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]. 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 [212]. 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 [1316]. In the last decades a remarkable progress has been made on this subject [116]. 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 [1723]. 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 [1824]. However, many investigations on the SHG focus on the incident plane wave or laser beam [1725]. 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]

Eext(xn)=lm{ik2anlmext,E×jl(k2r)Xlm(r^)+anlmext,Hjl(k2r)Xlm(r^)},
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]. The exact expressions for expansion coefficients anlmext,E and anlmext,H have been given in [7]. 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].

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)}
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,
Anlmout,H=8πωcGM,nlmjl(K1r)ε2(2ω)K2rjl(K1r)r[rhl(1)(K2r)]ε1(2ω)K1rhl(1)(K2r)r[rjl(K1r)]|r=a,
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], 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, 18, 27]. 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, 18]. 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]

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

Hence, the nonlinear source polarization for the surface is

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

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,
GM,nlm=χb,Mnlm,
GE,nlm=χb,Enlm.

The coefficients b,nlm, bnlm, b,Mnlm and b,Enlm are related to the nonlinear polarization vectors, which are given as [25]

bnlm=l1,m1l2,m2Anl1m1(1)Anl2m2(1)Cl1,m1,l2,m2,l,m(1,1),
bnlm=l1,m1l2,m2(Anl1m1(1)Anl2m2(1)Cl1,m1,l2,m2,l,m(1,1)+Anl1m1(0)Alnl2m2(0)Cl1,m1,l2,m2,l,m(0,0)),
b,Mnlm=2l1,m1l2,m2(Anl1m1(0)Anl2m2(1)Cl1,m1,l2,m2,l,m(1,1)+Anl1m1(1)Alnl2m2(1)Cl1,m1,l2,m2,l,m(0,1)),
b,Enlm=2il1,m1l2,m2(Anl1m1(0)Anl2m2(1)Cl1,m1,l2,m2,l,m(1,0)+Anl1m1(1)Anl2m2(1)Cl1,m1,l2,m2,l,m(1,1))
with

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

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].

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)}.

With the use of the addition theorem [29] and the boundary conditions, we obtain

nlm[δnnδllδmmTlE(Ωnlm,nlmEEAnlmsca,E+Ωnlm,nlmEHAnlmsca,H)]=nlmTlE[Ωnlm,nlmEEAnlmout,E+Ωnlm,nlmEHAnlmout,H],
nlm[δnnδllδmmTlH(Ωnlm,nlmHEAnlmsca,E+Ωnlm,nlmHHAnlmsca,H)]=nlmTlH[Ωnlm,nlmEEAnlmout,E+Ωnlm,nlmEHAnlmout,H],
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, 8]. 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 [68], 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)}
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^}.

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 [68]. 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.

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Figure 1 (a) and (d) display the FF loss probabilities (solid lines) and induced photon emission probabilities (dashed lines) for 0.7c electrons moving near a set of spheres as shown in the insets, while the corresponding SH loss probabilities are given in Fig. 1 (b) and (e), the SH photon emission probabilities are shown in Fig. 1 (c) and (f). Here we assume that all spheres are made of the same material. The permittivity of the material is described by the Drude model ε1(ω)=1ωp2/(ω(ω+iν)). Here, ωp and ν are the plasma and damping frequency, respectively. As specific values for these parameters we chooseωp=1.35×1016rad/s and ν=0.03ωp, which correspond to the metal Ag [30]. Due to the SHG is dominated by the surface component for the present system, in our calculations we take the surface second-order susceptibility χ^=2.79×1018m/V,χ^=χ^=3.98×1020m/V and χ^=0 [31]. For the FF spectra, the surface plasmon-polariton modes and couple features among the spheres exhibit clearly, which are agreement with those of the previous investigations [18]. In contrast to the FF case, negative loss probabilities are observed in the SH spectra although the photon emission probabilities are always positive. This means that the SH field cannot only cause the energy loss of moving electrons, but also it can increase the energy of the moving electron at some frequencies. The above results are for the case of the permittivity of the material being taken as Drude model. We have also performed calculations with different form of the permittivity such as those given in [32], the similar phenomena can also be observed.

