Abstract

We propose an exactly solvable electrodynamical model for surface plasmon amplification by stimulated emission of radiation (spaser). The gain medium is described in terms of the nonlinear permittivity with negative losses. The model demonstrates the main feature of a spaser: a self-oscillating state (spasing) arising without an external driving field if the pumping exceeds some threshold value. In addition, it properly describes synchronization of a spaser by an external field within the Arnold tongue and the possibility of compensating for Joule losses when the pumping is below threshold. The model also gives correct qualitative dependencies of spaser characteristics on pumping.

©2013 Optical Society of America

1. Introduction

Thanks to recent achievements of nanotechnology, quantum plasmonics has become one of the most rapidly growing areas of optics (see [16] and references therein). It is expected that nanoplasmonics will lead to a breakthrough in miniaturization of optical circuits, creating subwavelength optical components and on-chip integration of nanophotonics. The remarkable properties of nanoplasmonic devices are due to the ability of surface plasmons (SPs) to concentrate the optical energy at the space of much smaller size than the free space wavelength. However, the full development of nanoplasmonic devices is hindered by unacceptable level of Joule losses in metal nanostructures. The problem of Ohmic losses also slows down creation of numerous attractive applications based on metamaterials – man-made composites with negative dielectric permittivity and magnetic permeability (for review see [7,8]). To overcome this obstacle it was suggested introducing gain particles into plasmonic metallodielectric structures [1,3,5,917]. In these systems, a combination of metal nanoparticles (NP) with a nanoscale active medium having the population inversion results in the emergence of a nanoplasmonic counterpart of the laser – surface plasmon amplification by stimulated emission of radiation (spaser) first proposed by Bergman and Stockman [18] and considered as dipole nanolaser in [19]. This system was realized experimentally in [2024].

The spaser is a quantum device aimed at enhancing the near field of SPs excited on a metal NP by a quantum system with population inversion, e.g. a quantum dot or dye molecules. Since our consideration is qualitative, below for simplicity we consider a generic two-level system (TLS). The TLS is excited by non-coherent pumping. Then it relaxes and excites SPs at the NP. The physical principle of spaser operation is similar to that of a laser. The role of photons confined to the Fabri-Perot resonator is played by SPs confineded to a NP [18,2527]. Since the NP and the TLS are placed near to each other, the SPs excited on the NP trigger stimulated transition at the TLS, which in turn excites more SPs. This process is inhibited by losses in the NP, which together with pumping results in undamped stationary oscillations of the spaser dipole moment in the absence of external electromagnetic field (spasing) [6]. The spasing may occur only when the pumping exceeds some threshold value so that the energy supplied by pumping exceeds losses.

The main difference between spasers and lasers is that a spaser nonradiativly generates and amplifies the plasmonic mode localized on the NP in contrast to the travelling wave of a conventional laser. The SP amplification occurs due to radiationless energy transfer from the TLS to the NP. This process originates from the dipole-dipole (or any other near field [28]) interaction between the TLS and the plasmonic NP. This physical mechanism is highly efficient because the probability of SP excitation is approximately (kL)3 times larger than the probability of radiative emission [29], where L is the distance between the centers of the NP and the TLS and k is the free space wavenumber.

The decrease of the system size leads to the increase of the ratio of the Joule losses to the rediation ones [30]. In fact, this ratio is determined by the same factor (kL)3. The size of the system considered in the present communication is chosen such that radiation losses can be disregarded in comparison to the Joule losses.

The spaser is a non-linear self-oscillating system, and therefore, its response to an external harmonic field principally differs from the response of isolated metal inclusions. Unlike an isolated metal particle, the spaser can be synchronized with the external field only when the field amplitude exceeds a certain threshold value which depends upon the frequency of the field [16]. The domain of the field amplitude and frequency at which the synchronization occurs is called the Arnold tongue. Outside the Arnold tongue, the external field drives the spaser into stochastic oscillations. Inside the Arnold tongue, the energy supplied by pumping may compensate for Joule losses in the metal NPs [17] resulting in harmonic response of spaser on external field. Such compensation can be achieved even for values of pumping below the spasing threshold [31, 32].

Most of the results for spasers have been obtained in the framework of quantum-mechanical description (see, e.g., [6,12,14,1619,27,3235]). This description is reduced to the modified Maxwell-Bloch equations, in which quantum-mechanical operators are substituted by c-numbers. This approach is commonly used in the description of a single mode laser [36]. The same assumptions are made for the description of a gain medium in terms of permittivity with negative losses [37,38]. Thus, one can describe the TLS as a particle made of an amplifying medium, which has permittivity with negative losses. This approach is utilized in works [3942], in which authors describe gain medium using a linear permittivity with a negative imaginary part. Although such an approach correctly demonstrates the lasing threshold of a nanolaser, it fails to reproduce nonlinear features of spasers, such as a stationary state with spasing, the spaser’s behavior when pumping is above threshold, and the change in the population inversion by an external electromagnetic field. The authors of the cited articles realized the necessity of taking nonlinear effects into account, but attempts of doing so have a form of qualitative evaluations [43].

In this paper, we suggest an exactly solvable electrodynamical (more specifically, quasi-static) model of a spaser. The gain medium is assumed to be composed of dye molecules. The model is governed by the equations of classical electrodynamics and reflects the main features of spaser physics including the threshold transition to spasing. The quantum-mechanical effects are taking into account by introducing nonlinear permittivity. The model gives correct dependence of the SP amplitude on pumping and predicts existing of the region, in which a spaser may be synchronized by external wave, the Arnold tongue. The model also predicts compensation for losses by a spaser operating below the pumping threshold.

2. Theoretical model

2.1 Gain medium

The core of our consideration is the description of an amplifying (gain) medium in terms of dielectric permittivity. An expression for the permittivity may be deduced from the Maxwell-Bloch equations [37,38], which are commonly used in semiclassical descriptions of lasers [36,44] and spasers [19]. In the framework of this approach, the evolution of electric field E is related to the macroscopic polarization P of a gain subsystem via classical Maxwell’s equation:

××E+ε0c22Et2=4πc22Pt2,
where ε0 is the dielectric constant of the host matrix and c is the speed of light in vacuum. Gain molecules embedded into the host medium are modeled as two-level systems with the transition dipole moment μ spread in the host matrix. Dynamics of the polarization and the population inversion per unit volume n is governed by the equations following from the density matrix formalism [44]:
2Pt2+2τpPt+ω02P=2ω0|μ|2nE,
nt+1τn(nn0)=1ω0Re(E*Pt),
where ω0 is the transition frequency of the two-level system, τp and τn are relaxation times for polarization and inversion, respectively, and n0 stands for pumping of active atoms. Assuming a harmonic time dependence of the electric field and the polarization and excluding the population inversion from this system we obtain the relation between the polarization and the electric field inside the medium, resulting in the following expression for nonlinear permittivity of a gain medium with an anti-Lorentzian profile [37,38]:
εgain(ω)=ε0+D0ω0ωi+(ω2ω02)/2ωΓ1+β|E(ω)|2+[(ω2ω02)/2ωΓ]2,
where D0=4πμ2τpn0/ is a dimensionless parameter accounting for the stationary value of population inversion n0 of active atoms, β=μ2τnτp/2, and Γ=1/τp with typical values of polarization and population relaxation times for organic dyes being equal to τn100ps and τp10fs, respectively [45]. Below, we use the value of D0 as a characteristic of pumping of the gain medium.

In the current study, we focus on the interaction between lossy and gain media and their scattering properties rather than on the description of an amplifying medium. For this reason, we take some realistic values for ε0, Γ, ω0, and D0, and use them throughout the paper. Namely, we set ε0=4, Γ=0.05eV, ω0=2eV, and D00.25. The electric field is measured in units of β1/2. For our model, the estimated value of β1/2 is β1/2=1/τnτp/μ~10MV/m.

2.2 Core-shell spaser

Consistent consideration of spaser of arbitrary design requires finding a solution of a nonlinear Laplace problem, which generally leads to very cumbersome calculations. As a model of a spaser we consider a spherical gain core of radius rcore coated with a silver plasmonic shell of radius rshell (Fig. 1). This geometry allows us to exatly solve the nonlinear problem analytically. Indeed, in this system, the uniform field inside the core is a solution to the nonlinear Laplace equation. The radii used in our calculations are rshell=20nm and rcore=0.85rshell. For silver permittivity we use the Drude formula ε(ω)=εωpl2/ω/(ω+iγ). In order to fit actual experimental data [46], we use the parameters corresponding to silver: ε=4.9, ωpl=9.5eV, and γ=0.05eV. The plasmon resonance of a core-shell nanoparticle with such geometry splits into two resonances [47] having frequencies ω1=2.0eV and ω2=4.2eV. Recalling that the emission frequency of the gain medium is set to ω0=2eV, one would anticipate SP amplification and spasing in the vicinity of frequency ω1.

 

Fig. 1 A schematic drawing of a core-shell spaser.

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Spasing is characterized by self-oscillation of spaser supported by pumping. For our model this correponds to nonzero dipole moment of the core-shell NP at zero applied external field. To find this state we fix the potential φcore(r)=Einr inside the gain core, which corresponds to the uniform field Ecore(r)=φcore=Ein. Note that in this case Δφcore=0 even with nonlinear permittivity of the core. The potential inside the shell is chosen in the form φshell=bEinr+сEinr/r3, where r is the distance from the center of nanoparticle to the given point. It gives the field in the metallic shell as a linear response to the core field:

Eshell(r)=bEincEin/r3+3c(Einn)n/r3,
with Eshell(r)n=bEinn+2cEinn/r3 and Eshell(r)×n=bEin×ncEin×n/r3, where n = r/r is a unit vector. The potential outside the core-shell nanoparticle must vanish at the infinity and can be represented as φout=aEinr/r3. This gives the following expression for the field
Eout(r)=aEin/r3+3a(Einn)n/r3,
with Eout(r)n=2aEinn/r3 and Eout(r)×n=aEin×n/r3. Note that the chosen form of the electrostatic potential φout corresponds to the electric dipole moment of the NP, d=aEin. The three unknown coefficients, a, b, and c, can be found from the boundary conditions for the electromagnetic field at the inner and outer surfaces of the metallic shell [48]:
Dcoren|rcore=Dshelln|rcore,Ecore×n|rcore=Eshell×n|rcore,Dshelln|rshell=Doutn|rshell,Eshell×n|rshell=Eout×n|rshell,
where D=εE. The condition of spasing corresponds to the non trivial solution for Ein. Substituting Eqs. (4) and (5) into the boundary conditions (6) and assuming non-zero value of Ein, we arrive at the following system of equations:
{εgain(ω,Ein,D0)=(b+2c/rcore3)εshell,1=bc/rcore3,(b+2c/rshell3)εshell=2a/rshell3,bc/rshell3=a/rshell3.
Excluding unknown coefficients a, b, and c from system (7) we arrive at the following equation
εgain(ω,Ein,D0)=εshell2(εshell(ω)1)rcore32(2+εshell(ω))rshell32(εshell(ω)1)rcore3+(2+εshell(ω))rshell3.
Since εgain, and εshell are complex values, Eq. (8) is a system of two equations for three real quantitites D0, Ein, and ω. Thus, Eq. (8) implicitly defines the dependence Ein(D0).

3. Transition to spasing

Our immediate goal is finding the value of pumping, D0, for which the field inside the core is not zero. Substituting Eq. (3) into Eq. (8) we obtain

ε0+D0ω0ωi+(ω2ω02)/2ωΓ1+β|Ein|2+[(ω2ω02)/2ωΓ]2=F(ω),
where we introduce the notation
F(ω)=εshell2(εshell(ω)1)rcore32(2+εshell(ω))rshell32(εshell(ω)1)rcore3+(2+εshell(ω))rshell3.
The left hand side of Eq. (9) depends on the frequency ω, on the absolute value of the field inside the gain core, Ein, and on the gain D0, while the right hand side, F(ω), depends on the frequency only. After some algebra one can see that ImF(ω) is negative for a lossy shell. For any positive D0, the imaginary part of εgain is negative as well and it is equal to zero for D0 = 0 (see Eq. (3)). Thus, there is always an interval, 0 < D0 < Dth, in which ImF(ω)<Imεgain(ω) so that Eq. (9) does not have solutions. The minimal value of the gain, for which Eq. (9) holds, can be considered as the threshold pumping for spasing. The corresponding frequency, at which Eq. (9) can be satisfied, is the spasing frequency, ωsp. Note that for D0 = Dth and Ein = 0.

