THz radiation from two electron-beams interaction within a bi-grating and a sub-wavelength holes array composite sandwich structure

Yaxin Zhang; Y. Zhou; L. Dong

doi:10.1364/OE.21.021951

1. Introduction

Terahertz (THz) radiation, which is a very attractive electromagnetic (EM) spectral region for advanced applications such as security checking, biomedical analysis, wireless communication, and other areas, has attracted considerable attention during the last decade. The development of intense, compact, and tunable coherent THz-wave sources is of great interest for these applications [1–5]. Great efforts have been made for THz generation, and numerous papers covering various aspects of this topic have been reported. As a promising alternative, a linear electron-beam- (e-beam) driven source provides a good way to realize THz radiation sources. In recent years, significant progress has been made [6–10]. However, there are still many problems limiting the development of e-beam-driven THz sources, such as starting current density, efficiency, power level, and so on. Nowadays, the successful exploration of the sub-wavelength holes array (SHA) [11–15] may lead to a clearer perspective of this kind of THz radiation source. The SHA can support slow propagating surface waves called mimicking surface plasmons (MSP) [11,16–20], which gives the possibility of electron beam–MSP interaction that is able to generate THz waves. What is important is that the SHA can also support multi-e-beam working, so a few papers have presented studies on THz radiation from two e-beams interacting in an SHA [9,16,21]. Nevertheless, the SHA is an open structure, the starting current density is still at a high-level for 0.3 THz frequency (>50 A/cm²), and the efficiency is not likely to be ideal. Meanwhile, we know that as a traditional and effective slow-wave structure, a bi-grating structure is very suitable for one-electron-beam interaction at the THz region for surface wave coupling between the two gratings [21,22]. However, for bi-grating, output power generated from e-beam interaction should be improved. In order to improve the output power and efficiency of e-beam-driven THz sources, multi-beam interaction within a new structure seems to be a sufficient option. Therefore, combination of the bi-grating and SHA may bring their respective and individual advantages into THz generation from multi-e-beam-wave interaction.

In this paper, two e-beams interacting within a sandwich structure composed of a bi-grating and a sub-wavelength holes array is proposed and studied. The results show that the surface wave of the gratings can couple with the MSPs of the SHA and the two e-beams can be coupled with each other through the holes array in this system. Due to wave coupling and e-beam coupling, the modulation depth and radiation intensity have been enhanced significantly at the THz region. Furthermore, the starting current density is close to 10 A/cm², which can be realized by a non-convergent e-beam gun, and the efficiency can reach 4% at a 0.3THz working frequency.

2. Analysis of the EM modes in the composite sandwich structure

The composite structure shown in Fig. 1 consists of a bi-grating and an SHA structure. The SHA is located at the center between the two gratings as a sandwich, so we called it a composite sandwich structure (CSS).

Fig. 1 (a) 3-D Composite structure; (b) the section structure in 2-D.

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As we know, the SHA can support the MSP waves that have similar electromagnetic characteristics with surface plasmons; meanwhile, the gratings can support surface waves. Both MSP waves and surface waves are typical evanescent ones outside the surface of structure. In a CSS, the waves on the surfaces of the bi-grating and SHA will couple with each other so that the mode in this structure would not be an evanescent wave. By utilizing a commercial finite integration technique (FIT) solver (CST’s Microwave Studio), the field distribution has been studied. Figure 2 shows the electric field distribution of EM waves in the SHA, grating, and the CSS at the same condition independently. First, Fig. 2(a) describes the electric field distributions of the MSP in the SHA and the surface wave of the bi-grating, respectively. For the MSP of the SHA (solid red line in [Fig. 2(a)], it can be found that in the holes array the field intensity is stable and lossless, and in the outside space the field decays rapidly with an increase of the distance. The same as for the evanescent wave characteristics, the electric field intensity of surface wave on the bi-grating is shown as the blue dashed line in Fig. 2(a). However, in a CSS, in the gap between the SHA and gratings, which is called e-beam channel, the surface wave and the MSP will influence and couple with each other strongly to form the coupling mode in synchronicity. In this paper, we describe this mode as a composite mimicking surface plasmon mode (CMSP). Figure 2(b) shows the field distribution of the CMSP in the CSS. It is clear that this mode is not a monotonic evanescent wave. Although the EM field in the center of the channel is the smallest, the field intensity of such a location is still very strong. Compared to Figs. 2(a) and 2(b), the field intensity of the CSS is much larger than that of the grating and SHA at the same position. As we know in the beam-wave interaction, the intensity of the field at the e-beam location is a key factor that decides the efficiency and radiation intensity. Thus from Fig. 2, we can suppose that the e-beam interaction in this structure may be attractive.

