1. Introduction

Nanoparticle lasers (NPLs) attract more and more attention in recent years arising from the potential applications in many active fields such as biosensing, data storage, optical communications and medical applications [1–4]. This kind of nanoparticle laser is called “spaser” (short for surface plasmon amplification by stimulated emission of radiation) due to the fact that the lasing is based on the strong interactions between nanosized emitters and localized surface plasmons (LSPs), and can generate coherent plasmonic fields in the subwavelength regions [5]. The theoretical proposal of the spaser was recently demonstrated experimentally [6] with nanoparticles containing a gold core and a dye-doped silica shell, from which it can be understood that the core-shell nanoparticle itself serves as a resonator (or a resonant cavity) and the adjacent gain medium delivers energy to the SP mode. For spasers, the main challenge is to decrease the high gain threshold due to the strong metallic losses particularly at optical frequencies, because a spaser with a large or unattainable lasing threshold is undesirable for practical use. Recently, Li and Yu [7] achieved low-threshold gain by embedding Au-nanoparticle lasers into the medium with high refractive index. Pan et al. [8] introduced low-threshold plasmonic lasing based on the high-Q dipole void mode in a metallic nanoshell with large dimensions. Calander et al [9] studied the characteristics of the plasmonic core-shell nanoparticle lasers, and found that a laser made of Ag material possesses lower gain threshold than that of Au.Nanorod spaser [10, 11], symmetry-broken plasmonic core-shell spaser [12], magneto-optical spaser [13], and electric spaser [14] have also been studied.

For the preparation of nano-devices, there is also a demand for NPLs with extremely small volume [5–7]. Once the dimension of NPLs is reduced to the level of tens-of-nanometers, nonlocal theory should be adopted because of its more accurate description of electromagnetic (EM) properties in comparison with local theories [15, 16]. The nonlocality arises from the electron-electron interactions in the dielectric response of metals [17] and leads to the spatial dispersion of metals. As a consequence, it plays a further role in the metal’s SPs. Actually, the nonlocality (or spatial dispersion) was found to diminish the impact caused by geometric imperfections [18], reduce the field enhancement of the nanostructures, and make the plasmon resonant frequency blueshifted [19–26]. For NPLs composed of metal with very small sizes, their lasing properties will be affected by the nonlocality [27]. In this paper, we propose a coated nanoparticle composed of a nonlocal metallic shell around the dielectric gain core as a new type of spaser. Unlike NPLs consisted of a metallic core coated by the gain, this kind of spaser exhibits two lasing states corresponding to the hybridized antibonding and bonding modes.

It is well known that plasmonic nanoshells exhibit two resonant modes resulting from the hybridization of the inner and outer localized surface plasmons or, equivalently, the interaction between a cavity mode and a sphere mode [28]. Silver has smaller dissipation than gold due to the absence of interband transitions. Once the interband transitions in the response of the metal shell are taken into account, these resonances would be damped, especially for the high-energy resonant mode. Though it is not shown here, we conclude that the high-energy (antibonding) mode is dramatically suppressed due to the interband transitions by our numerical results. Hence, we only have one resonant mode (bonding mode) for gold nanoshell. A similar conclusion was made for an ultrathin nanotube [29]. In this paper, we employ nonlocal silver as our plasmonic shell material in order to investigate the lasing generation, as well as the threshold gain, corresponding to these two different resonant modes.

2. Model and theory

First, we consider a nanolaser consisting of a gain core with the inner radius $r_{\text{c}}$, coated by a nonlocal plasmonic shell with outer radius $r_{\text{s}}$. Without loss of generality, the nanolaser is assumed to be embedded in a host medium with the permittivity${\epsilon }_b=n_b^2$. In addition, the nonlocal metal is described with the Drude permittivity${\epsilon }_{\text{T}}$ for the transverse electric fields and a spatially dispersive permittivity ${\epsilon }_{\text{L}}$ for the longitudinal electric fields, which are written as [27, 30, 31],

For the incident plane wave, the incident electric field in the host medium can be written as,

The electric fields excited in the nonlocal shell contain both the transverse and longitudinal waves, respectively read as,

