# Abstract

Cherenkov radiation in natural transparent materials is generally forward-propagating, owing to the positive group index of radiation modes. While negative-index metamaterials enable reversed Cherenkov radiation, the forward photon emission from a swift charged particle is prohibited. In this work, we theoretically investigate emission behaviours of a swift charged particle in the nanometallic layered structure. Our results show that Cherenkov photons are significantly enhanced by longitudinal plasmon modes resulting from the spatial nonlocality in metamaterials. More importantly, longitudinal Cherenkov photons can be directed either forward or backward, stringently depending on the particle velocity. The enhanced flexibility to route Cherenkov photons holds promise for many practical applications of Cherenkov radiation, such as novel free-electron radiation sources and new types of Cherenkov detectors.

## 1 Introduction

Cherenkov radiation refers to the physical effect that a charged particle with a velocity exceeding the phase velocity of light in a medium can emit coherent photons at a constant angle [1], [2]. According to the theory developed by I. Frank and I. Tamm in 1937, the radiation angle *v* as *n*) [3].

A wide controllability of radiation angles is preferable for many practical applications of Cherenkov radiation, e. g., improving the flexibility to direct photon emissions from free-electron radiation sources or enhancing the sensitivity of Cherenkov detectors [4], [5], [6], [7], [8], [9], [10], [11], [12]. However, Cherenkov radiation in natural transparent materials is generally forward-propagating, because of the positive group velocity of radiation modes. To overcome this problem, researchers come up with numerous negative-index metamaterials to manipulate Cherenkov radiation [13], [14], [15], [16], [17], [18]. The recent theoretical and experimental studies show that radiation directions can be reversed when a swift charged particle interacts with left-handed metamaterials, hyperbolic metamaterials, or plasmonic waveguides [19], [20], [21]. Even so, the radiation angle of Cherenkov photons is still limited in a relatively narrow angular band, i. e., negative angles relative to the particle velocity. A novel mechanism that can enable to direct Cherenkov photons in a broader range, e. g., from a positive angle to a negative angle, is highly desirable.

In this work, we theoretically demonstrate that the spatial nonlocality provides an opportunity to flexibly control emission behaviours of a swift charged particle in a nanometallic layered structure. In the nonlocal treatment, the quantum-sized metallic structure supports longitudinal plasmon modes above the bulk plasmon frequency of metals [22], [23]. These longitudinal modes originate from the nonlocal electron screening in metals. The swift charged particle in the nonlocal layered structure enables excitation of longitudinal modes at resonant frequencies, and emits longitudinal Cherenkov photons. Interestingly, the sign of effective group index of longitudinal modes is dependent on the particle velocity in the nonlocal layered structure. With the consideration of realistic material loss, the effective group index is negative if the particle velocity is sufficiently large, while the effective group index is positive if the particle velocity is sufficiently small. Consequently, the variation of particle velocity enables us to route longitudinal Cherenkov photons in backward-propagating direction or forward-propagating direction at will.

## 2 Modelling Cherenkov radiation in the nanometallic layered structure

We begin with the theoretical model of Cherenkov radiation in the nanometallic layered structure. Without the loss of generality, we consider a charged particle with the velocity of *d*_{1}, the gap width between metal stabs is *d*_{2}, and thus the pitch of layered structure is *P* = *d*_{1} + *d*_{2}. In this work, we define the radiation angle *y*-axis and corresponding Poynting power vector of radiation modes.

### Figure 1:

In the cylinder coordinate (*ρ*, *φ*, *z*), the current density of swift charged particle

Here *ω* is the angular frequency, *k*_{0} = *ω*/*c, k*_{z} = *ω*/*v*, and

Comparing Eq. (1) and (2) with Eq. (3) and (4), we find that the potential of a swift charged particle along the *z*-axis takes the form of

where the wavevector component

To illustrate the impact of nonlocal electron screening in metals on emission behaviours of the swift charged particle, we compare Cherenkov radiations in the local layered structure and nonlocal layered structure. The local description of light-matter interactions assumes that the induced surface charges only precisely reside on the boundary of metals. And as a result, the free-electron response of metals is transversal, given by Drude model as *ω*_{p} is the bulk plasmon frequency, *γ* is the collision frequency. In the local treatment, we calculate the fields in all regions using potentials with decompositions of transverse magnetic field and transverse electric field. In the region *j*, the potentials of transverse magnetic field and transverse electric field take the form as

where *j*. The transverse radiation fields

Under the local description, the boundary conditions are [26]

The method of transfer matrix is employed for the calculation of scattering coefficients [27].

