {\hat{a}}_{j}\right\rangle $ and ${c}_{j}\equiv \left\langle {\hat{c}}_{j}\right\rangle $ describe the average values of the annihilation operators corresponding to the cavity and plasmonic modes, respectively. κ and Γ are, respectively, the decay rates of the cavity and the plasmonic modes. We have made the transformation (${a}_{j}(t)\to {a}_{j}(t)\exp \left[-{\rm{i}}\left({\omega }_{a}+J\right)t\right]$) in deriving the equations (2) and (3). When we consider the dynamics of the system in the absence of nonlinearity, i.e. g₀=0, equation (2) decouples into(4)$\begin{eqnarray}{\dot{a}}_{j}=-\left({\rm{i}}{\omega }_{a}+\kappa \right){a}_{j}+{\rm{i}}\displaystyle \frac{J}{2}\left({a}_{j+1}+{a}_{j-1}\right).\end{eqnarray}$

Figure 1.

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Figure 1.(a) Schematic diagram of the considered complex system of the cavity and MNP, where a MNP is embedded in a single-mode microcavity. (b) Schematic of the interaction in the MNP-microcavity complex. (c) The one-dimensional array of the MNP-microcavity complex.


By using the ansatz ${a}_{j}=\zeta \exp [{\rm{i}}({kj}-\omega t)]$ with k, ω denotes the wave vector and frequency of the collective mode. Combining with equation (4), the dispersion relation $\omega =J[1-\cos k]$ can be obtained.

For convenience, we assume that the one-dimensional array of the MNP-microcavity complex arranges along the z direction. Then we defined that the j-th MNP-microcavity unit is located at z_j=jl with l indicating the gap between any two adjacent units. In the following discussion, we set l to a unit for the simplicity of presentation. When the number of units N is large enough, the one-dimensional array can be regarded as a continuum. Thus we can use a(z, t) and c(z, t) defined on a continuum spatial domain to describe the cavity and plasmonic modes. The evolution of the modes a(z, t) and c(z, t) can be described with a pair of nonlinear partial differential equations:(5)$\begin{eqnarray}\begin{array}{rcl}{\partial }_{t}a(z,t) & = & -\kappa a(z,t)+{\rm{i}}{g}_{0}a(z,t)\left(c(z,t)\right.\\ & & \left.+{c}^{* }(z,t\right)+{\rm{i}}\displaystyle \frac{J}{2}{\partial }_{{zz}}a(z,t),\end{array}\end{eqnarray}$(6)$\begin{eqnarray}{\partial }_{t}c(z,t)=-\left({\rm{i}}{\omega }_{c}+{\rm{\Gamma }}\right)c(z,t)+{\rm{i}}{g}_{0}{\left|a(z,t)\right|}^{2},\end{eqnarray}$where the high-order terms have been neglected, i.e. ${a}_{j+1}+{a}_{j-1}-2{a}_{j}\approx {\partial }_{{zz}}a(z,t)$. In equation (6), the dipole–dipole coupling between adjacent MNPs G has been safely ignored because of $| G| \ll | {g}_{0}| $. Assuming c(z, 0)=0, we can obtain the formal solution of equation (6) yields(7)$\begin{eqnarray}\begin{array}{rcl}c(z,t) & = & {\rm{i}}{g}_{0}{\displaystyle \int }_{0}^{t}\exp \left[-\left({\rm{i}}{\omega }_{c}+{\rm{\Gamma }}\right)\right.\\ & & \left.\times \left(t-{t}^{{\prime} }\right)\right]{\left|a\left(z,{t}^{{\prime} }\right)\right|}^{2}{\rm{d}}{t}^{{\prime} }.\end{array}\end{eqnarray}$

