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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 25 — Dec. 5, 2011
  • pp: 24849–24857
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Forced synchronization of spaser by an external optical wave

E. S. Andrianov, A. A. Pukhov, A. V. Dorofeenko, A. P. Vinogradov, and A. A. Lisyansky  »View Author Affiliations


Optics Express, Vol. 19, Issue 25, pp. 24849-24857 (2011)
http://dx.doi.org/10.1364/OE.19.024849


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Abstract

We demonstrate that when the frequency of the external field differs from the lasing frequency of an autonomous spaser, the spaser exhibits stochastic oscillations at low field intensity. The plasmon oscillations lock to the frequency of the external field only when the field amplitude exceeds a threshold value. We find a region of values of the external field amplitude and the frequency detuning (the Arnold tongue) for which the spaser synchronizes with the external wave.

© 2011 OSA

1. Introduction

Joule (ohmic) losses in metallic nano-inclusions embedded in a host metamaterial medium are the main obstacle to realizing a practical “perfect” lens with subwavelength resolution as well as to other applications of metamaterials [1

1. W. Cai and V. Shalaev, Optical Metamaterials (Springer, 2010).

]. Losses can be compensated for, however, in amplifying media. The authors of [2

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

,3

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

] proposed making a stackable lens with alternating metamaterial and gain layers This can be accomplished by directly introducing gain inclusions such as molecules or quantum dots (QDs) into the matrix [4

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

6

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

]. The latter scheme has recently been realized experimentally [7

7. M. A. Noginov, G. Zhu, V. P. Drachev, and V. M. Shalaev, “Surface plasmons and gain media,” in Nanophotonics with Surface Plasmons, V. M. Shalaev and S. Kawata, eds. (Elsevier, 2007).

10

10. N. Meinzer, M. Ruther, S. Linden, C. M. Soukoulis, G. Khitrova, J. Hendrickson, J. D. Olitzky, H. M. Gibbs, and M. Wegener, “Arrays of Ag split-ring resonators coupled to InGaAs single-quantum-well gain,” Opt. Express 18(23), 24140–24151 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-23-24140. [CrossRef] [PubMed]

] demonstrating that it is possible to compensate for losses.

The combination of a gain medium and metallic nanoparticles (NPs) results in the emergence of a spaser (Surface Plasmon Amplification by Stimulated Emission of Radiation) first suggested by Bergman and Stockman [11

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

] and realized experimentally in [12

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

]. Spasers, as well as nanoplasmonics overall, is a new fast growing field with tremendous promise. Schematically, the spaser is a system of inversely excited two-level QDs surrounding metal nanoparticles [11

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

,13

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

]. Its principles of operation are analogous to those for a laser with the role of photons played by surface plasmons (SPs) localized at a NP that serves as the resonator [11

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

,14

14. M. I. Stockman, “Ultrafast nanoplasmonics under coherent control,” New J. Phys. 10(2), 025031 (2008). [CrossRef]

,15

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

]. In other words, in a spaser, near-fields of the NP are generated and amplified. The amplification of the SPs occurs due to non-radiative energy transfer from QDs. This process takes place due to the dipole-dipole (or any other near-field [16

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

]) interaction between the QD and the plasmon NP. The efficiency of such a mechanism depends on the probability of the non-radiative excitation of the surface plasmon, which is (kr)3 greater than the radiation of the photon [17

17. V. V. Klimov, “Nanoplasmonics,” Phys. Usp. 51(8), 839–844 (2008). [CrossRef]

], where ris the distance between the centers of the QD and the NP and k is the photon wavenumber. Stimulated radiation from the QD into the plasmon mode results in spasing. The excitation of the plasmon mode is carried out via the excitation of the QD.

