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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 13 — Jul. 1, 2013
  • pp: 15335–15349
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Design optimization of spasers considering the degeneracy of excited plasmon modes

Chanaka Rupasinghe, Ivan D. Rukhlenko, and Malin Premaratne  »View Author Affiliations


Optics Express, Vol. 21, Issue 13, pp. 15335-15349 (2013)
http://dx.doi.org/10.1364/OE.21.015335


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Abstract

We model spaser as an n-level quantum system and study a spasing geometry comprising of a metal nanosphere resonantly coupled to a semiconductor quantum dot (QD). The localized surface plasmons are assumed to be generated at the nanosphere due to the energy relaxation of the optically excited electron-hole pairs inside the QD. We analyze the total system, which is formed by hybridizing spaser’s electronic and plasmonic subsystems, using the density matrix formalism, and then derive an analytic expression for the plasmon excitation rate. Here, the QD with three nondegenerate states interacts with a single plasmon mode of arbitrary degeneracy with respect to angular momentum projection. The derived expression is analyzed, in order to optimize the performance of a spaser operating at the triple-degenerate dipole mode by appropriately choosing the geometric parameters of the spaser. Our method is applicable to different resonator geometries and may prove useful in the design of QD-powered spasers.

© 2013 OSA

1. Introduction

The emerging era of nanoplasmonics is expected to improve the speed and efficiency of optical devices, by allowing miniaturization beyond the diffraction limit using surface plasmons in circuits [1

1. M. Premaratne and G. P. Agrawal, Light Propagation in Gain Media: Optical Amplifiers(Cambridge University, 2011) [CrossRef] .

3

3. S. A. Maier and H. A. Atwater, “Plasmonics: Localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys. 98, 011101 (2005) [CrossRef] .

]. Despite their advantages in miniaturization, surface plasmons which excite at metal-dielectric interfaces are highly dissipative and vanish in a few wavelengths when traveling [1

1. M. Premaratne and G. P. Agrawal, Light Propagation in Gain Media: Optical Amplifiers(Cambridge University, 2011) [CrossRef] .

, 4

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

]. Therefore, the energy must be transferred from an external source to the surface plasmon wave to sustain its existence in nanoplasmonic circuits [1

1. M. Premaratne and G. P. Agrawal, Light Propagation in Gain Media: Optical Amplifiers(Cambridge University, 2011) [CrossRef] .

]. The spaser, which is the nanoplasmonic counterpart of a conventional laser is the prime device for generating these surface plasmon waves and amplifying them during propagation [5

5. 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, 027402 (2003) [CrossRef] [PubMed] .

, 6

6. R. F. Oulton, “Plasmonics: Loss and gain,” Nature Photon. 6, 219–221 (2012) [CrossRef] .

].

Bergman and Stockman proposed the theory of the spaser [5

5. 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, 027402 (2003) [CrossRef] [PubMed] .

] by claiming that a nanosystem with an excited active medium can transfer energy nonradiatively to a closely located plasmonic resonator and excite localized fields inside it. This energy transfer takes place due to the interaction between the resonator and active medium through the near field. It was stated that, electronic transitions in active medium are stimulated by the surface plasmons which already exist in the system resulting in a multiplication of the plasmon population, very much resembling the coherent feedback strategy used in conventional laser cavities [5

5. 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, 027402 (2003) [CrossRef] [PubMed] .

, 7

7. M. I. Stockman, “Spasers explained,” Nature Photon. 2, 327–329 (2008) [CrossRef] .

].

Since its theoretical formulation, many experimental efforts have been carried out to fabricate a spaser. Seidel et al.[ 8

8. J. Seidel, S. Grafström, and L. Eng, “Stimulated emission of surface plasmons at the interface between a silver film and an optically pumped dye solution,” Phys. Rev. Lett. 94, 177401 (2005) [CrossRef] [PubMed] .

] proved the possibility of the stimulated emission of surface plasmon by amplifying surface plasmons at the interface between a flat continuous silver film and a liquid containing organic dye molecules. The first demonstration of a spaser was done by Noginov et al.[ 9

9. 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, 1110–1112 (2009) [CrossRef] [PubMed] .

] using a gold nanosphere with a 7 nm radius, surrounded by the active medium, which is a 15 nm thick silica shell containing dye molecules. When the dye is optically pumped, it releases energy to the gold nanosphere causing excitation of localized surface plasmons. Another realization of a spaser was reported by Flynn et al.[ 10

10. R. A. Flynn, C. S. Kim, I. Vurgaftman, M. Kim, J. R. Meyer, A. J. Mäkinen, K. Bussmann, L. Cheng, F. S. Choa, and J. P. Long, “A room-temperature semiconductor spaser operating near 1.5 μm,” Opt. Express 19, 8954–8961 (2011) [CrossRef] [PubMed] .

] who sandwiched a gold-film plasmonic waveguide between optically pumped InGaAs quantum wells.

Several other structures have also been proposed for spasers, including a V-shaped metallic nanoparticle attached to quantum dots (QDs) [5

5. 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, 027402 (2003) [CrossRef] [PubMed] .

], a two-dimensional array of split-ring shape plasmonic resonators supported by a substrate acting as the active medium [11

11. N. Zheludev, S. Prosvirnin, N. Papasimakis, and V. Fedotov, “Lasing spaser,” Nature Photon. 2, 351–354 (2008) [CrossRef] .

], a bowtie-shaped metallic structure, in which QDs are placed in the bowtie gap and multiple quantum wells are located in the substrate [12

12. S. W. Chang, C. Y. A. Ni, and S. L. Chuang, “Theory for bowtie plasmonic nanolasers,” Opt. Express 16, 10580–10595 (2008) [CrossRef] [PubMed] .

], and a metal groove with QDs placed at its bottom [13

13. A. Lisyansky, I. Nechepurenko, A. Dorofeenko, A. Vinogradov, and A. Pukhov, “Channel spaser: Coherent excitation of one-dimensional plasmons from quantum dots located along a linear channel,” Phys. Rev. B 84, 153409 (2011) [CrossRef] .

