1. Introduction

Recently, terahertz (THz) science and technology have been extensively studied from various viewpoints [1]. Optical techniques for a generation and detection of the THz radiation usually require ultrafast short-pulsed lasers such as femtosecond lasers. As an alternative way, techniques using electron-beam (e-beam) have been attracting much attention to develop compact THz radiation source. Especially, the technique based on Smith-Purcell (SP) effect [2], radiation due to the interaction between e-beam and induced surface waves on the metal grating, has been expected as a promising technology to realize a next-generation table-top THz free electron laser (THz-FEL) [3–5]. On the other hand, recent progress in the researches on metamaterials is quite significant and various novel optical materials and devices have been proposed and demonstrated, such as artificial magnetic resonance, negative index materials (NIMs) and optical cloakings [6–8]. Since Pendry et al. put forward a concept of “spoof surface plasmon polariton” (or spoof SPP) in 2005 [9,10], many researchers have studied on the spoof SPP in the THz frequency range. Gan et al. reported a mechanism for slowing down THz surface waves based on metallic grating structures with graded depths (graded grating or GG) such as shown in Fig. 1(a), in which dispersion curves and cutoff frequencies of the spoof SPP can be arbitrarily designed with varying groove depth [11].

 

Fig. 1 (a) Schematic representation of the analyzed 2D system and definitions of dimensions of the graded grating. The Ag graded grating is placed at the center of the bottom of analyzed domain in vacuum. The total area of the analyzed domain has a dimension of approximately 20 mm × 40 mm. The grating period (Λ) and groove width (s) are 170 μm and 60 μm, respectively. The number of grooves of the grating (N) is set to be 35. The groove depth (d) is gradually made deeper or shallower and the shallowest and the deepest groove depths (d_s and d_d) are variable parameters. Δd is the groove depth variation. A 20-μm-wide bunched e-beam with Gaussian charge distribution was sent 20-μm (w) above the grating. (b) Dispersion relations of induced surface waves on periodic grating with d = 100, 168, and 236 μm, along with that of the e-beam (beam line).

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In this letter, we report on the numerical analysis of the e-beam induced THz radiation from the graded grating. 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 SP effect. Our findings may lead to the development of novel e-beam based THz radiation source.

2. Numerical analysis

The particle-in-cell finite-difference time-domain (PIC-FDTD) method is widely used to study underlying physical mechanism of the SP effect and to develop efficient table-top type of THz-FELs [12–14]. In this study, we have used simplified PIC-FDTD method [15] for the sake of saving computational time and memory. In our simplified model, the electron-bunch is treated as one particle, and the movement of electron-bunch is restricted only in one (x-y) plane, and only transverse electric mode, with E_x, E_y, H_z fields, has been analyzed. Although quite simplified, our simulation code works well enough to successfully reproduce previously reported SP superradiance [16].

In Fig. 1(a), schematic representation of the analyzed two-dimensional (2D) system and definitions of dimensions of the graded grating are shown. The graded grating is placed at the bottom of the analyzed domain in vacuum. The total area of the analyzed domain has a dimension of approximately 20 mm × 40 mm. The graded grating was assumed to be consisted of Ag, and the Drude model was adopted in order to model the dispersion characters of the dielectric function of it. The plasma frequency and the collision frequency were set to be 2.2 × 10³ and 5.4 THz, respectively [17]. The grating period (Λ) and the groove width (s) are 170 μm and 60 μm, respectively. The number of grooves of the grating (N) is set to be 35. The groove depth (d) is gradually made deeper or shallower and the shallowest and the deepest groove depths (d_s and d_d) are variable parameters. Here, we introduce another parameter, the groove depth variation (Δd), to uniquely define each analyzed graded gratings, but since in this study the number of grooves of the grating (N) is fixed, the Δd is uniquely determined once both d_s and d_d are set. The positive Δd corresponds to the increasing the groove depth towards the right as shown in Fig. 1. Each graded grating can be characterized by a set of these variables [d_s, d_d, Δd], here these are in μm, and we represent each analyzed system as GG[d_s, d_d, Δd].

In this study, we analyzed the systems of GG[100, 168, 2], GG[100, 168, −2], GG[168, 236, 2], and GG[100, 236, 4], as discussed in the following sections. In Fig. 1(b), the dispersion relations of induced surface waves on the periodic grating [4] with depths d of 100, 168, and 236 μm corresponding to the d_s or d_d in our systems are shown. That of e-beam (beam line) is also shown. The wave number k is normalized by K = 2π/Λ. It has been well recognized that the frequency of the e-beam induced surface wave on the grating is determined by an intersection point between the dispersion curve of the surface wave and a beam line of the e-beam [4]. Therefore, according to Fig. 1(b), the frequencies of the induced surface waves on the gratings of d = 100, 168, and 236 μm are expected to be about 0.40, 0.35, and 0.27 THz, respectively. We expect radiation around these frequencies from our GG systems. However, we should note that the dispersion relations in Fig. 1(b) and the expected values of the frequencies of the surface waves are calculated for the conventional periodic gratings.