In theoretical studies of a metamaterial with compensated losses, it is assumed that electromagnetic wave propagation can be described by the Maxwell equations with real valued effective electric permittivity and magnetic permeability. In its turn, the use of effective permittivities and refraction coefficients assumes that the dipole moment of a NP oscillates with the frequency of the external field and the amplitude of this oscillation is determined by the external field. In the absence of the external field, there should be no oscillations of a dipole moment. The classical linear description treats the gain medium as a system with a negative imaginary part of the permittivity [18

18. L.-M. Si, T. Jiang, K. Chang, T.-C. Chen, X. Lv, L. Ran, and H. Xin, “Active microwave metamaterials incorporating ideal gain devices,” Materials 4(1), 73–83 (2011). [CrossRef]

21

21. A. Govyadinov, V. Podolskiy, and M. A. Noginov, “Active metamaterials: Sign of refractive index and gain-assisted dispersion management,” Appl. Phys. Lett. 91(19), 191103 (2007). [CrossRef]

].

However, when spasers are used as gain inclusions, their dipole moments are excited not only by the external field but also by the radiation produced by QDs, which actually tends to compensate for loss. Unlike the classical linear description [18

18. L.-M. Si, T. Jiang, K. Chang, T.-C. Chen, X. Lv, L. Ran, and H. Xin, “Active microwave metamaterials incorporating ideal gain devices,” Materials 4(1), 73–83 (2011). [CrossRef]

21

21. A. Govyadinov, V. Podolskiy, and M. A. Noginov, “Active metamaterials: Sign of refractive index and gain-assisted dispersion management,” Appl. Phys. Lett. 91(19), 191103 (2007). [CrossRef]

], the semi-classical analysis shows that the spaser in the presence of pumping is a self-oscillating system. The dipole moment of the spaser’s NP oscillates autonomically even in the absence of the external field. The autonomic frequency of this self-oscillation is determined by the plasmon frequency, transition frequency of the gain inclusion and characteristic times of relaxation in NP and excitation of gain inclusions [15

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

]. Therefore, in order to develop a correct description of metamaterials with spasers, it is necessary to study the interaction of the spaser with the external field in details.

In this paper, we study analytically and numerically the operation of a spaser driven by an external optical wave. We demonstrate that the pumping drastically changes the spaser’s behavior in the optical field in comparison with the behavior of a passive QD-NP pair. In particular, when the frequency of the external field is detuned from the autonomous frequency of the spaser, even infinitesimally weak field drives the spaser to stochastic oscillations. The spaser can be synchronized to the external field only when the field amplitude exceeds a threshold value.

2. Equations of motion of the spaser

The simplest model of the spaser consists of a two-level QD of size rTLSwhich is positioned at a distance r from a metallic NP of size rNP [11

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

]. The whole system is immersed into a solid dielectric or semiconductor matrix with dielectric permittivity εM. The dipole moment of a typical QD of size rTLS 10 nm is μTLSclass~20 Debye [22

22. A. Muller, Q. Q. Wang, P. Bianucci, C. K. Shih, and Q. K. Xue, “Determination of anisotropic dipole moments in self-assembled quantum dots using Rabi oscillations,” Appl. Phys. Lett. 84(6), 981–983 (2004). [CrossRef]

]. For a NP with Ag-core and SiO2-shell with radius rNPr10nm, the classical dipole moment near the frequency of the plasmon resonance, ω, can be estimated to be μNPclass~200Debye. The energy of the NP-QD quasistatic dipole-dipole interaction is estimated to be V=ΩR~μNPclassμTLSclass/r3, which gives ΩR51012c1~102ω. Thus, the Rabi frequency (the coupling constant) is two orders of magnitude smaller than the generation frequency. This allows for the use of the slowly varying envelope approximation in the investigation of the dynamics of the spaser.

The transition processes of the NP and two-level QD interacting in a spaser can be described by the model Hamiltonian
H^=H^SP+H^TLS+V^+Г^,
(1)
where
H^SP=ωSPa˜^(t)a˜^(t),
(2)
H^TLS=ωTLSσ˜^(t)σ˜^(t)
(3)
describe the non-interacting NP and QD, respectively [11

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

,23

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

,24

24. R. H. Pantell and H. E. Puthoff, Fundamentals of Quantum Electronics (Wiley, 1969).

], the operator V^=μ^TLSE^NP determines their interaction. To take into account the interaction of the spaser, which is an open system, with surroundings (pumping, phonons in metal, etc.) following [24

