本論文主要研究克爾型非線性光波導結構的特性及應用。非線性光波導是指光波導結構中含有折射率會隨電場強度改變的介質。在非線性光波導結構的特性方面，利用模態定理分析覆層、導波層與基層均為克爾型非線性介質的三層光波導，並提出一組電場的一般式。此一般式可通用於其他克爾型非線性三層光波導的結構，並且求得其電場分佈及色散曲線。
在非線性光波導結構的應用方面，以馬赫詹德波導干涉儀結構為基礎，利用介質的不對稱性與結構的不對稱性來設計出全光式開關與全光式高密度波長多工器。數值結果顯示，所提出的光波導結構的確可作為全光式開關及全光式高密度分波多工器的應用。

In this thesis, the characteristics and the applications of Kerr-like nonlinear optical waveguide structures have been studied. The nonlinear optical waveguide is a medium whose refractive index changes with the electric field intensity. In the characteristics of Kerr-like nonlinear optical waveguide structures, we propose a general method for analyzing the three-layer optical waveguide structure with all nonlinear layers by using modal theory. Based on this method, the analysis of transforming arbitrary nonlinear layer into linear layer can be achieved easily by modifying nonlinear coefficient. All kinds of the transverse electric field distributions and the dispersion relation in the three-layer Kerr-like nonlinear optical waveguide structure have been obtained.
In the application of Kerr-like nonlinear optical waveguide structures, the Mach-Zehnder waveguide interferometer structure will be discussed. Based on the asymmetric medium and asymmetric construction, the new all-optical router switching device and dense wavelength division multiplexing device have been proposed. The numerical results show that the proposed structures could function as all-optical switch devices and all-optical dense wavelength division multiplexing device.

CONTENTS
Acknowledgement i
Abstract ii
Contents v
Figure Captions viii
List of Symbols xvi
Chapter 1 Introduction 1
Chapter 2 Basic Theory and Numerical
Method 4
2.1 Wave Equation 4
2.2 Modes of the Three-Layer Planar Waveguide 6
2.3 Third Order Nonlinear All-Optical Devices 10
2.3.1 Optical Kerr Effect 10
2.3.2 Self-Phase Modulation 12
2.3.3 Self-Focusing 13
2.3.4 Spatial Optical Soliton 13
2.3.5 All-Optical Devices 16
2.4 Beam Propagation Method 17
Figures 22
Chapter 3 A General Method for Analyzing the
TE-polarized Waves in the Three-layer
Nonlinear Waveguide Structure 31
3.1 Introduction 31
3.2 Analysis 34
3.3 Numerical Results 46
3.3.1 Three-Layer Nonlinear Structure 47
3.3.2 Nonlinear Cladding and Nonlinear
Substrate 48
3.3.3 Nonlinear Guiding Film 49
3.4 Summary 50
Figures 51
Chapter 4 New All-Optical Router Switching by
Using the Nonlinear Asymmetric
Mach-Zehnder Waveguide Structure 72
4.1 Introduction 72
4.2 Device Structure and Basic Analysis 73
4.2.1 Overall Structure 73
4.2.2 Nonlinear Phase Shift 74
4.3 Numerical Results 76
4.4 Summary 78
Figures 79
Chapter 5 New Nonlinear Asymmetric Mach-Zehnder
Waveguide Interferometer for DWDM
Nanometer-wave applications 89
5.1 Introduction 89
5.2 Device Structure and Basic Analysis 90
5.2.1 Overall Structure 90
5.2.2 Nonlinear Phase Shift 91
5.3 Numerical Results 93
5.3.1 Numerical Parameters 93
5.3.2 Local Nonlinear Asymmetric MZWI 94
5.3.3 Entire Nonlinear Asymmetric MZWI 96
5.4 Summary 98
Figures 99
Chapter 6 Conclusions 113
References 115

References
[1] Y. D. Wu, “Nonlinear all-optical switching device by using the spatial soliton collision,” Fiber and Integrated Optics, Vol. 23, pp. 387-404 (2004).
