## Design optimization of spasers considering the degeneracy of excited plasmon modes |

Optics Express, Vol. 21, Issue 13, pp. 15335-15349 (2013)

http://dx.doi.org/10.1364/OE.21.015335

Acrobat PDF (1047 KB)

### 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

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

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] .

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

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

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

*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] .

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*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 *μ*m,” Opt. Express **19**, 8954–8961 (2011) [CrossRef] [PubMed] .

**90**, 027402 (2003) [CrossRef] [PubMed] .

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

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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] .

*Light Propagation in Gain Media: Optical Amplifiers*(Cambridge University, 2011) [CrossRef] .

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. 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] .

**90**, 027402 (2003) [CrossRef] [PubMed] .

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

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

*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] .

*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] .

## 2. Spaser model

**90**, 027402 (2003) [CrossRef] [PubMed] .

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

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] .

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

*R*

_{1}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] .

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

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

*R*

_{1}is greater than

*v*(where

_{F}/ω*v*is Fermi velocity and

_{F}*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] .

*R*

_{2}must be chosen such that the shell thickness is large enough to the QD be entirely embedded within the dielectric (i.e.

*R*

_{2}−

*R*

_{1}> QD diameter).

*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. 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] .

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

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] .

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

*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] .

*l*and its projection

_{p}*m*, where

_{p}*ω*

_{lp}is the angular frequency of the plasmon mode

*l*. The stationary electric field is given by the expression where

_{p}*j*(

_{l}*kr*),

*y*(

_{l}*kr*),

*h*(

_{l}*kr*) for

*ν*= 1, 2, 3,

*k*is the wavenumber, and

*a*,

_{l}*d*,

_{l}*g*and

_{l}*f*are constants assuring the field’s continuity at boundaries.

_{l}*h̄ω*

_{lp}, 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

*a*,

_{l}*d*,

_{l}*g*and

_{l}*f*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

_{l}*l*th mode surface plasmon resonance: where

*s*

_{1}= −

*ψ*(

_{l}*k*

_{m}R_{1}),

*s*

_{2}=

*ψ*(

_{l}*k*

_{d}R_{1}),

*s*

_{3}= −

*χ*(

_{l}*k*

_{d}R_{1}),

*k*

_{2}= −

*ψ′*(

_{l}*k*

_{d}R_{2}),

*k*

_{3}=

*χ′*(

_{l}*k*

_{d}R_{2}),

*m*

_{1}=

*ξ*(

_{l}*k*

_{0}

*R*

_{2}),

*m*

_{2}= −

*ψ*(

_{l}*k*

_{d}R_{2}) and

*m*

_{3}=

*χ*(

_{l}*k*

_{d}R_{2}). Here the Reccati-Bessel functions are

*ψ*(

_{l}*x*) =

*xj*(

_{l}*x*),

*χ*(

_{l}*x*) = −

*xy*(

_{l}*x*);

*k*,

_{m}*k*, and

_{d}*k*

_{0}are the wavenumbers of the electromagnetic field in the metal, dielectric, and ambient, respectively.

*h̄ω*is a function of

_{l}*R*

_{1},

*R*

_{2},

*ε*

_{1},

*ε*

_{2}, and

*ε*

_{3}, 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

*a*,

_{l}*d*,

_{l}*g*in terms of

_{l}*f*as

_{l}*f*as a normalization constant, we follow the procedure of secondary quantization for dispersive media [41

_{l}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] .

*h̄ω*, Here

_{l}*V*implies the volume integral over three dimensional Euclidean space. We can note that owing to this equality,

*f*implicitly depends on the spaser shell parameters

_{l}*R*

_{1}and

*R*

_{2}.

### 2.2. Active medium

*〉 and two excited levels by |1*

_{e}*〉 and |2*

_{e}*〉. 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*

_{e}*R*, confined by an infinite potential barrier (i.e. potential

_{q}*V*(

*r*) = 0 for

*r*<

*R*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

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

*n*,

_{q}*l*,

_{q}*m*are principal, azimuthal and magnetic quantum numbers describing the states,

_{q}*ξ*

_{nqlq}is the

*n*th root of the spherical Bessel function of the first kind (i.e

_{q}*j*(

*ξ*

_{nqlq}) = 0), and

*ψ*

_{nq,lq,mq}gives the excitable energy levels of an electron-hole pair:

*ℰ*is the bandgap of the QD material. These energy levels denoted by

_{g}*ℰ*

_{nq,lq}are 2

*l*+ 1 times degenerate. For a transition from an initial state |

_{q}*s*〉 with quantum numbers

_{i}*s*= (

_{i}*n*) to a final state |

_{i}, l_{i}, m_{i}*s*〉 with quantum numbers

_{f}*s*= (

_{f}*n*,

_{f}*l*,

_{f}*m*), the absorbed energy from the system will be Δ

_{f}*ℰ*

_{sf,si}=

*ℰ*

_{nf,lf}−

*ℰ*

_{ni,li}. Energy will be released in case this quantity is negative.