On the other hand, the electric filed inside the gain core is given by

To obtain the unknown coefficients, the continuity boundary conditions of the tangential components for the electric and magnetic fields would be applied on the inner and outer interfaces. Besides that, an additional boundary condition (ABC) is required due to the excitation of the additional longitudinal mode inside the nonlocal metallic shell. Hence, we choose the vanishing normal component of the exciton polarization vector at the interfaces as ABC [31]. After some tedious manipulations, we can obtain the scattering coefficients $a_n^{\text{R}}$ and $b_n^{\text{R}}$, which are related to electric and magnetic scatterings,

In our case, the SP resonance amplification results from the interaction among the incident light, optical gain core, and the coated sphere. The gain core can provide energy for the SP resonance amplification, and the SP mode provides a means of strength-related feedback for photons in the gain core [10]. In general, gain material is phenomenologically described by a simple parameter: a complex refractive index $n_{\text{G}}$($n_{\text{G}}=\sqrt{{\epsilon }_{\text{G}}}$), and the gain coefficient is defined as $G=-4\pi \mathrm{Im}(n_G)/\lambda$. When the gain coefficient reaches the gain threshold, the light amplification and dissipation of the composite system reach a dynamic balance with $Q_{ext}=0$ which is just the spaser generation condition. In this situation, the strength of the SP resonance is significantly enhanced by the energy transferred form the gain media. As a consequence, one may get the gain threshold numerically from Eq. (10).

For nonmagnetic nanolasers, $a_n^{\text{R}}$dominates the polarizability of coated nanoparticle. Mathematically, electric resonances occur when the denominator of scattering coefficient $a_n^{\text{R}}$ vanishes. For compact nanolasers, the total size of the coated particle considered is much smaller than the incident wavelength, and the electric dipole resonances ($n=1$) dominate the total scattering property. Hence, the condition for the spaser generation can be determined by $D_1=0$. In the long-wavelength limit ($k_{\text{b}}{r}_{\text{s}}<<1$, $k_{\text{T}}{r}_{\text{s}}<<1$and$k_{\text{G}}{r}_{\text{c}}<<1$), one can reduce $D_1=0$ to the following relation,

3. Numerical results

We are now in a position to present the numerical results on the nonlocal nanolasers. We first consider the condition without any gain effect in the core, i.e., $\mathrm{Im}(n_{\text{G}})=0$. Figure 1 shows the calculated optical spectra of extinction efficiency $Q_{ext}$ for the gain core-Ag shell nanoparticle for zero gain coefficient. For such nanoshells, it is evident that both antibonding and bonding modes exist due to the hybridization of the sphere and the cavity modes (see the black solid line). When the nonlocality is taken into account, it mainly results in the blue-shift of the antibonding and bonding resonant peaks, while keep the magnitude of the resonant peaks almost unchanged. It is known that the nonlocality becomes much important when the dimensions of the particle spacers are small. On the other hand, when the size effect is considered, the damping constant in the plasmonic shell should be modified $\gamma ={\gamma }_{\infty }+\text{A}{\upsilon }_{\text{F}}/(r_{\text{s}}-r_{\text{c}})$, here${\gamma }_{\infty }$is the damping for the bulk metal, ${\upsilon }_{\text{F}}$is the Fermi velocity, and$\text{A}$ is the fitting constant factor. If one neglects the size-dependent damping by setting $\gamma ={\gamma }_{\infty }$, both resonant peaks are largely enhanced (cf, the difference between the red and green curves in Fig. 1).

 

Fig. 1 Extinction efficiency of the coated nanolasers with three different descriptions: nonlocal plasmonic shell with size-dependent damping (red), nonlocal plasmonic shell without size-dependent damping (green), and local plasmonic shell with size-dependent damping (black). The insert in the middle gives the schematic of the core-shell nanolaser with outer radius$r_s=20\mathrm{nm}$ and inner radius $r_s/r_c=2.01$. $n_{\text{G}}$and$n_{\text{b}}$are chosen as 1.5 and 1 respectively. The inserts around the two peaks show the corresponding near-field patterns

