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Abstract
The demand for terahertz (THz) sources that can be used in practical implementations has not yet been fully met. Through the development of small-scale free-electron lasers (FEL), we investigated a solution to overcome the low lasing gain of the FEL, which is caused by a small-current accelerator and a short-length undulator. To enhance the FEL gain, the FEL interaction between the electron beam and radiation was increased by reducing the cross-sectional area of the FEL oscillator mode. We developed a waveguide for the FEL oscillator, which has a remarkably small eye-shaped cross section with low wave loss. The mode cross-sectional area was calculated to be just 4 mm2 (full width at half maximum), which is considerably smaller than those of the free-space Gaussian mode and other waveguide modes. Using COMSOL Multiphysics simulation code, we calculated and analyzed the attenuation loss and mode cross-section area for waveguides of different shapes and sizes, and we observed that the dielectric coated eye-shaped waveguide has the lowest attenuation loss, that is, less than 2.5% for 1-m propagation at an operating wavelength of 300–600 µm. These results are in good agreement with our requirements for the tabletop THz FEL. Finally, we showed that the calculated operating wavelength range of the waveguide-mode THz FEL is 300–600 µm using the machine parameters of an accelerator and undulator that were developed by considering the dispersion relations of the eye-shaped waveguide and undulator radiation.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
There is an increasing demand in the market for small-sized powerful radiation sources in the terahertz (THz) spectral range for applications related to security inspection, remote monitoring of drones and gases, and medical inspection [1–8]. Various types of THz sources have been developed; however, they do not satisfy the performance requirements of the market for such applications. THz free-electron lasers (FELs) offer advantages such as a wide tuning range, narrow bandwidth, and high peak and average power [9–12]; however, they are also limited by their large size and high cost.
Efforts have been made to reduce the size of THz FELs to a small size that can be used in field applications [13,14]. A THz FEL with a free-space resonator has a large mode cross section and undulator gap, owing to its long wavelength, which are factors that reduce the FEL small-signal gain. A short undulator, which is used to realize a small THz FEL, also greatly lowers the FEL gain [13–16]. In FEL, waveguides have been used for generating long-wavelength coherent light [17–26]. Mainly circular, rectangular, or parallel plate waveguides have been used depending on the polarization of the FEL radiation.
A waveguide-mode FEL resonator with a smaller cross section than that of a free-space- mode resonator may mitigate these limitations because the small-signal gain of the FEL is inversely proportional to the mode cross-sectional area [15]. To reduce the waveguide cross section, it is necessary to use a two-dimensional (2D) waveguide. However, in the 2D waveguide, there may be a wall with a large field strength, and wave loss comparable to the FEL gain occurs, owing to the finite conductivity of the wall. We used a one-dimensional (1D) parallel-plate waveguide with a low wave loss for a previously developed THz FEL [27,28]. Research on reducing the mode cross-section in 1D waveguides has led to efforts to devise a curved parallel-plate waveguide [25]. It has been successfully demonstrated that wide-band, high-efficiency generation is possible with an FEL with the curved parallel-plate waveguide [26]. However, the mode of the 1D waveguide is still too large for a table-top THz FEL operating in the wavelength range of 300–600 µm, which is newly developed for the field application of security inspection [15].
To develop the tabletop high-power THz FEL, we devised a waveguide with a special eye-shaped cross section. This waveguide has a very small cross-sectional area of 4 mm2 at full-width at half maximum (FWHM) and a very low wave loss of less than 2.5% for 1-m propagation at an operating wavelength of 300–600 µm. In fact, the cross section of the eye-shaped waveguide comprises two facing arcs of the same radius. It consists of simple two-cylinder sections that are simple to machine. The waveguide is made of metal, and the inner surface is coated with a dielectric with a thickness of more than several tens of micrometers. When aluminum is used as the metal, such a surface can be easily realized through the anodizing process. We used COMSOL Multiphysics code to calculate the waveguide loss as a function of the wavelength and structure [29]. The mode distribution and wave loss for rectangular and hexagonal waveguides were calculated for comparison. In particular, in the case of the rectangular waveguide, the method was validated by comparing the results obtained from the well-known formula with those from the simulation. Moreover, the hexagonal waveguide provided an important intermediate step to obtain intuition for devising the eye-shaped waveguide.