]. Even though most of these spaser designs employ QDs in the active medium, it is possible to have optically pumped rare-earth ions, dyes or bulk electrical injection as the excitable gain element [1

1. M. Premaratne and G. P. Agrawal, Light Propagation in Gain Media: Optical Amplifiers(Cambridge University, 2011) [CrossRef] .

]. QDs are widely used in many lasing setups. As the charge carriers in a QD are confined in all three dimensions to a very small size, its density of states almost resembles a delta function, mimicking an atomistic behavior with a well defined spectral response. They also promise a better stability over temperature variations [14

14. M. Grundmann, J. Christen, N. N. Ledentsov, J. Böhrer, D. Bimberg, S. S. Ruvimov, P. Werner, U. Richter, U. Gösele, J. Heydenreich, V. M. Ustinov, A. Y. Egorov, A. E. Zhukov, P. S. Kop’ev, and Z. I. Alferov, “Ultra-narrow luminescence lines from single quantum dots,” Phys. Rev. Lett. 74, 4043–4046 (1995) [CrossRef] [PubMed] .

17

17. A. V. Fedorov, A. V. Baranov, I. D. Rukhlenko, T. S. Perova, and K. Berwick, “Quantum dot energy relaxation mediated by plasmon emission in doped covalent semiconductor heterostructures,” Phys. Rev. B 76, 045332 (2007) [CrossRef] .

].

Although ample research has been done on spasers, it is seen that results depend on the model. Furthermore, most of these studies very much focus on the role of the active medium. In this paper, instead of only considering the states of the active medium, we analyze the electronic and plasmonic states of the whole spaser as a single quantum system. Degeneracy of plasmon modes is also considered. This is a more general approach to describe the spaser quantum mechanically compared to available literature. We rigorously describe our model using a simple spaser geometry comprising of a spherical metal nanoparticle and a QD. A major importance of the spaser is its ability to be effectively used in nanoplasmonic circuits, to generate surface plasmon waves. This paper is based on studying the capability of a spaser to excite these surface plasmons and finding all the parameters affecting the excitation rate. We also consider dissipations to the environment and pay attention on all the material properties influencing the spaser operation. The presenting work also provides clear design guidelines for a spherical spaser. These guidelines allow one to select its operating wavelength, possible values for the size parameters of the spaser components, and optimum QD placement.

Among the quantum mechanical models of the spaser, the work of Stockman [5

5. 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, 027402 (2003) [CrossRef] [PubMed] .

, 18

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

, 19

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

] and Protsenko et al.[ 20

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

] are noteworthy. However, they only analyze the quantum states of the active medium assuming it as a TLS. We improve this model by considering the whole spaser as a single n-level quantum system with a 3-level active medium. Stockman considers a third level [18

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

] but assumes that its population is negligible and electron-hole pairs rapidly relax to the second level. In our model, there is a finite third level population and electron-holes pairs relax to the second level by interacting with the environment at a defined rate. Having a three level system, a designer has more control on choosing a suitable pump frequency such that it doesn’t overlap with any surface plasmon resonance frequency of the resonator. In addition, we consider dissipations through the generalized master equation in the form of interaction with the system’s bath. Here, all the major dissipation rates at each level are well represented. Since our intention is to find all the parameters affecting the plasmon excitation rate, it is required to consider all the forms of losses.

In our analysis, we first find the electric field and the eigenfrequencies of the localized surface plasmon modes in the resonator, then characteristics of the active medium, followed by the derivation of the spaser Hamiltonian, which is needed to analyze the spaser kinetics using the density matrix theory. Then we obtain an expression for the plasmon excitation rate of the spaser, and study how the geometric parameters affect the operation of a dipole spaser.

2. Spaser model

A spaser consists of a plasmonic resonator (or cavity), which supports surface plasmon modes, and an active medium, which amplifies the surface plasmons [5

5. 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, 027402 (2003) [CrossRef] [PubMed] .

]. Figure 1 shows a schematic diagram of our spaser model where the resonator is a metal nanosphere surrounded by dielectric shell, and a QD is embedded in the shell as the active medium. Spasing occurs as a consequence of the nonradiative energy transfer from the QD to the nanosphere, exciting localized surface plasmon modes inside it.

Fig. 1 Spaser model under consideration. R1 is the nanosphere radius, R2 is the radius of the dielectric shell’s outer boundary. A QD is located at the position r0 with respect to the nanosphere’s center. The parameters ε1, ε2 and ε3 denote the permittivities of the nanosphere, dielectric shell, and ambient, respectively.

The spherical core-shell structure is one of the most studied geometries for surface plasmon resonance [32

32. D. Sarid and W. Challener, Modern Introduction to Surface Plasmons: Theory, Mathematica Modeling, and Applications(Cambridge University, 2010) [CrossRef] .

34

34. K. L. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, “The optical properties of metal nanoparticles: The influence of size, shape, and dielectric environment,” J. Phys. Chem. B 107, 668–677 (2003) [CrossRef] .

], because it is one of the few geometries where Maxwell’s equations can be analytically solved [35

35. A. L. Aden and M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951) [CrossRef] .

, 36

36. C. Bohren and D. Huffman, Absorption and scattering of light by small particles(Wiley, 1983).

]. Suppose the inner radius of the metal nanosphere is R1 and it is smaller than the skin depth of the metal [37

37. E. Sondheimer, “The mean free path of electrons in metals,” Adv. Phys. 1, 1–42 (1952) [CrossRef] .

], which is usually around 25 nm for noble metals. The latter assumption is necessary to allow the localized surface plasmon modes to penetrate everywhere within the nanosphere and excite coherent electron cloud oscillations [18

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

, 38

38. M. I. Stockman, “Nanoplasmonics: The physics behind the applications,” Phys. Today 64, 39–44 (2011) [CrossRef] .

]. Hence, the nanosphere radius is very smaller compared to the wavelength of incident light. Further, we assume that R1 is greater than vF (where vF is Fermi velocity and ω is the surface plasmon frequency), which is about 1 nm for noble metals, to avoid the effects of Landau damping [38

38. M. I. Stockman, “Nanoplasmonics: The physics behind the applications,” Phys. Today 64, 39–44 (2011) [CrossRef] .

]. The outer radius of the shell, R2 must be chosen such that the shell thickness is large enough to the QD be entirely embedded within the dielectric (i.e. R2R1 > QD diameter).

