Enhanced coherent terahertz Smith-Purcell superradiation excited by two electron-beams

Yaxin Zhang; Liang Dong

doi:10.1364/OE.20.022627

1. Introduction

Recently, Terahertz (THz) science and technology has become a subject of a broad interest due to its potential applications in security checking, nondestructive testing, imaging and wireless communication etc. The strong demand of THz applications intrigues great interest in the development of intense, compact and tunable THz sources. As a promising alternative in the development of THz radiation sources, linear electron-beam driven source provides a good way. Such kind of radiation sources, including backward-wave oscillators (BWO), extended interaction oscillators (EIO) and orotron etc, has attracted many attentions. However, their performances are limited by the increased threshold current along with the increase of the radiation frequency. The performances can be improved by using high-harmonic radiation. It is well known that in an open diffraction grating, incoherent spontaneous Smith-Purcell (SP) radiation [1–5] is created as an electron passing close over the grating. The coherent radiation sources called Smith-Purcell free-electron lasers had been developed based on such phenomenon together with the continuous electron beam [6–9]. It is known that the spontaneous SP radiation contains a broad continuous frequency band and the radiation wavelength ( $λ$ ) corresponding to the observed angle ( $θ$ ) and period of grating ( $L$ ) can be described as $λ = L / | n | (1 / β - \cos θ)$ . Meanwhile the periodic electron-bunches contain frequency harmonics components. If such electron bunches with the harmonic frequency equivalent to the SP frequency pass through the grating, coherent monochromatic superradiation at relevant frequency and angle can be observed. The SP superradiation [10–12] provides a promising way to realize the compact, tunable radiation source which can work at high harmonic. In these years, there are many papers focusing on this topic. In 2007, there were two papers that proposed the THz SP superradiation excited by counter-streaming electron beams passing over the sub-wavelength holes array [13,14] and in 2010 there was a paper that presented the model of THz SP superradiation sources with two-section structure [15].

In this paper, we studied the enhanced SP superradiation excited by two pre-bunched electron-beams passing through 1-D subwavelength holes array (SHA) [16–23]. The results show that the two electron beams can be coupled with each other through the holes array to enhance the intensity of the superradiation. About twice higher supperradiation compared with that excited by one electron beam can be obtained. Moreover, the required current density for generation of such radiation at 0.62 THz is as low as 20A/cm².

2. The model and interaction

In this paper we extend the structure of reference [15] as shown in Fig. 1(a) and Fig. 1(b). The two bi-gratings structure acts as the modulation area. The two longitudinal direct-current (DC) electron-beams (e-beam) emitted from the cathodes will inject, synchronize and interact with the electromagnetic (EM) modes of the bi-gratings to form the two pre-bunched electron beams. The 1-D open sub-wavelength holes array acts as the radiation area.

Fig. 1 (a) The 3-D geometry. (b) the 2-D sketch map.(c) the Smith-Purcell radiation when two electron bunches pass close over the holes array.(d) the comparison of theoretical SP law and simulation results.

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As discussed in the previous section, when an electron bunch passes through the surface of the open grating, the broad band Smith-Purcell radiation can be excited. The relationship between the excitation wavelength and the observation angle (θ) as well as the period of structure (L) can be formulated as:

λ = L / | n | (1 / β - \cos θ)

Where n is the space harmonic order of electron bunch (n = 0, ± 1, ± 2, ± 3….).

In this structure, the 1-D sub-wavelength holes array has been applied instead of grating. So, the first step of simulation is to verify the possibility of the generation of SP radiation upon this 1-D sub-wavelength holes array. In this paper, the 3-D simulation has been performed with a fully electromagnetic particle-in-cell (PIC) [24] code Chipic which is a already mature and ready for commercial code developed based on the finite-difference time-domain (FDTD) [25] method to simulate the interaction and radiation in this structure. Chipic is a time-domain code for studying nonlinear interactions among electrons and electromagnetic fields, which can self-consistently calculate the motions and collective effects of a large number of charged particles and time-domain electromagnetic fields [26].