By means of equation (7), we can rewrite equation (5) as(8)$\begin{eqnarray}\begin{array}{rcl}0 & = & {\rm{i}}{\partial }_{t}a(z,t)+\displaystyle \frac{J}{2}{\partial }_{{zz}}a(z,t)+{\rm{i}}\kappa a(z,t)+2{g}_{0}^{2}a(z,t)\\ & & \times {\displaystyle \int }_{0}^{t}\sin \left[({\omega }_{c}-{\rm{i}}{\rm{\Gamma }})\left(t-{t}^{{\prime} }\right)\right]{\left|a\left(z,{t}^{{\prime} }\right)\right|}^{2}{\rm{d}}{t}^{{\prime} }.\end{array}\end{eqnarray}$

The first term in the right side of the above equation describes the photon hopping between adjacent units, which leads to dispersion of the cavity field; the second term describes the dissipation of the cavity field, and the final term describes the MNPs induced the shift of the resonant frequency of the microcavity mode brings in nonlinearity. By using the slowly varying approximation, i.e. $\left|{\partial }_{t}\right|a(z,t)| | \ll ({\omega }_{c}-{\rm{i}}{\rm{\Gamma }})| a(z,t)| $, the term ${\left|a\left(z,{t}^{{\prime} }\right)\right|}^{2}$ can be replaced by ${\left|a\left(z,t\right)\right|}^{2}$. Then equation (8) can be rewritten as(9)$\begin{eqnarray}\begin{array}{l}{\rm{i}}{\partial }_{t}a(z,t)+\displaystyle \frac{J}{2}{\partial }_{{zz}}a(z,t)+{\rm{i}}\kappa a(z,t)\\ \ \ +\ \displaystyle \frac{2{g}_{0}^{2}}{({\omega }_{c}-{\rm{i}}{\rm{\Gamma }})}| a(z,t){| }^{2}a(z,t)=0.\end{array}\end{eqnarray}$

If a reasonable and realistic set of parameters can be found so that both the dissipation rates of the microcavity and MNPs are much smaller than the resonant frequency of the MNP, i.e. Γ, κ≪ω_c, we can obtain the standard NLSE(10)$\begin{eqnarray}\begin{array}{l}{\rm{i}}{\partial }_{t}a(z,t)+\displaystyle \frac{J}{2}{\partial }_{{zz}}a(z,t)\\ \ \ +\ \displaystyle \frac{2{g}_{0}^{2}}{{\omega }_{c}}| a(z,t){| }^{2}a(z,t)=0,\end{array}\end{eqnarray}$where the nonlinear term depends on the coupling between the microcavity and MNP. Equation (10) admits solutions describing bright solitons, including high-order solitons for bright solitons. By defining ξ=(z−V_gt)/2z₀ (V_g=Jk denotes group velocity with k indicating the wave vector, and z₀ being the envelope width), η=t/2L_D, then equation (10) can be rewritten in the dimensionless form corresponding to the standard NLSE governing the cavity fields evolution,(11)$\begin{eqnarray}{\rm{i}}\displaystyle \frac{\partial u}{\partial \eta }+\displaystyle \frac{{\partial }^{2}u}{\partial {\xi }^{2}}+2| u{| }^{2}u=0,\end{eqnarray}$where u=a/U₀, ${L}_{D}=2{z}_{0}^{2}/J$ is the characteristic dispersion length, and ${L}_{\mathrm{NL}}={\omega }_{c}/(2{U}_{0}^{2}{g}_{0}^{2})$ is the nonlinear length with ${U}_{0}=(1/{z}_{0}){[J{\omega }_{c}/4{g}_{0}^{2}]}^{1/2}$. In equation (11) we have assumed that the characteristic dispersion length L_D is equal to the characteristic nonlinear length L_NL of the system. The above equation (11) supports the exact bright soliton solutions as(12)$\begin{eqnarray}u=2\beta {\rm{{\rm{sech}} }}[2\beta (\xi -\xi (0)+4\delta \xi )]{{\rm{e}}}^{-4{\rm{i}}{\beta }^{2}\eta },\end{eqnarray}$with β, δ, and ξ(0) being real parameters which determine the amplitude, velocity, and initial position of the soliton, respectively. By taking β=1/2 and δ=0, equation (12) can be reduced to $u={\rm{sech}} (\xi -\xi (0))\exp (-{\rm{i}}\eta )$ or, in terms of the cavity field,(13)$\begin{eqnarray}a={U}_{0}{\rm{{\rm{sech}} }}\ \left[\displaystyle \frac{1}{{z}_{0}}(z-z(0)-{V}_{g}t)\right]{{\rm{e}}}^{-{\rm{i}}t/2{L}_{D}}.\end{eqnarray}$

