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Optical Materials Express

Optical Materials Express

  • Editor: David J. Hagan
  • Vol. 1, Iss. 6 — Oct. 1, 2011
  • pp: 1077–1089
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Complex modes and near-zero permittivity in 3D arrays of plasmonic nanoshells: loss compensation using gain [Invited]

Salvatore Campione, Matteo Albani, and Filippo Capolino  »View Author Affiliations


Optical Materials Express, Vol. 1, Issue 6, pp. 1077-1089 (2011)
http://dx.doi.org/10.1364/OME.1.001077


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Abstract

We report on the possibility of adopting active gain materials (specifically, made of fluorescent dyes) to mitigate the losses in a 3D periodic array of dielectric-core metallic-shell nanospheres. We find the modes with complex wavenumber in the structure, and describe the composite material in terms of homogenized effective permittivity, comparing results from modal analysis and Maxwell Garnett theory. We then design two metamaterials in which the epsilon-near-zero frequency region overlaps with the emission band of the adopted gain media, and we show that metamaterials with effective parameters with low losses are feasible, thanks to the gain materials. Even though fluorescent dyes embedded in the nanoshells’ dielectric cores are employed in this study, the formulation provided is general, and could account for the usage of other active materials, such as semiconductors and quantum dots.

© 2011 OSA

1. Introduction

Metamaterials have been proposed for several innovative applications and have allowed, for example, the design of “perfect lenses” [1

1. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000). [CrossRef] [PubMed]

], and invisibility cloaks [2

2. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006). [CrossRef] [PubMed]

]. In general, however, plasmonic-based metamaterial losses at optical frequencies have been found to be significantly large, and thus have limited the application scenario. However, at infrared, ultraviolet and optical frequencies, the use of plasmonics mixed with active photonic materials has been found to be promising due to the fact that the gain experienced through the emission of a gain medium is capable of counteracting the high attenuation experienced by the electromagnetic wave due to the presence of the metal. This may indeed lead to loss-mitigated metamaterials, enabling effective permeability or low permittivity parameters at optical frequencies.

One of the key points is designing the metamaterial such that the frequency region of interest overlaps with the emission spectrum of the adopted gain medium. Different gain sources, optically pumped, could be adopted for this purpose: fluorescent dyes (e.g., Rhodamine, Fluorescein, Coumarin), semiconductor materials and quantum dots (e.g., InGaAs-GaAs quantum dots), rare earth materials (e.g., erbium).

It has been reported that the usage of the gain medium with metamaterials can provide a larger effective gain than when used alone, due to the strong local field enhancement inside the metamaterials [3

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

,4

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

].

Positive net gain (i.e., the gain is larger than the losses) has been shown to be possible over macroscopic distances in a dielectric–metal–dielectric plasmonic waveguide, where the gain has been provided by an optically pumped layer of fluorescent conjugated polymer (known to have very large emission cross sections) adjacent to the metal surface [5

5. M. C. Gather, K. Meerholz, N. Danz, and K. Leosson, “Net optical gain in a plasmonic waveguide embedded in a fluorescent polymer,” Nat. Photonics 4(7), 457–461 (2010). [CrossRef]

]. Also, a direct measurement of gain in propagating plasmons using the long-range surface plasmon–polariton supported by a symmetric metal strip waveguide that incorporates optically pumped dye molecules in solution as the gain medium has been shown [6

6. I. De Leon and P. Berini, “Amplification of long-range surface plasmons by a dipolar gain medium,” Nat. Photonics 4(6), 382–387 (2010). [CrossRef]

]. Furthermore, optical loss compensation effects have been recently experimentally observed in [7

7. G. Strangi, A. De Luca, S. Ravaine, M. Ferrie, and R. Bartolino, “Gain induced optical transparency in metamaterials,” Appl. Phys. Lett. 98(25), 251912 (2011). [CrossRef]

], and [8

8. A. De Luca, M. P. Grzelczak, I. Pastoriza-Santos, L. M. Liz-Marzán, M. La Deda, M. Striccoli, and G. Strangi, “Dispersed and encapsulated gain medium in plasmonic nanoparticles: a multipronged approach to mitigate optical losses,” ACS Nano 5(7), 5823–5829 (2011). [CrossRef] [PubMed]

], where Coumarin C500 and Rhodamine 6G fluorescent dyes were encapsulated into the dielectric shell of randomly dispersed nanoshell particles.

Effective parameters of metamaterials made of nanoshells with active gain materials embedded in the dielectric core, designed to operate in the visible range of the spectrum between 400 nm and 700 nm, have been simulated in [9

9. J. A. Gordon and R. W. Ziolkowski, “CNP optical metamaterials,” Opt. Express 16(9), 6692–6716 (2008). [CrossRef] [PubMed]

] by artificially setting the imaginary part of the dielectric core to fixed ideal loss/gain conditions, i.e., realistic gain materials have not been considered. A detailed analysis observing the effects of the gain value in the nanoshells’ core and of the density of the inclusions has been provided in [9

9. J. A. Gordon and R. W. Ziolkowski, “CNP optical metamaterials,” Opt. Express 16(9), 6692–6716 (2008). [CrossRef] [PubMed]

] to investigate the tunability of such metamaterials. For 3D periodic arrays, the authors of [9

9. J. A. Gordon and R. W. Ziolkowski, “CNP optical metamaterials,” Opt. Express 16(9), 6692–6716 (2008). [CrossRef] [PubMed]

] concluded that the effective permittivity can be engineered to assume both positive and negative values by selecting appropriately the lattice period and the gain value in the core of the nanoshells. In this paper we confirm the results found in [9

9. J. A. Gordon and R. W. Ziolkowski, “CNP optical metamaterials,” Opt. Express 16(9), 6692–6716 (2008). [CrossRef] [PubMed]

], and moreover we show complex modes in the 3D lattice and utilize realistic parameters for the gain medium to analyze feasibility. Loss compensation of the intrinsic losses of metals at optical frequencies by using gain materials has also been proposed in [10

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

14

14. M. A. Noginov, G. Zhu, M. Bahoura, J. Adegoke, C. Small, B. A. Ritzo, V. P. Drachev, and V. M. Shalaev, “The effect of gain and absorption on surface plasmons in metal nanoparticles,” Appl. Phys. B 86(3), 455–460 (2007). [CrossRef]

]. In [15

15. R. Bardhan, N. K. Grady, J. R. Cole, A. Joshi, and N. J. Halas, “Fluorescence enhancement by Au nanostructures: nanoshells and nanorods,” ACS Nano 3(3), 744–752 (2009). [CrossRef] [PubMed]

], it has been shown that metallic nanoparticles (nanoshells and nanorods) influence the properties of adjacent fluorophores; in that paper, the authors have shown an improvement in the quantum yield (defined here in Sec. 2.3) of the fluorophore IR800 showing the potential for contrast enhancement in fluorescence-based bioimaging. Similarly, in [16

16. J. Zhang, I. Gryczynski, Z. Gryczynski, and J. R. Lakowicz, “Dye-labeled silver nanoshell-bright particle,” J. Phys. Chem. B 110(18), 8986–8991 (2006). [CrossRef] [PubMed]

], Ruby dyes were incorporated into the dielectric core of randomly dispersed nanoshell particles, and an emission enhancement has been observed with respect to the case in absence of the metallic shell.