Such a phenomenon originates from various excitations of the SH field when a fast electron moves along a straight-line trajectory passing near the nonlinear materials. When a fast electron moves with velocity v and passes near the object, it may engender the radiation with wide frequency spectrum. The energy loss suffered by the fast electron is related to the force exerted by the radiation electric field acting on it (see Eq. (21)). The interaction between the radiation field and the object consisting of the nonlinear materials can cause the generation of the SH field. In contrast to the FF field, the change of the SH field is more abundant. The direction of the SH electric field may be identical with or opposite to the moving direction of the electron beam, which depends on the characteristic of the nonlinear object and the frequency. When the direction of the SH electric field is identical with the moving direction of the electron beam, the radiation electric field acting on the electron beam results in positive energy loss (around 2ω=11.2eVin Fig. 1(b) or 2ω=11.5eV in Fig. 1(e)). In contrast, the negative energy-loss appears (around 2ω=10.0eVin Fig. 1(b) or 2ω=10.7eVin Fig. 1(e)). However, we would like to point out that the energy conservation still be preserved for such a case. This is similar to the case of the generation of the SH field from the nonlinear materials, which the energy conservation is preserved.

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) . 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]. 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.

 

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.

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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.

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Figure 3 illustrates the distribution of the photon emission probability in polar angle as a function of photon energy for finite chain of N aligned KLN spheres. Different values of N have been considered, as shown by labels. Figure 3 (a), (b) and (c) correspond to the case of the FF, whereas (d), (e) and (f) to the SH. Comparing them, we find that the SH emissions exhibit different dependence of the angle in comparison with those of the FF. For the case of the FF, the Smith-Purcell radiations produced by interaction of fast electron beams with finite and infinite strings of sphere have been discussed very well in the previous works [8, 15]. Our present results are agreement with them. However, the nonlinear Smith-Purcell radiation caused by the SH field has never been observed before. Our calculated results demonstrate such an effect and show the possibility of using this effect to produce tunable SH emissions. The above discussions focus on only the certain electron velocity and impact parameters. In fact, if we change them, similar phenomena can also be observed.

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]  

3. T. L. Ferrell and P. M. Echenique, “Generation of surface excitations on dielectric spheres by an external electron beam,” Phys. Rev. Lett. 55(14), 1526–1529 (1985). [CrossRef]   [PubMed]  

4. A. Rivacoba, N. Zabala, and P. M. Echenique, “Theory of energy loss in scanning transmission electron microscopy of supported small particles,” Phys. Rev. Lett. 69(23), 3362–3365 (1992). [CrossRef]   [PubMed]  

5. J. B. Pendry and L. Martín-Moreno, “Energy loss by charged particles in complex media,” Phys. Rev. B Condens. Matter 50(8), 5062–5073 (1994). [CrossRef]   [PubMed]  

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]  

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]  

10. T. Ochiai and K. Ohtaka, “Relativistic electron energy loss and induced radiation emission in two-dimensional metallic photonic crystals.I. Formalism and surface plasmon polariton,” Phys. Rev. B 69(12), 125106 (2004). [CrossRef]  

11. J. Xu, Y. Dong, and X. Zhang, “Electromagnetic interactions between a fast electron beam and metamaterial cloaks,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 78(4), 046601 (2008). [CrossRef]   [PubMed]  

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]  

15. 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]  

19. P. C. Ray, “Size and shape dependent second order nonlinear optical properties of nanomaterials and their application in biological and chemical sensing,” Chem. Rev. 110(9), 5332–5365 (2010). [CrossRef]   [PubMed]  

20. S. Kujala, B. K. Canfield, M. Kauranen, Y. Svirko, and J. Turunen, “Multipole interference in the second-harmonic optical radiation from gold nanoparticles,” Phys. Rev. Lett. 98(16), 167403 (2007). [CrossRef]   [PubMed]  

21. Y. Pu, R. Grange, C.-L. Hsieh, and D. Psaltis, “Nonlinear optical properties of core-shell nanocavities for enhanced second-harmonic generation,” Phys. Rev. Lett. 104(20), 207402 (2010). [CrossRef]   [PubMed]  

22. J. U. Furst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, C. Marquardt, and G. Leuchs, “Naturally phase-matched second-harmonic generation in a whispering-gallery-mode resonator,” Phys. Rev. Lett. 104(15), 153901 (2010). [CrossRef]   [PubMed]  