24. R. H. Pantell and H. E. Puthoff, Fundamentals of Quantum Electronics (Wiley, 1969).

] we introduce the non-Hermitian operator Г^ that describes relaxation and pumping processes. Here a˜^(t)is the annihilation operator of the dipole SP, μ^TLS=μTLS(σ˜^(t)+σ˜^(t)) is the operator for the dipole moment of the QD, σ˜^=|ge| is the transition operator between ground |g and excited |e states of the QD, μTLS=e|er|g is the QD dipole moment matrix element. Quantization of the plasmon field without taking into account losses is carried out in the standard way [25

25. P. W. Milonni, The Quantum Vacuum: An Introduction to Quantum Electrodynamics (Academic Press, 1994).

]. Thus, we obtain [11

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

,15

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

]
E^NP=qωq2Wq[a^qEq(r)exp(iωqt)+a^q+Eq(r)exp(iωqt)],
(4)
where
Wq=14π(Reεω2)ω2|ωqEqEqdV=18π(2Reε+ωReεω)|ωSPE1E1dV
(5)
is the normalization factor and Eq(r)=φq is the q-th eigenmode determined by the geometry of the problem [11

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

]. In the dipole approximation, in Eq. (4) we retain only one dipole mode (q=1) with the frequencyωSP=ω1. For a spherical NP this dipole mode is uniform inside the particle and has the form of the field of the dipole with a unitary dipole moment, μ1, so that E1=μ1r3+3(μ1r)rr5 and E1=μ1rNP3 outside and inside of the NP, respectively. Note, that even though μ1 is a unit vector, it is a dimensional quantity, therefore, below we will show |μ1| explicitly.

In the absence of the external field, for the NP near field we have (8π)1Reε|ωSPE1E1dV=0 [26

26. F. Wang and Y. R. Shen, “General properties of local plasmons in metal nanostructures,” Phys. Rev. Lett. 97(20), 206806 (2006). [CrossRef] [PubMed]

]. The near field is not equal to zero at the resonance frequency only. However, in this case, the field vanishes at the infinity and Reε|ωSPE1E1dV=Reε|ωSPφφdV=φ(Reε|ωSPφ)dV. The latter integral is equal to zero because 2φ=0both inside and outside the NP. Furthermore, ε/ω=0 outside of the particle. This leads to the following result
W1=18πvolumeofNPωReεω|ωSPE1E1dV=|μ1|26rNP3ωReεω|ωSP,
(6)
which gives E^NP=3rNP3/(|μ1|2Reε/ω)E1(r)(a˜^+a˜^+). Thus, the operator for the dipole moments of the NP is μ^NP=μNP(a˜^+a˜^), where μNP=3rNP3/(ReεM/ω)μ1/|μ1|

Assuming that the frequencies of the QD transition and the frequency of the dipole SP are close, ωSPωTLS, we will look for the time dependencies of a˜^(t) and σ˜^(t) in the form a˜^(t)=a^(t)exp(iωat) and σ˜^(t)=σ^(t)exp(iωat), where a^(t) and σ^(t) are the slowly changing amplitudes and ωa is the autonomous frequency of the spaser which we seek. In the rotating wave approximation [24

24. R. H. Pantell and H. E. Puthoff, Fundamentals of Quantum Electronics (Wiley, 1969).

] we can neglect rapidly oscillating terms, exp(±2iωat), and obtain the interaction operator V^ in the form of the Jaynes–Cummings Hamiltonian [27

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

]
V^=ΩR(a^σ^+σ^a^),
(7)
where ΩR=3rMP3/(Reε/ω)E1(r)μTLS/|μ1| is the coupling constant (see also [11

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

,28

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

]).

The commutation relations for operators a^(t) and σ^(t) are standard: [a^,a^]=1^ and [σ^,σ^]=D^, where the operator D^ describes the inversion of the occupancies of the ground and excited states of the QD,D^(t)=n^e(t)n^g(t), n^e=|ee| and n^g=|gg| are the occupancy operators for the ground and excited states, n^g+n^e=1^. Using the Hamiltonian, Eqs. (1)-(3), and the commutation relations for operators a^(t) and σ^(t)we obtain the Heisenberg equations of motion for the operators a^(t), σ^(t), and D^(t) [28

