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Abstract
The reversed Cherenkov radiation is one of the most attractive research topics because of its unique characteristics and promising applications. It was generally believed that reversed Cherenkov radiations exist only in left-handed metamaterials (double negative mediums). In the present paper, we demonstrated that they can also be generated on plasmonic thin-films. Reversed Cherenkov radiations in the terahertz region and in the visible light region were achieved on the metamaterial thin-film and the metal thin-film, respectively. Their radiation frequencies and directions, which are interdependent with each other, are controllable. For the normal Cherenkov radiation, the wavelength on the thin-film increases with the frequency; whereas for the reversed Cherenkov radiation, the opposite is true. Theoretical analyses and simulations revealed that the normal or reversed Cherenkov radiation is generated depending on whether the forward or backward surface modes are excited on the plasmonic thin-film. Requirements of these reversed Cherenkov radiations were found out.
© 2017 Optical Society of America
1. Introduction
Since the date of discovery, the Cherenkov radiation (CR) [1, 2] has been attractive to worldwide researchers for its significant applications in particle detections and diagnostics [3–5], high-power radiation sources [6], and cosmic-ray physics [7, 8] etc. In recent years, CRs in new mediums, such as photonic crystal [9], active mediums [10] and surface plasmonics [11], have drawn increasing attentions. Of particular interest is the CR in the left-handed metamaterials (LHM) [12, 13], also called double-negative metamaterials with negative permittivity and negative permeability. Since being theoretically expected by Veselago [14], LHMs have been efficiently achieved in experiments [15–19]. They show various interesting properties, such as the reverse Doppler effect and reverse Snell’s law. One of the most attractive features is the reversed CR: the radiating energy propagates in the backward direction of the moving particles [20, 21]. Compared with the normal CR, in which the EM energy propagates in the forward direction, the reversed CR enjoys more attractive applications [22]. For example, it can promisingly be developed as efficient light radiation sources since it innately (without external feedbacks) supports the accumulated interaction between the electron beam and EM waves [23, 24], just like a backward-wave-oscillator. [25] and [26] investigated the possible applications of the reversed CR in generating radiation in the microwave and terahertz regions, indicating that it is a promising alternative for terahertz wave generation.
It was generally believed that the reversed CR can only be generated in LHMs, optical parameters of which are in the third quadrant of the permittivity-permeability coordinate [27]. This is because that most of previous investigations considered the CR in bulk mediums. Recently, [28] proposed to generate reversed CR by letting an electron-beam pass through the vacuum gap between two metals, so-called metal-air-metal (MAM) structure. The metals worked as the plasmonics with negative permittivities and positive permeabilities. Yet in order to get the reversed CR, the vacuum gap should be less than tens of nanometers, which exerts a great difficult for electron beams to pass through. In the present paper, we demonstrate a modified reversed CR on plasmonic thin-films. It is a dispersive two-dimensional (2D) CR existing on metamaterial or metallic plasmonic thin-films. It can propagate in both forward and backward directions depending on the radiation frequency. In other words, both the normal CR and the reversed CR can be generated. Two models, which respectively operate in the terahertz region and the visible light region, are considered in detail. The analytic theories are developed and verified by fully-electromagnetic simulations. Mechanism and requirements of the reversed Cherenkov radiations are also discussed.
2. Model description and simulation
The scheme to be considered is shown in Fig. 1. The plasmonic thin-film is set up on a dielectric substrate. A charged particle (free-electron-beam) parallelly skims over the thin-film at a definite distance. When the particle velocity is greater than phase velocities of surface waves on the thin-film, the 2D CR will be excited and will propagate along the thin-film [29]. The radiation direction is denoted by θ shown in the figure. Physically, it can be explained as follows. As the electron-beam passes over the thin-film, the effective electric dipoles will be excited on the film-surface and will propagate along with the electron-beam [20]. If the dipole-velocity (beam-velocity) is greater than phase velocities of surface waves, these dipoles will induce outgoing propagating waves on the surface, just as the CR in 3D space. The radiation direction satisfies the Cherenkov relation. It should be noted that the radiation waves here are surface waves horizontally propagating along the thin-film. This why we call it the 2D CR. Characteristics of the 2D CR essentially depend on the optical properties of thin-film. In the terahertz region, the plasmonic thin-film can be achieved by artificial metamaterials; whereas in the visible light region it can be realized by noble metals. Both cases will be considered in the following.