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To find out the gain threshold, we calculate the peak quantity of optical cross section efficiencies for antibonding resonance as a function of the gain coefficient $G$ ($G=-4\pi \mathrm{Im}(n_G)/\lambda$). The results are shown in Fig. 2. It is seen that $Q_{ext}$, $Q_{sca}$and $Q_{abs}$ change sensitively with $G$. When the gain is small, the amplification of SPR due to gain medium further enhances the absorption by the metal and scattering by the total coated nanoparticle, resulting an increasing positive$Q_{ext}$. On the contrary, for a sufficiently large gain coefficient, the amplification of light overwhelms the absorption of the metal and scattering from the whole coated nanoparticle, which leads to a decreasing negative value of $Q_{ext}$ [35, 36]. Hence, there is a critical value for the gain coefficient ($G=G_{\text{th}}$) around which $Q_{ext}$changes its sign. This takes place right at the optical singularity. In this case, the threshold value for this critical gain coefficient $G_{\text{th}}$ at which $Q_{ext}=0$ can be regarded as the gain threshold or the lasing threshold. Therefore, for the antibonding mode, the lasing state for the gain core-nonlocal plasmonic shell structure illustrated in Fig. 1 occurs at the resonant wavelength$\lambda =327.1\mathrm{nm}$, and the gain threshold $G_{\mathrm{th}}=10.3\times {10}^4{\mathrm{cm}}^{-1}$. Similarly, as to the bonding mode, one could find that the corresponding lasing state occurs at $\lambda =401.6\mathrm{nm}$ with the gain threshold $G_{\mathrm{th}}=8.2\times {10}^4{\mathrm{cm}}^{-1}$. It should be verified that, for different $G$ [or different $\mathrm{Im}(n_G)$], the resonant wavelength is expected to move slightly. However, in our calculations, the resonant wavelength remains almost unchanged, especially in the peak region in Fig. 2. We note that to simplify the calculations and obtain the threshold gain in the lasing state only, we fixed the excitation wavelength at $327.1\mathrm{nm}$ in the calculations. More accurate calculations can be done for each $G$with corresponding resonant wavelength. The conclusions are expected to be qualitatively the same.

 

Fig. 2 Resonant optical scattering $Q_{sca}$ (red), absorption $Q_{abs}$ (blue), and extinction $Q_{ext}$(black) cross section efficiencies as a function of the gain coefficient $G$with an excitation wavelength of $327.1\mathrm{nm}$. Maximal value for $Q_{sca}$ takes place at $G=10.3\times {10}^4{\mathrm{cm}}^{-1}$. Parameters are the same as those in Fig. 1.

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On the basis of the above analysis, we find that for a gain core-nonlocal Ag shell nanoparticle, there exist two lasing states due to the two resonant modes (antibonding and bonding modes), where $Q_{sca}$can reach extremely large value. Compared to the bonding mode, the antibonding mode has lower quality factor, hence, the required threshold gain for vanishing $Q_{ext}$ is higher. Therefore, the key questions are which factors contribute to the gain threshold and how to reduce the gain threshold in each mode.

To answer these questions, we examine the effects by the refractive index of the gain core and host medium, as well as the radius ratio $\eta$ ($\eta =r_s/r_c$). Figure 3(a) illustrates that as the refractive index $n_{\text{b}}$of the host medium increases, the gain thresholds corresponding to these two different resonant modes (antibonding and bonding modes) have opposite changing tendencies. In detail, with increasing $n_{\text{b}}$, the gain threshold for antibonding (bonding) mode decreases (increases). Although the antibonding resonance is indeed a dark mode and the magnitude of $Q_{ext}$is generally much smaller than that of bonding mode, once the gain is introduced, one can still achieve the lasing state with high scattering and radiation efficiencies [see Fig. 2]. Furthermore, the gain threshold required for the antibonding mode is much lower than that for the bonding mode if the host medium is SiO₂ ($n_{\text{b}}=1.52$). Therefore, one could achieve tunable lasing states with different gain thresholds by varying the host medium refractive index. On the other hand, the gain threshold dependences on the real part of the gain refractive index is quite different. With the increase of $n_G$, the gain threshold of antibonding (bonding) mode increases (decreses), as shown in Fig. 3(b). In particular, high refractive index of gain core ($n_{\text{G}}=3.25$) dramatically reduces the required gain thresholds of the bonding mode, which may provide a novel approach to achieve low threshold lasing.

