## Abstract

The stimulated amplifications of surface plasmons (SPs) propagating along a single silver nanoring is theoretically investigated by considering the interactions between SPs and activated semiconductor quantum dots (SQDs). Threshold condition for the stimulated amplifications, the SP density as a function of propagation length and the maximum SP density are obtained. The SPs can be nonlinearly amplified when the pumping rate of SQDs is larger than the threshold, and the maximum value of SP density increases linearly with the pumping rate of SQDs.

© 2010 OSA

## 1. Introduction

In recent years, there has been a great interest in studying plasmonics for various applications such as photonics and electronics [1–3], sensing [4, 5], nanolithography [6] and microscopy [7, 8]. Plasmonics as a discipline is concerned with the study of surface plasmons and using the properties of surface plasmons (SPs) to manipulate light at the nanoscale. SPs can be strongly localized in metal nanopartilces and nanoholes, and can also propagate in metal nanowires and thin films. The nanosized optical dipole emitters can strongly couple to both localized and nonlocalized SPs.

Based on the strong interaction between nanosized emitters and localized SPs, Bergman and Stockman first proposed a project of Surface Plasmon Amplification by Stimulated Emission of Radiation (SPASER) [9]. Associated with the idea of SPASER, several theoretical schemes and experimental studies about amplification of SPs have been reported [10–22].

The interaction between a nanosized emitter and propagating SPs in metal nanowires was systematically studied by Chang *et al*. [23, 24]. By optimizing nanostructures, the nanosized optical dipole can efficiently couple to the propagating SPs. Single plasmons propagating in Ag nanowire was successfully demonstrated [25]. Very recently, the Ag nanorings with smooth surface and bi-crystal were successfully synthesized, which can support standing waves of SPs and were free from the end-loss of SPs in nanowires [26, 27]. Semiconductor quantum dots (SQDs) with their potential applications in quantum computation and quantum information, have been studied widely in the past years [28–31]. These theoretical and experimental achievements motivate us to study the amplification of propagating SPs in Ag nanorings through strong coupling of exciton and plasmon.

## 2. Theoretical model and analysis

Figures 1(a)
and 1(b) illustrate the structure of our nanosystem consisting of an Ag nanoring and the surrounding active SQDs. The Ag nanoring and SQDs are embedded in a dielectric with dielectric constant *ε*
_{1}. *d* is the center-to-center distance between a SQD and the Ag nanowire, and all the SQDs have the same distance to the Ag nanowire. The SQDs spatial distribution is designed as the largest linear density surrounding the Ag nanoring for a given distance *d*. Assuming the largest linear population densities of SQDs is *η*, which depends on the structure parameters of the system with the relationship *η* = $2\pi d/(\pi {R}_{SQD}^{2})$, where *R*
_{SQD} is the radius of the sphere SQD. Figure 1(c) shows the energy-level diagram of the two-level SQD coupling with SPs. *Γ*
_{rad} and *Γ*
_{nonrad} are the rates of spontaneous emission into free space and the non-radiative emission into Ag nanoring, respectively. ${\Gamma}_{\text{sp}}^{(\text{SpE})}$ and ${\Gamma}_{\text{sp}}^{(\text{StE})}$ denote the spontaneous emission rate and stimulated emission rate into SPs, ${\Gamma}_{\text{sp}}^{(\text{StA})}$ is the stimulated absorption rate and *W*
_{01} is the pumping rate of the SQD. In the following part, we first discuss the stimulated interactions of a single SQD and SPs. Then we consider the interactions of many SQDs with SPs, and discuss the steady state solutions of this nano-system.

First, we consider the coupling between a single two-level SQD and SPs. The ring circumference is considered to be much longer than the wire radius, so the ring can be approximately taken as a straight cylinder wire to get the electromagnetic field. Only a fundamental mode of the electromagnetic field is allowed due to the nanowire limit [24]. The fundamental mode can be expressed as ${\mathbf{E}}_{i}(\mathbf{r},t)=a{\mathbf{Q}}_{i}(\mathbf{r})\mathrm{exp}(i{k}_{//}z-i{\omega}_{\text{sp}}t),$ where *i* = 1, 2 represent the electric fields outside and inside the cylinder, respectively.

