臺灣博碩士論文加值系統

摘要
有限差分法係一簡單且有效率之數值分析工具，本論文採用以有限差
分公式為基礎之全向量模態分析法來研究光波導的傳播特性。由於數值計算採用均勻格點分割，因此很容易可對任意幾何形狀的光波導結構截面進行切割。本論文採用折射率等效法處理曲形介質接面得以穩定數值計算並加速數值收斂。此外，為分析洩漏模態問題，例如計算波導?amp;#63841;侷限損耗，我們在有限差分公式推導中加入完全匹配層作為吸收邊界，文中並探討折射率等效法對洩漏模態問題分析的影響。求解所建立的特徵值問題係採用移位反冪次法(SIPM)。本論文探討一維與二維波導問題，包括平板波導、抗諧振反射光波導、步階式光纖、矩形埋入式波導、各向異性埋入式鈮酸鋰積體光波導以及微結構光纖。並與其他方法的數值結果詳加比較。

Abstract
Due to its simplicity and efficiency, a full-vectorial mode solver based on a finite difference scheme is applied to investigate the propagation characteristics of optical waveguides. Since uniform meshes are used in the numerical implementation, it is very easy to divide the computational window of any arbitrary cross-sectional geometries of the waveguides. An index averaging technique is employed to deal with curved dielectric interfaces for stabilizing the numerical calculation and accelerating convergence. In addition, for solving leaky-mode problems, such as the investigation of waveguide confinement loss, the perfectly matched layer (PML) absorbing boundary condition is incorporated into our finite difference formulations. The influence of the index averaging technique on the leaky-mode analysis is also discussed. We employ the shift inverse power method (SIPM) for solving the formulated eigenvalue problems. In this work, both one-dimensional and two-dimensional problems are considered, including the slab waveguide, the antiresonant reflecting optical waveguide (ARROW), the step-index optical fiber, the rectangular channel waveguide, the anisotropic embedded-channel LiNbO3 integrated optical waveguide, and microstructured optical fibers (MOFs). Comparsion of our calculation with other methods is discussed.

1 Introduction . . . . . . . . . . . . . . . . . . . . . . .1
1.1 Motivations . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Chapter Outline . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 The Finite Di®erence Waveguide Mode Solver and Related
Techniques 4
2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Formulations . . . . . . . . . . . . . . . . . . . . . . 4
2.3 Boundary Conditions . . . . . . . . . . . . . . . .. . . . . . . .12
2.4 The Perfectly Matched Layer . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.5 The Shift Inverse Power Method . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.6 Index Averaging Method . . . . . . . . . . . . . . . . . . . . . . . . . 18
3 Numerical Results for One-Dimensional Problems. . . . 21
3.1 Slab Waveguides . . . . . . . . . . . . . . . . . . . . . . . 21
3.2 ARROW Waveguides . . . . . . . . . . . . . . . . . . . . . . . 22
4 Numerical Results for Two-Dimensional Problems 39
4.1 Step-Index Fibers . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.2 Rectangular Channel Waveguides . . . . . . . . . . . .. . . . . . . . . . . 41
2
4.3 Anisotropic Embedded-Channel LiNbO3 Integrated Optical
Waveguides . . . . . . . . . . . . . . . . . . . . . . . 42
5 Microstructured Optical Fibers . . . . . . . . . . . . 61
5.1 Air-Hole-Assisted Optical Fibers . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.2 Photonic Crystal Fibers . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.2.1 Triangular Holey Fibers . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.2.2 Honeycomb Fibers . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.2.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . 66
6 Conclusion . . . . . . . . . . . . . . . . . . . . . . 87

[1] Ahmed, G. P., and P. Daly, \Finite-element method for inhomogineous
waveguides," Inst. Elec. Eng. Proc.-J, vol. 116, pp. 1661{1664, 1969.
[2] Ansbro, A. P., and I. Montrosset, \Vectorial finite difference scheme
for isotropic dielectric waveguidesL transverse electric field prepresentation," Inst. Elec. Eng. Proc-J., vol. 140, pp. 253-259, 1993.