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

,29

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

]
D^·=2iΩR(a^σ^σ^a^)D^D^0τD,
(8)
σ^·=(iδTLS1τσ)σ^+iΩRa^D^,
(9)
a^·=(iδSP1τa)a^iΩRσ^,
(10)
where δTLS=ωaωTLS and δSP=ωaωSP are the frequency differences. The Markovian interaction with the reservoir is described by the Liouvillian (non-Hermitian operator) term Г^ determining the decay rates with relaxation times τD1,τσ1, andτa1, which account for the relaxation processes for the SP annihilation operator, the QD polarization and the occupancy operators, respectively [27

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

]. The operator for the occupancy inversion D^0 describes the pumping [24

24. R. H. Pantell and H. E. Puthoff, Fundamentals of Quantum Electronics (Wiley, 1969).

,27

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

]. We now neglect quantum fluctuations and correlations and consider a^(t), σ^(t) and D^(t) as complex quantities (c-numbers), so that we can use complex conjugation instead of Hermitian conjugation [11

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

,28

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

,30

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

,31

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

]. The difference between the occupancies of the upper and lower levels D(t) must be a real valued quantity because the respective operator is Hermitian. The quantities σ(t) and a(t) are the complex amplitudes of the dipole oscillations of the QD and SP, respectively.

For D0<Dth the system of Eqs. (8)-(10) has a trivial stationary solution a=σ=0, D=D0only, while for D0>Dth there are two stationary solutions. The trivial unstable solution corresponds to the absence of SPs, while the stable one corresponds to laser generation:
a=eiφ2(D0Dth)τaτD,σ=eiψ2(D0Dth)(δSP2+1/τa2)τaΩR2τD,D=Dth,ω=ωSPτa+ωTLSτστa+τσ,
(11)
where exp(i(ψφ))=(δSP+i/τa)/(δSP2+1/τa2)1/2. The unstable solution appears only when the pumping level reaches the threshold value Dth=(1+δSP2τa2)/(ΩR2τaτσ) (see Fig. 1
Fig. 1 Stationary amplitudes a and σ are shown by dash-double dotted and dash-dotted lines, respectively. The stable solution for D is shown by the solid line. The unstable solution appearing for D>Dth is shown by the dashed line. For D0=D0 (also shown in Fig. 2) the stable and unstable solutions of D are marked by red dots.
). The stationary values of a, σ, and D shown in Fig. 1.

3. The spaser in the external electromagnetic field

Let us consider the dynamics of the NP and QD in the field of the external optical wave, E(t)=Ecosωft. Assuming that the external field is classical and taking into account the dipole interaction only, we can write the Hamiltonian in the form
H^ef=H^+Ω1(a˜^+a˜^)(exp(iωft)+exp(iωft))+Ω2(σ˜^+σ˜^)(exp(iωft)+exp(iωft)),
(12)
where H^ is defined by Eq. (2), Ω1=μNPE/, and Ω2=μTLSE/. As before, we use the Heisenberg equations for operators a^, σ^, and D^ to obtain equations of motion for “slowly varying” amplitudes,

D^·=2iΩR(a^σ^σ^a^)+2iΩ2(σ^σ^)D^D^0τD,
(13)
σ^·=(iΔTLS1τσ)σ^+iΩRa^D^+iΩ2D^,
(14)
a^·=(iΔSP1τa)a^iΩRσ^iΩ1.
(15)

Below we assume for simplicity that ωTLS=ωSP. Hence ΔSP=ωfωSP=ΔTLS=ωfωTLS=Δ. Note that without the QD, the polarization of the NP corresponding to the stationary solution of the system of Eqs. (13)-(15), a^˙=σ^˙=0, has the form
αrNP3=32(ΔSP+i/τa)(εNP(ω)ω)1,
(16)
where εNP(ω) is the permittivity of the metal NP and αis its polarizability. Generally, εNP(ω) is an effective permittivity of the NP in the matrix. However, for large enough NPs it can be considered as the bulk permittivity of the NP’s metal. In the slowly varying amplitude approximation, i.e. for a small detuning ΔSP1, this expression coincides with the classical one αclassrNP3=[εNP(ω)εM]/[εNP(ω)+2εM] (see also [30

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

]).

