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

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    [CrossRef]

2012

R. F. Oulton, “Plasmonics: Loss and gain,” Nature Photon.6, 219–221 (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. B85, 035405 (2012).
[CrossRef]

J. B. Khurgin and G. Sun, “How small can nano be in a nanolaser?” Nanophotonics1, 3–8 (2012).
[CrossRef]

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

J. B. Khurgin and G. Sun, “Injection pumped single mode surface plasmon generators: threshold, linewidth, and coherence,” Opt. Express20, 15309–15325 (2012).
[CrossRef] [PubMed]

2011

2010

D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nature Photon.4, 83–91 (2010).
[CrossRef]

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, 127401 (2010).
[CrossRef] [PubMed]

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

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

A. Ridolfo, O. Di Stefano, N. Fina, R. Saija, and S. Savasta, “Quantum plasmonics with quantum dot-metal nanoparticle molecules: influence of the fano effect on photon statistics,” Phys. Rev. Lett.105, 263601 (2010).
[CrossRef]

2009

A. Rosenthal and T. Ghannam, “Dipole nanolasers: A study of their quantum properties,” Phys. Rev. A79, 043824 (2009).
[CrossRef]

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

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. Transfer110, 1490–1501 (2009).
[CrossRef]

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]

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,” Nature460, 1110–1112 (2009).
[CrossRef] [PubMed]

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. Express17, 17570–17581 (2009).
[CrossRef] [PubMed]

2008

S. W. Chang, C. Y. A. Ni, and S. L. Chuang, “Theory for bowtie plasmonic nanolasers,” Opt. Express16, 10580–10595 (2008).
[CrossRef] [PubMed]

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. Express16, 19785–19798 (2008).
[CrossRef] [PubMed]

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

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

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]

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]

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

2007

A. K. Sarychev and G. Tartakovsky, “Magnetic plasmonic metamaterials in actively pumped host medium and plasmonic nanolaser,” Phys. Rev. B75, 085436 (2007).
[CrossRef]

W. Kwak, T. Kim, W. Chae, and Y. Sung, “Tuning the energy bandgap of CdSe nanocrystals via Mg doping,” Nanotechnology18, 205702 (2007).
[CrossRef]

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. B76, 045332 (2007).
[CrossRef]

2006

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. B110, 7238–7248 (2006).
[CrossRef] [PubMed]

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

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

2005

I. E. Protsenko, A. V. Uskov, O. A. Zaimidoroga, V. N. Samoilov, and E. P. O‘reilly, “Dipole nanolaser,” Phys. Rev. A71, 063812 (2005).
[CrossRef]

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]

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]

2003

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. V. Baranov, A. V. Fedorov, I. D. Rukhlenko, and Y. Masumoto, “Intraband carrier relaxation in quantum dots embedded in doped heterostructures,” Phys. Rev. B68, 205318 (2003).
[CrossRef]

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. B107, 668–677 (2003).
[CrossRef]

2002

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]

1999

1995

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]

1972

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B6, 4370–4379 (1972).
[CrossRef]

1957

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

1952

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

1951

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

Aden, A. L.

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

Agrawal, G. P.

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

Alferov, Z. I.

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]

Andrianov, E.

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. B85, 035405 (2012).
[CrossRef]

Ansel’m, A.

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

Atwater, H. A.

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]

Averitt, R.

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,” Nature460, 1110–1112 (2009).
[CrossRef] [PubMed]

Baranov, A.

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]

Baranov, A. V.

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. Express17, 17570–17581 (2009).
[CrossRef] [PubMed]

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. B76, 045332 (2007).
[CrossRef]

A. V. Baranov, A. V. Fedorov, I. D. Rukhlenko, and Y. Masumoto, “Intraband carrier relaxation in quantum dots embedded in doped heterostructures,” Phys. Rev. B68, 205318 (2003).
[CrossRef]

Baymuratov, A.

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,” Nature460, 1110–1112 (2009).
[CrossRef] [PubMed]

Bergman, D. J.

