低損耗的跨越波導(waveguide crossing)是發展微積體光路(integrated optics, planar light wave circuit, PLC)的重要環節，跨越波導的設計能夠節省各光學元件連結的佔用空間，讓積體光學系統縮小尺寸，所以如何設計低損耗的跨越波導在積體光學的應用中相當重要。
跨越波導是一個全方向全角度傳播的元件，分析的困難點在於精密計算波導跨越部分的場型與能量，探討不同設計的耦光效應(crossing coupling effect)與各方向的能量損耗，所以必須採用能準確計算大角度跨越或分歧波導的理論方法。此元件若是用光束傳播法(beam propagation method, BPM)模擬時，因為BPM法的限制，無法處理跨越波導這元件的大角度交錯之波導耦光效應，且無法計算反射的能量，同時在大角度彎曲、分歧波導結構和介電常數為大對比時，在分析上會因無法正確地模擬高階模態而造成不小的誤差，這將使得整個設計結果存在可觀的累計偏差數值。
本論文採用頻域有限差分法(finite-different frequency-domain method, FD-FD method)，配合自行開發出模態穿透邊界(layer-mode transparent boundary condition, LM-TBC)，可單一介面就達到完全的吸收效果，且不會有入射角度的限制，將能避免用時域有限差分法(finite-difference time-domain method, FD-TD method) 做計算。因為時域的後期訊號會含由非理想吸收邊界的反射的光場，這些反射光場的誤差，將使整個計算結果無法達到PLC設計中高精密度的需求。且FD-FD法在頻域上的解析度並無限制，對計算結果中相近的模態有很高的辨別能力，可分析各不同模態之間的轉換 (mode conversion)，所以能精確模擬跨越波導的耦光效應。
本論文用自行發展的頻域有限差分法與LM-TBC來分析跨越波導元件，以嚴謹的數值計算來達到PLC晶片設計中，對跨越波導元件高精密度的模擬要求，這對積體光學的設計與分析上有創新的貢獻。

Multiple dielectric crossing waveguides are indispensable in building a complex optical integrated circuit. Since each input/output waveguide will have many crossings, it is important to design a low-loss waveguide crossing to ensure the overall radiation loss is kept at a minimum.
The beam propagation method (BPM) is usually the method of choice for modeling large but low-index-contrast waveguide devices. BPM assumes one-way propagation and adopts the paraxial approximation. It is neither able to consider reflection of electromagnetic (EM) fields nor to perform wide angle propagation of forward fields. Therefore, it can not be used to analyze perpendicular dielectric crossing waveguides. At a maximum 0.5 dB power loss per crossing, the difficulty of simulation a waveguide crossing is how to compute the complex coupling waves with high enough precision.
In this thesis, two-dimensional planar integrated optical waveguide crossing is studied in detail for the through and cross power coupling coefficients with the finite-difference frequency-domain (FD-FD) method. By exploiting the dual symmetries: the “+” symmetry and the “X” symmetry in the perpendicular crossing waveguide, we are able to compute the EM fields and their power coefficients without using artificial absorbing boundary conditions (ABC) nor using the perfectly matching layer (PML). We develop the layer-mode based transparent boundary condition (LM-TBC) [1] for launching the fundamental incident mode as well as transmitting the reflected and scattered wave fields off the crossing area. Numerical results including the field distribution, power coefficients are carefully verified and the convergent comparisons are also studied in the thesis.

1. INTRODUCTION..............................................................1
2. FINITE-DIFFERENCE FREQUENCY-DOMAIN
METHOD...........................................................................2
2-1 Introduction of Wave Equation.................................7
2-2 Introduction of Differential Wave Equation..........12
3. HYBRID FD-FD METHOD............................................14
3-1 Layer-Mode Transparent Boundary Condition
Theory.........................................................................17
3-2 Index Averaging Theory.......................................... 22
3-3 Coefficients for Boundary Conditions..................27
3-4 Verify LM-TBC’s Performance................................33
4. MICROWAVE WAVEGUIDE CROSSINGS.................43
4-1 Symmetry of Microwave Waveguide Crossings.43
4-2 Simulation Results of Microwave Waveguide
Crossings..................................................................58
5. DIELECTRIC WAVEGUIDE CROSSINGS.................70
5-1 Symmetry of Dielectric Waveguide Crossings...71
5-2 Simulation Results of Dielectric Waveguide
Crossings.................................................................78
6. CONCLUSIONS.............................................................87
6-1 Summary...................................................................87
6-2 Future Works............................................................89
REFERENCES...................................................................90
PUBLICATION LIST...........................................................96
Appendix..............................................................................98
I. The Analysis of Modes in the Dielectric Waveguide.98
II. Power Reflection & Transmission Coefficients.....102

[1] H.W. Chang, W. C. Cheng and S.M. Lu, “Layer-mode transparent boundary condition for the hybrid FD-FD method”, Progress In Electromagnetics Research, PIER 94, 175-195, 2009.
