臺灣博碩士論文加值系統

摘要
本論文使用有限差分頻域法搭配適當邊界條件匹配處理來分析
不同截面之介電質光波導，並在程式中引入完美匹配層做為計算視窗
之吸收邊界以分析具有損耗的光波導。我們將不同結構的光波導分成
兩類來分析討論：線性介電質界面光波導與曲線介電質界面光波導。
線性介電質界面光波導包括方型埋入式波導、瘠形波導、斜面瘠形
波導。此外，數值範例的討論也包括多模態干涉現象的傳播問題。光
纖與光子晶體光纖被視為具有曲線介電質界面的光波導，光子晶體光
纖的特殊性質，如色散和雙折射性，以及單環與多環光子晶體光纖的
損耗特性都在此論文中詳加分析與討論。除了瞭解這些光波導不同的
特性外，我們也測試了有限差分頻域法的準確性和收斂性並且觀察到
不同網格規劃對計算結果之影響。我們比較使用適當邊界條件匹配與
使用傳統階梯式近似在分析處理不同波導上的效率來突顯前者具有
較高的處理效能。

Abstract In this research, the finite-difference ferquency-domain (FDFD) method with the proper boundary-condition (BC) matching scheme is adopted to analyze optical waveguides with different cross-sectional dielectric interfaces. The perfectly matched layer (PML) is incorporated into our algorithm as the absorbing boundary of the computing window for analyzing leaky waveguides. Several optical waveguides are analyzed and discussed by assorting them into two categories: optical waveguides with linear dielectric interfaces and those with curved ielectric interfaces. The optical waveguides with linear dielectric interfaces include the square channel waveguides,
rib waveguides, and sloped-side rib waveguides. The propagation problems of the multi-mode interference (MMI) phenomenon are also discussed and taken as numerical examples. The optical fibers and the photonic crystal fibers (PCFs) are considered as waveguides with curved dielectric interfaces. The novel properties of the PCFs, such as dispersion and birefringence are studied, and the leakage loss characteristics of the sigle- and multiple-ring of PCFs are investigated. Besides understanding various properties of these optical waveguides, we examine the accuracy and convergence characteristics of our FDFD method and observed the effect of the grid mesh arrangement on the obtained results. We compare the proper BC matching scheme with the conventional stair-case approximation in the analysis of different waveguides to mark advantages of the former scheme.

Contents
1 Introduction 1
1.1 Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Chapter Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2 The Finite-Difference Frequency-Domain Method 4
2.1 Centeral Difference Scheme . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Mode Solvers for 1-D Waveguide Problems . . . . . . . . . . . . . . . 5
2.2.1 The TE Polarized Wave . . . . . . . . . . . . . . . . . . . . . 5
2.2.2 The TM Polarized Wave . . . . . . . . . . . . . . . . . . . . . 7
2.3 Mode Solver for 2-D Waveguide Problems . . . . . . . . . . . . . . . 8
2.4 FDFD Method with Perfectly Matched Layers . . . . . . . . . . . . . 11
2.5 Improved Treatment for the Dielectric Interfaces . . . . . . . . . . . . 15
2.5.1 Index Average scheme . . . . . . . . . . . . . . . . . . . . . . 15
2.5.2 Proper Boundary Condition Matching . . . . . . . . . . . . . 15
3 2-D Optical Waveguides with Linear Dielectric Interfaces 24
3.1 Channel Waveguides . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.2 Rib waveguides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3 Sloped-Side Rib Waveguides . . . . . . . . . . . . . . . . . . . . . . . 27
3.4 Multi-Mode Interference Based on Self-Imaging . . . . . . . . . . . . 28
3.4.1 Self-Imaging Principle . . . . . . . . . . . . . . . . . . . . . . 28
3.4.2 Multi-Mode Waveguides . . . . . . . . . . . . . . . . . . . . . 28
3.4.3 Guided-Mode Propagation Analysis . . . . . . . . . . . . . . . 30
3.4.4 Multi-Mode Interference . . . . . . . . . . . . . . . . . . . . . 31
3.4.5 Self-Imagimg in Multi-mode Waveguides . . . . . . . . . . . . 34
4 2-D Optical Waveguides with Curved Dielectric Interfaces 56
4.1 Optical Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.2 Photonic Crystal Fibers . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.2.1 Photonic Crystal Fibers with Single- and Multiple-Ring of Air
Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.2.2 Single-Polarization Single-Mode PCFs . . . . . . . . . . . . . 60
4.3 Dispersion Design of Photonic Crystal Fibers . . . . . . . . . . . . . . 61
4.3.1 Ultra-Flattened-Dispersion Photonic Crystal Fibers . . . . . . 62
4.3.2 High Negative Dispersion Photonic Crystal Fibers . . . . . . . 63
5 Conclusion 86

Bibliography
[1] Ahmed, G. P., and P. Daly, “Finite-element method for inhomogeneous waveguides,”
Inst. Elec. Eng. Proc.-J, vol. 116, pp. 1661–1664, 1969.