A computational approach including rate equations has been presented in [17

17. A. Fang, Z. Huang, T. Koschny, and C. M. Soukoulis, “Overcoming the losses of a split ring resonator array with gain,” Opt. Express 19(13), 12688–12699 (2011). [CrossRef] [PubMed]

] and references therein, allowing for a self-consistent treatment of a split ring resonator (SRR) array with a gain layer underneath, showing numerically that the magnetic losses of the SRR can be compensated by the gain. Rate equations have also been used in [18

18. Y. Sivan, S. Xiao, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Frequency-domain simulations of a negative-index material with embedded gain,” Opt. Express 17(26), 24060–24074 (2009). [CrossRef] [PubMed]

]. A review regarding the management of loss and gain in metamaterials has been presented in [19

19. A. D. Boardman, V. V. Grimalsky, Y. S. Kivshar, S. V. Koshevaya, M. Lapine, N. M. Litchinitser, V. N. Malnev, M. Noginov, Y. G. Rapoport, and V. M. Shalaev, “Active and tunable metamaterials,” Laser Photon. Rev. 5(2), 287–307 (2011). [CrossRef]

], and references therein.

In this paper, we provide the analysis of a loss-compensated metamaterial at optical frequencies through optical pumping. In particular, we analyze a 3D periodic array of dielectric-core metallic-shell nanospheres, assuming fluorescent dyes encapsulated into the core of each spherical nanoparticle. Each nanoshell is modeled as a single electric dipole and by its polarizability, using the single dipole approximation (SDA) [20

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

22

22. S. Campione and F. Capolino, “Linear and planar periodic arrays of metallic nanospheres: fabrication, optical properties and applications,” in Selected Topics in Metamaterials and Photonic Crystals, A. Andreone, A. Cusano, A. Cutolo, and V. Galdi, eds. (World Scientific, 2011), pp. 141–194.

] and the metal permittivity is described by the Drude model. We compute the modes following the procedure described in [21

21. S. Steshenko and F. Capolino, “Single dipole approximation for modeling collections of nanoscatterers,” in Theory and Phenomena of Metamaterials, F. Capolino, ed. (CRC Press, 2009), p. 8.1.

,22

22. S. Campione and F. Capolino, “Linear and planar periodic arrays of metallic nanospheres: fabrication, optical properties and applications,” in Selected Topics in Metamaterials and Photonic Crystals, A. Andreone, A. Cusano, A. Cutolo, and V. Galdi, eds. (World Scientific, 2011), pp. 141–194.

]. Then, also by using Maxwell Garnett homogenization theory [23

23. A. Sihvola, Electromagnetic Mixing Formulas and Applications (IEEE Publishing, 1999).

,24

24. A. Sihvola, “Mixing rules,” in Theory and Phenomena of Metamaterials, F. Capolino, ed. (CRC Press, 2009), p. 9.1.

], we compute the relative effective permittivity εeff. Three interesting frequency regions can be outlined depending on its value: (i) one where εeff is rather large and positive; (ii) one where εeff is rather large and negative; and (iii) one where εeff is close to zero (either positive or negative), also called the epsilon-near-zero (ENZ) frequency region, which has been proposed as a viable way for a number of applications including cloaking, tunneling, high directivity radiators, optic nanocircuits, etc, as reported for example in [25

25. M. G. Silveirinha, A. Alu, B. Edwards, and N. Engheta, “Overview of theory and applications of epsilon-near-zero materials,” in URSI General Assembly (Chicago, IL, 2008).

] and references therein. Certainly, high losses hinder the interesting properties in such frequency regions, and loss mitigation mechanisms are inherently required to overcome this issue. In this paper, we are interested in showing a formulation for loss compensation and then specifically reducing losses in the ENZ frequency region. Therefore, we design metamaterials such that the effective ENZ region overlaps with the emission spectrum of the considered dyes, and we observe that loss-compensation is feasible. Notice however that the analysis here reported does not limit the usage of gain materials to overcome the losses in other frequency regions.

The structure of the paper is as follows. Mode analysis, Maxwell Garnett theory and modeling of the active gain material are introduced in Sec. 2. Then, in Sec. 3, we use two different fluorescent dyes (Rhodamine 6G and Rhodamine 800) to mitigate the losses for two particular metamaterials’ designs. Conclusions are reported in Sec. 4.

2. Simulation model

2.1 Modal analysis for periodic arrays of plasmonic nanoshells

We model each nanoshell as a single electric dipole at optical frequencies. As such, for a plasmonic spherical particle the induced electric dipole moment is
p=αeeEloc,
(1)
where αee is the electric polarizability of the nanoshell, Eloc is the local field produced by all the nanoshells of the array except the considered nanoshell plus the external incident field to the array, and bold letters refer to vector quantities. According to the Clausius-Mossotti approximation, the electric polarizability of a nanoshell is [20

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

,23

23. A. Sihvola, Electromagnetic Mixing Formulas and Applications (IEEE Publishing, 1999).

,31

31. K. Tanabe, “Field enhancement around metal nanoparticles and nanoshells: a systematic investigation,” J. Phys. Chem. C 112(40), 15721–15728 (2008). [CrossRef]

,32

32. A. Vallecchi, M. Albani, and F. Capolino, “EM characterization of Raspberry-like nanocluster metamaterials,” in Antennas and Propagation Society International Symposium (Toronto, Canada, 2010).

]
αee1=14πεhε0r23(ε2+2εh)(ε1+2ε2)+2β(ε2εh)(ε1ε2)(ε2εh)(ε1+2ε2)+β(2ε2+εh)(ε1ε2)ik36πεhε0,
(2)
where εh is the relative permittivity of the host medium (which can be vacuum, glass, water, or any other solvent), ε0 is the absolute permittivity of free space, k=ωεh/c0=k0εh is the host medium wavenumber, with k0 denoting the free space wavenumber and c0 the speed of light in free space, ε1 is the relative permittivity of the core (with radius r1), and ε2 is the relative permittivity of the shell (with outer radiusr2). Furthermore, β=ρ3, with ρ=r1/r2. The last imaginary term in Eq. (2) has been introduced to account for particle radiation [20

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

,21

21. S. Steshenko and F. Capolino, “Single dipole approximation for modeling collections of nanoscatterers,” in Theory and Phenomena of Metamaterials, F. Capolino, ed. (CRC Press, 2009), p. 8.1.

]. According to Mie theory, instead, the polarizability of a nanoshell is [20

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

]
αee=6πiεhε0k3ψ1(kr2)Am2ψ1(kr2)Bξ1(kr2)Am2ξ1(kr2)B,A=ψ1(m2kr2)Cχ1(m2kr2),
(3)
and
B=ψ1(m2kr2)Cχ1(m2kr2),C=m2ψ1(m2kr1)ψ1(m1kr1)m1ψ1(m2kr1)ψ1(m1kr1)m2χ1(m2kr1)ψ1(m1kr1)m1χ1(m2kr1)ψ1(m1kr1),
(4)
with ψ1(ρ)=ρj1(ρ)=sinρ/ρcosρ, ξ1(ρ)=ρh1(1)(ρ)=(i/ρ1)eiρ and χ1(ρ)=ρy1(ρ)=cosρ/ρ+sinρ the Riccati-Bessel functions [33

33. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover Publications, 1965).