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]  

25. 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. 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]  

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

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

30. M. A. Ordal, R. J. Bell, R. W. Alexander Jr, L. L. Long, and M. R. Querry, “Optical properties of fourteen metals in the infrared and far infrared: Al, Co, Cu, Au, Fe, Pb, Mo, Ni, Pd, Pt, Ag, Ti, V, and W,” Appl. Opt. 24(24), 4493–4499 (1985). [CrossRef]   [PubMed]  

31. D. Krause, C. W. Teplin, and C. T. Rogers, “Optical surface second harmonic measurements of isotropic thin-film metals: Gold, silver, copper, aluminum, and tantalum,” J. Appl. Phys. 96(7), 3626–3634 (2004). [CrossRef]  

32. E. D. Palik, Handbook of optical constants of solids (Academic, Orlando, 1985).

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

References

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  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]
  3. T. L. Ferrell and P. M. Echenique, “Generation of surface excitations on dielectric spheres by an external electron beam,” Phys. Rev. Lett. 55(14), 1526–1529 (1985).
    [CrossRef] [PubMed]
  4. A. Rivacoba, N. Zabala, and P. M. Echenique, “Theory of energy loss in scanning transmission electron microscopy of supported small particles,” Phys. Rev. Lett. 69(23), 3362–3365 (1992).
    [CrossRef] [PubMed]
  5. J. B. Pendry and L. Martín-Moreno, “Energy loss by charged particles in complex media,” Phys. Rev. B Condens. Matter 50(8), 5062–5073 (1994).
    [CrossRef] [PubMed]
  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]
  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]
  10. T. Ochiai and K. Ohtaka, “Relativistic electron energy loss and induced radiation emission in two-dimensional metallic photonic crystals.I. Formalism and surface plasmon polariton,” Phys. Rev. B 69(12), 125106 (2004).
    [CrossRef]
  11. J. Xu, Y. Dong, and X. Zhang, “Electromagnetic interactions between a fast electron beam and metamaterial cloaks,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 78(4), 046601 (2008).
    [CrossRef] [PubMed]
  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]
  15. 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]
  19. P. C. Ray, “Size and shape dependent second order nonlinear optical properties of nanomaterials and their application in biological and chemical sensing,” Chem. Rev. 110(9), 5332–5365 (2010).
    [CrossRef] [PubMed]
  20. S. Kujala, B. K. Canfield, M. Kauranen, Y. Svirko, and J. Turunen, “Multipole interference in the second-harmonic optical radiation from gold nanoparticles,” Phys. Rev. Lett. 98(16), 167403 (2007).
    [CrossRef] [PubMed]
  21. Y. Pu, R. Grange, C.-L. Hsieh, and D. Psaltis, “Nonlinear optical properties of core-shell nanocavities for enhanced second-harmonic generation,” Phys. Rev. Lett. 104(20), 207402 (2010).
    [CrossRef] [PubMed]
  22. J. U. Furst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, C. Marquardt, and G. Leuchs, “Naturally phase-matched second-harmonic generation in a whispering-gallery-mode resonator,” Phys. Rev. Lett. 104(15), 153901 (2010).
    [CrossRef] [PubMed]
  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]
  25. 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. 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]
  28. D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of Angular Momentum (World Scientific, Singapore, 1988).
  29. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, (Dover, New York, 1972), p. 363.
  30. M. A. Ordal, R. J. Bell, R. W. Alexander, L. L. Long, and M. R. Querry, “Optical properties of fourteen metals in the infrared and far infrared: Al, Co, Cu, Au, Fe, Pb, Mo, Ni, Pd, Pt, Ag, Ti, V, and W,” Appl. Opt. 24(24), 4493–4499 (1985).
    [CrossRef] [PubMed]
  31. D. Krause, C. W. Teplin, and C. T. Rogers, “Optical surface second harmonic measurements of isotropic thin-film metals: Gold, silver, copper, aluminum, and tantalum,” J. Appl. Phys. 96(7), 3626–3634 (2004).
    [CrossRef]
  32. E. D. Palik, Handbook of optical constants of solids (Academic, Orlando, 1985).
  33. D. N. Nikogosyan, Nonlinear optical crystals: A complete survey, Springer New York, USA (2003).