[2] Ando, T., H. Nakayama, S. Numata, J Yamauchi, and H. Nakano, “Eigenmode
analysis of optical waveguides by a Yee-mesh-based imaginary-distance propagation
method for an arbitrary dielectric interface,” J Lightwave Technol., vol.
20, pp. 1627–1634, 2002.
[3] Auguste, J. L., J. M. Blondy, J. Maury, J. Marcou, B. Dussardier, G. Monnom,
R. Jindal, K. Thyagarajan, and B. P. Pal, “Conception, realization, and
characterization of a very high negative chromatic dispersion fiber,” Opt. Fiber
Technol., vol. 8, pp. 89–105, 2002.
[4] Bachmann, M., M. K. Smit, L. B. Soldano, P. A. Besse, E. Gini, and H. Melchior,
“Polarization-insensitive low-voltage optical waveguide switch using In-
GaAsP/InP four-port Mach-Zehnder interferometer,” in Proc. Conf. Opt. Fiber
Commun. (OFC), San Jos´e, CA, pp. 23–33, 1993.
[5] Barkou, S. E., J. Broeng, and A. Bjarklev, “Silica-air photonoc crystal fiber
design that permits waveguiding by a true photonic band gap effect,” Opt.
Lett., vol. 24, pp. 46–48, 1999.
[6] Bierwirth, K., N. Schulz, and F. Arndt, “Finite-difference analysis of rectangular
dielectric waveguides by a new finite difference method,” J. Lightwavw
Technol., vol. 34, pp. 1104–1113, 1986.
[7] Brixner, B., “Refractive-index interpolation for fused silica,” J. Opt. Soc.
Amer., vol. 57, pp. 674–676, 1967.
[8] Broeng, J., D. Mogilevstev, S. E. Barkou, and A. Bjarklev, “Photonic crystal
fibers: a new class of optical waveguides,” Opt. Fiber Technol., vol. 5, pp. 305–
330, 1999.
[9] Bryngdahl, O., “Image formation using self-imaging techniques,” J. Opt. Soc.
Amer., vol. 63, pp. 416–419, 1973.
[10] Buck, J. A., Fundamentals of Optical Fibers, Wiley-Interscience (2004).
[11] 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.
[12] Chaudhuri, P. R., V. Paulose, C. Zhao, and C. Lu, “Near-elliptic core
polarization-maintaining photonic crystal fiber: Modeling birefringence characteristics
and realization,” IEEE Photon. Technol. Lett., vol. 16, pp. 1301–1303,
2004.
[13] Chew, W. C., and W. H. Weedon, “A 3D perfectly matched medium from modified
Maxwell’s equations with stretched coordinate,” IEEE Microwave Opt.
Technol. Lett., vol. 7, pp. 599–604, 1994.
[14] Chiang, P. J., C. P. Yu, and H. C. Chang, “Robust calculation of chromatic dispersion
coefficients of optical fibers from numerical determined effective indices
using Chebyshev-Lagrange interpolation polynomials,” J. Lightwave Technol.,
vol. 24, pp. 4411–4416, 2006.
[15] Cregan, R. F., B. J. Mangan, J. C. Knight, T. A. Birks, P. S. J.P. St. J. Russell,
P. J. Roberts, and D. C. Allan, “Single-mode photonic band gap guidance of
light in air,” Science, vol. 282, pp. 1479–1479, 1999.
[16] Ditkowski, A., J. S. Hesthaven, and C. H. Teng, “Modeling dielectric interfaes
in the FDTD-method: A comparative study,” in 2000 Progress in Electromagnetics
Research (PIERS 2000) Proceedings, Cambrige, Massachusetts, 2000.
[17] Dridi, K. H., J. S. Hesthaven, and A. Ditkowski, “Staircase-free finite-difference
time-domain formulation for general materials in complex geimetries,” IEEE
Trans. Antennas Propagat., vol. 49, pp.749–756, 2001.
[18] Ferrando, A., E. Silvestre, J. J. Miret, P. Andr´es, and M. V. Andr´es, “Fullvector
analysis of a realistic photonic crystal fiber,” Opt. Lett., vol. 24, pp.