[2] T. T. Shi, and S. Chi, “Nonlinear photonic switching by using the spatial soliton collision,” Optics Letters, Vol. 15, pp. 1123-1125 (1990).
[3] Y. D. Wu, “New all-optical wavelength auto-router based on spatial solitons,” Optics Express, Vol. 12, pp. 4172-4177 (2004).
[4] A. Villeneuve, K. A. Hemyari, J. U. Kang, C. N. Ironside, J. S. Aitchison, and G. I. Stegeman, “Demonstration of all-optical demultiplexing at 1555nm with an AlGaAs Directional Coupler,” Electronics Letters, Vol. 29, pp. 721-722 (1993).
[5] D. H. Auston et al., Nonlinear Photonics, Springer-Verlag Berlin Heidelberg, pp. 185-201 (1990).
[6] D. R. Heatley, E. M. Wright, J. Ehrlich, and G. I. Stegeman, “Nonlinear directional coupler with a diffusive Kerr-type nonlinearity,” Optics Letters, Vol. 13, pp. 419-421 (1988).
[7] Jian-Guo Ma, “TE-mode propagation properties of the coupled planar Kerr-like nonlinear waveguides,” IEEE MTT-S Digest, pp. 1985-1988 (2002).
[8] Y. Silberberg, and B. G. Sfez, “All-optical phase- and power-controlled switching in nonlinear waveguide junctions,” Optics Letters, Vol. 13, pp. 1132-1134 (1988).
[9] W. K. Burns, A F. Milton, “Mode conversion in planar-dielectric separating waveguides,” IEEE Journal of Quantum Electronics, Vol. QE-11, pp. 32-39 (1975).
[10] M. Izutsu, Y. Nakai, T. Sueta, “Operation mechanism of the single-mode optical-waveguide Y junction,” Optics Letters, Vol. 7, pp.136-138 (1982).
[11] K. Kawano, and T. Kitoh, Introduction to Optical Waveguide Analysis, John Wiley & Sons, pp. 165-230 (2001).
[12] P. F. Liao, and P. L. Kelley, Theory of Dielectric Optical Waveguides, American Telephone and Telegraph Company, pp.19-49 (1991).
[13] R. Syms, and J. Cozens, Optical Guided Waves and Devices, New York: McGraw-Hill, pp. 217-221 (1992).
[14] R. G. Hunsperger, Integrated Optics: Theory and Technology, Springer-Verlag Berlin Heidelberg, pp. 16-20. (1991).
[15] G. I. Stegeman, E. M. Wright, N. Finalyson, R. Zanoni, and C. T. Seaton, “Third order nonlinear integrated optics,” Journal of Lightwave Technology, Vol. 6, pp. 953-970 (1988).
[16] T. T. Shi, and S. Chi, “Beam propagation method analysis of transverse-electric waves propagating in a nonlinear tapered planar waveguide,” J. Opt. Soc. Am. B, Vol. 8, pp. 2318-2323 (1991).
[17] J. P. Torres, L. Torner, “Universal diagrams for TE waves guided by thin films bounded by saturable nonlinear media,” IEEE Journal of Quantum Electronics, Vol. 29, pp. 917-925 (1993).
[18] A. D. Boardman, P. Egan, “S-polarized waves in a thin dielectric film asymmetrically bounded by optically nonlinear media,” IEEE Journal of Quantum Electronics, Vol. QE-21, pp.1701-1713 (1985).
[19] Z. Jakubczyk, H. Jerominek, S. Patels, R. Tremblay, and C. Delisle, “Power-dependent attenuation of TE waves propagating in optical nonlinear waveguiding structures,” IEEE Journal of Quantum Electronics, Vol. QE-23, pp.1921-1928 (1987).
[20] S. Chelkowski, and J. Chrostowski, “Scaling rules for slab waveguides with nonlinear substrate,” Applied Optics, Vol. 26, pp.3681-3686 (1987).
[21] M. Fontaine, “Scaling rules for nonlinear thin film optical waveguides,” Applied Optics, Vol. 29, pp. 3891-3899 (1990).
