1. Introduction

Artificially structured metamaterials, firstly introduced by Veselago [1], exhibit some fascinating electromagnetic properties, which are not found in naturally occurring materials and composites, such as the reversal of Snell’s law, Doppler effect, and Cherenkov radiation. Since Smith and Shelby et al. demonstrated the realization of the first double-negative metamaterials (DNMs) [2,3] based on the pioneering work of Pendry et al. [4,5], there has been a lot of interest to understand the exotic behavior of different metamaterials and to explore their potential applications to components, devices, antennas, and so on. Typical examples are the advanced lenses and optics [6,7] as well as invisible cloaks [8,9].

Cherenkov radiation (CR) [10,11] is extensively employed in high-energy particle physics, cosmic-ray physics, and high-power radiation sources [12]. The research interest has arisen in reversed Cherenkov radiation (RCR) in DNMs due to the “reversed” behavior and easy control of effective electromagnetic parameters as well as potential applications to particle detectors and microwave/millimeter wave generation [13–19]. Although these publications [20,21] have claimed indirect observation of RCR by use of either free electrons or a phased electromagnetic dipole array to model moving charged particles, RCR has not been really observed using a charged particle beam. We have theoretically investigated the RCR in a circular waveguide either fully [22] or partially [23] loaded with the anisotropic DNMs using a single charged particle model and found that the total radiated energy even at high frequencies, such as terahertz band, is too small due to the large loss of DNMs to be easily detected. In order to greatly enhance the total radiated energy, in this paper, we propose a charged particle beam bunch method for the direct observation of RCR.

2. Theoretical analysis of RCR

We use the electromagnetic theory to analyze the field components generated by a charged particle beam bunch with a total charge qmoving through the channel in a circular waveguide partially loaded with DNMs (Fig. 1 ). There are two sets of cylindrical coordinates, one (${\rho }^{\prime },{\theta }^{\prime },z^{\prime }$) for the source and one $(\rho ,\theta ,z)$for the point where fields are being calculated. The anisotropic DNMs can be characterized by the permittivity and permeability tensors [22], the elements of which are described by the formulae (1) and (2) in [3], respectively. With the assumption that the space charge effect is negligible and the charged particles are uniformly distributed, the charged particle beam bunch with the radius ${\rho }_0$ and the axial length $2z_0$ travels uniformly along the z-axis with a constant speed $\overline{\upsilon }$. Note that the beam bunch length should satisfy the condition $2z_0<{\lambda }_{cp}$, with ${\lambda }_{cp}=\upsilon /f$ being the charged particle wavelength and f being the operating frequency. The charge density ${\rho }_P$ and the current density $\overline{J}$ can be written in the following forms:

 

Fig. 1 A schematic diagram of a charged particle beam bunch moving uniformly with a constant speed $\overline{\upsilon }$ in a circular waveguide partially filled with anisotropic DNMs. ${\rho }_0$ and $2z_0$ are the radius and the axial length of the charged particle bunch, respectively.

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where$T(t,z^{\prime })=1,(z^{\prime }\in (\upsilon t-z_0,\upsilon t+z_0))$, $q_0$is the electric charge of a single charged particle, and N is the charged particle number.

The Fourier transform of the current density in the frequency domain is given by

Here, the cross section of the circular waveguide is divided into three layers, the first is filled with charged particle beam bunch ($0<\rho \le {\rho }_0$), the second with vacuum (${\rho }_0<\rho \le a$), and the third with DNMs ($a<\rho \le b$), as shown in Fig. 1.

For the first layer, we use the Maxwell’s curl equations in phasor form, adopt the method of the separation of variables, follow the electromagnetic potentials approach [22,23], and carry out a series of tedious manipulations. Finally, we arrive at the scalar wave equation

where the radial wave number $k_{\rho }$ has this form $k_{\rho }^2={\omega }^2{\epsilon }_0{\mu }_0-{\omega }^2/{\upsilon }^2$, $A_z$ is the component of the vector potential along the z-axis, and ${\epsilon }_0$ and ${\mu }_0$ denote the permittivity and permeability respectively in vacuum.

The analytical solution to the above wave equation is [24]

where $G\left(\rho ,{\rho }^{\prime },z,z^{\prime },\omega \right)$ is the Green’s function for the inhomogeneous Helmholtz Eq. (3), namely, the solution corresponding to a unit point source satisfies the following equation [22,23]:

and the integration is carried out in the volume V in which the source $p\left(z^{\prime },\omega \right)$ is distributed.

Once obtaining the solution to (3), we derive the analytical expressions of field components based on the electromagnetic potentials approach.