A waveguide with a small cross-sectional area and low loss plays a very important role in the development of smaller devices in electron-based light sources such as vacuum electronic devices as well as FELs. And it can be used in fields such as THz integrated devices. The novel waveguide structure introduced in our study is expected to greatly contribute to the development of THz sources and activation of THz applications.
2. Gain and loss of waveguide-mode FEL oscillator
A compact long-wavelength FEL with a low electron-beam current (I) and a small number of undulator periods (N) suffers a lack of small-signal lasing gain (G), which has the following relation with these parameters:
where S is the cross-sectional area of the radiation mode in the FEL resonator. To enhance the gain, we need a waveguide that has a smaller cross-sectional mode than that of the free-space mode. Generally, in a waveguide with a small cross-sectional area, a relatively large wave loss occurs at the inner wall of the waveguide, owing to its finite conductivity. In an FEL driven by a planar undulator, transverse electric (TE) modes concentrated in the center axis of the waveguide interact most effectively with the electron beam. Among the modes, those with a waveguide loss that is sufficiently smaller than the small-signal gain can survive competitively during the FEL lasing process. Our aim is to find a low-loss 2D waveguide mode that has a considerably smaller cross-sectional area than that of the parallel-plate (1D) waveguide mode that we previously developed. The mode cross-sectional area of the 1D waveguide is estimated to be 15–23 mm2 (FWHM) for a wavelength range of 300–600 µm, a vertical gap of 3 mm, and an FEL resonator length of 1 m, which is much smaller than that of the free-space Gaussian mode. The average mode cross section of the Gaussian mode is estimated to be approximately 300–700 mm2 under the same conditions.The macropulse duration of the electron beam is approximately 5 µs, and we require a minimum net small-signal gain of 10% for every round trip of the wave in the resonator to reach FEL oscillation saturation within 3 µs. Therefore, we set the maximum allowable waveguide loss of the mode up to 5% per round trip, which corresponds to approximately 0.025 m-1 for the attenuation constant value, α, defined as follows:
where Pin and Pout are the input and output radiation powers, respectively, and the medium thickness is L. In our case, the length of the undulator and waveguide is approximately 1 m, and the corresponding round-trip length is 2 m.In the case of a 2D waveguide, the mode area is determined independently of the wavelength. We confirmed the practicality of our simulation method using a rectangular waveguide, which is the simplest and most well-known 2D waveguide with TE modes. Subsequently, the mode area and attenuation constant were calculated for waveguides that have hexagonal and eye-shaped cross sections. These were deformed from the rectangular shape, in which the TE-mode distribution is concentrated at the waveguide cross-sectional center. The vertical size of the waveguides was determined to be 3 mm by considering the betatron motion of the electrons in the vertical direction by the planar undulator [30]. In particular, an eye-shaped waveguide with a cross section comprising two arcs that face each other exhibits a much smaller cross-sectional area and attenuation constant than those of the other two waveguides. We made minor modifications to the waveguide but did not obtain better properties. Therefore, we finally chose an eye-shaped waveguide for the table-top THz FEL oscillator.
2.1 Rectangular waveguide
We calculated the mode distribution and waveguide loss depending on wavelength and geometry using the electromagnetic module of the COMSOL Multiphysics simulation code. To check the precision of the simulation result, we compared it to an analytic calculation for a rectangular waveguide because its characteristics are easily available in textbooks [31,32]. We used the simulation tool to calculate the attenuation loss of the TE01 mode in the wavelength range of 300–600 µm for a rectangular waveguide with a cross-sectional dimension of 10 mm × 3 mm, as shown in Fig. 1.
Fig. 1. Cross-sectional shape and TE01-mode distribution of a rectangular waveguide expressing the relative strength of electric field with colors.
We also calculated the attenuation constant of the rectangular waveguide analytically using the following analytical relation [30]:
where Rs is the surface resistivity of the metal wall, β is the propagation constant, k is the wave number, and $\eta $ is the intrinsic impedance of the waveguide.Figure 2(a) shows the calculated results of the attenuation constant of the TE01 mode of the rectangular waveguide with a cross-sectional dimension of 10 mm × 3 mm in the wavelength range of 300–600 µm using the simulation and theoretical formulae. From the figure, we can clearly observe that there is no significant difference between the simulated and analytical values. The difference between the values is less than 10%, and their dependence on wavelength is almost identical. Therefore, we can use the simulation code to further optimize the waveguide shape for a table-top THz FEL oscillator.