Fig. 1 Diagram of the scheme. θ denotes the propagation direction of the CR on the thin-film. The inset shows the x–z section of the scheme.
In the microwave-to-terahertz region, metamaterials have been achieved by micro-structures arrays. And their relative permittivity and permeability can be respectively expressed as [26]:
and where ω is the angular frequency, ωpe is the effective electric plasma frequency, ω0 denotes the magnetic resonance frequency, F is the filling factor of the micro-structures, γe and γm indicate the electric loss and magnetic loss, respectively. The above equations show that metamaterials are dispersive mediums with electromagnetic properties depending on frequency. In the present paper, typical metamaterial parameters in the terahertz region [30] are used and shown in model-1 of Table. 1. Calculated relative permittivity and relative permeability (the real parts) versus frequency are illustrated in Fig. 2(a). One can see that the double negative is only found in a narrow frequency band (0.22–0.25 THz), while in a much larger frequency band (0.1–0.2 THz and 0.3–0.5 THz), the metamaterial is a plasmonic medium with negative permittivity and positive permeability.Fig. 2 (a) Dependencies of the real part of the relative permittivity and the relative permeability of the metamaterial on the frequency in the terahertz region. (b) Dependency of the real part of the relative permittivity of the silver on the frequency in the visible light region.
Table 1. Parameters of the plasmonic thin-films.
In the visible light region, the plasmonic thin-film can be achieved by noble metals. According to the modified Drude model, the dielectric function of metals can be expressed as [31]:
where ε∞ is a constant representing the influence of metal atoms, ωp is the effective bulk plasma frequency of the free electrons in the metal, and γ indicates the electrons’ collision frequency. The relative permeability of the metal is unity. In the present paper, we apply silver thin-film with parameters [32] shown in model-2 of Table. 1. The calculated relative permittivity is given in Fig. 2(b), which shows that the metal thin-film is a plasmonic film when frequency is from 500 THz to 900 THz.We would like to show the FDTD simulation [33, 34] results at first. In simulations, the electromagnetic properties of thin-films are described by Eqs. (1–3). The dielectric and structure parameters follow that given in Table. 1. The charged particle is considered as a Gaussian electron-beam-pulse with definite electron-energy (beam-velocity). For model-1 in the terahertz region, the electron-energy is set as 80 keV, and the dielectric index of substrate is 2.25. Figure 3 shows the simulation obtained contour map of the electric field (Ez) on the film surface when the electron-beam is flying over the thin-film. One can see that the beam-induced waves propagate away from the beam path, and waves with different wavelengths (frequencies) propagate in different directions, forming a unique radiation pattern, which remarkably differs from that of the ordinary CR [35]. More interestingly, both the forward and backward waves are excited as shown in the figure. Figure 4 further illustrates the contour maps of the electric field (Ez) with different frequencies, i.e., the fields are shown in the frequency domain. The wave-vectors (k) are shown by red arrows, and the EM energy flux vectors are blue arrows. When the frequency changes from 0.11 THz to 0.15 THz, both the wave-vector and EM energy flux vector are in outgoing directions; whereas as the frequency is from 0.4 THz to 0.44 THz, the the wave-vectors turn to be in the ingoing directions, opposite to the EM energy flux direction. These are similar to that of the CR in 3D LHMs presented by [36–38], which demonstrated that the outgoing and ingoing waves indicate the normal CR and reversed CR, respectively. Thus we actually have achieved the reversed CR on the plasmonic thin-films. We designate the power flux direction as the radiation direction, denoted by aforementioned θ, which is less than 90° for the normal CR and greater than 90° for the reversed CR.
Fig. 3 Simulation obtained contour maps of the Ez field on the surface of the metamaterial thin-film when the electron-beam is flying over the thin-film.
Fig. 4 Contour maps of the Ez field on the surface of the metamaterial thin-film for different frequencies in the frequency domain.