[3] B¶erenger, J.-P., A perfectly matched layer for the absorption of electromagnetic waves," J. Comput. Phys., vol. 114, pp. 185{200, 1994.
[4] Bierwirth, K., N. Schulz, and F. Arndt, Finite-difference analysis of
rectangular dielectric waveguides by a new finite difference method," J.
Lightwave Technol., vol. 34, pp. 1104{1113, 1986.
[5] Birks, T. A., P. J. Roberts, P. St. J. Russell, D. M. Atkin, and T. J.
Shepherd, Full 2-D photonic bandgaps in silica/air structures," Electron. Lett., vol. 31, pp. 1941-1943, 1995.
[6] Birks, T. A., J. C. Knight, and P. St. J. Russell, Endlessly single-mode
photonic crystal fiber," Opt. Lett., vol. 22, pp. 961{963, 1997.
[7] Bodewig, E., Matrix Calculus. Amsterdam: North Holland Pub. Co.,
1956.
[8] Brechet, F., J. Marcou, D. Pagnoux, and P. Roy, Complete analysis
of the characteristics of propagation into photonic crystal fibers, by the finite element method," Opt. Fiber Technol., vol. 6, pp. 181-191, 2000.
[9] Brixner, B., Refractive-index interpolation for fused silica," J. Opt.Soc. Amer., vol. 57, pp. 674-676, 1967.
[10] Broderick, N. G. R., T. M. Monro, P. J. Bennett, and D. J. Richardson,
Nonlinearity in holey optical fibers: measurement and future opportunities," Opt. Lett., vol. 24, pp. 1395-1397, 1999.
[11] Burden, R. L., and J. D. Faires, Numerical Analysis. Boston, MA:
PWSKENT, 1989.
[12] Cendes, Z. J., and P. Silvester, Numerical solution of dielectric loaded
waveguides: I-Finite-Element analysis," IEEE Trans. Microwave Theory
Tech., vol. 18, pp. 1124-1131, 1970.
[13] Chew, W. C., and W. H. Weedon, A 3D perfectly matched medium
from modified Maxwell''s equations with stretched coordinates," Mi-
crowave Opt. Technol. Lett., vol. 7, pp. 599-604, 1994.
[14] Chiang, Y. C., Y. P. Chiou, and H. C. Chang, Improved full-vectorial
finite-difference mode solver for optical waveguides with step-index profiles," J. Lightwave Technol., vol. 20, pp. 1609-1618, 2002.
[15] Chiou, Y. P., 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. Lightwave Technol., vol. 18, pp.
243-251, 2000.
[16] Chung, Y., and N. Dagli, \Analysis of z-invariant and z-variant semi-
conductor rib waveguides by explict finite difference beam propagation
method with nonuniform mesh configuration," IEEE J. Quantum Electron., vol. 27, pp. 2296-2305, 1991.
[17] Deng, J.-J., and Y.-T. Huang, A novel hybrid coupler based on antiresonant reflecting optical waveguides," J. Lightwave Technol., vol. 16, pp.
1062-1068, 1998.
[18] Dong, H., A. Chronopoulos, and J. Zou, Vectorial integrated finite-
difference analysis of dielectric waveguides," J. Lightwave Technol., vol.
11, pp. 1559-1564, 1993.
[19] Dridi, K. H., J. S. Hesthaven, and A. Ditkowski, Staircase-free finite-
difference time-domain formulation for general materials in complex geometries," IEEE Trans. Antennas Propagat., vol. 49, pp. 749-756, 2001.
[20] Duguay, M. A., Y. Kokubun, and T. L. Koch, Antiresonant reflecting
optical waveguides in SiO2-Si multilayer structures," Appl. Phys. Lett.,
vol. 49, pp. 13-15, 1986.
[21] Ferrando, A., E. Silvestre, J. J. Miret, P. Andr¶es, and M. V. Andr¶es,
Full-vector analysis of a realistic photonic crystal fiber," Opt. Lett., vol.
24, pp. 276-278, 1999.