Fig. 2 Electric field distribution in different structures: (a) electric field distribution of MSP of SHA and surface wave of grating respectively; (b) electric field distribution of the mode CMSP in the CSS.

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Next, the theoretical study of the dispersion relation of the coupling-mode CMSP in this system has been carried out. This structure is a symmetric one and we can divide the space into three regions as shown in Fig. 1(b). Then Maxwell’s equations together with the periodic boundary conditions lead to the following expressions of the EM fields in each regions.

In region I (fields in the SHA), the EM waves in the SHA can be expressed as:

{\begin{cases} E_{z}^{I} = A e^{- j k_{x}^{I} x} \cos (k_{y}^{I} y) \\ H_{y}^{I} = - \frac{k_{x}^{I}}{ω μ_{0}} A e^{- j k_{x}^{I} x} \cos (k_{y}^{I} y) \end{cases},

where

k_{x}^{I}^{2} = k^{2} - k_{y}^{I}^{2}

,

k = ω / c

,

k_{y}^{I} = π / a

, a is the width of the hole, and c is light velocity.

In region III (fields in the grating), the fundamental mode TEM mode in the grating is considered, so the fields take the form of

{\begin{cases} E_{z}^{I I I} = D \frac{\sin k (g + h_{2} - x)}{\sin k h_{2}} \\ H_{y}^{I I I} = \frac{j k}{ω μ} D \frac{\cos k (g + h_{2} - x)}{\sin k h_{2}} \end{cases} .

In region II (fields in the e-beam channel), the EM field expressions in the periodic direction can be transformed into a spatial harmonics series by using the Floquet theorem for periodic structures. Considered wave coupling in this region, the fields are expressed as follows:

{\begin{cases} E_{z}^{I I} = \sum_{n = - \infty}^{\infty} (B_{n} e^{- j k_{x n}^{I I} x} + C_{n} e^{j k_{x n}^{I I} x}) e^{- j k_{z n} z} \\ H_{y}^{I I} = \sum_{n = - \infty}^{\infty} \frac{- ω ε}{k_{x n}^{I I}} (B_{n} e^{- j k_{x n}^{I I} x} - C_{n} e^{j k_{x n}^{I I} x}) e^{- j k_{z n} z} \end{cases},

where

k_{x n}^{I I}^{2} = k^{2} - k_{z n}^{2}

,

k_{z n} = k_{z 0} + 2 n π / L

,

k_{z 0} = ω / v_{0}

, n is the spatial harmonic number, and v₀ is beam velocity. The parameters A, B, C, D, which describe the characteristics of the EM wave in this structure, are determined by the boundary conditions.

Together with the boundary conditions, the dispersion relation of the coupling mode CMSP in the CSS can be obtained as:

\begin{array}{l} [\frac{d}{L} \sum_{n = - \infty}^{\infty} \sin c^{2} (\frac{k_{z n} d}{2}) \frac{j \cot k_{x n}^{I I} g}{k_{x n}^{I I}} + \frac{j \cot k h_{2}}{k}] [\frac{d}{L} \sum_{n = - \infty}^{\infty} \sin c^{2} (\frac{k_{z n} d}{2}) \frac{j \cot k_{x n}^{I I} g}{k_{x n}^{I I}} + \frac{k_{x}^{I}}{k^{2}}] - \\ {\frac{d}{L} \sum_{n = - \infty}^{\infty} \sin c^{2} (\frac{k_{z n} d}{2}) \frac{j}{2 k_{x n}^{I I} \sin k_{x n}^{I I} g}}^{2} = 0. \end{array}