When the nanosphere radius and the shell thickness are fixed, the resonator supports a series of plasmon modes (dipole, quadrupole, etc.) with unique energies [32

32. D. Sarid and W. Challener, Modern Introduction to Surface Plasmons: Theory, Mathematica Modeling, and Applications(Cambridge University, 2010) [CrossRef] .

, 33

33. P. K. Jain, K. S. Lee, I. H. El-Sayed, and M. A. El-Sayed, “Calculated absorption and scattering properties of gold nanoparticles of different size, shape, and composition: Applications in biological imaging and biomedicine,” J. Phys. Chem. B 110, 7238–7248 (2006) [CrossRef] [PubMed] .

]. These plasmon modes may overlap with the QD in different degrees, and hence experience gain in different amounts. The plasmon mode receiving the highest gain survives and becomes the dominant spaser mode [9

9. 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, 1110–1112 (2009) [CrossRef] [PubMed] .

, 38

38. M. I. Stockman, “Nanoplasmonics: The physics behind the applications,” Phys. Today 64, 39–44 (2011) [CrossRef] .

]. The gain to the spasing mode is received as a result of the electronic transitions in the QD causing recombination of an electron-hole pair. To ensure continuous spasing, we facilitate population inversion of the two energy levels pertaining to these transitions using a suitable pump source. Here we opt for optical pumping for the sake of design simplicity. Even though QD is the target of the pumping field, it may also excite the resonator in the vicinity [39

39. J. B. Khurgin, G. Sun, and R. Soref, “Practical limits of absorption enhancement near metal nanoparticles,” Appl. Phys. Lett. 94, 071103–071103 (2009) [CrossRef] .

]. To minimize this extraneous effect, we intentionally select the optical pumping frequency with considerable detuning from the surface plasmon resonances of the resonator. Therefore, the excited field is much weaker compared with the spaser mode which overlaps with QD emission lines. However, we have not made any restrictive assumption here because one may completely avoid this situation by adopting a different pumping mechanism such as direct electric injection to QD as suggested in [40

40. J. B. Khurgin, G. Sun, and R. Soref, “Electroluminescence efficiency enhancement using metal nanoparticles,” Appl. Phys. Lett. 93, 021120–021120 (2008) [CrossRef] .

].

2.1. Localized surface plasmon modes of the resonator

To find the electric field and eigenfrequencies of the localized surface plasmon modes, we assume that QD’s presence does not perturb the electric field in the system because it is very small compared to the nanosphere and therefore the resultant permittivity change of the shell is insignificant. We adopt a standard spherical coordinate system (r,θ,ϕ) where the origin coincides with the center of the spherical shell (see Fig. 1). The supported electromagnetic modes can be found by solving the vector Helmholtz equation for the spherical shell following the method of Debye potentials [35

35. A. L. Aden and M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951) [CrossRef] .

, 36

36. C. Bohren and D. Huffman, Absorption and scattering of light by small particles(Wiley, 1983).

]. The solution gives a series of localized surface plasmon modes denoted by the angular momentum number lp and its projection mp,
E(r,t)=lp=1mp=lplpElpmp(r)eiωlpt,
(1)
where ωlp is the angular frequency of the plasmon mode lp. The stationary electric field is given by the expression
Elm(r)={dlNlm(1)forrR1,glNlm(1)+flNlm(2)forR1<rR2,alNlm(3)forr>R2,
(2)
where Nlm(ν)=(1/k)××[cosϕPlm(cosθ)zl(ν)(kr)]. Here, zl(ν)(kr) is one of the spherical Bessel functions jl(kr), yl(kr), hl(kr) for ν = 1, 2, 3, Plm are associated Legendre polynomials, k is the wavenumber, and al, dl, gl and fl are constants assuring the field’s continuity at boundaries.

Plasmon modes expressed by Eq. (1) possess a unique energy determined by h̄ωlp, which can be found by solving the dispersion relation of the resonator. In order to obtain the dispersion relation, we first write a set of equations expressing the constants al, dl, gl and fl in Eq. (2) by equating the tangential components of the field at the metal-dielectric and dielectric-ambient interfaces. This results in a homogeneous system of linear equations in which we apply the condition for a nontrivial solution and obtain the following dispersion relation for the lth mode surface plasmon resonance:
m1[s1(k3q2k2q3)q1(k3s2k2s3)]=k1[s1(m3q2m2q3)q1(m3s2m2s3)],
(3)
where q1=ε2ψl(kmR1), q2=ε1ψl(kdR1), q3=ε1χl(kdR1), s1 = −ψl(kmR1), s2 = ψl(kdR1), s3 = −χl(kdR1), k1=ε2ξl(k0R2), k2 = −ψ′l(kdR2), k3 = χ′l(kdR2), m1 = ξl(k0R2), m2 = −ψl(kdR2) and m3 = χl(kdR2). Here the Reccati-Bessel functions are ψl(x) = xjl(x), ξl(x)=xhl(1)(x) and χl(x) = −xyl(x); km, kd, and k0 are the wavenumbers of the electromagnetic field in the metal, dielectric, and ambient, respectively.