The quantities defined by Eqs. (13)-(15) differ from those in Eqs. (9)-(11). Without the external field the frequency ωa in the substitutions a˜^(t)=a^(t)exp(iωat) and σ˜^(t)=σ^(t)exp(iωat) is found from the solution of stationary Eqs. (8)-(10). When the external field is present, we use the representations a˜^(t)=a^(t)exp(iωft) and σ˜^(t)σ^(t)exp(iωft). We look for a stationary solution of the system of Eqs. (13)-(15). The stationary values of D satisfy a cubic equation and are shown in Fig. 2
Fig. 2 The stationary values of D as a function of the amplitude of the external field for (a) zero (Δ=0) and (b) non-zero (Δ=1011s1) detuning. For both graphs τa=1014s, τσ=1011s, τD=0.51014s, ΩR=1013s1, D0=D0=0.55.
as a function of the amplitude of the external field E. For Δ=0 (Fig. 2а) in the absence of the external field E=0 and D0Dth two points, stable and unstable, correspond to the points marked by dots in Fig. 1.

4. Spaser synchronization

Only the lower branch of the synchronization region corresponds to a stable solution. The stationary solutions at zero external field and with no detuning (δTLS=δSP=Δ=0) are shown in Fig. 1. When detuning is present (we still assume that δSP=δTLS=0 but Δ0), the stable stationary solution exists for E>ESynch(Δ) only (Fig. 2b). This result is confirmed by our numerical simulation shown in Fig. 3
Fig. 3 The dependence of the plasmon dipole moment on the amplitude of the external field and the frequency detuning Δ. The speckle structure at low values of E corresponds to the chaotic behavior of the dipole moment.
. Such behavior of a self-oscillating system is referred to as synchronization by an external periodic influence [32

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

]. The region in which the synchronization takes place is known as the Arnold tongue. To obtain an analytical estimate of the Arnold tongue boundary, we find the solution of Eqs. (13)-(15) in the first approximation with respect to the field. For the case δSP=0, using the substitution a=|a|exp(iφ) and σ=|σ|exp(iψ)in Eq. (15) and equating its imaginary parts, we obtain

φ·=ΔΩR|σ||a|cos(ψφ)Ω|a|cosφ.
(17)

Since E(t)=Ecosωft, φ represents the phase shift between the external field and spaser oscillations. For |a|, |σ|, and cos(ψφ) we can use their values obtained from Eq. (11) in the zeroth order with respect to the field approximation. As the result, we have
φ·=Δξcosφ,
(18)
where ξ=Ω1/|a| (in this approximation cos(ψφ)=0). If one introduces a function U(φ)=Δ[φ+(ξsinφ)/Δ], Eq. (18) takes the form of Adler’s equation [32

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

]:

φ·=U(φ)φ.
(19)

The phase dynamics can be viewed as the motion of a particle sliding along the potential profile U(φ) (Fig. 4
Fig. 4 The potential U(φ) for |ξ/Δ|<1 (dashed blue line) and |ξ/Δ|>1 (solid red line).
) in a viscous liquid. For a small field and/or large detuning, |ξ/Δ|<1, the phase difference of the system and the external field increases monotonically. For |ξ/Δ|>1, the “particle” should be trapped in one of the minima of the potential function. In this regime the oscillation of the system is synchronized and the phase difference is time independent. In the approximation considered, the width of the synchronization region is proportional to the field amplitude. Within the Arnold tongue, if the energy of the interaction of the NP with the external field is smaller than the QD-NP interaction, the amplitude of the auto-oscillations depends weakly upon the amplitude of the external field, which in this case plays the role of a synchronizer.

When Δ0 the threshold field value Esynch is determined by the equation (μNPESynch/)2=(D0Dth)Δ2(τa/4τD). For large values of the detuning, Esynch becomes independent of Δ (see Fig. 3). Taking into account that τa1014s1 and τσ1011s1 for plasmonic NPs, one can estimate the asymptotic value of the Arnold tongue boundary, ESynch*. Our numerical calculations show that ESynch(Δ) tends to a plateau for Δ21011s1, which gives ESynch*3103V/m. This agrees with calculations of [3