To find the value of ωsp we rewrite Eq. (9) as

D01+β|Ein|2+[(ω2ω02)/2ωΓ]2=F(ω)ε0i+(ω2ω02)/2ωΓωω0.
Since the left hand side of this equation is real-valued, we obtain an equation that determines ωsp:
ImF(ωsp)ε0i+(ωsp2ω02)/2ωspΓ=0.
Equation (12) can be solved numerically to any desired accuracy. When Eq. (12) is satisfied, Eq. (11) is reduced to a single real-valued equation which determines the relation between the internal field Ein and D0. Indeed, setting the frequency to ωsp in Eq. (11) we obtain:
D01+β|Ein(D0)|2+[(ωsp2ω02)/2ωspΓ]2=ωspω0ReF(ωsp)ε0i+(ωsp2ω02)/2ωspΓ.
In the previous paragraph we argued that Ein = 0 and D0 = Dth satisfy Eq. (9). An increase of D0 accompanied with an increase of Ein does not change the right hand side of Eq. (13). Thus,
D01+β|Ein(D0)|2+[(ωsp2ω02)/2ωspΓ]2=Dth1+0+[(ωsp2ω02)/2ωspΓ]2.
Hence, Ein, as well as the dipole moment of the NP, depends on D0Dth:
Ein=(D0Dth)/βDth1+[(ωsp2ω02)/2ωspΓ]2,d=aEin.
where a=[(1+2εshell)(εshellεgain)r3+(εshell1)(2εshell+εgain)R3](9εshell)1. This is similar to the bifurcation behavior in regular lasers and to that predicted for spasers [16,49]. Indeed, for D0 = Dth the trivial stable point D = D0, d = 0 becomes unstable and a new stable point D = Dth, d ≠ 0 arises with square-root dependence of d on (D0Dth).

Once ωsp is found numerically or analytically, one obtains threshold pumping from Eq. (13):

Dth=ωspω0(1+[(ωsp2ω02)/2ωspΓ]2)ReF(ωsp)ε0i+(ωsp2ω02)/2ωspΓ.

Thus, our classical model predicts the threshold behavior for spasing, gives correct dependence of the amplitude of the spaser dipole oscillations on pumping, as well as expresses the values of threshold pumping and the intrinsic frequency of spasing via the parameters of the system. For the parameters specified above, the calculated spasing frequency and threshold are ωsp1.987eV and Dth0.25.

4. Loss compensation

As we mentioned in Introduction, including a gain medium into a metamaterial made of plasmonic NPs turns metamaterial into a matrix filled with spasers which inject energy into the system in order to compensate for loss. The general goal of such compensation is to construct a lossless metamaterial in which the electromagnetic response mimics the response of an ordinary composite without loss. Such metamaterial should be characterized by an effective dielectric permittivity. Such a description implies that the spaser response should monotonically depend on the amplitude of the external field and should oscillate with the frequency of the driving field. When the exact loss compensation occurs, the imaginary part of the spaser dipole moment is equal to zero [17].

In works [14,31,50] the possibility of loss compensation was studied by computer simulation. In order to be able to use the effective permittivity to describe the spaser response, the authors considered very short pulses of the external wave. It was assumed that during the pulse, the population inversion does not change. Thus, despite using nonlinear equations, the authors obtain results of the linear theory.

The present toy model of the spaser allows us to consider nonlinear response of the spaser. Above pumping threshold, a spaser is a self-oscillating system with fixed frequency and amplitude. Therefore, in this regime spasers are not very convenient for loss compensation for a wide range of frequencies [51,52]. Even though such a spaser can be synchronized by external optical field, so that it oscillates with the frequency of that field [16] and losses can be compensated at certain frequencies and amplitudes of that external field [17], the amplitude of spaser dipole oscillations weakly depends on the amplitude of the external field. The value of this amplitude is about the same as the amplitude of oscillations of a non-driven spaser.

The response of a spaser operating below pumping threshold is more suitable for compensating losses in a metamaterial matrix. Indeed, below pumping threshold a spaser does not oscillate without an external field. Such a spaser is always synchronized by the external field. The amplitude of the dipole oscillations nonlinearly depends on the strength of the external field. The question remains whether the driven below-threshold spaser could actually compensate for losses. The quantum-mechanical consideration of this problem is done in [32].

In order to calculate the response of a driven below-threshold spaser on the external incident field, we should include a respective term Eext=xEin into system of Eqs. (7). Assuming that the NP dipole moment oscillates with the frequency of the external field ω, we obtain:

{εgain(ω,Ein,D0)=(b+2c/rcore3)εshell,1=bc/rcore3,(b+2c/rshell3)εshell=x+2a/rshell3,bc/rshell3=xa/rshell3.
Numerical solution of Eqs. (17) gives the values of the NP dipole moment d=aEin for a given frequency and the amplitude of the external field. Indeed, once frequency and internal field Ein are fixed, from system (17) we obtain all the unknown coefficients, in particular, a and x which define the dipole moment d and the incident field Eext, respectivelly.

In Fig. 2 we show parametric plots of the dipole moment of the driven spaser versus external field for two regimes: below- and above-threshold spaser operation and for two different frequencies ω<=1.98eV<ωsp and ω>=1.99eV>ωsp. One can see that below the threshold, there is a range of external field amplitudes in which the imaginary part of the total dipole moment is negative (see Fig. 2(b)). In this regime, with the help of spasers, energy of pumping is transferred to the system and exceeds losses. For a certain amplitude, the imaginary part of dipole moment turnes to zero. At this point, losses in the composite NP are exactly compensated by energy transferred to the system by the spaser operating below spasing threshold. The external field amplitude for which exact loss compensation occurs depends upon the field frequency forming a curve of the exact compensation. As one can see from Fig. 2(a), there is no loss compensation for frequencies smaller that the spasing frequency ωsp. This agrees with the results of the quantum-mechanical approach [32], which shows that loss compensation at frequencies ω<ωsp is impossible. As one can see from Fig. 2(d), the field required for loss compensation can be estimated as 0.005β1/2, so that the typical loss compensation field is of the order of ~50 kV/m which is much smaller than the breakdown field of a dielectric (~10 ÷ 100 MV/m).

 

Fig. 2 Spaser response on external harmonic field in (top panels) below- and (bottom panels) above-threshold regimes. Solid and dashed lines show real and imaginary parts of the dipole moment, respectively. Frequencies of incident field are (a), (c) ω< = 1.98 eV < ωsp and (b), (d) ω> = 1.99 eV < ωsp. The red and green circles in the bottom panels show the exact loss compensation when the imaginary part of the dipole moment turns to zero.

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Our model shows that loss compensation can be achieved in the regime of above-threshold pumping as well (marked by a green circle in Fig. 2(d)). This is also in agreement with the quantum-mechanical consideration [16,17,53]. However, as we discuss above, in this regime the spaser’s response to the external field is strongly non-linear which makes it less attractive for applications. Finally, as one can see from Fig. 2(c), above the threshold loss compensations may formally occur in the region of frequencies ω < ωsp (marked by a red circle in Fig. 2(c)). However, as shown in the following section, this solution is unstable.

Recently, Khurgin and Sun (KS) [54,55] raised concerns of possibility of loss compensation by spasers and their practical realization. As KS discussed, in the injection pumped spasers, the threshold current density may reach unsustainable values. Here, we would like to emphasize the difference between our model and the one considered by KS. In contrast to papers [54,55], in which semiconductor optical devices with current pumping are studied, we consider organic dye which is pumped optically. Semiconductor and dye-based lasers belong to different classes of lasers [56]. These classes are characterized by relations between relaxation times. The population inversion relaxation time of organic dye is of the order of τn~1010s. Thus, the Purcell factor in our case is significantly smaller than one considered in Refs [54,55]. For typical relaxation times of the plasmon mode, τa~1014s, the polarization, τp~1014s, and the Rabi frequency, ΩR~1013s1, the Purcell factor is of the order of 1+ΩR2τaτp~1. This is by two orders of magnitude smaller than the evaluation obtained by KS. Moreover, KS studied the regime of developed spasing. This implies that the mean number of plasmons is greater than unity (npl1). In this paper, we consider a spaser operating near or below threshold. In this case [34], and local fields and pumping energy have reasonable values.

The pump intensity can be estimated as the net power dissipated in the active medium and the plasmonic NP near threshold: Wpump~ω0neN/τn+ωspnpl/τa, where is the population of the excited state of the gain medium at threshold, which is of the order of 1. Therefore, for core radius of ~20 nm and gain molecules density of ρ~1019cm3 we obtain that inside the active core of spaser there are approximately N~rcore3ρ~100 of dye molecules. For value of τn=1010s, this yields the pump power Wpump105W. The energy flux density is then given by Wpump/λ2, where accounts for the resonant cross-section of the spaser interaction with the pumping optical wave (note that the interaction cross-section is not the squared spaser’s radius, , but the squared wavelength, ). Since the Pointing vector of a plane wave is S=c2ε0|E|2 (where ε0 is the electric constant), we find an estimation for the electric field strength for the pumping wave Epump~Wpump/(ε0c2λ2)~10V/m.

5. Spaser synchronization and the Arnold tongue

Above threshold spaser is a self-oscillating system. As it is known from the general theory of nonlinear oscillators [57], response of a self-oscillating system on an external harmonic perturbation may be either periodic or stochastic, i.e. oscillations of above-threshold spaser in an external optical field may be unstable. To shed light on this issue, we consider equations describing the temporal evolution of the dipole moment of a spaser driven by the external oscillating field.

To investigate the time evolution of the stationary state we cannot consider the external wave as a plane wave with a constant amplitude, but should consider a slowly varying long pulse of the external field Eext(t)=Eslow(t)exp(iωt) and corresponding dipole moment, d(t)=dslow(t)exp(iωt), induced in a core-shell spaser. Here, dslow(t) and Eslow(t) are slowly varying envelopes, which Fourier transformations include only frequencies that are much smaller than the central frequency ω. The external field and dipole moment are related via the nonlinear operator,

α^1(t,Ein(t))d(t)=Eext(t).

The explicit form of the operator α^1(Ein) is not known because it depends on the field inside the gain medium, Ein(t), which is a long pulse as well, whereas permittivity (3) is obtained for a harmonic field with the amplitude E(ω). To make Eqs. (18) and (3) consistent with each other, we define the operator α^1(Ein) through its action on the harmonic field as follows:

α1(ω,Ein(ω))d˜(ω)=E˜ext(ω),
where Ein(ω) is the value that should be put into Eq. (3) and ‘~’ indicates the Fourier transformation of the corresponding quantity. In this case α1 is simply the relation between incident field and dipole moment and can be obtained from the system (17): α1=Eext/d=x/a.

Applying the Fourier transformation to Eq. (19) we get:

Eext(t)=α1(Ω,Ein)d˜(Ω)eiΩtdΩ=eiωtα1(ω+ν,Ein)d˜slow(ν)eiνtdν.
Since the main contribution to integral (20) is given by harmonics having νω, we obtain
Eext(t)eiωt[α1(ω,Ein)+νdα1(ω,Ein)dω]d˜slow(ν)eiνtdν==eiωtα1(ω,Ein)d˜slow(ν)eiνtdν+eiω0tdα1(ω,Ein)dωνd˜slow(ν)eiνtdν=eiωt[α1(ω,Ein)dslow(t)+idα1(ω,Ein)dωddslow(t)dt],
where the term with d/dt accounts for small broadening of dipole moment spectra (see also [58,59]). Cancelling the oscillating factor eiωt at both sides of Eq. (21) we arrive at the desired equation describing temporal evolution of the spaser dipole moment in slowly varying external field with the central frequency ω:

idα1dωddtdslow(t)+α1dslow(t)=Eslow(t).

Provided Eslow(t)E is constant, Eq. (22) has a stationary solution dslow=αE. In order to study how small perturbations of this solution evolve with time, let us consider a perturbation in the form dslow=αE+δdeΛt. Substituting this into Eq. (22) we arrive at

idα1dωδdΛ+α1δd=0.
The instability growth rate Λ is then given by

Λ=iα1(ω,Ein)(dα1(ω,Ein)/dω)1.

The stationary solution of Eq. (22) becomes unstable when ReΛ>0. In Fig. 3 we plot the real part of the instability growth rate for both below- and above-threshold regimes of a driven core-shell spaser for the incident field frequency ω<=1.98eV.

 

Fig. 3 Real parts of the instability growth rate Λ for below- (solid line) and above-threshold (dashed line) regimes of a driven spaser at the frequency of incident field ω< = 1.98 eV. For below-threshold spaser the instability growth rate is always negative so that spaser oscillations in external field are always stable. Above threshold, spaser is synchronized by strong external field, while its oscillations in weak external field are unstable and may show stochastic dynamics [16].

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In the below-threshold regime,ReΛ is negative for any field. Thus, the steady-state oscillations are stable with respect to small perturbations. When pumping increases, so that D0 exceeds Dth, a region in which ReΛ>0 arises. In this region, the steady-state solution becomes unstable. In the above-threshold regime without external field, the spaser oscillates with its spasing frequency, which is obtained from Eq. (12). When the external field is applied, depending on its amplitude and frequency, the spaser may or may not oscillate with the frequency of this field. When it has the same frequency as the external field, the steady-state solution is stable, and spaser is synchronized with the external field. The region in which the synchronization takes place is known the Arnold tongue [60]. This region is shown in Fig. 4 for D0 = 0.25. When the frequency of the external field is tuned to the spasing frequency, ωsp, the synchronization occurs for vanishingly small amplitude of the external field.

 

Fig. 4 Stable (filled) and unstable regions of a driven above-threshold spaser for D0 = 0.25. The region of stability corresponds to steady-state solutions with ReΛ<0. The two vertical lines correspond to the frequencies ω> and ω<, which are also used in Fig. 2. The red and green circles correspond to those in Fig. 2 (c) and 2(d). The red circle, in which Im[d]=0, lies in the instability region, so loss compensation does not occur, while the green circle lies in the stable region and compensation does take place. The red and green curves show the compensation curves lying in unstable and stable regions, respectively, along which the imaginary part of dipole moment turns to zero.

