The Cherenkov radiation is a radiation emission induced by external charged particles traveling in transparent medium [1, 2]. This radiation takes place when the velocity of the particles is faster than light velocity in the medium. It has a conical distribution (so-called Cherenkov cone) in the emission angle and has a broad spectrum. Since its first experimental observation in 1934, the Cherenkov radiation has played a major role in detection of high-energy charged particles. Its playgrounds range from subatomic physics to cosmology and biology.

In ordinary medium, the Cherenkov cone, which is formed by relevant wave number vectors, emerges with angle θ _CR = cos⁻¹(c/(nv)) in the forward direction of the particle trajectory. Here, n,c, and v refer to the refractive index of the medium, light velocity in vacuum, and the particle velocity, respectively. In 1968, Veselago argued that the Cherenkov cone appears in the backward direction, in a medium with simultaneously negative permittivity and permeability, called double-negative material [3]. This prediction has stimulated theoretical studies of the Cherenkov radiation in artificially photonic structures [4–8], after an experimental realization of the double-negative material [9]. In particular, certain two-dimensional photonic crystals are shown to exhibit the backward Cherenkov radiation [4]. Experimental evidences of the backward Cherenkov radiation were reported recently [10].

Here, we propose an alternative. It imitates the backward Cherenkov radiation via the interaction between charged particles and an artificial photonic structure of wire type. As we will see, it provides a simple model of the backward radiation of conical shape and can tailor the radiation pattern outside the artificial photonic structure. A remarkable difference from the previous studies is that we consider traveling charged particles in vacuum, outside the artificial structure of finite cross section. Most previous studies deal with charged particles traveling inside artificial photonic structures that have infinite spatial extent in the direction perpendicular to the particle trajectory. In our setting, the resulting radiation is thus not of the Cherenkov radiation, but of the Smith-Purcell radiation [11]. A simple kinetics of the Smith-Purcell radiation and a resonant excitation of certain photonic wire modes enable us to design a conical radiation in the backward direction of the particle trajectory. We should note that a similar Smith-Purcell radiation from aligned nano-particles was studied theoretically [12].

Let us consider a photonic wire structure with one-dimensional periodicity with period a in the dielectric function. We assume that the photonic wire has the circular symmetry with respect to the wire axis. Therefore, the photonic band modes are classified according to the angular momentum m ∈ Z. There is a double degeneracy between m and −m, except for m = 0, unless optical activity is negligible. Such photonic wires include linear chain of spherical nano-particles [12, 13], distributed Bragg reflector (DBR) of circular pillar [14], and fiber Bragg grating [15]. Then, we consider a charged particle traveling outside the wire, in parallel to the wire axis. A schematic illustration of the system under study is shown in Fig. 1, for a wire with aligned spherical particles inside.

 

Fig. 1. Schematic illustration of the system under study. It consists of a periodic arrangement of spherical air holes with lattice constant a, embedded in a infinitely-long circular cylinder. A charged particle travels with impact parameter b, outside the cylinder. The particle trajectory is parallel to the cylindrical axis (taken to be the z axis). The charged particle can induce the radiation emission that can be detected at a far-field observation point of solid angle Ω = (θ,ϕ).

Download Full Size | PDF

The traveling charged particle accompanies the evanescent radiation wave whose dispersion relation is given by ω = vk_z, called v-line. This line is outside the light cone ω = c|k_z| in the frequency- momentum space, and thus the charged particle cannot emit propagating radiation by itself. However, the evanescent wave is scattered by the photonic wire, acquiring the Umklapp shift to the initial momentum k_z. The resulting radiation, whose dispersion relation is given by ω = v(k_z+G) being G a reciprocal lattice, can enter inside the light cone. Therefore, the induced radiation can be propagating. This is the so-called Smith-Purcell effect which provides a different mechanism of charged-particle-induced radiation. Besides, the scattering by the photonic wire can show a significant enhancement owing to a resonance in the photonic wire. Let us express the dispersion curve of the photonic band modes with angular momentum m by ω = ω_m(k_z). The photonic bands extend in the entire first Brillouin zone (−π/a < k_z < π/a). If the shifted v-line ω = v(k_z + G) intersects the dispersion curve, the charged particle excites resonantly the photonic band mode at the intersection point [16]. When the point is inside the light cone, a strong signal of the Smith-Purcell radiation emerges. The emission (polar) angle of the Smith-Purcell radiation is determined kinetically by

We should stress here that the polar angle θ_G can be greater than 90°, in a striking contrast to the ordinary Cherenkov radiation with θ _CR < cos⁻¹(1/n) < 90°. Hence, the emission angle can be in the backward direction. Moreover, the radiation has the dominant angular momentum of ±m, depending on the mode to be excited.

