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Smith-Purcell radiation is emitted when an electron passes above the surface of a metallic grating. Its mechanism can be explained with Huygens’ principle by the radiation of a moving oscillating dipole, which is formed by the moving charge and its image in the metallic grating. Here, an alternative way is presented to enhance the THz Smith-Purcell radiation. By drilling a hole in the fins of a grating as an effective electron channel, the oscillation dipole happens in two dimensions here, instead of one dimension. As a result, the Smith-Purcell radiation power is ten times more than the case in which the electron passes very close to the grating surface. This method is expected to improve the efficiency of the devices which are based on the Smith-Purcell radiation.
© 2017 Optical Society of America
Corrections
10 May 2017: A correction was made to Fig. 1.
1. Introduction
In 1953, Smith and Purcell used an electron beam of about 300 keV travelling along the surface of a diffraction grating to observe visible light radiation [1]. Based on the experimental results, the dependence of the wavelength on beam velocity, grating period and radiation angle could be quantitatively confirmed,
Where denotes the wavelength of the radiation produced at angle with respect to the beam, D is the grating period, and m is an integer. We write , where denotes the electron’s velocity, and c the speed of light. The radiation mechanism can be simply explained with Huygens’ principle by a moving radiating dipole, which is formed by the moving electron and its image charge in the metallic grating. When the electron moves along the grating, the distance between the electron and its image charge has a periodical variation, and it emits radiation [2–4]. Smith-Purcell (SP) radiation has many applications, especially in THz radiation source [5–10]. As we know, SP radiation is not coherent. But it becomes coherent through electron beam bunching, which leads to a new kind of free electron laser. It is a compact and tunable radiation source, and has potential as a THz radiation source [11–15].With the development of nano-technology, the SP radiation in photonic crystal [16–18], subwavelength hole array [19–22] and nano-scale periodical structure are investigated recently [23,24], it gives a great opportunity to develop THz radiation sources based on SP radiation mechanism. Especially, the exploration of the SP radiation enhancement attracts people’s great attentions. For example, the surface plasmon polaritons in noble metal [25–28] or in graphene [29,30] were transformed into radiation to enhance the SP radiation, and the surface modes [9,21] or resonant modes [23,24], which obtained the energy from the moving electrons, thus enhanced the SP radiation.
In many previous works on SP radiation from a metallic grating, the electron beam channels are above the grating. The radiation becomes strong when the moving electron is close to the grating surface [2–4]. Here, we propose a grating structure with a hole drilled into the fins of the grating (hole-grating structure), and this hole is an electron beam channel. The radiation power is dramatically enhanced, but the SP radiation characteristic remain unchanged. The new method cannot only be applied for enhanced SP radiation, but also for the devices based on SP effects, such as the free electron laser.
2. The proposed model and its dispersion relation
In the traditional SP radiation model, the electron beam is moving above the grating, and the distance between the electron and its image charge has a periodical variation like a dipole oscillation, then it emits radiation. If there is a hole in the fins of a grating, when the moving electron pass through the hole, there are four the dipole oscillations occurring in the cross section. This mechanism is shown in Fig. 1.
Fig. 1 (a) The electron is moving above the grating. From the time t0 to t1, the distances of the electron and its image charge are variable, shown in (b) and (c), and it forms an oscillation dipole. (d) The electron is moving through the hole in the structure of a hole in the fins of a grating, and there are four dipole oscillations forming, shown as (e) and (f). At the time of t1, there is no metal in the X direction and the positive Y direction, it means the location of the image charge is infinite.
The structure is proposed as shown in Fig. 2, partly based on this mechanism. A grating is cut into a metal substrate and then a hole is drilled in the fins of the grating. The hole serves not only as a channel for the electron beam, but also as a tool to enhance the interaction between the surface wave and the electron beam.
When the electron beam is bunched by the interaction with surface wave, the SP radiation becomes coherent. The surface wave excited by the moving electron beam is a local field and supported by the periodic structure. It is a TM mode and the Ez component will interact with the electron beam.
Here, we drill a hole in the fins of the grating. If the electrons pass through the hole, the TM waveguide modes of the rectangular hole and the surface wave are both excited. When the lowest cutoff frequency of the TM waveguide mode (the cutoff frequency of TM11 mode is the lowest in all TM modes for rectangular hole) is higher than the surface wave frequency, the hole is cut off for the surface wave. The lowest-frequency dispersion curve is related to the grating surface wave, and is determined by the grating period and depth. The dispersion relations of the surface wave in grating can be deduced by the Maxwell equation and boundary conditions [31,32],
Where is the wave vector in z direction, , , is electron beam velocity, is the angular frequency, D is the grating period, a is the width of the groove, and h is the depth of the groove.For the grating with period 160 μm and the groove depth 240 μm, the frequency of the surface wave is 261 GHz for the beam energy 20 keV () based on the Eq. (2), so we need to guarantee the lowest cutoff frequency of the TM waveguide mode of the rectangular hole to be larger than the grating surface wave frequency (261 GHz). Here, we select the hole size 120 μm by 120 μm.
