本研究探討了使用金屬和介電材質之週期性波導，產生類似雷射輻射的緊湊型桌面自由電子雷射 (FEL) 的理論研究。我們使用石墨光電陰電子槍，通過熱輔助多光子光電效應產生電子束。此電子槍的陰極設計具有20毫米的曲率半徑，不僅能聚焦電子束，還可以用作反向波振盪 (BWO) 輻射反射的腔體。
為了在圓柱銅管內製造週期性波導，我們提出了一種使用銅製 M1.6 公制螺紋作為螺旋波導，並由 30-keV 電子束激發。CST數值模擬結果顯示在 0.2616 THz 頻率有顯著峰值，這與理論色散曲線預測的 0.2666 THz 的高階 TM-TM 耦合模式一致。為了進一步分析螺旋波導的性能，我們提出線性矩形波紋且沒有方位變化的圓柱型光柵波導。在相同的電子束條件下，圓柱型光柵波導的結果顯示出更好的性能。這是因為螺旋波導破壞了方位對稱性，由一個稱為折疊數（m_B）的參數定義，這導致兩個 TE 模式、兩個 TM 模式和 TE-TM 模式的非零耦合係數。相比之下，圓柱型光柵波導保持方位對稱，摺疊數為零，導致 TE 和 TM 模式之間的耦合係數為零。當 TE 和 TM 模式耦合時，部分 TM 模式的能量轉移到 TE 模式，降低了能量交換的整體效率。因此，圓柱型光柵波導在維持能量傳輸效率方面表現更佳。接下來我們通過改變槽深度以提升圓柱型光柵波導的性能，增加槽深度可提高耦合效率，此外，圓柱型光柵波導的槽可作為共振腔，我們證實了在光柵深度與波長比值 D/λ=0.27 時，圓柱型光柵波導達到共振腔條件，達到最高的功率和效率，與四分之一波長的理論最佳共振條件非常接近。這些分析表明精確結構設計在優化圓柱型金屬週期性波導性能中的重要性。
在最後一章中，我們提出兩種介電材質波導結構，分別為矽圓孔波導和矽方孔波導，並在布拉格共振條件下運行，以實現高效率能量轉換。由於圓孔的平滑的邊界條件，矽圓孔波導相比矽方孔波導展現了更均勻的模式場分佈和更高的橫向磁場強度，這有助於穩定波傳輸和有效模式耦合，並在能量儲存方面和整體性能更具有優勢。
此研究對螺旋和線性波紋波導之不同結構產生兆赫波輻射進行了全面比較，有助於更高效率和緊湊的自由電子激光器 (FELs)的發展。

This study explores the theoretical development of compact, tabletop-size free-electron lasers (FELs) that generate laser-like radiation using metallic and dielectric periodic waveguides. We utilized a graphite-based photocathode gun to generate an electron beam through thermal-assisted multi-photon photoemission. The cathode with a radius of curvature of 20 mm, not only focuses the beam but also serves as the cavity for reflecting the backward-wave oscillation (BWO) radiation.
To fabricate the metallic periodic waveguides within the cylindrical copper tube, we proposed a copper M1.6 metric thread as a helically corrugated waveguide and excited by a 30-keV electron beam. The numerical CST simulation result shows a narrow radiation peak at a frequency of 0.2616 THz, which is consistent with the higher-order TM-TM coupled modes at 0.2666 THz predicted by the theoretical dispersion curves. To further analyze the performance of a helical structure, we introduced a cylindrical grating waveguide with linearly rectangular corrugations and no azimuthal variation. Under the same input electron beam conditions, the result of the cylindrical grating waveguide demonstrated better performance. This is because the helically corrugated waveguide breaks azimuthal symmetry, defined by a parameter called the fold number (m_(B )), leading to non-zero coupling coefficients for two TE, two TM, and one TE-TM mode. In contrast, the cylindrical grating waveguide maintains azimuthal symmetry, with a fold number always zero, resulting in zero coupling coefficients between TE and TM modes. When TE and TM modes are coupled, part of the energy from the TM mode is transferred to the TE mode, reducing the overall efficiency of energy exchange between the electrons and the wave. As a result, the cylindrical grating waveguide performs better in maintaining energy transfer efficiency. Therefore, we continually enhance the performance of the cylindrical grating waveguide by varying the depth of the grooves, increasing the groove depth leads to enhancement in the coupling efficiency. Additionally, the grooves of the cylindrical grating waveguides can be treated as resonators, we verified that the resonance condition of cylindrical grating waveguides is achieved when the grating depth-to-wavelength ratio D/λ=0.27. This condition yields the highest power and energy transfer efficiency, aligning closely with the theoretical optimal resonator condition of a quarter-wavelength depth. These analyses demonstrate the importance of precise structural design in optimizing the performance of cylindrical metallic periodic waveguides for advanced applications.
