1. Introduction

The magnetic response of naturally occurring materials is negligible in the visible-IR regime for fundamental reasons [1]. This severely limits the functionality of optics in that regime: were a strong magnetic response possible, many applications, such as optical cloaking [2–4] and perfect lensing [5, 6], could be achievable. It is possible to make the magnetic response stronger by using artificial metal nanostructures [7–11]. While great strides have been made towards achieving a stronger magnetic response in the visible regime [12–14], still further work is required to reduce the resonance wavelength, to increase the strength of the response and to reduce material losses. To aid in this continuing effort, it is highly desirable to have a tool to probe the magnetic response of metal nanostructures at the nanometer scale.

Electron energy loss spectroscopy (EELS) has long been used to probe the response of matter [15–18]. For example, the experimental measurement of a surface plasmon is commonly performed by EELS [19–21]. More recently, EELS has allowed for mapping out the electronic response of metal nanostructures, allowing for the probing of the distribution of localized surface plasmons [22–27]. Of particular interest, EELS can probe dark modes that are not coupled to radiation and cannot be measured with conventional methods [28].

Here, we propose using the well-established tool of electron energy loss scattering, but with the twist of vortex electron beams, or vortex-EELS, to map out the magnetic response of materials at the nanometer scale. Vortex electron beams have recently been demonstrated using diffractive phase-plates in transmission electron microscope setups [29, 30]. These electron beams possess orbital angular momentum (OAM), which has proven to be extremely useful in the field of optics [31, 32].

In this work, we show that these vortex electron beams provide an opportunity for mapping out the near-field magnetic response of metal nanostructures. First we present a semi-classical theory that shows how the spiraling electron trajectory creates a moving effective magnetic charge, which in turn can be used to probe the magnetic response of metal nanostructures at the nanometer scale. We present example calculations for the well-known split ring resonator (SRR) metal nanostructure, which show that the scattering signal achieved from vortex-EELS is within an order of magnitude of conventional EELS. These findings are extremely promising for future work to map out the local magnetic response at the nanometer scale.

2. Vortex-EELS Theory

The vortex-EELS theory is similar to past approaches to EELS that consider the energy loss from a moving point electron charge [33], except that here we consider a spiraling trajectory. This spiraling trajectory creates a magnetic current, that can be thought of as arising from a moving magnetic charge. Here we show how the magnetic charge and magnetic current density is related to the angular momentum and the trajectory of the electron beam. We then show how the energy loss spectrum can be obtained from the magnetic Green function that is induced from the magnetic charge.

2.1. Effective Magnetic Charge

The electron spiral has radius of a, while moving with constant velocity in the z-direction of v_z. The total electron velocity is

2.2. Vortex-EELS Scattering Loss Probability

The two differences between the present vortex-EELS theory and the conventional EELS theory are that: 1) the effective magnetic charge replaces the electric charge, and 2) the duality of the electric and the magnetic fields allows any calculation involving electric fields, permittivities and electric field Green functions to be replaced with magnetic fields, permeabilities and magnetic Green functions. The energy loss, ΔE, from a scattering object can be found from the Poynting theorem in the usual way [35], allowing for the inclusion of magnetic current:

Inserting for j_m(r, t) into Eq. (10), we have

The magnetic field can be obtained from the magnetic Green function (or tensor), Ḡ^H, through

Noticing that z_e(t) = v_zt, the time integral of Eq. (15) can be expressed as an integral along magnetic charge trajectory, through

3. Vortex-EELS for a Split Ring Resonator using FDTD

To quantify the vortex-EELS probability for a nanometric structure, we consider a U-shaped SRR. Figure 1 shows a schematic of the SRR under consideration. The dimensions shown are chosen to match a past work [14].

 

Fig. 1 Schematic of U-shaped split ring resonator showing dimensions and axes.

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3.1. FDTD Analysis of the SRR

Figure 2 shows transmission and reflection of an array of the SRRs (see Fig. 1), with periodicity 315 nm along the x-direction and 330 nm along the y-direction. The transmission and reflection was calculated using FDTD simulations with a normally-incident plane wave source of x-polarization. The response of gold is taken from experiments [38]. We use a uniform grid around the nanostructure with mesh size 4 nm in all directions, Bloch boundaries in the x- and y-directions and perfectly matched layer boundaries in the positive and negative z-direction. Figure 2 agrees quantitatively with past calculations using the finite element calculation method for the same structure [14].

 

Fig. 2 Normal incidence transmission and reflection spectra for horizontal polarization.