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

,10

10. N. Meinzer, M. Ruther, S. Linden, C. M. Soukoulis, G. Khitrova, J. Hendrickson, J. D. Olitzky, H. M. Gibbs, and M. Wegener, “Arrays of Ag split-ring resonators coupled to InGaAs single-quantum-well gain,” Opt. Express 18(23), 24140–24151 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-23-24140. [CrossRef] [PubMed]

] in which wave propagation in the system of spasers was considered. In [3

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

,10

10. N. Meinzer, M. Ruther, S. Linden, C. M. Soukoulis, G. Khitrova, J. Hendrickson, J. D. Olitzky, H. M. Gibbs, and M. Wegener, “Arrays of Ag split-ring resonators coupled to InGaAs single-quantum-well gain,” Opt. Express 18(23), 24140–24151 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-23-24140. [CrossRef] [PubMed]

] the amplitude of the incident wave was a few orders of magnitude larger than ESynch*. Such a field is comparable with the near field inside the spaser. In this case, spaser becomes synchronized with the external field for any value of detuning; it ceases to be an autonomous system and responds linearly to the external field as can be seen in Fig. 3. For such strong fields, losses cannot be compensated exactly. Moreover, our numerical calculations show that for a very strong field, the population inversion of the QD decreases, the excitation of SPs by QDs is inhibited, and the spaser becomes just a lossy NP. Thus, when detuning is substantial, spasers cannot be used as active inclusions for the perfect lens because both loss and amplification destroy the perfect image, so that the exact compensation of loss is required.

For applications it is important to know not only a response of a single spaser to an external electromagnetic wave but also the behavior of a system of interacting spasers randomly or regularly distributed in the dielectric matrix. The interaction between spasers may lead to a cooperative behavior of spasers. This problem is a subject of our future research.

Acknowledgments

The authors are indebted to Yu. E. Lozovik and C. Z. Ning for useful discussions. This work was supported by RFBR Grants Nos. 10-02-91750 and 11-02-92475 and by a PSC-CUNY grant.

References and links

1.

W. Cai and V. Shalaev, Optical Metamaterials (Springer, 2010).

2.

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

3.

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

4.

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

5.

A. K. Popov and V. M. Shalaev, “Compensating losses in negative-index metamaterials by optical parametric amplification,” Opt. Lett. 31(14), 2169–2171 (2006). [CrossRef] [PubMed]

6.

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

7.

M. A. Noginov, G. Zhu, V. P. Drachev, and V. M. Shalaev, “Surface plasmons and gain media,” in Nanophotonics with Surface Plasmons, V. M. Shalaev and S. Kawata, eds. (Elsevier, 2007).

8.

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

9.

E. Plum, V. A. Fedotov, P. Kuo, D. P. Tsai, and N. I. Zheludev, “Towards the lasing spaser: controlling metamaterial optical response with semiconductor quantum dots,” Opt. Express 17(10), 8548–8551 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-10-8548. [CrossRef] [PubMed]

10.

N. Meinzer, M. Ruther, S. Linden, C. M. Soukoulis, G. Khitrova, J. Hendrickson, J. D. Olitzky, H. M. Gibbs, and M. Wegener, “Arrays of Ag split-ring resonators coupled to InGaAs single-quantum-well gain,” Opt. Express 18(23), 24140–24151 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-23-24140. [CrossRef] [PubMed]

11.

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

12.

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

13.

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

14.

M. I. Stockman, “Ultrafast nanoplasmonics under coherent control,” New J. Phys. 10(2), 025031 (2008). [CrossRef]

15.

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

16.

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

17.

V. V. Klimov, “Nanoplasmonics,” Phys. Usp. 51(8), 839–844 (2008). [CrossRef]

18.

L.-M. Si, T. Jiang, K. Chang, T.-C. Chen, X. Lv, L. Ran, and H. Xin, “Active microwave metamaterials incorporating ideal gain devices,” Materials 4(1), 73–83 (2011). [CrossRef]

19.

A. D. Boardman, Y. G. Rapoport, N. King, and V. N. Malnev, “Creating stable gain in active metamaterials,” J. Opt. Soc. Am. B 24(10), A53–A61 (2007). [CrossRef]

20.