Based on the dispersion in Eq. (4), the dispersion relations for the modes in the CSS can be calculated. As mentioned above, there is wave coupling between the MSP of the SHA and the surface wave of the bi-grating. So, first of all, we studied the dispersion relations of the SHA and the bi-grating, respectively, as shown in Fig. 3(a). In this figure, there are overlapping regions between the dispersion line of the MSP (red dashed line) and the surface wave of the bi-grating (black solid line). In the overlapping regions the frequencies and phases are the same, so we call them coupling regions. As shown in Fig. 3(b), in the CSS, because of the wave coupling the dispersion lines have been distorted around the overlapping regions and then separated. Next, dispersion lines of the modes in the CSS with different widths of e-beam channels have been discussed as shown in Fig. 3(c). It can be found that with the increase of width, the wave-coupling strength is decreased so that the mode separation and deformation of the dispersion line is weaker. Thus in Fig. 3(c), in the case of g=50 um, the mode separation is the best with the largest deformation. In a word, the numerical calculation of dispersion lines of the modes verifies the wave coupling between the MSP of the SHA and the surface wave of the bi-grating. Then, the simulation study was carried out to analyze the mode distribution. The simulation results shown in Fig. 4 also demonstrate that the surface waves of the bi-grating can couple with the MSP of the SHA very well to act as a whole. The same as in the theoretical study, as the distance between the gratings and the SHA decreases the coupling strength and field intensity increases. Moreover, it shows, in the e-beam channel, that the fields of the CMSP are not monotonic evanescent fields but are stable and balanced fields.

Fig. 3 Dispersion relation of the modes: (a) the dispersion lines of bi-grating and SHA respectively; (b) the dispersion line of the CMSP mode in the CSS; (c) the dispersion line of the CMSP mode with different size of e-beam channel.

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Fig. 4 Simulation results of the modes in the CSS with different e-beam channel size g: (a) g=100 um; (b) g=150 um; (c) g=200 um.

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Both theoretical study and simulation illustrate that due to the wave coupling in the CSS, the dispersion relations and EM characteristics of the modes have been changed and the field intensity has been enhanced.

3.Two electron beams interaction within the CSS

The model of the two e-beams interaction is shown in Fig. 5. First, two direct current (dc) e-beams with the same parameters are injected into the system and pass through the e-beam channel to excite the CMSP mode. Second, in the case of synchronization, the interaction between EM fields of the CMSP mode and e-beams occurs. During the interaction, the dc e-beams exchange energy with the EM fields to generate the THz radiation. In this part, theoretical study and 3-D particle in cell (PIC) simulation have been carried out to study the physical process of interaction.

Fig. 5 Sketch map of the interaction: (a) composite structure with two e-beams; (b) the section structure in 2-D. Region III, $g_{e} < x < g_{e} + p$ , is the e-beam.

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First, linear theory [23] is applied to analyze the interaction process. In the linear theory, the higher-order disturbed quantity is neglected. So for the e-beams, the physics parameters are a linear superposition of the steady-state part and disturbed part, which is caused by the interaction as [23]

{\begin{cases} ρ = ρ_{0} + ρ_{1} e^{j (ω t - k_{z} z)} \\ v_{z} = v_{0} + v_{1} e^{j (ω t - k_{z} z)} \\ J_{z} = J_{0} + J_{1} e^{j (ω t - k_{z} z)} \end{cases} .

Based on Eq. (5), according to the theorem continuity and Maxwell equations, the nonhomogeneous Helmholtz equation for the interaction between e-beams and EM waves can be written as a homogeneous Helmholtz equation of the space charge wave of the electron beam [23,24]:

\nabla_{T}^{2} E_{z}^{b e a m} + (k^{2} - k_{z}^{2}) [1 - \frac{ω_{p}^{2}}{{(ω - k_{z} v_{z})}^{2}}] E_{z}^{b e a m} = 0,

where

E_{z}^{b e a m}

is the space charge wave of the e-beam, which is caused by the interaction between the e-beam and the CMSP mode. So in the linear theory, the space charge wave can reflect the interaction process.

E_{z}^{b e a m}

can be obtained by solving Eq. (6):

E_{z}^{b e a m} = \sum_{n = - \infty}^{\infty} (D_{n} e^{- j k_{x n}^{I I I} x} + E_{n} e^{j k_{x n}^{I I I} x}) e^{- j k_{z n} z},

where

k_{x n}^{I I I} = \sqrt{(k^{2} - k_{z n}^{2}) (1 - \frac{ω_{p}^{2}}{{(ω - k_{z n} v_{0})}^{2}})}

, and ω_p and v₀ are the plasma frequency and velocity of the e-beams, respectively. In this model, two e-beams with the same parameters are applied. Thus, it is a typical symmetrical system. Based on this consideration, the space charge waves of the two e-beams have the same expression. Matching the boundary conditions, the interaction equation can be obtained as:

(M + \frac{j \cot k h_{2}}{k}) (M + \frac{k_{x}^{I}}{k^{2}}) - N^{2} = 0,

where

\begin{array}{l} M = \frac{d}{L} \sum_{n = - \infty}^{\infty} \sin c^{2} (\frac{k_{z n} d}{2}) \frac{1}{k_{x n}^{I I}} \frac{β_{n}^{-} γ_{n}^{+} e^{- j k_{x n}^{I I I} p} - β_{n}^{+} γ_{n}^{-} e^{- j k_{x n}^{I I I} p}}{e^{- j k_{x n}^{I I I} p} β_{n}^{-}^{2} - e^{j k_{x n}^{I I I} p} β_{n}^{+}^{2}}, \\ N = \frac{d}{L} \sum_{n = - \infty}^{\infty} \sin c^{2} (\frac{k_{z n} d}{2}) \frac{1}{k_{x n}^{I I}} \frac{β_{n}^{+} γ_{n}^{+} + β_{n}^{-} γ_{n}^{-}}{e^{- j k_{x n}^{I I I} p} β_{n}^{-}^{2} - e^{j k_{x n}^{I I I} p} β_{n}^{+}^{2}}, \\ β_{n}^{-} = 2 \cos k_{x n}^{I I} g_{e} - \frac{1}{\sqrt{α_{n}}} j 2 \sin k_{x n}^{I I} g_{e}, β_{n}^{+} = 2 \cos k_{x n}^{I I} g_{e} + \frac{1}{\sqrt{α_{n}}} j 2 \sin k_{x n}^{I I} g_{e}, \\ γ_{n}^{-} = - \frac{1}{\sqrt{α_{n}}} 2 \cos k_{x n}^{I I} g_{e} - j 2 \sin k_{x n}^{I I} g_{e}, γ_{n}^{+} = \frac{1}{\sqrt{α_{n}}} 2 \cos k_{x n}^{I I} g_{e} - j 2 \sin k_{x n}^{I I} g_{e}, α_{n} = 1 - \frac{ω_{p}^{2}}{{(ω - k_{z n} v_{0})}^{2}} . \end{array}

In case the velocity of the electron beam is equal to the phase velocity of the CMSP mode, the e-beams keep synchronization with the CMSP modes and then the interaction occurs. According to Eq. (8), the growth rate can be calculated. In general, by making use of the growth rate we can evaluate the interaction region and optimize the voltage of the e-beams and parameters of the structure. Moreover, the value of the growth rate directly corresponds to the electronic energy conversion efficiency. In the numerical calculation, the optimized parameters of the CSS are acquired: d=0.08 mm, a=0.48 mm, L=0.16 mm, h₁=0.2 mm, g_e=0 mm, p=0.2 mm, h₂=0.2 mm, and the energy of the e-beams is 15 keV. Figure 6(a) shows that the growth rate of the two e-beams’ interaction within this system is significant and the value is relatively high, which means that the interaction is efficient. Moreover, the comparison between one-beam and two-beam interaction has been carried out. Figure 6(b) clearly demonstrates that the value of the growth rate of the two-beam interaction is much higher, which means the coupling between the two beams can enhance the efficiency of the interaction.

Fig. 6 Growth rate of the beam-wave interaction, where ω_p is the plasma frequency of the e-beams, which is used to normalize the growth rate ω_i. (a) The scheme of electron beam–CMSP interaction around the intersection between the dispersion line of the e-beam and the CMSP mode; the e-beams synchronize with the CMSP. (b) Comparison of the growth rates of one-beam interaction and two-beam interaction.

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Secondly, a 3-D simulation has been performed with a fully electromagnetic PIC code developed based on the FDTD method. The physical process of interaction has been simulated as shown in Fig. 7. In PIC simulation, the parameters of the CSS and the e-beams are the same as the optimized ones obtained by theoretical study. E-beams with a thickness of 0.2 mm and energy of 15keV are employed to pass over 30 periods to interact with the CMSP modes.

Fig. 7 Simulation results of interaction: (a) phase space of the e-beams; (b) contour map of E_z; (c) field intensity and frequency spectrum; (d) energy distribution.