[2] Z.W. Lin, “Analysis of 2D optical waveguide structures using frequency-domain finite-difference method,” Master Thesis, Institute Elctro-Optical Engineering, National Sun Yat-sen University, Taiwan, 2004.
[3] H.W. Chang, S.M. Wang, “Large-Scale Hybrid FD-FD Method for Micro-Ring Cavities”, Frontiers in Optics/Laser Science conference, Oct. 2005.
[4]. Lin, C.F., Optical Components for Communications, Kluwer Academic Publishing, Boston, 2004.
[5]. Pavesi, L. and G. Guillot, Optical Interconnects, the Silicon Approach, Springer-Verlag, Berlin, 2006.
[6]. Ishimaru, A., Electromagnetic Propagation, Radiation, and Scattering Prentice Hall, Englewood Clifffs, N. J., 1991.
[7]. Chew, W.-C, Waves and Fields in Inhomogeneous Media, New York: Van Norstrand Reinhold , 1990.
[8]. Taflove, A, S. Hagness, Compuatational Electrodynamics: The Finite-Difference Time-Domain Method, Artech House, Norwood, MA, 2000.
[9]. Zhang Y., Ding W., and Liang C. H., “Study on the Optimum Virtual Topology for MPI based Parallel Conformal FDTD Algorithm on PC Clusters”, J. of Electromagn. Waves and Appl., Vol. 19, No. 13, 1817–1831, 2005.
[10]. Zheng, G., A. A. Kishk, A. W. Glisson, and A. B. Yakovlev, “A novel implementation of modified maxwell's equations in the periodic Finite-Difference Time-Domain method”, Progress In Electromagnetics Research, PIER 59, 85–100, 2006.
[11]. Yamauchi, J., Propagating Beam Analysis of Optical Waveguides, Research Studies Press, Baldock, England, 2003.
[12]. Yee, K.S., “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas and Propagation, Vol. 14, pp. 302-307, 1966.
[13]. Taflove, A. and M. Brodwin, “Numerical solution of steady-state electromagnetic scattering problem using the time-dependent Maxwell’s equations,” IEEE Trans. Microwave Theory and Techniques, Vol. 23, pp. 623-630, 1975.
[14]. Taflove, A. and M. Brodwin, “Computation of the electromagnetic fields and induced temperatures within a model of the microwave-irradiated human eye,” IEEE Trans. Microwave Theory and Techniques, Vo l. 23, pp. 888-896, 1975.
[15]. Holland, R., “ Threde: a free-field EMP coupling and scattering code,” IEEE Trans. Nuclear Science, Vol. 24, pp. 2416-2421, 1977.
[16]. Kunz, K. S. and K. M. Lee, “A three-dimensional finite-difference solution of the external response of an aircraft to a complex transient EM environment I: The method and its implementation,” IEEE Trans. Electromagnetic Compatibility, Vol. 20, pp. 328-333, 1978.
[17]. Huang, Y. C., “Study of Absorbing Boundary Conditions for Finite Difference Time Domain Simulation of Wave Equations,” Master Thesis, Graduate Institute of Communication Engineering, National Taiwan University, June, 2003.
[18]. Mur, G., “Absorbing boundary conditions for the finite difference approximation of the time domain electromagnetic field equation,” IEEE Trans. Electromagnetic Compat., EMC-23, Vol. 4, pp. 377-382, 1981.
[19]. Lindman, E. L., “ ‘Free-space’ boundary conditions for the time dependent wave equation,” J. Comp. Phys., Vol. 18, pp 66-78, 1975.