*〉 and |2*

_{e}*〉, we select two lowest energy levels [i.e (*

_{e}*n*) = (1, 0), (1, 1)], assuming that probability of populating the higher energy levels are very small, and |0

_{q}, l_{q}*〉 is mapped to the ground level. This mapping also enables us to calculate the QD radius,*

_{e}*R*required for an efficient energy transfer to the resonator where the energy received by the spaser modem,

_{q}*h̄ω*

_{lp}matches the energy released by the QD, −Δ

*ℰ*

_{s0e,si}, giving the resonance QD radius: 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

*〉, |1*

_{e}*〉, and |2*

_{e}*〉, as described in Section 2.2. We assume that active medium strongly interacts only with the plasmon mode*

_{e}*l*, which is the spaser mode. Here we note that this assumption is not valid for higher modes when frequency spacing between them become smaller than the QD emission linewidth. Let us define the state |0

_{p}*〉 with zero plasmons, and 2*

_{pl}*l*+ 1 states

_{p}*n*= 2

*l*+ 4 product states defined as |1

_{p}*〉 ≡ |0*

_{s}*〉 |0*

_{e}*〉, |2*

_{pl}*〉 ≡ |1*

_{s}*〉 |0*

_{e}*〉, |3*

_{pl}*〉 ≡ |2*

_{s}*〉 |0*

_{e}*〉, |4*

_{pl}*〉 ≡ |0*

_{s}*〉*

_{e}*〉 ≡ |0*

_{s}*〉*

_{e}*〉, |2*

_{s}*〉 and |3*

_{s}*〉 are associated with zero plasmons and |4*

_{s}*〉, |5*

_{s}*〉,...,|*

_{s}*n*〉 possess one plasmon of the spaser mode. The |1

_{s}*〉 → |3*

_{s}*〉 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 |2*

_{s}*〉 to one of the states |4*

_{s}*〉, |5*

_{s}*〉,...,|*

_{s}*n*〉 is the driving force for the phenomena of spasing because they excite plasmon modes in the resonator. Some transitions may occur from the state |

_{s}*j*〉 to |

_{s}*i*〉 due to the interaction with the bath. They can be considered as dissipations. Having this model, we analyze the kinetics of the

_{s}*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

*H*of the spaser contains the non interacting electronic and plasmonic Hamiltonians,

*H*and

_{e}*H*, and Hamiltonian,

_{pl}*H*, of the interacting subsystems: where

_{i}*H*=

_{e}*h̄ω*

_{1e}|1

*〉 〈1*

_{e}*| +*

_{e}*h̄ω*

_{2e}|2

*〉 〈2*

_{e}*|, and*

_{e}*b*

_{lp,mp}are the creation and annihilation operators of the surface plasmons corresponding to the quantum numbers

*l*and

_{p}*m*.

_{p}*H*can be decomposed to represent the interactions between the electron-hole pairs and pump light as

_{i}*H*and the interactions between electron-hole pairs and surface plasmons as

_{e,L}*H*: where Here

_{e,pl}*V*

_{sf,si}is the matrix element for the transition |

*i*〉 → |

*f*〉,

*i, f*= {0

*, 1*

_{e}*, 2*

_{e}*},*

_{e}*V*is the normalization volume,

_{n}*φ*(

*t*) and

*ω*are the envelope function and frequency of the pump light, and c.c. represents the complex conjugate [44

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

*〉 to |1*

_{e}*〉 is taken into account through the relaxation constant*

_{e}*ξ*

_{23}in spaser kinetics. The matrix element

*u*〉 and |

_{c}*u*〉 are the Bloch functions, |

_{v}*s*〉 and |

_{i}*s*〉 are the envelope wavefunctions of the initial and final electronic states characterized by the sets of quantum numbers

_{f}*s*and

_{i}*s*and

_{f}**r̃**is the displacement vector of the electron-hole pairs. Assuming the equality

**E**

_{lpmp}=

*E*

_{lpmp}

**e**

_{lpmp}, we may write The first matrix element in Eq. (12) can be expressed through the Kane’s parameter

*P*[47, 48] as,

*V*,

_{QD}**r**=

**r**

_{0}+

**r′**is the electron position inside the quantum dot, and

**r**

_{0}is radius vector of the quantum dot’s center.