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6. Discussion

The model presented here reproduces general features of the quantum description of a spaser including the pumping threshold for spasing, the bifurcation behavior at the transition point, correct dependencies of spaser characteristics on pumping, and the existence of a region of spaser synchronization with an external field – the Arnold tongue. Our semiclassical model also predicts the possibility of loss compensation by a spaser operating below threshold.

Our model also reveals inconsistencies in linear models of nanolasers. In particular, the authors of [39] consider a metal-coated nanolaser and report a lasing turn-off above the threshold. This result is in disagreement with the experimental observation of spasing in core-shell nanolasers and with general theory of lasers [44]. The authors make a suggestion that the on/off behavior of lasing in coated nanoparticles is caused by detuning of the resonance when the gain is added. However, in our study we show that the spasing frequency does not depend on gain and is a function of nanolaser geometry only (see Eq. (12)).

Using our model, it is interesting to look at the discussion concerning the possibility of loss compensation in plasmonic systems with gain (see [52] and comments to this work). In paper [52], Stockman argues that in a resonant plasmonic structure Ohmic losses are compensated for by gain when spasing occurs. Indeed, this argument is valid for a closed system in which there is no incoming and outgoing radiation. In this case, loss compensation and lasing simply coincide.

In an open system coupled with the radiation, it is necessary to compensate for both Ohmic and radiation losses for spasing to occur. In this case, as we show above for a spaser operating below threshold, lossless scattering of an incoming wave may occur when the system does not spase. This happens because the magnitude of dipole oscillations is smaller than that in the above-threshold spaser and the pumping energy is, therefore, sufficient to compensate for the loss. This situation is analogous to the scheme suggested in [9], in which Ohmic losses in the illuminated photonic crystal composed of alternating metallic and dielectric amplifying layers are compensated below the lasing threshold.

Acknowledgments

The authors would like to thank A. A. Pukhov for helpful discussion. They also are indebted to R. E. Noskov for helpful discussion of Eq. (22). This work was supported by RFBR Grants Nos. 12-02-01093, 13-02-00407 by a PSC-CUNY grant and by the Dynasty Foundation.

References and links

1. V. M. Shalaev and S. Kawata, eds., Nanophotonics with Surface Plasmons (Elsevier, 2007).

2. S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).

3. S. I. Bozhevolniy, ed., Plasmonic. Nanoguides and Circuits (Pan Stanford Publishing, 2009).

4. S. V. Gaponenko, Introduction to Nanophotonics (Cambridge University, 2010).

5. Z. M. Wang and A. Neogi, eds., Nanoscale Photonics and Optoelectronics (Springer, 2010).

6. M. I. Stockman, “Nanoplasmonics: past, present, and glimpse into future,” Opt. Express 19(22), 22029–22106 (2011). [CrossRef]   [PubMed]  

7. R. Marques, F. Martin, and M. Sorolla, Metamaterials with Negative Parameters: Theory, Design and Microwave Applications (Wiley, 2008).

8. W. Cai and V. Shalaev, Optical Metamaterials: Fundamentals and Applications (Springer, 2010).

9. S. A. Ramakrishna and J. Pendry, “Removal of absorption and increase in resolution in a near-field lens via optical gain,” Phys. Rev. B 67(20), 201101 (2003). [CrossRef]  

10. M. A. Noginov, G. Zhu, M. Bahoura, J. Adegoke, C. E. Small, B. A. Ritzo, V. P. Drachev, and V. M. Shalaev, “Enhancement of surface plasmons in an Ag aggregate by optical gain in a dielectric medium,” Opt. Lett. 31(20), 3022–3024 (2006). [CrossRef]   [PubMed]  

11. A. K. Popov and V. M. Shalaev, “Compensating losses in negative-index metamaterials by optical parametric amplification,” Opt. Lett. 31, 2169–2171 (2006). [CrossRef]   [PubMed]  

12. A. K. Sarychev and G. Tartakovsky, “Magnetic plasmonic metamaterials in actively pumped host medium and plasmonic nanolaser,” Phys. Rev. B 75(8), 085436 (2007). [CrossRef]  

13. N. I. Zheludev, S. L. Prosvirnin, N. Papasimakis, and V. A. Fedotov, “Lasing spaser,” Nat. Photonics 2(6), 351–354 (2008). [CrossRef]  

14. A. Fang, T. Koschny, and C. M. Soukoulis, “Lasing in metamaterial nanostructures,” J. Opt. 12(2), 024013 (2010). [CrossRef]  

15. S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466(7307), 735–738 (2010). [CrossRef]   [PubMed]  

16. E. S. Andrianov, A. A. Pukhov, A. V. Dorofeenko, A. P. Vinogradov, and A. A. Lisyansky, “Forced synchronization of spaser by an external optical wave,” Opt. Express 19(25), 24849–24857 (2011). [CrossRef]   [PubMed]  

17. E. S. Andrianov, A. A. Pukhov, A. V. Dorofeenko, A. P. Vinogradov, and A. A. Lisyansky, “Dipole response of spaser on an external optical wave,” Opt. Lett. 36(21), 4302–4304 (2011). [CrossRef]   [PubMed]  

18. D. J. Bergman and M. I. Stockman, “Surface plasmon amplification by stimulated emission of radiation: quantum generation of coherent surface plasmons in nanosystems,” Phys. Rev. Lett. 90(2), 027402 (2003). [CrossRef]   [PubMed]  

19. I. E. Protsenko, A. V. Uskov, O. A. Zaimidoroga, V. N. Samoilov, and E. P. O’Reilly, “Dipole nanolaser,” Phys. Rev. A 71(6), 063812 (2005). [CrossRef]  

20. M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460(7259), 1110–1112 (2009). [CrossRef]   [PubMed]  

21. R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461(7264), 629–632 (2009). [CrossRef]   [PubMed]  

22. R.-M. Ma, R. F. Oulton, V. J. Sorger, G. Bartal, and X. Zhang, “Room-temperature sub-diffraction-limited plasmon laser by total internal reflection,” Nat. Mater. 10(2), 110–113 (2011). [CrossRef]   [PubMed]  

23. C.-Y. Wu, C.-T. Kuo, C.-Y. Wang, C.-L. He, M.-H. Lin, H. Ahn, and S. Gwo, “Plasmonic green nanolaser based on a metal-oxide-semiconductor structure,” Nano Lett. 11(10), 4256–4260 (2011). [CrossRef]   [PubMed]  

24. Y.-J. Lu, J. Kim, H.-Y. Chen, C. Wu, N. Dabidian, C. E. Sanders, C.-Y. Wang, M.-Y. Lu, B.-H. Li, X. Qiu, W.-H. Chang, L.-J. Chen, G. Shvets, C.-K. Shih, and S. Gwo, “Plasmonic nanolaser using epitaxially grown silver film,” Science 337(6093), 450–453 (2012). [CrossRef]   [PubMed]  

25. K. Li, X. Li, M. I. Stockman, and D. J. Bergman, “Surface plasmon amplification by stimulated emission in nanolenses,” Phys. Rev. B 71(11), 115409 (2005). [CrossRef]  

26. M. I. Stockman, “Spasers explained,” Nat. Photonics 2(6), 327–329 (2008). [CrossRef]  

27. M. I. Stockman, “The spaser as a nanoscale quantum generator and ultrafast amplifier,” J. Opt. 12(2), 024004 (2010). [CrossRef]  

28. I. R. Gabitov, B. Kennedy, and A. I. Maimistov, “Coherent amplification of optical pulses in metamaterials,” IEEE J. Sel. Top. Quantum Electron. 16(2), 401–409 (2010). [CrossRef]  

29. V. V. Klimov, Y. N. Istomin, and Y. A. Kosevich, “Plasma phenomena in nanostructures and neutron stars,” Phys. Usp. 51(8), 839–859 (2008). [CrossRef]  

30. L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2006).

31. S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Overcoming Losses with Gain in a Negative Refractive Index Metamaterial,” Phys. Rev. Lett. 105(12), 127401 (2010). [CrossRef]   [PubMed]  

32. E. S. Andrianov, D. G. Baranov, A. A. Puhov, A. V. Dorofeenko, A. P. Vinogradov, and A. A. Lisyansky, “Loss compensation by spasers in metamaterials,” http://arxiv.org/abs/1209.0422.

33. A. S. Rosenthal and T. Ghannam, “Dipole nanolasers: A study of their quantum properties,” Phys. Rev. A 79(4), 043824 (2009). [CrossRef]  

34. E. S. Andrianov, A. A. Pukhov, A. V. Dorofeenko, A. P. Vinogradov, and A. A. Lisyansky, “Rabi oscillations in spasers during nonradiative plasmon excitation,” Phys. Rev. B 85(3), 035405 (2012). [CrossRef]  

35. A. N. Lagarkov, A. K. Sarychev, V. N. Kissel, and G. Tartakovsky, “Superresolution and enhancement in metamaterials,” Phys. Usp. 52(9), 959–967 (2009). [CrossRef]  

36. M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University Press, 1997).

37. A. A. Zyablovsky, A. V. Dorofeenko, A. A. Pukhov, and A. P. Vinogradov, “Lasing in a gain slab as a consequence of the causality principle,” J. Commun. Technol. Electron. 56(9), 1139–1145 (2011). [CrossRef]  

38. S. Solimeno, B. Crosignani, and P. Di Porto, Guiding, Diffraction, and Confinement of Optical Radiation (Academic, 1986).

39. J. A. Gordon and R. W. Ziolkowski, “The design and simulated performance of a coated nano-particle laser,” Opt. Express 15(5), 2622–2653 (2007). [CrossRef]   [PubMed]  

40. A. Mizrahi, V. Lomakin, B. A. Slutsky, M. P. Nezhad, L. Feng, and Y. Fainman, “Low threshold gain metal coated laser nanoresonators,” Opt. Lett. 33(11), 1261–1263 (2008). [CrossRef]   [PubMed]  

41. X. F. Li and S. F. Yu, “Design of low-threshold compact Au-nanoparticle lasers,” Opt. Lett. 35(15), 2535–2537 (2010). [CrossRef]   [PubMed]  

42. A. Veltri and A. Aradian, “Optical response of a metallic nanoparticle immersed in a medium with optical gain,” Phys. Rev. B 85(11), 115429 (2012). [CrossRef]  

43. N. M. Lawandy, “Localized surface plasmon singularities in amplifying media,” Appl. Phys. Lett. 85(21), 5040–5042 (2004). [CrossRef]  

44. R. H. Pantell and H. E. Puthoff, Fundamentals of Quantum Electronics (John Wiley & Sons Inc, 1969).

45. P. Sperber, W. Spangler, B. Meier, and A. Penzkofer, “Experimental and theoretical investigation of tunable picosecond pulse generation in longitudinally pumped dye laser generators and amplifiers,” Opt. Quantum Electron. 20(5), 395–431 (1988). [CrossRef]  

46. P. B. Johnson and R. W. Christy, “Optical Constants of the Noble Metals,” Phys. Rev. B 6(12), 4370–4379 (1972). [CrossRef]  

47. E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302(5644), 419–422 (2003). [CrossRef]   [PubMed]  

48. C. F. Boheren and D. R. Huffman, Absorption and Scattering of Light by Small Particles, (Wiley, 1983)

49. A. P. Vinogradov, E. S. Andrianov, A. A. Pukhov, A. V. Dorofeenko, and A. A. Lisyansky, “Quantum plasmonics of metamaterials: loss compensation using spasers,” Phys. Usp. 55(10), 1046–1053 (2012). [CrossRef]  

50. M. Wegener, J. L. García-Pomar, C. M. Soukoulis, N. Meinzer, M. Ruther, and S. Linden, “Toy model for plasmonic metamaterial resonances coupled to two-level system gain,” Opt. Express 16(24), 19785–19798 (2008). [CrossRef]   [PubMed]  

51. M. Premaratne and G. P. Agrawal, Light Propagation in Gain Medium (Cambridge University, 2011).

52. M. I. Stockman, “Spaser action, loss compensation, and stability in plasmonic systems with gain,” Phys. Rev. Lett. 106(15), 156802 (2011). [CrossRef]   [PubMed]  

53. A. A. Lisyansky, E. S. Andrianov, A. V. Dorofeenko, A. A. Pukhov, and A. P. Vinogradov, “Forced spaser oscillations,” Proc. SPIE 8457, 84570X (2012). [CrossRef]  

54. J. B. Khurgin and G. Sun, “Practicality of compensating the loss in the plasmonic waveguides using semiconductor gain medium,” Appl. Phys. Lett. 100(1), 011105 (2012). [CrossRef]  

55. J. B. Khurgin and G. Sun, “Injection pumped single mode surface plasmon generators: threshold, linewidth, and coherence,” Opt. Express 20(14), 15309–15325 (2012). [CrossRef]   [PubMed]  

56. P. W. Milonni and J. H. Eberly, Laser Physics (Wiley, 2010).

57. A. H. Nayfeh and D. T. Mook, Nonlinear Oscillations (Wiley-VCH, 1995).

58. A. A. Zharov, R. E. Noskov, and M. V. Tsarev, “Plasmon-induced terahertz radiation generation due to symmetry breaking in a nonlinear metallic nanodimer,” J. Appl. Phys. 106(7), 073104 (2009). [CrossRef]  

59. R. E. Noskov, P. A. Belov, and Y. S. Kivshar, “Subwavelength modulational instability and plasmon oscillons in nanoparticle arrays,” Phys. Rev. Lett. 108(9), 093901 (2012). [CrossRef]   [PubMed]  

60. A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization. A Universal Concept in Nonlinear Sciences (Cambridge University, 2001).