[22] M. Fontaine, “Scaling rules for transverse magnetic waves propagation in nonlinear thin-film optical waveguides,” Applied Optics, Vol. 31, pp. 1244-1251 (1992).
[23] K. Porsezian, and V. C. Kuriakose, Optical Solitons: Theory and Experimental Challenges, Springer-Verlag Berlin Heidelberg, pp.33-104 (2003).
[24] B. E. A. Saleh, and M. C. Teich, Fundamentals of Photonics, John wiley & Sons, pp. 737-796 (1991).
[25] G. I. Stegeman, D. N. Christodoulides, and M. Segev, “Optical spatial solitons: historical perspectives,” IEEE Journal on Selected Topics in Quantum Electronics, Vol. 6, pp. 1419-1427 (2000).
[26] A. W. Snyder, D. J. Mitchell, and Y. S. Kivshar, “Unification of linear and nonlinear wave optics,” Modern Physics Letters B, Vol. 23, pp. 1479-1506 (1995).
[27] Y. Baek, R. Schiek, G. Krijnen, G. I. Stegeman, I. Baumann, and W. Sohler, “All-optical intefrated Mach-Zehnder switching in lithium niobate waveguides due to cascade nonlinearities,” CLEO’96, pp.297-298 (1996).
[28] R. G. Hunsperger, Integrated Optics: Theory and Technology, Springer-Verlag Berlin Heidelberg, pp. 124-246. (1991).
[29] M. N. Islam, Ultrafast Fiber Switching Devices and Systems, AT&T, pp. 1-52 (1992).
[30] K. Hayashi, M. Koshiba, Y. Tsuji, S. Yoneta, and R. Kaji, “Combination of beam propagation method and mode expansion propagation method for bidrectional optical beam propagation analysis,” Journal of Lightwave Technology, Vol. 16, pp.2040-2045 (1998).
[31] D. I. Kovsh, S. Yang, D. J. Hangan, And E. W. Van Stryland, “Nonlinear optical beam propagation for optical limiting,” Applied Optics, Vol. 38, pp. 5168-5180 (1999).
[32] P. Danielsen, and D. Yevick, “Improved analysis of the propagating beam method in longitudinally perturbed optical waveguides,” Applied Optics, Vol. 21, pp. 4188-4189 (1982)
[33] B. Hermansson, D. Yevick, and P. Danielsen, “Propagation beam analysis of multimode waveguide tapers,” IEEE Journal of Quantum Electronics, Vol. QE-19, pp. 1246-1251 (1983).
[34] D. Yevick, and B. Hermansson, “Efficient beam propagation techniques,” IEEE Journal of Quantum Electronics, Vol. QE-26, pp. 109-112 (1990).
[35] Feit, M. D., and J. A. Fleck, Jr. “Light propagation in graded-index optical fibers,” Applied Optics, Vol. 17, pp. 3990-3998 (1978).
[36] R. Scarmozzino, and R. M. Osgood, “Comparison of finite difference and Fourier transform solutions of the parabolic wave equation with emphasis on integrated-optics applications,” J. Opt. Soc. Am. A, Vol. 8, pp.724-731 (1991).
[37] Chung, Y., and N. Dagli “Explicit finite difference beam propagation method: applications to semiconductor rib waveguide Y-junction analysis,” Electronics Letters, Vol. 26, pp. 711-713 (1990).
[38] W. Huang, C. Xu, S. T. Chu, S. K. Chaudhuri, “The finite-difference vector beam propagation method: analysis and assessment,” Journal of Lightwave Technology, Vol. 10, pp. 295-305 (1992).
[39] Y. Tsuji, M. Koshiba, and T. Shiraishi, “Finite element beam propagation method for three-dimensional optical waveguide structures,” Journal of Lightwave Technology, Vol. 15, pp. 1728-1734 (1997).
[40] B. M. Nyman, P. R. Prucnal, “The modified beam propagation method,” Journal of Lightwave Technology, Vol. 7, pp. 931-936 (1989).
[41] H. Kogelnik, and V. Ramaswamy, “Scaling rules for thin-film optical waveguides,” Applied Optics, Vo. 13, pp. 1857-1862 (1974).
