1. Introduction

Recently there has been a great deal of interest in studying electromagnetic invisibility cloaks, which can be created by various schemes such as coordinate transformation techniques [1–12], dipolar scattering cancellation [13], and anomalous localized resonance [14]. The coordinate transformation method proposed in [1,2] has received much attention because it is especially powerful for designing invisibility cloaks that can, in principle, completely shield enclosed objects from electromagnetic illumination. For such an ideal invisibility cloak, some of the material parameters have infinite values at the interior surface of the cloak, which makes it very difficult to fabricate experimentally. Thus, the effect of imperfect material parameters on the performance of the cloak has also been discussed [9–12]. However, all of these discussions focus on the point source or plane wave. In fact, a moving electron beam or electrifiable wire can also radiate an electromagnetic field. It is natural to ask whether or not a cloak can be designed to shield the radiation of a moving electron beam or electrifiable wire. For example, when a moving electron beam or electrifiable wire is put inside a cylindrical invisibility cloak (ideal or nonideal), what phenomena will happen?

It is well known that the interaction between a moving electron beam and dielectric materials has been investigated extensively [15–17]. The interaction between a moving electron and microstructures gives rise to the emission and excitation of radiation. The motivation of these investigations lies in applications of scanning transmission electron microscopy (STEM) and electron energy loss spectroscopy (EELS) [16–27]. STEM has proved to be a powerful technique for determining different microstructures of the nanometer scale [15]. EELS in STEM is also a useful tool to investigate both surface and bulk excitations of the sample [16–28]. Recently the electromagnetic interaction between an external electron beam and a spherical cloak has also been studied by us [12]. The results have shown that the efficiency of a nonideal electromagnetic spherical cloak can be exhibited in EELS. However, the electromagnetic radiation of a moving electron beam cannot be confined inside a spherical cloak. It is very interesting to design a cloak to confine the radiation of a moving electron beam, which prevents it from producing any effect on the exterior environment

In this work we study the interaction of a cylindrical cloak (ideal and nonideal) with external and internal electron beams and test the shielding efficiency of the cloak to see whether or not the electromagnetic radiation of the moving electron beam can be confined inside the cylindrical cloak. The exact solution for the energy loss suffered by a fast electron moving inside or outside a cylindrical cloak is established within a fully relativistic approach. The energy loss spectra and the photon emission for such a structure with different combinations of electron velocity and impact parameter will be discussed in detail. We find that radiation can be shielded very well by such cloaks when an electron moves inside or outside of them. The efficiency of a nonideal cloak and the effect of various nonideal parameters on cloak invisibility can be exhibited in the spectra of energy loss and radiation emission.

2. Theory

We consider a fast electron moving with constant velocity v parallel to the cylindrical cloak shell with outer radius R ₂ and inner radius R ₁ + δ, as shown in Fig. 1.

 

Fig. 1. Schematic picture of the geometry depicting a moving electron in vacuum with velocity v in the interaction of a nonideal invisibility cloak cylinder. The outer boundary is still fixed at R ₂; the inner boundary is at R ₁ + δ, where δ is a positive number.

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The cloak layer (R ₁ + δ < r <R₂) is a specific anisotropic and inhomogeneous medium characterized by [10, 11]

where ε _r, ε_θ, and ε _z are the permittivity along the r, θ, and z directions, respectively. μ _r, μ_θ, and μ _z are the corresponding permeability. Following the approach in [4], the permittivity and permeability tensor components for the cloak shell can be given as

where ε ₃ = ε ₀ and μ ₃ = μ ₀ correspond to an ideal case. Here, ε ₀ and ε ₀ are the permittivity and permeability in vacuum. Without loss of generality, the electron trajectory will be chosen parallel to the z axis with impact parameter b with respect to the origin of coordinates (the center of the cloak shell). In a source-free medium, the field ($\overrightarrow{E}$ and $\overrightarrow{H}$) can be decomposed into TE and TM modes with respect to ẑ, respectively, as

where ψ ^E and ψ ^M represent the scalar potentials. For the above cloak structure, we consider two kinds of cases.