*E*

_{lpmp}(

**r**) can be derived from Eq. (2) and

*ψ*

_{sf}is taken from Eq. (5). In case QD is very small compared to the nanosphere, it is reasonable to assume that

*E*

_{lpmp}(

**r**) is approximately constant over the QD’s volume, therefore the integral in Eq. (13) can be simplified to

*V*

_{s2e,s0e}, electric field term

*E*

_{lpmp}in Eq. (11) should be replaced by the electric field caused by pump light.

### 3.2. Plasmon excitation rate of the spaser

*n*states system comprises of the product states |1

*〉, |2*

_{s}*〉,...,|*

_{s}*n*〉, using the density matrix formalism. We define the populations of those states by

_{s}*ρ*

_{11},

*ρ*

_{22},...,

*ρ*and assume that the system has a short-term memory [49

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

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

*γ*is the population relaxation rate of the state |

_{μμ}*μ*〉,

_{s}*γ*= (

_{μν}*γ*+

_{μμ}*γ*)

_{νν}*/*2 +

*γ̂*is the coherence relaxation rate between the states |

_{μν}*μ*〉 and |

_{s}*ν*〉,

_{s}*γ̂*is the pure dephasing rate, and

_{μν}*ξ*is the transition rate from state |

_{νκ}*κ*〉 to state |

_{s}*ν*〉 due to interaction with the bath. We assume that the lifetime of the ground state |1

_{s}*〉 is very large, by setting*

_{s}*γ*

_{11}= 0. The parameters

*γ*

_{44},

*γ*

_{55},...,

*γ*define the dissipation of degenerated states of the plasmon mode denoted by

_{nn}*l*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

_{p}, m_{p}*γ*.

_{pl}*ρ*

_{44},

*ρ*

_{55},...,

*ρ*represent the excitation rates for the plasmon modes

_{nn}*l*= (

_{p}, m_{p}*l*−

_{p},*l*), (

_{p}*l*, −

_{p}*l*+ 1)

_{p}*,*...,(

*l*) respectively. The sum of the plasmon populations of

_{p}, l_{p}*l*th mode,

_{p}*ℛ*

_{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: where Δ

_{L}_{3}=

*ω*−

_{L}*ω*

_{3}, Δ

_{2}

*=*

_{j}*ω*

_{2}−

*ω*when

_{j}*ω*

_{2},

*ω*

_{3},

*ω*and

_{j}*ω*are the energies of the states |2

_{L}*〉, |3*

_{s}*〉, |*

_{s}*j*〉 and pump light respectively. As all the degenerate plasmon modes have the same energy,

_{s}*ω*=

_{j}*ω*

_{lp}∀

*j*and therefore Δ

_{2}

*= Δ*

_{j}_{2}

*. We can assume that detuning of the energy of the pump light with the energy of the state |3*

_{p}*〉, Δ*

_{s}

_{L}_{3}≪

*γ*

_{13}and hence,

*γ*=

_{jj}*γ*and

_{pl}*γ*

_{2}

*=*

_{j}*γ*

_{2}

*∀*

_{p}*j*. With these simplifications, Eq. (16) can be further simplified to 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

**64**, 39–44 (2011) [CrossRef] .

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

*γ*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

_{pl}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] .

*ω*

_{lp}depends on resonator’s size parameters (i.e.

*R*

_{1}and

*R*

_{2}) 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,

*ℛ*

_{lp}mainly depends on the matrix elements for the electron-hole pair-plasmon interaction,

_{2}

*because we assume that matrix element for the pump light-QD interaction*

_{p}*V*

_{s2e,s0e}is constant under CW operation. In the case of exact resonance (i.e.

*R*is given by Eq. 6), Δ

_{q}_{2}

*= 0 and we achieve the highest plasmon excitation rate.*

_{p}*R*

_{1},

*R*

_{2}, there is a unique spaser mode energy

*h̄ω*

_{lp}and

*R*may be chosen according to

_{q}*ω*

_{lp}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

*l*= 1) is amplified by the active medium. We construct this spaser using gold for the nanosphere, and coating silica (SiO

_{p}_{2}) 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.

*ε*

_{2}= 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. R. Averitt, S. Westcott, and N. Halas, “Linear optical properties of gold nanoshells,” J. Opt. Soc. Am. B **16**, 1824–1832 (1999) [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] .

*ω*is the bulk plasma frequency of gold, Γ is the electron collision frequency with damping defined by Γ =

_{p}*γ*+

_{b}*v*2

_{F}/*R*

_{1},

*γ*is the bulk electron collision frequency of gold,

_{b}*v*is the Fermi velocity, and

_{F}*ε*

_{∞}is the contribution from the interband transitions obtained by fitting

*ε*

_{1}(

*ω*,

*R*

_{1})|

_{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] .