It is worth noting that we have used the slow-varying approximation in obtaining equations (5)–(6), i.e. $\left|{\partial }_{t}\right|a(z,t)| | \ll ({\omega }_{c}-{\rm{i}}{\rm{\Gamma }})| a(z,t)| $. Based on equation (13), we can obtain the relation(14)$\begin{eqnarray}\begin{array}{rcl}| \displaystyle \frac{{\partial }_{t}a(z,t)}{a(z,t)}| & = & | \displaystyle \frac{\tanh \left[\tfrac{1}{{z}_{0}}(z-z(0)-{V}_{g}t)\right]{V}_{g}}{{z}_{0}}\\ & & -\displaystyle \frac{{\rm{i}}}{2{L}_{D}}|\leqslant \sqrt{\displaystyle \frac{{V}_{g}^{2}}{{z}_{0}^{2}}+\displaystyle \frac{1}{4{L}_{D}^{2}}}.\end{array}\end{eqnarray}$

Thus we can identify the existing conditions of the fundamental soliton in the present system, i.e. ${z}_{0}\gg 1$, $k\ll 1$, and ${\left(\tfrac{{J}^{2}{k}^{2}}{{z}_{0}^{2}}+\tfrac{{J}^{2}}{16{z}_{0}^{4}}\right)}^{1/2}\ll {\omega }_{c}$. Checking our assumption, that leads to equation (10) being indeed practical. Below we give a practical example for a realistic one-dimensional array of the MNP-microcavity complex. For the photon crystal cavity or micropillar cavity, it has been shown that it is possible to obtain the quality factor Q=2500 with resonance frequency ω_a=2.3 eV [27]. The dissipation rate of the microcavity mode is about κ=ω_a/Q≈1meV. Considering a gold nanoparticle, the resonance frequency of the dipolar plasmonic mode can be calculated with the Drude model (ω₁=2.3 eV). The total dissipation including radiation decay and Ohmic loss is about Γ=2.65 meV according to the following [28]. As we know, the generation of solitons results from the balance between dispersion and nonlinearity. We should note that the physical mechanism leading to nonlinearity here is quite different from that of the Kerr-like response; here, the dynamic processes are caused by the interaction between the plasmonic mode and microcavity mode. With the parameters above, the existing conditions of the fundamental soliton can be well characterized, and hence the existence of bright solitons in the present system is supported.

3. Results

Based on equation (11), we show in figure 2(a) the result of numerical simulation on the soliton wave shape $| a/{U}_{0}{| }^{2}$ versus dimensionless time t/2L_D and position z/z₀ with the parameter settings outlined above by taking equation (13) as an initial condition. It can be seen that in this case that the soliton is fairly stable in the present system, which mainly results from the balance between dispersion and nonlinearity. We note that the present result is still valid even with the decay of the cavity and plasmonic modes. In figure 2(b), we also perform additional numerical simulations of the cavity field distribution $| {a}_{j}/{U}_{0}{| }^{2}$ starting directly from equations (5) and (6) via the Runge–Kutta method without using any approximation. The initial value of a_j(0) is chosen as the fundamental soliton solution at time t=0, and the initial value of the plasmonic mode is chosen as c_j(0)=0 for simplicity. It can be found in figure 2(b), except for small ripples appearing on its peak of the soliton wave shape due to the decay of the cavity and plasmonic modes. For direct insight into the influence of the decay of the cavity and plasmonic modes on the cavity field distribution, we present a comparison in figures 2(c) and (d) between the cavity field distribution a/U₀ and the fundamental soliton solution for two cases, (c) t=2L_D and (d) t=4L_D. Direct comparison of figures 2(c) and (d) implies that the result of the numerical simulation in figure 2(b) is in great agreement with the exact soliton solution in equation (13), thus the full model in equations (5) and (6) supports a nearly shape-preserving soliton. Since the dipole–dipole interaction between MNPs is mentioned in the original Hamiltonian, we will focus on discussing the influence of different values of G on the evolution of the soliton. In figure 3, we directly simulate the cavity field distribution $| {a}_{j}/{U}_{0}{| }^{2}$ from equations (2) and (3) by taking a_j(0) and c_j(0)=0 as initial conditions. In can be found in figure 3, that for several different values of G, the evolution of cavity field distribution $| {a}_{j}/{U}_{0}{| }^{2}$ is consistent with G=0. According to the above results, we can safely neglect the dipole–dipole interaction between MNPs in the present model.