 

Fig. 3 Extinction efficiencies of the coated particles with: (a) a refractive index $n_{\text{G}}\text{=}1.52$ for the gain core and various host medium $n_{\text{b}}=1$(black), $n_{\text{b}}=1.2$(red) and $n_{\text{b}}=1.52$(blue); (b) a host medium $n_{\text{b}}=1.52$ and various gain refractive index of gain core $n_{\text{G}}\text{=}1.52$(blue) and $n_{\text{G}}\text{=3}\text{.25}$(pink) at the zero gain. The inserted data denote the gain thresholds (unit:${\mathrm{cm}}^{-1}$) for each resonant modes correspondingly. All the outer radii are fixed as $r_s=20\text{nm}$ and the radii ratios$\eta =2.01$.

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It is well known that the antibonding and bonding modes strongly depend on the radii ratio $\eta$, due to the fact that the thickness of the metallic shell plays an important role in the hybridization of the inner and outer localized surface plasmons. As a consequence, different $\eta$ would lead to different lasing wavelength, as well as different gain thresholds. However, if we account this new degree of freedom, the situation will become rather complicated because the way to find the gain thresholds we introduced previously is nothing but a numerical ‘testing’ method. In order to give a clear picture about the relation among these physical parameters, we shall employ the rigorous analytical derivations Eq. (11) for the lasing generation conditions of the nonlocal coated nanoparticle lasers. As a matter of fact, Eq. (11) determines the condition for the electric dipole spaser generation in the nonlocal case. Therefore, here we solve Eq. (11) numerically to get the required refractive index of the gain medium$n_{\text{G}}=\sqrt{{\epsilon }_{\text{G}}}$, and the threshold $G_{\text{th}}=-4\pi \mathrm{Im}(n_G)/\lambda$, simultaneously.

Figure 4 plots the complex $n_{\text{G}}$obtained by Eq. (11) as functions of the incident wavelength $\lambda$ and the radii ratio $\eta$. Notice that we mark the positions of $\mathrm{Re}(n_{\text{G}})\sim 1.52$ and $\mathrm{Re}(n_{\text{G}})\sim 3.25$ since appropriate materials may be found to support the refractive index in this range experimentally. For small $n_{\text{b}}$ [see Fig. 4(a) and (b)], there are two resonant areas of $\mathrm{Re}(n_{\text{G}})$ where lasing occurs, corresponding to the antibonding band and the bonding band. In fact, situations are quite similar for large $n_{\text{b}}$ [see Fig. 4(c)], but the bonding band in this case is far from our interested frequency regime, and the gain threshold of this band is quite high. Note that, although we have employed different estimations for the spaser condition and gain threshold, they are essentially the same with a relative error below $2\%$. The parameters used in Fig. 3 are just at the same specific points found in Fig. 4, and the features of Fig. 3 are also well illustrated in Fig. 4. It is clearly shown that the gain threshold $G_{\text{th}}$decreases in the antibonding band when $n_{\text{b}}$ becomes larger, however in the bonding region $G_{\text{th}}$goes increasing [see Figs. 4(d), 4(e) and 4(f)]. Moreover, gain core with larger refractive index generally increases $G_{\text{th}}$ in the antibonding band and reduces $G_{\text{th}}$in the bonding band [see Fig. 4(e), the blue arrows denote the same results shown in Fig. 3(b)].

 

Fig. 4 Images of $\mathrm{Re}(n_{\text{G}})$ and $G_{\text{th}}$in the plane of $\lambda$and $\eta$ with $n_{\text{b}}=1$ [(a) and (d)], $n_{\text{b}}=1.52$[(b) and (e)] and $n_{\text{b}}=3.25$ [(c) and (f)] respectively. The positions where $\mathrm{Re}(n_{\text{G}})\sim 1.52$and $\mathrm{Re}(n_{\text{G}})\sim 3.25$are indicated by black dots.