Fig. 2. (a) Attenuation constants of a rectangular waveguide with a cross-sectional dimension of 10 mm × 3 mm calculated using a simulation (blue dots) and analytic formulae (red solid line). (b) Simulated attenuation constant of the 600-µm wave and mode cross-sectional area of the rectangular waveguides for a horizontal dimension of 4 to 12 mm.
2.2 Hexagonal waveguide
We aim to minimize waveguide loss and the mode cross-sectional area by exploiting a hexagonal shape. Figure 3 shows the cross-sectional shape and center-focused high-order TE-mode distribution of the hexagonal waveguide with dimensions of 10 mm × 3 mm.
Fig. 3. Calculated attenuation constant of a 600-µm wave and mode cross-sectional area of hexagonal waveguides for a horizontal dimension of 7 to 15 mm. Inset is the Cross-sectional shape and mode distribution of a hexagonal waveguide with a cross-sectional dimension of 10 mm × 3 mm.
The cross-sectional area of the TE-mode of the hexagonal waveguide with the above-mentioned dimension is 6.9 mm2, which is almost 2.2 time that of a rectangular waveguide of the same dimensions. We calculated the attenuation constant and the mode cross-sectional area using the simulation code as the waveguide horizontal length changed from 7 to 15 mm, as shown in Fig. 3. The wavelength used in for the simulation was the same as that of the rectangular waveguide, that is, 600 µm. When the horizontal length of the waveguide is reduced, the mode cross-sectional area is proportionally reduced, but the mode attenuation increases almost six times compared with that of the rectangular waveguide, which may hinder table-top THz FEL lasing. The reason for such a high attenuation loss is the sharp edges of the hexagonal waveguide shape, which prevents a smooth change in the mode distribution. The loss occurs owing to the high electric field strength on the inclined walls.
While we could achieve a smaller mode cross-sectional area with the hexagonal waveguide, the attenuation loss is substantially higher than the required value for FEL oscillation. Removing the sharp edges from the hexagonal shape may minimize the attenuation loss and achieve a more optimized mode distribution for the table-top THz FEL. To achieve this, we chose an eye-shaped waveguide, which was made by combining two facing arcs of the same radius. This shape is very easy to manufacture, which presents an additional advantage.
2.3 Eye-shaped waveguide
The limitation caused by a small mode area and low attenuation loss is not completely mitigated by the above-mentioned waveguide shapes. Using an eye-shaped waveguide, we aim to achieve the optimized shape of a waveguide that has low attenuation loss and a low mode cross-sectional area for the table-top THz FEL. Figure 4 presents the cross-sectional shape and center-focused high-order TE-mode distribution of an eye-shaped waveguide that has a height of 3 mm and two facing arcs with a 9-mm radius.
Fig. 4. Calculated attenuation constant of a 600-µm wave and mode cross-sectional area of eye-shaped waveguides for an arc radius of 7 to 15 mm while maintaining a center-part height of 3 mm. Inset is the Cross-sectional shape and mode distribution of an eye-shaped waveguide.
The horizontal size of the eye-shaped waveguide mode is 3.4 mm in FWHM and The TE mode cross-sectional area of the eye-shaped waveguide with the above-mentioned dimension is 4 mm2, which is approximately one-third that of the rectangular waveguide and approximately 1.65 times lower than that of the hexagonal waveguide. The mode cross-sectional area can be further decreased by reducing the radius of the eye-shaped waveguide arcs. We calculated the attenuation constant and mode cross-sectional area using COMSOL Multiphysics simulation code. The waveguide arc radius was changed from 7 to 15 mm while maintaining a center-part height of 3 mm, as shown in Fig. 4. The wavelength used in the simulation was 600 µm. When the arc radius of the waveguide is reduced, the mode cross-sectional area is proportionally reduced, and the mode attenuation loss increases slightly; however, it is still considerably smaller than that of the other two waveguides. In Fig. 5, we compare the results of the mode cross-sectional area for horizontal lengths of 7–15 mm and attenuation losses for the three waveguides. It is clear from the figure that the eye-shaped waveguide has the smallest mode area and the lowest attenuation loss among the waveguides. Modifications of the eye-shaped waveguide using aspherical and complex shapes were simulated; however, they did not provide improved results on the cross-sectional area and attenuation loss.