Figure 2.

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Figure 2.(a) Surface plot of the cavity field mode $| a/{U}_{0}{| }^{2}$ versus dimensionless time t/2L_D and position z/z₀ obtained by numerically solving equation (11) with the initial condition given in equation (13). (b) Surface plot of the cavity mode $| {a}_{j}/{U}_{0}{| }^{2}$ versus dimensionless time t/2L_D and position z/z₀ obtained by numerically solving equations (5) and (6) without using any approximation. (c) and (d) The system parameters are chosen as κ/2π=3.8 MHz, J/2π=−95 GHz, ω_c=2.3 eV, g₀=−2.9 meV, N=100, Γ=2.65 meV.

Figure 3.

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Figure 3.Comparison of the cavity field distribution $| {a}_{j}/{U}_{0}{| }^{2}$ for several different values of G. (a) t=2L_D and (b) t=4L_D, for G=0 (solid red line); G=−0.9 meV (dotted green line), and G=−1.5 meV (dashed blue line). The other parameters are the same as those shown in figure 2.


The collision property between two solitons is one of the most intriguing aspects of soliton dynamics. To obtain further confirmation on the soliton solution obtained above and check its stability, we perform the simulations of the soliton collision by solving equations (5) and (6) with the initial distribution of the field consisting of a soliton pair so that(15)$\begin{eqnarray}\begin{array}{rcl}u(\xi ,0) & = & {\rm{{\rm{sech}} }}[\xi -\xi (0)]\exp ({\rm{i}}\varphi )\\ & & +q{\rm{{\rm{sech}} }}[q\xi +q\xi (0)],\end{array}\end{eqnarray}$where q is the relative amplitude of the two solitons, and φ is the relative phase. Figure 4 shows the evolution of a soliton pair for several values of the parameters q and φ. Obviously, soliton collision depends sensitively on both the relative phase φ and the amplitude ratio q. In the case of equal-amplitude solitons (q=1), figure 4(a) displays the two solitons attracting each other in the in-phase case ($\varphi =0$) such that they collide periodically along with the array of the MNP-microcavity complex. However, for the cases of φ=π/4 and φ=π/2, the solitons repel each other, and their separation increases with the evolution. From the view of design and application, repulsion and separation are not acceptable. It would lead to instability and jitter in the arrival time of the solitons because the relative phase of neighboring solitons is not likely to remain well controlled. It is known that the increase of the initial separation ξ(0) can be used to avoid soliton interaction. However, the large separation between the two solitons limits the bit rate in the application of the communication. As shown in figure 4(d), the separation for two in-phase solitons almost remains unchanged for an initial soliton separating as small as 70 units length in the present array of the MNP-microcavity complex if their initial amplitudes differ from 10% (q=1). In addition, one can also find that the peak intensity of the two solitons deviates by only 1%. Note that such small deviations in the peak intensity are of little significance for the maintenance of solitons.

Figure 4.

