本論文研究重點為多模光波導內自我成像特性。波導內自我成像的距離與波導寬度、長度及折射率分佈有關。自我成像特性主要可用於訊號的傳輸或聚光等功能。透過使用多模波導上的繞射特性，我們提出了一個方法來設計在固定的寬度上縮短自我成像的距離。縮短自我成像距離的好處可使得元件的尺寸能更小且加快傳輸速度，節省更多材料，也能在積體光路板上做更多有關光的應用。使用電腦模擬生成的平面全息圖(computer-generated planar holograms, CGPHs)，調整導引模態間的耦合係數以生成所需的繞射特性。在計算方面我們用模態耦合理論(coupled-mode theory)作為基礎。設計方面使用矽絕緣體(silicon-on-insulator, SOI)做為平台。此外再利用光束傳播模擬法(beam propagation method simulation , BPM simulation)來驗證模態耦合理論。

This thesis focuses on the self-imaging properties of multimode waveguides . The self-imaging of waveguides is related to their widths, lengths, and refractive index distributions. Self-imaging properties can be used to signal transmission or aggregate light. By engineering the multimode waveguide diffraction properties, we proposed a method to design fixed-width waveguides with shortened self-imaging lengths. The benefits of shortened self-imaging lengths can be made smaller and speed up the transfer speed to save more material, but also in the integrated optical circuits board to do more about the application of light. Using the computer-generated planar holograms, we adjust the coupling coefficients between the guided modes to generate the desired diffraction properties. In the calculations, we use the coupled-mode theory as a basis. Devices are designed based on a silicon-on-insulator(SOI)platform.
In addition, we use beam propagation simulation method to verify the coupled-mode theory.

1.王健,導波光學,(清華大學出版社,北京,2010).
2.K. Okamoto, Fundamentals of Optical Waveguides. (Academic, Burlington, MA, 2006).
3.D. Mendlovic, and H. M. Ozaktas, 「Fractional Fourier transforms and their optical implementation: I,」 J. Opt. Soc. Am. A 10(9), 1875–1881 (1993).
4.O. Bryngdahl, 「Image formation using self-imaging techniques,」 J. Opt. Soc. Am. 63(4), 416–419 (1973).
5.R. Ulrich, 「Image formation by phase coincidence in optical waveguides,」 Opt. Commun. 13(3), 259–264(1975).
6.L. B. Soldano, and E. C. M. Pennings, 「Optical multi-mode interference devices based on self-imaging: principles and applications,」 J. Lightwave Technol. 13(4), 615–627(1995).
7. P. A. Besse, M. Bachmann, H. Melchior, L. B. Soldano, and M. K.Smit, 「Optical bandwidth and fabrication tolerances of multimode interference couplers,」 J. Lightwave Technol. 12(6), 1004–1009(1994).
8. R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg,and Y.Silberberg, 「Experimental observation of linear and nonlinear optical bloch oscillations,」 Phys. Rev. Lett. 83(23), 4756–4759 (1999).
9. D. N. Christodoulides, F. Lederer, and Y. Silberberg, 「Discretizing light behaviour in linear and nonlinear waveguide lattices,」 Nature 424(6950), 817–823 (2003).
10. R. Gordon, 「Harmonic oscillation in a spatially finite array waveguide,」 Opt. Lett. 29(23), 2752–2754 (2004).
11. N. K. Efremidis, and D. N. Christodoulides, 「Revivals in engineered waveguide arrays,」 Opt. Commun. 246(4-6), 345–356 (2005).
12. T. W. Mossberg, 「Planar holographic optical processing devices,」 Opt. Lett. 26(7), 414–416 (2001).
13. T. Hashimoto, T. Saida, I. Ogawa, M. Kohtoku, T. Shibata, and H. Takahashi, 「Optical circuit design based on a wavefront-matching method,」 Opt. Lett. 30(19), 2620–2622 (2005).
14. A. Yariv, 「Coupled-mode theory for guided-wave optics,」 IEEE J. Quantum Electron. 9(9), 919–933 (1973).
15. S.-Y. Tseng, Y. Kim, C. J. K. Richardson, and J. Goldhar, 「Implementation of discrete unitary transformations by multimode waveguide holograms,」 Appl. Opt. 45(20), 4864–4872 (2006).
16. S.-Y. Tseng, C. Fuentes-Hernandez, D. Owens, and B. Kippelen, 「Variable splitting ratio 2 x 2 MMI couplers using multimode waveguide holograms,」 Opt. Express 15(14), 9015–9021 (2007).
17. S.-Y. Tseng, S. Choi, and B. Kippelen, 「Variable-ratio power splitters using computer-generated planar holograms on multimode interference couplers,」 Opt. Lett. 34(4), 512–514 (2009).
18. M. Bachmann, M.K. Smit, L.B. Soldano, P.A. Besse, E. Gini, and H. Melchior, 「Polarization-insensitive low-voltage optical waveguide switch using InGaAsP/InP four-port Mach-Zehnder interferometer,」 in Proc. Conf. Opt. Fiber Commun.(OFC), CA, pp.32-33, 1993.
19. M.J.N. van Stralen, R. van Roijen, E.C.M. Pennings, J.M.M. van der Heijden, T. van Dongen, and B.H. Verbeek, 「Design and fabrication of integrated InGaAsP ring lasers with MMI-outcouplers,」 in Proc. European Conf. Integrated Optics(ECIO), Switzerland, pp. 2.24-2.25, 1993.
20. R. van Roijen, E.C.M. Pennings, M.J.N. van Stralen, T. Van Dongen, B.H. Verbeek, and J.M.M. van der Heijden, 「Compact InP-based ring lasers employing multimode interference couplers and combiners,」 Appl. Phys. Lett., vol. 64, no. 14, pp. 1753-1755, 1994.
21. M. Bachmann, P. A. Besse, and H. Melchior, 「General self-imaging properties in N×N multimode interference couplers including phase relations,」 Appl. Opt., vol. 33, pp. 3905-3911, June 1994.
22. Elmar G Sauter and Ajoy K Ghatakt, 「The harmonic oscillator problem and the parabolic index optical waveguide: II. Quantum mechanical and wave optical analyses」, Eur. J . Phys. 10 (1989) 143-150.
23. K. Kawano, and T. Kitoh, Introduction to Optical Waveguide Analysis (Wiley, New York, 2001).
24. G. R. Hadley, 「Wide-angle beam propagation using Pade approximant operators,」 Opt. Lett. 17(20), 1426–1428(1992).
25. M. T. Chu, and G. H. Golub, Inverse Eigenvalue Problems: Theory,Algorithms, and Applications, (Oxford, 2005).