276–278, 1999.
[19] Ferrando, A., E. Silvestre, J. J. Miret, and P. Andres, “Nearly zero ultraflattened
dispersion in photonic crystal fibers,” Opt. Lett., vol. 25, pp. 790-792,
2000.
[20] Ferrando, A., E. Silvestre, P. Andres, J. J. Miret, and M. V. Andres, “Designing
the properties of dispersionflattened photonic crystal fibers,” Opt. Express, vol.
9, pp. 687–697, 2001.
[21] G´er¨ome, F., J.-L. Auguste, and J.-M. Blondy, “Design of dispersioncompensating
fibers based on a dual-concentric-core photonic fibers,” Opt. Lett.,
vol. 29, pp. 2725–2727, 2004.
[22] Gruner-Nielsen, L., S. N. Knudsen, B. Edvold, T. Veng, D. Magnussen, C.
C. Larsen, and H. Damsgaard, “Dispersion compensating fibres,” Opt. Fiber
Technol., vol. 6, pp. 164–180, 2000.
[23] Gr¨uner-Nielsen, L., M. Wandel, P. Kristensen, C. Jorgensen, L. V. Jorgensen,
B. Edvold, B. Palsdottir and D. Jakobsen, “Dispersion-compensating fibers,”
IEEE J. Lightwave Technol., vol. 23, pp. 3566–3579, 2005.
[24] Gundu, K. M., M. Kolesik, J. V. Moloney, and K. S. Lee, “Ultra-flatteneddispersion
selectively liquid-filled photonic crystal fibers,” Opt. Express, vol.
14, pp. 6870–6878, 2006.
[25] Hadley, G. R., “High-accuracy finite-element equations for dielectric waveguide
analysis II: dielectric corners,” J. Lightwave Technol., vol. 20, pp.1219–1231,
2002.
[26] Hansen, T. P., J. Broeng, S. E. B. Libori, E. Knuders, 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.
[27] Knight, J. C., T. A. Birks, P. St. J. Russel, and D. M. Atkin, “All-silica singlemode
optical fiber with photonic crystal cladding,” Opt. Lett., vol. 21, pp.
1547–1549, 1996.
[28] Knight, J. C., T. A. Birks, and P. St. J. Russel, “Endlessly single-mode photonic
crystal fiber,” Opt. Lett., vol. 22, pp. 961–963, 1997.
[29] Knight, J. C., J. Broeng, T. A. Birks, and P. St. J. Russell, “Photonic band
gap guidance in optical fibers,” Science, vol. 282, pp. 1476–1478, 1998.
[30] Koshiba, M., and K. Inoue, “Simple and efficient finite-element analysis of
microwave and optical waveguides,” IEEE Trans. Microwave Theory Tech., vol.
40, pp. 371–377, 1992.
[31] Koshiba, M. and K. Saitoh, “Numerical verification od degeneracy in hexagonal
photonic crystal fibers,” IEEE Photon. Technol. Lett., vol. 13, pp. 1313–1318,
2001.
[32] Koshiba, M., S. Maruyama, and K. Hirayama, “A vector finite element mehtod
with high-order mixed-interpolation-type triangilar elements for optical waveguide
problems,” J. Lightwave Technol., vol. 12, pp. 495–502, 1994.
[33] Kubota, H., S. Kawanishi, S. Koyanagi, M. Tanaka, and S. Yamaguchi, “Absolutely
single polarization photonic crystal fiber,” IEEE Photon. Technol. Lett.,
vol. 16, pp. 182–184, 2004.
[34] Lee, J, F., D. K. Sun, and Z. J. Cendes, “Full-wave analysis of dielectric waveguides
using tangential vector finite elements,” IEEE Trans. Microwave Theory
Tech., vol. 39, pp. 1262–1271, 1991.
[35] Li, D. U., and H. C. Chang, “An efficient full-vectorial finite element modal
analysis of dielectric waveguides incorporating in homogenous elements across
dielectric discontinuities,” IEEE J. Quantum Electron., vol. 36, pp. 1251–1261,
2000.
[36] Li, Y. F., K. Iizuka, and J. W. Y. Lit, “Equivalent-layer method for optical
waveguides with a multiple quantum well structure,” Opt. Lett., vol. 17, pp.
273–275, 1992.
[37] Lui, W. W., C. L. Xu, W. P. Huang, K. Yokoyama, and S. Seki, “Full-vectorial
mode analysis with considerations of fueld singularities at corners of optical
waveguides,” J. Lightwave Technol., vol. 17, pp. 1509–1513, 1999.