2010 (6)

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

P. C. Ray, “Size and shape dependent second order nonlinear optical properties of nanomaterials and their application in biological and chemical sensing,” Chem. Rev. 110(9), 5332–5365 (2010).
[CrossRef] [PubMed]

Y. Pu, R. Grange, C.-L. Hsieh, and D. Psaltis, “Nonlinear optical properties of core-shell nanocavities for enhanced second-harmonic generation,” Phys. Rev. Lett. 104(20), 207402 (2010).
[CrossRef] [PubMed]

J. U. Furst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, C. Marquardt, and G. Leuchs, “Naturally phase-matched second-harmonic generation in a whispering-gallery-mode resonator,” Phys. Rev. Lett. 104(15), 153901 (2010).
[CrossRef] [PubMed]

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]

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]

2009 (2)

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]

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]

2008 (1)

J. Xu, Y. Dong, and X. Zhang, “Electromagnetic interactions between a fast electron beam and metamaterial cloaks,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 78(4), 046601 (2008).
[CrossRef] [PubMed]

2007 (1)

S. Kujala, B. K. Canfield, M. Kauranen, Y. Svirko, and J. Turunen, “Multipole interference in the second-harmonic optical radiation from gold nanoparticles,” Phys. Rev. Lett. 98(16), 167403 (2007).
[CrossRef] [PubMed]

2004 (4)

T. Ochiai and K. Ohtaka, “Relativistic electron energy loss and induced radiation emission in two-dimensional metallic photonic crystals.I. Formalism and surface plasmon polariton,” Phys. Rev. B 69(12), 125106 (2004).
[CrossRef]

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

D. Krause, C. W. Teplin, and C. T. Rogers, “Optical surface second harmonic measurements of isotropic thin-film metals: Gold, silver, copper, aluminum, and tantalum,” J. Appl. Phys. 96(7), 3626–3634 (2004).
[CrossRef]

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]

2003 (1)

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]

2001 (1)

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]

2000 (1)

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]

1999 (2)

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]

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]

1998 (1)

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]

1994 (1)

J. B. Pendry and L. Martín-Moreno, “Energy loss by charged particles in complex media,” Phys. Rev. B Condens. Matter 50(8), 5062–5073 (1994).
[CrossRef] [PubMed]

1992 (2)

A. Rivacoba, N. Zabala, and P. M. Echenique, “Theory of energy loss in scanning transmission electron microscopy of supported small particles,” Phys. Rev. Lett. 69(23), 3362–3365 (1992).
[CrossRef] [PubMed]

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]

1985 (3)

M. A. Ordal, R. J. Bell, R. W. Alexander, L. L. Long, and M. R. Querry, “Optical properties of fourteen metals in the infrared and far infrared: Al, Co, Cu, Au, Fe, Pb, Mo, Ni, Pd, Pt, Ag, Ti, V, and W,” Appl. Opt. 24(24), 4493–4499 (1985).
[CrossRef] [PubMed]

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]

T. L. Ferrell and P. M. Echenique, “Generation of surface excitations on dielectric spheres by an external electron beam,” Phys. Rev. Lett. 55(14), 1526–1529 (1985).
[CrossRef] [PubMed]

1953 (1)

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]

Ahorinta, R.

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]

Aktsipetrov, O. A.

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]

Albers, W. M.

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]

Alexander, R. W.

Andersen, U. L.

J. U. Furst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, C. Marquardt, and G. Leuchs, “Naturally phase-matched second-harmonic generation in a whispering-gallery-mode resonator,” Phys. Rev. Lett. 104(15), 153901 (2010).
[CrossRef] [PubMed]

Araya, K.

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]

Bachelier, G.

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]

Bell, R. J.

Benichou, E.

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]

Brevet, P.-F.

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]

Butet, J.

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]

Canfield, B. K.

S. Kujala, B. K. Canfield, M. Kauranen, Y. Svirko, and J. Turunen, “Multipole interference in the second-harmonic optical radiation from gold nanoparticles,” Phys. Rev. Lett. 98(16), 167403 (2007).
[CrossRef] [PubMed]

Dadap, J. I.

Dong, Y.