According to Eq. (3), the energy of the plasmon mode, h̄ωl is a function of R1, R2, ε1, ε2, and ε3, which implies that only the size parameters of the resonator determine the energy of the spaser mode when the spaser materials are chosen. Furthermore, this nontrivial solution of the boundary conditions allows us to express the coefficients al, dl, gl in terms of fl as
al=s1(k2q3k3q2)q1(k2s3k3s2)k1(q2s1q1s2)fl,dl=q3s2q2s3q2s1q1s2fl,andgl=q1s3q3s1q2s1q1s2fl.
To find an expression for fl as a normalization constant, we follow the procedure of secondary quantization for dispersive media [41

41. I. D. Rukhlenko, D. Handapangoda, M. Premaratne, A. V. Fedorov, A. V. Baranov, and C. Jagadish, “Spontaneous emission of guided polaritons by quantum dot coupled to metallic nanowire: Beyond the dipole approximation,” Opt. Express 17, 17570–17581 (2009) [CrossRef] [PubMed] .

, 42

42. L. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Course of Theoretical Physics Vol 8: Electrodynamics of Continuous Media (Elsevier, 2004).

], and equate the total energy of the electric field of the plasmon mode to h̄ωl,
14πVd(ωε)dω|Elm(r)|2dr=h¯ωl.
(4)
Here V implies the volume integral over three dimensional Euclidean space. We can note that owing to this equality, fl implicitly depends on the spaser shell parameters R1 and R2.

2.2. Active medium

Characteristics of the active medium determine the strength of its electron-hole pairs’ interaction with the plasmon modes in the resonator and the amount of amplification that each plasmon mode receives. The characteristics of the QD includes its wavefunction and the energy levels of the electron-hole pairs. Electronic transitions between these levels fuel spasing and we assume that, as in the case of lasers, those transitions representing recombination of electron-hole pairs would excite the plasmon modes in the resonator. Normally, an excitable QD like this can be effectively described using three energy levels to account for pumping and stimulated transitions. We denote the ground level of the QD by state vector |0e〉 and two excited levels by |1e〉 and |2e〉. To assign energies to these three levels, we have to take the quantum confinement effects into account. To do this, we approximate the QD by a sphere with a radius Rq, confined by an infinite potential barrier (i.e. potential V(r) = 0 for r < Rq and it is infinite otherwise). The resulting time-independent Schrodinger equation is separable in spherical coordinates and possesses a solution similar to the hydrogen atom and given by [43

43. M. Ventra, S. Evoy, and J. Heflin, Introduction to Nanoscale Science and Technology, Nanostructure Science and Technology (Springer, 2004) [CrossRef] .

]
ψnq,lq,mq(r,θ,ϕ)=2Rq3jlq(ξnqlqr/Rq)j(lq+1)(ξnqlq)Ylqmq(θ,ϕ),
(5)
where nq, lq, mq are principal, azimuthal and magnetic quantum numbers describing the states, ξnqlq is the nqth root of the spherical Bessel function of the first kind (i.e j(ξnqlq) = 0), and Ylqmq are spherical harmonics.

The corresponding eigenvalue of the wave function ψnq,lq,mq gives the excitable energy levels of an electron-hole pair: nq,lq=g+h¯22μRq2ξnq,lq2, where μ=me*mh*/(me*+mh*) is the reduced mass of an electron-hole pair, me* and mh* are effective masses of an electron and a hole, and g is the bandgap of the QD material. These energy levels denoted by nq,lq are 2lq + 1 times degenerate. For a transition from an initial state |si〉 with quantum numbers si = (ni, li, mi) to a final state |sf〉 with quantum numbers sf = (nf, lf, mf), the absorbed energy from the system will be Δsf,si = nf,lfni,li. Energy will be released in case this quantity is negative.

Having this knowledge, we map three QD energy levels. For |1e〉 and |2e〉, we select two lowest energy levels [i.e (nq, lq) = (1, 0), (1, 1)], assuming that probability of populating the higher energy levels are very small, and |0e〉 is mapped to the ground level. This mapping also enables us to calculate the QD radius, Rq required for an efficient energy transfer to the resonator where the energy received by the spaser modem, h̄ωlp matches the energy released by the QD, −Δs0e,si, giving the resonance QD radius:
Rq=h¯ξnili2μ(h¯ωlpg).
(6)
Although achieving a perfectly matched resonance may be difficult in practice, having a closer value to the quantity in Eq. (6) will be adequate for spasing. This analysis of QD reveals how its geometrical parameters affect the spaser operation.

3. Spaser kinetics

Fig. 2 (a) Electronic subsystem of the spaser representing the states of the QD’s electron-hole pairs. (b) Plasmonic subsystem of the spaser representing the degenerate localized surface plasmon modes in the resonator. (c) Total system of the spaser with n = 2lp + 4 levels. γpl is the plasmon decay rate and ξij denotes the rate of transition from the state |js〉 to |is〉 due to the interaction with bath.

The product states |1s〉, |2s〉 and |3s〉 are associated with zero plasmons and |4s〉, |5s〉,...,|ns〉 possess one plasmon of the spaser mode. The |1s〉 → |3s〉 transition, which is the excitation of ground electron-hole pairs to the highest energy level in our model, occurs due to the electron-hole pairs’ interactions with the pump light which we analyze classically. Transitions from the state |2s〉 to one of the states |4s〉, |5s〉,...,|ns〉 is the driving force for the phenomena of spasing because they excite plasmon modes in the resonator. Some transitions may occur from the state |js〉 to |is〉 due to the interaction with the bath. They can be considered as dissipations. Having this model, we analyze the kinetics of the n-level system by first constructing its Hamiltonian, and then deriving the density matrix equations to find the corresponding active state populations.

3.1. Hamiltonian of the spaser

Hamiltonian H of the spaser contains the non interacting electronic and plasmonic Hamiltonians, He and Hpl, and Hamiltonian, Hi, of the interacting subsystems:
H(t)=He+Hpl+Hi(t),
(7)
where He = h̄ω1e |1e〉 〈1e| + h̄ω2e |2e〉 〈2e|, and Hpl=mp=lplph¯ωlpblp,mpblp,mp. Here blp,mp, blp,mp are the creation and annihilation operators of the surface plasmons corresponding to the quantum numbers lp and mp.