A 20-μm-wide bunched e-beam with Gaussian charge distribution was sent 20-μm (w) above the grating at the relativistic speed. The maximum current density at the center of the e-beam was assumed to be 1.0 × 10⁶ A/m² and the acceleration energy of the e-beam is 30 keV, which is comparable to the recent experimental condition [3]. In order to make the underlying physical mechanism clear, here we have assumed that the e-beam is short-pulsed. The 20-μm longitudinal profile of a bunched e-beam corresponds to the pulse duration of ~200 fs. Such a subpicosecond pulsed e-beam had been achieved in the real experimental measurements.

3. Results and discussion

3.1 THz radiation from graded grating system

Figure 2(a) shows the snapshot of magnetic field H_z after the e-beam passed over GG[100, 168, 2]. It is clearly observed that the surface waves are induced on the grating, and a relatively intense far-field radiation is emitted only from the left-end of the grating (backward direction). The other far-field radiation from the right-end of or above the grating is relatively weak, and it is not simple cylindrical waves or their interference. As depicted in Fig. 1(b), the dispersion relations of the surface waves are not the same for different groove depth, therefore for the graded grating, the frequencies of the surface waves should be different from point to point on the grating [11]. Therefore the relatively wide-band surface wave can be produced on the graded grating as a superposition of these surface waves with different frequencies, as discussed above. In addition, since the dispersion curves of the surface waves are not the same for each groove of the graded grating, the cutoff frequencies of the surface waves are different at different positions on the grating. The frequency and the cutoff frequency of the surface wave are lower at deeper groove depth of the grating, and therefore the induced surface wave cannot propagate toward the grating with the deeper groove depth, and propagate only toward the grating with the shallower groove depth. In GG[100, 168, 2], since the induced surface wave can propagate only to the left, most of the far-field radiations from the grating are emitted from the left-end of the grating. On the contrary, for a radiation from a periodic grating, namely for traditional SP effect, the frequency of the induced surface wave is determined to have a particular single value, and therefore the overall emission from the grating has simple cylindrical waves and their interference. Moreover, since the cutoff frequencies of the surface wave are identical everywhere in the grating, the surface wave can propagate back and forth from side to side on all over the grating, and therefore the far-field radiation can be emitted from both ends of the grating [12].

 

Fig. 2 (a) Snapshot of H_z field after the e-beam passed over GG[100, 168, 2]. (b) FFT spectra of near field (surface wave) H_z monitored at several positions 10 μm above each groove with d = 160, 150, 140, 130, 120, 110, or 100 μm, along with that of the far-field radiation monitored at the observation point P (from top to bottom), in GG[100, 168, 2]. (c) Spatial distributions of H_z fields long after the exciting quasi-monochromatic electromagnetic pulse has been damped when the frequency of the mode is 0.314 (A), 0.329 (B), and 0.347 (C) THz (from top to bottom) in GG[100, 168, 2]. Each mode and its name (A, B, C) correspond to the peaks in the far-field radiation spectrum in the bottom panel of (b).

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To investigate the radiation characteristics from the graded grating, we set an observation point at 2000 μm from the left-end and 10 μm from the top of the grating (observation point P) and monitored the temporal response of H_z. In the bottom panel of Fig. 2(b), a FFT spectrum of H_z field in GG[100, 168, 2] monitored at the observation point P is shown. It can be seen that the radiation spectrum is wide-band and are consisted of several sharp peaks. Each frequency component of the radiation might be attributed to the induced surface waves with different frequency originated at different points on the graded grating, and therefore should be estimated from the operation points shown in Fig. 1(b). However, agreement of these frequencies is not so well. The surface mode in the graded grating is sensitive to the local environment of the groove and should be different from the conventional periodic grating system. Therefore, only a rough estimate is valid. On the other hand, for a periodic grating, the radiation at a particular radiation angle has only a particular frequency and its harmonics that can be uniquely determined with an acceleration of the e-beam and a period of the grating. Therefore it is a unique property for the interaction of e-beam with the graded grating that the radiation spectrum observed at a particular position has several sharp peaks and is wide-band. In Fig. 2(b) (except a bottom figure), FFT spectra of near field (surface wave) H_z monitored at several positions 10 μm above each groove with d = 160, 150, 140, 130, 120, 110, or 100 μm (from top to bottom) in GG[100, 168, 2] are shown. As the observation point moves toward the left, the higher frequency components and therefore more peaks start to appear. These results confirm that the each induced surface wave cannot propagate toward the right with the deeper groove depth and can propagate only toward the left. The spectrum of H_z monitored at the left-end of the grating (d = 100 μm) is almost the same with the far-field radiation spectrum (bottom). This confirms that the far-field radiation is attributed to an emitted surface wave from the left-end of the grating. In order to investigate positions where each mode appeared in the radiation spectrum originates, a quasi-monochromatic exciting electromagnetic pulse with narrow-band spectra correspond to each peak in the far-field radiation spectrum is launched in overall the analyzed domain in GG[100, 168, 2] and the temporal responses are monitored. The frequency of the exciting pulse is 0.314 (A), 0.329 (B), and 0.347 (C) THz, where each mode and its name (A, B, C) correspond to the peaks in the radiation spectrum in Fig. 2(b). In Fig. 2(c), the spatial distributions of H_z fields long after the exciting pulse has been damped are shown (from top to bottom). It is seen that each mode originates at different locations of the grating, and as the frequency of the mode gets higher the location moves toward the left with shallower groove depth. This result supports the fact discussed above that the frequency of the surface wave are lower at deeper groove depth of the grating.