Fig. 5. (a) Calculated attenuation loss of the center-focused TE modes of the three waveguides with horizontal and vertical lengths of 10 and 3 mm, respectively, over a wavelength range of 300–600 µm. (b) Calculated mode cross-sectional area of the waveguide TE modes depending on the horizontal length from 7 to 15 mm. Their maximum vertical lengths are fixed to 3 mm, and the wavelength of the modes is 600 µm.
Thus, the above results clearly show that a simple eye-shaped waveguide is optimal for the table-top THz FEL. However, we must further reduce the attenuation constant of the eye-shaped waveguide to meet the desired value for our case, that is, less than 0.025 m-1.
3. Reducing the eye-shaped waveguide loss with dielectric coating on the inner surface
To reduce the attenuation constant of the eye-shaped waveguide to the desired value of less than 0.025 m-1, we introduce a dielectric coating on the inner walls of the waveguide by exciting a hybrid mode that has the same cross-sectional distribution as the TE mode. We chose an alumina layer on the aluminum surface to create a relatively thick and mechanically strong dielectric coating on the metal surface. We calculated the effect of alumina coating on the attenuation loss of the eye-shaped waveguide for different coating thicknesses ranging from 10 to 100 µm in the wavelength range of 300–600 µm using the COMSOL Multiphysics simulation code. Figure 6 presents the calculated attenuation constant of the hybrid mode for different coating thicknesses in the specified wavelength range. The required attenuation constant for the eye-shaped waveguide, that is, less than 0.025 m-1, can be achieved by applying an alumina coating with a thickness of more than 50 µm on the inner surface of the eye-shaped waveguide, which is marked by the dotted line in the figure. Finally, we selected an alumina-coated eye-shaped waveguide composed of two arcs with a curvature of 9 mm and a center height of 3 mm as an optical resonator for the table-top THz FEL. The thickness of the alumina layer was determined to be greater than 50 µm.
Fig. 6. Calculated attenuation constant of a center-focused hybrid mode of an alumina coated eye-shaped waveguide for a coating thickness range of 10–100 µm and a mode wavelength range of 300–600 µm. The dotted line indicates the optimized coating thickness that is required to obtain the desired attenuation constant, that is, less than 0.025 m-1 for the eye-shaped waveguide. The maximum horizontal and vertical waveguide sizes are 10 and 3 mm, respectively.
Alumina was selected for the dielectric layer material, owing to the ease of implementing a uniform coating via the anodizing process, which can be done without any special expertise. Additionally, it has strong adhesion and a low relative permittivity, that is, ∼9.3. A low value is recommended because the relative permittivity is directly proportional to the attenuation constant.
As a preliminary study to verify the coating uniformity of the alumina, we applied an alumina coating of 60 µm on a 1-m-long aluminum slab and measured its thickness depending on the position. The measured thickness of the alumina layer is shown in Fig. 7. Even though no special effort was put into the anodizing process, the thickness deviation was at a satisfactory level of less than 5%. The measurement was performed using an electronic coating thickness gauge (PosiTector 6000 probe, DeFelsko, New York, USA).
Fig. 7. Measured thickness of alumina layer according to the longitudinal position on a 1-m aluminum slab. The measurements were taken every 2 cm on the slab, and the target thickness of the alumina layer was 60 µm. The measured thickness is 60 ${\pm} $ 1.5 µm.
4. Operating wavelengths of the eye-shaped waveguide FEL
In general, the FEL operating wavelength in waveguide mode is different from that in free-space mode, even with the same accelerator and undulator parameters. We consider the dispersion relation of the eye-shaped waveguide to obtain the operating wavelength of the waveguide-mode THz FEL. The easiest approach is to obtain wavelength values that simultaneously satisfy the dispersion relations of the FEL and waveguide. We begin with the FEL dispersion relation. An electron that travels along the z-axis and enters a planar undulator has a sinusoidal or wiggling motion with the same period as the undulator (λu). The radiation generated by the movement travels faster than the average z-axis velocity of the electron (vz). When the difference between the electron and the radiation after one cycle of motion in the undulator is equal to the wavelength of the radiation, the radiation of that wavelength satisfies the coherent condition. If the time required for the electron to move one cycle of the wiggling motion is T, the above condition satisfies the following relation:
where vp is the phase velocity of the radiation, and λ is the wavelength of the radiation. That is, as in the above relation, when the phase of the emitted radiation advances further by the span of the wavelength during the time that the electron proceeds one cycle, the power of that wavelength increases compared to those of other wavelengths, owing to the coherent condition. The relation can be expressed using the wavenumbers of the radiation (k = 2π/λ) and undulator (ku = 2π/λu) as follows:The average longitudinal velocity of the electron in the undulator is as follows:
Here, γ is the Lorentz factor, and K (${\cong} 0.934{B_u}{\lambda _u})$ is the undulator deflection parameter, where ${B_u}$ is the magnetic-field strength of the undulator.