The interacting Hamiltonian Hi can be decomposed to represent the interactions between the electron-hole pairs and pump light as He,L and the interactions between electron-hole pairs and surface plasmons as He,pl:
Hi(t)=He,L(t)+He,pl,
(8)
where
He,L(t)=φ(t)Vs2e,s0e2eiωLt|2e0e|+c.c.,
(9)
He,pl=ih¯glpmp=lplpVs1e,s0elp,mpblp,mp|0e1e|+c.c.
(10)
Here Vsf,si is the matrix element for the transition |i〉 → |f〉, i, f = {0e, 1e, 2e}, glp=ωlp/2ε2Vn, where Vn is the normalization volume, φ(t) and ωL are the envelope function and frequency of the pump light, and c.c. represents the complex conjugate [44

44. A. Fedorov, A. Baranov, and Y. Masumoto, “Coherent control of optical-phonon-assisted resonance secondary emission in semiconductor quantum dots,” Opt. Spectrosc. 93, 52–60 (2002) [CrossRef] .

46

46. A. Fedorov and I. Rukhlenko, “Study of electronic dynamics of quantum dots using resonant photoluminescence technique,” Opt. Spectrosc. 100, 716–723 (2006) [CrossRef] .

]. Relaxation process from the state |2e〉 to |1e〉 is taken into account through the relaxation constant ξ23 in spaser kinetics. The matrix element Vsf,silp,mp corresponding to plasmon’s interaction with electron-hole pairs can be written as
Vsf,silp,mp=f|Elpmp.r˜|i=sf|uc|Elpmp.r˜|uv|si,
(11)
where |uc〉 and |uv〉 are the Bloch functions, |si〉 and |sf〉 are the envelope wavefunctions of the initial and final electronic states characterized by the sets of quantum numbers si and sf and is the displacement vector of the electron-hole pairs. Assuming the equality Elpmp = Elpmp elpmp, we may write
Vsf,silp,mp=uc|elpmp.r˜|uvsf|Elpmp|si.
(12)
The first matrix element in Eq. (12) can be expressed through the Kane’s parameter P[47

47. A. Ansel’m, Introduction to Semiconductor Theory (Mir, 1981).

, 48

48. D. Bimberg, R. Blachnik, P. Dean, T. Grave, G. Harbeke, K. Hübner, U. Kaufmann, W. Kress, and O. Madelung, Physics of Group IV Elements and III–V Compounds / Physik der Elemente der IV. Gruppe und der III–V Verbindungen, v. 17 (Springer, 1981).

] as, uc|elpmp.r˜|uv=2P/g and the second matrix element, sf|Elpmp|si=ϒsf,silp,mp, can be expressed by
ϒsf,silp,mp=VQDψsf*(r)Elpmp(r)ψsi(r)dr.
(13)
where the integration is evaluated over the quantum dot’s volume VQD, r = r0 + r′ is the electron position inside the quantum dot, and r0 is radius vector of the quantum dot’s center. Elpmp (r) can be derived from Eq. (2) and ψsf is taken from Eq. (5). In case QD is very small compared to the nanosphere, it is reasonable to assume that Elpmp (r) is approximately constant over the QD’s volume, therefore the integral in Eq. (13) can be simplified to ϒsf,silp,mp=Elpmp(r0)δsf,si. Hence, from Eq. (12), the matrix element for the spaser mode’s interaction with electron-hole pairs can be given by
Vsf,silp,mp=2Pgϒsf,silp,mp.
(14)
Since this quantity determines the contribution to the total Hamiltonian by the interaction of the spaser mode with the active medium, it also defines the strength of spasing. If the QD is moved away from the nanosphere this matrix element becomes very much smaller. Therefore, the closer the QD stronger the spasing. When calculating the Vs2e,s0e, electric field term Elpmp in Eq. (11) should be replaced by the electric field caused by pump light.

3.2. Plasmon excitation rate of the spaser

Having the Hamiltonian calculated, we analyze the n states system comprises of the product states |1s〉, |2s〉,...,|ns〉, using the density matrix formalism. We define the populations of those states by ρ11, ρ22,...,ρnn and assume that the system has a short-term memory [49

49. U. Fano, “Description of states in quantum mechanics by density matrix and operator techniques,” Rev. Mod. Phys. 29, 74–93 (1957) [CrossRef] .

] and coupled to a bath which is the reservoir for system’s dissipations. Then the relaxation superoperator, which is added to the commuted Hamiltonian and density operator to incorporate dissipations, consists of a set of constants that define the relaxation kinetics of diagonal and off-diagonal elements of the reduced density matrix [44

44. A. Fedorov, A. Baranov, and Y. Masumoto, “Coherent control of optical-phonon-assisted resonance secondary emission in semiconductor quantum dots,” Opt. Spectrosc. 93, 52–60 (2002) [CrossRef] .

, 49

49. U. Fano, “Description of states in quantum mechanics by density matrix and operator techniques,” Rev. Mod. Phys. 29, 74–93 (1957) [CrossRef] .

]. Using the Markov and secular approximations [50

50. K. Blum, Density Matrix Theory and Applications(Springer, 2010).

], master equation for the system can be given by
ρμν(t)t=1ih¯[H(t),ρ(t)]μνγμνρμν(t)+δμνκνξνκρκκ(t),
(15)
where γμμ is the population relaxation rate of the state |μs〉, γμν = (γμμ + γνν)/2 + γ̂μν is the coherence relaxation rate between the states |μs〉 and |νs〉, γ̂μν is the pure dephasing rate, and ξνκ is the transition rate from state |κs〉 to state |νs〉 due to interaction with the bath. We assume that the lifetime of the ground state |1s〉 is very large, by setting γ11 = 0. The parameters γ44, γ55,...,γnn define the dissipation of degenerated states of the plasmon mode denoted by lp, mp quantum numbers. However, since the energy and dielectric properties are common for all the degenerated plasmon states, we assume that all these plasmon dissipation constants are equal and can be denoted by γpl.