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With regard to the influence of the radii ratios on the gain threshold, it is found that larger $\eta$ generally leads to lower $G_{\text{th}}$ in the antibonding band. In contrast, $G_{\text{th}}$in the bonding band only slightly increases as the radii ratio $\eta$ becomes large. This is because the nonlocal metallic shell has size-dependent damping and thick shell present low damping for the metal. Hence, the required gain thresholds should be reduced. On the other hand, the LSP at the inner interface between the gain core and the metal shell dominate the antibonding band, while the bonding band is associated with the LSP at the outer interface. Therefore, this effect is more obvious in the antibonding band. Nevertheless, in the view point of designing such kind of coated nanospaser with low gain threshold, these two bands still have large parameter space for low$G_{\text{th}}$, which could be used accordingly. More specifically, Fig. 4 provides us two strategies to achieve such coated nanospasers with low $G_{\text{th}}$ corresponding to the usage of the antibonding and bonding bands respectively. High $n_{\text{b}}$ and generally low $\mathrm{Re}(n_{\text{G}})$contribute to the low $G_{\text{th}}$in the antibonding band [see Fig. 4(f)]. Otherwise, low $n_{\text{b}}$ and high $\mathrm{Re}(n_{\text{G}})$ lead to low $G_{\text{th}}$ in the bonding band [see Fig. 4(d)]. Although, each of them has a flexible radii ratio $\eta$, as well as the lasing wavelength. For the purpose of realistic fabrications, the bonding band in Fig. 4(d) seems to have a broader region for low gain threshold. We note that since much smaller coated nanolasers than those in Ref [8] are considered here, by introducing a single metallic nanoshell around a dielectric gain core, the laseing thresholds here are large, as expected.

Finally, to illustrate the nonlocal effects on the gain threshold and the real part of $n_{\text{G}}$, it is helpful to define a set of new parameters $\delta =(G_{\text{th}}-G_{\text{th-local}})/G_{\text{th-local}}$ and $\xi =[\mathrm{Re}(n_{\text{G}})-\mathrm{Re}(n_{\text{G-local}})]/\mathrm{Re}(n_{\text{G-local}})$ to make comparisons between the nonlocal case $G_{\text{th}}$[$\mathrm{Re}(n_{\text{G}})$] and the local case $G_{\text{th-local}}$[$\mathrm{Re}(n_{\text{G-local}})$] for the gain-assisted core-shell nanoparticle spaser. Here, $n_{\text{G-local}}$is yielded by Eq. (12) for the local case. Generally, both $\mathrm{Re}(n_{\text{G}})$and $G_{\text{th}}$ are larger than $\mathrm{Re}(n_{\text{G-local}})$ and $G_{\text{th-local}}$within the realistic operating region. There are only several small portions of the parameter space where $\delta$and $\xi$ become negative [see Figs. 5(a) and 5(e)]. Moreover, $\mathrm{Re}(n_{\text{G}})$in the antibonding band is more strongly affected by the nonlocality than in the banding band, as shown in the upper panels of Fig. 5. The situation is quite similar for $\mathrm{Re}(n_{\text{G}})$ with regard to the nonlocal effects, expect for the case of $n_{\text{b}}=3.25$ in Fig. 5(f), where the nonlocal effects have less influence on the$G_{\text{th}}$in the antibonding band. Consequently, one should be cautious to explore the significant nonlocal effects if these regions are concerned in the design. In addition, we find the regions with low gain threshold shown in Fig. 4 are less affected by the nonlocality [see the bonding band in Fig. 5(d) and the antibonding band in Fig. 5(f)].

 

Fig. 5 Images of $\delta$ and $\xi$in the plane of $\lambda$and $\eta$ with $n_{\text{b}}=1$[(a) and (d)], $n_{\text{b}}=1.52$[(b) and (e)] and $n_{\text{b}}=3.25$[(c) and (f)] respectively.

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4. Conclusion

In conclusion, we have suggested a nonlocal coated nanospaser containing a nonlocal Ag shell and a gain core. Based on the full-wave nonlocal Mie theory, we establish the spaser generation condition for compact plasmonic nanolasers in the long-wavelength limit. For such small coated nanolasers, two lasing states arising from the hybridized antibonding and bonding modes exist. It is demonstrated that high refractive index of the host medium and generally low refractive index of gain core dramatically decrease the gain thresholds for the antibonding lasing mode. In contrast, low gain thresholds for the bonding lasing mode would benefit from the host medium with low refractive index and the gain core with high refractive index. For the purpose of realistic fabrications, the bonding lasing mode has a broader range of radii ratio as well as the lasing wavelength to achieve low gain threshold. In general, the gain threshold and required refractive index for the nonlocal spaser are found to be larger than those for the local one, and the nonlocal effects are more obvious in the antibonding mode than in the bonding mode. We expect that these results can be useful for prospective applications in the design of the ultrasmall nanoparticle spaser with low gain threshold.