Parameters used in this study.
We have numerically analyzed an electron beam (e-beam)-induced directional terahertz (THz) radiation from metamaterials. Here, we used metallic grating structures with graded depths, in which only one-way surface mode can be supported based on the spoof surface plasmon polariton (spoof SPP) concept and gives unique directional THz radiation. For numerical analysis, we used a simplified particle-in-cell (PIC) finite-difference time-domain (FDTD) method. First, we describe our simplified PIC-FDTD method in detail. Then, we show our results on the e-beam-induced directional THz radiation from graded grating with graded depths. By passing pulsed (bunched) e-beam along the grating surface, directional THz radiations are obtained from one side of the grating with shallower grooves. The direction of these radiations can be switched backward or forward by making the groove depth deeper or shallower. The spectra of these directional radiations are wideband and contain multiple sharp peaks. The deepest and the shallowest groove depths determine the lowest and the highest frequency of the radiation band, respectively. These unique radiation characteristics cannot be explained by the conventional Smith-Purcell radiation and should be attributed to the spoof SPP that originates from different locations on the graded grating.
- graded grating
- electron beam
- Smith-Purcell radiation
- particle-in-cell finite-difference time-domain
Recently, terahertz (THz) science and technology have been extensively studied from various viewpoints [1, 2]. The THz frequency range is generally considered to be the range of 0.1–10 THz. The electromagnetic (EM) waves that fall in the THz range can be utilized for various types of applications such as spectroscopy, nondestructive inspection, security, and information and communications technology. Optical techniques for a generation and detection of the THz radiation usually require ultrafast short-pulsed lasers. As an alternative way, vacuum electronic-based techniques have attracted much attention to develop next-generation table-top THz radiation sources . It has been known as Smith-Purcell radiation (SPR) since the 1950s that EM radiation can be obtained by passing electron beam (e-beam) accelerated at a relativistic speed along the surface of periodically corrugated metallic grating . The wavelength
On the other hand, quite significant progress has been made in the researches on metamaterials in recent years, and various novel optical effects have been proposed and demonstrated, such as negative refraction, superlensing, and optical cloakings [11, 12, 13]. Based on metamaterials’ concept, one can design a rich variety of optical materials with unique dispersion characters which cannot be obtained in nature. The metamaterials’ concept also offers new designing freedom of surface waves. It has been believed that surface waves like surface plasmon polaritons (SPPs) in the visible or near infrared cannot be supported in longer wavelength range like in THz because metals tend to behave as a perfect electric conductor (PEC). However, Pendry et al. showed that surface waves like SPPs could be supported even on PECs provided that there were arrays of corrugations or holes on metals [14, 15]. The dispersion relations of such surface waves resemble those of SPPs, and the surface waves introduced by Pendry et al. are usually called
Here, we show an e-beam-induced THz radiation from such graded grating based on PIC-FDTD method [3, 17]. We have obtained THz radiation with unique characteristics such as arbitrarily chosen bandwidth and unique directionality, which cannot be expected from the conventional theory developed for the SPR. Our findings may be utilized to develop novel e-beam-based THz radiation sources.
2. Numerical simulation
In this section, numerical simulation techniques employed in this study, simplified PIC-FDTD method, is described in detail, and parameters for our simulations are summarized.
2.1. Overview of simplified PIC-FDTD approach for the analysis of e-beam-induced THz radiation
The PIC-FDTD method has been widely used to study underlying physical mechanism of the Smith-Purcell superradiance [8, 9, 10]. To save computational time and memory, we have used simplified version of PIC-FDTD method [17, 18]. In our simplified model, the electron-bunch is treated as one negatively charged particle, the movement of the particle is restricted only in two-dimensional (2D) (x-y) plane, and only transverse electric (TE) mode, with
Figure 2 shows a typical Yee’s 2D uniform rectangular grid for TE mode.
The dielectric properties of metals are strongly dispersive; therefore, we utilized recursive convolution (RC) approach  to model metallic grating. By adopting Drude model, frequency dependence of dielectric permittivity of metal can be expressed as follows:
Since the Fourier-transformed electric susceptibility χ(τ) of Drude type of dispersion satisfies the condition for a recursive computation, the convolution in Eq. (15) can be solved in a recursive manner.
In the PIC-FDTD method, time-dependent Maxwell’s equations are coupled with the equation of motion of relativistic charged particles driven by the inertia and the Lorentz force and solved in a leapfrog manner similar to the main FDTD algorithm. In our simplified version, we assume the electron-bunch as one negatively charged particle with the following Gaussian spatial charge distribution:
In order to include the movement of the electron-bunch in the FDTD formalism, a current source term is added to Ampere’s law:
Figure 4 schematically summarizes our simplified PIC-FDTD simulation scheme. The solution of the time-dependent Maxwell’s equations gives spatial counter maps of EM fields and their time evolution. The solution of the equation of motion of relativistic electron-bunch gives its trajectory, and the continuity equation gives the current and charge densities required for Maxwell’s equations. The flowchart of our PIC-FDTD simulation is shown in Figure 5.