*k*

_{i⊥}and

*k*

_{//}, are transverse and longitudinal wave vectors,

*J*

_{0}and

*H*

_{0}are Bessel function and Hankel function of the first kind, respectively.

*a*is a constant. $\widehat{r}$and $\widehat{z}$denote the orientation. The SP is quantized using the method of harmonic oscillator quantization [32]. For simplicity, here we neglect the damping of SPs. Under the rotating wave approximation, the exciton-plasmon interaction Hamiltonian is written as

*ω*

_{sp}is the frequency of SPs, which is approximately equal to the dipole transitions frequency

*ω*

_{10}of the SQD under the resonant excitation condition

*ω*

_{sp}≈

*ω*

_{10}≈

*ω*.

*g*is the coupling factor of the SQDs and SPs.

*g*=

*ω*

^{1/2}

**μ**

_{10}·

**Q**

_{1}(

**r**) / (2

*cћ*

^{1/2}) with

**μ**

_{10}being the transition matrix element of SQDs. The SQDs emission rate into SPs is obtained from the Fermi's golden rule, ${\Gamma}_{\text{SP}}({\omega}_{10})={\Gamma}_{\text{SP}}^{\text{(StE)}}+{\Gamma}_{\text{SP}}^{\text{(SpE)}}=2\pi {\left|g\right|}^{2}(n+1){D}_{\text{SP}}({\omega}_{\text{SP}}),$where ${\Gamma}_{\text{SP}}^{(\text{StE})}=2\pi {\left|g\right|}^{2}n{D}_{sp}(\omega )$ and ${\Gamma}_{\text{SP}}^{(\text{SpE})}=2\pi {\left|g\right|}^{2}{D}_{sp}(\omega ).$

*D*

_{SP}(

*ω*) is the SP density of states, and (

*n*+1) is the total number of SPs attributed to the stimulated and spontaneous emissions of SQDs. The value of ${\Gamma}_{\text{sp}}^{(\text{StA})}$ is equal to that of ${\Gamma}_{\text{sp}}^{(\text{StE})}$. It should be noted that in our discussion the spontaneous emission into free space

*Γ*

_{rad}and the non-radiative emission into Ag nanoring

*Γ*

_{nonrad}are ignored. In Ref. 24 a Purcell factor is defined by $P\text{'}={\Gamma}_{\text{sp}}^{(\text{SpE})}/({\Gamma}_{\text{rad}}+{\Gamma}_{\text{nonrad}})$ to denote the efficiency of spontaneous emission into SPs, and the dependences of ${P}^{\prime}$as functions of wire radius and SQD position are discussed in details. In our system we define a modified Purcell factor as $P=({\Gamma}_{\text{sp}}^{\text{(SpE)}}+{\Gamma}_{\text{sp}}^{\text{(StE)}})/({\Gamma}_{\text{rad}}+{\Gamma}_{\text{nonrad}})$ to include the contribution of stimulated emissions, so

*P*= (

*n*+ 1) ${P}^{\prime}$. For our nanosystem, the wire diameter is generally ~100nm, and the distance

*d*between the SQD and the Ag nanowire is ~60nm, which means ${P}^{\prime}$ ~10

^{0}. When

*n*>>1,

*P*is very large and

*Γ*

_{rad}and

*Γ*

_{nonrad}can be ignored in the SP rate equations.

We now discuss the interactions of many SQDs with SPs. According to the energy-level diagram, the rate equations for the nanosystem can be written as follows,

*N*

_{sp}(

*l*) is the number of SPs per unit length after the SPs propagate the distance

*l*along the ring. The total number of the SPs satisfies

*n*=

*N*

_{sp}(

*l*)

*L*', where

*L*' is the quantization length, which is assumed to be equal to the circumference

*L*of the ring.

*υ*

_{sp}is the group velocity of the SPs.