[42] Jian-Guo Ma, Ingo wolff, “Propagation characteristics of TE-wave guided by thin films bounded by nonlinear media,” IEEE Transactions on Microwave Theory and Techniques, Vol. 43, pp. 790-795 (1995).
[43] T. Rozzi, “Modal analysis of nonlinear propagation in dielectric slab waveguide,” Journal of Lightwave Technology, Vol. 14, pp. 229-234 (1996).
[44] G. I. Stegeman, E. M. Wright, C. T. Seaton, J. V. Moloney, T. P. Shen, A. A. Maradudin, and R. F. Wallis, “Nonlinear slab-guided waves in non-Kerr-like media,” IEEE Journal of Quantum Electronics, Vol. QE-22, pp.977-983 (1986).
[45] G. I. Stegeman, C. T. Seaton, J. Chilwell, and S. D. Smith, “Nonlinear waves guided by thin films,” Appl. Phys. Lett., Vol. pp. 830-832 (1984).
[46] R. W. Micallef, Y. S. Kivshar, J. D. Love, D. Burak, and R. Binder, “ Generation of spatial solitons using non-linear guided modes,” Optical and Quantum Electronics, Vol. 30, pp.751-770 (1998).
[48] C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, and S. D. Smith, “Calculations of nonlinear TE waves guided by thin dielectric films bounded by nonlinear media,” IEEE Journal of Quantum Electronics, Vol. QE-21, pp.774-783 (1985).
[48] G. I. Stegeman, C. T. Seaton, J. Chilwell, and S. D. Smith, “Nonlinear waves guided by thin films,” Appl. Phys. Lett., Vol. 44, pp.830-832 (1984).
[49] J. G. Ma, and I. Wolff, “TE wave properties of slab dielectric guide bounded by nonlinear non-Kerr-like media,” IEEE Transactions on Microwave Theory and Techniques, Vol. 44, pp. 730-738 (1996).
[50] A. D. Boardman, P. Egan, “Optically nonlinear waves in thin films,” IEEE Journal of Quantum Electronics, Vol. QE-22, pp.319-324 (1986).
[51] W. Chen, A. A. Maradudin, “S-polarized guided and surface electromagnetic waves supported by a nonlinear dielectric film,” J. Opt. Soc. Am. B, Vol. 5, pp. 529-538 (1988).
[52] S. J. Al-Bader, H. A. Jamid, “Nonlinear waves in saturable self-focusing thin films bounded by linear media,” IEEE Journal of Quantum Electronics, Vol. 24, pp. 2052-2058 (1988).
[53] Jian-Guo Ma, Z. Chen, “Numerically determining the dispersion relations of nonlinear TE slab-guided waves in non-Kerr-like media,” IEEE Transactions on Microwave Theory and Techniques, Vol. 45, pp. 1113-1117 (1997).
[54] J. D. Begin, and M. Cada, “Exact analytic solutions to the nonlinear wave equation for a saturable Kerr-like medium: modes of nonlinear optical waveguides and couplers,” IEEE Transactions on Quantum Electronics, Vol. 30, pp. 3006-3016 (1994).
[55] R. A. Sammut, and C. Pask, “Gaussian and equivalent-step-index approximations for nonlinear waveguides,” J. Opt. Soc. Am. B, Vol. 8, pp. 395-402 (1991).
[56] S. W. Kang, “Optical slab waveguides of Kerr material with linear profile,” Optics Communications, Vol. 174, pp. 127-131 (2000).
[57] T. Rozzi, and F. Chiaraluce, and L. Zappelli, “Phase-Plane approach to nonlinear propagation in dielectric slab waveguide,” IEEE Transactions on Microwave Theory and Techniques, Vol. 40, pp. 102-111 (1992).
[58] J. G. Ma, and Z. Chen, “Solitary electromagnetic waves in non-Kerr-like nonlinear media,” IEEE Transactions on Magnetics, vol. 33, pp.1560-1563 (1997).
