在這篇論文裡，我們整理了有限差分在光學系統模擬上的一系列應用，包含了有限差分法、有限差分光束傳播法，還有時域分析中的時域有限差分法與時域有限差分光束傳播法，它們都可以應用到次波長的結構模擬上。除了專書之外，我們還研讀了許多相關論文，而將它們的發展歷史、理論推導…等等作有系統的整理，而且也完成了大部分的程式，並且也去驗證了論文上的結果。最後並實際地運用在一些典型的波導結構的計算上。

In this thesis, we focus on simulation and analysis of optical system with subwavelength scale using finite difference scheme which include finite difference method (FDM), finite difference beam propagation method (FD-BPM), finite difference time domain method (FDTDM), and time-domain beam propagation method (TD-BPM). We review many related papers and study carefully how the wave equation and finite difference form is derived. We also implemented most of them base on Matlab. Finally, we use these programs to simulate some waveguide structures and validate out code by checking the result with published data.

[1-1] R. März, Integrated Optics, ARTECH HOUSE, Boston, 1995.
[1-2] R.G. Hunsperger, Integrated optics: theory and technology, Spriger-Verlag, New York, 1992.
[1-3] 張智星，MATLAB程式設計與應用，清蔚科技，2000。
[1-4] 蒙以正，MATLAB5專業的設計技巧，碁峰資訊，1998。
[2-1] K. Kawano and T. Kitoh, Introduction to Optical Waveguide Analysis, Wiley, New York, 2001.
[2-2] A.W. Snyder and J.D. Love, Optical waveguide theory,
Chapman and Hall, London, 1983.
[2-3] R. März, Integrated Optics, ARTECH HOUSE, Boston, 1995.
[2-4] Y.P Chiou, Y.C. Chiang and H.C. Chang, “Improved three-point formulas considering the interface conditions in the finite-difference analysis of step index optical devices,” J. of Lightwave Technol., vol.18, pp.243-251, 2000.
[2-5] C.L. Xu, W.P. Huang and S.K. Chaudhuri, "Efficient and accurate vector mode calculation by beam propagation method," J. of Lightwave Technol., vol.11, pp.1209-1215,1993.
[2-6] E. Snitzer, “Cylindrical dielectric waveguide modes,” J. Opt. Soc. Am., vol.51, pp.491-498, 1961.
[2-7] D. Gloge, “Weakly guiding fibers,” Appl. Opt., vol.10, pp. 2252-2258, 1971.
[3-1] R. Scarmozzino, A. Gopinath, R. Pregla and S. Helfert, “ Numerical techniques for modeling guided-wave photonic devices,” IEEE J. of Sel. Top. In Q. Electr., Vol.6, No.1, pp.150-162, 2000.
[3-2] T.M. Benson, P. Sewell, P.C. Kendall and S. Sujecki, “Finite difference methods in optoelectronic simulation,” Inter. conf. On transparent optical networks, pp.47-48, 1999.
[3-3] C. Vassallo, “1993-1995 optical mode solvers,” Opt. Quantum Electron., Vol.29, pp.95-114, 1997.
[3-4] K.S. Chiang, “Review of numerical and approximate method for the modal analysis of general optical dielectric waveguides,” Opt. Quantum Electron., Vol.26, pp.113-134, 1994.
[3-5] M.J. Robertson, S. Ritchie and P. Dayan, “Semiconductor waveguides: analysis of optical propagation in single rib structure and directional couplers,” IEE Proc. J., vol.132, pp.336-342, 1985.
[3-6] M. Stern, “Semivectorial polarized finite difference methods optical waveguides with arbitrary index profiles,” IEE Proc. J., vol. 135, pp.56-63, 1988.
[3-7] M. Stern, “Semivectorial polarized H field solutions for dielectric waveguides with arbitrary index profiles,” IEE Proc. J., vol. 135, pp.333-338, 1988.
[3-8] C.L. Xu, W.P. Huang, M.S. Stern and S.K. Chaudhuri, “Full-vectorial mode calculations by finite difference method,” IEE Proc. J., vol. 141, pp.281-286, 1994.
[3-9] C. Vassallo and J. Michiel, “Comparison of a few transparent boundary conditions for finite-difference optical mode-solvers,” J. Lightwave Technol., vol.15, pp.397-402, 1997.
[3-10] D.R. Heatley, G. Vitrant and A. Kevorkian, “Simple finite-difference algorithm for calculating waveguide modes,” Opt. Quantum Electron., vol.26, pp.151-163, 1994.
[3-11] A.W. Snyder and J.D. Love, Optical waveguide theory,
Chapman and Hall, London, 1983.
