1. Introduction

Cherenkov radiation (CR) [1, 2] is important in high-energy particle physics, cosmic-ray physics, high-power radiation sources, and so on [3–6]. Typical examples are the discoveries of the anti-proton [7] and the J-particle [8]. In 1967, the unusual electromagnetic properties of materials with negative permittivity and negative permeability were first predicted [9]. These materials which are called double-negative metamaterials (DNMs) possess unique properties such as the negative refractive index, the reversed CR, and the reversed Doppler effect. After a thin wires structure with negative permittivity [10] and a split-ring resonator (SRR) [11] or a “Swiss Roll” [12] structure with negative permeability were proposed and the first experimental verification of the DNMs was reported [13], the research on CR in the DNMs was awakened. For examples, the CR by a point charge in unbounded isotropic DNMs [14] was first studied, the CR by an electron bunch that moves in a vacuum above an isotropic DNM was theoretically investigated [15] and then a novel design and experimental observation for the reversed CR were reported [16, 17]. In addition, Cherenkov radiation in the context of photonic crystals and metamaterials was also investigated [18, 19].

Currently, a DNM is composite medium with different materials such as the metallic strips for the SRRs and rods and dielectric materials for holding the strips. Therefore, the DNMs are in essence anisotropic rather than isotropic. A charged particle in a waveguide fully filled with such an anisotropic DNM will lose energy by polarization radiation in addition to CR. The polarization losses are actually responsible for the greater part of the energy loss and the particle is quickly brought to stop. Hence, it might be better to leave a vacuum channel in the center to allow the particle to pass. As a result, the polarization losses can be avoided. In this paper, a general CR theory in anisotropic DNMs is presented, especially for the case of an empty waveguide partially filled with anisotropic DNMs.

2. Theoretical analysis

The electromagnetic properties of an anisotropic DNM are characterized by both diagonal permittivity (ε̿) and permeability tensors (µ̿). Their elements are described by the Drude [10] and Lorentz [11] models, respectively. In cylindrical coordinates (ρ,θ, z), the elements can be expressed as

where ω is the excitation angular frequency, ω_ρp the effective plasma frequency in the ρ̂ direction, γ_ρe the collision frequency representing “electronic” dissipation, and

where γ_mρ is the collision frequency accounting for the “magnetic” loss, ω _0ρ the magnetic resonance frequency, and F_ρ the filling fraction in the ρ̂ direction. Similarly, the other elements ε_rθ(ω),ε_rz(ω),µ_rθ(ω), and µ_rz(ω) are obtained by replacing subscript ρ with θ or z in the above formulae.

We consider a charge q moving in an anisotropic DNM with a constant velocity ῡ=ẑυ. Using the vector potential method (B̄=∇×Ā=∇×ẑA_z), we obtain a vector wave equation as follows:

where ϕ is the scalar potential and J the current density formed by the charge. After separating the vector wave equation (3) into three scalar equations and letting A_z=g(ρ)µ_θq/(2π)exp(iωz/υ), we derive a scalar wave equation for g(ρ) from Eq. (3):

where k_ρ is the radial wave number and δ(ρ) the Dirac delta function. The solution to Eq. (4) can be represented as the following forms (Table 1) where ξ, ζ, η, and γ are the unknown coefficients, J ₀(k_ρρ) and N ₀(k_ρρ) are Bessel functions of the first and second kinds respectively, H ⁽²⁾ ₀(k_ρρ) Hankel function of the second kind, I ₀(s_ρ;ρ) and K ₀(s_ρρ) modified Bessel functions of the first and second kinds respectively, β=υ/c with c being the light velocity of free space, Re{} the real part operator, and CRC denotes Cherenkov radiation condition.

Table 1. Different solutions of g(ρ) for different regions.

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Thus, the electric and magnetic field components can be expressed in terms of g(ρ) as follows:

The unknown coefficients can be solved by matching the boundary conditions. From the expressions (5), (6), and (7), we can clearly see that CR should be reversed and wave vector k=-ρ̂Re(k_ρ)+ẑk_z(k_z=ω/υ) and time-averaged Poynting vector <S̄>=1/2 Re(Ē×H̄^*), generally speaking, not exactly anti-parallel.

This theory can be used for some useful cases such as the CR in the waveguide filled with multi-layer DNMs, half-space CR problem, and the CR in linear periodic structures. In what follows, we focus on a detailed theoretical analysis of CR in a cylindrical waveguide partially filled with anisotropic DNMs.

We consider a cylindrical waveguide of radius b in such a way that there is an empty cylindrical channel of radius a, and suppose a charge move along the axis shown in Fig. 1.

 

Fig. 1. A schematic diagram of a charge moving in a waveguide partially filled with anisotropic DNMs.