Populations ρ44, ρ55,...,ρnn represent the excitation rates for the plasmon modes lp, mp = (lp,lp), (lp, −lp + 1),...,(lp, lp) respectively. The sum of the plasmon populations of lpth mode, lp gives the number of plasmons excited in the spaser at a given time. Hence, this quantity can be referred as the effective plasmon excitation rate of the spaser. We solve the system of partial differential equations given in the Eq. (15) for the continuous wave (CW) operation assuming that φ(t) = 1 and obtain an expression for the total plasmon excitation rate:
lp=j=4nωlp22ε2h¯ξ23γ22γ33γjjγ13γ132+ΔL32γ2jγ2j2+Δ2j2|Vs2e,s0e|2|Vs1e,s0elp,jn+lp|2,
(16)
where ΔL3 = ωLω3, Δ2j = ω2ωj when ω2, ω3, ωj and ωL are the energies of the states |2s〉, |3s〉, |js〉 and pump light respectively. As all the degenerate plasmon modes have the same energy, ωj = ωlpj and therefore Δ2j = Δ2p. We can assume that detuning of the energy of the pump light with the energy of the state |3s〉, ΔL3γ13 and hence, γ13γ132ΔL321γ13. Further, we also assume that all the degenerate states of the plasmon modes decay equally and therefore γjj = γpl and γ2j = γ2pj. With these simplifications, Eq. (16) can be further simplified to
lp=ξ23ωlp22ε2h¯γ13γ22γ33γplγ2pγ2p2+Δ2p2|Vs2e,s0e|2j=4n|Vs1e,s0elp,jn+lp|2,
(17)
The system interacting with the bath introduces various relaxations to the electronic subsystem. We assume that these relaxation rates and the decay rate of the dominant spaser mode depend only on the materials of the spaser [38

38. M. I. Stockman, “Nanoplasmonics: The physics behind the applications,” Phys. Today 64, 39–44 (2011) [CrossRef] .

]. Since the nanosphere is much smaller than the wavelength, we can neglect the radiation losses [51

51. F. Wang and Y. R. Shen, “General properties of local plasmons in metal nanostructures,” Phys. Rev. Lett. 97, 206806 (2006) [CrossRef] [PubMed] .

] and only consider the nonradiative decay terms. However, this is not an overly restrictive assumption because in case radiation losses are not negligible, the decay rate γpl of the spaser mode can also incorporate the resultant of both radiative and nonradiative decay rates. In addition, decay rates could potentially be different for other plasmon modes [52

52. G. Sun, J. B. Khurgin, and C. Yang, “Impact of high-order surface plasmon modes of metal nanoparticles on enhancement of optical emission,” Appl. Phys. Lett. 95, 171103–171103 (2009) [CrossRef] .

], which are not taken into account assuming that they weakly overlap with the QD emission spectrum. As discussed in Section 2.1, energy of the spaser mode ωlp depends on resonator’s size parameters (i.e. R1 and R2) when the materials of the spaser components are chosen. Therefore, it can be observed from Eq. (17) that the total plasmon excitation rate of the spaser, lp mainly depends on the matrix elements for the electron-hole pair-plasmon interaction, Vs1e,s0elp,mp for each degenerate state and the detuning Δ2p because we assume that matrix element for the pump light-QD interaction Vs2e,s0e is constant under CW operation. In the case of exact resonance (i.e. Rq is given by Eq. 6), Δ2p = 0 and we achieve the highest plasmon excitation rate.

For fixed materials and size parameters R1, R2, there is a unique spaser mode energy h̄ωlp and Rq may be chosen according to ωlp to achieve a higher plasmon excitation rate. Then the position of the QD plays a major role deciding the amount of amplification of the plasmons as the electric field of the spaser mode changes with the location according to Eq. (2). To analyze these factors, we investigate the spaser’s behavior with respect to spaser size parameters and the QD’s location in the following section taking a dipole spaser as a case study.

4. Case study: A dipole spaser

Let us consider a spaser whose dipole mode (lp = 1) is amplified by the active medium. We construct this spaser using gold for the nanosphere, and coating silica (SiO2) over it, making up a dielectric shell. We embed a CdSe QD in this shell. Once the materials for spaser components are chosen, we are ready to investigate the dipole spaser’s operation according to its geometric parameters. Here, we use the analytical results obtained in Sections 2 and 3 on the resonator’s plasmonic properties and the characteristics of the resonator-QD interactions.

The knowledge of the frequency dependence of the permittivities are required to calculate the electric field of the spaser mode and the plasmon decay rate. For silica, we assume a non-dispersive permittivity, ε2 = 2.15 in contrast to gold in which we assume a frequency dependent permittivity. Furthermore, since our gold nanosphere is very small, we have to consider the size dependency modification of the permittivity as well [53

53. J. Lim, A. Eggeman, F. Lanni, R. D. Tilton, and S. A. Majetich, “Synthesis and single-particle optical detection of low-polydispersity plasmonic-superparamagnetic nanoparticles,” Adv. Mater. 20, 1721–1726 (2008) [CrossRef] .

, 54

54. R. Averitt, S. Westcott, and N. Halas, “Linear optical properties of gold nanoshells,” J. Opt. Soc. Am. B 16, 1824–1832 (1999) [CrossRef] .

]. In order to incorporate this size effect, we use the model [54

54. R. Averitt, S. Westcott, and N. Halas, “Linear optical properties of gold nanoshells,” J. Opt. Soc. Am. B 16, 1824–1832 (1999) [CrossRef] .

], ε1(ω,R1)=εωp2ω2+iωΓ(R1), where ωp is the bulk plasma frequency of gold, Γ is the electron collision frequency with damping defined by Γ = γb + vF/2R1, γb is the bulk electron collision frequency of gold, vF is the Fermi velocity, and ε is the contribution from the interband transitions obtained by fitting ε1(ω, R1)|R1→∞ to the experimental data published by Johnson and Christy [55

55. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972) [CrossRef] .