Y.-Y. Yu, D.-Z. Lin, L.-S. Huang, and C.-K. Lee, “Effect of subwavelength annular aperture diameter on the nondiffracting region of generated Bessel beams,” Opt. Express 17(4), 2707–2713 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-4-2707. [CrossRef] [PubMed]

21.

A. Govyadinov, V. Podolskiy, and M. A. Noginov, “Active metamaterials: Sign of refractive index and gain-assisted dispersion management,” Appl. Phys. Lett. 91(19), 191103 (2007). [CrossRef]

22.

A. Muller, Q. Q. Wang, P. Bianucci, C. K. Shih, and Q. K. Xue, “Determination of anisotropic dipole moments in self-assembled quantum dots using Rabi oscillations,” Appl. Phys. Lett. 84(6), 981–983 (2004). [CrossRef]

23.

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

24.

R. H. Pantell and H. E. Puthoff, Fundamentals of Quantum Electronics (Wiley, 1969).

25.

P. W. Milonni, The Quantum Vacuum: An Introduction to Quantum Electrodynamics (Academic Press, 1994).

26.

F. Wang and Y. R. Shen, “General properties of local plasmons in metal nanostructures,” Phys. Rev. Lett. 97(20), 206806 (2006). [CrossRef] [PubMed]

27.

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

28.

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

29.

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

30.

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

31.

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

32.

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

OCIS Codes
(240.4350) Optics at surfaces : Nonlinear optics at surfaces
(240.6680) Optics at surfaces : Surface plasmons
(350.4238) Other areas of optics : Nanophotonics and photonic crystals

ToC Category:
Optics at Surfaces

History
Original Manuscript: October 20, 2011
Revised Manuscript: November 14, 2011
Manuscript Accepted: November 14, 2011
Published: November 21, 2011

Citation
E. S. Andrianov, A. A. Pukhov, A. V. Dorofeenko, A. P. Vinogradov, and A. A. Lisyansky, "Forced synchronization of spaser by an external optical wave," Opt. Express 19, 24849-24857 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-25-24849


