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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 23 — Nov. 7, 2011
  • pp: 22999–23007
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Negative electron energy loss and second-harmonic emission of nonlinear nanoparticles

Jinying Xu and Xiangdong Zhang  »View Author Affiliations


Optics Express, Vol. 19, Issue 23, pp. 22999-23007 (2011)
http://dx.doi.org/10.1364/OE.19.022999


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Abstract

A fast and general technique to investigation the interaction between a fast electron and nonlinear materials consisting of centrosymmetric spheres is presented by means of multiple scattering of electromagnetic multipole fields. Two kinds of new effect, the negative electron energy loss caused by the second-harmonic field and the second-harmonic Smith-Purcell radiation using finite chain of nonlinear spheres, are predicted for the first time. It is shown that these new effects can be probed by the electron energy loss spectrum, suggesting their possible applications in tunable light sources for the second-harmonic generation.

© 2011 OSA

Electron-energy-loss spectroscopy (EELS) in combination with scanning transmission electron microscopy (STEM) has proved to be a powerful technique for determining different microstructures of nanometer scale [1

1. F. J. García de Abajo, “Optical excitations in electron microscopy,” Rev. Mod. Phys. 82(1), 209–275 (2010). [CrossRef]

]. The interaction between the moving electron and the microstructures gives rise to the emission and excitation of the radiation (e.g. Smith-Purcell radiation). Thus, the EELS in the STEM is a useful tool to investigate both surface and bulk excitations of the samples [2

2. R. García-Molina, A. Gras-Marti, and R. H. Ritchie, “Excitation of edge modes in the interaction of electron beams with dielectric wedges,” Phys. Rev. B Condens. Matter 31(1), 121–126 (1985). [CrossRef] [PubMed]

12

12. J. Xu and X. Zhang, “Cloaking radiation of moving electron beam and relativistic energy loss spectra,” Opt. Express 17(6), 4758–4772 (2009). [CrossRef] [PubMed]

]. The radiation induced by the moving electron is scalable in frequency, and thus can be used as a novel radiation source without limitation of frequency [13

13. S. J. Smith and E. M. Purcell, “Visible light from localized surface charges moving across a grating,” Phys. Rev. 92(4), 1069–1069 (1953). [CrossRef]

16

16. N. Yamamoto, K. Araya, and F. J. Garcia de Abajo, “Photon emission from silver particles induced by a high-energy electron beam,” Phys. Rev. B 64(20), 205419 (2001). [CrossRef]

]. In the last decades a remarkable progress has been made on this subject [1

1. F. J. García de Abajo, “Optical excitations in electron microscopy,” Rev. Mod. Phys. 82(1), 209–275 (2010). [CrossRef]

16

16. N. Yamamoto, K. Araya, and F. J. Garcia de Abajo, “Photon emission from silver particles induced by a high-energy electron beam,” Phys. Rev. B 64(20), 205419 (2001). [CrossRef]

]. However, all studies so far are limited in the microstructures consisting of linear materials.

On the other hand, nonlinear optical responses of small particles have been subject of intensive studies due to the development of nanofabrication techniques [17

17. T. F. Heinz, in Nonlinear Surface Electromagnetic Phenomena, edited by H.-E. Ponath and G. I. Stegeman (North-Holland, Amsterdam, 1991).

23

23. J. Butet, G. Bachelier, I. Russier-Antoine, C. Jonin, E. Benichou, and P.-F. Brevet, “Interference between selected dipoles and octupoles in the optical second-harmonic generation from spherical gold nanoparticles,” Phys. Rev. Lett. 105(7), 077401 (2010). [CrossRef] [PubMed]

]. In particular, second harmonic generation (SHG) can be employed to probe numerous physical and chemical processes occurring at the surfaces of small particles and its application to colloids provides complementary or unique tools in medicine and technology [18

18. J. I. Dadap, J. Shan, and T. F. Heinz, “Theory of optical second-harmonic generation from a sphere of centrosymmetric material: small-particle limit,” J. Opt. Soc. Am. B 21(7), 1328–1347 (2004). [CrossRef]

24

24. V. K. Valev, A. V. Silhanek, N. Verellen, W. Gillijns, P. Van Dorpe, O. A. Aktsipetrov, G. A. E. Vandenbosch, V. V. Moshchalkov, and T. Verbiest, “Asymmetric optical second-harmonic generation from chiral G-shaped gold nanostructures,” Phys. Rev. Lett. 104(12), 127401 (2010). [CrossRef] [PubMed]

]. However, many investigations on the SHG focus on the incident plane wave or laser beam [17

17. T. F. Heinz, in Nonlinear Surface Electromagnetic Phenomena, edited by H.-E. Ponath and G. I. Stegeman (North-Holland, Amsterdam, 1991).