For model-2 in the visible light region, we change the electron-energy to be 40 keV and let structure parameters follow that given in Table. 1. Simulation results are presented in Fig. 5 and Fig. 6, which show the field maps in the time domain and the frequency domain, respectively. Similarly to that in model-1, both the normal CR and reversed CR are generated on the metal thin-film. The former is in the frequency region of 670–750 THz; while the latter is in the 880–900 THz region. Thus we have achieved the reversed 2D CR on the plasmonic thin-films in both the terahertz and the visible light regions by simulations. Detailed analyses and discussions will be given in the following sections.
Fig. 5 Simulation obtained contour maps of the Ez field on the surface of the metal thin-film when the electron-beam is flying over the thin-film.
Fig. 6 Contour maps of the Ez field on the surface of the metal thin-film for different frequencies in the frequency domain.
3. Formula derivation
In this section, we derive the analytic formulas that govern this radiation. The whole space can be partitioned into three regions: the upper vacuum space, the thin-film, and the dielectric substrate as shown in the inset of Fig. 1. We deal with the electromagnetic fields in each region respectively and then fit them to the boundary conditions. In the upper vacuum space (region-I), the electric field and the magnetic field should respectively satisfy following two non-homogeneous equations:
Here ε0 and μ0 are respectively the permittivity and permeability of the vacuum. ρ and J⃗ are the charge density and current density of the source. For a charged particle moving in the z-direction, they can be respectively expressed as: in which q and ve are respectively the charge quantity and velocity of the beam, δ is the Dirac delta function, and x0 denotes the x-directional position of the beam.Based on Maxwell equations, all other electromagnetic field components can be expressed in terms of the y-directional electric field component Ey and magnetic field component Hy (see Eqs. (21) and (22)). Thus, we will first deal with Ey and Hy components. Considering the y-direction components of the electric and magnetic fields in region-I, Eq. (4) and Eq. (5) turn to:
Applying the Fourier transformation to above equations (here ω is the angular frequency and ky is the wave vector in y-direction), we get following non-homogeneous Helmholtz equations: in which is the tangential Poisson operator, k0 = ω/c is the wave vector in the vacuum, c is the light speed in the vacuum, Ey (ky, ω), Hy (ky, ω), ρ(ω), Jz (ω) are Fourier transformations of Ey, Hy, ρ and Jz, respectively. Solutions of above two equations are composed by two parts: the special solution which represents the incident wave from the charged particle and the general solution signifying reflected waves from the thin-film. Solving Eqs. (11) and (12), we obtain the incident waves of the particle in the frequency domain [39]: where β = ve/c, , kz = ω/ve, , sign is the sign-function.Then we consider general solutions. To satisfy the radiation boundary condition of the upper space, general solutions (the reflection wave) in region-I should be expressed as:
where , , A1 and B1 are coefficients to be determined by boundaries.In the thin-film and the substrate, fields can be obtained by solving the homogeneous wave equations—Eq. (11) and Eq. (12) with source components in the right side of equations being zero. In order to consider electromagnetic properties of the metamaterial thin-film or the metallic nano-film, ε0 and μ0 in those equations should be replaced by the ε = εr ε0 and μ = μr μ0 (μr = 1 for metal), respectively. Here εr and μr are respectively the relative permittivity and relative permeability discussed in section 2. In the thin-film (region-II), waves will be reflected by both interfaces of the thin-film, as a result of which the general solution the y-directional electric field and that of the magnetic field should be respectively expressed as:
in which , , A2,3 and B2,3 are coefficients to be determined by boundary conditions. The first and the second term in above equations denote upward wave and downward wave, respectively.In the substrate (region-III), waves will propagate in the downward direction without reflection, such that the y-directional electric field and magnetic field should respectively be expressed as:
in which , , εrs is the dielectric constant of the substrate, A4 and B4 are constants to be determined by boundary conditions.In each region, all other field components can be expressed in terms of Ey and Hy by applying the following equations [40]:
At the boundaries (both interfaces of the thin-film), the tangential (x-directional and y-directional) electromagnetic fields should satisfy following continuous conditions: To satisfy the above boundary conditions, the tangential wave-numbers in all regions should match: kyIII = kyII = kyI = ky, kzIII = kzII = kzI = kz. Substituting field expressions into above boundary conditions, we can get eight equations with eight variables (A1, ..., A4, B1, ..., B4): where X is a 8 × 8 coefficient matrix determined by boundary conditions, Y is the vector of unknowns [A1, ..., A4, B1, ..., B4]T, and Z is the vector representing the incident fields from the charged particle. After tedious but straightforward mathematical treatments, all unknowns (coefficients of fields) can be solved, and fields in all regions are then obtained.We mainly consider the EM waves propagating horizontally along the thin-film. According to Poynting’s theorem, the z-directional and y-directional radiation power density within the thin-film should be respectively expressed as:
and The radiation direction (θ) of the 2D CR on the thin-film can then be obtained by which is greater than 90° when Sz is negative (in the opposite direction of ve), indicating the reversed CR is generated. According to Eq. (28), Sz could be negative in cases that εr and μr are both negative, or one of them is negative. This is the theoretical base of the reversed CR in the LHMs and plasmonic mediums.Letting the determinant of coefficient matrix X of Eq. (27) to be zero (|X| = 0) and applying the relation of , one can get the dispersion equation, which governs the EM waves propagating along the thin-film:
in which , , .4. Numerical calculations
In this section, we present the calculation results based on equations obtained section 3. Applying the following reverse Fourier transformation:
we can get the field distributions in the structure for each frequency component, i.e., field maps in the frequency domain. The calculated contour maps of the Ez field on the thin-film (x=0) are shown in Fig. 7 and Fig. 8, which denote model-1 in the terahertz region and model-2 in the visible light region, respectively. For model-1, the 2D CR is in the forward direction (both wave-vector and EM energy flux are in the outgoing direction with θ < 90°) as the frequency is from 0.09 THz to 0.18 THz; while it is in the backward direction (wave-vector is opposite to EM energy flux direction and θ > 90°) as the frequency changes from 0.38 THz to 0.44 THz. For model-2, the 2D CR is in the forward direction (θ < 90°) as the frequency is from 670 THz to 750 THz, and it is in the backward direction (θ > 90°) as the frequency changes from 885 THz to 905 THz. The radiation angles agree well with the simulations results.Fig. 7 Analytic calculated contour maps of the Ez field on the surface of the metamaterial thin-film (model-1) for different frequencies in the frequency domain.
Fig. 8 Analytic calculated contour maps of the Ez field on the surface of the metal thin-film (model-2) for different frequencies in the frequency domain.
Figure 9 further illustrates the dependency of the radiation frequency on the direction θ, calculated based on Eq. (30). One can see that, for both model-1 and model-2, the radiation frequency increases monotonously with θ, while it changes quite slowly after θ is large enough (θ > 150° shown in the insets). Its reason will be given in the next section. We note that the radiation direction of the 2D CR differs remarkably from that of the 3D CR in the bulk metamaterials illustrated in [41]. It is because that the latter dealt with the radiation in infinite mediums, in which the radiation pattern is only determined by optical properties of the medium. Whereas for the 2D CR in the present paper, according to Eqs.(28–30), the radiation direction depends not only on the electromagnetic properties (εr and μr) of mediums but also on the structure boundaries, which determines the values of A2,3 and B2,3.
Fig. 9 Dependency of the frequency on the direction of the 2D CR (a) on the metamaterial thin-film and (b) on the metallic thin-film.
From the contour maps, we also note that, for the normal CR, wavelength on the thin-film decreases as the frequency increases, whereas for the reversed CR, the very opposite is true: wavelength increases with frequency. Analyses of these results will be given in the next section.