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References

  1. W. Cai and V. Shalaev, Optical Metamaterials (Springer, 2010).
  2. S. A. Ramakrishna and J. B. Pendry, “Removal of absorption and increase in resolution in a near-field lens via optical gain,” Phys. Rev. B67(20), 201101 (2003). [CrossRef]
  3. A. Fang, T. Koschny, and C. M. Soukoulis, “Lasing in metamaterial nanostructures,” J. Opt.12(2), 024013 (2010). [CrossRef]
  4. M. A. Noginov, G. Zhu, M. Bahoura, J. Adegoke, C. E. Small, B. A. Ritzo, V. P. Drachev, and V. M. Shalaev, “Enhancement of surface plasmons in an Ag aggregate by optical gain in a dielectric medium,” Opt. Lett.31(20), 3022–3024 (2006). [CrossRef] [PubMed]
  5. A. K. Popov and V. M. Shalaev, “Compensating losses in negative-index metamaterials by optical parametric amplification,” Opt. Lett.31(14), 2169–2171 (2006). [CrossRef] [PubMed]
  6. A. K. Sarychev and G. Tartakovsky, “Magnetic plasmonic metamaterials in actively pumped host medium and plasmonic nanolaser,” Phys. Rev. B75(8), 085436 (2007). [CrossRef]
  7. M. A. Noginov, G. Zhu, V. P. Drachev, and V. M. Shalaev, “Surface plasmons and gain media,” in Nanophotonics with Surface Plasmons, V. M. Shalaev and S. Kawata, eds. (Elsevier, 2007).
  8. N. I. Zheludev, S. L. Prosvirnin, N. Papasimakis, and V. A. Fedotov, “Lasing spaser,” Nat. Photonics2(6), 351–354 (2008). [CrossRef]
  9. E. Plum, V. A. Fedotov, P. Kuo, D. P. Tsai, and N. I. Zheludev, “Towards the lasing spaser: controlling metamaterial optical response with semiconductor quantum dots,” Opt. Express17(10), 8548–8551 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-10-8548 . [CrossRef] [PubMed]
  10. N. Meinzer, M. Ruther, S. Linden, C. M. Soukoulis, G. Khitrova, J. Hendrickson, J. D. Olitzky, H. M. Gibbs, and M. Wegener, “Arrays of Ag split-ring resonators coupled to InGaAs single-quantum-well gain,” Opt. Express18(23), 24140–24151 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-23-24140 . [CrossRef] [PubMed]
  11. D. J. Bergman and M. I. Stockman, “Surface plasmon amplification by stimulated emission of radiation: quantum generation of coherent surface plasmons in nanosystems,” Phys. Rev. Lett.90(2), 027402 (2003). [CrossRef] [PubMed]
  12. 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(7259), 1110–1112 (2009). [CrossRef] [PubMed]
  13. M. I. Stockman, “Spasers explained,” Nat. Photonics2(6), 327–329 (2008). [CrossRef]
  14. M. I. Stockman, “Ultrafast nanoplasmonics under coherent control,” New J. Phys.10(2), 025031 (2008). [CrossRef]
  15. M. I. Stockman, “The spaser as a nanoscale quantum generator and ultrafast amplifier,” J. Opt.12(2), 024004 (2010). [CrossRef]
  16. I. R. Gabitov, B. Kennedy, and A. I. Maimistov, “Coherent amplification of optical pulses in metamaterials,” IEEE J. Sel. Top. Quantum Electron.16(2), 401–409 (2010). [CrossRef]
  17. V. V. Klimov, “Nanoplasmonics,” Phys. Usp.51(8), 839–844 (2008). [CrossRef]
  18. L.-M. Si, T. Jiang, K. Chang, T.-C. Chen, X. Lv, L. Ran, and H. Xin, “Active microwave metamaterials incorporating ideal gain devices,” Materials4(1), 73–83 (2011). [CrossRef]
  19. A. D. Boardman, Y. G. Rapoport, N. King, and V. N. Malnev, “Creating stable gain in active metamaterials,” J. Opt. Soc. Am. B24(10), A53–A61 (2007). [CrossRef]
  20. Y.-Y. Yu, D.-Z. Lin, L.-S. Huang, and C.-K. Lee, “Effect of subwavelength annular aperture diameter on the nondiffracting region of generated Bessel beams,” Opt. Express17(4), 2707–2713 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-4-2707 . [CrossRef] [PubMed]
  21. A. Govyadinov, V. Podolskiy, and M. A. Noginov, “Active metamaterials: Sign of refractive index and gain-assisted dispersion management,” Appl. Phys. Lett.91(19), 191103 (2007). [CrossRef]
  22. A. Muller, Q. Q. Wang, P. Bianucci, C. K. Shih, and Q. K. Xue, “Determination of anisotropic dipole moments in self-assembled quantum dots using Rabi oscillations,” Appl. Phys. Lett.84(6), 981–983 (2004). [CrossRef]
  23. L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University Press, 2006).
  24. R. H. Pantell and H. E. Puthoff, Fundamentals of Quantum Electronics (Wiley, 1969).
  25. P. W. Milonni, The Quantum Vacuum: An Introduction to Quantum Electrodynamics (Academic Press, 1994).
  26. F. Wang and Y. R. Shen, “General properties of local plasmons in metal nanostructures,” Phys. Rev. Lett.97(20), 206806 (2006). [CrossRef] [PubMed]
  27. M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University Press, Cambridge, 1997).
  28. I. E. Protsenko, A. V. Uskov, O. A. Zaimidoroga, V. N. Samoilov, and E. P. O’Reilly, “Dipole nanolaser,” Phys. Rev. A71(6), 063812 (2005). [CrossRef]
  29. K. Li, X. Li, M. I. Stockman, and D. J. Bergman, “Surface plasmon amplification by stimulated emission in nanolenses,” Phys. Rev. B71(11), 115409 (2005). [CrossRef]
  30. A. S. Rosenthal and T. Ghannam, “Dipole nanolasers: A study of their quantum properties,” Phys. Rev. A79(4), 043824 (2009). [CrossRef]
  31. A. N. Lagarkov, A. K. Sarychev, V. N. Kissel, and G. Tartakovsky, “Superresolution and enhancement in metamaterials,” Phys. Usp.52(9), 959–967 (2009). [CrossRef]
  32. A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization. A Universal Concept in Nonlinear Sciences (Cambridge University Press, 2001).

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