], and m1=ε1/εh, m2=ε2/εh are the core and shell relative refractive indexes. Notice that a prime in Eqs. (3) and (4) refers to the first derivative of the function with respect to its argument.

In this paper we consider dielectric-core metallic-shell particles, with ε1=εr and ε2=εm, where εr is the relative permittivity of the chosen dielectric material and the metal permittivity εm is described through the Drude model as
εm=εωp2ω(ω+iγ),
(5)
where ωp is the plasma angular frequency, γ the damping term, and ε is a “high frequency” permittivity determined to match experimental data in the visible region.

Consider now a 3D periodic array of nanoshells, immersed in a homogeneous background, with relative permittivity εh, for which each nanoshell is placed at positions rn=r0+dn, where nn1,n2,n3=0,±1,±2,..., is a triple index, and dn=n1ax^+n2by^+n3cz^, r0=x0x^+y0y^+z0z^(where a caret on top of a bold letter refers to unit vector quantities), and a, b and c are the periodicities along x-, y- and z-direction, respectively [21

21. S. Steshenko and F. Capolino, “Single dipole approximation for modeling collections of nanoscatterers,” in Theory and Phenomena of Metamaterials, F. Capolino, ed. (CRC Press, 2009), p. 8.1.

,22

22. S. Campione and F. Capolino, “Linear and planar periodic arrays of metallic nanospheres: fabrication, optical properties and applications,” in Selected Topics in Metamaterials and Photonic Crystals, A. Andreone, A. Cusano, A. Cutolo, and V. Galdi, eds. (World Scientific, 2011), pp. 141–194.

,34

34. A. L. Fructos, S. Campione, F. Capolino, and F. Mesa, “Characterization of complex plasmonic modes in two-dimensional periodic arrays of metal nanospheres,” J. Opt. Soc. Am. B 28(6), 1446–1458 (2011). [CrossRef]

]. Suppose that the array is then excited by a plane wave or by a quasi-periodic excitation with wavevector kB=kxx^+kyy^+kzz^. The formulation here adopted has been described elsewhere [21

21. S. Steshenko and F. Capolino, “Single dipole approximation for modeling collections of nanoscatterers,” in Theory and Phenomena of Metamaterials, F. Capolino, ed. (CRC Press, 2009), p. 8.1.

,22

22. S. Campione and F. Capolino, “Linear and planar periodic arrays of metallic nanospheres: fabrication, optical properties and applications,” in Selected Topics in Metamaterials and Photonic Crystals, A. Andreone, A. Cusano, A. Cutolo, and V. Galdi, eds. (World Scientific, 2011), pp. 141–194.

,35

35. S. Campione, S. Steshenko, M. Albani, and F. Capolino, “Complex modes in three dimensional periodic arrays of plasmonic nanospheres,” Opt. Express (to be submitted).

]. Briefly, mode analysis in the 3D periodic array is performed by computing the eigensolutions of the homogeneous version of
A_(kB)p0=αeeEinc(r0),  A_(kB)=I_αeeG_(r0,r0,kB),
(6)
i.e., when no impressed excitation is present Einc(r0)=0, and a bar under a bold letter refers to dyadic quantities. The term G_(r0,r0,kB) represents the regularized Green’s function [35

35. S. Campione, S. Steshenko, M. Albani, and F. Capolino, “Complex modes in three dimensional periodic arrays of plasmonic nanospheres,” Opt. Express (to be submitted).

], and provides the field contribution evaluated at r0 produced by all the nanoshells but the one at r0, and I_ is the identity dyad. In other words, the complex mode wavenumbers kB are computed by solving for the complex zeroes of the determinant of A_(kB). In the following, we assume that the modes travel along the z direction with wavenumber kz (for the sake of brevity modes with oblique propagation direction are not considered in this feasibility study).

2.2 Effective parameters

In general, Maxwell Garnett theory [23

23. A. Sihvola, Electromagnetic Mixing Formulas and Applications (IEEE Publishing, 1999).

,24

24. A. Sihvola, “Mixing rules,” in Theory and Phenomena of Metamaterials, F. Capolino, ed. (CRC Press, 2009), p. 9.1.

] can be applied to retrieve the effective parameters of a composite medium as
εeff=εh+εhND1[ε0εhαee1+ik36π]13,     μeff=1+1ND1[αmm1+ik36π]13,
(7)
where ND=f/VN, with f the filling fraction f=VN/VL (assuming simple cubic lattices), VN is the nanoshell volume and VL is the unit cell volume, and αmm is the magnetic polarizability of a nanoshell. Then, the effective refractive index can be calculated as neff=εeffμeff. Moreover, as we will do in the following, the effective refractive index is also computed by using mode analysis as
neff=kzk0,
(8)
where kz is the wavenumber of the “dominant” mode (assuming there is one) computed from mode analysis.

2.3 Modeling of the gain material

We assume to model the gain material made of fluorescent dye molecules as a four level atomic system [17

17. A. Fang, Z. Huang, T. Koschny, and C. M. Soukoulis, “Overcoming the losses of a split ring resonator array with gain,” Opt. Express 19(13), 12688–12699 (2011). [CrossRef] [PubMed]

,36

36. A. Siegman, Lasers (University Science Books, 1986).

,37

37. A. S. Nagra and R. A. York, “FDTD analysis of wave propagation in nonlinear absorbing and gain media,” IEEE Trans. Antenn. Propag. 46(3), 334–340 (1998). [CrossRef]

], as also proposed in [38

38. D. Faubert, S. L. Chin, M. Cormier, and M. Boloten, “Numerical analysis of short laser pulse superposition in a fluorescent dye medium,” Can. J. Phys. 57(2), 160–167 (1979). [CrossRef]

,39

39. G. Grönninger and A. Penzkofer, “Determination of energy and duration of picosecond light pulses by bleaching of dyes,” Opt. Quantum Electron. 16(3), 225–233 (1984). [CrossRef]

], with occupation density Ni(r,t) of the gain medium in the i-th state, i=0,1,2,3, with N0(r)+N1(r)+N2(r)+N3(r)=N¯0, where N¯0 is the total dye concentration, which corresponds to the N0(r) concentration in absence of pumping (i.e., N1(r)=N2(r)=N3(r)=0, thus N0(r)=N¯0). Furthermore, τij is the lifetime for the transition from state i to the lower state j, and Γpump is the pumping rate from level 0 to level 3. According to [36

36. A. Siegman, Lasers (University Science Books, 1986).

,37

37. A. S. Nagra and R. A. York, “FDTD analysis of wave propagation in nonlinear absorbing and gain media,” IEEE Trans. Antenn. Propag. 46(3), 334–340 (1998). [CrossRef]

], locally, the polarization density at the emission frequency band Pe(r,t) obeys to the semi-classical “equation of motion”
2t2Pe(r,t)+ΔωatPe(r,t)+ωa2Pe(r,t)=σaΔN(r,t)E(r,t),
(9)
where Δωa is the bandwidth of the dye transition at the emitting angular frequency ωa, ΔN(r,t)=N2(r,t)N1(r,t) is the population inversion, and σa the coupling strength of Pe(r,t) to the electric field E(r,t). The expression of σa has been provided, for example, in [36

36. A. Siegman, Lasers (University Science Books, 1986).

] (pages 221-222, chapter 5) as σa=6πε0c03γrad/(εrωa2) (see also [40

40. S.-H. Chang and A. Taflove, “Finite-difference time-domain model of lasing action in a four-level two-electron atomic system,” Opt. Express 12(16), 3827–3833 (2004). [CrossRef] [PubMed]

]), where γrad is the radiative transition rate from level 2 to level 1, whose value can be obtained from the quantum yield γrad/(γrad+γnonrad)=γradτ21, obtained experimentally and readily available for the dyes later considered.

