1. Introduction

Negative refraction metamaterial, commonly known for its reversal of Snell’s law of refraction, has aroused great interest in scientific communities [1–4]. It is found that negative refraction can be realized using an chiral material [5–8]. In some recent literatures [9–10], it is shown that negative refraction and backward propagation can be supported for one of internal eigenwaves in the isotropic chiral medium with the sufficiently strong chirality.

When a charged particle travels inside a medium, it can drive the medium to emit coherent electromagnetic energy called Cerenkov radiation (CR) [11]. There are three key characteristics for CR in a conventional material: it occurs only when the particle’s velocity exceeds the medium’s phase velocity, the energy propagates only in the forward direction, and there is a forward-pointing conical wavefront. One possible source of unusual CR is in a negative refraction metamaterial in which CR is predicted to flow backward; i.e., opposite to the particle velocity [1, 12]. Another possibility exists near a periodic structure. For a metallic grating [13] and one-dimensionally periodic multilayer stacks [14], simple Bragg scattering of light can give rise to radiation without any velocity threshold. A variety of CR patterns can occur in a single photonic crystal under different particle-velocity regimes [15], such as backward-propagating CR in one velocity range.

In this paper we analyze the CR emitted from a charged particle traveling with a constant velocity in strong lossless chiral media. A formal solution will be obtained for the electromagnetic field components. We reveal a variety of CR patterns that can occur in strong chiral medium under different particle-velocity regimes.

2. General formulations

According to Maxwell equations and the constitutive relation for isotropic chiral media [16]

we obtain the following dispersion relation for the wavenumber $k_{\pm }=\omega \left(\sqrt{\mu \epsilon }\pm \kappa \right)$, where “+” and “-” represent two different eigenwaves. In above expression, κ indicates the chirality, which is assumed to be positive in this paper. Similar dual conclusions can be easily expanded to the negative chirality. ε and µ are the permittivity and permeability of the chiral medium, respectively.

It has been shown that light wave propagation in such media possesses two circularly polarized eigenmodes, a left- and a right- circularly polarized (LCP and RCP) wave with two unequal characteristic phase velocities ν₊ and ν_-, respectively:

Traditionally, it is regarded as a natural limit to all chiral media for $\kappa >\sqrt{\mu \epsilon }$, however in the recent research, we know that strong chiral medium with $\kappa <\sqrt{\mu \epsilon }$ can exist at least at or near the resonant frequency of the permittivity of a chiral medium (called chiral nihility [17]). When $\kappa >\sqrt{\mu \epsilon }$, we have $k_{-}=\omega \left(\sqrt{\mu \epsilon }-\kappa \right)<0$, and then one of the two circularly polarized eigenwaves will be backward wave [7, 9–10]. It is worth noting that strong chirality does not result in a strong spatial dispersion, strong chirality parameter leads to positive energy without any frequency-band limitation in the weak spatial dispersion [18].

Consider a particle with charge q moving at a velocity ν⃑ in a unbounded lossless isotropic strong chiral material described by Eq. (1) and Eq. (2). We assume that ν is a constant in the direction z⃑. In a cylindrical coordinate system with z axis pointed in the direction of particle’s motion, the current density J⃑ of the moving charge is J𠇑(r𠇑,t)=z𠇑(qν/2πρ)δ(ρ)δ(z-ν t), where δ denotes the stand Dirac function, and ρ is related to the coordinates x and y via ρ ²=x ²+y ². Then we can Fourier transform J𠇑(r𠇑,t) to the frequency domain and get

Considering Eq. (4), the constitutive relations of Eq. (1) and Eq. (2), and Maxwell’s equations, the governing equations for the emitted electric field Ẽ and magnetic field H̃ become

where Ẽ and H̃ represent the Fourier transforms of E⃑(t) andH⃑(t), respectively. Since Eq. (5) is linear, its solution can be expressed as the following:

where V′ is the source region occupied by J̃(r⃑′), and $\widehat{\Gamma }$ (r,r′) is the dyadic Green’s function of the observation point r⃑ and of the source point r⃑′ in the unbounded chiral medium [19].

Substituting (4) into (7), using $\widehat{\Gamma }(\rho ,\varphi ,z;{\rho }^{\prime },{\varphi }^{\prime },\omega /v)={\displaystyle {\int }_{-\infty }^{+\infty }dz\text{'}\Gamma (r,r^{\prime })e^{i\omega z^{\prime }/v}}$ and knowing that the field is independent of the coordinate φ due to isotropy of the chiral media, we can get

Through calculating, $\widehat{\Gamma }$ (ρ,φ,z;ρ′,φ′,ω/ν) can be expressed as [19]:

where

${\chi }_{\pm }=\mp \frac{k^2-k_{\pm }^2}{4{\omega }^2\sqrt{\mu \epsilon }\kappa },M_{\pm }\left(k_{\pm }\right)=[(1⁄k_{\pm }^2)\nabla \nabla +U\pm (1⁄k_{\pm })U\pm \nabla ],$

Ĝ _±(ρ,φ,z;ρ′,φ′,ω/ν=(i/4)H ⁽¹⁾ ₀(γ _± R)e^iβz,

with $k=\omega \sqrt{\mu \epsilon }$,β=ω/ν,${\gamma }_{\pm }=\sqrt{k_{\pm }^2-{\beta }^2}$,$R=\sqrt{{\rho }^2+{\rho \prime }^2-2\rho \rho \prime \cos \left(\phi -\phi \prime \right)}$, and H() denotes the Hankel functionU is the three-dimensional unit dyadic.