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From the simulation results, it is found that while the e-beams pass through the channel, the CMSP wave can be excited [Fig. 7(b)]. When the phase velocity of the CMSP mode is equal to that of the e-beams, they can synchronize and interact with each other. During the two-beam interaction, the upper e-beam exchanges energy with the EM fields of CMSP in the upper channel. Then because of the transmission characteristics of the SHA, the EM fields in the upper channel penetrate into the holes to propagate to the lower space and vice versa for the EM fields in the lower channel. Therefore, the energy has been coupled between the upper and lower e-beams. Figure 7(a) shows that because of coupling the interaction is sufficient and the e-beams bunch very well. Figure 7(c) shows that the amplitude of E_z at the surface of the SHA in the CSS can reach 7.1 kV/mm, and the interaction frequency 0.302 THz agrees with the theoretical expectation (0.3 THz) very well. In Fig. 7(d), it demonstrates the phase space and modulation depth of the e-beams. The modulation depth is defined as the ratio between the highest energy loss and the original energy of the e-beam. As we know, during the interaction process the energy of the e-beam will converted to the radiation; so, in the interaction, the e-beam will lose its energy. The more energy it loses, the higher THz radiation can be obtained. Furthermore, the modulation depth is proportional to the efficiency. Figure 7(d) shows the modulation depth is 22%, and according to the field intensity of the radiation shown in Fig. 7(c), the efficiency is about 4%. So the modulation depth and conversion efficiency of such interaction is very attractive.

Figure 8 gives comparisons of the results among the two e-beams’ interactions in the CSS, SHA, and bi-grating with the same parameters of e-beams at 0.3 THz working frequency, respectively. Figure 8(a) shows the comparison of longitudinal electrical field intensity of the THz radiation generated from the interactions in these three structures independently. It can be found that the field amplitude of THz radiation in the CSS is much larger than that of the others, which means the radiation intensity from this source is the strongest. Figure 8(b) demonstrates that the modulation depth of such interaction is much higher than that of the other two schemes, which verifies the energy conversion efficiency of this system is at a very high-level.

Fig. 8 Comparison of the results among the interactions in bi-grating, SHA, and CSS with the same parameters of e-beams at 0.3 THz working frequency. (a) Comparison of longitudinal electrical field intensity during the interaction process; (b) comparison of modulation depth of the e-beams.

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It is well known that the starting current density is a crucial factor that restricts the development of linear e-beam-driven THz sources. For the traditional slow-wave structure the required starting current density is usually very high at a THz frequency [25,26]. In this composite MSP–e-beams interaction, the starting current density is relatively low. Fixing the electron energy at 15 keV, we sweep the beam current density as shown in Fig. 9. The region between the starting point and the saturating point is the linear growing region where the radiation intensity from the interaction linearly increases with the current density increase. It can be found that the starting current density is only about 12 A/cm², and the saturation point is about 110 A/cm².

Fig. 9 Starting current density relation.

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Moreover, with regard to manufacturing the CSS, the grating can be produced using low-speed wire-cutting electrical discharge machining (LWEDM) with precision machining equipment. The SHA can be fabricated by lithography galvano-forming ab-forming (LIGA), a kind of micro-electro-mechanical systems (MEMS) processing. Then gratings and SHA can be assembled into a system with an accurate fixture. The two planar e-beams of more than 10 A/cm² can be emitted by a scandate dispenser cathode. Thus, constructing such a device is technically possible.

4.Conclusion

The composite sandwich structure, which is composed of a bi-grating and a sub-wavelength holes array, is suggested in this paper. It shows that the wave coupling between the surface wave of grating and the MSP of the SHA forms the CMSP mode. The fields of such modes are not monotonic evanescent waves and the field intensity is larger than that of bi-gratings and SHAs at the same condition.

Two e-beams’ interaction in this composite structure is proposed to generate THz radiation. Theoretical study and simulation show that two e-beams can be coupled with each other through the holes array and can interact with the CMSP modes efficiently. Compared with the interaction in the traditional grating structure and the SHA, the modulation depth and the radiation intensity have been enhanced significantly. The modulation depth and the efficiency can reach 22% and 4%, respectively, and the starting current density is only 12A/cm². Relatively high energy conversion efficiency, low starting current density, and two e-beams working––the characteristics of this radiation system may provide good opportunities for development of multi-beam-driven THz radiation sources.

Acknowledgments

This work is supported by the National Natural Science Foundation of China under Contract Nos. 61370011 and 61001031, the National High-tech Research and Development Projects 2011AA010204, and the Fundamental Research Funds for the Central Universities under contract No ZYGX2012J056.

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THz radiation from two electron-beams interaction within a bi-grating and a sub-wavelength holes array composite sandwich structure

Abstract

1. Introduction

2. Analysis of the EM modes in the composite sandwich structure

3.Two electron beams interaction within the CSS

4.Conclusion

Acknowledgments

References and links

Cited By

Figures (9)

Equations (9)

Optics Express