3.2 Influence of varying grating parameter on THz radiation characteristics

Figure 3(a) shows a snapshot of H_z field observed in GG[100, 168, −2] which is a left-right reversal of GG[100, 168, 2]. In this system, it can be seen that the relatively intense far-field radiation is emitted only from the right-end of the grating (forward direction). This agrees well with the fact discussed above that the induced surface wave cannot propagate toward the deeper groove depth. We set an observation point Q at 2000 μm from the right-end and 10 μm from the top of the grating and monitored the temporal response of the far-field radiation. In Fig. 3(b), the FFT spectrum of H_z field at the point Q in GG[100, 168, −2] (dashed line), along with that monitored at the point P in GG[100, 168, 2] (i.e. reproduction of the bottom spectrum in Fig. 2(b)) (solid line), is shown. It can be seen that both spectra are almost identical. These results indicate that either left- or right-end of the grating can be chosen as the emitting edge by appropriately designing the groove depth variation (Δd).

 

Fig. 3 (a) Snapshot of H_z field after the e-beam passed over GG[100, 168, −2]. (b) FFT spectrum of H_z field monitored at the observation point Q in GG[100, 168, −2] (dashed line), along with that monitored at the point P in GG[100, 168, 2] (solid line). (c) FFT spectra of H_z fields monitored at the observation point P in GG[100, 168, 2] (solid line) and GG[168, 236, 2] (dashed line). (d) FFT spectra of H_z fields monitored at the observation point P in GG[100, 236, 4].

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In order to further investigate how the grating parameters [d_s, d_d, Δd] affect the THz radiation characteristics, we have analyzed the FFT spectra of H_z fields monitored at the observation point P in GG[100, 168, 2] (solid line) and GG[168, 236, 2] (dashed line) as shown in Fig. 3(c). Although the shallowest and the deepest groove depths (d_s and d_d) are changed, they show common general characteristics, that is, the far-field radiation spectrum is wide-band with several sharp peaks as can be seen in Fig. 2(b). It is shown that with deepening the overall groove depths (d) through the grating the radiation spectrum shifts toward lower frequency, and since the d_d of GG[100, 168, 2] is equal to the d_s of GG[168, 236, 2], both spectra do not overlap each other. Since the overall variations of the groove depths through the gratings Δd × N are exactly same for these two systems, it can be seen that the bandwidths of the spectra are almost the same. We had also analyzed GG[100, 236, 4] of which Δd is twice of above two systems and of which Δd × N is a summation of those of the above two systems. In Fig. 3(d), the FFT spectra of H_z fields monitored at the observation point P in GG[100, 236, 4] are shown. It can be seen that similarly the far-field radiation spectrum is wide-band and has several sharp peaks. In addition, the bandwidth of the spectrum is broadened due to the large range of the total variation of the groove depths through the grating Δd × N, and approximately the spectral range is a superposition of those of the above two systems GG[100, 168, 2] and GG[168, 236, 2]. These results demonstrate that the deepest and the shallowest groove depths determine the lowest and the highest frequencies of the radiation, respectively. Therefore the desired THz far-field radiation that has the wide band spectrum with several sharp peaks can be obtained by appropriately designing the graded grating structure to support desired spoof SPP and may lead to a development of the novel THz radiation source. In addition, another grating parameters such as grating period (Λ) or width (s) also changes the dispersion curves of the surface waves on the grating [4], thus alters radiation characteristics. Changing the acceleration energy of the e-beam also shifts the operation points and radiation characteristics should be tuned. Detailed analysis of the influence of these parameters on the radiation characteristics is currently under study and will be reported elsewhere.

4. Conclusions

In conclusions, we have numerically analyzed, based on a simplified PIC-FDTD method, an e-beam induced THz radiation from a graded grating with monotonically varied depth. Upon exciting with e-beam, directional THz radiations with wide-band spectrum containing several sharp peaks are obtained only from the one of the edge of the grating, which cannot be expected from the conventional theory of SP radiation. It was clarified that each modes originate from different locations on the graded grating and can propagate toward only the shallower groove as a surface wave, reflecting different dispersion characteristics and the cutoff frequencies of spoof SPP at each locations, and all of these modes eventually emitted from the one of the edge of the graded grating with the shallowest groove. These directional radiations can be directed toward either backward or forward by making the groove depth deeper or shallower, namely appropriately designing the groove depth variation (Δd). The lowest and the highest frequencies of the radiation can be chosen by appropriately changing the deepest and the shallowest groove depths, respectively. These unique radiations cannot be obtained from the uniformly grooved grating. Our findings may open the way for a development of novel THz radiation source based on the spoof SPP on the appropriately designed metallic grating structure or metasurfaces.