In Chapter 5, we further proposed dielectric waveguide structures, which are the silicon circular hole-grating waveguide and the silicon square hole-grating waveguide, operating at Bragg resonance conditions to achieve efficient energy transfer. The silicon circular hole-grating waveguide demonstrates a more uniform mode field distribution and higher transverse magnetic field strength compared to the silicon square hole-grating waveguide due to the smooth boundaries of the circular holes, which facilitate stable wave transmission and enhance effective mode coupling. This uniformity and strength make the silicon circular hole waveguide advantageous for energy storage and overall performance.
This study provides a comprehensive comparison of THz radiation from electron-driven helically and linearly corrugated waveguides, contributing to the development of more efficient and compact FELs.

[1] Huang, Z. and K.-J. Kim, Review of x-ray free-electron laser theory. Physical Review Special Topics-Accelerators and Beams, 2007. 10(3): p. 034801.
[2] Mehdi, I., et al., THz diode technology: Status, prospects, and applications. Proceedings of the IEEE, 2017. 105(6): p. 990-1007.
[3] Hafez, H., et al., Intense terahertz radiation and their applications. Journal of Optics, 2016. 18(9): p. 093004.
[4] Milton, S.V., et al., Exponential Gain and Saturation of a Self-Amplified Spontaneous Emission Free-Electron Laser. Science, 2001. 292: p. 2037 - 2041.
[5] Doria, A., et al., Coherent emission and gain from a bunched electron beam. IEEE Journal of Quantum Electronics, 1993. 29: p. 1428-1436.
[6] Madey, J.M., Stimulated emission of bremsstrahlung in a periodic magnetic field. Journal of Applied Physics, 1971. 42(5): p. 1906-1913.
[7] Elias, L.R., et al., Observation of stimulated emission of radiation by relativistic electrons in a spatially periodic transverse magnetic field. Physical Review Letters, 1976. 36(13): p. 717.
[8] Pierce, J.R. and L.M. Field, Traveling-wave tubes. Proceedings of the IRE, 1947. 35(2): p. 108-111.
[9] Cherenkov, P.A., Visible light from clear liquids under the action of gamma radiation. Comptes Rendus (Doklady) de l'Aeademie des Sciences de l'URSS, 1934. 2(8): p. 451-454.
[10] Anderhub, H., et al., Design and operation of FACT–the first G-APD Cherenkov telescope. Journal of Instrumentation, 2013. 8(06): p. P06008.
[11] Doro, M., et al., Dark matter and fundamental physics with the Cherenkov Telescope Array. Astroparticle Physics, 2013. 43: p. 189-214.
[12] Ren, H., et al., Nonlinear Cherenkov radiation in an anomalous dispersive medium. Physical review letters, 2012. 108(22): p. 223901.
[13] Cherenkov, A., V. Lobanov, and M. Gorodetsky, Dissipative Kerr solitons and Cherenkov radiation in optical microresonators with third-order dispersion. Physical Review A, 2017. 95(3): p. 033810.
[14] Glaser, A.K., et al., Optical dosimetry of radiotherapy beams using Cherenkov radiation: the relationship between light emission and dose. Physics in Medicine & Biology, 2014. 59(14): p. 3789.
[15] Shaffer, T.M., E.C. Pratt, and J. Grimm, Utilizing the power of Cerenkov light with nanotechnology. Nature nanotechnology, 2017. 12(2): p. 106-117.
[16] Smith, S.J. and E. Purcell, Visible light from localized surface charges moving across a grating. Physical Review, 1953. 92(4): p. 1069.