5. Analyses and discussions
In this section, we make discussions about the mechanism and requirements of the reversed CR illustrated in above sections. The 2D CRs on the thin-films are surface waves, characteristics of which are essentially determined the dispersion property of the structure. Based on Eq. (31), dispersion curves are calculated and illustrated in Fig. 10, in which subplots (a) and (b) show model-1 and model-2, respectively. In the figure, beam lines with electron energies of 80 keV (ve = 0.502c) and 40 keV (ve = 0.374c) are also depicted. The dispersion curves lying on the right side of the beam-line represent surface waves with phase velocity being less than beam velocity (vp < ve), indicating that the CR condition is satisfied. For model-1, three surface modes could be excited by the 80 keV electron-beam: the lowest mode (mode-I in Fig. 10(a)), the highest mode (mode-II in Fig. 10(a)), and the resonant mode. The lowest mode is the forward wave with the phase velocity and group (EM energy) velocity being in the same direction, indicating that the EM radiation is in the forward direction, which leads to the normal CR. While for the highest mode, dispersion curves lying on the right side of the beam-line are backward waves with the phase velocity and group velocity being in opposite directions. Thus the EM energy propagates in the backward direction, which is exactly the reason of the reversed CR. The resonant mode is caused by the magnetic resonance of the metamaterial at frequency of 0.219 THz. For comparison, dispersion curves of the case that the substrate is replaced by the metamaterial, which were widely investigated in previous literatures, are also shown. Under this condition, only the forward wave and the resonant mode exist, and the backward wave does not exist, indicating that the reversed 2D CR can not be generated on the surface of a bulk plasmonics medium. For model-2, two surface plasmonic modes could be excited by the 40 keV electron-beam. Similarly to that of model-1, the lower mode is the forward model (mode-I in Fig. 10(b)), which generates the normal CR, and the higher mode is the backward mode (mode-II in Fig. 10(b)), leading to the reversed CR. Thus whether normal CR or reversed CR is generated depends on whether forward mode or backward mode excited by the charged particle satisfies the CR condition. We note that, for model-1 and model-2 of the present paper, both normal CR and reversed CR will be excited on the thin-films since both forward and backward modes satisfy the CR condition. We also notice that the negative slopes of the backward dispersion curves are quite small when curves are close to the intersection (point B) with beam-line. At the intersection ‘B’, waves are exactly in the backward direction with θ = 180°. That explains why the frequency changes slowly for large radiation angles (150° < θ < 180°) as shown in Fig. 9.
Fig. 10 Calculated dispersion curves for (a) metamaterial thin-film scheme and (b) metallic thin-film scheme.
Then we look at the radiation wavelength depending on frequency. Based on the relation between the wavelength and phase velocity of surface waves (CRs) on the thin-film: λ = 2πvp/ω, we can get the dependence of wavelength on frequency:
For the forward wave, dkr/dω > 0, which leads to dλ/dω < 0, indicating that the radiation wavelength decreases as frequency increases. Whereas for the backward wave, dkr/dω < 0, leading to dλ/dω > 0, such that the radiation wavelength increases with the frequency. These are exactly what we have obtained in previous simulations and calculations.Now we consider the structure requirements of the reversed CR on the thin-film, which can also be obtained by examining the dispersion characteristics. Dependence of the dispersion curves on the thickness of the thin-film is shown Fig. 11, in which only backward waves are depicted. Subplots (a) and (b) illustrate model-1 and model-2, respectively. For both models, the negative slope (backward-wave) of the dispersion curves decreases gradually and finally disappears as the film thickness increases, see d=150μm in Fig. 11 (a) and d=50nm in Fig. 11 (b). It can be explained that the backward surface wave is originated from the coupling of the free-conduction-electrons on both interfaces of the thin-film [42]. As the film thickness increases, the coupling is gradually weaken and finally eliminated. Thus, only when the thickness is less than a certain value can the reversed 2D CR be generated.
Fig. 11 Calculated dispersion curves dependency on the thickness of the thin-film. (a) metamaterial thin-film case, (b) metallic thin-film case. Here only the high order mode with backward dispersion property is presented.
Lastly, we look forward to the applications of the reversed 2D CR. Like reversed 3D CRs, it is an effective way for particle detection and charged-beam diagnostic. Compared with 3D CR, the 2D CR is guided wave propagating along the surface, which means that it could be easier to be detected and measured. Another promising application is in near-field optics and integrated optics. The reversed 2D CR can be manipulated and steered on the thin-film with subwavelength transverse size, beyond the diffraction limit. This means that it could probably be developed as integrated light sources or devices on chip.
6. Conclusion
To summarize, we have revealed a reversed Cherenkov radiation on plasmonic thin-films in the terahertz region and in the visible light region. The mechanism and requirements of this radiation have been found out theoretically and verified by simulations. This two-dimensional reversed Cherenkov radiation may lead to attractive applications in physics and optics.
Funding
Natural Science Foundation of China (61471332, U1632150); Anhui Provincial Natural Science Foundation (1508085QF113); Chinese Universities Scientific Fund (WK2310000059).
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