25

25. Y. Pavlyukh and W. Hubner, “Nonlinear Mie scattering from spherical particles,” Phys. Rev. B 70(24), 245434 (2004). [CrossRef]

]. The question is whether or not the SHG can be realized by the moving electron beam? Whether or not the nonlinear properties of the materials can be explored by the EELS? To the best of our knowledge, these problems have not been investigated so far.

In this work we present a fast and general technique to investigate the interaction between a fast electron and the nonlinear material consisting of centrosymmetric spheres and explore the relativistic EELS and radiation emission in the nonlinear materials. We consider a fast electron moving along a straight-line trajectory with constant velocity v and passing near a cluster of spheres consisting of the second-order nonlinear material with relative permittivity ε1(ω) and relative permeability μ1(ω), placed in a medium having real, frequency-independent permittivity ε2 and permeability μ2. The radiation field of the moving electron can be expanded in terms of the vector spherical waves as [7

7. F. J. García de Abajo, “Relativistic energy loss and induced photon emission in the interaction of a dielectric sphere with an external electron beam,” Phys. Rev. B 59(4), 3095–3107 (1999). [CrossRef]

]
Eext(xn)=lm{ik2anlmext,E×jl(k2r)Xlm(r^)+anlmext,Hjl(k2r)Xlm(r^)},
(1)
where xn=rrn,rn is the center coordinate of the nth sphere, k2 is the wave vector of the background, jl is the first-kind spherical Bessel function and Xlm are vector spherical harmonics which is given in [26

26. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).

]. The exact expressions for expansion coefficients anlmext,E and anlmext,H have been given in [7

7. F. J. García de Abajo, “Relativistic energy loss and induced photon emission in the interaction of a dielectric sphere with an external electron beam,” Phys. Rev. B 59(4), 3095–3107 (1999). [CrossRef]

]. As the nonlinear effect of the material is neglected, a self-consistent linear multiple scattering equation for the fundamental frequency (FF) has been derived, which can be found in [8

8. F. J. García de Abajo, “Interaction of radiation and fast electrons with clusters of dielectrics: A multiple scattering approach,” Phys. Rev. Lett. 82(13), 2776–2779 (1999). [CrossRef]

].

As the nonlinear effect of the material is considered, the radiation of the moving electron beam can cause the nonlinear polarization of the materials and produce the second harmonic (SH) field. The total SH field at a space point can be viewed as consisting of two distinct components: a radiation field without being scattered by any of the spheres, and the scattered field being scattered by at least one of the spheres. For example, the SH source field outside the nth sphere is expressed as
Eout(2ω)(xn)=lm{Anlmout,EHElm(2ω)(xn)+Anlmout,HHHlm(2ω)(xn)}
(2)
with
Anlmout,E=4πiGE,nlmr[rjl(K1r)]4πl(l+1)Gr,nlmjl(K1r)ε1(2ω)K2jl(K1r)r[rhl(1)(K2r)]ε2(2ω)K2hl(1)(K2r)r[rjl(K1r)]|r=a,
(3)
Anlmout,H=8πωcGM,nlmjl(K1r)ε2(2ω)K2rjl(K1r)r[rhl(1)(K2r)]ε1(2ω)K1rhl(1)(K2r)r[rjl(K1r)]|r=a,
(4)
where HElm(2ω)(r)=(i/K2)×hl(K2r)Xlm(r^) and HHlm(2ω)(r)=hl(K2r)Xlm(r^), c is the light velocity in vacuum, hl is the first-kind spherical Hankel function, K1=2ωcε1(2ω)μ1(2ω) and K2=2ωcε2(2ω)μ2(2ω). Here Gr,nlm, GM,nlm and

GE,nlmare the expansion coefficient of the nonlinear polarization vector P(2ω)(r)=Ps(2ω)+Pb(2ω)=lmGr,nlmYlm(θ,φ)r^+GM,nlmXlm(θ,φ)+GE,nlmr^×Xlm(θ,φ), Ps(2ω)=χs(2):E(ω)(r)E(ω)(r)δ(ra) and Pb(2ω)=χb(2):E(ω)(r)E(ω)(r) represent surface and bulk nonlinear polarization vectors [17

17. T. F. Heinz, in Nonlinear Surface Electromagnetic Phenomena, edited by H.-E. Ponath and G. I. Stegeman (North-Holland, Amsterdam, 1991).