2.1 Electron moving outside the cylindrical cloak

When a fast electron moves outside the cylindrical cloak, the scalar potential of the incident field ($({\psi }^{M(E),\mathrm{inc}}(\overrightarrow{r}))$, the scattered field $({\psi }^{M(E),\mathrm{ind}}(\overrightarrow{r}))$, the internal field $({\psi }^{M(E),\mathrm{int}}(\overrightarrow{r}))$ and the field of the cloak layer $({\psi }^{M(E),c}(\overrightarrow{r}))$ can be expanded in terms of multipoles as

where J_m, H_m, and N_m represent the mth order Bessel function, the Hankel function, and the Neumann function of the first kind, respectively. Here $k_{r1}=\sqrt{k_1^2-k_z^2}$ with $k_1=\frac{\omega \sqrt{{\epsilon }_1{\mu }_1}}{c}$, $k_{r2}=\sqrt{k_2^2-k_z^2}$ with $k_2=\frac{\omega \sqrt{{\epsilon }_2{\mu }_2}}{c}$, $k_{r3}=\sqrt{k_3^2-k_z^2}$ with $k_3=\frac{\omega \sqrt{{\epsilon }_3{\mu }_3}}{c}$, and k_z is the wave vector along the z direction. ψ_m ^M(E),inc, ψ_m ^M(E),ind, ψ_m ^M(E),int, ψ_m^M(E),c1, and ψ_m^M(E),c2 are the corresponding expanded coefficients, which can be determined by the boundary conditions.

2.2 Electron moving inside the cylindrical cloak

When a fast electron moves inside the cylindrical cloak, the scalar potential of the incident field $({\phi }^{M(E),\mathrm{inc}}(\overrightarrow{r}))$, the scattered field $({\phi }^{M(E),\mathrm{ind}}(\overrightarrow{r}))$, the external field $({\phi }^{M(E),\mathrm{ext}}(\overrightarrow{r}))$, and the field of the cloak layer $({\phi }^{M(E),c}(\overrightarrow{r}))$ can be expanded in terms of multipoles as

where ϕ_m ^M(E),inc, ϕ_m ^M(E),ind, ϕ_m ^M(E),ext, ϕ_m^M(E),c1, and ϕ_m^M(E),c2 are the corresponding expanded coefficients.

An electron moving inside an infinite, homogeneous medium j along a trajectory parallel to the z axis and described by r = b, φ = 0, and z = vt will set up an electric field that reads, in frequency space ω [26],

where r_< =min{r,b), r_> = max{r,b}, and k_z = ω/v are the momentum components of the electron field parallel to the direction of motion.

Inserting ${\overrightarrow{E}}^{\mathrm{inc}}$ in Eq. (3) by using Eqs. (5)–(12), we can obtain unknown coefficients corresponding to the radiation field of the electron moving outside or inside the cloak shell in the following form:

The radiation field of the moving electron beam may engender collective excitations of the medium (including the cloak shell) that act back upon the electron, which leads to the energy loss of the moving electron. The energy loss can be related to the force exerted by the induced electric field E ^ind acting on it as [26]

where L is the length of the trajectory, and

represents the electron energy loss probability per unit of path length. Only the z component of the induced field is needed. We can obtain

where ψ_m ^E,ind and ϕ_m ^E,ind ind represent the induced scattered field, which can be obtained from Mie’s scattering theory. The explicit forms for these induced scattered fields are given in the Appendix.

Corresponding to the energy loss, the coupling of the electron with radiation modes of the system gives rise to radiation emission, which can be expressed as [25,28]

where r̂ points to the surface of the large cylinder, and the integral over the time has been included. Expressing the field in terms of their frequency components, one finds

where

is the probability of emitting a photon of energy ω per unit energy range and unit solid angle around direction θ. If k_r1² > 0, only the field outside the cloak contributes to the radiation in the r → ∞ limit. Integrating over angles, one finds

where ψ_m ^m(E),ind and ϕ_m ^M(E),ext can be obtained from Mie’s scattering theory, as shown in the Appendix. Based on Eqs. (17) and (21), the relativistic energy loss and the radiation emission can be obtained by the numerical calculation.