J. Xu, Y. Dong, and X. Zhang, “Electromagnetic interactions between a fast electron beam and metamaterial cloaks,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 78(4), 046601 (2008).
[CrossRef] [PubMed]

Doucas, G.

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]

Echenique, P. M.

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]

A. Rivacoba, N. Zabala, and P. M. Echenique, “Theory of energy loss in scanning transmission electron microscopy of supported small particles,” Phys. Rev. Lett. 69(23), 3362–3365 (1992).
[CrossRef] [PubMed]

T. L. Ferrell and P. M. Echenique, “Generation of surface excitations on dielectric spheres by an external electron beam,” Phys. Rev. Lett. 55(14), 1526–1529 (1985).
[CrossRef] [PubMed]

Elser, D.

J. U. Furst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, C. Marquardt, and G. Leuchs, “Naturally phase-matched second-harmonic generation in a whispering-gallery-mode resonator,” Phys. Rev. Lett. 104(15), 153901 (2010).
[CrossRef] [PubMed]

Ferrell, T. L.

T. L. Ferrell and P. M. Echenique, “Generation of surface excitations on dielectric spheres by an external electron beam,” Phys. Rev. Lett. 55(14), 1526–1529 (1985).
[CrossRef] [PubMed]

Furst, J. U.

J. U. Furst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, C. Marquardt, and G. Leuchs, “Naturally phase-matched second-harmonic generation in a whispering-gallery-mode resonator,” Phys. Rev. Lett. 104(15), 153901 (2010).
[CrossRef] [PubMed]

Garcia de Abajo, F. J.

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]

García de Abajo, F. J.

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

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]

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]

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]

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]

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]

García-Molina, R.

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]

Gillijns, W.

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]

Grange, R.

Y. Pu, R. Grange, C.-L. Hsieh, and D. Psaltis, “Nonlinear optical properties of core-shell nanocavities for enhanced second-harmonic generation,” Phys. Rev. Lett. 104(20), 207402 (2010).
[CrossRef] [PubMed]

Gras-Marti, A.

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]

Heinz, T. F.

Howie, A.

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]

Hsieh, C.-L.

Y. Pu, R. Grange, C.-L. Hsieh, and D. Psaltis, “Nonlinear optical properties of core-shell nanocavities for enhanced second-harmonic generation,” Phys. Rev. Lett. 104(20), 207402 (2010).
[CrossRef] [PubMed]

Hubner, W.

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

Jonin, C.

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]

Kauranen, M.

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]

S. Kujala, B. K. Canfield, M. Kauranen, Y. Svirko, and J. Turunen, “Multipole interference in the second-harmonic optical radiation from gold nanoparticles,” Phys. Rev. Lett. 98(16), 167403 (2007).
[CrossRef] [PubMed]

Kimmitt, M. F.

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]

Krause, D.

D. Krause, C. W. Teplin, and C. T. Rogers, “Optical surface second harmonic measurements of isotropic thin-film metals: Gold, silver, copper, aluminum, and tantalum,” J. Appl. Phys. 96(7), 3626–3634 (2004).
[CrossRef]

Kujala, S.

S. Kujala, B. K. Canfield, M. Kauranen, Y. Svirko, and J. Turunen, “Multipole interference in the second-harmonic optical radiation from gold nanoparticles,” Phys. Rev. Lett. 98(16), 167403 (2007).
[CrossRef] [PubMed]

Lassen, M.

J. U. Furst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, C. Marquardt, and G. Leuchs, “Naturally phase-matched second-harmonic generation in a whispering-gallery-mode resonator,” Phys. Rev. Lett. 104(15), 153901 (2010).
[CrossRef] [PubMed]

Leuchs, G.

J. U. Furst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, C. Marquardt, and G. Leuchs, “Naturally phase-matched second-harmonic generation in a whispering-gallery-mode resonator,” Phys. Rev. Lett. 104(15), 153901 (2010).
[CrossRef] [PubMed]

Long, L. L.

Marquardt, C.

J. U. Furst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, C. Marquardt, and G. Leuchs, “Naturally phase-matched second-harmonic generation in a whispering-gallery-mode resonator,” Phys. Rev. Lett. 104(15), 153901 (2010).
[CrossRef] [PubMed]

Martín-Moreno, L.