However, a more accurate description of light-matter interactions takes the nonlocal electron screening in metals into account [28]. Owing to the quantum repulsion of electrons, the surface charge densities have a finite penetration inside metals. Such a spatial dispersion gives rise to a longitudinal response of metals, which can be theoretically described by the hydrodynamic model as *j* as

where *j*. Thus, the longitudinal electric field can be obtained as

Especially, due to the existence of longitudinal electric field, an additional boundary condition, i. e., the longitudinal continuity of electric field, is applied at the boundary of metals

We employ the method of transfer matrix to calculate all the scattering coefficients in each region of this configuration [31], [32]. Finally, Cherenkov radiation in the nonlocal layered structure is the superposition of transverse electric field, transverse magnetic field and longitudinal electric field.

In addition, to qualify the photon emission enhanced by longitudinal plasmon modes, we compute the energy loss density and photon extraction efficiency of a swift charged particle. The energy loss density is determined by employing power dissipation formula as [21]

where

where

## 3 Photon emission enhanced by longitudinal plasmon modes

In this section, we demonstrate that the photon emission from a swift charged particle is significantly enhanced by longitudinal plasmon modes in the quantum-sized layered structure. As a concrete example, we select the silver (Ag) as the metal, where *ω*_{p} = 9.0 eV, *γ* = 0.032 eV and *v*_{F} = 136 × 10^{6} m/s. The thickness of metal slab is *d*_{1} = 2 nm, and the pitch of the layered structure is *P* = 6 nm. The number of the period in calculation is *n* = 10. In this configuration, the spatial nonlocality is strong as *λ*_{TF} ∼ 5 Å is the Tomas–Fermi screening length. When the particle velocity *λ*_{0} = 129 nm, the photon emission is prohibited inside the local layered structure but appears only when *λ*_{p} = (2*πc*)/*ω*_{p} = 138 nm (Figure 2A) [34]. This is because, under the local description, the metal is no longer a plasmonic material as *d*_{1} ≫ *λ*_{TF}, the enhancement of photon emission is relatively weak [29].

### Figure 2:

## 4 Radiation angles of longitudinal Cherenkov photons

To reveal radiation angles of longitudinal Cherenkov photons, we investigate the isofrequency contours of bulk Block modes in the nonlocal layered structure [35]. The isofrequency contours in the lossless case show that the bulk Block modes appear only in a finite range of wavevectors (Figure 3A). When the wavevector is sufficiently small, the isofrequency contour is hyperboloid-shaped, indicating that the nonlocal layered structure enables negative refraction. When the wavevector is relatively large, the isofrequency becomes flat. As a consequence, the energy flow of bulk Block modes is normal relative to the interface. When the wavevector is sufficiently large, the isofrequency contour closes due to the decoupling of longitudinal plasmon modes in metal slabs. Beyond the wavevector cutoff, the bulk Block modes disappear. However, isofrequency contours of bulk Block modes are smoothen by taking the realistic material loss into account (Figure 3B). In the limit of small wavevectors, the isofrequency contour remains hyperbolic, while it becomes elliptical when the wavevector is sufficiently large. The exotic isofrequency contours indicate that the effective group index can be controlled in a flexible way. Taking the bulk Block mode at 129 nm for example, we find that the sign of effective group index is stringently dependent on the *z*-component wavevectors

### Figure 3:

Properly engineering isofrequency contours enables us to sensitively control the radiation angles of longitudinal Cherenkov photons in a broad range of particle velocities, while the photon emission remains relatively intensive. The radiation angle here is calculated using *λ*_{0} = 129 nm, there is a relatively wide controllability of radiation angles (Figure 3A). When the particle velocity