Assuming time harmonic polarization density and electric field Pe(r,t)=Re[Pe(r)eiωt], E(r,t)=Re[E(r)eiωt], and constant population inversion ΔN (i.e., steady state), Eq. (9) becomes

Pe(r)=ε0χeE(r),χe=1ε0σaΔNω2+iΔωaωωa2.
(10)

According to chapter 2 in [36

36. A. Siegman, Lasers (University Science Books, 1986).

], the total displacement might be written as
D(r)=ε0E(r)+Pr(r)+Pe(r)=ε0εrE(r)+Pe(r),
(11)
where Pr(r) is the polarization contribution due to the dielectric medium hosting the gain material, and Pe(r) is the polarization contribution due to the dispersed gain material itself, from which the effective absolute permittivity of the gain medium is
εg=ε0εr+σaΔNω2+iΔωaωωa2.
(12)
The model shown in Eqs. (9)(12) is detailed in standard laser textbooks (such as [36

36. A. Siegman, Lasers (University Science Books, 1986).

]). Here it has been just briefly summarized to guide the reader and to introduce every dye molecular parameter that will be adopted in the following sections. One should notice that as mentioned in [36

36. A. Siegman, Lasers (University Science Books, 1986).

] (pages 104-105, chapter 2), D(r) and Pe(r) could be expressed in two slightly different ways; to avoid ambiguities and misunderstandings, we report in Eqs. (10) and (11) the expressions we used.

Under stationary regime assumption (constant electron densities in any state) with time harmonic polarization, the population inversion is
ΔN=(τ21τ10)Γpump1+(τ32+τ21+τ10)ΓpumpN¯0,
(13)
where we have assumed that the electric field is small enough to neglect nonlinear saturation terms (as also discussed in [41

41. A. Fang, T. Koschny, M. Wegener, and C. M. Soukoulis, “Self-consistent calculation of metamaterials with gain,” Phys. Rev. B 79(24), 241104 (2009). [CrossRef]

]) that may induce depletion of the population inversion [36

36. A. Siegman, Lasers (University Science Books, 1986).

]. The relation of Eq. (13) to the pump light intensity is not straightforward when the gain medium is enclosed by a nanoshell or close to other nano scatterers, whereas it is rather simple for an open gain medium. As a first approximation, one can consider the relation in open gain media where Γpump=σabsIpump/(hf30), where σabs is the absorption cross section, Ipump is the pump intensity, h is the Planck constant, and f30 is the pump frequency [36

36. A. Siegman, Lasers (University Science Books, 1986).

]. A further improvement of the model (which will be performed in a future study) is based on analyzing the electrodynamic system at the pump frequency and on determining local pump fields and thus local absorptions. From Eqs. (12) and (13) the linearized active constitutive permittivity of the gain material in the nanoparticle cores is

εg=ε0εr+σaω2+iΔωaωωa2(τ21τ10)Γpump1+(τ32+τ21+τ10)ΓpumpN¯0.
(14)

3. Results of epsilon near zero composite materials with mitigated loss

Two cases are assumed, the first one made by silver shells (Fig. 1(a)), the second one made by gold shells (Fig. 1(b)), thus exhibiting resonance at lower frequency.

3.1 Case with silver shells

We assume that Rhodamine 6G (R6G) dye molecules are into the dielectric core as in Fig. 1(a). According to [42

42. O. Svelto, Principles of Lasers (Kluwer Academic Plenum Publishers, 1998).

45

45. D. Magde, R. Wong, and P. G. Seybold, “Fluorescence quantum yields and their relation to lifetimes of rhodamine 6G and fluorescein in nine solvents: improved absolute standards for quantum yields,” Photochem. Photobiol. 75(4), 327–334 (2002). [CrossRef] [PubMed]

], R6G has the following parameters: center emission frequency fa=526 THz (570 nm) (an homogeneously broadened Lorentzian lineshape is assumed), wavelength linewidth is Δλa=30 nm, and consequently Δωa=2πΔνa, with frequency linewidth Δνa=c0Δλa/λa2=27.7 THz. Assuming a four level system, the pumping wavelength (between level 0 and level 3) is 531 nm (565 THz), and the decay rate from level 2 to level 1 is 1/τ21, where τ21=3.99 ns. Moreover, the other decay rates from level 3 to level 2 and from level 1 to level 0 are 1/τ32=1/τ10, where τ32=τ10=100 fs. Also, we consider two different densities of the dye molecules as N¯0=3×1018 cm3, and N¯0=6×1018 cm3 corresponding to about 5 mM and 10 mM, respectively. The coupling constant σa in Eq. (9) is evaluated as discussed in Sec. 2.3 (and also shown in chapter 5 in [36

36. A. Siegman, Lasers (University Science Books, 1986).

]), for which σa=6.55×108 C2/kg, having used γrad=2.38×108s1 as in [45

45. D. Magde, R. Wong, and P. G. Seybold, “Fluorescence quantum yields and their relation to lifetimes of rhodamine 6G and fluorescein in nine solvents: improved absolute standards for quantum yields,” Photochem. Photobiol. 75(4), 327–334 (2002). [CrossRef] [PubMed]

] (assuming a quantum yield of 0.95, defined at the end of Sec. 2.3).

3.1.1 Mode analysis and effective parameters computation

In this section, we adopt the structure in Fig. 1(a). The outer shell radius is r2=25 nm, the dielectric core and environment are made by a material with εr=εh=2.25, r1=20 nm(ρ=r1/r2=0.8), and a=b=c=75 nm. The shell is made of silver, with relative permittivity ε2=εm, with εm as in Eq. (5), whose Drude model parameters are ε=5, ωp=1.37×1016rad/s  and γ=27.3×1012s1 [46

46. A. Alù, A. Salandrino, and N. Engheta, “Negative effective permeability and left-handed materials at optical frequencies,” Opt. Express 14(4), 1557–1567 (2006). [CrossRef] [PubMed]

,47

47. I. El-Kady, M. M. Sigalas, R. Biswas, K. M. Ho, and C. M. Soukoulis, “Metallic photonic crystals at optical wavelengths,” Phys. Rev. B 62(23), 15299–15302 (2000). [CrossRef]

]. The core has a relative permittivity ε1=εg/ε0, with εg as in Eq. (14) with the parameters for R6G. We use the nanoshell Mie electric polarizability expression in Eq. (3) for the results in Fig. 2
Fig. 2 Wavenumber dispersion diagram versus frequency for T-pol for the structure in Fig. 1(a), using the polarizability in Eq. (3). (a) Real part and (b) imaginary part of the wavenumber kz=βz+iαz, for lossy, lossless and loss-compensated cases.
, where we show the modes in the 3D lattice, for transverse polarization, traveling along the z direction, for three cases: (i) accounting for metal losses, (ii) ideal lossless case (i.e., γ=0 in Eq. (5)), and (iii) in presence of gain (10 mM of R6G optically pumped at 531 nm with pumping rate Γpump=1.5×109 s1).