As a numerical demonstration, we consider a metallic wire with periodic arrangement of spherical air holes inside. We choose this system simply because numerical calculation of high accuracy is available by employing vector cylindrical- and spherical-wave expansions along with the photonic Korringa-Kohn-Rostoker formalism [17, 18]. The dielectric constant of the wire is modeled with the free-electron metal, ε(ω) = 1−ω ² _p/ω/(ω + iη), and the radius of the wire is taken to be 0.5a. The air-hole radius is fixed to be 0.3a.

Figure 2 shows the angular-momentum-resolved photonic band structure of the photonic wire. These bands are basically the zone-folded ones of the surface plasmon polariton of the homogeneous metal wire, and thus the band diagram becomes dense around ω = ω_p/√2 (we choose ω_pa/2πc = 1). In reality, the dense region is not favored because if we take account of dissipation, photonic bands there are readily merged, turning into a broad peak of the photonic density of states. Therefore, we are forced to utilize the bands at low frequencies.

 

Fig. 2. The photonic band diagram of the metallic wire with periodic arrangement of air holes inside. The radius and the dielectric constant of the wire are given by 0.5a and ε = 1−(ω_p/ω)², respectively, where ω_pa/(2πc) is fixed to be 1. The air-hole radius is taken to be 0.3a. The photonic bands are classified according to angular momentum m (indicated in the legend) with respect to the wire axis. The light line ω = c|k_z| and the shifted v-line ω = v(k_z + G) with v = 0.7c are also shown by dashed and solid lines, respectively.

Download Full Size | PDF

Let us consider the radiation spectrum induced by an electron traveling with velocity v=0.7c and impact parameter b = a. By overlaying the shifted v-line to the photonic band diagram, we can find a sequence of the intersection points, which represent the excitation of the photonic band modes. Figure 3 shows the resulting radiation spectra Γ(ω) with and without dissipation in the metal wire. To be precise, the net radiation energy W obtained in far-field over all solid angles is expressed as

per unit trajectory length. The quantity Γ(ω)dω represents the probability of finding the propagating photon in the frequency interval between ω and ω + dω, over all solid angles in far-field.

In Fig. 3 the spectra have the low-frequency cut off at ωa/2πc = v/(c+v) and decreases exponentially from ωa/2πc = 0.75 to 1.0, owing to the absence of the photonic band modes. The spectra above ωa/2πc = 1.0 (ω = ω_p) are less than 10⁻⁷ of the vertical axis (not shown). It is remarkable that a sequence of sharp peaks are obtained in the radiation spectrum even if the dissipation is taken into account. The peaky spectra show a striking contrast to the flat spectrum of the Cherenkov radiation,

 

Fig. 3. The Smith-Purcell radiation spectra Γ(ω) for dissipation-less (η = 0, black curve) and dissipative (ηa/(2πc) = 0.01, red curve) metallic wire. The following parameters are used: v = 0.7c and b = a. For comparison, the Cherenkov radiation spectrum Γ_CR(ω) is also shown (by blue curve) for a medium with refractive index n = 2.

Download Full Size | PDF

provided that n is dispersion-free. Here, µ ₀,e, and h(= 2πh̄) stand for the vacuum permeability, electron charge, and Planck’s constant, respectively. The lowest (in frequency) three peaks, for instance, are attributed to m = 0 (ωa/(2πc) = 0.447), m = ±1 (0.508) and m = ±2 (0.580). Each peak has the certain height and width, from which we can evaluate the probability of finding photon around the peak frequency. For instance, in the lowest peak, the probability is evaluated as 1.52×10⁻⁴ per electron and per spherical hole. Here, we approximated the peak by a rectangular one of width δωa/(2πc) = 0.01 and height Γ = 0.01µ ₀ e ²/(2h). This small probability can be increased with the multiple factor of N ² _e by using bunched electrons of number N_e. Another way to increase the probability is to decrease impact parameter b.

The azimuthal-angle distribution dΓ/dϕ of the induced radiation at the peak frequencies is shown in Fig. 4. Here, the polar-angle distribution is given by a delta-function δ (θ − θ_G), where θ_G ~ 144°,123°, and 107° for the lowest three peaks. It is remarkable that the angular distribution at ωa/2πc = 0.447 is almost circular symmetric, while the other two exhibits two-fold and four-fold rotational symmetries. In general, a 2|m|-fold symmetric angular distribution is obtained by exciting the mode with ±m. These two angular-momentum components dominate in the radiation field E. However, the cross term in the radiation intensity |E|² between m and −m gives rise to the ϕ-dependence of exp(±2imϕ), having the 2|m|-fold symmetry.