[59] S. Okafuji, “Propagation characteristics of TE0 waves in three-layer optical waveguides with self focusing and self defocusing nonlinear layers,” IEEE Proceedings-J, Vol. 140, pp. 127-136 (1993).
[60] K. S. Chiang, and R. A. Sammut, “Effective-index method for spatial solitons in planar waveguides with Kerr-type nonlinearity,” J. Opt. Soc. Am. B, Vol. 10, pp.704-708 (1993).
[61] H. W. Schürmann,” On the theory of TE-polarized waves guided by a nonlinear three-layer structure,” Z. Phys. B, Vol. 97, pp.515-522 (1995).
[62] T. T. Shi, S. Chi, “Nonlinear TE-wave propagation in a symmetric, converging, single-mode Y-junction waveguide,” J. Opt. Soc. Am. B, Vol. 9, pp. 1338-1340 (1992).
[63] R. A. Sammut, C. Pask, and Q. Y. Li, “Theoretical study of spatial solitons in planar waveguides,” J. Opt. Soc. Am. B, Vol. 10, pp. 485-491 (1993).
[64] S. Y. Shin, E. M. Wright, and G. I. Stegeman, “Nonlinear TE waves of coupled waveguides bounded by nonlinear media,” Journal of Light Technology, Vol. 6, pp. 977-983 (1988).
[65] T. Sakakibara, N. Okamoto, “Nonlinear TE waves in a dielectric slab waveguide with two optically nonlinear layers,” IEEE Journal of Quantum Electronics, Vol. QE-23, pp.2084-2088 (1987).
[66] J. S. Jeong, C. H. Kwak, and E. H. Lee, “Optical properties of TE nonlinear waves in planar waveguides with two nonlinear bounding layers,” Optics Communications, Vol. 116, pp. 351-357 (1995).
[67] U. trutschel, F. Lederer, and M. Golz, “Nonlinear guided waves in multilayer systems,” IEEE Journal of Quantum Electronics, Vol. 25, pp.194-200 (1989).
[68] S. She, S. Zhang, “Analysis of nonlinear TE waves in a periodic refractive index waveguide with nonlinear caldding,” Optics Communications, Vol. 161, pp. 141-148 (1999).
[69] M. H. Chen, Y. D. Wu, “A general method for analyzing the multi-layer planar waveguide,” Fiber and Integrated Optics, Vol.11, pp.159-176 (1992).
[70] Y. D. Wu, M. H. Chen, and H. J. Tasi, “Analyzing multilayer optical waveguides with nonlinear cladding and substrates,” J. Opt. Soc. Am. B, Vol. 19, pp. 1737-1745 (2002).
[71] Y. D. Wu, “Analyzing multilayer optical waveguides with a localized arbitrary nonlinear guiding film,” IEEE Journal of Quantum Electronics, Vol. 40, pp. 529-540 (2004).
[72] A. D. Polyanin, and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations, Chapman & Hall/CRC, pp. 301-304 (2003).
[73] M. r. Spiegel, and J. Liu, Mathematical Handbook of Formulas and Tables, McGraw-hill, pp. 195-201 (1999).
[74] Y. D. Wu, “Coupled-soliton all-optical logic device with two parallel tapered waveguides,” Fiber and Integrated Optics, Vol. 23, pp. 405-414 (2004).
[75] C. N. Ironside, M. O’Neill, “Guided wave all-optical logic devices”, IEE Colloquium on, pp. 15/1 – 15/4 (1988).
[76] A. Villeneuve, K. A. Hemyari, J. U. Kang, C. N. Ironside, J. S. Aitchison, and G. I. Stegeman, “Demonstration of all-optical demultiplexing at 1555nm with an AlGaAs Directional Coupler,” Electronics Letters, Vol. 29, pp. 721-722 (1993).
[77] S. M. Jensen, “The nonlinear coherent coupler,” IEEE Journal of Quantum Electronics, Vol. QE-18, pp. 1580-1583 (1982).
[78] L. Thylen, “Beam-propagation method analysis of a nonlinear directional coupler,” Optics Letters, Vol. 11, pp. 739-741 (1986).