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3.2. Vortex-EELS Maps for the SRR

Figure 3 shows the vortex-EELS scattering probability for a 100 keV beam with R₀ = (0, 90 nm), as calculated using Eq. (17). The Green tensor component, $G_{zz}^{\text{ind}}$, was calculated using a z-oriented magnetic dipole source in FDTD with perfectly matched layer boundary conditions, and monitoring the z component of the magnetic field. Simulations were carried out for 30 positions of the magnetic dipole source positions with a 4 nm spacing. For dipoles within the lossy material, FDTD properly accounts for regularization of the Green tensor [39]. The radius of the vortex beam is taken to be 1 nm, which has been shown in recent experiments [30, 40]. The topological charge is taken as n = 1. The peak of the electron loss occurs at 0.863 eV, or 1437 nm, which is around the magnetic resonance in Fig. 2.

 

Fig. 3 Calculated vortex electron beam energy loss scattering probability of the SRR structure in Fig. 1 for 100 keV beam with transverse co-ordinate R₀ = (0, 90 nm).

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Figure 4 shows the vortex-EELS probability map for the energy loss peak at 0.863 eV with a 100 keV beam. The location of the maximum scattering probability is in the middle of the SRR, close to the metal edge. An additional video (Media 1) is provided showing the evolution of the map approaching the resonance.

 

Fig. 4 Map of vortex-EELS probability as a function of position for the 0.863 eV loss peak and a 100 keV beam. A video (Media 1) showing the vortex-EELS for different loss energies is provided online.

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3.3. EELS Maps for the SRR

Figure 5 shows the conventional EELS scattering probability map of the same SRR structure for its maximum scattering loss energy of 0.8 eV. This figure was calculated using standard EELS theory [41] and by moving an electric dipole source over the trajectory of the beam. This figure is provided to show the different distribution of the EELS and vortex-EELS maps. As expected, the EELS map of Fig. 5 shows the same distribution and a similar energy resonance as previously found for the lowest energy EELS spectrum for a similar SRR structure [42]. This corresponds to the lowest order plasmonic resonance [43].

 

Fig. 5 Map of EELS probability as a function of position for the 0.8 eV loss energy and a 100 keV beam. A video (Media 2) showing the EELS for different loss energies is provided online.

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For the EELS map, the maximum scattering occurs at the ends of the arms of the SRR structure. It should be noted that we have validated the quantitative EELS values provided by our calculation method by reproducing past scattering values, as shown in the Appendix. Although in many cases having a smaller mesh size will increase the accuracy of FDTD, the local density of states inside a lossy material is necessarily grid size dependent. This dependence is related to the breakdown of the dipole approximation. A physically reasonable grid size to choose then is the comparable to the size of the physical source, in this case the vortex electron beam. In this way, FDTD calculations can correctly regularize the devergent Green function due to the finite sized grid inside the lossy structure [39]. Similar techniques are commonly employed for computing the spontaneous emission rate of a photon source in a lossy medium; one must model finite size dipoles to have a meaningful (and non-divergent) answer.

4. Discussion

The most encouraging result of the calculations presented in this theoretical work is the finding that vortex-EELS is within an order of magnitude of the conventional EELS for the same structure. Since regular EELS has been used extensively to map out the electric response of metal nanoparticles, this finding is promising for the near-future use of vortex-EELS to map out the magnetic response. For the SRR, the maximum magnetic response occurs in the region of the highest magnetic field of the so-called “magnetic plasmon resonance,” which is consistent with past works on SRR structures [11].

From a theoretical point of view, the magnetic and electric Green functions are actually not independent. Indeed, they are related through the expression [44]:

It is worthwhile to mention that the vortex electron trajectory will also have the usual electric contribution. The vortex-EELS contribution can be separated out by noting the spatial-spectral vortex-EELS response; that is, the resonances of Figs. 4 and 5 occur at different energies. Another technique would be to filter the scattered electron beam by its OAM (for example, by using diffractive or magnetic prisms) if it is has transferred OAM to the SRR, then it will no longer have OAM.

It should be noted that our calculations of vortex-EELS assumed n = 1 and a = 1 nm, which is consistent with recent experiments showing subnanometer focusing of vortex beams [45]. It is possible, however, to increase the effective magnetic charge either by increasing the topological charge n (linear dependence) or decreasing the beam size a (inverse square dependence). Since the magnetic charge is squared in the energy loss calculations, any improvements will be squared there as well. The increased n has been demonstrated in experiment [40]. It is possible that improved electron optics will lead to reduced a as well.

Finally, we comment that should vortex-EELS be successfully demonstrated, it will prove to be a powerful tool to quantify the local magnetic response of metal nanostructures. This is of interest to many applications; for example, the development of negative index metamaterials with a stronger response, a lower loss and a shorter wavelength resonance.

5. Conclusions

We have presented the theory of vortex-EELS to map out the magnetic response of a metal nanostructure at the nanometer scale. Our findings show that with reasonable parameters based on past experiments, the vortex-EELS signal from a SRR should be within an order of magnitude of conventional EELS. Since conventional EELS is performed routinely on metal nanostructures, this is a promising result for future works aimed at improving the magnetic response of metal nanostructures in the visible regime by obtaining a microscopic picture.