References

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  1. V. M. Shalaev and S. Kawata, eds., Nanophotonics with Surface Plasmons (Elsevier, 2007).
  2. S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).
  3. S. I. Bozhevolniy, ed., Plasmonic. Nanoguides and Circuits (Pan Stanford Publishing, 2009).
  4. S. V. Gaponenko, Introduction to Nanophotonics (Cambridge University, 2010).
  5. Z. M. Wang and A. Neogi, eds., Nanoscale Photonics and Optoelectronics (Springer, 2010).
  6. M. I. Stockman, “Nanoplasmonics: past, present, and glimpse into future,” Opt. Express 19(22), 22029–22106 (2011).
    [Crossref] [PubMed]
  7. R. Marques, F. Martin, and M. Sorolla, Metamaterials with Negative Parameters: Theory, Design and Microwave Applications (Wiley, 2008).
  8. W. Cai and V. Shalaev, Optical Metamaterials: Fundamentals and Applications (Springer, 2010).
  9. S. A. Ramakrishna and J. Pendry, “Removal of absorption and increase in resolution in a near-field lens via optical gain,” Phys. Rev. B 67(20), 201101 (2003).
    [Crossref]
  10. M. A. Noginov, G. Zhu, M. Bahoura, J. Adegoke, C. E. Small, B. A. Ritzo, V. P. Drachev, and V. M. Shalaev, “Enhancement of surface plasmons in an Ag aggregate by optical gain in a dielectric medium,” Opt. Lett. 31(20), 3022–3024 (2006).
    [Crossref] [PubMed]
  11. A. K. Popov and V. M. Shalaev, “Compensating losses in negative-index metamaterials by optical parametric amplification,” Opt. Lett. 31, 2169–2171 (2006).
    [Crossref] [PubMed]
  12. A. K. Sarychev and G. Tartakovsky, “Magnetic plasmonic metamaterials in actively pumped host medium and plasmonic nanolaser,” Phys. Rev. B 75(8), 085436 (2007).
    [Crossref]
  13. N. I. Zheludev, S. L. Prosvirnin, N. Papasimakis, and V. A. Fedotov, “Lasing spaser,” Nat. Photonics 2(6), 351–354 (2008).
    [Crossref]
  14. A. Fang, T. Koschny, and C. M. Soukoulis, “Lasing in metamaterial nanostructures,” J. Opt. 12(2), 024013 (2010).
    [Crossref]
  15. S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466(7307), 735–738 (2010).
    [Crossref] [PubMed]
  16. E. S. Andrianov, A. A. Pukhov, A. V. Dorofeenko, A. P. Vinogradov, and A. A. Lisyansky, “Forced synchronization of spaser by an external optical wave,” Opt. Express 19(25), 24849–24857 (2011).
    [Crossref] [PubMed]
  17. E. S. Andrianov, A. A. Pukhov, A. V. Dorofeenko, A. P. Vinogradov, and A. A. Lisyansky, “Dipole response of spaser on an external optical wave,” Opt. Lett. 36(21), 4302–4304 (2011).
    [Crossref] [PubMed]
  18. D. J. Bergman and M. I. Stockman, “Surface plasmon amplification by stimulated emission of radiation: quantum generation of coherent surface plasmons in nanosystems,” Phys. Rev. Lett. 90(2), 027402 (2003).
    [Crossref] [PubMed]
  19. I. E. Protsenko, A. V. Uskov, O. A. Zaimidoroga, V. N. Samoilov, and E. P. O’Reilly, “Dipole nanolaser,” Phys. Rev. A 71(6), 063812 (2005).
    [Crossref]
  20. M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460(7259), 1110–1112 (2009).
    [Crossref] [PubMed]
  21. R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461(7264), 629–632 (2009).
    [Crossref] [PubMed]
  22. R.-M. Ma, R. F. Oulton, V. J. Sorger, G. Bartal, and X. Zhang, “Room-temperature sub-diffraction-limited plasmon laser by total internal reflection,” Nat. Mater. 10(2), 110–113 (2011).
    [Crossref] [PubMed]
  23. C.-Y. Wu, C.-T. Kuo, C.-Y. Wang, C.-L. He, M.-H. Lin, H. Ahn, and S. Gwo, “Plasmonic green nanolaser based on a metal-oxide-semiconductor structure,” Nano Lett. 11(10), 4256–4260 (2011).
    [Crossref] [PubMed]
  24. Y.-J. Lu, J. Kim, H.-Y. Chen, C. Wu, N. Dabidian, C. E. Sanders, C.-Y. Wang, M.-Y. Lu, B.-H. Li, X. Qiu, W.-H. Chang, L.-J. Chen, G. Shvets, C.-K. Shih, and S. Gwo, “Plasmonic nanolaser using epitaxially grown silver film,” Science 337(6093), 450–453 (2012).
    [Crossref] [PubMed]
  25. K. Li, X. Li, M. I. Stockman, and D. J. Bergman, “Surface plasmon amplification by stimulated emission in nanolenses,” Phys. Rev. B 71(11), 115409 (2005).
    [Crossref]
  26. M. I. Stockman, “Spasers explained,” Nat. Photonics 2(6), 327–329 (2008).
    [Crossref]
  27. M. I. Stockman, “The spaser as a nanoscale quantum generator and ultrafast amplifier,” J. Opt. 12(2), 024004 (2010).
    [Crossref]
  28. I. R. Gabitov, B. Kennedy, and A. I. Maimistov, “Coherent amplification of optical pulses in metamaterials,” IEEE J. Sel. Top. Quantum Electron. 16(2), 401–409 (2010).
    [Crossref]
  29. V. V. Klimov, Y. N. Istomin, and Y. A. Kosevich, “Plasma phenomena in nanostructures and neutron stars,” Phys. Usp. 51(8), 839–859 (2008).
    [Crossref]
  30. L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2006).
  31. S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Overcoming Losses with Gain in a Negative Refractive Index Metamaterial,” Phys. Rev. Lett. 105(12), 127401 (2010).
    [Crossref] [PubMed]
  32. E. S. Andrianov, D. G. Baranov, A. A. Puhov, A. V. Dorofeenko, A. P. Vinogradov, and A. A. Lisyansky, “Loss compensation by spasers in metamaterials,” http://arxiv.org/abs/1209.0422 .
  33. A. S. Rosenthal and T. Ghannam, “Dipole nanolasers: A study of their quantum properties,” Phys. Rev. A 79(4), 043824 (2009).
    [Crossref]
  34. E. S. Andrianov, A. A. Pukhov, A. V. Dorofeenko, A. P. Vinogradov, and A. A. Lisyansky, “Rabi oscillations in spasers during nonradiative plasmon excitation,” Phys. Rev. B 85(3), 035405 (2012).
    [Crossref]
  35. A. N. Lagarkov, A. K. Sarychev, V. N. Kissel, and G. Tartakovsky, “Superresolution and enhancement in metamaterials,” Phys. Usp. 52(9), 959–967 (2009).
    [Crossref]
  36. M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University Press, 1997).
  37. A. A. Zyablovsky, A. V. Dorofeenko, A. A. Pukhov, and A. P. Vinogradov, “Lasing in a gain slab as a consequence of the causality principle,” J. Commun. Technol. Electron. 56(9), 1139–1145 (2011).
    [Crossref]
  38. S. Solimeno, B. Crosignani, and P. Di Porto, Guiding, Diffraction, and Confinement of Optical Radiation (Academic, 1986).
  39. J. A. Gordon and R. W. Ziolkowski, “The design and simulated performance of a coated nano-particle laser,” Opt. Express 15(5), 2622–2653 (2007).
    [Crossref] [PubMed]
  40. A. Mizrahi, V. Lomakin, B. A. Slutsky, M. P. Nezhad, L. Feng, and Y. Fainman, “Low threshold gain metal coated laser nanoresonators,” Opt. Lett. 33(11), 1261–1263 (2008).
    [Crossref] [PubMed]
  41. X. F. Li and S. F. Yu, “Design of low-threshold compact Au-nanoparticle lasers,” Opt. Lett. 35(15), 2535–2537 (2010).
    [Crossref] [PubMed]
  42. A. Veltri and A. Aradian, “Optical response of a metallic nanoparticle immersed in a medium with optical gain,” Phys. Rev. B 85(11), 115429 (2012).
    [Crossref]
  43. N. M. Lawandy, “Localized surface plasmon singularities in amplifying media,” Appl. Phys. Lett. 85(21), 5040–5042 (2004).
    [Crossref]
  44. R. H. Pantell and H. E. Puthoff, Fundamentals of Quantum Electronics (John Wiley & Sons Inc, 1969).
  45. P. Sperber, W. Spangler, B. Meier, and A. Penzkofer, “Experimental and theoretical investigation of tunable picosecond pulse generation in longitudinally pumped dye laser generators and amplifiers,” Opt. Quantum Electron. 20(5), 395–431 (1988).
    [Crossref]
  46. P. B. Johnson and R. W. Christy, “Optical Constants of the Noble Metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
    [Crossref]
  47. E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302(5644), 419–422 (2003).
    [Crossref] [PubMed]
  48. C. F. Boheren and D. R. Huffman, Absorption and Scattering of Light by Small Particles, (Wiley, 1983)
  49. A. P. Vinogradov, E. S. Andrianov, A. A. Pukhov, A. V. Dorofeenko, and A. A. Lisyansky, “Quantum plasmonics of metamaterials: loss compensation using spasers,” Phys. Usp. 55(10), 1046–1053 (2012).
    [Crossref]
  50. M. Wegener, J. L. García-Pomar, C. M. Soukoulis, N. Meinzer, M. Ruther, and S. Linden, “Toy model for plasmonic metamaterial resonances coupled to two-level system gain,” Opt. Express 16(24), 19785–19798 (2008).
    [Crossref] [PubMed]
  51. M. Premaratne and G. P. Agrawal, Light Propagation in Gain Medium (Cambridge University, 2011).
  52. M. I. Stockman, “Spaser action, loss compensation, and stability in plasmonic systems with gain,” Phys. Rev. Lett. 106(15), 156802 (2011).
    [Crossref] [PubMed]
  53. A. A. Lisyansky, E. S. Andrianov, A. V. Dorofeenko, A. A. Pukhov, and A. P. Vinogradov, “Forced spaser oscillations,” Proc. SPIE 8457, 84570X (2012).
    [Crossref]
  54. J. B. Khurgin and G. Sun, “Practicality of compensating the loss in the plasmonic waveguides using semiconductor gain medium,” Appl. Phys. Lett. 100(1), 011105 (2012).
    [Crossref]
  55. J. B. Khurgin and G. Sun, “Injection pumped single mode surface plasmon generators: threshold, linewidth, and coherence,” Opt. Express 20(14), 15309–15325 (2012).
    [Crossref] [PubMed]
  56. P. W. Milonni and J. H. Eberly, Laser Physics (Wiley, 2010).
  57. A. H. Nayfeh and D. T. Mook, Nonlinear Oscillations (Wiley-VCH, 1995).
  58. A. A. Zharov, R. E. Noskov, and M. V. Tsarev, “Plasmon-induced terahertz radiation generation due to symmetry breaking in a nonlinear metallic nanodimer,” J. Appl. Phys. 106(7), 073104 (2009).
    [Crossref]
  59. R. E. Noskov, P. A. Belov, and Y. S. Kivshar, “Subwavelength modulational instability and plasmon oscillons in nanoparticle arrays,” Phys. Rev. Lett. 108(9), 093901 (2012).
    [Crossref] [PubMed]
  60. A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization. A Universal Concept in Nonlinear Sciences (Cambridge University, 2001).

