本篇論文的主要目的在提供一個嚴謹的橫向模態積分方程式來分析電介質波導的TE與TM向量場，這個方法主要可以分為以下兩個部份。首先是建立一個獨立的TE與TM積分方程式對於凹凸的電磁波導與電介質平板波導接面的分析，並且透過介面未知場量的波導模態選擇，我們可以減少核心矩陣的大小來降低計算量而得到相同的精確度。
第二個部份則是對於二維類矩陣的電介質波導，建立一個全向量TE與TM耦合的積分方程式來分析其橫向模態的特性，我們藉由垂直或水平切割的方式，將二維的結構分成多個片狀的一維多層介質，此一維多層介質的模態較為容易建立，而整個波導的電磁場則是以片狀之間的一維y方向介面未知場來架構，藉由這些未知的介面電場可以建立一個向量耦合橫向場的積分方程式，其中適當地選擇未知場量的基底，如同第一部份不但減少核心矩陣與計算量，並且可以使結果快速收歛以及改善Gibb’s的介面現象(Gibb’s phenomenon)。

The subject of this dissertation is to develop a rigorous transverse-mode integral equation formulation for analyzing TE/TM electromagnetic mode field solutions for dielectric waveguides. The main topics are composed of two related parts. The first part deals with scalar problems. In which we propose a transverse-mode integral-equation formulation for problems such as mode solutions of the ridged microwave waveguides. This same technique also applies to EM waves scattering off the facet of dielectric slab waveguides terminating in free space. For both problems we constructed a specifically chosen basis for the unknown tangential field functions, and we were able to reduce the kernel matrix size by more than one half without noticeable degradation of the field solutions.
In the second part of the thesis, we apply a full-vector integral-equation formulation to analyze modal characteristics of the complex, two-dimensional, rectangular-like dielectric waveguide that is divisible into vertical slices of one-dimensional layered structures. The entire electromagnetic vector mode field solution is completely determined by the y-component electric and magnetic field functions on the interfaces between slices. These interfacial functions are governed by a system of vector-coupled transverse-mode integral equations (VCTMIE) whose kernels are made of orthonormal sets of both TE-to-y and TM-to-y modes from each slice. Finally, we show the numerical results to present the stable and quick convergence of this method as well as to improve the Gibb’s phenomenon in the recreation of the transverse fields.

1. INTRODUCTION 1
1.1 Overview 1
1.2 Research Motivations and Objectives 4
1.3 Summary 5

2. LAYERED MODES 9
2.1 Mode Solving Techniques 9
2.2 CE-CH Method 21

3. TRANSVERSE-MODE INTEGRAL EQUATIONS 27
3.1 Transverse Electric and Magnetic Modes 27
3.2 Transverse-Mode Integral Equations 29
3.3 Modal Analysis of Convex and Concave Ridged Waveguides 33
3.4 Dielectric Slab Waveguides with an Abrupt Termination 36

4. VECTORIAL COUPLED TRANSVERSE-MODE INTEGRAL EQUATIONS 53
4.1 Derivation of the TE-y and TM-y Sturm-Liouville Form 53
4.2 Vectorial Coupled Transverse-Mode Integral Equations 57
4.3 Nonlinear Matrix Formulation 75

5. NUMERICAL RESULTS AND DISCUSSIONS 89
5.1 Convex and Concave Ridged Waveguides 89
5.2 Dielectric Slab Waveguides with an Abrupt Termination 91
5.3 Modal Analysis of Dielectric Rectangular-Like Waveguides 97
5.4 Summary 111

