## 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 [2–12]. 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 [13–16]. In the last decades a remarkable progress has been made on this subject [1–16]. 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 [17–23]. 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 [18–24]. However, many investigations on the SHG focus on the incident plane wave or laser beam [17–25]. 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 ${\epsilon}_{1}(\omega )$ and relative permeability ${\mu}_{1}(\omega )$, placed in a medium having real, frequency-independent permittivity ${\epsilon}_{2}$ and permeability ${\mu}_{2}$. The radiation field of the moving electron can be expanded in terms of the vector spherical waves as [7]

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

${G}_{E,nlm}$are the expansion coefficient of the nonlinear polarization vector ${\overrightarrow{P}}^{(2\omega )}\left(\overrightarrow{r}\right)={\overrightarrow{P}}_{s}^{(2\omega )}+{\overrightarrow{P}}_{b}^{(2\omega )}\text{=}{\displaystyle \sum _{lm}{\text{G}}_{r,nlm}{Y}_{lm}(\theta ,\phi )\widehat{r}+{G}_{M,nlm}{\overrightarrow{X}}_{lm}(\theta ,\phi )+{G}_{E,nlm}\widehat{r}\times {\overrightarrow{X}}_{lm}(\theta ,\phi )}$, ${\overrightarrow{P}}_{s}^{(2\omega )}={\overleftrightarrow{\chi}}_{s}^{(2)}:{\overrightarrow{E}}^{(\omega )}(\overrightarrow{r}){\overrightarrow{E}}^{(\omega )}(\overrightarrow{r})\delta (r-a)$ and ${\overrightarrow{P}}_{b}^{(2\omega )}={\overleftrightarrow{\chi}}_{b}^{(2)}:{\overrightarrow{E}}^{(\omega )}(\overrightarrow{r})\nabla {\overrightarrow{E}}^{(\omega )}(\overrightarrow{r})$ represent surface and bulk nonlinear polarization vectors [17], respectively, ${\overleftrightarrow{\chi}}_{s}^{(2)}$ and ${\overleftrightarrow{\chi}}_{b}^{(2)}$ 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 ${\overleftrightarrow{\chi}}_{s}^{(2)}$ then has three nonvanishing and independent elements: ${\chi}_{\perp \perp \perp}$,${\chi}_{\perp \parallel \parallel}$, ${\chi}_{\parallel \perp \parallel}={\chi}_{\parallel \parallel \perp}$,where $\perp $ and $\parallel $ refer to the local spatial components perpendicular and parallel to the surface. The susceptibility ${\overleftrightarrow{\chi}}_{s}^{(2)}$ can be written, in terms of the unit vectors for the spherical coordinate system, as the triadic [18]

Hence, the nonlinear source polarization for the surface is

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

The coefficients ${b}_{\perp \perp \perp ,}^{nlm}$, ${b}_{\perp \parallel \parallel}^{nlm}$, ${b}_{\parallel \perp \parallel ,\text{M}}^{nlm}$ and ${b}_{\parallel \perp \parallel ,E}^{nlm}$ are related to the nonlinear polarization vectors, which are given as [25]

Here ${k}_{1}$ is the wave vector inside the sphere. The coefficients ${C}_{{l}_{1},{m}_{1},{l}_{2},{m}_{2},l,m}^{(\alpha ,\beta )}$ 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 ${\overrightarrow{E}}^{(\omega )}(\overrightarrow{r})$ represents the FF field, $\overrightarrow{n}$ is a vector normal to the surface, $a$ is the radius of the sphere. As for the SH scattered field (${\overrightarrow{E}}_{sca}^{(2\omega )}\left({\overrightarrow{x}}_{n}\right)$) of the $nth$ sphere, it can be expanded as

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

Similar to the case of the linear scattering [6–8], the SH loss probability ${\Gamma}^{loss}(2\omega )$ for an electron moving with velocity $v$ can be obtained from the induced SH electric field (${\overrightarrow{E}}^{(2\omega )}$) acting on the electron as

Here ${\overrightarrow{H}}^{(2\omega )}$ 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 [6–8]. 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 .

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 ${\epsilon}_{1}(\omega )=1-{\omega}_{p}^{2}/(\omega (\omega +i\nu ))$. Here, ${\omega}_{p}$ and $\nu $ are the plasma and damping frequency, respectively. As specific values for these parameters we choose${\omega}_{p}=1.35\times {10}^{16}rad/s$ and $\nu =0.03{\omega}_{p}$, 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 ${\widehat{\chi}}_{\perp \perp \perp}=2.79\times {10}^{-18}m/V$,${\widehat{\chi}}_{\parallel \parallel \perp}={\widehat{\chi}}_{\parallel \perp \parallel}=3.98\times {10}^{-20}m/V$ and ${\widehat{\chi}}_{\perp \parallel \parallel}=0$ [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 [1–8]. 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 $2\omega =11.2eV$in Fig. 1(b) or $2\omega =11.5eV$ in Fig. 1(e)). In contrast, the negative energy-loss appears (around $2\omega =10.0eV$in Fig. 1(b) or $2\omega =10.7eV$in 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 K_{3}Li_{2}Nb_{5}O_{15} (KLN). The dielectric constant of the KLN is ${\epsilon}_{1}=1+3.708{\omega}^{2}/[{\omega}^{2}-0.04601\times {(2\pi c)}^{2}]$ and the surface second-order susceptibility ${\chi}_{\perp \perp \perp}=12pm/V$ and ${\chi}_{\parallel \perp \parallel}={\chi}_{\perp \parallel \parallel}=11.8pm/V$ [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.

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 .

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.

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