], respectively, χs(2) and χb(2) are the corresponding surface and the bulk second-order susceptibility tensors. In general, to describe a second-order nonlinear optical response at the interface between two centrosymmetric media, it is useful to divide the matter into three regions, the interfacial zone and two bulk regions. For most systems, the nonlinear optical response occurs at the interfacial zone within a distance of a few Angstroms [17

17. T. F. Heinz, in Nonlinear Surface Electromagnetic Phenomena, edited by H.-E. Ponath and G. I. Stegeman (North-Holland, Amsterdam, 1991).

, 18

18. J. I. Dadap, J. Shan, and T. F. Heinz, “Theory of optical second-harmonic generation from a sphere of centrosymmetric material: small-particle limit,” J. Opt. Soc. Am. B 21(7), 1328–1347 (2004). [CrossRef]

, 27

27. F. X. Wang, F. J. Rodríguez, W. M. Albers, R. Ahorinta, J. E. Sipe, and M. Kauranen, “Surface and bulk contributions to the second-order nonlinear optical response of a gold film,” Phys. Rev. B 80(23), 233402 (2009). [CrossRef]

]. Here we treat the nonlinearity of the interface as a sheet of current or, equivalently, as a sheet of nonlinear source polarization according to [17

17. T. F. Heinz, in Nonlinear Surface Electromagnetic Phenomena, edited by H.-E. Ponath and G. I. Stegeman (North-Holland, Amsterdam, 1991).

, 18

18. J. I. Dadap, J. Shan, and T. F. Heinz, “Theory of optical second-harmonic generation from a sphere of centrosymmetric material: small-particle limit,” J. Opt. Soc. Am. B 21(7), 1328–1347 (2004). [CrossRef]

]. If we make no assumptions about the nature of the interface other than that it exhibits isotropic symmetry with a mirror plane perpendicular to the interface, the nonlinear susceptibility tensor χs(2) then has three nonvanishing and independent elements: χ,χ, χ=χ,where and refer to the local spatial components perpendicular and parallel to the surface. The susceptibility χs(2) can be written, in terms of the unit vectors for the spherical coordinate system, as the triadic [18

18. J. I. Dadap, J. Shan, and T. F. Heinz, “Theory of optical second-harmonic generation from a sphere of centrosymmetric material: small-particle limit,” J. Opt. Soc. Am. B 21(7), 1328–1347 (2004). [CrossRef]

]

χs(2)=χr^r^r^+χr^(θ^θ^+φ^φ^)+χ(θ^r^θ^+φ^r^φ^+θ^θ^r^+φ^φ^r^).
(5)

Hence, the nonlinear source polarization for the surface is

Ps(2ω)=r^(χEr(ω)Er(ω)+χEt(ω)Er(ω))+2χEr(ω)Et(ω).
(6)

Here the subscripts r and t refer, respectively, to components perpendicular and parallel to the surface of the sphere. Thus, the expansion coefficients of the nonlinear polarization vector are expressed as

Gr,nlm=χbnlm+χbnlm,
(7)
GM,nlm=χb,Mnlm,
(8)
GE,nlm=χb,Enlm.
(9)

Anlm(1)=1k1ad[rjl(k1r)]dr|r=aanlmext,E,
(14)
Anlm(0)=jl(k1a)anlmext,H,
(15)
Anlm(1)=ε1l(l+1)k1ajl(k1a)anlmext,E.
(16)

Here k1 is the wave vector inside the sphere. The coefficients Cl1,m1,l2,m2,l,m(α,β) result from the expansion of scalar (vector) products of the vector spherical harmonics, comprise products of 6j (9j) coefficients and two Clebsch-Gordan coefficients [28

28. D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of Angular Momentum (World Scientific, Singapore, 1988).

].