[4-1] M.D. Feit and J.A. Fleck, Jr., “Light propagation in grade-index optical fibers,” Appl. Opt., vol. 17, pp.3990-3998, 1978.
[4-2] Y. Chung and N. Dagli, “Assessment of finite difference beam propagation,” IEEE J. Quantum Electron., vol.26, pp.1335-1339, 1990.
[4-3] M. Koshica and Y. Tsuji, “A wide-angle finite element beam propagation method,” IEEE Photon. Technol. Lett., vol.8, pp.1208-1210, 1996.
[4-4] T.M. Benson, P. Sewell, A. Vukovic and D.Z. Djurdjevic, “Advances in finite difference beam propagation method,” Inter. conf. On transparent optical networks, pp.36-41, 2001.
[4-5] R. Scarmozzino, A. Gopinath, R. Pregla and S. Helfert, “ Numerical techniques for modeling guided-wave photonic devices,” IEEE J. of Sel. Top. In Q. Electron., Vol.6, pp.150-162, 2000.
[4-6] D. Yevick, “A guide to electric field propagation techniques for guided-wave optics,” Opt. Quantum Electron., vol.26, pp.185-197, 1994.
[4-7] W.P. Huang, C.L. Xu, S.T. Chu and S.K. Chaudhuri, “A vector beam propagation method for guided-wave optics,” IEEE Photon. Technol. Lett., vol.3, pp.910-913, 1991.
[4-8] W.P. Huang, C.L. Xu and S.K. Chaudhuri, “A finite-difference vector beam propagation method based on H-fields,” IEEE Photon. Technol. Lett., vol.3, pp.1117-1120, 1991.
[4-9] W.P. Huang, C.L. Xu and S.K. Chaudhuri, “A finite-difference vector beam propagation method for three-dimensional waveguide structures,” IEEE Photon. Technol. Lett., vol.4, pp.148-151, 1992.
[4-10] W.P. Huang, C.L. Xu, S.T. Chu and S.K. Chaudhuri, “The finite-difference vector beam propagation method: analysis and assessment,” J. Lightwave Technol., vol.10, pp.295-305, 1992.
[4-11] W.P. Huang and C.L. Xu, “Simulation of three-dimensional optical waveguides by a full-vector beam propagation method,” IEEE J. of Quantum Electron., vol.29, pp.2639-2649, 1993.
[4-12] C.L. Xu, W.P. Huang, S.K. Chaudhuri and J. Chrostowski, “An unconditionally stable vectorial beam propagation method for 3-D structures,” IEEE Photon. Technol. Lett., vol.6, pp.549-551, 1994.
[4-13] G.R. Hadley, “Wide-angle beam propagation using Padé approximant operators,” Opt. Lett., vol. 17, pp. 1426-1428, 1992.
[4-14] G.R. Hadley, “Multistep Method for wide-angle beam propagation,” Opt. Lett., vol. 17, pp. 1743, 1992.
[4-15] F. Ma, C.L. Xu and W.P. Huang, “Wide-angle full vectorial beam propagation method,” IEE Proc. J., vol.143, pp.139-143, 1996.
[4-16] G.R. Hadley, “Transparent boundary condition for beam propagation method,” Opt. Lett., vol. 16, pp. 624-626, 1991.
[4-17] G.R. Hadley, “Transparent boundary condition for the beam propagation method,” IEEE J. of Quantum Electron., vol.28, pp.363-370, 1992.
[4-18] J.P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Computational Phys., vol.114, pp.185-200, 1994.
[4-19] W.P. Huang, C.L. Xu, W. Lui and K. Yokoyama, “Perfectly matched layer(PML) boundary condition for the beam propagation method,” IEEE Photon. Technol. Lett., vol.8, pp.649-651, 1996.
[4-20] C.L. Xu, W.P. Huang and S.K. Chaudhuri, “Efficient and accurate vector mode calculations by beam propagation method,” J. of Lightwave Technol., vol.11, pp.1209-1215, 1993.
[4-21] S. Jüngling and J.C. Chen, “A study and Optimization of eigenmode calculations using the imaginary distance beam propagation method,” IEEE J. of Quantum Electron., vol.30, pp.2098-2105, 1994.
[4-22] J. Yamauchi, T. Ando and H. Nakano “Beam-propagation analysis of optical fibers by alternating direction implicit method,” Eletron. Lett., vol.27, pp.1663-1665, 1991.
[4-23] P.L. Liu and B.J. Li “Semivectorial beam-propagation method for analyzing polarized modes of rib waveguides,” IEEE J. of Quantum Electron., vol.28, pp.778-782, 1992.