Appendix

Starting from the full Maxwell equations, assuming μ_r = 1, one can show

(19)$$\nabla \times \nabla \times \mathit{E}\left(\mathit{r},\omega \right)-k_0^2{\mathit{\epsilon }}_r\left(\mathit{r},\omega \right)\mathit{E}\left(\mathit{r},\omega \right)={\epsilon }_0{k}_0^2{\mathit{P}}_e\left(\mathit{r},\omega \right)=i\omega {\mu }_0{\mathit{j}}_e\left(\mathit{r},\omega \right),$$

where P_e and k₀ are the electric polarization and wavenumber. Using an electric-dipole polarization source in the Maxwell equations, it is easy to derive the electric-field Green function equation:

(20)$$\nabla \times \nabla \times {\overline{\mathit{G}}}^{\text{E}}\left(\mathit{r},\omega \right)-k_0^2\mathit{\epsilon }\left(\mathit{r},\omega \right){\overline{\mathit{G}}}^{\text{E}}\left(\mathit{r},\omega \right)={\epsilon }_0{k}_0^2\delta \left(\mathit{r}-{\mathit{r}}^{\prime }\right),$$

so that $\mathit{E}\left(\mathit{r},\omega \right)=\frac{1}{{\epsilon }_0}\int d{\mathit{r}}^{\prime }{\overline{\mathit{G}}}^{\text{E}}\left(\mathit{r},{\mathit{r}}^{\prime },\omega \right)\cdot {\mathit{P}}_e\left({\mathit{r}}^{\prime },\omega \right)$, is in term of the E-field current density:

(21)$$\mathit{E}\left(\mathit{r},\omega \right)=\frac{i}{{\epsilon }_0\omega }\int d{\mathit{r}}^{\prime }{\overline{\mathit{G}}}^{\text{E}}\left(\mathit{r},{\mathit{r}}^{\prime },\omega \right)\cdot {\mathit{j}}_e\left({\mathit{r}}^{\prime },\omega \right).$$

One can also construct a wave equation for the H field:

(22)$$\begin{array}{lll}\nabla \times \frac{1}{{\mathit{\epsilon }}_r\left(\mathit{r},\omega \right)}\nabla \times \mathit{H}\left(\mathit{r},\omega \right)-k_0^2\mathit{H}\left(\mathit{r},\omega \right)\hfill & =\hfill & \nabla \times \frac{-i\omega }{{\mathit{\epsilon }}_r\left(\mathit{r},\omega \right)}{\mathit{P}}_e\left(\mathit{r},\omega \right)=\nabla \times \frac{1}{{\mathit{\epsilon }}_r\left(\mathit{r},\omega \right)}{\mathit{j}}_e\left(\mathit{r},\omega \right)\hfill \\ \hfill & \equiv \hfill & {\mu }_0{k}_0^2{\mathit{P}}_m\left(\mathit{r},\omega \right)\equiv i\omega {\epsilon }_0{\mathit{j}}_m\left(\mathit{r},\omega \right),\hfill \end{array}$$

where we have introduced a magnetic-field polarization, P_m, and a magnetic-field current density. Equating coefficients then gives

(23)$${\mathit{j}}_m\left(\mathit{r},\omega \right)=\frac{-i}{\omega {\epsilon }_0}\nabla \times \frac{1}{{\mathit{\epsilon }}_r\left(\mathit{r},\omega \right)}{\mathit{j}}_e\left(\mathit{r},\omega \right).$$

This gives Eq. (3). Note that the $k_0^2$ difference between Eq. (16) and Eq. (22) requires suitable normalization of dipole source in calculations. We choose a different form for Eq. (16) since it is more conventional.

We have repeated past calculation of EELS scattering by a silver disk for 100 keV beam to ensure that our numerical procedure was giving good quantitative agreement with those works [33]. The FDTD parameters chosen were the same as those in Section 3.3. Figure 6 (a) shows a schematic of the disk structure. Figure 6 (b) shows the scattering probability as a function of energy and the radial displacement. These results are close to those found in a past work using the boundary element method [33]. In particular, the scattering probability of the peak at 2.5 eV is 2%/eV (close to edge of disk), and the scattering probability of the peak at 3.4 eV is 2%/eV, whereas the values found in that past work were around 2.5%/eV.

 

Fig. 6 (a) Schematic of silver disk, (b) Electron energy loss scattering probability for a silver disk, for comparison with Fig. 2(e) of Ref. [33].

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Acknowledgments

This work was supported by a National Sciences and Engineering Research Council of Canada (NSERC) Strategic Project Grant.

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