] for bulk material. Also we assume ωp = 1.36 × 1016 s−1, γb = 3.33 × 1013 s−1, vF = 1.4 × 106 m/s, and ε = 9.84 [54

54. R. Averitt, S. Westcott, and N. Halas, “Linear optical properties of gold nanoshells,” J. Opt. Soc. Am. B 16, 1824–1832 (1999) [CrossRef] .

,56

56. K. Kolwas, A. Derkachova, and M. Shopa, “Size characteristics of surface plasmons and their manifestation in scattering properties of metal particles,” J. Quant. Spectrosc. Radiat. Transfer 110, 1490–1501 (2009) [CrossRef] .

]. Spaser’s outer boundary is assumed to be in free space and hence ε3 = 1.

Using these permittivities, we solve the dispersion relation given in Eq. (3) for dipole mode and then plot energy of the spaser mode as a function of the nanosphere radius and shell thickness as shown in Fig. 3(a). The contour plot shows that there can be many (R1, R2) pairs which can result in the same spaser mode energy. For example, if the spaser mode energy is 2.385 eV (i.e. equivalent wavelength is approximately 520 nm), it traces a curve on the contour plot as marked. It is important to note that, although we refer energies in electron volts for convenience, they should be converted to SI units when substituted to equations.

Fig. 3 Plots of (a) energy of the spaser mode and (b) normalized plasmon excitation rate with respect to nanosphere radius and shell thickness when QD’s location is fixed to the middle of the dielectric shell, (c) plasmon excitation rate with respect to QD radius and shell thickness when the nanosphere radius is 10 nm and QD’s location is fixed to the middle of the dielectric shell, (d) plasmon excitation rate with respect to QD’s location for different R1, R2 pairs, (e) threshold gain with respect to nanosphere radius and shell thickness.

To find the appropriate QD radius to match the operating point, we substitute the corresponding spaser mode energy in Eq. (6). For the previously set operating point, 2.385 eV, the substitution gives resonant QD radius to be 2.4 nm, when the QD material parameters are Eg = 1.74 eV, me*=0.13, and mh*=0.45[57

57. L. Liu, Q. Peng, and Y. Li, “An effective oxidation route to blue emission cdse quantum dots,” Inorg. Chem. 47, 3182–3187 (2008) [CrossRef] [PubMed] .

, 58

58. W. Kwak, T. Kim, W. Chae, and Y. Sung, “Tuning the energy bandgap of CdSe nanocrystals via Mg doping,” Nanotechnology 18, 205702 (2007) [CrossRef] .

]. Calculating the energy levels of the QD with this radius, and considering the fact that dipole mode is triply degenerate with mp = −1, 0 and 1, we redraw the total system diagram given in Fig. 2(c) for this dipole spaser to attain the system illustrated in Fig. 4. The corresponding electronic levels |0e〉, |1e〉 and |2e〉 of the QD in this system posses energies 0, 2.385 and 3.059 eV respectively. The resulting total system has n = 6 states denoted by |1s〉, |2s〉,...,|6s〉.

Fig. 4 The total system diagram for a dipole spaser deduced from Fig. 2.

Evaluating this 6 state system of the dipole spaser for the highest plasmon excitation rate gives the optimum size parameters. Just opting an arbitrary R1, R2 pair on the preferred spaser mode energy curve may not result in the highest plasmon generation. To investigate the behavior of our dipole spaser, we follow the results derived in Section 3 by substituting lp = 1. As we discussed there, the total plasmon excitation rate of the dipole spaser is represented by the sum of the populations of the states |4s〉, |5s〉, and |6s〉, 1 = ρ44 + ρ55 + ρ66. We use the expression for lp given in Eq. (17) to study the plasmon excitation rate of the dipole spaser. This equation contains some relaxation constants, γ13, γ22, γ33, γ2p, and ξ23, which depend on the environment and used materials, but do not depend on spaser’s geometry. Therefore, we keep them constant and compare the plasmon excitation rate by investigating the normalized plasmon population in the relevant plots, as our objective is to study how the spaser’s geometrical parameters result in relatively high or low plasmon populations. While calculating the matrix elements, for the sake of simplicity, we assume that QD is located on the nanosphere’s dipole axis in θ = 0 direction.

The only relaxation rate influenced by spaser’s geometry is the decay rate of surface plasmons, γpl is a function of the spaser mode energy given by γplIm[ε1(ω)]/ωRe[ε1(ω)]|ω=ωlp[19

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

]. Since the nanosphere is very much smaller than the wavelength, here we consider only nonradiative decay assuming that its radiation loss is negligible [51

51. F. Wang and Y. R. Shen, “General properties of local plasmons in metal nanostructures,” Phys. Rev. Lett. 97, 206806 (2006) [CrossRef] [PubMed] .

]. By substituting the model for nanosphere’s permittivity in this expression, the plasmon decay rate can be simplified to γplΓ2[1+(Γ/ωlp)2]. Since both the quantities Γ and ωlp depend on size parameters R1 and R2, γpl also depends on them for a given material. Continuing with these simplifications, we plot the plasmon excitation rate, 1 of the dipole spaser with respect to nanosphere radius and shell thickness, as shown in Fig. 3(b), when the QD’s location is fixed to the middle of the dielectric shell and Rq is fixed to the resonant radius given by Eq. (6). It can be noted that plasmon excitation rate is higher for smaller nanosphere radii and shell thicknesses. It monotonically decreases when the total volume increases. This plot helps us to figure out the (R1, R2) pair that gives the highest plasmon excitation rate for a preferred spaser mode energy marked on Fig. 3(a). Hence, a designer has the freedom to optimize the spaser geometry by tuning either nanosphere radius or dielectric shell thickness.