6. CONCLUSIONS 135



REFERENCES 137
PUBLICATION LIST 143

1. E. A. Marcatili, “Dielectric rectangular waveguide and dielectric coupler for integrated optics,” Bell Syst. Tech. J., 2071-2103 (1969).
2. Y. H. Cheng and W. G. Lin, “Investigation of rectangular dielectric waveguides: an iteratively equivalent index method,” IEE Proc. Opto-Electron. 137, 323-329 (1990).
3. J. S. Lee and S. Y. Shin, “On the validity of the effective-index method for rectangular dielectric waveguides,” J. Lightwave Technol. 11, 1320-1324 (1993).
4. Y. Cai, T. Mizumoto, and Y. Naito, “Improved perturbation feedback method for the analysis of rectangular dielectric waveguides,” J. Lightwave Technol. 9, 1231-1237 (1991).
5. J. E. Goell, “A circular-harmonic computer analysis of rectangular dielectric waveguides,” Bell Syst. Tech. J., 2133-2160 (1969).
6. G. H. Brooke and M. M. Z. Kharadly, “Scattering by Abrupt Discontinuities on Planar Dielectric Waveguides,” IEEE Trans. Microwave Theory Tech., Vol. MTT-30, No. 5, pp. 760--770, 1982.
7. C. Vassallo, “Reflectivity of multidielectric coatings deposited on the end facet of a weakly guiding dielectric slab waveguide,” J. Opt. Soc. Am. A, 5, No.11, pp. 1918--1928, Nov. 1988.
8. P. C. Kendall, D. A. Roberts, P. N. Robson, M. J. Adams, M.J. Roberson, “Semiconductor laser facet reflectivities using free-space radiation modes,” IEE Proceedings-J, 140, No.1, pp.49--55, Feb. 1993.
9. I. G. Tigelis and T. G. Theodoropoulos, “Radiation properties of an abruptly terminated five-layer symmetric slab waveguide,” J. Opt. Soc. Amer. A, vol. 14, no. 6, pp. 1260-1267, Jun. 1997.
10. H. Derudder, D. De Zutter and F. Olyslager, “Analysis of waveguide discontinuities using perfectly matched layers,” Electronic Letters 29th, 34, No.22, pp. 2138--2140, Oct. 1998.
11. M. Öz and R. R. Krchnavek, “Power Loss Analysis at a Step Discontinuity of a Multimode Optical Waveguide,” J. of Lightwave Techno., 16, No. 12, pp. 2451--2457, Dec. 1998.
12. I. G. Tigelis, “Abrupt trasition coupling between two-layered symmetric slab waveguides,” J. Opt. Soc. Amer. A, 15, no. 1, pp. 84-91, Jan. 1998.
13. R. Mittra, Y. L. Hou, and V. Jannejad, “Analysis of open dielectric waveguides using mode-matching technique and variational methods,” IEEE Trans. Microwave Theory Tech. 28, 36-43 (1980).
14. D. Marcuse, “Radiation losses of tapered dielectric slab waveguides,” Bell Syst. Tech. J., 49, pp. 273-289, Feb. 1970.
15. A. W. Snyder, “Coupled mode theory for optical fibers,” J. Opt. Soc. Am. 62,pp. 1267-1277, 1972.
16. A. Yariv, “Coupled mode theory for guided wave optics,” IEEE J. Quantum Electron. QE-9, pp. 919-933, 1973.
17. W. K. Burns, “Mode coupling in optical waveguide horns,” IEEE J. Quantum Electron. QE-13, pp. 828-835, 1977.
18. D. Marcuse, “Radiation losses of step-tapered waveguides,” Appl. Opt. 19, pp. 3676-3681, 1981.
19. P. G. Suchoski, Jr., and V. Ramaswamy, “Exact numerical technique for the analysis of step discontinuities and tapers in optical dielectric waveguides,” J. Opt. Soc. Amer. A, 3, no. 2, pp. 194-203, Feb. 1986.
20. P. Clarricoats and A. Sharpe, “Modal matching applied to a discontinuity in a planar surface waveguide,” Electron. Lett., 8, pp. 28-29, Jan. 1972.
