本論文研究光在矩形截面光子晶體環形波導運行的特性。環形波導形狀設定為矩形是為了易於後續以電路板進行製程。我們利用圓柱座標有限時域差分法計算光在結構中的運行，改變矩形截面的寬度與厚度，找出讓光能量能夠侷限在結構之中的參數。我們改變環形波導中心寬度，利用快速傅立葉轉換觀察模態有效折射率的變化。接著我們將中心寬度固定，改變連續波射入波導之中，並研究光波頻率與色散的關係。在優化光運行的特性之後，矩形截面的厚度定為電路板的2倍，矩形截面寬度為0.45a (a為光子晶體晶格常數)。當矩形厚度為電路板的3倍時，矩形寬度為0.33a的參數進行模擬。經波導傳播耗損的計算，兩種結構的波導傳播耗損分別為-0.746 dB⁄cm以及-0.736 dB⁄cm。我們觀察到兩結構最低的背向散射光強度僅有-28.56 dB。

This study explores the properties of light propagation in periodically arranged torus photonic crystal waveguides, designed with rectangular cross-sections to simplify fabrication processes. Using the cylindrical-coordinate finite-difference time-domain method, we systematically varied the width and thickness of the rectangular cross-section to obtain a higher energy confinement within the structure. Two configurations were selected for further simulation. 1. The thickness of the rectangular cross-section is twice that of the PCB. The width of the rectangular cross-section is 0.45a where a is the lattice constant of the photonic crystal structure. 2. The thickness of the rectangular cross-section is three times that of the PCB. The width of the rectangular cross-section is 0.33a. The effective index neff of the waveguide was analyzed using fast Fourier transform while varying the width of the central waveguide. We studied the dispersion relation of the waveguide. The propagation loss calculations were investigated for both configurations. The simulated propagation losses were -0.746 dB⁄cm, and -0.736 dB⁄cm, respectively. Additionally, the minimum Back scattering was as low as -28.56 dB.

1. Yablonovitch, E., Inhibited Spontaneous Emission in Solid-State Physics and Electronics. Physical Review Letters, 1987. 58(20): p. 2059-2062.
2. John, S., Strong localization of photons in certain disordered dielectric superlattices. Physical Review Letters, 1987. 58(23): p. 2486-2489.
3. Chiu, W.-Y., et al., A photonic crystal ring resonator formed by SOI nano-rods. Optics Express, 2007. 15(23): p. 15500-15506.
4. Shih-Shou, L., et al., Fabricating a hollow optical waveguide for optical communication applications. Journal of Microelectromechanical Systems, 2006. 15(3): p. 584-587.
5. Chen, C.-C., et al., Photonic crystal directional couplers formed by InAlGaAs nano-rods. Optics Express, 2005. 13(1): p. 38-43.
6. Taflove, A. and S. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd edition. Vol. 2062. 2005.
7. Taflove, A., A. Oskooi, and S.G. Johnson, Advances in FDTD computational electrodynamics : photonics and nanotechnology / Allen Taflove, editor; Ardavan Oskooi and Steven G. Johnson, Coeditors. Artech House antennas and propagation library. 2013, Boston: Artech House.
8. Decleer, P., et al., Nonuniform and Higher-order FDTD Methods for the Schrödinger Equation. Journal of Computational and Applied Mathematics, 2021. 381: p. 113023.
9. Tsai, Y.-L., et al., Optical confinement using a doughnut waveguide. Journal of Physics D: Applied Physics, 2010. 43(24): p. 245103.
10. Dib, N., et al., Analysis of cylindrical transmission lines with the finite-difference time-domain method. IEEE Transactions on Microwave Theory and Techniques, 1999. 47(4): p. 509-512.
11. Liu, J., et al., An Effective CFS-PML Implementation for Cylindrical Coordinate FDTD Method. IEEE Microwave and Wireless Components Letters, 2012. 22(6): p. 300-302.
12. Lo, S.-S. and C.-C. Chen, 1×2 Multimode interference couplers based on semiconductor hollow waveguides formed from omnidirectional reflectors. Optics Letters, 2007. 32(13): p. 1803-1805.
13. Lo, S.-S., M.-S. Wang, and C.-C. Chen, Semiconductor hollow optical waveguides formed by omni-directional reflectors. Optics Express, 2004. 12(26): p. 6589-6593.
14. Shih-Shou, L., et al., Fabricating low-loss hollow optical waveguides via amorphous silicon bonding using dilute KOH solvent. IEEE Photonics Technology Letters, 2005. 17(12): p. 2592-2594.
15. Chiu, H.-K., et al., Sharply bent hollow optical waveguides formed by an omni-directional reflector. Journal of Physics D: Applied Physics, 2009. 42(19): p. 195106.
16. Lo, S.-S. and C.-C. Chen, Air-core hollow optical waveguides with omnidirectional reflectors. Optical Engineering, 2006. 45(4): p. 044601.
17. Chiu, H.-K., et al., Compact and low-loss bent hollow waveguides with distributed Bragg reflector. Optics Express, 2008. 16(19): p. 15069-15073.
18. Cai, D.-P., et al., Liquid crystal infiltrated waveguide with distributed Bragg reflectors. Optical Materials Express, 2011. 1(8): p. 1471-1477.
19. Cai, D.-P., et al., Electrically tunable liquid crystal waveguide attenuators. Optics Express, 2011. 19(12): p. 11890-11896.