In order to examine how the location of the QD affects the plasmon excitation rate of the spaser, we fix R1 and R2, and plot 1 for the case of resonant QD radius (i.e. Rq is also fixed). Here we vary QD’s location r0 within the dielectric shell from the innermost to the outermost position with respect to the nanosphere’s center. Such plots for four different R1, R2 pairs are shown in Fig. 3(d). It can be observed from these plots that the plasmon excitation rate rapidly decreases when the QD is moved away from the nanosphere. This happens because the interactions between plasmon modes and electron-hole pairs in QD gets weaker towards the shell’s outer boundary as the matrix element for interactions, V1e,0elp,mp decreases in the radial direction according to Eqs. (2), (13) and (14). The decreasing rate is higher for smaller nanosphere radii. Especially, when there are many QDs, their contribution to the spasing mode won’t be uniform. Hence, when designing a multiple QD spaser, we must set the optimum size parameters such that a large number of QDs are closely located to the nanosphere center.

Thus far, we have only considered the case where the QD radius is tuned according to Eq. (6) such that its emission energy exactly equals to the energy of the spaser mode. To investigate the influence of the QD radius on spasing, we vary Rq within an interval of 1–3 nm and plot the plasmon excitation rate, as shown in Fig. 3(c), for different shell thicknesses keeping the nanosphere radius fixed to 10 nm. According to this plot, highest plasmon excitation rate is observable in the case of exact resonance and it rapidly decreases when the Rq deviates from the resonant QD radius. For example, if the the QD radius deviates much (i.e. about 0.5 nm) from the resonant value, the resultant plasmon excitation rate will tend to zero.

The case study which we performed throughout this section provides us design guidelines on choosing the optimum size parameters of spaser components and where to place QDs to achieve a maximum plasmon excitation rate. In the process of designing a spaser, one might choose its operating wavelength as the first step, then pick possible values for the size parameters R1, R2 and Rq by examining the resulting in plasmon excitation rate. Then, it is necessary to figure out where to embed the QD. Placing the QD closer to the nanosphere will give a higher plasmon excitation rate. However, if multiple QDs are to be placed, then one need to consider about the precise placement because not all QDs can be placed close to the nanosphere’s boundary.

Although we studied a spaser realization with spherical structure, the derivations done in Section 3 is valid for a spaser with any geometry. It might not be possible to obtain the electric field and the energies of plasmon modes analytically for most of the other resonator geometries as we did in Section 2.1. But numerical solutions for the field can be admitted into the expressions derived in Section 3 to analyze the spaser kinetics and investigate the plasmon excitation rate which characterize its performance.

5. Conclusions

We have theoretically modeled spaser as an n-level quantum system formed by amalgamating spaser’s electronic and plasmonic subsystems. With this model, we studied a simple spaser geometry consists of a metal nanosphere resonantly coupled to a QD. The energy transfer between the QD and the nanosphere accompanied the relaxation of electron–hole pairs resonantly generated inside the QD by a continuous-wave laser. By employing the density matrix theory, we analytically found the excitation rate of surface plasmons where three nondegenerate electron–hole pair states are coupled to a single plasmon mode of arbitrary angular momentum. The obtained expression was then examined numerically for the special case of a spaser operating at the triple-degenerate dipole mode. It was shown that the plasmon excitation rate can be significantly enhanced by appropriately choosing the geometric parameters of the spaser.

Acknowledgments

The work of C. Rupasinghe is supported by the Monash University Institute of Graduate Research. The work of I. D. Rukhlenko and M. Premaratne is supported by the Australian Research Council, through its Discovery Early Career Researcher Award DE120100055 and Discovery Grant DP110100713, respectively.

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F. Wang and Y. R. Shen, “General properties of local plasmons in metal nanostructures,” Phys. Rev. Lett. 97, 206806 (2006) [CrossRef] [PubMed] .

52.

G. Sun, J. B. Khurgin, and C. Yang, “Impact of high-order surface plasmon modes of metal nanoparticles on enhancement of optical emission,” Appl. Phys. Lett. 95, 171103–171103 (2009) [CrossRef] .

53.

J. Lim, A. Eggeman, F. Lanni, R. D. Tilton, and S. A. Majetich, “Synthesis and single-particle optical detection of low-polydispersity plasmonic-superparamagnetic nanoparticles,” Adv. Mater. 20, 1721–1726 (2008) [CrossRef] .

54.

R. Averitt, S. Westcott, and N. Halas, “Linear optical properties of gold nanoshells,” J. Opt. Soc. Am. B 16, 1824–1832 (1999) [CrossRef] .

55.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972) [CrossRef] .

56.

K. Kolwas, A. Derkachova, and M. Shopa, “Size characteristics of surface plasmons and their manifestation in scattering properties of metal particles,” J. Quant. Spectrosc. Radiat. Transfer 110, 1490–1501 (2009) [CrossRef] .

57.

L. Liu, Q. Peng, and Y. Li, “An effective oxidation route to blue emission cdse quantum dots,” Inorg. Chem. 47, 3182–3187 (2008) [CrossRef] [PubMed] .

58.

W. Kwak, T. Kim, W. Chae, and Y. Sung, “Tuning the energy bandgap of CdSe nanocrystals via Mg doping,” Nanotechnology 18, 205702 (2007) [CrossRef] .

59.

I. E. Protsenko, “Quantum theory of dipole nanolasers,” J. Russ. Laser Res. 33, 559–577 (2012) [CrossRef] .

OCIS Codes
(140.3430) Lasers and laser optics : Laser theory
(230.0230) Optical devices : Optical devices
(250.5403) Optoelectronics : Plasmonics
(250.5590) Optoelectronics : Quantum-well, -wire and -dot devices

ToC Category:
Optics at Surfaces

History
Original Manuscript: May 8, 2013
Revised Manuscript: June 12, 2013
Manuscript Accepted: June 12, 2013
Published: June 19, 2013

Citation
Chanaka Rupasinghe, Ivan D. Rukhlenko, and Malin Premaratne, "Design optimization of spasers considering the degeneracy of excited plasmon modes," Opt. Express 21, 15335-15349 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-13-15335


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