*ρ*

_{eff}=

*ρ*

_{1}-

*ρ*

_{0}is the population inversion with

*ρ*

_{1}being the linear population densities of SQDs in the excited states and

*ρ*

_{0}in the ground states. The linear population density of SQDs is

*η = ρ*

_{1}+

*ρ*

_{0}. In Eq. (2) ${\Gamma}_{\text{sp}}^{(\text{SpE})}$ is neglected because ${\Gamma}_{\text{sp}}^{(\text{SpE})}<<{\Gamma}_{\text{sp}}^{(\text{StE})}$. For simplification, the interactions between the SQDs are ignored and the orientation of each SQD dipole moment is assumed to be random, so we use the average of ${\left|g\right|}^{2}$denoted by$\u3008{\left|g\right|}^{2}\u3009$instead of ${\left|g\right|}^{2}$in the following calculation.

The steady state solution for the population inversion *ρ*
_{eff} can be obtained from Eq. (3),

*N'*is a constant for a given

*W*

_{01}in our system with the dimension of linear population density. Defining the gain coefficient

*G'*=

*dN*

_{sp}(

*l*)/[

*N*(

_{sp}*l*)

*dl*] and substituting Eq. (4) into Eq. (2) yields the gain coefficient ${G}^{\prime}={\Gamma}_{\text{sp}}^{\text{(SpE)}}L{\rho}_{\text{eff}}/[{\upsilon}_{\text{sp}}(1+{N}_{\text{sp}}(l)/{N}^{\prime})]$, which shows that

*G'*is decreasing with the increasing of

*N*

_{sp}(

*l*). Under small-signal condition, namely

*N*

_{sp}(

*l*) <<

*N'*, the small-signal gain coefficient is written as ${G}^{0}={\Gamma}_{\text{sp}}^{\text{(SpE)}}L{\rho}_{\text{eff}}^{\text{0}}/{\upsilon}_{\text{sp}}.$ Taking the loss of SPs into account with the loss coefficient

*α*= 2Im

*k*

_{//}, the net gain coefficient is given by

The small-signal gain coefficient is modified as *G*
^{0}-*α.* It should be noted that *G*
^{0} should at least satisfy *G*
^{0} >*α* to realize the amplification of SPs, thus *G*
^{0} = *α* is interpreted as the threshold condition. The solution of *W*
_{01} from the equation *G*
^{0} = *α* is the threshold value of pumping rate denoted by ${W}_{01}^{\text{c}}.$ Substituting *dN*
_{sp}(*l*)/[*N*
_{sp}(*l*)*dl*] into the left-hand side of Eq. (5), and integrating from 0 to *l* yields the variation of *N*
_{sp}(*l*) with propagating length:

*G*decreases with the increasing of

*N*

_{sp}(

*l*), and when

*N*

_{sp}(

*l*) reaches to a certain value,

*G*turns to be almost zero, which means that the number of SPs becomes saturated and reaches the maximum. The maximum value of SP number per unit length denoted by ${N}_{\text{sp}}^{\mathrm{max}}$ could be obtained from Eq. (5)

Under the condition of *G*
^{0} >>*α*, Eq. (7) can be approximately written as ${N}_{\text{sp}}^{\mathrm{max}}$ = *η*(-${\Gamma}_{\text{sp}}^{(\text{SpE})}$ +*W*
_{01})/(2*αυ*
_{sp}), where it is assumed that the SP dispersion relation satisfies *ω* = *υ*
_{sp}
*k*
_{//}, and it shows clearly the relationships between ${N}_{\text{sp}}^{\mathrm{max}}$ and the parameters of the system.