20. Rodenbeck, C.T., et al., Terrestrial Microwave Power Beaming. IEEE Journal of Microwaves, 2022. 2(1): p. 28-43.
21. Takahashi, T., et al. Development of phased array for high accurate microwave power transmission. in 2011 IEEE MTT-S International Microwave Workshop Series on Innovative Wireless Power Transmission: Technologies, Systems, and Applications. 2011.
22. Shinohara, N., H. Matsumoto, and K. Hashimoto, Phase-controlled magnetron development for SPORTS: Space power radio transmission system. URSI Radio Science Bulletin, 2004. 2004(310): p. 29-35.
23. Balanis, C.A., Antenna theory: analysis and design. 2016: John wiley & sons.
24. Glaser, P.E., Power from the Sun: Its Future. Science, 1968. 162(3856): p. 857-861.
25. Glaser, P.E., F.P. Davidson, and K.I. Csigi, Solar Power Satellites: A Space Energy System for Earth. 1998: Wiley.
26. Lalanne, P. and E. Silberstein, Fourier-modal methods applied to waveguide computational problems. Optics Letters, 2000. 25(15): p. 1092-1094.
27. Dulkeith, E., et al., Group index and group velocity dispersion in silicon-on-insulator photonic wires. Optics Express, 2006. 14(9): p. 3853-3863.
28. Khurgin, J.B., Optical buffers based on slow light in electromagnetically induced transparent media and coupled resonator structures: comparative analysis. Journal of the Optical Society of America B, 2005. 22(5): p. 1062-1074.
29. Lo, S.-S. and C.-C. Chen, High finesse of optical filter by a set Fabry-Perot cavity. Journal of the Optical Society of America B, 2007. 24(8): p. 1853-1856.
30. Chiu, H.-K., et al., Wavelength-selective filter based on a hollow optical waveguide. Applied Optics, 2011. 50(2): p. 227-230.
31. Kasapi, A., et al., Electromagnetically Induced Transparency: Propagation Dynamics. Physical Review Letters, 1995. 74(13): p. 2447-2450.
32. Bigelow, M.S., N.N. Lepeshkin, and R.W. Boyd, Observation of Ultraslow Light Propagation in a Ruby Crystal at Room Temperature. Physical Review Letters, 2003. 90(11): p. 113903.
33. Ku, P.-C., et al., Slow light in semiconductor quantum wells. Optics Letters, 2004. 29(19): p. 2291-2293.
34. Altug, H. and J. Vučković, Experimental demonstration of the slow group velocity of light in two-dimensional coupled photonic crystal microcavity arrays. Applied Physics Letters, 2005. 86(11).
35. Notomi, M., E. Kuramochi, and T. Tanabe, Large-scale arrays of ultrahigh-Q coupled nanocavities. Nature Photonics, 2008. 2(12): p. 741-747.
36. Klein, M., et al., Slow light in a 2D semiconductor plasmonic structure. Nature Communications, 2022. 13(1): p. 6216.
37. Cheng, W.-C., et al., Measurement-based delay, angular dispersion and propagation loss characteristics of outdoor propagation in beam domain and multi-beam operation at 38 GHz for 5-G communication systems. IET Microwaves, Antennas & Propagation, 2022. 16(5): p. 257-271.
38. Lim, J., et al., Experimental Demonstration of Germanium-on-Silicon Slot Waveguides at Mid-Infrared Wavelength. IEEE Photonics Journal, 2022. 14(3): p. 1-9.
39. Yang, B., et al., Fabrication and Characterization of Small Optical Ridge Waveguides Based on SU-8 Polymer. Journal of Lightwave Technology, 2009. 27(18): p. 4091-4096.
40. Lombardet, B., et al., Propagation loss measurements and Fabry–Pérot mode analysis using out-of-plane light scattering in photonic crystal waveguides. Applied Physics Letters, 2005. 86(11): p. 111111.
41. Feuchter, T. and C. Thirstrup, High precision planar waveguide propagation loss measurement technique using a Fabry-Perot cavity. IEEE Photonics Technology Letters, 1994. 6(10): p. 1244-1247.
42. Ma, Y., et al., Direct measurement of propagation losses in silver nanowires. Optics Letters, 2010. 35(8): p. 1160-1162.
43. Hu, J., et al., Fabrication and testing of planar chalcogenide waveguide integrated microfluidic sensor. Optics Express, 2007. 15(5): p. 2307-2314.
44. Bourk, T.R., M.M.Z. Kharadly, and J.E. Lewis, Measurement of waveguide attenuation by resonance methods. Electronics Letters, 1968. 4(13): p. 267-268.
45. Corporation, R. AD1000™ Laminates Data Sheet. 2022 2022-07-28]; Available from: https://rogerscorp.com/-/media/project/rogerscorp/documents/advanced-connectivity-solutions/english/data-sheets/ad1000-laminates-data-sheet.pdf.
46. Kane, Y., Numerical solution of initial boundary value problems involving maxwell's equations in isotropic media. IEEE Transactions on Antennas and Propagation, 1966. 14(3): p. 302-307.
47. Kawano, K. and T. Kitoh, Introduction to optical waveguide analysis: Solving maxwell's equation and the schrödinger equation. 2001: John wiley & sons.
48. Berenger, J.-P., A perfectly matched layer for the absorption of electromagnetic waves. Journal of Computational Physics, 1994. 114(2): p. 185-200.
49. Ablowitz, M.J., Nonlinear Dispersive Waves: Asymptotic Analysis and Solitons. Cambridge Texts in Applied Mathematics. 2011, Cambridge: Cambridge University Press.
50. Oubrerie, K., et al., Controlled acceleration of GeV electron beams in an all-optical plasma waveguide. Light: Science & Applications, 2022. 11(1): p. 180.