As stated in Sec. 2.2, by using Eq. (8), it is possible to compute the effective refractive index of the homogenized array from the wavenumber information reported in Fig. 2. Then, we observed by using Eq. (7) with the magnetic polarizability of a nanoshell that the effective permeability μeff1 for the entire frequency region analyzed (the maximum deviation from 1 is at 100 THz (3 µm), whereμeff0.994, with imaginary part less than 2.5×104). Therefore, we compute the effective relative permittivity as εeffneff2, reported in Fig. 3
Fig. 3 (a) Real part and (b) imaginary part of the effective relative permittivity for the structure in Fig. 1(a) computed in three different ways: by Maxwell Garnett, by using the polarizability in (2) (MG – CM); the polarizability in Eq. (3) (MG – Mie); and mode analysis.
. Notice how the different methods agree in the entire frequency region analyzed; around the resonance, they differ because mode analysis includes field retardation effects not accounted for in Maxwell Garnett formulation. Also, there is a frequency shift towards lower frequencies using MG with Mie theory with respect to MG with quasi static polarizability (Clausius Mossotti) as discussed in [48

48. M. Meier and A. Wokaun, “Enhanced fields on large metal particles: dynamic depolarization,” Opt. Lett. 8(11), 581–583 (1983). [CrossRef] [PubMed]

] and [21

21. S. Steshenko and F. Capolino, “Single dipole approximation for modeling collections of nanoscatterers,” in Theory and Phenomena of Metamaterials, F. Capolino, ed. (CRC Press, 2009), p. 8.1.

] for example.

3.1.2 Epsilon-near-zero region for silver shells

In this section, we are interested in alleviating the losses in a frequency region in which the real part of epsilon is close to zero (either negative or positive). By using the results shown in Sec. 3.1.1 for the structure in Fig. 1(a), we observe that the permittivity retrieved through mode analysis in Fig. 3(a) crosses zero at around 526 THz (570 nm), which overlaps well with the emission frequency of R6G. We then better analyze the frequency region 515-535 THz, and we consider 5 mM and 10 mMof R6G dye molecules in the dielectric core, optically pumped at 531 nm, with pumping rate Γpump=1.5×109 s1. The real and the imaginary parts of the relative effective permittivity, obtained from mode analysis, are reported in Fig. 4
Fig. 4 (a) Real and (b) imaginary parts of the relative effective permittivity for the case in Fig. 3, obtained from mode analysis with polarizability in Eq. (3), in the epsilon-near-zero region around 526 THz.
. Notice from Fig. 4(b) how the imaginary part in presence of the gain material is reduced (red and green curves) with respect to the case in absence of the gain material (blue curve). Indeed, this demonstrates that the presence of R6G dye molecules allows for the alleviation of the losses, reaching a reduction of the extinction coefficient αz/k0 from 0.16 to 0.01 at f = 526 THz, in the epsilon-near-zero frequency band, for the highest concentration considered.

3.2 Case with gold shells

3.2.1 Mode analysis and effective parameters computation

As in Sec. 3.1.1, the effective refractive index of the homogenized array is retrieved from the wavenumber values reported in Fig. 5. The effective permeability (computed using Eq. (7)) is μeff1 for the entire frequency region analyzed (the maximum deviation from 1 is at around 100 THz, whereμeff0.99, with imaginary part less than 1.7×103). Therefore, we compute again the effective relative permittivity as εeffneff2, reported in Fig. 6
Fig. 6 (a) Real part and (b) imaginary part of the effective relative permittivity for the structure in Fig. 1(b) computed in three different ways: by Maxwell Garnett, by using the polarizability in Eq. (2) (MG – CM); the polarizability in Eq. (3) (MG – Mie); and mode analysis.
. Notice again the agreement between the different methods in the entire frequency region analyzed; around the resonance, they differ because mode analysis includes field retardation effects not accounted for in Maxwell Garnett formulation.

3.2.2 Epsilon-near-zero region for gold shells

We are again interested in alleviating propagation losses in a frequency region in which the real part of epsilon is close to zero (either negative or positive), as we attempted for the case with silver shell. By using the results shown in Sec. 3.2.1 for the structure in Fig. 1(b), we observe that the permittivity obtained from mode analysis in Fig. 6(a) crosses zero at around 421 THz (712 nm), which overlaps well with the emission frequency of R800. We then focus on the frequency region 400-440 THz, and we consider concentrations relative to 5 mM and 10 mM of R800 dye molecules in the dielectric core, optically pumped at 680 nm with pumping rate Γpump=6.5×109 s1. The real and the imaginary parts of the relative effective permittivity, obtained from mode analysis, are reported in Fig. 7
Fig. 7 (a) Real and (b) imaginary parts of the relative effective permittivity for the case in Fig. 5, obtained from mode analysis with polarizability in Eq. (3), in the epsilon-near-zero region around 421 THz.
. Similarly to what described for the silver shell case in Sec. 3.1, it can be observed in Fig. 7(b) how the imaginary part in presence of the gain material is greatly reduced (red and green curves) with respect to the case in absence of the gain material (blue curve), showing that the presence of R800 dye molecules allows for the alleviation of the losses, reaching a reduction of the extinction coefficient αz/k0 from 0.37 to 7×104 at f = 422 THz, in the epsilon-near-zero frequency band, for the highest concentration considered.

4. Conclusion

We have reported on the possibility of designing loss-compensated metamaterials, made of a 3D lattice of nanoshells, that exhibit epsilon near zero with moderate losses at optical frequencies by using optically pumped fluorescent dye molecules in the cores of the metamaterial constituent nanoshells. Indeed, we have been able to reduce the extinction coefficient αz/k0 from 0.16 to 0.01 in the epsilon-near-zero frequency band using dielectric-core silver-shell nanospheres in Fig. 1(a), and from 0.37 to 7×104 in the epsilon-near-zero frequency band using dielectric-core gold-shell nanospheres in Fig. 1(b), by using realistic parameters to model the emission of the dye molecules. However, high concentrations of dye molecules may impact in the overall compensation due to the presence of fluorescence quenching and other non-radiative phenomena. We assumed that the collection of the fluorescent dyes within the core can be approximated as an effective homogeneous material with gain that electrodynamically interacts with the metal nanoparticle. The quenching effect leads to a reduction of the gain in the system, and a treatment for each individual molecule-nanoparticle and molecule-molecule interaction is needed to estimate its real impact. Therefore, future work shall be devoted to optimization and also to experimental verification.