The Fig. 1(c) shows the contour map of radiated magnetic field Bx. As shown in Fig. 1(c) the radiation excited by two electron-bunches in the 1-D sub-wavelength holes array is just SP-like radiation. Figure 1(d) shows the comparison between the simulation and the theoretical calculation. As shown in Fig. 1(d), the distribution of such radiation (the red dotted line) shows good agreement with that calculated by using Eq. (1) (the black solid line). These results reveal that the SP radiation can be excited by two-electron bunches passing through upper and lower sides of the surface. So it can be assumed that when the two pre-bunched beams pass through the surface of such structure, the coherent SP superradiation could be generated.

The second step is to study the dispersion characteristics of the bi-grating. The layout of the designed structure has been spatially subscribed into three regions as shown in Fig. 2(a) . The EM fields inside each region can be expressed as:

Fig. 2 (a) the structure of bi-grating (b) dispersion Brillouin diagram: the blue line is the calculated dispersion curve of the bi-grating, the red cross line is the beam line with beam energy 50kV and the point P is the interaction point of the dispersion curve. The area between P1 and P2 is the radiation area of the SHA.

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In region I

E_{z}^{Ι} {=Q}^{Ι} \frac{\sin k_{c x} (\frac{A}{2} + D_{1} - x)}{\sin k_{c x} (\frac{A}{2} + D_{1})} \sin (k_{y} y); H_{y}^{Ι} = \frac{j ω ε}{k_{c x}} Q^{Ι} \frac{\cos k_{c x} (\frac{A}{2} + D_{1} - x)}{\sin k_{c x} (\frac{A}{2} + D_{1})} \sin (k_{y} y)

In region II

{\begin{cases} E_{z}^{Ι Ι} = \sum_{n = - \infty}^{\infty} [Q^{Ι Ι}_{n} \cos h (k_{x n} x) + P^{Ι Ι}_{n} \sin h (k_{x n} x)] \sin (k_{y} y) e^{- j k_{z n} z} \\ H_{y}^{Ι Ι} = \sum_{n = - \infty}^{\infty} \frac{j ω ε k_{x n}}{k^{2}_{z n} - k^{2}} [Q^{Ι Ι}_{n} \sin h (k_{x n} x) + P^{Ι Ι}_{n} \cos h (k_{x n} x)] \sin (k_{y} y) e^{- j k_{z n} z} \end{cases}

In region III

E_{z}^{Ι Ι Ι} {=Q}^{Ι Ι Ι} \frac{\sin k_{c x} (\frac{A}{2} + D_{1} + x)}{\sin k_{c x} (\frac{A}{2} + D_{1})} \sin (k_{y} y); H_{y}^{Ι Ι Ι} =- \frac{j ω ε}{k_{c x}} Q^{Ι Ι Ι} \frac{\cos k_{c x} (\frac{A}{2} + D_{1} + x)}{\sin k_{c x} (\frac{A}{2} + D_{1})} \sin (k_{y} y)

By applying the boundary conditions the dispersion relation can be obtained as:

\frac{\cot (k_{0} D_{1})}{k_{0}} = \frac{W}{L_{1}} \sum_{n = - \infty}^{\infty} \sin c^{2} (\frac{k_{z n} W}{2}) \frac{k_{y n}}{k^{2}_{z n} - k^{2}_{0}} \tan h (\frac{k_{y n} A}{2})

where

k_{z n} = k_{z 0} + 2 n π / L_{1}

,

k^{2}_{x n} = k^{2}_{0} - k^{2}_{z n} - k^{2}_{y}

,

k_{y} = \frac{π}{B}

,

k_{0} = 2 π f / c

,c is the velocity of the light.