J. B. Pendry and L. Martín-Moreno, “Energy loss by charged particles in complex media,” Phys. Rev. B Condens. Matter 50(8), 5062–5073 (1994).
[CrossRef] [PubMed]

Moshchalkov, V. V.

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]

Mulvey, J. H.

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]

Ochiai, T.

T. Ochiai and K. Ohtaka, “Relativistic electron energy loss and induced radiation emission in two-dimensional metallic photonic crystals.I. Formalism and surface plasmon polariton,” Phys. Rev. B 69(12), 125106 (2004).
[CrossRef]

Ohtaka, K.

T. Ochiai and K. Ohtaka, “Relativistic electron energy loss and induced radiation emission in two-dimensional metallic photonic crystals.I. Formalism and surface plasmon polariton,” Phys. Rev. B 69(12), 125106 (2004).
[CrossRef]

Omori, M.

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]

Ordal, M. A.

Pavlyukh, Y.

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

Pendry, J. B.

J. B. Pendry and L. Martín-Moreno, “Energy loss by charged particles in complex media,” Phys. Rev. B Condens. Matter 50(8), 5062–5073 (1994).
[CrossRef] [PubMed]

Psaltis, D.

Y. Pu, R. Grange, C.-L. Hsieh, and D. Psaltis, “Nonlinear optical properties of core-shell nanocavities for enhanced second-harmonic generation,” Phys. Rev. Lett. 104(20), 207402 (2010).
[CrossRef] [PubMed]

Pu, Y.

Y. Pu, R. Grange, C.-L. Hsieh, and D. Psaltis, “Nonlinear optical properties of core-shell nanocavities for enhanced second-harmonic generation,” Phys. Rev. Lett. 104(20), 207402 (2010).
[CrossRef] [PubMed]

Purcell, E. M.

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]

Querry, M. R.

Ray, P. C.

P. C. Ray, “Size and shape dependent second order nonlinear optical properties of nanomaterials and their application in biological and chemical sensing,” Chem. Rev. 110(9), 5332–5365 (2010).
[CrossRef] [PubMed]

Ritchie, R. H.

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]

Rivacoba, A.

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]

A. Rivacoba, N. Zabala, and P. M. Echenique, “Theory of energy loss in scanning transmission electron microscopy of supported small particles,” Phys. Rev. Lett. 69(23), 3362–3365 (1992).
[CrossRef] [PubMed]

Rodríguez, F. J.

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]

Rogers, C. T.

D. Krause, C. W. Teplin, and C. T. Rogers, “Optical surface second harmonic measurements of isotropic thin-film metals: Gold, silver, copper, aluminum, and tantalum,” J. Appl. Phys. 96(7), 3626–3634 (2004).
[CrossRef]

Russier-Antoine, I.

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]

Shan, J.

Silhanek, A. V.

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]

Sipe, J. E.

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]

Smith, S. J.

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]

Strekalov, D. V.

J. U. Furst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, C. Marquardt, and G. Leuchs, “Naturally phase-matched second-harmonic generation in a whispering-gallery-mode resonator,” Phys. Rev. Lett. 104(15), 153901 (2010).
[CrossRef] [PubMed]

Svirko, Y.

S. Kujala, B. K. Canfield, M. Kauranen, Y. Svirko, and J. Turunen, “Multipole interference in the second-harmonic optical radiation from gold nanoparticles,” Phys. Rev. Lett. 98(16), 167403 (2007).
[CrossRef] [PubMed]

Teplin, C. W.

D. Krause, C. W. Teplin, and C. T. Rogers, “Optical surface second harmonic measurements of isotropic thin-film metals: Gold, silver, copper, aluminum, and tantalum,” J. Appl. Phys. 96(7), 3626–3634 (2004).
[CrossRef]

Turunen, J.

S. Kujala, B. K. Canfield, M. Kauranen, Y. Svirko, and J. Turunen, “Multipole interference in the second-harmonic optical radiation from gold nanoparticles,” Phys. Rev. Lett. 98(16), 167403 (2007).
[CrossRef] [PubMed]

Valev, V. K.

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]

Van Dorpe, P.

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]

Vandenbosch, G. A. E.

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]

Verbiest, T.

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]

Verellen, N.

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]

Walsh, J.

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]

Wang, F. X.