Acknowledgments

The authors acknowledge partial support from National Science Foundation (NSF)-CMMI award 1101074, and from the European Commission FP7/2008, “Nanosciences, Nanotechnologies, Materials and New Production Technologies NMP −2008-2.2-2,” grant “METACHEM,” no. 228762. The authors are also grateful to Dr. Ashod Aradian and Dr. Serge Ravaine, Centre de Recherche Paul Pascal, France, and to Prof. Giuseppe Strangi, University of Calabria, Italy, for useful discussions. S. Campione acknowledges also support by a Grant-in-Aid of Research from Sigma Xi, The Scientific Research Society and by an SPIE scholarship in Optics and Photonics.

References and links

1.

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000). [CrossRef] [PubMed]

2.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006). [CrossRef] [PubMed]

3.

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

4.

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

5.

M. C. Gather, K. Meerholz, N. Danz, and K. Leosson, “Net optical gain in a plasmonic waveguide embedded in a fluorescent polymer,” Nat. Photonics 4(7), 457–461 (2010). [CrossRef]

6.

I. De Leon and P. Berini, “Amplification of long-range surface plasmons by a dipolar gain medium,” Nat. Photonics 4(6), 382–387 (2010). [CrossRef]

7.

G. Strangi, A. De Luca, S. Ravaine, M. Ferrie, and R. Bartolino, “Gain induced optical transparency in metamaterials,” Appl. Phys. Lett. 98(25), 251912 (2011). [CrossRef]

8.

A. De Luca, M. P. Grzelczak, I. Pastoriza-Santos, L. M. Liz-Marzán, M. La Deda, M. Striccoli, and G. Strangi, “Dispersed and encapsulated gain medium in plasmonic nanoparticles: a multipronged approach to mitigate optical losses,” ACS Nano 5(7), 5823–5829 (2011). [CrossRef] [PubMed]

9.

J. A. Gordon and R. W. Ziolkowski, “CNP optical metamaterials,” Opt. Express 16(9), 6692–6716 (2008). [CrossRef] [PubMed]

10.

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

11.

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

12.

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

13.

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]

14.

M. A. Noginov, G. Zhu, M. Bahoura, J. Adegoke, C. Small, B. A. Ritzo, V. P. Drachev, and V. M. Shalaev, “The effect of gain and absorption on surface plasmons in metal nanoparticles,” Appl. Phys. B 86(3), 455–460 (2007). [CrossRef]

15.

R. Bardhan, N. K. Grady, J. R. Cole, A. Joshi, and N. J. Halas, “Fluorescence enhancement by Au nanostructures: nanoshells and nanorods,” ACS Nano 3(3), 744–752 (2009). [CrossRef] [PubMed]

16.

J. Zhang, I. Gryczynski, Z. Gryczynski, and J. R. Lakowicz, “Dye-labeled silver nanoshell-bright particle,” J. Phys. Chem. B 110(18), 8986–8991 (2006). [CrossRef] [PubMed]

17.

A. Fang, Z. Huang, T. Koschny, and C. M. Soukoulis, “Overcoming the losses of a split ring resonator array with gain,” Opt. Express 19(13), 12688–12699 (2011). [CrossRef] [PubMed]

18.

Y. Sivan, S. Xiao, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Frequency-domain simulations of a negative-index material with embedded gain,” Opt. Express 17(26), 24060–24074 (2009). [CrossRef] [PubMed]

19.

A. D. Boardman, V. V. Grimalsky, Y. S. Kivshar, S. V. Koshevaya, M. Lapine, N. M. Litchinitser, V. N. Malnev, M. Noginov, Y. G. Rapoport, and V. M. Shalaev, “Active and tunable metamaterials,” Laser Photon. Rev. 5(2), 287–307 (2011). [CrossRef]

20.

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

21.

S. Steshenko and F. Capolino, “Single dipole approximation for modeling collections of nanoscatterers,” in Theory and Phenomena of Metamaterials, F. Capolino, ed. (CRC Press, 2009), p. 8.1.

22.

S. Campione and F. Capolino, “Linear and planar periodic arrays of metallic nanospheres: fabrication, optical properties and applications,” in Selected Topics in Metamaterials and Photonic Crystals, A. Andreone, A. Cusano, A. Cutolo, and V. Galdi, eds. (World Scientific, 2011), pp. 141–194.

23.

A. Sihvola, Electromagnetic Mixing Formulas and Applications (IEEE Publishing, 1999).

24.

A. Sihvola, “Mixing rules,” in Theory and Phenomena of Metamaterials, F. Capolino, ed. (CRC Press, 2009), p. 9.1.

25.

M. G. Silveirinha, A. Alu, B. Edwards, and N. Engheta, “Overview of theory and applications of epsilon-near-zero materials,” in URSI General Assembly (Chicago, IL, 2008).

26.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972). [CrossRef]

27.

L. Landau and E. M. Lifschitz, Electrodynamics of Continuous Media (Pergamon Press, 1984), Chap. IX.

28.

K. Ohta and H. Ishida, “Comparison among several numerical integration methods for Kramers-Kronig transformation,” Appl. Spectrosc. 42(6), 952–957 (1988). [CrossRef]

29.

K. E. Peiponen and E. M. Vartiainen, “Kramers-Kronig relations in optical data inversion,” Phys. Rev. B Condens. Matter 44(15), 8301–8303 (1991). [CrossRef] [PubMed]

30.

S. A. Ramakrishna and T. M. Grzegorczyk, Physics and Applications of Negative Refractive Index Materials (CRC Press and SPIE Press, 2009).

31.

K. Tanabe, “Field enhancement around metal nanoparticles and nanoshells: a systematic investigation,” J. Phys. Chem. C 112(40), 15721–15728 (2008). [CrossRef]

32.

A. Vallecchi, M. Albani, and F. Capolino, “EM characterization of Raspberry-like nanocluster metamaterials,” in Antennas and Propagation Society International Symposium (Toronto, Canada, 2010).

33.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover Publications, 1965).

34.

A. L. Fructos, S. Campione, F. Capolino, and F. Mesa, “Characterization of complex plasmonic modes in two-dimensional periodic arrays of metal nanospheres,” J. Opt. Soc. Am. B 28(6), 1446–1458 (2011). [CrossRef]

35.

S. Campione, S. Steshenko, M. Albani, and F. Capolino, “Complex modes in three dimensional periodic arrays of plasmonic nanospheres,” Opt. Express (to be submitted).

36.

A. Siegman, Lasers (University Science Books, 1986).

37.

A. S. Nagra and R. A. York, “FDTD analysis of wave propagation in nonlinear absorbing and gain media,” IEEE Trans. Antenn. Propag. 46(3), 334–340 (1998). [CrossRef]

38.

D. Faubert, S. L. Chin, M. Cormier, and M. Boloten, “Numerical analysis of short laser pulse superposition in a fluorescent dye medium,” Can. J. Phys. 57(2), 160–167 (1979). [CrossRef]

39.

G. Grönninger and A. Penzkofer, “Determination of energy and duration of picosecond light pulses by bleaching of dyes,” Opt. Quantum Electron. 16(3), 225–233 (1984). [CrossRef]

40.

S.-H. Chang and A. Taflove, “Finite-difference time-domain model of lasing action in a four-level two-electron atomic system,” Opt. Express 12(16), 3827–3833 (2004). [CrossRef] [PubMed]

41.