The E(ω)(r) represents the FF field, n is a vector normal to the surface, a is the radius of the sphere. As for the SH scattered field (Esca(2ω)(xn)) of the nth sphere, it can be expanded as
Esca(2ω)(xn)=lm{Anlmsca,EHElm(2ω)(xn)+Anlmsca,HHHlm(2ω)(xn)},17)
where Anlmsca,E and Anlmsca,H are the expansion coefficients. The total incident SH field (Eloc(2ω)(xn)) of the nth sphere can be viewed as consisting of two distinct components: a source field from other spheres without being scattered by any of the spheres, and the sum of all the SH scattered wave from other spheres. It can be expressed as

Eloc(2ω)(xn)=nnlm{(Anlmout,E+Anlmsca,E)HElm(2ω)(xn)+(Anlmout,H+Anlmsca,H)HHlm(2ω)(xn)}.
(18)

With the use of the addition theorem [29

29. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, (Dover, New York, 1972), p. 363.

] and the boundary conditions, we obtain
nlm[δnnδllδmmTlE(Ωnlm,nlmEEAnlmsca,E+Ωnlm,nlmEHAnlmsca,H)]=nlmTlE[Ωnlm,nlmEEAnlmout,E+Ωnlm,nlmEHAnlmout,H],
(19)
nlm[δnnδllδmmTlH(Ωnlm,nlmHEAnlmsca,E+Ωnlm,nlmHHAnlmsca,H)]=nlmTlH[Ωnlm,nlmEEAnlmout,E+Ωnlm,nlmEHAnlmout,H],
(20)
where Ωnlm,nlmPP(P,P=E,H) represents the free-space propagator functions and TlE(H)are the Mie coefficients for a single sphere, which their explicit forms can be found elsewhere [7

7. F. J. García de Abajo, “Relativistic energy loss and induced photon emission in the interaction of a dielectric sphere with an external electron beam,” Phys. Rev. B 59(4), 3095–3107 (1999). [CrossRef]

, 8

8. F. J. García de Abajo, “Interaction of radiation and fast electrons with clusters of dielectrics: A multiple scattering approach,” Phys. Rev. Lett. 82(13), 2776–2779 (1999). [CrossRef]

]. The Eqs. (19) and (20) are the basic equations for the present nonlinear multiple-scattering system. Based on them, the SH fields can be obtained exactly by numerical calculations. The main steps in the calculations are as follows. First, the FF scattering field is determined by solving the linear multiple-scattering equation. Such a field is used to compute the total nonlinear polarization at the SH. Once one knows the total nonlinear polarization at the SH one can determine the source coefficients, Anlmout,E and Anlmout,H. Then, the scattering coefficients at the SH are determined from the Eqs. (19) and (20). Finally, the SH field can be obtained from the sum of the source field and all the SH scattered field.

Similar to the case of the linear scattering [6

6. F. J. García de Abajo and A. Howie, “Relativistic electron energy loss and electron-induced photon emission in inhomogeneous dielectrics,” Phys. Rev. Lett. 80(23), 5180–5183 (1998). [CrossRef]

8

8. F. J. García de Abajo, “Interaction of radiation and fast electrons with clusters of dielectrics: A multiple scattering approach,” Phys. Rev. Lett. 82(13), 2776–2779 (1999). [CrossRef]

], the SH loss probability Γloss(2ω) for an electron moving with velocity v can be obtained from the induced SH electric field (E(2ω)) acting on the electron as
Γloss(2ω)=1πϖdtRe{ei2ωtvE(2ω)(rt,2ω)}
(21)
where rt describes the electron trajectory. The corresponding probability of emitting a photon of energy ω per unit energy range and unit solid angle (Ω) around the direction r:

Γrad(2ω,Ω)=limrr24π2kRe{[E(2ω)(2ω)×H(2ω)(2ω)]r^}.
(22)

Here H(2ω) is the corresponding SH magnetic field. From Eqs. (21) and (22), we can obtain the same concrete forms to those of the linear system as presented in [6

6. F. J. García de Abajo and A. Howie, “Relativistic electron energy loss and electron-induced photon emission in inhomogeneous dielectrics,” Phys. Rev. Lett. 80(23), 5180–5183 (1998). [CrossRef]