The calculated dispersion line is shown in Fig. 2(b) (blue line is for the mode in bi-grating) and the parameters of the simulation are shown in Table 1 . The electron bunch with a fixed velocity induces a set of evanescent waves that contain plenty of frequency components. According to the Bloch-Floquet theorem, these waves should be expanded into infinite space harmonics corresponding to the n in Eq. (1) in order to match the boundary condition of periodic structure. The Smith-Purcell radiation region is confined between the positive and negative light lines as illustrated in Fig. 2(b). Only the space harmonics that falls into this region can radiate out from the surface of the SHA. In the Fig. 2(b), it can be found that there are two red lines, one is for the DC e-beams, and the other is for the first negative space harmonic (n = −1) of electron bunch. As shown in this figure, the the SP radiation corresponding frequency region (from P1 to P2) for the first negative space harmonic (n = −1) bunch is from 0.3 THz to 0.75 THz. Moreover, the SP radiation frequency region can be also obtained by calculation of the Eq. (1).

Table 1. Main parameters of the simulation

View Table

When the longitudinal direct-current (DC) e-beam is injected in the modulation area, the fundamental mode of the symmetric gratings will be excited. The synchronization and interaction will occur when the phase velocity of the mode matches the velocity of the e-beam (labeled as interaction point in Fig. 2(b)).

During the interaction, the DC e-beams exchange energy with the mode so that the velocity and density of the DC e-beams will be modulated and pre-bunched. A bundle of frequency harmonics components (lω₀, l = 1,2,3,4…) is included in the pre-bunched electron beam.

As shown in Fig. 2(b), the 2nd frequency harmonics component of pre-bunched e-beams is located at the radiation point within the SP radiation frequency band where superradiation can be excited.

Secondly, the simulation of the interaction in the modulation area has been carried out as shown in Fig. 3 . During the interaction process, the two DC e-beams are found to synchronize and interact with the mode and then are gradually bunched as shown in Fig. 3(a). It is clear that the interaction frequency agrees very well with the theory ((Fig. 3(b) and Fig. 3(c)). Moreover, in the bi-grating structure that waves along the upper and downward gratings can be coupled so that the coupling impedance and intensity of the wave excited by the e-beam are so strong as to insure the formation of the well-bunched electron beam (Fig. 3(d)) .

Fig. 3 Simulation of interaction in bi-grating structure.(a)The e-beams phasespace in bi-grating. (b)The time evolution of the electric field E_z(t). (c)The associated FFT of the E_z(t). (d)The contour map of E_z.

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3. The enhanced coherent THz superradiation excited by two pre-bunched electron beams

In the section above, the study on the interaction in the modulation area has been carried out. In order to study the superradiation excited by the two pre-bunched e-beams, in this section the simulation of radiation excited by ideal periodical electron bunches with fundamental frequency 0.31THz has been carried out as shown in Fig. 4 .

Fig. 4 The simulation model of superradiation excited by ideal periodical electron bunches

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The simulation model consists of three parts: absorbing boundary, cathodes and 1-D subwavelength holes array.

The theoretically calculated emission angle for the supperradiation is 60.5°, as shown in Fig. 5(a) . The blue curve represents the simulated Bx field as a function of the observation angle. The observation angle of the peak Bx field is in good agreement with that at the supperradiation point. Figure 5(b) and 5(c) show the contour map and intensity of the radiated fields. The results illustrate that the radiation out from the surface of the designed structure at almost the same angle as the one calculated theoretically.

Fig. 5 The simulation results of ideal electron beam bunches superradiation.(a)Distribution of superradiation frequency and it’s Bx(t) field amplitude.(b)Contour map when the periodic electron bunches passing over the 1-D holes array.(c)Time evolution of the Ez(t) field at the radiation angle and associated FFT.

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Moreover, the frequency spectrum shows the intensity of 2nd harmonics (0.62 THz) is much higher than that of fundamental harmonic (0.31THz) as shown in Fig. 5(c). These results confirm that ideal periodic electron bunches can excite the superradiation when passing through the 1-D holes array.