## 3. Numerical results

We now calculate the amplification properties in a realistic system with the parameters *D*
_{W}=100nm, the wavelength in vacuum *λ*
_{0}=1000nm, *d*=60nm, *L*=5*λ*
_{sp}= 2.99μm. *ε*
_{1}=2, *ε*
_{2}= −50+0.6*i* and *μ*
_{10} = 1.9×10^{17}
*esu* [9]. The Ag permittivity is taken from Ref [33]. The corresponding loss coefficient α is calculated to be 0.033μm^{−1}. We take *R*
_{SQD}=5nm for the SQD radius, and then we obtain *η* = 4.8 × 10^{3}μm^{−1}, *ρ*
_{eff}
^{0}=1.4×10^{1}μm^{−1}, ${\Gamma}_{\text{SP}}^{(\text{SpE})}=2\pi {\left|g\right|}^{2}{D}_{\text{sp}}(\omega )=2.8215\times {10}^{11}{\text{s}}^{-1},$where ${D}_{\text{sp}}(\omega )=(L/2\pi )(d{k}_{//}/d\omega )=L/(2\pi {\upsilon}_{\text{sp}}).$ The threshold pumping rate is ${W}_{01}^{\text{c}}=1.00582{\Gamma}_{\text{sp}}^{(\text{SpE})}=2.8379\times {10}^{11}{\text{s}}^{-1}.$The number of SPs per unit at the starting point *N*
_{sp}(0) is assumed to be far smaller than *N'* to ensure that the small signal approximation is satisfied at the starting point. Here we assume *N*
_{sp}(0)/ *N'* = 0.01 at ${W}_{10}=9{W}_{01}^{\text{c}}$, namely, *N*
_{sp}(0) = 0.033μm^{−1}.

Figure 2
shows the change of SP number per unit length *N*
_{sp}(*l*) within a very short propagating length *l*. Figure 2 (a) shows that when the pumping rate is larger than the pumping rate threshold ${W}_{01}^{\text{c}}$, N_{sp}(*l*) will increase exponentially with the propagating length, which means that the SPs amplification is realized. The larger the pumping rate, the faster the SPs number increases. For the pumping rate ${W}_{10}={W}_{01}^{\text{c}}$, the number of SPs keeps unchanged, i.e., SPs is neither amplified nor reduced. For ${W}_{10}<{W}_{01}^{\text{c}}$, the number of SPs decreases with the propagating length, and the amplification cannot be realized, as shown in Fig. 2(b).

When ${W}_{10}>{W}_{01}^{\text{c}}$, *N*
_{sp}(*l*) at first increases dramatically with the propagating length *l*, which leads to the decrease of the gain coefficient *G*, then *N*
_{sp}(*l*) turns to be saturated when the gain equals to the loss, as shown in Fig. 3(a)
. Therefore, at first the SP field is first amplified as propagating along the nanoring, then reaches to its saturated value after circling for several rounds and at last the SP field in the nanoring is equivalent everywhere. As the pumping rate increases, the maximum value of SP number becomes larger. Figure 3(b) shows the maximum value of *N*
_{sp}(*l*) as a function of the pumping rate *W*
_{01}. It shows that when ${W}_{10}>{W}_{01}^{\text{c}}$, ${N}_{\text{sp}}^{\mathrm{max}}$ is linearly increased with the pumping rate *W*
_{01}.

## 4. Conclusion

In conclusion, we have investigated the stimulated emission of SQDs into SPs propagating along an Ag nanoring, and obtained the steady-state solution for SPs amplification by using the rate equations. We obtained the threshold condition for amplification, SPs numbers per unit length as a function of propagation length, and the maximum SPs number per unit length, respectively. Numerical calculations show that when the pumping rate of SQDs *W*
_{01} is larger than the pumping threshold ${W}_{01}^{\text{c}}$, at first the SPs number first increases quickly within short propagating length, and then becomes saturated after long propagating distance. The saturated or maximum value of the SPs number per unit length is linearly dependent on the pumping rate. We note that the maximum value of SPs number is proportional to the pumping rate. There are many prospective applications in plasmonic devices for Ag nanoring SPs amplification.

## Acknowledgments

This work was supported by the Natural Science Foundation of China (10534030, 10874134), the National Program on Key Science Research (2006CB921504 and 2007CB935300), Key Project of Ministry of Education (708063) and the fund of Anhui province for young teachers under Grant No.2010SQRL037ZD.

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