A. Fang, T. Koschny, M. Wegener, and C. M. Soukoulis, “Self-consistent calculation of metamaterials with gain,” Phys. Rev. B 79(24), 241104 (2009). [CrossRef]

42.

O. Svelto, Principles of Lasers (Kluwer Academic Plenum Publishers, 1998).

43.

M. A. R. C. Alencar, G. S. Maciel, C. B. de Araújo, R. Bertholdo, Y. Messaddeq, and S. J. L. Ribeiro, “Laserlike emission from silica inverse opals infiltrated with Rhodamine 6G,” J. Non-Cryst. Solids 351(21-23), 1846–1849 (2005). [CrossRef]

44.

D. Magde, G. E. Rojas, and P. G. Seybold, “Solvent dependence of the fluorescence lifetimes of xanthene dyes,” Photochem. Photobiol. 70(5), 737–744 (1999). [CrossRef]

45.

D. Magde, R. Wong, and P. G. Seybold, “Fluorescence quantum yields and their relation to lifetimes of rhodamine 6G and fluorescein in nine solvents: improved absolute standards for quantum yields,” Photochem. Photobiol. 75(4), 327–334 (2002). [CrossRef] [PubMed]

46.

A. Alù, A. Salandrino, and N. Engheta, “Negative effective permeability and left-handed materials at optical frequencies,” Opt. Express 14(4), 1557–1567 (2006). [CrossRef] [PubMed]

47.

I. El-Kady, M. M. Sigalas, R. Biswas, K. M. Ho, and C. M. Soukoulis, “Metallic photonic crystals at optical wavelengths,” Phys. Rev. B 62(23), 15299–15302 (2000). [CrossRef]

48.

M. Meier and A. Wokaun, “Enhanced fields on large metal particles: dynamic depolarization,” Opt. Lett. 8(11), 581–583 (1983). [CrossRef] [PubMed]

49.

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

50.

S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Overcoming losses with gain in a negative refractive index metamaterial,” Phys. Rev. Lett. 105(12), 127401 (2010). [CrossRef] [PubMed]

51.

N. K. Grady, N. J. Halas, and P. Nordlander, “Influence of dielectric function properties on the optical response of plasmon resonant metallic nanoparticles,” Chem. Phys. Lett. 399(1-3), 167–171 (2004). [CrossRef]

52.

S. Campione, A. Vallecchi, and F. Capolino, “Closed form formulas and tunability of resonances in pairs of gold-dielectric nanoshells,” Proc. SPIE 7757, 775738 (2010). [CrossRef]

53.

S. Campione, S. Steshenko, and F. Capolino, “Description and characterization of the complex modes in a linear chain of gold metal nanospheres,” Proc. SPIE 7946, 79461V (2011). [CrossRef]

OCIS Codes
(160.1245) Materials : Artificially engineered materials
(160.3918) Materials : Metamaterials
(250.5403) Optoelectronics : Plasmonics

ToC Category:
Metamaterials

History
Original Manuscript: July 20, 2011
Revised Manuscript: August 25, 2011
Manuscript Accepted: August 26, 2011
Published: September 6, 2011

Virtual Issues
Nanoplasmonics and Metamaterials (2011) Optical Materials Express

Citation
Salvatore Campione, Matteo Albani, and Filippo Capolino, "Complex modes and near-zero permittivity in 3D arrays of plasmonic nanoshells: loss compensation using gain [Invited]," Opt. Mater. Express 1, 1077-1089 (2011)
http://www.opticsinfobase.org/ome/abstract.cfm?URI=ome-1-6-1077