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]

Xu, J.

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]

J. Xu, Y. Dong, and X. Zhang, “Electromagnetic interactions between a fast electron beam and metamaterial cloaks,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 78(4), 046601 (2008).
[CrossRef] [PubMed]

Yamamoto, N.

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]

Zabala, N.

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]

A. Rivacoba, N. Zabala, and P. M. Echenique, “Theory of energy loss in scanning transmission electron microscopy of supported small particles,” Phys. Rev. Lett. 69(23), 3362–3365 (1992).
[CrossRef] [PubMed]

Zhang, X.

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]

J. Xu, Y. Dong, and X. Zhang, “Electromagnetic interactions between a fast electron beam and metamaterial cloaks,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 78(4), 046601 (2008).
[CrossRef] [PubMed]

Appl. Opt. (1)

Chem. Rev. (1)

P. C. Ray, “Size and shape dependent second order nonlinear optical properties of nanomaterials and their application in biological and chemical sensing,” Chem. Rev. 110(9), 5332–5365 (2010).
[CrossRef] [PubMed]

J. Appl. Phys. (1)

D. Krause, C. W. Teplin, and C. T. Rogers, “Optical surface second harmonic measurements of isotropic thin-film metals: Gold, silver, copper, aluminum, and tantalum,” J. Appl. Phys. 96(7), 3626–3634 (2004).
[CrossRef]

J. Opt. Soc. Am. B (1)

Opt. Express (1)

Phys. Rev. (1)

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]

Phys. Rev. B (6)

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]

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]

T. Ochiai and K. Ohtaka, “Relativistic electron energy loss and induced radiation emission in two-dimensional metallic photonic crystals.I. Formalism and surface plasmon polariton,” Phys. Rev. B 69(12), 125106 (2004).
[CrossRef]

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]

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]

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

Phys. Rev. B Condens. Matter (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]

J. B. Pendry and L. Martín-Moreno, “Energy loss by charged particles in complex media,” Phys. Rev. B Condens. Matter 50(8), 5062–5073 (1994).
[CrossRef] [PubMed]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

J. Xu, Y. Dong, and X. Zhang, “Electromagnetic interactions between a fast electron beam and metamaterial cloaks,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 78(4), 046601 (2008).
[CrossRef] [PubMed]

Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics (1)

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]

Phys. Rev. Lett. (10)

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]

S. Kujala, B. K. Canfield, M. Kauranen, Y. Svirko, and J. Turunen, “Multipole interference in the second-harmonic optical radiation from gold nanoparticles,” Phys. Rev. Lett. 98(16), 167403 (2007).
[CrossRef] [PubMed]

Y. Pu, R. Grange, C.-L. Hsieh, and D. Psaltis, “Nonlinear optical properties of core-shell nanocavities for enhanced second-harmonic generation,” Phys. Rev. Lett. 104(20), 207402 (2010).
[CrossRef] [PubMed]

J. U. Furst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, C. Marquardt, and G. Leuchs, “Naturally phase-matched second-harmonic generation in a whispering-gallery-mode resonator,” Phys. Rev. Lett. 104(15), 153901 (2010).
[CrossRef] [PubMed]

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]

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]

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]

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]

T. L. Ferrell and P. M. Echenique, “Generation of surface excitations on dielectric spheres by an external electron beam,” Phys. Rev. Lett. 55(14), 1526–1529 (1985).
[CrossRef] [PubMed]

A. Rivacoba, N. Zabala, and P. M. Echenique, “Theory of energy loss in scanning transmission electron microscopy of supported small particles,” Phys. Rev. Lett. 69(23), 3362–3365 (1992).
[CrossRef] [PubMed]

Rev. Mod. Phys. (1)

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

Other (6)

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

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

E. D. Palik, Handbook of optical constants of solids (Academic, Orlando, 1985).

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

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

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

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Figures (3)

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.

Fig. 2
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.

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.