8

8. F. J. García de Abajo, “Interaction of radiation and fast electrons with clusters of dielectrics: A multiple scattering approach,” Phys. Rev. Lett. 82(13), 2776–2779 (1999). [CrossRef]

]. Based on them, we can calculate the SH energy loss and photon emission probabilities of a fast electron passing through any cluster of nonlinear spheres. The calculated results are plotted in Fig. 1
Fig. 1 (a) and (d): Energy loss probabilities (solid lines) and photon-emission probabilities (dashed lines) without nonlinear effects for a moving electron with v = 0.7c passing at b = 1.1a from Ag spheres with a = 10nm and d = 22nm. (a) one sphere; (d) four spheres. (b) and (e) correspond to energy loss probabilities caused by the SH field; (c) and (f) to photon-emission probabilities caused by the SH field.
.

The magnitude of the negative energy loss depends on features of the nonlinear object. For example, the loss probabilities caused by the SH field in Fig. 1 are very weak. However, the situation can be changed by using nonlinear sphere chain as shown in Fig. 2 (a)
Fig. 2 Energy loss and photon-emission probabilities for an electron moving with v = 0.7c parallel to a finite periodic chain of 15 aligned spheres consisting of the KLN, as shown in (a). Here a = 10nm, d = 22nm and b = 1.1a. (b) Energy loss probability for the FF field (the same curve for the emission probability without absorptions); (c) Energy loss probability caused by the SH field; (d) photon-emission probabilities caused by the SH field.
. The energy loss probabilities for an electron moving with v = 0.7c parallel to a finite periodic chain with N = 15 are plotted in Fig. 2 (b) and (c) for the FF and the SH fields, respectively, while the corresponding SH photon emission probability is shown in Fig. 2 (d). Here N represents the number of spheres in the chain and all spheres consist of the nonlinear K3Li2Nb5O15 (KLN). The dielectric constant of the KLN is ε1=1+3.708ω2/[ω20.04601×(2πc)2] and the surface second-order susceptibility χ=12pm/V and χ=χ=11.8pm/V [33

33. D. N. Nikogosyan, Nonlinear optical crystals: A complete survey, Springer New York, USA (2003).

]. The negative electron energy loss probability is observed again from the figure (see Fig. 2 (c)). For the energy loss probability caused by the FF field, we find that there is the same order for the 15-sphere chain in comparison with the corresponding single sphere. However, the energy loss and photon emission probabilities caused by the SH field are improved 2-order due to quasi-phase matching effect. If we add the number of the sphere in the chain, for example, as N = 30, 3-order improvement can be realized. In such a case, the loss probability caused by the SH field is much bigger than that by the FF field at the corresponding frequency. This means that the nonlinear effect can be explored by the STEM.

At the same time, strong SH emissions can also be obtained by the moving the electron beam. The above calculations about the emissions have been integrated over all directions. In fact, the SH emissions strongly depend on the polar angle. This can be seen more clearly from Fig. 3
Fig. 3 Probability of photon emission for an electron moving parallel to a finite chain of N aligned spheres as a function of photon energy and polar angle θ. (a), (b) and (c) correspond to the cases for the FF, while (d), (e) and (f) to those of the SH. The values of N under consideration are 1, 5 and 15, respectively. Here a = 50nm and d = 120nm. The other parameters are identical with those in Fig. 2.
.

In summary, we have presented a general technique to investigate the interaction between a fast electron and nonlinear materials consisting of centrosymmetric spheres. Based on such a first-principles multiple scattering technique, we have observed negative electron energy losses caused by the SH field and SH Smith-Purcell radiations using finite chains of nonlinear spheres. We have also demonstrated that these new effects can be probed by the electron energy loss spectrum. Our finding contains two aspects of implication. On the one hand, it may thus open a new way to investigate the nonlinear effect of the materials by using the STEM. On the other hand, it is evident that nonlinear Smith-Purcell emissions can be used as a tunable light source for the SHG.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant No.10825416) and the National Key Basic Research Special Foundation of China under Grant 2007CB613205.

References and links

1.

F. J. García de Abajo, “Optical excitations in electron microscopy,” Rev. Mod. Phys. 82(1), 209–275 (2010). [CrossRef]

2.