Substituting (9) with ρ′=0 into (8), then after a lengthy mathematical manipulation, Ẽ can be expressed as the following:

where $\Omega =q\sqrt{\mu ⁄\epsilon }⁄16\pi$,τ _±=γ _± ρ ρ,̂ and $\widehat{\phi }$ are the unit vectors along ρ and φ directions, respectively. After a similar manipulation, the solution for Eq. (6) (the corresponding magnetic field H̃) can be obtained:

3. Discussions

In strong chiral media, there are two different characteristic phase velocities of two different eigenwaves for their state of polarization which are given in Eq. (3). Therefore, it appears that in such media the CR condition, where the particle’s velocity must be greater than the medium’s phase velocity, can be met for two different values of particle’s velocity. We can identify three regimes of the charge velocity with three different CR patterns in strong chiral media.

1)|ν_±|<ν.

In this case, each frequency component of the radiated field in the CR consists of two cylindrical waves Ẽ ₊ (for the wave number k ₊) and Ẽ _- (for the wave number k _-). Since we are interested in radiation from the charge, we use the asymptotic values of H ⁽¹⁾ ₀(τ _±) and H ⁽¹⁾ ₁(τ _±) to find the far-field solutions.

Using the large-argument (|τ _±|≫1) asymptotic expression for the Hankel functions, $H_0^{\left(1\right)}\left({\tau }_{\pm }\right)\approx \sqrt{(2⁄\pi {\tau }_{\pm })}{e}^{i{\tau }_{\pm }-i\pi ⁄4}$, $H_1^{\left(1\right)}\left({\tau }_{\pm }\right)\approx \sqrt{(2⁄\pi {\tau }_{\pm })}{e}^{i{\tau }_{\pm }-i3\pi ⁄4}$, Eq. (10) and Eq. (11) can be modified as follows:

where $\eta =(q⁄16\pi )\sqrt{2\mu ⁄\pi \rho \epsilon }$. We can rewrite Ẽ _±, H̃ _± in the following way:

For the far electric and magnetic fields, the corresponding Poynting vectors are written as:

And the Poynting vectors for the left- and right- polarized waves are written as:

And the total energies per unit area radiated out in $\widehat{\rho }$ and ẑ direction for the left- and right-polarized waves are written as:

For

strong chiral media $\kappa >\sqrt{\mu \epsilon }$,W _+z>0,W _+ρ>0,W _-z<0,W _-ρ>0. Then it is found that CR in this case is associated with forward and backward directions of emission. Somewhat Similar phenomenon is observed in photonic crystal [15] and left-handed medium with weakly loss [12]. CR in this case is shown in Fig. 1(a). The constant phase fronts of Ẽ ₊ and Ẽ _- form cones around the zẑ direction. The direction θ _± that k⃑ _± makes with zẑ is determined from Eq. (15.1). Thus there are two cones of radiation for two cylindrical waves (Ẽ ₊ and Ẽ _-) of the radiated field. The line connecting points A to B forms the phase front of the radiation which is propagating with the wave vector k⃑ ₊, and the line connecting points A′ to B forms the phase front of the radiation which is propagating with the wave vector k⃑ _-. 2)ν ₊<ν<|ν _-|.

In this case, we see that γ ₊ is real and γ _- is imaginary. Then Ẽ ₊,H̃ ₊,S̃ ₊,W̃ ₊ are the same with that in case 1, and the right-polarized waves are evanescent in the $\widehat{\rho }$ direction. We can find a forward-propagating CR. CR in this case is shown in Fig. 1(b). The constant phase front of Ẽ ₊ forms a cone around the zẑ direction. The direction θ ₊ that k⃑ ₊ makes with zẑ is determined from cosθ ₊=β/k ₊. Thus there is a single cone of radiation for the left-polarized waves (Ẽ ₊) of the radiated field. The line connecting points A to B forms the phase front of the radiation which is propagating with the wave vector k⃑ ₊.

 

Fig. 1. CR in strong chiral media$\kappa >\sqrt{\mu \epsilon }$. (a): case 1(|ν_±|<ν); (b): case 2(ν₊<ν<ν_-|).

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3)ν<|ν_±|.

In this case, both γ ₊ and γ _- are imaginary. The fields decrease exponentially in the $\widehat{\rho }$ direction, and there is no CR field in strong chiral media.

In the former discussions the materials are considered being lossless. If losses exist in strong chiral media, we have to consider the constitutive parameters ε,µ and κ are complex [7]. Then the wavenumbers k _± are complex. Considering the analysis in Ref. 20, the condition for Cerenkov radiation is now ν²>(ω/Re(k _±))². The argument of the Hankel functions is complex, but the solutions of Eq. (5) and Eq. (6) are unchanged. Using the analysis method similar to that in Ref. 20, we can see that the direction of power radiation is determined by the arguments of ε,µ and κ. There are still backward power for the right-polarized waves and forward power for the left-polarized waves in strong chiral medium. In addition we see that the directions of k⃑ _± are different from that of S̃ _±. If the losses are small, there is almost no difference between the direction of k⃑ ₊ and the direction of S̃ ₊, and the direction of k⃑ _- is almost opposite to that of S̃ _-.

4. Conclusions

In summary, we have analyzed the problem of CR emitted from a charged particle traveling with a constant velocity in strong isotropic lossless chiral media. The analytical expressions for the electromagnetic field components have be given. We showed that there are a variety of CR patterns in strong chiral medium under different particle-velocity regimes. In each case, the characteristics of the CR have been discussed. And in one velocity range, a radiation pattern with double cone of propagation can be expected, and the radiation is associated with forward and backward directions of emission. CR has offered many applications in a variety of fields such as particle physics, high-energy physics, and cosmic-ray physics [21]. The characteristics of CR in strong chiral medium can find application in velocity-sensitive particle detection and radiation generation.