Equations (22)

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E ext ( x n )= lm { i k 2 a nlm ext,E × j l ( k 2 r ) X lm ( r ^ )+ a nlm ext,H j l ( k 2 r ) X lm ( r ^ ) } ,
E out (2ω) ( x n )= lm { A nlm out,E H Elm (2ω) ( x n )+ A nlm out,H H Hlm (2ω) ( x n ) }
A nlm out,E = 4πi G E,nlm r [ r j l ( K 1 r) ]4π l(l+1) G r,nlm j l ( K 1 r) ε 1 (2ω) K 2 j l ( K 1 r) r [r h l (1) ( K 2 r)] ε 2 (2ω) K 2 h l (1) ( K 2 r) r [r j l ( K 1 r)] | r=a ,
A nlm out,H = 8πω c G M,nlm j l ( K 1 r) ε 2 (2ω) K 2 r j l ( K 1 r) r [r h l (1) ( K 2 r)] ε 1 (2ω) K 1 r h l (1) ( K 2 r) r [r j l ( K 1 r)] | r=a ,
χ s (2) = χ r ^ r ^ r ^ + χ r ^ ( θ ^ θ ^ + φ ^ φ ^ )+ χ ( θ ^ r ^ θ ^ + φ ^ r ^ φ ^ + θ ^ θ ^ r ^ + φ ^ φ ^ r ^ ).
P s (2ω) = r ^ ( χ E r (ω) E r (ω) + χ E t (ω) E r (ω) )+2 χ E r (ω) E t (ω) .
G r,nlm = χ b nlm + χ b nlm ,
G M,nlm = χ b ,M nlm ,
G E,nlm = χ b ,E nlm .
b nlm = l 1 , m 1 l 2 , m 2 A n l 1 m 1 (1) A n l 2 m 2 (1) C l 1 , m 1 , l 2 , m 2 ,l,m (1,1) ,
b nlm = l 1 , m 1 l 2 , m 2 ( A n l 1 m 1 (1) A n l 2 m 2 (1) C l 1 , m 1 , l 2 , m 2 ,l,m (1,1) + A n l 1 m 1 (0) A ln l 2 m 2 (0) C l 1 , m 1 , l 2 , m 2 ,l,m (0,0) ) ,
b ,M nlm =2 l 1 , m 1 l 2 , m 2 ( A n l 1 m 1 (0) A n l 2 m 2 (1) C l 1 , m 1 , l 2 , m 2 ,l,m (1,1) + A n l 1 m 1 (1) A ln l 2 m 2 (1) C l 1 , m 1 , l 2 , m 2 ,l,m (0,1) ) ,
b ,E nlm =2i l 1 , m 1 l 2 , m 2 ( A n l 1 m 1 (0) A n l 2 m 2 (1) C l 1 , m 1 , l 2 , m 2 ,l,m (1,0) + A n l 1 m 1 (1) A n l 2 m 2 (1) C l 1 , m 1 , l 2 , m 2 ,l,m (1,1) )
A nlm (1) = 1 k 1 a d[r j l ( k 1 r)] dr | r=a a nlm ext,E ,
A nlm (0) = j l ( k 1 a) a nlm ext,H ,
A nlm (1) = ε 1 l(l+1) k 1 a j l ( k 1 a) a nlm ext,E .
E sca (2ω) ( x n )= lm { A nlm sca,E H Elm (2ω) ( x n )+ A nlm sca,H H Hlm (2ω) ( x n ) } ,
E loc (2ω) ( x n )= n n lm { ( A nlm out,E + A n lm sca,E ) H Elm (2ω) ( x n )+( A nlm out,H + A n lm sca,H ) H Hlm (2ω) ( x n ) } .
n l m [ δ n n δ l l δ m m T l E ( Ω nlm, n l m EE A n l m sca,E + Ω nlm, n l m EH A n l m sca,H ) ] = n l m T l E [ Ω nlm, n l m EE A n l m out,E + Ω nlm, n l m EH A n l m out,H ] ,
n l m [ δ n n δ l l δ m m T l H ( Ω nlm, n l m HE A n l m sca,E + Ω nlm, n l m HH A n l m sca,H ) ] = n l m T l H [ Ω nlm, n l m EE A n l m out,E + Ω nlm, n l m EH A n l m out,H ] ,
Γ loss (2ω)= 1 πϖ dtRe{ e i2ωt v E (2ω) ( r t ,2ω)}
Γ rad (2ω,Ω)= lim r r 2 4 π 2 k Re{[ E (2ω) (2ω)× H (2ω) (2ω)] r ^ }.

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