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]

Berwick, K.

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. B76, 045332 (2007).
[CrossRef]

Bimberg, D.

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]

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

Blachnik, R.

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

Blum, K.

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

Bohren, C.

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

Böhrer, J.

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).
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[CrossRef]

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[CrossRef] [PubMed]

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Figures (4)

Fig. 1
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.

Fig. 2
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.

Fig. 3
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.

Fig. 4
Fig. 4

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

Equations (19)

Equations on this page are rendered with MathJax. Learn more.

E ( r , t ) = l p = 1 m p = l p l p E l p m p ( r ) e i ω l p t ,
E l m ( r ) = { d l N l m ( 1 ) for r R 1 , g l N l m ( 1 ) + f l N l m ( 2 ) for R 1 < r R 2 , a l N l m ( 3 ) for r > R 2 ,
m 1 [ s 1 ( k 3 q 2 k 2 q 3 ) q 1 ( k 3 s 2 k 2 s 3 ) ] = k 1 [ s 1 ( m 3 q 2 m 2 q 3 ) q 1 ( m 3 s 2 m 2 s 3 ) ] ,
a l = s 1 ( k 2 q 3 k 3 q 2 ) q 1 ( k 2 s 3 k 3 s 2 ) k 1 ( q 2 s 1 q 1 s 2 ) f l , d l = q 3 s 2 q 2 s 3 q 2 s 1 q 1 s 2 f l , and g l = q 1 s 3 q 3 s 1 q 2 s 1 q 1 s 2 f l .
1 4 π V d ( ω ε ) d ω | E l m ( r ) | 2 d r = h ¯ ω l .
ψ n q , l q , m q ( r , θ , ϕ ) = 2 R q 3 j l q ( ξ n q l q r / R q ) j ( l q + 1 ) ( ξ n q l q ) Y l q m q ( θ , ϕ ) ,
R q = h ¯ ξ n i l i 2 μ ( h ¯ ω l p g ) .
H ( t ) = H e + H p l + H i ( t ) ,
H i ( t ) = H e , L ( t ) + H e , p l ,
H e , L ( t ) = φ ( t ) V s 2 e , s 0 e 2 e i ω L t | 2 e 0 e | + c . c . ,
H e , p l = i h ¯ g l p m p = l p l p V s 1 e , s 0 e l p , m p b l p , m p | 0 e 1 e | + c . c .
V s f , s i l p , m p = f | E l p m p . r ˜ | i = s f | u c | E l p m p . r ˜ | u v | s i ,
V s f , s i l p , m p = u c | e l p m p . r ˜ | u v s f | E l p m p | s i .
ϒ s f , s i l p , m p = V Q D ψ s f * ( r ) E l p m p ( r ) ψ s i ( r ) d r .
V s f , s i l p , m p = 2 P g ϒ s f , s i l p , m p .
ρ μ ν ( t ) t = 1 i h ¯ [ H ( t ) , ρ ( t ) ] μ ν γ μ ν ρ μ ν ( t ) + δ μ ν κ ν ξ ν κ ρ κ κ ( t ) ,
l p = j = 4 n ω l p 2 2 ε 2 h ¯ ξ 23 γ 22 γ 33 γ j j γ 13 γ 13 2 + Δ L 3 2 γ 2 j γ 2 j 2 + Δ 2 j 2 | V s 2 e , s 0 e | 2 | V s 1 e , s 0 e l p , j n + l p | 2 ,
l p = ξ 23 ω l p 2 2 ε 2 h ¯ γ 13 γ 22 γ 33 γ p l γ 2 p γ 2 p 2 + Δ 2 p 2 | V s 2 e , s 0 e | 2 j = 4 n | V s 1 e , s 0 e l p , j n + l p | 2 ,
g t h = ω l p c ε 2 Im ε 1 ( ω l p ) [ Re ( ε 2 ε 2 ε 1 ( ω l p ) ) ] 1 1 .

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