2.2. Analyzed models and parameters
In Figure 1, the analyzed 2D system and definitions of dimensions of the graded grating are schematically shown. In Table 1, parameters used in this study are summarized. The graded grating was assumed to be consisted of Ag, and the Drude model was adopted in order to model the dispersion characters of its dielectric function and solved by using RC scheme as discussed above. The plasma frequency (
|Grating period (
|Groove width (
|Groove depth (
|Number of grooves (
|Plasma frequency of Ag (
||2.2 × 103 THz|
|Collision frequency of Ag (
|Electron-bunch energy||30 keV|
|Half width of electron-bunch (
|Bunch-grating distance (
A 20-μm-wide (
3. Results and discussions
In the e-beam-induced radiations from conventional periodic grating, there are two mechanisms. One is the so-called Smith-Purcell radiation emitted while the e-beam is passing over the grating. The radiation angle and its frequency satisfy Eq. (1). The other is the scattering of surface waves at both ends of the grating long after the e-beam moved away from the grating. These long-lived surface waves can propagate back and forth on the grating surface and can be emitted repeatedly even long after the e-beam has moved away from the grating as long as the surface waves can live. The frequency of the scattered surface waves is determined by the intersection of the dispersion curves of the surface wave and the beam line. Here, we are interested in the second mechanism long after the e-beam has moved away from the gratings, but the groove depths are gradually graded, and, therefore, the dispersion curves of the surface waves induced by an e-beam cannot be uniquely determined.
Figure 6(a) and (b) shows snapshot contour maps of the
As discussed by Gan et al. , the dispersion relations of the surface waves on these graded gratings are different at each location on the grating, and thus the frequencies of the e-beam-induced surface waves should also be different at different locations. These surface modes with different frequency components originated from different locations on the graded gratings can propagate toward the side with shallower groove depth due to the cutoff nature as reported by Gan et al. , which may give a mechanism of the directional radiation obtained only from the shallow end of the graded grating.
Figure 7(a) shows the time-domain
In order to clarify the nature of the surface modes, we have also investigated near fields on the different locations on the graded gratings. Figure 8 shows Fourier-transformed spectra of the near-field (surface wave)
In order to reveal from where each mode originate in the graded grating, we excited the system with quasi-monochromatic EM pulse and monitored long enough until the initial pulse damped and only long-lived surface modes survive. Figure 9 shows spatial distributions of
The fact that the dispersion characters and frequency of the e-beam-induced surface mode is quite sensitive to the local environment of the grooves suggests that one can design the radiation frequency of the directional radiation from graded gratings by appropriately choosing groove parameters of the grating. Figure 10 shows the Fourier-transformed spectra of the far-field radiation from graded gratings with different groove parameters: GG[100, 304, 6], GG[100, 236, 4], GG[100, 168, 2], GG[168, 236, 2], GG[50, 186, 4], and GG[50, 118, 2]. Roughly speaking, the deepest and shallowest grooves determine the lowest and highest frequencies of the radiation, respectively. This can be confirmed by comparing spectra for GG[100, 304, 6], GG[100, 236, 4], and GG[100, 168, 2], for example. The highest frequency of these radiations is almost the same ∼0.40 THz and determined by their common shallowest groove depth
We have numerically analyzed the e-beam-induced directional THz radiation from metallic grating structures with graded depths. We used a simplified PIC-FDTD method for numerical analysis to save computational time and memory, and the detailed description of our method is given here. In our simplified model, the electron-bunch is treated as one negatively charged particle with Gaussian charge distribution, and its movement is restricted only in 2D space, and only TE mode, with
Our results show unique directional THz radiation from graded gratings. By passing pulsed (bunched) e-beam along the grating surface, directional THz radiations are obtained from one side of the grating with shallower grooves. The direction of these radiations can be switched backward or forward by making the groove depth deeper or shallower. The spectra of these directional radiations are wideband and contain multiple sharp peaks. The deepest and the shallowest groove depths determine the lowest and the highest frequency of the radiation band, respectively. These unique radiation characteristics cannot be explained by the conventional Smith-Purcell radiation and should be attributed to the spoof SPP that originates from different locations on the graded grating. The unique e-beam-induced radiation from metamaterials based on spoof SPP’s concept may open a way for a development of novel types of THz radiation sources.
This work is partly supported from Okasan-Kato Foundation. The presented works have been carried out with graduate students who formerly belonged and currently belong to our research group, Okajima, Omura, and Yoshida.
Conflict of interest
There is no conflict of interest.
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