Based on the results above, the next step of the simulation is to analyze the superradiation excited by two pre-bunched electron beams. Figure 6 shows the simulation model and Fig. 7 shows the simulated results of the radiation process. It is clearly that the well pre-bunched e-beams have passed over the 1-D subwavelength holes array in Fig. 7(a). As shown in Fig. 7(b), it can be found that the contour map of the radiated fields is nearly the same as those shown in Fig. 5(b). Moreover, the spectrum and intensity of the radiated fields ((Fig. 7(d) and 7(c)) demonstrate that the 2nd harmonic frequency component dominate the radiation. These simulation results show that in such system, the superradiation can be excited by the two pre-bunched e-beams with a fixed angle at the 2nd harmonic frequency.

Fig. 6 The simulation model of superradiation in the whole system

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Fig. 7 The simulation results of two pre-bunched electron beams superradiation. (a).The process of two well bunched electron beam passing over the 1-D holes array. (b)The contour map at 4.245ns when bunched electron beam passing over the holes array.(c)Time evolution of the Ey(t) field at the radiation angle and associated FFT.

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As we known, in the holes array the field can penetrate from the upper space to the lower space, so the two-beams can be coupled with each other through the 1-D holes array as shown in Fig. 8(a) . Consequently intensity of the superradiation is influenced by the coupling of the field. The study on the comparison between the intensities of the longitudinal electronic fields of the superradiation excited by one pre-bunched e-beam and two pre-bunched e-beams has been carried out. From the comparison, it can be found that the intensity of the radiated fields excited by two e-beams is about twice higher than that excited by a single e-beam (Fig. 8(b)). This result suggests significant enhancement of the coherent superradiation by using the two beams. Moreover, the spectrum analysis in Fig. 8(c) shows that the intensity of 1st harmonics (the evanescent wave) is still very strong in the one e-beam case, which indicates the intensity of the supperadiation is not strong enough to suppress the 1st harmonics. However, the coupling of the two beams enhances the superradiation so that the intensity of supperadiation is much stronger than that of 1st harmonic as shown in Fig. 8(d). These results confirm that enhanced coherent THz superradiation can be obtained in this system.

Fig. 8 (a) the comparison of the intensites of Ez(t) field excited by one pre-bunched beam and two beams (b) contour map of the field in the holes array (c) the frequency spectrum of radiation excited by one beam (d) the frequency spectrum of radiation excited by two beams

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4. Conclusions

The enhanced coherent THz Smith-Purcell superradiation is discussed in this paper. The simulation results suggest that the two bi-gratings structure provides beam-wave interaction and modulation to from the well pre-bunched e-beams. The coherent THz Smith-Purcell superradiation is significantly enhanced when these well pre-bunched beams pass through the 1-D subwavelength holes array. Because of the coupling between the two e-beams, the intensity of radiated field is twice higher than that excited by one e-beam due to the coupling between the two e-beams. Moreover, as low as 20A/cm² current density is required to generate 0.62THz radiation with relative high power. Along with this concept, high-power and compact THz radiation sources can be realized by SP superradiation excited by multiple-pre-bunched e-beams.

Acknowledgments

This work is supported by National Natural Science Foundation of China under Contract No. 61001031 and National High-tech Research and Development Projects 2011AA010204.

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Parameter	Symbol	Value	Unit	Parameter	Symbol	Value	Unit
Modulation Area				Radiation Area
Grating Period	L1	0.3	mm	Hole Period	L2	0.25	mm
Grating Depth	D1	0.2	mm	Hole Depth	h	0.25	mm
Grating Width	W1	0.15	mm	Hole Width	a	0.15	mm
Cavity Width	B	1.1	mm	Hole Length	b	0.6	mm
Number of Period	N1	50	——	Number of Period	N2	35	——
Electron Energy				50kV
Beam Current				20A/cm²

Enhanced coherent terahertz Smith-Purcell superradiation excited by two electron-beams

Abstract

1. Introduction

2. The model and interaction

3. The enhanced coherent THz superradiation excited by two pre-bunched electron beams

4. Conclusions

Acknowledgments

References and links

Cited By

Figures (8)

Tables (1)

Equations (5)

Optics Express