[17] Yamamoto, N., F. Javier García de Abajo, and V. Myroshnychenko, Interference of surface plasmons and Smith-Purcell emission probed by angle-resolved cathodoluminescence spectroscopy. Physical Review B, 2015. 91(12): p. 125144.
[18] Ye, Y., et al., Deep-ultraviolet smith–purcell radiation. Optica, 2019. 6(5): p. 592-597.
[19] Johnson, H.R., Backward-wave oscillators. Proceedings of the IRE, 1955. 43(6): p. 684-697.
[20] Harris, S., Proposed backward wave oscillation in the infrared. Applied Physics Letters, 1966. 9(3): p. 114-116.
[21] Shibata, Y., et al., Coherent Smith-Purcell radiation in the millimeter-wave region from a short-bunch beam of relativistic electrons. Physical Review E, 1998. 57(1): p. 1061.
[22] Andrews, H. and C. Brau, Gain of a Smith-Purcell free-electron laser. Physical Review Special Topics—Accelerators and Beams, 2004. 7(7): p. 070701.
[23] Andrews, H.L., et al., Dispersion and attenuation in a Smith-Purcell free electron laser. Physical Review Special Topics—Accelerators and Beams, 2005. 8(5): p. 050703.
[24] Fowler, R.H., The analysis of photoelectric sensitivity curves for clean metals at various temperatures. Physical review, 1931. 38(1): p. 45.
[25] DuBridge, L.A., Theory of the energy distribution of photoelectrons. Physical Review, 1933. 43(9): p. 727.
[26] Bechtel, J., W. Smith, and N. Bloembergen, Four-photon photoemission from tungsten. Optics Communications, 1975. 13(1): p. 56-59.
[27] Damascelli, A., et al., Multiphoton electron emission from Cu and W: An angle-resolved study. Physical Review B, 1996. 54(9): p. 6031.
[28] Sprangle, P. and R.A. Smith, Theory of free-electron lasers. Physical Review A, 1980. 21(1): p. 293.
[29] Gilmour Jr, A.S. and A. Gilmour, Klystrons, traveling wave tubes, magnetrons, crossed-field amplifiers, and gyrotrons. 2011: Artech House.
[30] Radley, D., The theory of the Pierce type electron gun. International Journal Of Electronics, 1958. 4(2): p. 125-148.
[31] Tydex. THz Materials. Available from: http://www.tydexoptics.com/ru/products/thz_optics/thz_materials/.
[32] Zhang, L., et al., Multi-mode coupling wave theory for helically corrugated waveguide. IEEE transactions on microwave theory and techniques, 2011. 60(1): p. 1-7.
[33] Donaldson, C.R., et al., Fivefold helically corrugated waveguide for high-power W-band gyro-devices and pulse compression. IEEE Transactions on Electron Devices, 2021. 69(1): p. 347-352.
[34] Pan, Y. and A. Gover, Spontaneous and stimulated emissions of a preformed quantum free-electron wave function. Physical Review A, 2019. 99(5): p. 052107.
[35] Gover, A., et al., Superradiant and stimulated-superradiant emission of bunched electron beams. Reviews of Modern Physics, 2019. 91(3): p. 035003.
[36] Liu, W., et al., Free electron terahertz wave radiation source with two-section periodical waveguide structures. Journal of Applied Physics, 2012. 111(6).
[37] Zhang, Y.-X., et al., Coherent terahertz radiation from high-harmonic component of modulated free-electron beam in a tapered two-asymmetric grating structure. Applied Physics Letters, 2012. 101(12).
[38] S. Liu, H.L., W. Wang, and Y. Mo, Introduction of Microwave Electronics. 1985, National Defense Industrial.
[39] Gentili, C., Microwave Amplifiers and Oscillators. 1987, McGraw-Hill, New York.
[40] Szczepkowicz, A., L. Schächter, and R.J. England, Frequency-domain calculation of Smith–Purcell radiation for metallic and dielectric gratings. Applied Optics, 2020. 59(35): p. 11146-11155.
[41] Liu, W. and Z. Xu, Special Smith–Purcell radiation from an open resonator array. New Journal of Physics, 2014. 16(7): p. 073006.