### Figure 4:

## 5 Applications

The feasibility to direct Cherenkov photons in a broad range of angles facilitates many practical applications of Cherenkov radiation. On the one hand, an electron beam traveling in the nanometallic layered structure can be employed for an integrated free-electron radiation source by coupling longitudinal Cherenkov photons from the nanometallic layered structure to the far field. The novel free-electron radiation source is capable of generating forward-propagating photons and backward-propagating photons, switched by the particle velocity. On the other hand, since the photon emission is enhanced in a broad range of particle velocities, to which the radiation angle of longitudinal Cherenkov photons is sensitive, our configuration extends the momentum range for the sensitive determination of particle velocity or particle identification. For example, the calculated radiation angle *p* and the rest masses of charged particles (Figure 5). When *p* = 0.04 GeV/*c*, electron, proton, pion, and kaon would emit longitudinal Cherenkov photons in angles of −52.2°, −33.6°, −11.8°, and 5.1°, respectively. The giant differences among radiation angles indicate that the broad controllability of longitudinal Cherenkov photons enhances the sensitivity to determine particle velocities and discriminate charged particles in an extended momentum range.

### Figure 5:

## 6 Conclusion

To conclude, we theoretically demonstrate that the spatial nonlocality in metamaterials provides a new degree of freedom to direct Cherenkov photons. Particularly, longitudinal plasmon modes resulting from the spatial nonlocality in nanometallic layered structure enhance the photon emission for four orders in a broad range of particle velocities above the bulk plasmon wavelength of metals, where Cherenkov photons are generally believed to be prohibited in the local approximation. In contrast to emission behaviours of a swift charged particle in natural transparent materials and negative-index metamaterials, the nonlocal layered structure could direct longitudinal Cherenkov photons in both the forward-propagating direction and backward-propagating direction. Meanwhile, our calculation results show that the radiation angle of longitudinal Cherenkov photons is sensitively controlled by the particle velocity. In addition, owing to the available large mode refractive index for longitudinal modes, the threshold velocity for longitudinal Cherenkov photons is relatively small, facilitating the particle identification or determination of particle velocity in an extended momentum range. This work enriches the theory of Cherenkov radiation and offers a new insight to control emission behaviours of a swift charged particle [36], [37], [38], [39], [40], [41].

**Funding source: **Singapore Ministry of Education

**Award Identifier / Grant number: **MOE2018-T2-2-189 (S)

**Award Identifier / Grant number: **MOE2017-T1-001-239 (RG91/17 (S))

**Award Identifier / Grant number: **MOE2016-T2-2-159

**Award Identifier / Grant number: **MOE2018-T2-1-176

**Funding source: **A*STAR AME

**Award Identifier / Grant number: **A18A7b0058

**Funding source: **National Research Foundation Singapore

**Award Identifier / Grant number: **NRF-CRP18-2017-02

**Funding source: **National Natural Science Foundation of China

**Award Identifier / Grant number: **11504252

# Acknowledgments

Y. Luo was partly sponsored by Singapore Ministry of Education under Grant No. MOE2018-T2-2-189 (S), MOE2017-T1-001-239 (RG91/17 (S)); A*STAR AME programmatic Grant No. A18A7b0058; Q. J. Wang was partly sponsored by funding from the Ministry of Education, Singapore (Grant Nos. MOE2016-T2-2-159 and MOE2018-T2-1-176), the National Research Foundation Singapore, Competitive Research Program (No. NRF-CRP18-2017-02). D. Gao was partly sponsored by National Natural Science Foundation of China (Grant No. 11504252).

**Competing interests**: The authors declare no competing financial interests.**Data and materials availability**: All data needed to evaluate the conclusions in the paper are present in the paper.

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**Received:**2020-02-20

**Accepted:**2020-04-21

**Published Online:**2020-05-23

© 2020 Hao Hu et al., published by De Gruyter, Berlin/Boston

This work is licensed under the Creative Commons Attribution 4.0 International License.