Finally, the energy density and the Poynting vector flow of the radiation field at the lowest peak frequency (ωa/(2πc) = 0.447) are shown in Fig. 5. We can find that the Poynting vector flow exhibits the radiation emission in backward directions in far-field. In near-field, however, it is directed to forward directions (not-shown because of a strong scale contrast of the Poynting vector between near- to far-field). This is reasonable because the relevant photonic band mode has the positive group velocity in the z direction. In the intermediate region 1.4 < x/a < 2.0 an optical eddy is formed connecting from the near- to the far-field configurations. Besides, the maximal energy density is found on the surface of the metal wire. This property indicates that the photonic band mode concerned comes from the surface plasmon polariton of the homogeneous metal wire. Note that the above configuration is not of a time-snapshot, but of a particular time-Fourier component. Therefore, the wavefront is hidden there. We will see a clear wavefront in a time-domain calculation.

 

Fig. 4. The azimuthal angle distribution of the induced radiation emission at the lowest three peaks in Fig. 3. Black, red, and blue curves stand for ωa/(2πc) = 0.447,0.508, and 0.580, respectively.

Download Full Size | PDF

Let us discuss in detail our radiation in comparison to the ordinary Cherenkov radiation. The spectrum of our radiation is peaky, practically regarded as a set of monochromatic waves having different frequencies. Each monochromatic wave has the fixed polar angle, and the intensity distribution of the azimuthal angle oscillates with exp(±2imϕ). Two extreme cases should be noted: m = 0 and m→±∞. Both of the cases give rise to a conical radiation with the circular symmetry. We should comment, however, that such circular-symmetric distribution can take place even at finite m, when the degeneracy between m and −m is lifted. The lifting occurs if the substances have nonzero magneto-optical effect and a static magnetic field is applied parallel to the wire axis. The polar angle changes with changing the velocity of the particle and the physical parameters of the photonic wire. The physical parameters, such as the air-hole radius and wire radius in Fig. 1, affect the photonic band structure and thus the resonance frequencies. In this way, we can control the emission angle.

Although we have demonstrated the novel radiation in a certain photonic wire structure, such a radiation can take pace also in other types of photonic wires. The key ingredients are the one-dimensional periodicity and the circular symmetry. The photonic wire structures having such properties include aligned spherical particles without the outer wire, DBR of pillar type, optical fiber with periodic array of circular grooves, and fiber Bragg grating. The first example exhibits the photonic band structure whose origin is the coupling among the Mie resonance of each sphere. This resonance is characterized by three-dimensional angular momentum l, and there is a 2l +1-fold degeneracy concerning the azimuthal angular momentum m. This degeneracy is lifted by the coupling, resulting in a rather dense band structure of lifted bands. The dense bands tend to mix with each other if the dissipation is non-negligible. The previous study on the Smith-Purcell radiation in aligned nano-particles indicates the above features, but nearly circular-symmetric angular distribution was obtained at a resonance frequency [12]. The DBR pillar of circular cross section will be the most promising, because a sophisticated lithography technique is available to fabricate it. The pillar radius must be on the order of relevant wavelength to obtain clear resonance signals. The optical fiber with periodic grooves and fiber Bragg grating are also favorable. However, owing to their rather weak index contrast, strong signals of the radiation emission will be absent. This is because the Smith-Purcell effect diminishes in the limit of the vanishing groove or vanishing index contrast.

 

Fig. 5. The radiation energy density and the Poynting vector flow in the zx plane (y = 0) at ωa/2πc = 0.447. The Poynting vector flow below x/a = 1.4 is omitted and the maximal energy density is normalized to be 1. The electron trajectory is at (x,y) = (a,0), and is indicated by the green arrow.

Download Full Size | PDF

In summary, we have presented a simple model to imitate the Cherenkov radiation in backward directions. This radiation is caused by the Smith-Purcell effect via the interaction between charged particles and a photonic wire structure with a one-dimensionally periodic dielectric function. The key ingredients are the circular symmetry and the one-dimensional periodicity of the photonic wire. The excitation of the photonic band modes of zero angular momentum can yield a conical radiation emitted in backward directions. The excitation of the modes with nonzero angular momentum m results in peculiar radiation patterns with 2|m|-fold rotational symmetries. However, they can be conical if optical activity is non-negligible. We have also discussed possible experimental realizations using more realistic photonic wires.