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Figure 4.Surface plot of the solitary wave versus dimensionless time t/2L_D and position z/z₀ for collision between two solitons under four different choices of amplitude ratio and relative phase, i.e. (a) q=1, φ=0; (b) q=1, φ=π/4; (c) q=1, φ=π/2, and (d) q=1.1, φ=0. Initially the separation of the solitons is 70 units length, corresponding to ξ(0)=3.5. The time is 20 times the dispersion length, corresponding to 20L_D. The system parameters are chosen as κ/2π=3.8 MHz, J/2π=−95 GHz, ω_c=2.3 eV, g₀=−2.9 meV, G=0, N=100, Γ=2.65 meV.


As illustrated in figures 2 and 4, the nonlinearity induced by LSPR-microcavity interaction can compensate the dispersion caused by photon hopping of the adjacent microcavity, and thus the optical soliton can be achievable in the present array of the MNP-microcavity complex. However, it is not difficult to see that a decrease in the peak intensity of the soliton due to the decay of the cavity and plasmonic modes would produce soliton broadening because the reduced peak intensity weakens the nonlinear effect necessary to compensate for the dispersion. Interestingly enough, the decay effect of the system can be compensated by choosing the appropriate radius of the MNP. Figure 5 shows that the coupling strength g₀ between the LSPR and microcavity depends on the radius of the MNP. We now turn to the management of the decay through amplifying the peak intensity of the soliton by modulating the radius of the MNP. Figure 6 shows the evolution of a fundamental soliton for two different radii of MNPs by solving equations (5) and (6). It can be found that the soliton is preserved better for R=50 nm than that for R=30 nm. The radius dependence of the soliton in figure 6 can readily be understood according to equation (10). The nonlinearity coefficient $2{g}_{0}^{2}/{\omega }_{c}$ increases when the radius of the MNP increases. The balance between the enhanced nonlinear and the dispersion caused by photon hopping of the adjacent microcavity is reasonably well satisfied even when the decay of the system is included, and thus the soliton is able to sustain itself in the present system.

Figure 5.

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Figure 5.Coupling strength g₀ between the MNP and cavity as a function of the radius R of the MNP with ϵ_g=4. The mode volume of the cavity mode is chosen as V_a=1 μm³. For maximum overlap, i.e. $| f({r}_{0})| =1$.

Figure 6.

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Figure 6.Surface plot of the solitary wave versus dimensionless time t/2L_D and position z/z₀ under two different radius of the MNP, i.e. (a) R=30 nm; (b) R=50 nm. The other parameters are the same as in figure 2.