[22] Ferrando, A., E. Silvestre, J. J. Miret, and P. Andr¶es, Nearly zero
ultraflattened dispersion in photonic crystal fibers," Opt. Lett., vol. 25,
pp. 790-792, 2000.
[23] Gander, M. J., R. McBride, J. D. C. Jones, D. Mogilevtsev, T. A. Birks,
J. C. Knight, and P. St. J. Russell, Experimental measurement of group
velocity dispersion in photonic crystal fibres," Electron. Lett., vol. 35,
pp. 63-65, 1999.
[24] Hadley, G. R., Transparent boundary condition for the beam propagation method," IEEE J. Quantum Electron., vol. 28, pp. 963-970, 1992.
[25] Hadley, G. R., and R. E. Smith, Full-vector waveguide modeling using
an iterative finite-difference method with transparent boundary conditions," J. Lightwave Technol., vol. 13, pp. 465-469, 1995.
[26] Hadley, G. R., High-accuracy finite-difference equations for dielectric
waveguide analysis I: Uniform regions and dielectric interfaces," J. Light-
wave Technol., vol. 20, pp. 1210-1218, 2002a.
[27] Hadley, G. R., High-accuracy finite-difference equations for dielectric
waveguide analysis II: Dielectric corners," J. Lightwave Technol., vol.
20, pp. 1219-1231, 2002b.
[28] Hansen, T. P., J. Broeng, S. E. B. Libori, E. Knudsen, A. Bjarklev, J. R.
Jensen, and H. Simonsen, Highly birefringent index-guiding photonic
crystal fibers," IEEE Photon. Technol. Lett., vol. 13, pp. 588-590 ,2001.
[29] Hasegawa, T., E. Sasaoka, M. Onishi, M. Nishimura, Y. Tsuji, and M.
Koshiba, Hole-assisted lightguide fiber for large anomalous dispersion
and low optical loss," Opt. Express, vol. 9, pp. 681-686, 2001.
[30] Huang, W. P., M. Shubair, A. Nathan, and Y. L. Chow, The modal
Characteristics of ARROW structures," J. Lightwave Technol., vol. 10,
pp. 1015-1022, 1992.
[31] Itoh, T. ed., Numerical Techniques for Microwave and Millimeter-Wave
Passive Structure. New York: Wiley, 1989.
[32] Jennings, A., Matrix Computation for Engineers and Scientists. New
York: Wiley, 1977.
[33] Jiang, W., J. Chrostowski, and M. Fontaine, Analysis of ARROW
waveguides," Opt. Commun., vol. 72, pp. 180-185, 1989.
[34] Knight, J. C., T. A. Birks, P. St. J. Russell, and D. M. Atkin, All-silica
single-mode optical fiber with photonic crystal cladding," Opt. Lett.,
vol. 21, pp. 1547-549, 1996.
[35] Knight, J. C., J. Arriaga, T. A. Birks, A. Ortigosa-Blanch, W. J.
Wadsworth, and P. St. J. Russell, Anomalous dispersion in photonic
crystal fiber," IEEE Photon. Technol. Lett. vol. 12, pp. 807-809, 2000.
[36] Koshiba, M, and K. Inoue, Vectorial finite-element formulation without
spurious modes for dielectric waveguides," Electron. Lett., vol. 20, pp.
409-410, 1984.
[37] Kubica, J., D. Uttamchandani, and B. Culshaw, Modal propagation
within ARROW waveguides," Opt. Commun., vol. 78, pp. 133-136,
1990.
[38] Kubica, J., J. Gazecki, and G. K. Reeves, Multimode operation of
ARROW waveguides," Opt. Commun., vol. 102, pp. 217-220, 1993.
[39] Kukubun, Y., T. Baba, and T. Sakaki, Low-loss antiresonant reflecting
optical waveguide on Si substrate in visible-wavelength region," Electron. Lett., vol. 22, pp. 892-893, 1986.
[40] Lee, J. F., D. K. Sun, and Z. J. Cendes, Full-wave analysis of dielec-
tricwaveguides using tangential vector finite elememts," IEEE Trans.