[79] T. Pertsch, U. Peschel, and F. Lederer, “All-optical switching in quadratically nonlinear waveguide arrays,” Optics Letters, Vol. 28, pp. 102-104 (2003).
[80] G. V. Treyz, “Silicon Mach-Zehnder waveguide interferometers operating at 1.3μm,” Electronics Letters, Vol.27, pp. 118-120 (1991).
[81] K. Al-hemyari, C. N. Ironside, and J. S. Aitchison, “Resonant nonlinear optical properties of GaAs-GaAlAs single quantum-well waveguide and an integrated asymmetric Mach-Zehnder interferometer,” IEEE Journal of Quantum Electronics, Vol. 28, pp. 2051-2056 (1992).
[82] Y. Baek, R. Schiek, G. Krijnen, G. I. Stegeman, I. Baumann, and W. Sohler, “All-optical intefrated Mach-Zehnder switching in lithium niobate waveguides due to cascade nonlinearities,” CLEO’96, pp.297-298 (1996).
[83] N. S. Lagali, M. R. Paiasm, R. I. Macdonald, K. Worhoff, and A. Driessen, “Analysis of generalized Mach-Zehnder interometers for variable-ration power splitting and optimized switching,” Journal of Lightwave Technology, Vol. 17, pp. 2542-2550 (1999).
[84] Y. H. Pramono, and Endarko, “Nonlinear waveguides for optical logic and computation,” Journal of Nonlinear Optical Physics & Materials, Vol. 10, pp. 209-222 (2001).
[85] T. Yabu, M. Geshiro, T. kitamura, K. Nishida, and S. Sawa, “All-optical logic gates containing a two-mode nonlinear waveguide,” IEEE Journal of Quantum Electronics, Vol. 38, pp. 37-46 (2002).
[86] Q. Wang, G. Zhu, H. Chen, J. Jaques, J. Leuthold, A. B. Piccirilli, and N. K. Dutta, “Studey of all-optical XOR using Mach-Zehnder interferometer and differential scheme,” IEEE Journal of Quantum Electronics, Vol. 40, pp. 703-710 (2004).
[87] A. L. Lattes, H. A. Haus, F. J. Leonberger, and E. P. Ippen, “An ultrafast all-optical gate,” IEEE Journal of Quantum Electronics, Vol. QE-19, pp. 1718-1723 (1983).
[88] H. Kawaguchi, “Proposal for a new all-optical waveguide functional device,” Optics Letters, Vol. 10, pp. 411-413 (1985).
[89] L. Thylen, N. Finlayson, C. T. Seaton, and G. I. Stegeman, “All-optical guided-wave Mach-Zehnder switching device,” Appl. Phys. Lett., Vol. 51, pp. 1304-1306 (1987).
[90] J. S. Aitchison, A. H. Kean, G. N. Ironside, A. Villeneuve, and G. I. Stegeman, “Ultrafast all-optical switching in AlGaAs directional coupler in 1.55μm spectral region,” Electronics Letters, Vol. 27, pp. 1709-1710 (1991).
[91] K. Al-hemyari, J. S. Aitchison, C. N. Ironside, G. T. Kennedy, R. S. Grant, and W. Sibbett, “Ultrafast all-optical switching in GaAlAs integrated interferometer in 1.55μm spectral region,” Electronics Letters, Vol. 28, pp. 1090-1092 (1992).
[92] Y. D. Wu, “All-optical logic device using bent nonlinear tapered Y-junction waveguide Structure,” Fiber and Integrated Optics, Vol. 20, pp.517-524, (2001).
[93] T. T. Shi, and S. Chi, “Nonlinear TE-wave propagation in a symmetric, converging, single-mode Y-junction waveguide,” J. Opt. Soc. Am. B, Vol. 9, pp. 1338-1340 (1992).
[94] K. Ogusu, “Experimental studey of dielectric waveguide Y-junctions for millimeter-wave integrated circuits,” IEEE Transactions on Microwave Theory and Techniques, Vol. MTT-33, pp. 506-509 (1985).