*ω*= 1.36 × 10

_{p}^{16}s

^{−1},

*γ*= 3.33 × 10

_{b}^{13}s

^{−1},

*v*= 1.4 × 10

_{F}^{6}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. 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] .

*ε*

_{3}= 1.

*R*

_{1},

*R*

_{2}) 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.

*E*= 1.74 eV,

_{g}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] .

*m*= −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 |0

_{p}*〉, |1*

_{e}*〉 and |2*

_{e}*〉 of the QD in this system posses energies 0, 2.385 and 3.059 eV respectively. The resulting total system has*

_{e}*n*= 6 states denoted by |1

*〉, |2*

_{s}*〉,...,|6*

_{s}*〉.*

_{s}*R*

_{1},

*R*

_{2}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

*l*= 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 |4

_{p}*〉, |5*

_{s}*〉, and |6*

_{s}*〉,*

_{s}*ℛ*

_{1}=

*ρ*

_{44}+

*ρ*

_{55}+

*ρ*

_{66}. We use the expression for

*ℛ*

_{lp}given in Eq. (17) to study the plasmon excitation rate of the dipole spaser. This equation contains some relaxation constants,

*γ*

_{13},

*γ*

_{22},

*γ*

_{33},

*γ*

_{2}

*, and*

_{p}*ξ*

_{23}, 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.

*γ*is a function of the spaser mode energy given by

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

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

*ω*

_{lp}depend on size parameters

*R*

_{1}and

*R*

_{2},

*γ*also depends on them for a given material. Continuing with these simplifications, we plot the plasmon excitation rate,

_{pl}*ℛ*

_{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

*R*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 (

_{q}*R*

_{1},

*R*

_{2}) 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.

*R*

_{1},

*R*

_{2}pair which results in the highest plasmon excitation rate. In our example, parameters (

*R*

_{1},

*R*

_{2}) = (6 nm, 11 nm) offer the highest spaser mode amplification for the 2.385 eV energy curve. Here we emphasize that, although we obtained this result for a single QD dipole spaser, a better spaser configuration may consists of many QDs to achieve a higher gain. If this dipole spaser consists of

*N*QDs, then

*R*

_{1}and

*R*

_{2}, and plot

*ℛ*

_{1}for the case of resonant QD radius (i.e.

*R*is also fixed). Here we vary QD’s location

_{q}*r*

_{0}within the dielectric shell from the innermost to the outermost position with respect to the nanosphere’s center. Such plots for four different

*R*

_{1},

*R*

_{2}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,

*R*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

_{q}*R*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.

_{q}*g*. Threshold gain can be found by applying the condition of population inversion:

_{th}*ρ*

_{22}≥

*ρ*

_{11}. It can be shown that it does not depend on spaser geometry, and is a function of the dielectric constants of the spaser materials and the spaser mode frequency given by [19

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

*R*

_{1}and

*R*

_{2}and nanosphere’s permittivity is a function of

*R*

_{1}, threshold gain also becomes a function of these two geometrical parameters as we plot in Fig. 3(e). According to the graph, the gain does not vary much with the size parameters but the required threshold gain for spasing is little higher when the nanosphere is smaller. We mark the 2.385 eV curve, which we used in the previous example, on threshold pump power plot and it provides an idea of choosing the correct geometrical parameters. It should be noted that, the expression for threshold gain is derived assuming that stimulated emission dominates in the spaser. However, there can be cases where spontaneous emission dominates spaser kinetics when the number of plasmons is one. In such situations, the semiclassical concept of threshold is not applicable and plasmon population in the resonator increases linearly with the pump rate [59

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

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] .

*R*

_{1},

*R*

_{2}and

*R*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.

_{q}## 5. Conclusions

*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

## References and links

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11. | N. Zheludev, S. Prosvirnin, N. Papasimakis, and V. Fedotov, “Lasing spaser,” Nature Photon. |

12. | S. W. Chang, C. Y. A. Ni, and S. L. Chuang, “Theory for bowtie plasmonic nanolasers,” Opt. Express |

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 |

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. |

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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. |

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55. | P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B |

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 |

57. | L. Liu, Q. Peng, and Y. Li, “An effective oxidation route to blue emission cdse quantum dots,” Inorg. Chem. |

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59. | I. E. Protsenko, “Quantum theory of dipole nanolasers,” J. Russ. Laser Res. |

**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

Sort: Year | Journal | Reset

### References

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