2012 (8)

Y.-J. Lu, J. Kim, H.-Y. Chen, C. Wu, N. Dabidian, C. E. Sanders, C.-Y. Wang, M.-Y. Lu, B.-H. Li, X. Qiu, W.-H. Chang, L.-J. Chen, G. Shvets, C.-K. Shih, and S. Gwo, “Plasmonic nanolaser using epitaxially grown silver film,” Science 337(6093), 450–453 (2012).
[Crossref] [PubMed]

E. S. Andrianov, A. A. Pukhov, A. V. Dorofeenko, A. P. Vinogradov, and A. A. Lisyansky, “Rabi oscillations in spasers during nonradiative plasmon excitation,” Phys. Rev. B 85(3), 035405 (2012).
[Crossref]

A. Veltri and A. Aradian, “Optical response of a metallic nanoparticle immersed in a medium with optical gain,” Phys. Rev. B 85(11), 115429 (2012).
[Crossref]

A. P. Vinogradov, E. S. Andrianov, A. A. Pukhov, A. V. Dorofeenko, and A. A. Lisyansky, “Quantum plasmonics of metamaterials: loss compensation using spasers,” Phys. Usp. 55(10), 1046–1053 (2012).
[Crossref]

A. A. Lisyansky, E. S. Andrianov, A. V. Dorofeenko, A. A. Pukhov, and A. P. Vinogradov, “Forced spaser oscillations,” Proc. SPIE 8457, 84570X (2012).
[Crossref]

J. B. Khurgin and G. Sun, “Practicality of compensating the loss in the plasmonic waveguides using semiconductor gain medium,” Appl. Phys. Lett. 100(1), 011105 (2012).
[Crossref]

J. B. Khurgin and G. Sun, “Injection pumped single mode surface plasmon generators: threshold, linewidth, and coherence,” Opt. Express 20(14), 15309–15325 (2012).
[Crossref] [PubMed]

R. E. Noskov, P. A. Belov, and Y. S. Kivshar, “Subwavelength modulational instability and plasmon oscillons in nanoparticle arrays,” Phys. Rev. Lett. 108(9), 093901 (2012).
[Crossref] [PubMed]

2011 (7)

M. I. Stockman, “Spaser action, loss compensation, and stability in plasmonic systems with gain,” Phys. Rev. Lett. 106(15), 156802 (2011).
[Crossref] [PubMed]

A. A. Zyablovsky, A. V. Dorofeenko, A. A. Pukhov, and A. P. Vinogradov, “Lasing in a gain slab as a consequence of the causality principle,” J. Commun. Technol. Electron. 56(9), 1139–1145 (2011).
[Crossref]

R.-M. Ma, R. F. Oulton, V. J. Sorger, G. Bartal, and X. Zhang, “Room-temperature sub-diffraction-limited plasmon laser by total internal reflection,” Nat. Mater. 10(2), 110–113 (2011).
[Crossref] [PubMed]

C.-Y. Wu, C.-T. Kuo, C.-Y. Wang, C.-L. He, M.-H. Lin, H. Ahn, and S. Gwo, “Plasmonic green nanolaser based on a metal-oxide-semiconductor structure,” Nano Lett. 11(10), 4256–4260 (2011).
[Crossref] [PubMed]

E. S. Andrianov, A. A. Pukhov, A. V. Dorofeenko, A. P. Vinogradov, and A. A. Lisyansky, “Forced synchronization of spaser by an external optical wave,” Opt. Express 19(25), 24849–24857 (2011).
[Crossref] [PubMed]

E. S. Andrianov, A. A. Pukhov, A. V. Dorofeenko, A. P. Vinogradov, and A. A. Lisyansky, “Dipole response of spaser on an external optical wave,” Opt. Lett. 36(21), 4302–4304 (2011).
[Crossref] [PubMed]

M. I. Stockman, “Nanoplasmonics: past, present, and glimpse into future,” Opt. Express 19(22), 22029–22106 (2011).
[Crossref] [PubMed]

2010 (6)

A. Fang, T. Koschny, and C. M. Soukoulis, “Lasing in metamaterial nanostructures,” J. Opt. 12(2), 024013 (2010).
[Crossref]

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466(7307), 735–738 (2010).
[Crossref] [PubMed]

X. F. Li and S. F. Yu, “Design of low-threshold compact Au-nanoparticle lasers,” Opt. Lett. 35(15), 2535–2537 (2010).
[Crossref] [PubMed]

S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Overcoming Losses with Gain in a Negative Refractive Index Metamaterial,” Phys. Rev. Lett. 105(12), 127401 (2010).
[Crossref] [PubMed]

M. I. Stockman, “The spaser as a nanoscale quantum generator and ultrafast amplifier,” J. Opt. 12(2), 024004 (2010).
[Crossref]

I. R. Gabitov, B. Kennedy, and A. I. Maimistov, “Coherent amplification of optical pulses in metamaterials,” IEEE J. Sel. Top. Quantum Electron. 16(2), 401–409 (2010).
[Crossref]

2009 (5)

A. S. Rosenthal and T. Ghannam, “Dipole nanolasers: A study of their quantum properties,” Phys. Rev. A 79(4), 043824 (2009).
[Crossref]

A. N. Lagarkov, A. K. Sarychev, V. N. Kissel, and G. Tartakovsky, “Superresolution and enhancement in metamaterials,” Phys. Usp. 52(9), 959–967 (2009).
[Crossref]

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460(7259), 1110–1112 (2009).
[Crossref] [PubMed]

R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461(7264), 629–632 (2009).
[Crossref] [PubMed]

A. A. Zharov, R. E. Noskov, and M. V. Tsarev, “Plasmon-induced terahertz radiation generation due to symmetry breaking in a nonlinear metallic nanodimer,” J. Appl. Phys. 106(7), 073104 (2009).
[Crossref]

2008 (5)

M. Wegener, J. L. García-Pomar, C. M. Soukoulis, N. Meinzer, M. Ruther, and S. Linden, “Toy model for plasmonic metamaterial resonances coupled to two-level system gain,” Opt. Express 16(24), 19785–19798 (2008).
[Crossref] [PubMed]

N. I. Zheludev, S. L. Prosvirnin, N. Papasimakis, and V. A. Fedotov, “Lasing spaser,” Nat. Photonics 2(6), 351–354 (2008).
[Crossref]

V. V. Klimov, Y. N. Istomin, and Y. A. Kosevich, “Plasma phenomena in nanostructures and neutron stars,” Phys. Usp. 51(8), 839–859 (2008).
[Crossref]

M. I. Stockman, “Spasers explained,” Nat. Photonics 2(6), 327–329 (2008).
[Crossref]

A. Mizrahi, V. Lomakin, B. A. Slutsky, M. P. Nezhad, L. Feng, and Y. Fainman, “Low threshold gain metal coated laser nanoresonators,” Opt. Lett. 33(11), 1261–1263 (2008).
[Crossref] [PubMed]

2007 (2)

J. A. Gordon and R. W. Ziolkowski, “The design and simulated performance of a coated nano-particle laser,” Opt. Express 15(5), 2622–2653 (2007).
[Crossref] [PubMed]

A. K. Sarychev and G. Tartakovsky, “Magnetic plasmonic metamaterials in actively pumped host medium and plasmonic nanolaser,” Phys. Rev. B 75(8), 085436 (2007).
[Crossref]

2006 (2)

2005 (2)

K. Li, X. Li, M. I. Stockman, and D. J. Bergman, “Surface plasmon amplification by stimulated emission in nanolenses,” Phys. Rev. B 71(11), 115409 (2005).
[Crossref]

I. E. Protsenko, A. V. Uskov, O. A. Zaimidoroga, V. N. Samoilov, and E. P. O’Reilly, “Dipole nanolaser,” Phys. Rev. A 71(6), 063812 (2005).
[Crossref]

2004 (1)

N. M. Lawandy, “Localized surface plasmon singularities in amplifying media,” Appl. Phys. Lett. 85(21), 5040–5042 (2004).
[Crossref]

2003 (3)

D. J. Bergman and M. I. Stockman, “Surface plasmon amplification by stimulated emission of radiation: quantum generation of coherent surface plasmons in nanosystems,” Phys. Rev. Lett. 90(2), 027402 (2003).
[Crossref] [PubMed]

S. A. Ramakrishna and J. Pendry, “Removal of absorption and increase in resolution in a near-field lens via optical gain,” Phys. Rev. B 67(20), 201101 (2003).
[Crossref]

E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302(5644), 419–422 (2003).
[Crossref] [PubMed]

1988 (1)

P. Sperber, W. Spangler, B. Meier, and A. Penzkofer, “Experimental and theoretical investigation of tunable picosecond pulse generation in longitudinally pumped dye laser generators and amplifiers,” Opt. Quantum Electron. 20(5), 395–431 (1988).
[Crossref]

1972 (1)

P. B. Johnson and R. W. Christy, “Optical Constants of the Noble Metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[Crossref]

Adegoke, J.

Ahn, H.

C.-Y. Wu, C.-T. Kuo, C.-Y. Wang, C.-L. He, M.-H. Lin, H. Ahn, and S. Gwo, “Plasmonic green nanolaser based on a metal-oxide-semiconductor structure,” Nano Lett. 11(10), 4256–4260 (2011).
[Crossref] [PubMed]

Andrianov, E. S.

E. S. Andrianov, A. A. Pukhov, A. V. Dorofeenko, A. P. Vinogradov, and A. A. Lisyansky, “Rabi oscillations in spasers during nonradiative plasmon excitation,” Phys. Rev. B 85(3), 035405 (2012).
[Crossref]

A. P. Vinogradov, E. S. Andrianov, A. A. Pukhov, A. V. Dorofeenko, and A. A. Lisyansky, “Quantum plasmonics of metamaterials: loss compensation using spasers,” Phys. Usp. 55(10), 1046–1053 (2012).
[Crossref]

A. A. Lisyansky, E. S. Andrianov, A. V. Dorofeenko, A. A. Pukhov, and A. P. Vinogradov, “Forced spaser oscillations,” Proc. SPIE 8457, 84570X (2012).
[Crossref]

E. S. Andrianov, A. A. Pukhov, A. V. Dorofeenko, A. P. Vinogradov, and A. A. Lisyansky, “Forced synchronization of spaser by an external optical wave,” Opt. Express 19(25), 24849–24857 (2011).
[Crossref] [PubMed]

E. S. Andrianov, A. A. Pukhov, A. V. Dorofeenko, A. P. Vinogradov, and A. A. Lisyansky, “Dipole response of spaser on an external optical wave,” Opt. Lett. 36(21), 4302–4304 (2011).
[Crossref] [PubMed]

Aradian, A.

A. Veltri and A. Aradian, “Optical response of a metallic nanoparticle immersed in a medium with optical gain,” Phys. Rev. B 85(11), 115429 (2012).
[Crossref]

Bahoura, M.

Bakker, R.

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460(7259), 1110–1112 (2009).
[Crossref] [PubMed]

Bartal, G.

R.-M. Ma, R. F. Oulton, V. J. Sorger, G. Bartal, and X. Zhang, “Room-temperature sub-diffraction-limited plasmon laser by total internal reflection,” Nat. Mater. 10(2), 110–113 (2011).
[Crossref] [PubMed]

R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461(7264), 629–632 (2009).
[Crossref] [PubMed]

Belgrave, A. M.

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460(7259), 1110–1112 (2009).
[Crossref] [PubMed]

Belov, P. A.

R. E. Noskov, P. A. Belov, and Y. S. Kivshar, “Subwavelength modulational instability and plasmon oscillons in nanoparticle arrays,” Phys. Rev. Lett. 108(9), 093901 (2012).
[Crossref] [PubMed]

Bergman, D. J.

K. Li, X. Li, M. I. Stockman, and D. J. Bergman, “Surface plasmon amplification by stimulated emission in nanolenses,” Phys. Rev. B 71(11), 115409 (2005).
[Crossref]

D. J. Bergman and M. I. Stockman, “Surface plasmon amplification by stimulated emission of radiation: quantum generation of coherent surface plasmons in nanosystems,” Phys. Rev. Lett. 90(2), 027402 (2003).
[Crossref] [PubMed]

Chang, W.-H.

Y.-J. Lu, J. Kim, H.-Y. Chen, C. Wu, N. Dabidian, C. E. Sanders, C.-Y. Wang, M.-Y. Lu, B.-H. Li, X. Qiu, W.-H. Chang, L.-J. Chen, G. Shvets, C.-K. Shih, and S. Gwo, “Plasmonic nanolaser using epitaxially grown silver film,” Science 337(6093), 450–453 (2012).
[Crossref] [PubMed]

Chen, H.-Y.

Y.-J. Lu, J. Kim, H.-Y. Chen, C. Wu, N. Dabidian, C. E. Sanders, C.-Y. Wang, M.-Y. Lu, B.-H. Li, X. Qiu, W.-H. Chang, L.-J. Chen, G. Shvets, C.-K. Shih, and S. Gwo, “Plasmonic nanolaser using epitaxially grown silver film,” Science 337(6093), 450–453 (2012).
[Crossref] [PubMed]

Chen, L.-J.

Y.-J. Lu, J. Kim, H.-Y. Chen, C. Wu, N. Dabidian, C. E. Sanders, C.-Y. Wang, M.-Y. Lu, B.-H. Li, X. Qiu, W.-H. Chang, L.-J. Chen, G. Shvets, C.-K. Shih, and S. Gwo, “Plasmonic nanolaser using epitaxially grown silver film,” Science 337(6093), 450–453 (2012).
[Crossref] [PubMed]

Chettiar, U. K.

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466(7307), 735–738 (2010).
[Crossref] [PubMed]

Christy, R. W.

P. B. Johnson and R. W. Christy, “Optical Constants of the Noble Metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[Crossref]

Dabidian, N.

Y.-J. Lu, J. Kim, H.-Y. Chen, C. Wu, N. Dabidian, C. E. Sanders, C.-Y. Wang, M.-Y. Lu, B.-H. Li, X. Qiu, W.-H. Chang, L.-J. Chen, G. Shvets, C.-K. Shih, and S. Gwo, “Plasmonic nanolaser using epitaxially grown silver film,” Science 337(6093), 450–453 (2012).
[Crossref] [PubMed]

Dai, L.

R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461(7264), 629–632 (2009).
[Crossref] [PubMed]

Dorofeenko, A. V.