By simulating the dispersion relation of the eye-shaped waveguide, we can calculate the operating wavelength range of the waveguide-mode THz FEL. Here, we present the calculated wavelength range for the table-top THz FEL with different electron beam energies, that is, 3.5 and 4.5 MeV. The magnetic field strength of the undulator ranges from 4 to 11 kG, which are available parameters of the table-top FEL that we are developing. For a beam energy of 4.5 MeV and undulator magnetic strength of 7–11 kG, the calculated FEL wavelength range is ∼300–600 µm, as shown in Fig. 8(a). For an electron beam energy of 3.5 MeV and undulator magnetic strength of 4–8 kG, the FEL wavelength is ∼300–600 µm, as shown in Fig. 8(b). By varying the electron beam energy and magnetic strength of the undulator, we can easily obtain the desired wavelength value, i.e., 300–600 µm, which can be used for security inspection.
Fig. 8. (a) Calculated FEL wavelength for an electron beam energy of 4.5 MeV and undulator magnetic strength range of 7–11 kG, and for an electron beam energy of 3.5 MeV and undulator magnetic strength of 4–8 kG.
5. Conclusion
A table-top THz FEL is expected to satisfy the needs of the security inspection market with its high power and broad-band wavelength tenability. It is necessary to overcome technical limitations to realize a smaller FEL, and the most important of these is to obtain sufficient gain for lasing. If a mode with a small cross section is used, the FEL small-signal gain is increased in inverse proportion to its area. We devised a special eye-shaped waveguide with a center-focused TE-mode to compensate the gain decrease caused by the small-sized accelerator and undulator. The mode cross-sectional area is just 4 mm2 (FWHM), which is less than 1/75 of the free-space Gaussian mode and 1/4 of the 1D plane-parallel waveguide mode. By applying an alumina layer to the eye-shaped waveguide to maintain the cross-sectional distribution, we obtained considerably low wave loss of less than 2.5% for a 1-m propagation at the operating wavelength range of 300–600 µm. It was confirmed that FEL lasing in the wavelength range of 300–600 µm is possible with the specifications of the accelerator and undulator, which we are currently developing.
In an FEL resonator using a small cross-sectional area waveguide, the requirement for the trajectory envelope of the electron beam becomes more stringent than that in the free-space mode. This requires the development of an undulator with a more precise magnetic field distribution. However, that level is technically enough to realize. The horizontal size of the eye-shaped waveguide mode is 3.4 mm in FWHM, and the undulator we are developing has obtained a total electron envelope with a full width of 2.8 mm that satisfies the requirement sufficiently.
FEL is a high-power coherent light source that realizes perfect wavelength tunability. The development of the FEL, which was considered a large research facility, into a small device will be an important step to realize its industrial application. In particular, it provides a new opportunity in a field where development is being delayed due to the absence of an appropriate light source even with high applicability, such as that with THz wave. The outcomes of this study contribute to the development of a high-power table-top THz FEL for industrial applications, such as security inspection.
Funding
Ministry of Science and ICT, South Korea; A National Research Council of Science & Technology (NST) grant awarded by the Korean government (MSIT) (No. CAP-18-05-KAERI); YOUNG IN ACE Co., Ltd. through funded by the Commercializations Promotion Agency for R&D Outcomes grant funded by the Korean government (MSIT)..
Acknowledgments
The authors would like to thank the late Dr. Sergey Miginsky for helpful discussions and Analysis Technology Research Center of the YOUNG IN ACE Co., Ltd.
Disclosures
The authors declare no conflict of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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