[20]. Liao, Z. P., H. Wong, B. Yang and Y. F. Yuan, “A transmitting boundary for transient wave analysis,” Scientia Sinica (series A), Vol. 27, pp.1063-1076, 1984.
[21]. Grote, M. and J. Keller, “Nonreflecting boundary conditions for time dependent scattering,” J. Comp. Phys., Vol. 127, pp.52-81, 1996.
[22]. Engquist, B. and A. Majda, “Absorbing boundary conditions for the numerical simulation of waves,” Math Comp., Vol. 31, pp.629-651, 1971.
[23]. Engquist, B. and A. Majda, “Radiation boundary conditions for acoustic and elastic wave calculations,” J. Comp. Phys., Vol. 127, pp.52-81, 1996.
[24]. Hadley, G. R., “Transparent boundary conditions for the beam propagation method,” IEEE J. Quantum Electron, Vol. 28, pp.363-370, 1992.
[25]. Berenger, J. P., “A perfectly matched layer for the absorption of electromagnetic waves,” J. Com. Physics, Vol. 114, pp.185-200, 1994.
[26]. Berenger, J. P., “Perfectly matched layer for the FDTD solution of wave-structure interaction problems,” IEEE Trans. Ant. Propagat., Vol. 44, No.1. pp.110-117, 1996.
[27]. Zheng, K., W.-Y. Tam, D.-B. Ge, and J.-D. Xu, “Uniaxial PML absorbing boundary condition for truncating the boundary of dng metamaterials,” Progress In Electromagnetics Research Letters, Vol. 8, 125-134, 2009.
[28]. Juntunen, J. S., “Zero reflection coefficient in discretized PML,” IEEE Microwave and Wireless Compt. Letter, Vol. 11, No. 4, 2001.
[29]. Johnson, S. G., “Notes on perfectly matched layers (PMLs),” MIT 18.369 and 18.336 course note, July 2008.
[30]. Hwang, J.-N., “A compact 2-D FDFD method for modeling microstrip structures with nonuniform grids and perfectly matched layer,” Trans. on MTT., Vol. 53, 653-659, Feb. 2005.
[31]. Robertson, M. J., S. Ritchie, and P. Dayan, “Semiconductor waveguides: analysis of optical propagation in single rib structures and directional couplers,” Inst. Elec. Eng. Proc.-J. 132, 336–342 (1985).
[32]. Vassallo, C. “Improvement of finite difference methods for step-index optical waveguides,” Inst. Elec. Eng. Proc.-J. 139, 137–142 (1992).
[33]. Hua, Y., Q. Z. Liu, Y. L. Zou, and L. Sun, “A hybrid FE-BI method for EM scattering from dielectric bodies partially covered by conductors,” Journal of Electromagnetic Waves and Applications, Vol. 22, No. 2-3, 423-430, 2008.
[34]. Wang, S.-M., “Development of Large-Scale FDFD Method for Passive Optical Devices” Master Thesis (in Chinese), Institute of Elctro-Optical Engineering, National Sun Yat-sen University, June, 2005.
[35]. Chang, H.-W. and W.-C. Cheng, “Analysis of dielectric waveguide termination with tilted facets by analytic continuity method,” J. of Electromagn. Waves and Appl., Vol. 21, 1621–1630, 2007.
[36]. Cheng, W.-C. and H.-W. Chang, “Comparison of PML with layer-mode based TBC for FD-FD method in a layered medium,” International Conference on Optics and Photonics in Taiwan, P1- 170, Dec. 2008.
[37]. Magnanini, R. and F. Santosa,, “Wave propagation in a 2-d optical waveguide”, SIAM J. Appl. Math V. 61, No. 4. pp. 1237-1252, 2000.
[38]. Chang, H.-W. and M.-H. Sheng, “Field analysis of dielectric waveguide devices based on coupled transverse-mode integral equation - Mathematical and numerical formulations,” Progress In Electromagnetics Research, PIER 78, 329-347, 2008.
[39] W.C. Cheng, H.W. Chang, “FD-FD analysis of optical waveguide crossings”, OPT’07, Nov.2007.
[40] S.C. Lai, “Arbitrarily-oriented PEC/PMC-wall conforming boundary conditions for FD-FD method and its applications,” Master Thesis, Institute Elctro-Optical Engineering, National Sun Yat-sen University, Taiwan, 2008.