[4-24] P.L. Liu, S.L. Yang and D.M. Yuan “The semivectorial beam propagation method,” IEEE J. of Quantum Electron., vol.29, pp.1205-1211, 1993.
[4-25] Y.L. Hsueh, M.C. Yang and H.C. Chang, “Three-dimensional noniterative full-vectorial beam propagation method based on the alternating direction implicit method,” J. of Lightwave Technol., vol.17, pp.2389-2397, 1999.
[4-26] W.H. Press, S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery, Numerical Recipe, Cambridge University Press, New York, 1992.
[4-27] G.R. Hadley, “Wide-angle beam propagation using Padé approximant operators,” Opt. Lett., vol. 17, pp. 1426-1428, 1992.
[4-28] G.R. Hadley, “Multistep Method for wide-angle beam propagation,” Opt. Lett., vol. 17, pp. 1743, 1992.
[4-29] F. Ma, C.L. Xu and W.P. Huang, “Wide-angle full vectorial beam propagation method,” IEE Proc. J., vol.143, pp.139-143, 1996.
[4-30] Y.P. Chiou and H.C. Chang, “Efficient beam-propagation method based on Padé approximants in the propagation direction,” Opt. Lett., vol.22, pp.949-951, 1997.
[4-31] Y.P. Chiou, Y.C.Chiang, and H.C. Chang, "Improved three point
formulas considering the interface conditions in the finite-
difference analysis of step-index optical devices," Journal of
Lightwave Technology, vol.18, pp.243-251, 2000.
[5-1] K.S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,”IEEE Trans. Antennas Propagag., vol.AP-14,pp.302-307,1966.
[5-2] G. Mur, “Absorbing boundary conditions for the finite-difference time-domain approximation of the time domain electromagnetic field equations,”IEEE Trans. Electromagn. Compat., vol.EMC-23,pp.377-382,1981.
[5-3] J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Computat. Phys., vol.114, pp. 185-220,1994.
[5-4] A. Taflove and S.C. Hagness, Computational Electrodynamics, Artech House, 2000.
[5-5] K.L. Shlager, J.G. Maloney, S.L. Ray and A.F. Peterson, “Relative accuracy of several finite-difference time-domain methods in two and three dimensions,” IEEE Trans. Antennas Propagat., vol.41, pp.1732-1737, 1993.
[5-6] H.M. Masoudi and J.M. Arnold, “Parallel beam propagation methods,” IEEE Photon. Technol. Lett., vol.6, pp.848-850, 1994.
[5-7] W.P. Huang, C.L. Xu and J. Chrostowski, “A time-domain propagation scheme for simulation of dynamics of optical guided-wave devices,” IEEE Photon. Technol. Lett., vol.5, pp.1071-1073, 1993.
[5-8] R.Y. Chan and J.M. Liu, “Time-domain wave propagation in optical structures,” IEEE Photon. Technol. Lett., vol.6, 1001-1003, 1994.
[5-9] F. Horst, H.J.W.M. Hoekstra, A. Driessen and T.J.A. Popma, “Time domain beam propagation method for the simulation of temporal solitons in periodic media,” LEOS, vol.2, pp.27-28, 1995.
[5-10] P.L. Liu, Q. Zhao and F.S. Choa, “Slow-wave finite-difference beam propagation method,” IEEE Photon. Technol. Lett., vol.7, 890-892, 1995.
[5-11] F. Ma, “Slowly Varying envelope simulation of optical waves in time domain with transparent and absorbing boundary conditions,” J. of Lightwave Technol., vol.15, pp.1974-1985, 1997.
[5-12] G.H. Jin, J. Harari, J.P. Vilcot and Decoster, “An improved time-domain beam propagation method for integrated optics components,” IEEE Photon. Technol. Lett., vol.9, pp.348-350, 1997.
[5-13] H.M. Masoudi, M.A. Al-sunaidi and J.M. Arnold, “Time-domain finite-difference beam propagation method,” IEEE Photon. Lett., vol.11, pp.1274-1276, 1999.
[5-14] J. Shibayama, T. Takahashi, J. Yamauchi and H. Nakano, “Efficient time-domain finite-difference beam propagation methods for the analysis of slab and circularly symmetric waveguides,” J. of Lightwave Technol., vol.18, pp.437-442, 2000.
[5-15] H.M. Masoudi, M.A. Al-Sunaidi and J.M. Arnold, “Efficient time-domain beam propagation method for modeling integrated optical devices,” J. of Lightwave Technol., vol.19, pp.759-771, 2001.