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In the layer 1 (vacuum), obviously the CRC is not met, while in the layer 2 (DNM), the CRC is satisfied. As a result, g(ρ) for different layers can be derived:

Here the coefficients are determined by matching the boundary conditions at _ρ=a and _ρ=b,

where _p12=ε _z1 k _ρ2/(ε _z2 s _ρ1) and

Note that the subscripts 1 and 2 correspond to the layer 1 and 2, respectively. Thus the field components in the two layers are determined by the formulae (5)–(7). By definition [3], the total radiated energy per unit length of path can be calculated as:

where the quantity ρ ₀ is the minimum average distance to the field source for which classical electrodynamics still holds.

Now, we can theoretically discuss the influence of the loss of anisotropic DNMs on CR for two cases.

On the one hand, when the loss is not considered, the CR just happens at the poles of η, for which the following condition is satisfied

The total radiated energy becomes the summation of the residue of the discrete frequencies when the above guidance condition is satisfied. Thus the spectral density is discrete at frequencies determined by Eq. (13).

On the other hand, when the loss is taken into account, the CR occurs within the operating frequency band for which the CR is satisfied. It means that the spectral density is continuous when both dispersion and loss are considered due to the causality principle [20].

3. Numerical results and discussions

Currently, DNMs can be realized at optical frequencies. To understand the physics of the process, we work with microwave or terahertz DNMs because they are easier to manufacture [17].

As an example, CR takes place in operating frequency band ([4.002, 5.024]). Fig. 2 shows the spectral density for the partially and fully filled anisotropic DNM cases. Note that f_i represents the spectral density as a function of ω. We find that there exist different spectral density distributions, with the peaks corresponding to the guided modes for no loss. The value of the spectral density at a given frequency for the “partial” case is smaller than that for the “full” case on a whole. The corresponding total radiated energies per unit length are shown in Fig. 3. We conclude that the total radiated energy can be enhanced by choosing a suitable DNM and the total radiated energy for the “partial” case is two orders of magnitude smaller than that for the “full” case. The reason is that in the “partial” case, the CRC is not met in the layer 1, so there are evanescent waves, while in the “full” case, the CRC is satisfied, and hence there exist guided waves.

 

Fig. 2. Comparison between the spectral densities in the waveguides partially and fully filled with anisotropic DNMs.

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Fig. 3. Comparison between the total radiated energies in the waveguides partially and fully filled with anisotropic DNMs.

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Obviously, the total radiated energy in the “partial” case is smaller. In order to improve the total radiated energy, the DNMs need scale to higher frequencies such as the operating frequency band [100.05, 114.95] GHz. Here we take these parameters as follows: ω _pρ=2π×1.6×10¹¹ rad/s, ω _pz=2π×1.7×10¹¹ rad/s, ω _0θ=2π×10¹¹ rad/s, F_θ=0.65, a=0.3mm,γ _eρ=γ _ez=γ _mθ=γ=10⁹rad/s, β and b are unchanged. The total radiated energy can be enhanced about two orders. If the empty channel radius and the operating wavelength further decrease, then the total radiated energy can further be enhanced. This is due to the fact that there exist evanescent waves in the layer 1 and the operating frequency band can be expanded. Such backward emitted CR maybe detected for highly charged particles such as the Dirac monopoles or heavy ions produced for example at the Heavy Ion Collider at BNL. However, the total radiated energy per singly charged particle like an electron is still too small to be efficiently detected! Hence, an effective way to detect the reversed CR is to use an intense electron beam moving along the waveguide partially filled with the DNMs at microwave frequencies. It should be mentioned that we treat the metamaterials within the effective medium approximation in order to focus on a general theory of the CR in the DNMs and we will propose an actual metamaterial structure and investigate the CR in the waveguide partially loaded this metamaterials by using a rigorous EM theory [21] and the experimental method.

At last, we address the issue of the potential applications. The reversed CR has an advantage that the detectors for the particle and the detectors for the reversed CR are naturally separated in the forward and the backward regions respectively so their physical interference is minimized, and the detection sensitivity can be improved. Furthermore, optical DNMs would be perfect for particle detectors, which operate generally in the optical region. An amplifier (or oscillator) based on the reversed CR can employ the DNMs rather than the traditional periodic structure to guide the slow-waves. By tailoring the DNMs, we can easily obtain a slower phase velocity, which means the amplifier can operate at a low-voltage. Therefore, it is easier to synchronize the amplified electromagnetic waves with an intense electron beam, and the output power can be enhanced. Certainly, there are still some challenging works needed to be performed. For examples, the operating frequency band of the suggested amplifier is narrower than the traditional traveling-wave tube due to the resonance property, a DNM formed by the metals and dielectrics does not work well at a high-voltage, and the larger loss limits the output power.

4. Conclusion

In this paper, we have developed a general theory for CR in anisotropic DNMs where both dispersion and loss are simultaneously considered. The spectral density and the total radiated energy are demonstrated by using a typical example. The effective way to improve the total radiated energy is suggested and meanwhile the potential applications in particle detection and wave generation or amplification are also discussed. We expect these results to enable a new class of particle detectors or high-power sources.