Similarly, we also derive the field components for the second and the third layers by solving the homogenous equation corresponding to (3), respectively. Here, we just concentrate on the RCR in the third layer when the CR condition is satisfied. Note that the CR condition describes that the charged particle speed exceeds the phase velocity of radiated electromagnetic wave in the DNM [12]. Thus, matching the boundary conditions at$\rho ={\rho }_0$, $\rho =a$, and $\rho =b$, we derive the unknown coefficients for the different layers. As a result, all the field components for the three layers are entirely determined. Once knowing the field components, we can depict the reversed radiation behavior in the third layer according to the time-averaged Poynting vector $<\overline{S}>$. In addition, the total radiated energy per unit length can be derived. From the analytical expressions, we find that for the case with the beam length 2z ₀ much smaller than the RCR wavelength of interest, the total radiated energy is proportional to the square of charged particle number N. This result means that the total radiated energy of a short charged-particle beam bunch can be greatly enhanced to make the radiation readily observable. Here, we note that the aforementioned conclusion is made with the assumption that the charged particles are uniformly distributed in the first layer, and for simplicity, the space charge effect has not been taken into account but is smaller under the condition that the charged particle number is smaller.

3. Numerical results and discussions

In order to demonstrate the enhanced RCR in the present case which are now easily realized at microwave band for next experiment, we give a typical example as follows. The double-negative behavior of the anisotropic DNMs presented in the following paragraphs is for the frequency region (10.50, 10.99) GHz. The electron speed used is 0.8c, with c being the velocity of light in free space, ρ₀ and z₀ being 0.1 mm and 2.5 mm, respectively, and $N=5.0\times {10}^7$ which implies the space charge effect should be negligible. We note that the bunch length used here is smaller than the wavelength of the relevant radiated electromagnetic wave. The parameters a and b shown in Fig. 1 are set to 1 mm and 10 mm, respectively. Thus, the RCR occurs in the narrow bandwidth (10.50, 10.95) GHz. In the third layer, we show the reversed or backward radiation property according to $<\overline{S}>$, which at different positions are not parallel to each other due to anisotropy of the DNMs, as shown in Fig. 2 . The values of the emitted Cherenkov cones can be changed with the use of different DNMs or by varying the charged particle velocity. Poynting vectors at the discontinuity of two media reflect and transmit according to the usual Snell’s law, after the proper sign of the relative (positive or negative) index of refraction, n, has been taken.

 

Fig. 2 Power flux densities at different positions that clearly show the reversed behavior of CR in DNMs.

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The contribution to RCR is mainly from the peaks from the poles of the spectral density function in the spectral density$f_i$, as shown in Fig. 3 . The peak contribution increases with decreasing material loss. In other words, as the loss increases, the peak number decreases and the spectral density become smoother over the radiation frequency band. Thus, we obtain the maximum coherent radiation at a certain frequency corresponding to a peak when the electrons radiate in phase. This may provide a new method to produce a novel microwave or terahertz wave. In addition, numerical results show that the spectral density, as described in the unbounded DNMs [14], is not sensitive to the degree of anisotropy for the charged particle beam bunch model.

 

Fig. 3 Radiated energy spectral density over the radiation frequency band.

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The effect of the charged particle velocity and the material loss on the total radiated energy per unit length is illustrated in Fig. 4 . The total radiated energy is an increasing function of the charged particle velocity due to the fact that increasing the charged particle velocity results in expanding the operating frequency band in which the RCR occurs. As stated before in the spectral density, the total radiated energy naturally decreases with increasing loss.

 

Fig. 4 Total radiated energy as a function of the loss for different charged particle velocities.

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We next investigated the influence of the electron beam bunch radius ${\rho }_0$ and its length $2z_0$on the total radiated energy. Under the condition that the charge density of the electron beam bunch is kept constant for the latter case, the numerical results are shown in Figs. 5 and 6 , respectively. The fact that the total radiated energy increases with increasing${\rho }_0$can be understood as follows: when the electron beam bunch radius ${\rho }_0$ increases, it means the charged particle numberN increases. Thus, the total radiated energy can be greatly enhanced ($\propto N^2$), as stated before. In the same way, the total radiated energy decreases with increasing bunch length $2z_0$, which makes the radiation from the electrons in the bunch ever less coherent.

 

Fig. 5 Total radiated energy as a function of the radius of the electron beam bunch with constant charge density.

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Fig. 6 Total radiated energy as a function of the length of the electron beam bunch.

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Based on the present analysis reported here, the RCR for a short electron beam bunch is greatly enhanced in comparison with that for a single electron case [23]. For example, when N and γ are $5\times {10}^7$and $5\times {10}^8$ rad/s, respectively, the RCR for such an electron beam bunch with $z_0=$ 2.5 mm and ${\rho }_0=0.1$ mm is enhanced by a factor of $~2.3\times {10}^{15}$(close to$N^2$) times relative to the radiation intensity of a single electron. The radiation is so strong that it should be readily detectable by use of a spectrum analyzer. In addition, in order to further improve the total radiated energy, the DNMs can be scaled to higher frequencies such as the terahertz band, as described in [23].

4. Conclusion

In this paper, we have developed a charged particle beam bunch method for enhancing the RCR in a circular waveguide partially loaded by the double-negative metamaterials. This method, under the assumption that (a) the charged particles are uniformly distributed over a short bunch length and (b) the space charge effect is negligible, presents a fact that the total radiated energy is proportional to the square of the charged particle number. It means that the radiated energy can be greatly enhanced by increasing the charged particle number. It is important to use an electron beam model presented here to experimentally observe RCR, since RCR in DNMs has not been really observed so far due to many experimental limitations. Such an experiment is being carried out by our group and is expected to report the related results in the near future.