3. Numerical results and discussion

We first consider an electron moving outside a cylindrical cloak. The medium outside the cloak (r > R₂) is a vacuum (i.e. ε ₁ = ε ₀, μ ₁ = μ ₀), and the object inside the cloak (r < R₁ + δ) is a dielectric cylinder. In order to test the shielding efficiency of the cloak, we have calculated the field distribution for the moving electron beam in the presence of different objects. The result is plotted in Fig. 2. Figure 2(a) displays the field distribution in the xy plane when an electron moves with v = 0.7c passing near a dielectric object without the cloak. The radius (R₁) of the dielectric object is taken as 0.5λ ₀, ε ₂/ε ₀ =11.9 + 0.1i and μ ₂ = μ ₀. Here λ ₀ is the unit of the length, and the corresponding unit of the frequency is ω ₀ = 2πc/λ ₀. Due to the generality of λ ₀, the following calculated results are applicable to any frequency range of the radiation. The field in the figure is over a 1.5λ ₀ × 1.5λ ₀ region. Figures 2(b) and 2(c) show the corresponding result when the object is covered by a nonideal cloak with δ = 0.5R ₁ and δ = 0.01R ₁, respectively. Here R ₂ is taken as 1.0λ ₀. The corresponding field distributions in the xz planes for three kinds of cases are also plotted in Figs. 2(d), 2(e), and 2(f), respectively. Comparing them, we find an evident cloaking effect of the nonideal cloak to the moving electron beam. For example, the amplitude of the field inside the dielectric object at x = y = 0.001λ ₀ decreases 98.7% when a nonideal cloak with 8 = 0.5R ₁ is introduced [see Figs. 2(b) and 2(e)]. It becomes 99.9% when a nonideal cloak with δ = 0.01R ₁ is used [see Figs. 2(c) and 2(f)]. At the same time, the field distribution outside the cloak with δ =0.01R ₁ is near the case without any object. Such an efficiency of cloaking can also be exhibited in EELS.

 

Fig. 2. E_z field distributions for moving electron beam with v = 0.7c passing outside a dielectric cylinder (b = 1.1R ₂) with ε ₂/ε ₀ = 11.9 + 0.1i and μ ₂ = μ ₀. (a),(b) Correspond to the case without a cloak in the xy and xz planes, respectively. (c),(d) Correspond to the case with a cloak of δ = 0.5R ₁. (e),(f) Correspond to the case with a cloak of δ = 0.01R ₁. ε _t and μ _t of the cloak shell are taken according to Eq. (2), where ε ₃ = ε ₀ and μ ₃ = μ ₀. R ₁ = 0.5λ ₀ and R ₂ = 1.0λ ₀. Here x, y, and z are in unit of λ ₀.

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Fig. 3. Energy loss as a function of ω/ω ₀ for electron passing outside a dielectric cylinder and a cloak shell with δ = 0.01R ₁ and 0.8R ₁, respectively. ω ₀ = 2πc/λ ₀. The other parameters are taken the same as ones in Fig. 2.

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Figure 3 describes the energy loss probability per unit of path length as a function of frequency (ω/ω ₀) for an electron passing near the dielectric object and the cloak. The solid line corresponds to the case without a cloak, and the dashed line and the dotted line correspond to the case with a cloak of δ = 0.8R₁ and δ = 0.01R₁, respectively. It can be seen clearly that the energy loss decreases when a nonideal cloak is introduced. The decrease degree depends on the value of δ. For example, the maximum in the spectrum (see Fig. 3) decreases 73.2% when a nonideal cloak with δ = 0.8R ₁ is used. It decreases 99.96% for a nonideal cloak with δ = 0.01R ₁. That is to say, the energy loss is very sensitive to the δ. This can be seen more clearly in Fig. 4. Figure 4 shows the energy loss probability per unit of path length as a function of δ/R₁ for an electron passing near the cloak. When δ = 0 (perfect cloak), the loss probability is zero. With the increase of δ, it increases rapidly. If we change the electron velocity v and the impact parameter b, similar phenomena can be observed. This means that cloaking efficiency can exhibit very well in the energy loss spectra, which is similar to the case of a previous investigation on a spherical cloak [12].

 

Fig. 4. Energy loss as a function of δ for electron passing outside the cloak and cylinder. The other parameters are taken the same as those in Fig. 2.