In our simulation, we use a square-shaped e-beam to drive the structure. In this paper, the study of the lowest-frequency dispersion curve (here, it refers to the TM surface wave mode) has been carried out using a finite-difference time-domain (FDTD) method. By changing the parameters including the period, the groove depth, and the position and shape of the hole, the dispersion relations of the hole-grating structure are given in Fig. 3. In Figs. 3(a) and 3(b), it is shown that the shape and the position of the hole have little influence on the dispersion relations. The dispersion relation for the hole-grating structure given by simulation agrees with the theoretical prediction based on Eq. (2), which is given in black solid line in Fig. 3(a). In Figs. 3(c) and 3(d), it shows that the dispersion curve decreases as the period increases, and the dispersion increases as the grating groove depth increases. These two characteristics are the same for a grating without a hole channel. From Figs. 3(a)-3(d), it means that the lowest-frequency dispersion curve of the hole-grating structure has a similar character to that of the traditional grating structure [10–12]. Based on this result, this kind of structure can be used directly in many devices based on the SP effect, especially when there is a grating structure in these devices, such as in the SP free electron laser [25–29].
Fig. 3 The dispersion relation of the hole-grating structure. (a) the dispersion relation v.s. hole shape (hole width), (b) the dispersion relation v.s. hole position, which denotes the distance between the lowest position of the groove and the center position of the hole, (c) The dispersion relation v.s. grating period, and (d) the dispersion relation v.s. grating groove depth.
3. The enhanced Smith-Purcell radiation in the hole-grating structure
Next, the role of the hole as an electron beam channel is discussed. Figure 4(a) is the traditional structure, the emitter of the electron is above the grating. The distribution of Ez field in transverse and longitudinal views is given in Fig. 4(b). From the Fig. 4(b), it is seen that if the electron beam channel is above the grating, the main parts of the Ez field cannot be applied to the interaction between the field and the electron beam. From the field profile, it appears very natural to make a hole in the fins of the grating as an electron beam channel. Figure 4(c) is the grating with hole. The distribution of Ez field is given in Fig. 4(d). For comparison, the hole as an electron beam channel will improve the interaction efficiency between the electron beam moving along z direction and the Ez field in the grating.
Fig. 4 The electron beam channels are compared in two structure models. (a) A traditional grating structure, with the electron beam moving above it. (b) The corresponding Ez field distributions in transverse and longitudinal views. (c) A hole is drilled in the fins of the grating as an electron channel. (d) The corresponding Ez field distributions in transverse and longitudinal views. The e-beam channels are shown in red dot square in Figs. 4(b) and 4(d).
Then, the Particle-in-cell (PIC) simulations are carried out. The emitter is set in the left, and grating structure is at the bottom of the simulation area. When a square-shaped e-beam propagates through the holes, the contour map of Ez is given in Fig. 5(b). It is the same as the contour map of the Ez field excited by an electron bunch moving above the grating shown in Fig. 5(a). This means the SP radiation is also excited in the new structure, and the SP radiation is even stronger, as shown in the contour maps. From the contour maps, the surface wave is also observed. And it is a local filed and is mainly near the grating. In many SP devices, the interaction between the surface wave and the electron beam is used. Here, we compare the surface wave in the two structures. In order to compare their quality, the electrons are the same. A measurement line along the z direction is set at the center of the electron beam, as shown by dotted line in Figs. 5(a) and 5(b). The Ez field distribution at the measurement line is given in Fig. 5(c). It shows that for the hole-grating structure, the surface wave is stronger. An observation point at the grating surface is set to get the surface wave in time domain. The observation points are the same in the position of these two structures. In Fig. 6, the time domain results also show the surface wave in hole-grating structure is stronger.
Fig. 5 The PIC simulation results. (a) The Ez field contour map for the grating structure; (b) the Ez field contour map for the hole-grating structure; and (c) the comparison of surface waves at the measurement line.
The role of the hole in SP radiation is discussed in detail in Fig. 7. The contour map shown in Fig. 7(a) indicates that it is SP radiation. Because of the dipolar effect, the spacing of the contour lines in front of the moving electron beam is smaller than those behind the beam. In general, when an electron beam is moving above the grating, the SP radiation will increase as the distance between the electrons and grating decrease. Next, the comparison of SP radiation between grating and the hole-grating structure is carried out. For the traditional case, the electron beam is placed just above the surface of the grating so as to sample the strongest field. In the hole-grating structure case, the electron moves in the center of the hole. For these two cases, the shapes of the electron beam are the same. An observed point is set in the upper-space of the grating to analyze the radiation fields. The Ez fields are shown in Fig. 7(b), with the magnitude of the field in hole-grating structure being in red, and the magnitude of the field in the grating without a hole being in black. A more than three times enhancement of the Ez field are observed in the Fig. 7(b). The corresponding frequency spectrums are given in Fig. 7(c).