21. G. A. Hockham and A. B. Sharpe, “Dielectric waveguide discontinuities,” Electron. Lett., vol. 8, pp. 230-231, May 1972.
22. T. E. Rozzi, “Rigorous Analysis of the Step Discontinuity in a Planar Dielectric Waveguide,” IEEE Trans. Microwave Theory Tech., 26, No. 10, pp. 738--745, 1978.
23. S. T. Peng and A.A. Oliner, “Guidance and Leakage Properties of a Class of Open Dielectric Waveguides: Part I- Mathematical Formulations,” IEEE Trans. Microwave Theory Tech. 29, pp. 843-855 (1981).
24. E. W. Kolk, N. H. G. Baken, and H. Blok, “Domain integral equation analysis of integrated optical channel and ridgedwaveguides in stratified media,” J. Lightwave Technol. 38, pp. 78-85 (1990).
25. H. Y. Yang, J. A. Castaneda, and N. G. Alexopoulos, “An integral equation analysis of an infinite array of rectangular dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 38, pp. 873-880 (1990).
26. K. Sabetfakhri and L. P. B. Katehi, “An integral transform technique for analysis of planar dielectric structures,” IEEE Trans. Microwave Theory Tech. 42, pp. 1052-1062 (1994).
27. G. Athanasoulias and N. K. Uzunoglu, “An accurate and efficient entire-domain basis Galerkin’s method for the integral equation analysis of integrated rectangular dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 43, pp. 2794-2804 (1995).
28. V. A. Kalinin and B. K. J. C. Nauwelaers, “Free space dyadic Green’s function applied to the full-wave numerical analysis of planar transmission lines and dielectric waveguides,” IEE Proc. Microwave Antennas Propag. 143, pp. 328-334(1996).
29. T. Rasmussen, J. H. Povlsen, A. Bjarklev, O. Lumholt, B. Pedersen, and K. Rottwitt, “Detailed comparison of two approximate methods for the solution of the scalar wave equation for a rectangular optical waveguide,” J. Lightwave Technol. 11, pp. 429-433 (1993).
30. U. Rogge and R. Pregla, “Method of lines for the analysis of dielectric waveguides,” J. Lightwave Technol. 11, pp. 2015-2020 (1993).
31. R. Scarmozzino, A. Gopinath, R. Pregla, and S. Helfert, “Numerical techniques for modeling guided wave photonic devices,” IEEE J. of Selected Topics in Quantum Electron. 6, pp. 150-162 (2000).
32. A. S. Sudbo, “Film mode matching: a versatile numerical method for vector mode field calculations in dielectric waveguides,” Pure Appl. Opt. 2, pp. 211-233 (1993).
33. A. S. Sudbo, “Improved formulation of film mode matching method for mode field calculations in dielectric waveguides,” Pure Appl. Opt. 3, pp. 381-388 (1994).
34. B. M. Azizur Rahman and J. B. Davies, “Analysis of Optical Waveguide Discontinuities,” J. Lightwave Technol., 6, no. 1, pp. 52-57, Jan. 1988.
35. S. J. Chung and C. H. Chen, “A Partial Variational Approach for Arbitrary Discontinuities in Planar Dielectric Waveguides,” IEEE Trans. Microwave Theory Tech., 37, no. 1, pp. 208—214, Jan. 1989.
36. Q. H. Liu and W. C. Chew, “Analysis of Discontinuities in Planar Dielectric Waveguides: An Eigenmode Propagation Method,” IEEE Trans. Microwave Theory Tech., 39, No. 3, pp. 422--430, Mar. 1991.
37. D. U. Li and H. C. Chang, “Full-vectorial finite element modal analysis of bounded and unbounded waveguides,” in Proceedings of Asia Pacific Microwave Conference 2001, (National Taiwan University, 2001), pp. 376-379.