Microwave Theory Tech., vol. 39, pp. 1262-1271, 1991.
[41] Lizier, J. T., and G. E. Town, Splice losses in holey optical fibers,"
IEEE Photon. Technol. Lett., vol. 13, pp. 794-96, 2001.
[42] LÄusse, P., P. Stuwe, J. SchÄule, and H.-G. Unger, Analysis of vectorial
mode fields in optical waveguides by a new finite difference method," J.
Lightwave Technol., vol. 12, pp. 487{494, 1994.
[43] Mitchell, A. R., and D. F. Griffiths, The Finite Difference Method in
Partical Differential Equations. New York: Wiley, 1987.
[44] Mogilevtsev, D., T. A. Birks, and P. St. J. Russell, Localized function method for modeling defect modes in 2-D photonic crystals," J.
Lightwave Technol., vol. 17, pp. 2078-2081, 1999.
[45] Monro, T. M., P. J. Bennett, N. G. R. Broderick, and D. J. Richardson,
Holey fibers with random cladding distributions," Opt. Lett., vol. 25,
pp. 206-208, 2000.
[46] Mortensen, N. A. Effective area of photonic crystal fibers," Opt. Ex-
press., vol. 10, pp. 341-348, 2002.
[47] Pekel, ÄU., and R. Mittra, \A ‾nite-element method frequency-domain
application of the perfectly matched layer (PML) concept," Microwave
Opt. Technol. Lett., vol. 9, pp. 117-122, 1995.
[48] Rahman, B. M. A., and J. B. Davies, Finite-element analysis of optical
and microwave waveguide problems," IEEE. Trans. Microwvae Theory
Tech., vol. 32, pp. 20-28, 1984.
[49] Sacks, Z. S., D. M. Kinsland, R. Lee, and J. F. Lee,, A perfectly
matched anisotropic absorber for use as an absorbing boundary condition," IEEE Trans. Antennas Propagat., vol. 43, pp. 1460-1463, 1995.
[50] Saitoh, K., and M. Koshiba, Full-Vectorial Imaginary-Distance Beam
Propagation Method Based on a Finite Element Scheme: Application
to Photonic Crystal Fibers," IEEE J. Quantum Electron,, vol. 38, pp.
927-933, 2002.
[51] Stern, M. S., P. C. Kendall, and P. W. A. Mcllroy, \Analysis of the
spectral index method for vector modes of rib waveguides," Inst. Elec.
Eng. Proc.-J., vol. 137, pp. 21-26, 1990.
[52] Svedin, Jan A. M., \A modified finite-element method for dielectric
waveguides using an asymptotically correct approximation on infinite
elements," IEEE Tans. Microwave Theory Tech., vol. 39, pp. 258-266,
1991.
[53] Taflove, A., and S. C. Hagness, Computational Electromagnetics: The
Finite Difference Time Domain Method, Second Edition,. Boston, MA:
Artech House, 2000.
[54] Vandenbulcke, P., and P. E. Lagasse, Eigenmode analysis of anisotropic
optical fibers or integrated optical waveguides," Electron. Lett., vol.12,
pp. 120-121, 1976.
[55] Xu, C. L., W. P. Huang, M. S. Stern, and S. K. Chaudhuri, Full-
vectorial mode calculations by finite difference method," Inst. Elec. Eng.
Proc.-J., vol. 141, pp. 281-286, 1994.
[56] Yu, C.-P., and H.-C. Chang, Finite Difference Modal Analysis of Photonic Crystal Fibers," Proc. 2001 Progress in Electromagnetics Research
Symposium (PIERS 2001), p. 432, Osaka, Japan, July 18{22, 2001.
[57] Zhu, Z., and T. G. Brown, Multipole analysis of hole-assisted optical fibers," Opt. Commun., vol. 206, pp. 333-339, 2002a.
[58] Zhu, Z., and T. G. Brown, Full-vectorial finite-difference analysis of mi-
crostructured optical fibers," Opt. Express, vol. 10, pp. 853-864, 2002b.