Fig. 7 The comparison of SP radiation. (a) The Ez field contour map for the hole-grating structure, (b) the comparison of SP radiation field between the traditional grating and the hole-grating structure, and (c) the corresponding spectrum of the radiation field.
4. The enhanced THz coherent Smith-Purcell radiation based on the RF electron bunches with hole-grating structure
When the electron beam is injected in the hole-grating structure, the surface wave will be excited, which can be observed on the surface of the grating, it is a local field and decays quickly away from the grating. The intensity of the surface wave is larger than that of the SP radiation shown in Fig. 7(a). The synchronization and interaction will occur when the phase velocity of the surface wave matches the velocity of the electron beam. Then the velocity and the density of the electron beam can be modulated by the surface wave and the electron beam will be pre-bunched. The higher harmonics of the bunch frequency of electron beam can excite the coherent SP radiation. These characteristics are very useful to develop the vacuum device from microwave to terahertz [6].
The coherent SP radiation mechanism can be explained as below. It is known that the SP radiation consists of a wide continuous frequency band and the radiation wavelength depends on the observed angle and grating period as in Eq. (1). Based on Eq. (1), the radiation frequency region of the −1 space harmonic can be calculated as shown in Fig. 8(b), which ranges from point A (400GHz) to point B (700GHz). In Fig. 8(b), the blue line is the dispersion curve of the surface wave, and the red line, which has an intersection point with the blue line, is the electron beam line. This intersection point means the phase velocity of the surface wave matches the velocity of the electron beam. It determines the frequency of the surface wave, and the electron beam can also be pre-bunched with this frequency (253GHz). Many higher harmonics components will be included in the bunched electron beam. The frequency of the second harmonic of the bunched electron beam is 506GHz, which is located in the SP radiation region. So the second harmonic (506GHz) can be transformed into SP radiation, and the fixed frequency SP radiation lead to a fixed radiation angle based on Eq. (1). This radiation phenomenon is also called SP super-radiation.
Fig. 8 (a) The schematic of the grating with hole in the fingers excited by the well bunched electron beam, and (b) the SP super-radiation.
The bunched electron beam has harmonics, here we use a fully-bunched electron beam with bunch frequency f0 (253GHz) to excite the SP radiation in grating and hole-grating structure, the schematic is given in Fig. 8(a). From above analysis, the energy of the first harmonic of the bunched beam cannot be transformed into SP radiation, but the second harmonic can be transformed into SP radiation. The bunched electron beam is the same for the two cases, so the energy of the second harmonic is also the same. The enhancement of radiation means that a higher transformation efficiency can be achieved. In Fig. 9, the simulation of the new structure excited by a well-bunched electron beam is carried in CST. The contour map of Ez field is given in Fig. 9(a). The radiation has a fixed frequency 506GHz and a fixed angle of 90° as predicted in Fig. 8(b). The radiation fields for both cases are shown in Fig. 9(b), and their spectrums are given in Fig. 9(c). The hole-grating structure shows enhanced radiation power. This narrow single peak frequency in Fig. 9(c) means this is coherent SP radiation, and it is corresponding to frequency of the second harmonic of the pre-bunched electron beam. Radiation power is enhanced by more than an order of magnitude. This means that the transformation efficiency in the hole-grating structure has increased significantly.
Fig. 9 The simulation results of the SP radiation excited by a well-bunched electron beam. (a) The contour map of Ez fields, (b) the time domain of Ez field for grating and hole-grating structure, and (c) the corresponding spectrums.
5. Conclusion
In summary, an improvement for the SP radiation is presented by two-dimensional oscillation dipole in a metallic grating with holes. The results show the intensity of SP radiation can be significantly enhanced by 4 times with one electron bunch. Moreover, by utilizing RF electron bunches, 2nd harmonics THz SP super-radiation with 0.5THz working frequency can be achieved. The radiation intensity is 10 times larger than traditional one-dimensional oscillation dipole in the grating. This improvement may have promising prospect in developing high-power and efficient THz sources.
Funding
Natural Science Foundation of China (61501094, 61270011, 91438118); National Key Basic Research Program of China (2014CB339806); Fundamental Research Funds for the Central Universities of China (ZYGX2014J034).
Acknowledgments
This work is partly supported by China Scholarship Council.
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