E. S. Andrianov, A. A. Pukhov, A. V. Dorofeenko, A. P. Vinogradov, and A. A. Lisyansky, “Rabi oscillations in spasers during nonradiative plasmon excitation,” Phys. Rev. B 85(3), 035405 (2012).
[Crossref]

A. A. Lisyansky, E. S. Andrianov, A. V. Dorofeenko, A. A. Pukhov, and A. P. Vinogradov, “Forced spaser oscillations,” Proc. SPIE 8457, 84570X (2012).
[Crossref]

A. P. Vinogradov, E. S. Andrianov, A. A. Pukhov, A. V. Dorofeenko, and A. A. Lisyansky, “Quantum plasmonics of metamaterials: loss compensation using spasers,” Phys. Usp. 55(10), 1046–1053 (2012).
[Crossref]

A. A. Zyablovsky, A. V. Dorofeenko, A. A. Pukhov, and A. P. Vinogradov, “Lasing in a gain slab as a consequence of the causality principle,” J. Commun. Technol. Electron. 56(9), 1139–1145 (2011).
[Crossref]

E. S. Andrianov, A. A. Pukhov, A. V. Dorofeenko, A. P. Vinogradov, and A. A. Lisyansky, “Dipole response of spaser on an external optical wave,” Opt. Lett. 36(21), 4302–4304 (2011).
[Crossref] [PubMed]

E. S. Andrianov, A. A. Pukhov, A. V. Dorofeenko, A. P. Vinogradov, and A. A. Lisyansky, “Forced synchronization of spaser by an external optical wave,” Opt. Express 19(25), 24849–24857 (2011).
[Crossref] [PubMed]

Drachev, V. P.

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466(7307), 735–738 (2010).
[Crossref] [PubMed]

M. A. Noginov, G. Zhu, M. Bahoura, J. Adegoke, C. E. Small, B. A. Ritzo, V. P. Drachev, and V. M. Shalaev, “Enhancement of surface plasmons in an Ag aggregate by optical gain in a dielectric medium,” Opt. Lett. 31(20), 3022–3024 (2006).
[Crossref] [PubMed]

Fainman, Y.

Fang, A.

A. Fang, T. Koschny, and C. M. Soukoulis, “Lasing in metamaterial nanostructures,” J. Opt. 12(2), 024013 (2010).
[Crossref]

Fedotov, V. A.

N. I. Zheludev, S. L. Prosvirnin, N. Papasimakis, and V. A. Fedotov, “Lasing spaser,” Nat. Photonics 2(6), 351–354 (2008).
[Crossref]

Feng, L.

Gabitov, I. R.

I. R. Gabitov, B. Kennedy, and A. I. Maimistov, “Coherent amplification of optical pulses in metamaterials,” IEEE J. Sel. Top. Quantum Electron. 16(2), 401–409 (2010).
[Crossref]

García-Pomar, J. L.

Ghannam, T.

A. S. Rosenthal and T. Ghannam, “Dipole nanolasers: A study of their quantum properties,” Phys. Rev. A 79(4), 043824 (2009).
[Crossref]

Gladden, C.

R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461(7264), 629–632 (2009).
[Crossref] [PubMed]

Gordon, J. A.

Gwo, S.

Y.-J. Lu, J. Kim, H.-Y. Chen, C. Wu, N. Dabidian, C. E. Sanders, C.-Y. Wang, M.-Y. Lu, B.-H. Li, X. Qiu, W.-H. Chang, L.-J. Chen, G. Shvets, C.-K. Shih, and S. Gwo, “Plasmonic nanolaser using epitaxially grown silver film,” Science 337(6093), 450–453 (2012).
[Crossref] [PubMed]

C.-Y. Wu, C.-T. Kuo, C.-Y. Wang, C.-L. He, M.-H. Lin, H. Ahn, and S. Gwo, “Plasmonic green nanolaser based on a metal-oxide-semiconductor structure,” Nano Lett. 11(10), 4256–4260 (2011).
[Crossref] [PubMed]

Halas, N. J.

E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302(5644), 419–422 (2003).
[Crossref] [PubMed]

Hamm, J. M.

S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Overcoming Losses with Gain in a Negative Refractive Index Metamaterial,” Phys. Rev. Lett. 105(12), 127401 (2010).
[Crossref] [PubMed]

He, C.-L.

C.-Y. Wu, C.-T. Kuo, C.-Y. Wang, C.-L. He, M.-H. Lin, H. Ahn, and S. Gwo, “Plasmonic green nanolaser based on a metal-oxide-semiconductor structure,” Nano Lett. 11(10), 4256–4260 (2011).
[Crossref] [PubMed]

Herz, E.

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460(7259), 1110–1112 (2009).
[Crossref] [PubMed]

Hess, O.

S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Overcoming Losses with Gain in a Negative Refractive Index Metamaterial,” Phys. Rev. Lett. 105(12), 127401 (2010).
[Crossref] [PubMed]

Istomin, Y. N.

V. V. Klimov, Y. N. Istomin, and Y. A. Kosevich, “Plasma phenomena in nanostructures and neutron stars,” Phys. Usp. 51(8), 839–859 (2008).
[Crossref]

Johnson, P. B.

P. B. Johnson and R. W. Christy, “Optical Constants of the Noble Metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[Crossref]

Kennedy, B.

I. R. Gabitov, B. Kennedy, and A. I. Maimistov, “Coherent amplification of optical pulses in metamaterials,” IEEE J. Sel. Top. Quantum Electron. 16(2), 401–409 (2010).
[Crossref]

Khurgin, J. B.

J. B. Khurgin and G. Sun, “Practicality of compensating the loss in the plasmonic waveguides using semiconductor gain medium,” Appl. Phys. Lett. 100(1), 011105 (2012).
[Crossref]

J. B. Khurgin and G. Sun, “Injection pumped single mode surface plasmon generators: threshold, linewidth, and coherence,” Opt. Express 20(14), 15309–15325 (2012).
[Crossref] [PubMed]

Kildishev, A. V.

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466(7307), 735–738 (2010).
[Crossref] [PubMed]

Kim, J.

Y.-J. Lu, J. Kim, H.-Y. Chen, C. Wu, N. Dabidian, C. E. Sanders, C.-Y. Wang, M.-Y. Lu, B.-H. Li, X. Qiu, W.-H. Chang, L.-J. Chen, G. Shvets, C.-K. Shih, and S. Gwo, “Plasmonic nanolaser using epitaxially grown silver film,” Science 337(6093), 450–453 (2012).
[Crossref] [PubMed]

Kissel, V. N.

A. N. Lagarkov, A. K. Sarychev, V. N. Kissel, and G. Tartakovsky, “Superresolution and enhancement in metamaterials,” Phys. Usp. 52(9), 959–967 (2009).
[Crossref]

Kivshar, Y. S.

R. E. Noskov, P. A. Belov, and Y. S. Kivshar, “Subwavelength modulational instability and plasmon oscillons in nanoparticle arrays,” Phys. Rev. Lett. 108(9), 093901 (2012).
[Crossref] [PubMed]

Klimov, V. V.

V. V. Klimov, Y. N. Istomin, and Y. A. Kosevich, “Plasma phenomena in nanostructures and neutron stars,” Phys. Usp. 51(8), 839–859 (2008).
[Crossref]

Koschny, T.

A. Fang, T. Koschny, and C. M. Soukoulis, “Lasing in metamaterial nanostructures,” J. Opt. 12(2), 024013 (2010).
[Crossref]

Kosevich, Y. A.

V. V. Klimov, Y. N. Istomin, and Y. A. Kosevich, “Plasma phenomena in nanostructures and neutron stars,” Phys. Usp. 51(8), 839–859 (2008).
[Crossref]

Kuo, C.-T.

C.-Y. Wu, C.-T. Kuo, C.-Y. Wang, C.-L. He, M.-H. Lin, H. Ahn, and S. Gwo, “Plasmonic green nanolaser based on a metal-oxide-semiconductor structure,” Nano Lett. 11(10), 4256–4260 (2011).
[Crossref] [PubMed]

Lagarkov, A. N.

A. N. Lagarkov, A. K. Sarychev, V. N. Kissel, and G. Tartakovsky, “Superresolution and enhancement in metamaterials,” Phys. Usp. 52(9), 959–967 (2009).
[Crossref]

Lawandy, N. M.

N. M. Lawandy, “Localized surface plasmon singularities in amplifying media,” Appl. Phys. Lett. 85(21), 5040–5042 (2004).
[Crossref]

Li, B.-H.

Y.-J. Lu, J. Kim, H.-Y. Chen, C. Wu, N. Dabidian, C. E. Sanders, C.-Y. Wang, M.-Y. Lu, B.-H. Li, X. Qiu, W.-H. Chang, L.-J. Chen, G. Shvets, C.-K. Shih, and S. Gwo, “Plasmonic nanolaser using epitaxially grown silver film,” Science 337(6093), 450–453 (2012).
[Crossref] [PubMed]

Li, K.

K. Li, X. Li, M. I. Stockman, and D. J. Bergman, “Surface plasmon amplification by stimulated emission in nanolenses,” Phys. Rev. B 71(11), 115409 (2005).
[Crossref]

Li, X.

K. Li, X. Li, M. I. Stockman, and D. J. Bergman, “Surface plasmon amplification by stimulated emission in nanolenses,” Phys. Rev. B 71(11), 115409 (2005).
[Crossref]

Li, X. F.

Lin, M.-H.

C.-Y. Wu, C.-T. Kuo, C.-Y. Wang, C.-L. He, M.-H. Lin, H. Ahn, and S. Gwo, “Plasmonic green nanolaser based on a metal-oxide-semiconductor structure,” Nano Lett. 11(10), 4256–4260 (2011).
[Crossref] [PubMed]

Linden, S.

Lisyansky, A. A.

A. A. Lisyansky, E. S. Andrianov, A. V. Dorofeenko, A. A. Pukhov, and A. P. Vinogradov, “Forced spaser oscillations,” Proc. SPIE 8457, 84570X (2012).
[Crossref]

A. P. Vinogradov, E. S. Andrianov, A. A. Pukhov, A. V. Dorofeenko, and A. A. Lisyansky, “Quantum plasmonics of metamaterials: loss compensation using spasers,” Phys. Usp. 55(10), 1046–1053 (2012).
[Crossref]

E. S. Andrianov, A. A. Pukhov, A. V. Dorofeenko, A. P. Vinogradov, and A. A. Lisyansky, “Rabi oscillations in spasers during nonradiative plasmon excitation,” Phys. Rev. B 85(3), 035405 (2012).
[Crossref]

E. S. Andrianov, A. A. Pukhov, A. V. Dorofeenko, A. P. Vinogradov, and A. A. Lisyansky, “Forced synchronization of spaser by an external optical wave,” Opt. Express 19(25), 24849–24857 (2011).
[Crossref] [PubMed]

E. S. Andrianov, A. A. Pukhov, A. V. Dorofeenko, A. P. Vinogradov, and A. A. Lisyansky, “Dipole response of spaser on an external optical wave,” Opt. Lett. 36(21), 4302–4304 (2011).
[Crossref] [PubMed]

Lomakin, V.

Lu, M.-Y.

Y.-J. Lu, J. Kim, H.-Y. Chen, C. Wu, N. Dabidian, C. E. Sanders, C.-Y. Wang, M.-Y. Lu, B.-H. Li, X. Qiu, W.-H. Chang, L.-J. Chen, G. Shvets, C.-K. Shih, and S. Gwo, “Plasmonic nanolaser using epitaxially grown silver film,” Science 337(6093), 450–453 (2012).
[Crossref] [PubMed]

Lu, Y.-J.

Y.-J. Lu, J. Kim, H.-Y. Chen, C. Wu, N. Dabidian, C. E. Sanders, C.-Y. Wang, M.-Y. Lu, B.-H. Li, X. Qiu, W.-H. Chang, L.-J. Chen, G. Shvets, C.-K. Shih, and S. Gwo, “Plasmonic nanolaser using epitaxially grown silver film,” Science 337(6093), 450–453 (2012).
[Crossref] [PubMed]

Ma, R.-M.

R.-M. Ma, R. F. Oulton, V. J. Sorger, G. Bartal, and X. Zhang, “Room-temperature sub-diffraction-limited plasmon laser by total internal reflection,” Nat. Mater. 10(2), 110–113 (2011).
[Crossref] [PubMed]

R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461(7264), 629–632 (2009).
[Crossref] [PubMed]

Maimistov, A. I.

I. R. Gabitov, B. Kennedy, and A. I. Maimistov, “Coherent amplification of optical pulses in metamaterials,” IEEE J. Sel. Top. Quantum Electron. 16(2), 401–409 (2010).
[Crossref]

Meier, B.

P. Sperber, W. Spangler, B. Meier, and A. Penzkofer, “Experimental and theoretical investigation of tunable picosecond pulse generation in longitudinally pumped dye laser generators and amplifiers,” Opt. Quantum Electron. 20(5), 395–431 (1988).
[Crossref]

Meinzer, N.

Mizrahi, A.

Narimanov, E. E.

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460(7259), 1110–1112 (2009).
[Crossref] [PubMed]

Nezhad, M. P.

Ni, X.

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466(7307), 735–738 (2010).
[Crossref] [PubMed]

Noginov, M. A.

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460(7259), 1110–1112 (2009).
[Crossref] [PubMed]

M. A. Noginov, G. Zhu, M. Bahoura, J. Adegoke, C. E. Small, B. A. Ritzo, V. P. Drachev, and V. M. Shalaev, “Enhancement of surface plasmons in an Ag aggregate by optical gain in a dielectric medium,” Opt. Lett. 31(20), 3022–3024 (2006).
[Crossref] [PubMed]

Nordlander, P.

E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302(5644), 419–422 (2003).
[Crossref] [PubMed]

Noskov, R. E.