R. García-Molina, A. Gras-Marti, and R. H. Ritchie, “Excitation of edge modes in the interaction of electron beams with dielectric wedges,” Phys. Rev. B Condens. Matter 31(1), 121–126 (1985). [CrossRef] [PubMed]

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F. J. García de Abajo and A. Howie, “Relativistic electron energy loss and electron-induced photon emission in inhomogeneous dielectrics,” Phys. Rev. Lett. 80(23), 5180–5183 (1998). [CrossRef]

7.

F. J. García de Abajo, “Relativistic energy loss and induced photon emission in the interaction of a dielectric sphere with an external electron beam,” Phys. Rev. B 59(4), 3095–3107 (1999). [CrossRef]

8.

F. J. García de Abajo, “Interaction of radiation and fast electrons with clusters of dielectrics: A multiple scattering approach,” Phys. Rev. Lett. 82(13), 2776–2779 (1999). [CrossRef]

9.

F. J. García de Abajo, A. Rivacoba, N. Zabala, and P. M. Echenique, “Electron energy loss spectroscopy as a probe of two-dimensional photonic crystals,” Phys. Rev. B 68(20), 205105 (2003). [CrossRef]

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12.

J. Xu and X. Zhang, “Cloaking radiation of moving electron beam and relativistic energy loss spectra,” Opt. Express 17(6), 4758–4772 (2009). [CrossRef] [PubMed]

13.

S. J. Smith and E. M. Purcell, “Visible light from localized surface charges moving across a grating,” Phys. Rev. 92(4), 1069–1069 (1953). [CrossRef]

14.

G. Doucas, J. H. Mulvey, M. Omori, J. Walsh, and M. F. Kimmitt, “First observation of Smith-Purcell radiation from relativistic electrons,” Phys. Rev. Lett. 69(12), 1761–1764 (1992). [CrossRef] [PubMed]

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F. J. García de Abajo; “Smith-Purcell radiation emission in aligned nanoparticles,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 61(55B), 5743–5752 (2000). [CrossRef] [PubMed]

16.

N. Yamamoto, K. Araya, and F. J. Garcia de Abajo, “Photon emission from silver particles induced by a high-energy electron beam,” Phys. Rev. B 64(20), 205419 (2001). [CrossRef]

17.

T. F. Heinz, in Nonlinear Surface Electromagnetic Phenomena, edited by H.-E. Ponath and G. I. Stegeman (North-Holland, Amsterdam, 1991).

18.

J. I. Dadap, J. Shan, and T. F. Heinz, “Theory of optical second-harmonic generation from a sphere of centrosymmetric material: small-particle limit,” J. Opt. Soc. Am. B 21(7), 1328–1347 (2004). [CrossRef]

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23.

J. Butet, G. Bachelier, I. Russier-Antoine, C. Jonin, E. Benichou, and P.-F. Brevet, “Interference between selected dipoles and octupoles in the optical second-harmonic generation from spherical gold nanoparticles,” Phys. Rev. Lett. 105(7), 077401 (2010). [CrossRef] [PubMed]

24.

V. K. Valev, A. V. Silhanek, N. Verellen, W. Gillijns, P. Van Dorpe, O. A. Aktsipetrov, G. A. E. Vandenbosch, V. V. Moshchalkov, and T. Verbiest, “Asymmetric optical second-harmonic generation from chiral G-shaped gold nanostructures,” Phys. Rev. Lett. 104(12), 127401 (2010). [CrossRef] [PubMed]

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Y. Pavlyukh and W. Hubner, “Nonlinear Mie scattering from spherical particles,” Phys. Rev. B 70(24), 245434 (2004). [CrossRef]

26.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).

27.

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28.

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OCIS Codes
(160.4330) Materials : Nonlinear optical materials
(260.2110) Physical optics : Electromagnetic optics
(300.2140) Spectroscopy : Emission

ToC Category:
Materials

History
Original Manuscript: August 18, 2011
Revised Manuscript: September 30, 2011
Manuscript Accepted: October 10, 2011
Published: October 28, 2011

Citation
Jinying Xu and Xiangdong Zhang, "Negative electron energy loss and second-harmonic emission of nonlinear nanoparticles," Opt. Express 19, 22999-23007 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-23-22999


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