38. Y. C. Chiang, Y. P. Chiou, and H. C. Chang, “Improved full-vectorial finite-difference mode solver for optical waveguides with step-index profiles,” J. Lightwave Technol. 20, pp. 1609-1618 (2002).
39. G. Ronald Hadley, “High-accuracy finite-difference equations for dielectric waveguide analysis II- dielectric corners,” J. Lightwave Technol. 20, pp. 1219-1231 (2002).
40. N. Thomas, “Finite-Difference Methods for the Modal Analysis of Dielectric Waveguides with Rectangular Corners,” Ph. D Thesis, University of Nottingham, U.K. (2004).
41. M. Koshiba, S. Maruyama, and K.Hirayama, “A vector finite element method with the high-order mixed-interpolation-type triangular elements for optical waveguiding problems,” J. Lightwave Technol., vol.12, no.3, pp. 495-502, Mar. 1994.
42. J. Jin, The Finite Element Method in Electromagnetics, John Wiley & Sons, Inc., 1993.
43. M. S. Alam, M. S. Ali, M. S. Islam, and M. N. Islam, “On an efficient finite element method for the analysis of microwave and optical waveguides,” Proceedings of IECEC-2001, pp. 234-237, Jan. 2001.
44. L. R. Gomaa, “Beam Propagation Method Applied to a Step Discontinuity in Dielectric Planar Waveguide,” IEE Proceedings-J, 135, No.3, pp. 205--206, Jun. 1988.
45. Y. P. Chiou and H. C. Chang, “Analysis of Optical Waveguide Discontinuities Using the Padé Approximants,” IEEE Photon. Tech. Lett., 9, No. 7, pp. 964-966, Jul. 1997.
46. J. Van Bladel, Singular Electromagnetic Fields and Sources, Chap.4, IEEE/OUP Series on Electromagnetic Wave Theory. Oxford, U.K.: Oxford University Press, 1995.
47. R. E. Collin, Field Theory of Guided Waves, 2nd ed., Section 1.5, IEEE/OUP Series on Electromagnetic Wave Theory. Oxford, U.K.: Oxford University Press, 1991.
48. A. S. Sudbo, “Why are accurate computations of mode fields in rectangular dielectric waveguides difficult?,” J. Lightwave Technol. 10, pp. 418-419 (1992).
49. A. Ishimaru, Electromagnetic Wave Propagation, Radiation, and Scattering, Section 3.7, Prentice-Hall, Inc., Englewood Cliffs, NJ, U.S.A., 1991.
50. T. Itoh, Numerical Techniques for Microwave and Millimeter-Wave Passive Structure, Chap. 6, John Wiley & Sons, Inc., Singapore, 1989.
51. M. H. Sheng, “Rigorous Leaky Mode Analysis of Antiresonant Reflecting Optical Waveguides,” Chap. 3, Ph. D Thesis, NSYSU, 2005.
52. R. F. Harrington, Time-Harmonic Electromagnetic Fields, Section 4-4, McGrew-Hill, New York, 1961.
53. T. L. Wu and H. W. Chang, “Guiding mode expansion of a TE and TM transverse-mode integral equation for dielectric slab waveguides with an abrupt termination,” J. Opt. Soc. Am. A. 18, pp. 2823-2832 (2001).
54. T. L. Wu, Wen-Cheng He, and H. W. Chang, “Numerical Study of Convex and Concave Rectangular Ridged Waveguides with Large Aspect Ratios,” Proc. National Science Counc. ROC (A), 23, No.6, pp. 799-809. (1999).
55. Hung-Wen Chang, Tso-Lun Wu, and Meng-Huei Sheng, “Vectorial modal analysis of dielectric waveguides based on coupled transverse-mode integral equation: I – mathematical formulations,” J. Opt. Soc. Am. A. 23, pp. 1468-1477. (2006)
56. Hung-Wen Chang and Tso-Lun Wu, “Vectorial modal analysis of dielectric waveguides based on coupled transverse-mode integral equation: II – Numerical analysis,” J. Opt. Soc. Am. A. 23, pp. 1478-1487. (2006)