R. E. Noskov, P. A. Belov, and Y. S. Kivshar, “Subwavelength modulational instability and plasmon oscillons in nanoparticle arrays,” Phys. Rev. Lett. 108(9), 093901 (2012).
[Crossref] [PubMed]

A. A. Zharov, R. E. Noskov, and M. V. Tsarev, “Plasmon-induced terahertz radiation generation due to symmetry breaking in a nonlinear metallic nanodimer,” J. Appl. Phys. 106(7), 073104 (2009).
[Crossref]

O’Reilly, E. P.

I. E. Protsenko, A. V. Uskov, O. A. Zaimidoroga, V. N. Samoilov, and E. P. O’Reilly, “Dipole nanolaser,” Phys. Rev. A 71(6), 063812 (2005).
[Crossref]

Oulton, R. F.

R.-M. Ma, R. F. Oulton, V. J. Sorger, G. Bartal, and X. Zhang, “Room-temperature sub-diffraction-limited plasmon laser by total internal reflection,” Nat. Mater. 10(2), 110–113 (2011).
[Crossref] [PubMed]

R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461(7264), 629–632 (2009).
[Crossref] [PubMed]

Papasimakis, N.

N. I. Zheludev, S. L. Prosvirnin, N. Papasimakis, and V. A. Fedotov, “Lasing spaser,” Nat. Photonics 2(6), 351–354 (2008).
[Crossref]

Pendry, J.

S. A. Ramakrishna and J. Pendry, “Removal of absorption and increase in resolution in a near-field lens via optical gain,” Phys. Rev. B 67(20), 201101 (2003).
[Crossref]

Penzkofer, A.

P. Sperber, W. Spangler, B. Meier, and A. Penzkofer, “Experimental and theoretical investigation of tunable picosecond pulse generation in longitudinally pumped dye laser generators and amplifiers,” Opt. Quantum Electron. 20(5), 395–431 (1988).
[Crossref]

Popov, A. K.

Prodan, E.

E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302(5644), 419–422 (2003).
[Crossref] [PubMed]

Prosvirnin, S. L.

N. I. Zheludev, S. L. Prosvirnin, N. Papasimakis, and V. A. Fedotov, “Lasing spaser,” Nat. Photonics 2(6), 351–354 (2008).
[Crossref]

Protsenko, I. E.

I. E. Protsenko, A. V. Uskov, O. A. Zaimidoroga, V. N. Samoilov, and E. P. O’Reilly, “Dipole nanolaser,” Phys. Rev. A 71(6), 063812 (2005).
[Crossref]

Pukhov, A. A.

E. S. Andrianov, A. A. Pukhov, A. V. Dorofeenko, A. P. Vinogradov, and A. A. Lisyansky, “Rabi oscillations in spasers during nonradiative plasmon excitation,” Phys. Rev. B 85(3), 035405 (2012).
[Crossref]

A. P. Vinogradov, E. S. Andrianov, A. A. Pukhov, A. V. Dorofeenko, and A. A. Lisyansky, “Quantum plasmonics of metamaterials: loss compensation using spasers,” Phys. Usp. 55(10), 1046–1053 (2012).
[Crossref]

A. A. Lisyansky, E. S. Andrianov, A. V. Dorofeenko, A. A. Pukhov, and A. P. Vinogradov, “Forced spaser oscillations,” Proc. SPIE 8457, 84570X (2012).
[Crossref]

A. A. Zyablovsky, A. V. Dorofeenko, A. A. Pukhov, and A. P. Vinogradov, “Lasing in a gain slab as a consequence of the causality principle,” J. Commun. Technol. Electron. 56(9), 1139–1145 (2011).
[Crossref]

E. S. Andrianov, A. A. Pukhov, A. V. Dorofeenko, A. P. Vinogradov, and A. A. Lisyansky, “Dipole response of spaser on an external optical wave,” Opt. Lett. 36(21), 4302–4304 (2011).
[Crossref] [PubMed]

E. S. Andrianov, A. A. Pukhov, A. V. Dorofeenko, A. P. Vinogradov, and A. A. Lisyansky, “Forced synchronization of spaser by an external optical wave,” Opt. Express 19(25), 24849–24857 (2011).
[Crossref] [PubMed]

Pusch, A.

S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Overcoming Losses with Gain in a Negative Refractive Index Metamaterial,” Phys. Rev. Lett. 105(12), 127401 (2010).
[Crossref] [PubMed]

Qiu, X.

Y.-J. Lu, J. Kim, H.-Y. Chen, C. Wu, N. Dabidian, C. E. Sanders, C.-Y. Wang, M.-Y. Lu, B.-H. Li, X. Qiu, W.-H. Chang, L.-J. Chen, G. Shvets, C.-K. Shih, and S. Gwo, “Plasmonic nanolaser using epitaxially grown silver film,” Science 337(6093), 450–453 (2012).
[Crossref] [PubMed]

Radloff, C.

E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302(5644), 419–422 (2003).
[Crossref] [PubMed]

Ramakrishna, S. A.

S. A. Ramakrishna and J. Pendry, “Removal of absorption and increase in resolution in a near-field lens via optical gain,” Phys. Rev. B 67(20), 201101 (2003).
[Crossref]

Ritzo, B. A.

Rosenthal, A. S.

A. S. Rosenthal and T. Ghannam, “Dipole nanolasers: A study of their quantum properties,” Phys. Rev. A 79(4), 043824 (2009).
[Crossref]

Ruther, M.

Samoilov, V. N.

I. E. Protsenko, A. V. Uskov, O. A. Zaimidoroga, V. N. Samoilov, and E. P. O’Reilly, “Dipole nanolaser,” Phys. Rev. A 71(6), 063812 (2005).
[Crossref]

Sanders, C. E.

Y.-J. Lu, J. Kim, H.-Y. Chen, C. Wu, N. Dabidian, C. E. Sanders, C.-Y. Wang, M.-Y. Lu, B.-H. Li, X. Qiu, W.-H. Chang, L.-J. Chen, G. Shvets, C.-K. Shih, and S. Gwo, “Plasmonic nanolaser using epitaxially grown silver film,” Science 337(6093), 450–453 (2012).
[Crossref] [PubMed]

Sarychev, A. K.

A. N. Lagarkov, A. K. Sarychev, V. N. Kissel, and G. Tartakovsky, “Superresolution and enhancement in metamaterials,” Phys. Usp. 52(9), 959–967 (2009).
[Crossref]

A. K. Sarychev and G. Tartakovsky, “Magnetic plasmonic metamaterials in actively pumped host medium and plasmonic nanolaser,” Phys. Rev. B 75(8), 085436 (2007).
[Crossref]

Shalaev, V. M.

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466(7307), 735–738 (2010).
[Crossref] [PubMed]

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460(7259), 1110–1112 (2009).
[Crossref] [PubMed]

A. K. Popov and V. M. Shalaev, “Compensating losses in negative-index metamaterials by optical parametric amplification,” Opt. Lett. 31, 2169–2171 (2006).
[Crossref] [PubMed]

M. A. Noginov, G. Zhu, M. Bahoura, J. Adegoke, C. E. Small, B. A. Ritzo, V. P. Drachev, and V. M. Shalaev, “Enhancement of surface plasmons in an Ag aggregate by optical gain in a dielectric medium,” Opt. Lett. 31(20), 3022–3024 (2006).
[Crossref] [PubMed]

Shih, C.-K.

Y.-J. Lu, J. Kim, H.-Y. Chen, C. Wu, N. Dabidian, C. E. Sanders, C.-Y. Wang, M.-Y. Lu, B.-H. Li, X. Qiu, W.-H. Chang, L.-J. Chen, G. Shvets, C.-K. Shih, and S. Gwo, “Plasmonic nanolaser using epitaxially grown silver film,” Science 337(6093), 450–453 (2012).
[Crossref] [PubMed]

Shvets, G.

Y.-J. Lu, J. Kim, H.-Y. Chen, C. Wu, N. Dabidian, C. E. Sanders, C.-Y. Wang, M.-Y. Lu, B.-H. Li, X. Qiu, W.-H. Chang, L.-J. Chen, G. Shvets, C.-K. Shih, and S. Gwo, “Plasmonic nanolaser using epitaxially grown silver film,” Science 337(6093), 450–453 (2012).
[Crossref] [PubMed]

Slutsky, B. A.

Small, C. E.

Sorger, V. J.

R.-M. Ma, R. F. Oulton, V. J. Sorger, G. Bartal, and X. Zhang, “Room-temperature sub-diffraction-limited plasmon laser by total internal reflection,” Nat. Mater. 10(2), 110–113 (2011).
[Crossref] [PubMed]

R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461(7264), 629–632 (2009).
[Crossref] [PubMed]

Soukoulis, C. M.

Spangler, W.

P. Sperber, W. Spangler, B. Meier, and A. Penzkofer, “Experimental and theoretical investigation of tunable picosecond pulse generation in longitudinally pumped dye laser generators and amplifiers,” Opt. Quantum Electron. 20(5), 395–431 (1988).
[Crossref]

Sperber, P.

P. Sperber, W. Spangler, B. Meier, and A. Penzkofer, “Experimental and theoretical investigation of tunable picosecond pulse generation in longitudinally pumped dye laser generators and amplifiers,” Opt. Quantum Electron. 20(5), 395–431 (1988).
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Stockman, M. I.

M. I. Stockman, “Nanoplasmonics: past, present, and glimpse into future,” Opt. Express 19(22), 22029–22106 (2011).
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M. I. Stockman, “Spaser action, loss compensation, and stability in plasmonic systems with gain,” Phys. Rev. Lett. 106(15), 156802 (2011).
[Crossref] [PubMed]

M. I. Stockman, “The spaser as a nanoscale quantum generator and ultrafast amplifier,” J. Opt. 12(2), 024004 (2010).
[Crossref]

M. I. Stockman, “Spasers explained,” Nat. Photonics 2(6), 327–329 (2008).
[Crossref]

K. Li, X. Li, M. I. Stockman, and D. J. Bergman, “Surface plasmon amplification by stimulated emission in nanolenses,” Phys. Rev. B 71(11), 115409 (2005).
[Crossref]

D. J. Bergman and M. I. Stockman, “Surface plasmon amplification by stimulated emission of radiation: quantum generation of coherent surface plasmons in nanosystems,” Phys. Rev. Lett. 90(2), 027402 (2003).
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Stout, S.

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460(7259), 1110–1112 (2009).
[Crossref] [PubMed]

Sun, G.

J. B. Khurgin and G. Sun, “Injection pumped single mode surface plasmon generators: threshold, linewidth, and coherence,” Opt. Express 20(14), 15309–15325 (2012).
[Crossref] [PubMed]

J. B. Khurgin and G. Sun, “Practicality of compensating the loss in the plasmonic waveguides using semiconductor gain medium,” Appl. Phys. Lett. 100(1), 011105 (2012).
[Crossref]

Suteewong, T.

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460(7259), 1110–1112 (2009).
[Crossref] [PubMed]

Tartakovsky, G.

A. N. Lagarkov, A. K. Sarychev, V. N. Kissel, and G. Tartakovsky, “Superresolution and enhancement in metamaterials,” Phys. Usp. 52(9), 959–967 (2009).
[Crossref]

A. K. Sarychev and G. Tartakovsky, “Magnetic plasmonic metamaterials in actively pumped host medium and plasmonic nanolaser,” Phys. Rev. B 75(8), 085436 (2007).
[Crossref]

Tsakmakidis, K. L.

S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Overcoming Losses with Gain in a Negative Refractive Index Metamaterial,” Phys. Rev. Lett. 105(12), 127401 (2010).
[Crossref] [PubMed]

Tsarev, M. V.

A. A. Zharov, R. E. Noskov, and M. V. Tsarev, “Plasmon-induced terahertz radiation generation due to symmetry breaking in a nonlinear metallic nanodimer,” J. Appl. Phys. 106(7), 073104 (2009).
[Crossref]

Uskov, A. V.

I. E. Protsenko, A. V. Uskov, O. A. Zaimidoroga, V. N. Samoilov, and E. P. O’Reilly, “Dipole nanolaser,” Phys. Rev. A 71(6), 063812 (2005).
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Veltri, A.

A. Veltri and A. Aradian, “Optical response of a metallic nanoparticle immersed in a medium with optical gain,” Phys. Rev. B 85(11), 115429 (2012).
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Vinogradov, A. P.

A. P. Vinogradov, E. S. Andrianov, A. A. Pukhov, A. V. Dorofeenko, and A. A. Lisyansky, “Quantum plasmonics of metamaterials: loss compensation using spasers,” Phys. Usp. 55(10), 1046–1053 (2012).
[Crossref]

E. S. Andrianov, A. A. Pukhov, A. V. Dorofeenko, A. P. Vinogradov, and A. A. Lisyansky, “Rabi oscillations in spasers during nonradiative plasmon excitation,” Phys. Rev. B 85(3), 035405 (2012).
[Crossref]

A. A. Lisyansky, E. S. Andrianov, A. V. Dorofeenko, A. A. Pukhov, and A. P. Vinogradov, “Forced spaser oscillations,” Proc. SPIE 8457, 84570X (2012).
[Crossref]

A. A. Zyablovsky, A. V. Dorofeenko, A. A. Pukhov, and A. P. Vinogradov, “Lasing in a gain slab as a consequence of the causality principle,” J. Commun. Technol. Electron. 56(9), 1139–1145 (2011).
[Crossref]

E. S. Andrianov, A. A. Pukhov, A. V. Dorofeenko, A. P. Vinogradov, and A. A. Lisyansky, “Dipole response of spaser on an external optical wave,” Opt. Lett. 36(21), 4302–4304 (2011).
[Crossref] [PubMed]

E. S. Andrianov, A. A. Pukhov, A. V. Dorofeenko, A. P. Vinogradov, and A. A. Lisyansky, “Forced synchronization of spaser by an external optical wave,” Opt. Express 19(25), 24849–24857 (2011).
[Crossref] [PubMed]

Wang, C.-Y.