In addition, since small perturbations are most common in nature, we will focus on studying the evolution of initially small perturbations of the fundamental soliton in the standard framework by using the perturbation method. The analytical solutions of equations (5) and (6) can be written in a general form: a(z, t)=a₀(z, t)+δ a(z, t) and c(z, t)=c₀(z, t)+δ c(z, t), where δ a(z, t) and δ c(z, t) are the perturbations of the cavity field and plasmonic, respectively. Substitution of the solution into the evolution equations (5) and (6) and ignoring the high-order terms of δ a(z, t) and δ c(z, t) leads to the following linear equations:(16)$\begin{eqnarray}\begin{array}{rcl}2{\rm{i}}\displaystyle \frac{\partial \delta a}{\partial t} & = & -{\rm{i}}\kappa ({a}_{0}+\delta a)-2{g}_{0}\left[{a}_{0}(\delta c+\delta {c}^{* })\right.\\ & & \left.+\delta a({c}_{0}+{c}_{0}^{* })\right]+J\displaystyle \frac{{\partial }^{2}\delta a}{\partial {z}^{2}},\end{array}\end{eqnarray}$(17)$\begin{eqnarray}\displaystyle \frac{\partial \delta c}{\partial t}=-\left({\rm{i}}{\omega }_{c}+{\rm{\Gamma }}\right)\delta c+{\rm{i}}{g}_{0}({a}_{0}^{* }\delta a+{a}_{0}\delta {a}^{* }),\end{eqnarray}$which describe the evolutionary perturbations in position and time. The perturbations of the plasmonic can be adiabatically eliminated under the slowly varying approximation of the cavity field, viz. ${\rm{d}}\left|a(z,t)\right|/{\rm{d}}t\ll {\omega }_{c}\left|a(z,t)\right|$. In this case, the perturbations of the cavity field can be described by a linear partial differential equation:(18)$\begin{eqnarray}{\rm{i}}{\partial }_{t}\delta a+\displaystyle \frac{J}{2}{\partial }_{{zz}}\delta a+\displaystyle \frac{2{g}_{0}^{2}}{({\omega }_{c}-{\rm{i}}{\rm{\Gamma }})}\left({a}_{0}^{2}\delta {a}^{* }+| {a}_{0}{| }^{2}\delta a\right)=0.\end{eqnarray}$The stability of the evolutionary perturbation can also be confirmed numerically. The noise of the cavity and MNPs can be well described by the white Gaussian noise $\hat{a}$ and $\hat{{F}_{\mathrm{th}}}$ with $\begin{eqnarray*}\left\langle {\hat{a}}_{\mathrm{in}}(t){\hat{a}}_{\mathrm{in}}^{\dagger }({t}^{{\prime} })\right\rangle =\delta (t-{t}^{{\prime} })\end{eqnarray*}$$\left\langle {\hat{a}}_{\mathrm{in}}(t)\right\rangle =0$, $\left\langle {\hat{F}}_{\mathrm{th}}(t){\hat{F}}_{\mathrm{th}}^{\dagger }({t}^{{\prime} })\right\rangle ={\rm{\Gamma }}\int {{\rm{e}}}^{-{\rm{i}}\omega (t-{t}^{{\prime} })}$ $\begin{eqnarray*}\left[{\rm{\coth }}({\hslash }\omega /2{k}_{B}T)+1\right]{\rm{d}}\omega /2\pi {\omega }_{c}\end{eqnarray*}$ and $\begin{eqnarray*}\left\langle {\hat{F}}_{\mathrm{th}}(t)\right\rangle =0\end{eqnarray*}$ and the evolution of the cavity field distribution with the initial distribution equation (13) by including noise is shown in figure 7. As shown by the numerical results, the typical spatiotemporal structure is retained from the strong noise which confirms the stability of the fundamental soliton in the MNP-microcavity array.

Figure 7.

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Figure 7.The evolution of the cavity field distribution in the complex system under the initial distribution equation (13). (a) and (b) as comparison figures, indicate the evolution of the cavity field distribution in the MNP-microcavity array in the presence of white Gaussian noise under the same parameters. The white Gaussian noise is chosen with a SNR (signal to noise ratio) of about 20 dB here. The other parameters are the same as those outlined in figure 2.

4. Conclusions

In conclusion, we have shown that optical solitons are achievable in a one-dimensional array that consists of a chain of periodically spaced identical MNP-microcavity complex systems. We have demonstrated that the dynamics of the cavity fields distribution of the present complex system can be described by the standard NLSE under slow-varying approximation; results show no difference from the discrete model. It is worth noting that formation of the soliton originates from the good balanced between the nonlinear induced by LSPR-microcavity interaction and the dispersion caused by the photon hopping of the adjacent microcavity, which is different from the solitons which stem from the MNPs with a nonlinear Kerr-like response. In addition, we have analyzed the influence of the decay of the cavity and plasmonic modes on the soliton by numerically simulating the original dynamics equations of the cavity and plasmonic modes. Interestingly, we found that a decrease in the peak intensity of the soliton due to the decay of the cavity and plasmonic modes can be suppressed by varying the radius of the MNP, which may lead to many potential applications in optical communications and optical engineering. The present results provide a new possibility of utilizing the MNP-microcavity complex for light manipulation, as well as to guide the design of the photon switch and on-chip photon architecture.

Acknowledgments

The research is supported in part by the National Natural Science Foundation of China under Grant Nos. 11 774 054 and 12 075 036.

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