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Now let us turn to the case of a moving electron beam in medium 2 [inside the cloak, i.e.,(b < R₁ + δ)]. Here the dielectric constant of the medium outside the cloak(r > R₂) is taken as ε ₁/ε ₀ = 11.9 + 0.1i and μ ₁ = μ ₀. The medium inside the cloak (r < R₁ + δ) is vacuum (i.e. ε ₂ = ε ₀, μ ₂ = μ ₀). The ε ₃ and μ ₃ of the cloak in Eq. (2) are still taken as ε ₃ = ε ₀ and μ ₃ = μ ₀. The calculated results for the E_z field distribution in the xz plane induced by such a moving electron with v = 0.7c are plotted in Fig. 5. Figure 5(a) corresponds to the case without a cloak, and Figs. 5(b) and 5(c) correspond to the cloaking case with δ = 0.5R ₁ and δ = 0.01R ₁, respectively. The induced field in the xz plane is symmetric with respect to the z axis. The change of field distribution is also sensitive to the δ. With the decrease of δ, the field intensity outside the cloak decreases rapidly. When δ = 0, the field intensity outside the cloak is zero. This indicates that the induced field by the moving electron is completely shielded inside the cloak. It is interesting that the energy loss is also zero in such a case, although the induced scattering field is not zero at the path of the moving electron beam.

Figure 6 shows the corresponding results of energy-loss probability per unit of path length as a function of δ for the electron moving inside the cloak. The increase of energy loss with the increase of δ is observed again. This corresponds to the field distribution in Fig. 5. As δ → 0, the energy loss is also near zero. This is because the forces exerted by the induced scattering field acting on the moving electron cancel out each other due to the symmetry of the scattering field. That is to say, the cloaking efficiency is still observed by the energy loss spectra for the case of a moving electron beam inside a cloak, although the induced scattering field is not zero at the path of the moving electron beam.

In contrast to the spectra of energy loss, the radiation emission probability (Γ^rad) for the moving electron beam inside the above cloak is plotted as a dotted line in Fig. 7. Here the parameters ε ₁ = 11.9ε ₀, μ ₁ = μ ₀, δ = 0.5R ₁, ε ₃/ε ₀ = 5.0 + 0.2i, and μ ₃ = μ ₀ are taken. In order to compare them with Γ^loss more clearly, the corresponding energy loss probability is also presented as solid line in the figure. It is seen clearly that the radiation emission probability is always smaller than the energy loss probability due to the existence of absorption in the material. But, the feature between them is similar. Such a phenomenon is also similar to the case of a previous investigation on a spherical cloak [12]. That is to say, the efficiency of a cloak can also be explored by measuring the radiation emission.

 

Fig. 5. Induced E_z field distributions in the xz plane for the electron beam (a) with v = 0.7c moving along the z axis inside the cylindrical air hole without the cloak, (b) with a cloak of δ = 0.5R ₁, and (c) δ = 0.01R ₁. The ε _t and μ _t of the cloak shell are taken according to Eq. (2), where ε ₃ = ε ₀ and μ ₃ = μ ₀. The dielectric constant of the background or outside the cloak is taken as ε ₁/ε ₀ = 11.9 + 0.1i and μ ₁ = μ ₀.

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Fig. 6. Energy loss as a function of δ for electrons with v = 0.7c moving along the z axis inside the cylindrical air hole with a nonideal cylindrical cloak. The other parameters are taken identical with those in Fig. 5.

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Fig 7. Comparison between the loss probability (Γ = Γ^loss) and the radiation emission probability (Γ = Γ^rad) as a function of ω/ω ₀ for electron beam with v = 0. 7 c moving inside the cloak along the z axis with δ = 0.5R. The ε _t and μ _t of the cloak are taken according to Eq. (2), where ε ₃/ε ₀ = 5.0 + 0.2i and μ ₃ = μ ₀. The background outside the cloak (r > R₂)is a dielectric material with ε ₁/ε ₀ = 11.9 and μ ₁ = μ ₀. The other parameters are taken the same as those in Fig. 6.

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4. Conclusion

Based on the theory of classical electrodynamics, we have provided an exact solution for the energy loss and radiation emission caused by a fast electron moving inside or outside a cylindrical cloak (ideal and nonideal) within a fully relativistic approach. The effect of various imperfect parameters on the efficiency of the cloak has also been discussed. We have found that the radiation cannot only be shielded very well by the cylindrical cloak when an electron moves inside or outside of it, the efficiency of the nonideal cloak and the effect of various nonideal parameters on the cloak invisibility can also be exhibited in the spectra of electron energy loss and radiation emission. This means that the property of a cylindrical cloak can be explored by STEM.