Y.-J. Lu, J. Kim, H.-Y. Chen, C. Wu, N. Dabidian, C. E. Sanders, C.-Y. Wang, M.-Y. Lu, B.-H. Li, X. Qiu, W.-H. Chang, L.-J. Chen, G. Shvets, C.-K. Shih, and S. Gwo, “Plasmonic nanolaser using epitaxially grown silver film,” Science 337(6093), 450–453 (2012).
[Crossref] [PubMed]

C.-Y. Wu, C.-T. Kuo, C.-Y. Wang, C.-L. He, M.-H. Lin, H. Ahn, and S. Gwo, “Plasmonic green nanolaser based on a metal-oxide-semiconductor structure,” Nano Lett. 11(10), 4256–4260 (2011).
[Crossref] [PubMed]

Wegener, M.

Wiesner, U.

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460(7259), 1110–1112 (2009).
[Crossref] [PubMed]

Wu, C.

Y.-J. Lu, J. Kim, H.-Y. Chen, C. Wu, N. Dabidian, C. E. Sanders, C.-Y. Wang, M.-Y. Lu, B.-H. Li, X. Qiu, W.-H. Chang, L.-J. Chen, G. Shvets, C.-K. Shih, and S. Gwo, “Plasmonic nanolaser using epitaxially grown silver film,” Science 337(6093), 450–453 (2012).
[Crossref] [PubMed]

Wu, C.-Y.

C.-Y. Wu, C.-T. Kuo, C.-Y. Wang, C.-L. He, M.-H. Lin, H. Ahn, and S. Gwo, “Plasmonic green nanolaser based on a metal-oxide-semiconductor structure,” Nano Lett. 11(10), 4256–4260 (2011).
[Crossref] [PubMed]

Wuestner, S.

S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Overcoming Losses with Gain in a Negative Refractive Index Metamaterial,” Phys. Rev. Lett. 105(12), 127401 (2010).
[Crossref] [PubMed]

Xiao, S.

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466(7307), 735–738 (2010).
[Crossref] [PubMed]

Yu, S. F.

Yuan, H.-K.

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466(7307), 735–738 (2010).
[Crossref] [PubMed]

Zaimidoroga, O. A.

I. E. Protsenko, A. V. Uskov, O. A. Zaimidoroga, V. N. Samoilov, and E. P. O’Reilly, “Dipole nanolaser,” Phys. Rev. A 71(6), 063812 (2005).
[Crossref]

Zentgraf, T.

R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461(7264), 629–632 (2009).
[Crossref] [PubMed]

Zhang, X.

R.-M. Ma, R. F. Oulton, V. J. Sorger, G. Bartal, and X. Zhang, “Room-temperature sub-diffraction-limited plasmon laser by total internal reflection,” Nat. Mater. 10(2), 110–113 (2011).
[Crossref] [PubMed]

R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461(7264), 629–632 (2009).
[Crossref] [PubMed]

Zharov, A. A.

A. A. Zharov, R. E. Noskov, and M. V. Tsarev, “Plasmon-induced terahertz radiation generation due to symmetry breaking in a nonlinear metallic nanodimer,” J. Appl. Phys. 106(7), 073104 (2009).
[Crossref]

Zheludev, N. I.

N. I. Zheludev, S. L. Prosvirnin, N. Papasimakis, and V. A. Fedotov, “Lasing spaser,” Nat. Photonics 2(6), 351–354 (2008).
[Crossref]

Zhu, G.

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460(7259), 1110–1112 (2009).
[Crossref] [PubMed]

M. A. Noginov, G. Zhu, M. Bahoura, J. Adegoke, C. E. Small, B. A. Ritzo, V. P. Drachev, and V. M. Shalaev, “Enhancement of surface plasmons in an Ag aggregate by optical gain in a dielectric medium,” Opt. Lett. 31(20), 3022–3024 (2006).
[Crossref] [PubMed]

Ziolkowski, R. W.

Zyablovsky, A. A.

A. A. Zyablovsky, A. V. Dorofeenko, A. A. Pukhov, and A. P. Vinogradov, “Lasing in a gain slab as a consequence of the causality principle,” J. Commun. Technol. Electron. 56(9), 1139–1145 (2011).
[Crossref]

Appl. Phys. Lett. (2)

N. M. Lawandy, “Localized surface plasmon singularities in amplifying media,” Appl. Phys. Lett. 85(21), 5040–5042 (2004).
[Crossref]

J. B. Khurgin and G. Sun, “Practicality of compensating the loss in the plasmonic waveguides using semiconductor gain medium,” Appl. Phys. Lett. 100(1), 011105 (2012).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

I. R. Gabitov, B. Kennedy, and A. I. Maimistov, “Coherent amplification of optical pulses in metamaterials,” IEEE J. Sel. Top. Quantum Electron. 16(2), 401–409 (2010).
[Crossref]

J. Appl. Phys. (1)

A. A. Zharov, R. E. Noskov, and M. V. Tsarev, “Plasmon-induced terahertz radiation generation due to symmetry breaking in a nonlinear metallic nanodimer,” J. Appl. Phys. 106(7), 073104 (2009).
[Crossref]

J. Commun. Technol. Electron. (1)

A. A. Zyablovsky, A. V. Dorofeenko, A. A. Pukhov, and A. P. Vinogradov, “Lasing in a gain slab as a consequence of the causality principle,” J. Commun. Technol. Electron. 56(9), 1139–1145 (2011).
[Crossref]

J. Opt. (2)

A. Fang, T. Koschny, and C. M. Soukoulis, “Lasing in metamaterial nanostructures,” J. Opt. 12(2), 024013 (2010).
[Crossref]

M. I. Stockman, “The spaser as a nanoscale quantum generator and ultrafast amplifier,” J. Opt. 12(2), 024004 (2010).
[Crossref]

Nano Lett. (1)

C.-Y. Wu, C.-T. Kuo, C.-Y. Wang, C.-L. He, M.-H. Lin, H. Ahn, and S. Gwo, “Plasmonic green nanolaser based on a metal-oxide-semiconductor structure,” Nano Lett. 11(10), 4256–4260 (2011).
[Crossref] [PubMed]

Nat. Mater. (1)

R.-M. Ma, R. F. Oulton, V. J. Sorger, G. Bartal, and X. Zhang, “Room-temperature sub-diffraction-limited plasmon laser by total internal reflection,” Nat. Mater. 10(2), 110–113 (2011).
[Crossref] [PubMed]

Nat. Photonics (2)

M. I. Stockman, “Spasers explained,” Nat. Photonics 2(6), 327–329 (2008).
[Crossref]

N. I. Zheludev, S. L. Prosvirnin, N. Papasimakis, and V. A. Fedotov, “Lasing spaser,” Nat. Photonics 2(6), 351–354 (2008).
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Nature (3)

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Figures (4)

Fig. 1
Fig. 1 A schematic drawing of a core-shell spaser.
Fig. 2
Fig. 2 Spaser response on external harmonic field in (top panels) below- and (bottom panels) above-threshold regimes. Solid and dashed lines show real and imaginary parts of the dipole moment, respectively. Frequencies of incident field are (a), (c) ω< = 1.98 eV < ωsp and (b), (d) ω> = 1.99 eV < ωsp. The red and green circles in the bottom panels show the exact loss compensation when the imaginary part of the dipole moment turns to zero.
Fig. 3
Fig. 3 Real parts of the instability growth rate Λ for below- (solid line) and above-threshold (dashed line) regimes of a driven spaser at the frequency of incident field ω< = 1.98 eV. For below-threshold spaser the instability growth rate is always negative so that spaser oscillations in external field are always stable. Above threshold, spaser is synchronized by strong external field, while its oscillations in weak external field are unstable and may show stochastic dynamics [16].
Fig. 4
Fig. 4 Stable (filled) and unstable regions of a driven above-threshold spaser for D0 = 0.25. The region of stability corresponds to steady-state solutions with ReΛ<0 . The two vertical lines correspond to the frequencies ω > and ω < , which are also used in Fig. 2. The red and green circles correspond to those in Fig. 2 (c) and 2(d). The red circle, in which Im[d]=0 , lies in the instability region, so loss compensation does not occur, while the green circle lies in the stable region and compensation does take place. The red and green curves show the compensation curves lying in unstable and stable regions, respectively, along which the imaginary part of dipole moment turns to zero.

Equations (25)

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××E+ ε 0 c 2 2 E t 2 = 4π c 2 2 P t 2 ,
2 P t 2 + 2 τ p P t + ω 0 2 P= 2 ω 0 | μ | 2 nE,
n t + 1 τ n ( n n 0 )= 1 ω 0 Re( E * P t ),
ε gain (ω)= ε 0 + D 0 ω 0 ω i+( ω 2 ω 0 2 )/2ωΓ 1+β | E(ω) | 2 + [ ( ω 2 ω 0 2 )/2ωΓ ] 2 ,
E shell ( r )=b E in c E in / r 3 +3c( E in n)n/ r 3 ,
E out ( r )=a E in / r 3 +3a( E in n)n/ r 3 ,
D core n | r core = D shell n | r core , E core ×n | r core = E shell ×n | r core , D shell n | r shell = D out n | r shell , E shell ×n | r shell = E out ×n | r shell ,
{ ε gain ( ω, E in , D 0 )=( b+2c/ r core 3 ) ε shell , 1=bc/ r core 3 , (b+2c/ r shell 3 ) ε shell =2a/ r shell 3 , bc/ r shell 3 =a/ r shell 3 .
ε gain ( ω, E in , D 0 )= ε shell 2( ε shell (ω)1 ) r core 3 2( 2+ ε shell (ω) ) r shell 3 2( ε shell (ω)1 ) r core 3 +( 2+ ε shell (ω) ) r shell 3 .
ε 0 + D 0 ω 0 ω i+( ω 2 ω 0 2 )/2ωΓ 1+β | E in | 2 + [ ( ω 2 ω 0 2 )/2ωΓ ] 2 =F(ω),
F( ω )= ε shell 2( ε shell (ω)1 ) r core 3 2( 2+ ε shell (ω) ) r shell 3 2( ε shell (ω)1 ) r core 3 +( 2+ ε shell (ω) ) r shell 3 .
D 0 1+β | E in | 2 + [ ( ω 2 ω 0 2 )/2ωΓ ] 2 = F(ω) ε 0 i+( ω 2 ω 0 2 )/2ωΓ ω ω 0 .
Im F( ω sp ) ε 0 i+( ω sp 2 ω 0 2 )/2 ω sp Γ =0.
D 0 1+β | E in ( D 0 ) | 2 + [ ( ω sp 2 ω 0 2 )/2 ω sp Γ ] 2 = ω sp ω 0 Re F( ω sp ) ε 0 i+( ω sp 2 ω 0 2 )/2 ω sp Γ .
D 0 1+β | E in ( D 0 ) | 2 + [ ( ω sp 2 ω 0 2 )/2 ω sp Γ ] 2 = D th 1+0+ [ ( ω sp 2 ω 0 2 )/2 ω sp Γ ] 2 .
E in = ( D 0 D th )/β D th 1+ [ ( ω sp 2 ω 0 2 )/2 ω sp Γ ] 2 , d=a E in .
D th = ω sp ω 0 ( 1+ [ ( ω sp 2 ω 0 2 )/2 ω sp Γ ] 2 )Re F( ω sp ) ε 0 i+( ω sp 2 ω 0 2 )/2 ω sp Γ .
{ ε gain ( ω, E in , D 0 )=( b+2c/ r core 3 ) ε shell , 1=bc/ r core 3 , (b+2c/ r shell 3 ) ε shell =x+2a/ r shell 3 , bc/ r shell 3 =xa/ r shell 3 .
α ^ 1 ( t, E in (t) )d( t )= E ext ( t ).
α 1 ( ω, E in ( ω ) ) d ˜ ( ω )= E ˜ ext ( ω ),
E ext ( t )= α 1 ( Ω, E in ) d ˜ ( Ω ) e iΩt dΩ = e iωt α 1 ( ω+ν, E in ) d ˜ slow ( ν ) e iνt dν .
E ext ( t ) e iωt [ α 1 ( ω, E in )+ν d α 1 ( ω, E in ) dω ] d ˜ slow ( ν ) e iνt dν = = e iωt α 1 ( ω, E in ) d ˜ slow ( ν ) e iνt dν + e i ω 0 t d α 1 ( ω, E in ) dω ν d ˜ slow ( ν ) e iνt dν = e iωt [ α 1 ( ω, E in ) d slow ( t )+i d α 1 ( ω, E in ) dω d d slow ( t ) dt ],
i d α 1 dω d dt d slow ( t )+ α 1 d slow ( t )= E slow ( t ).
i d α 1 dω δdΛ+ α 1 δd=0.
Λ=i α 1 ( ω, E in ) ( d α 1 ( ω, E in )/dω ) 1 .

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