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References

  1. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett.85(18), 3966–3969 (2000). [CrossRef] [PubMed]
  2. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science314(5801), 977–980 (2006). [CrossRef] [PubMed]
  3. D. J. Bergman and M. I. Stockman, “Surface plasmon amplification by stimulated emission of radiation: quantum generation of coherent surface plasmons in nanosystems,” Phys. Rev. Lett.90(2), 027402 (2003). [CrossRef] [PubMed]
  4. M. I. Stockman, “Spasers explained,” Nat. Photonics2(6), 327–329 (2008). [CrossRef]
  5. M. C. Gather, K. Meerholz, N. Danz, and K. Leosson, “Net optical gain in a plasmonic waveguide embedded in a fluorescent polymer,” Nat. Photonics4(7), 457–461 (2010). [CrossRef]
  6. I. De Leon and P. Berini, “Amplification of long-range surface plasmons by a dipolar gain medium,” Nat. Photonics4(6), 382–387 (2010). [CrossRef]
  7. G. Strangi, A. De Luca, S. Ravaine, M. Ferrie, and R. Bartolino, “Gain induced optical transparency in metamaterials,” Appl. Phys. Lett.98(25), 251912 (2011). [CrossRef]
  8. A. De Luca, M. P. Grzelczak, I. Pastoriza-Santos, L. M. Liz-Marzán, M. La Deda, M. Striccoli, and G. Strangi, “Dispersed and encapsulated gain medium in plasmonic nanoparticles: a multipronged approach to mitigate optical losses,” ACS Nano5(7), 5823–5829 (2011). [CrossRef] [PubMed]
  9. J. A. Gordon and R. W. Ziolkowski, “CNP optical metamaterials,” Opt. Express16(9), 6692–6716 (2008). [CrossRef] [PubMed]
  10. N. M. Lawandy, “Localized surface plasmon singularities in amplifying media,” Appl. Phys. Lett.85(21), 5040–5042 (2004). [CrossRef]
  11. S. Anantha Ramakrishna and J. Pendry, “Removal of absorption and increase in resolution in a near-field lens via optical gain,” Phys. Rev. B67(20), 201101 (2003). [CrossRef]
  12. S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature466(7307), 735–738 (2010). [CrossRef] [PubMed]
  13. A. K. Sarychev and G. Tartakovsky, “Magnetic plasmonic metamaterials in actively pumped host medium and plasmonic nanolaser,” Phys. Rev. B75(8), 085436 (2007). [CrossRef]
  14. M. A. Noginov, G. Zhu, M. Bahoura, J. Adegoke, C. Small, B. A. Ritzo, V. P. Drachev, and V. M. Shalaev, “The effect of gain and absorption on surface plasmons in metal nanoparticles,” Appl. Phys. B86(3), 455–460 (2007). [CrossRef]
  15. R. Bardhan, N. K. Grady, J. R. Cole, A. Joshi, and N. J. Halas, “Fluorescence enhancement by Au nanostructures: nanoshells and nanorods,” ACS Nano3(3), 744–752 (2009). [CrossRef] [PubMed]
  16. J. Zhang, I. Gryczynski, Z. Gryczynski, and J. R. Lakowicz, “Dye-labeled silver nanoshell-bright particle,” J. Phys. Chem. B110(18), 8986–8991 (2006). [CrossRef] [PubMed]
  17. A. Fang, Z. Huang, T. Koschny, and C. M. Soukoulis, “Overcoming the losses of a split ring resonator array with gain,” Opt. Express19(13), 12688–12699 (2011). [CrossRef] [PubMed]
  18. Y. Sivan, S. Xiao, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Frequency-domain simulations of a negative-index material with embedded gain,” Opt. Express17(26), 24060–24074 (2009). [CrossRef] [PubMed]
  19. A. D. Boardman, V. V. Grimalsky, Y. S. Kivshar, S. V. Koshevaya, M. Lapine, N. M. Litchinitser, V. N. Malnev, M. Noginov, Y. G. Rapoport, and V. M. Shalaev, “Active and tunable metamaterials,” Laser Photon. Rev.5(2), 287–307 (2011). [CrossRef]
  20. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).
  21. S. Steshenko and F. Capolino, “Single dipole approximation for modeling collections of nanoscatterers,” in Theory and Phenomena of Metamaterials, F. Capolino, ed. (CRC Press, 2009), p. 8.1.
  22. S. Campione and F. Capolino, “Linear and planar periodic arrays of metallic nanospheres: fabrication, optical properties and applications,” in Selected Topics in Metamaterials and Photonic Crystals, A. Andreone, A. Cusano, A. Cutolo, and V. Galdi, eds. (World Scientific, 2011), pp. 141–194.
  23. A. Sihvola, Electromagnetic Mixing Formulas and Applications (IEEE Publishing, 1999).
  24. A. Sihvola, “Mixing rules,” in Theory and Phenomena of Metamaterials, F. Capolino, ed. (CRC Press, 2009), p. 9.1.
  25. M. G. Silveirinha, A. Alu, B. Edwards, and N. Engheta, “Overview of theory and applications of epsilon-near-zero materials,” in URSI General Assembly (Chicago, IL, 2008).
  26. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B6(12), 4370–4379 (1972). [CrossRef]
  27. L. Landau and E. M. Lifschitz, Electrodynamics of Continuous Media (Pergamon Press, 1984), Chap. IX.
  28. K. Ohta and H. Ishida, “Comparison among several numerical integration methods for Kramers-Kronig transformation,” Appl. Spectrosc.42(6), 952–957 (1988). [CrossRef]
  29. K. E. Peiponen and E. M. Vartiainen, “Kramers-Kronig relations in optical data inversion,” Phys. Rev. B Condens. Matter44(15), 8301–8303 (1991). [CrossRef] [PubMed]
  30. S. A. Ramakrishna and T. M. Grzegorczyk, Physics and Applications of Negative Refractive Index Materials (CRC Press and SPIE Press, 2009).
  31. K. Tanabe, “Field enhancement around metal nanoparticles and nanoshells: a systematic investigation,” J. Phys. Chem. C112(40), 15721–15728 (2008). [CrossRef]
  32. A. Vallecchi, M. Albani, and F. Capolino, “EM characterization of Raspberry-like nanocluster metamaterials,” in Antennas and Propagation Society International Symposium (Toronto, Canada, 2010).
  33. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover Publications, 1965).
  34. A. L. Fructos, S. Campione, F. Capolino, and F. Mesa, “Characterization of complex plasmonic modes in two-dimensional periodic arrays of metal nanospheres,” J. Opt. Soc. Am. B28(6), 1446–1458 (2011). [CrossRef]
  35. S. Campione, S. Steshenko, M. Albani, and F. Capolino, “Complex modes in three dimensional periodic arrays of plasmonic nanospheres,” Opt. Express (to be submitted).
  36. A. Siegman, Lasers (University Science Books, 1986).
  37. A. S. Nagra and R. A. York, “FDTD analysis of wave propagation in nonlinear absorbing and gain media,” IEEE Trans. Antenn. Propag.46(3), 334–340 (1998). [CrossRef]
  38. D. Faubert, S. L. Chin, M. Cormier, and M. Boloten, “Numerical analysis of short laser pulse superposition in a fluorescent dye medium,” Can. J. Phys.57(2), 160–167 (1979). [CrossRef]
  39. G. Grönninger and A. Penzkofer, “Determination of energy and duration of picosecond light pulses by bleaching of dyes,” Opt. Quantum Electron.16(3), 225–233 (1984). [CrossRef]
  40. S.-H. Chang and A. Taflove, “Finite-difference time-domain model of lasing action in a four-level two-electron atomic system,” Opt. Express12(16), 3827–3833 (2004). [CrossRef] [PubMed]
  41. A. Fang, T. Koschny, M. Wegener, and C. M. Soukoulis, “Self-consistent calculation of metamaterials with gain,” Phys. Rev. B79(24), 241104 (2009). [CrossRef]
  42. O. Svelto, Principles of Lasers (Kluwer Academic Plenum Publishers, 1998).
  43. M. A. R. C. Alencar, G. S. Maciel, C. B. de Araújo, R. Bertholdo, Y. Messaddeq, and S. J. L. Ribeiro, “Laserlike emission from silica inverse opals infiltrated with Rhodamine 6G,” J. Non-Cryst. Solids351(21-23), 1846–1849 (2005). [CrossRef]
  44. D. Magde, G. E. Rojas, and P. G. Seybold, “Solvent dependence of the fluorescence lifetimes of xanthene dyes,” Photochem. Photobiol.70(5), 737–744 (1999). [CrossRef]
  45. D. Magde, R. Wong, and P. G. Seybold, “Fluorescence quantum yields and their relation to lifetimes of rhodamine 6G and fluorescein in nine solvents: improved absolute standards for quantum yields,” Photochem. Photobiol.75(4), 327–334 (2002). [CrossRef] [PubMed]
  46. A. Alù, A. Salandrino, and N. Engheta, “Negative effective permeability and left-handed materials at optical frequencies,” Opt. Express14(4), 1557–1567 (2006). [CrossRef] [PubMed]
  47. I. El-Kady, M. M. Sigalas, R. Biswas, K. M. Ho, and C. M. Soukoulis, “Metallic photonic crystals at optical wavelengths,” Phys. Rev. B62(23), 15299–15302 (2000). [CrossRef]
  48. M. Meier and A. Wokaun, “Enhanced fields on large metal particles: dynamic depolarization,” Opt. Lett.8(11), 581–583 (1983). [CrossRef] [PubMed]
  49. P. Sperber, W. Spangler, B. Meier, and A. Penzkofer, “Experimental and theoretical investigation of tunable picosecond pulse generation in longitudinally pumped dye-laser generators and amplifiers,” Opt. Quantum Electron.20(5), 395–431 (1988). [CrossRef]
  50. S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Overcoming losses with gain in a negative refractive index metamaterial,” Phys. Rev. Lett.105(12), 127401 (2010). [CrossRef] [PubMed]
  51. N. K. Grady, N. J. Halas, and P. Nordlander, “Influence of dielectric function properties on the optical response of plasmon resonant metallic nanoparticles,” Chem. Phys. Lett.399(1-3), 167–171 (2004). [CrossRef]
  52. S. Campione, A. Vallecchi, and F. Capolino, “Closed form formulas and tunability of resonances in pairs of gold-dielectric nanoshells,” Proc. SPIE7757, 775738 (2010). [CrossRef]
  53. S. Campione, S. Steshenko, and F. Capolino, “Description and characterization of the complex modes in a linear chain of gold metal nanospheres,” Proc. SPIE7946, 79461V (2011). [CrossRef]

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