[38] L¨usse, P., P. Stuwe, J. Sch¨ule, and H.-G. Unger, “Analysisof vectorial mode
fields in optical waveguides by a new finite difference method,” J. Lightwave
Technol., vol. 12, pp. 487–494, 1994.
[39] Maliston, I. H., “Interspecimen comparison of the refractive index of fused
silica,” J. Opt. Soc. Amer., vol. 15, pp. 1205–1209, 1965.
[40] Mittra, R., and ¨U. Pekel, “A new look at the perfectly matched layer (PML)
concept for the reflectionless absorption of electromagnetic waves,” IEEE Microwave
and Guided Wave Lett., vol. 5, pp. 84–86, 1995.
[41] Ohke, S., T. Umeda, and Y. Cho, “Equivalent-layer method for optical waveguide
with a multiple quantum well structure: comment,” Opt. Lett., vol.18, pp.
1870–1872, 1993.
[42] Ortigosa-Blanch, A., J. C. Knight, W. J. Wadsworth, J. Arriaga, B. J. Mangan,
T. A. Birks, and P. S. J. Russell, “Highly birefringent photonic crystal fibers,”
Opt. Lett., vol. 25, pp. 1325–1327, 2000.
[43] Pekel, ¨U., and R. Mittra, “A finite-element method frequency-domain application
of perfectly matched layer (PML) concept,” Microwave Opt. Technol.
Lett., vol. 9, pp. 117–122, 1995.
[44] Pelosi, P. M., P. Vandenbulcke, C. D. W. Wilkinson, and R. M. D. L. Rue,
“Propagation characteristics of tapezoidal cross-section ridge opticl waveguides:
An experimental and theoretical investigation,” Appl. Opt., vol. 17, pp. 1187–
1192, 1978.
[45] Rappaport, C. M., “Perfectly matched absorbing boundary conditions based
on anisotropic lossy mapping of space,” IEEE Microwave Guided Wave Lett.,
vol. 5, pp. 90–92, 1995.
[46] Rahman, B. M. A., and J. B. Davies, “Finite-element analysis of optical and
microwave waveguide problems,” IEEE Trans. Microwave Theory Tech., vol.
32, pp. 20–28, 1984.
[47] Reeves, W. H., J. C. Knight, P. St. J. Russell, and P. J. Roberts, “Demonstration
of ultra-flattened dispersion in photonic crystal fibers,” Opt. Express, vol.
10, pp. 609–613, 2002.
[48] Rozzi, T., G. Gerri, M. N. Huasin, and L. Zappellil, “Variational analysis od the
dielectric rib waveguide using the concept of ”transition function” and including
edge sinularitier,” IEEE Trans. Microwave Theory Tech., vol 39, pp. 247–256,
1991.
[49] Saini, M., and E. K. Sharma, “Equivalent refractive index of MQW waveguides,”
IEEE J. Quantum Electron., vol. 32, pp. 1383–1390, 1996.
[50] Saitoh, K., and M. Koshiba, “Photonic bandgap fibers with high birefringence,”
IEEE Photon. Technol. Lett., vol. 14, pp. 1291–1293, 2002.
[51] Saitoh, K., and M. Koshiba, “Single-polarization single-mode photonic crystal
fibers,” IEEE Photon. Technol. Lett., vol. 15, pp. 1384–1386, 2003.
[52] Saitoh, K., and M. Koshiba, “Highly nonlinear dispersion-flattened photonic
crystal fibers for supercontinuum generation in a telecommunication window,”
Opt. Express, vol. 12, pp. 2027–2032, 2004.
[53] 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 Elencron., vol. 38, pp. 927–933, 2002.
[54] Saitoh, K., N. Florous, amd M. Koshiba, “Ultra-flattened chromatic dispersion
controllability using a defected-core photonic crystal fiber with low confinement
losses,” Opt. Express, vol. 13, pp. 8365–8371, 2005.
[55] Steel, M. J., T. P. White, C. Martijn de Sterke, R. C. McPhedran, and L.
C. Bottern, “Symmetry and degenracy in micristructured optical fibers,” Opt.
Lett., vol. 26, pp. 488–490, 2001.
[56] Stern, M. S., “Semivectorial polarized finite difference method for optical waveguides
with arbitrary index profiles,” Inst. Elec. Eng. Proc.-J., vol. 135, pp.
56–63, 1988.
[57] Stern, M. S., P. C. Kendall, and P. W. Mcllroy, “Analysis of the spectral index
method for vector modes of rib waveguides,” Inst. Elec. Eng. Proc.-J., vol. 137,
pp. 21–26, 1990
[58] Soldano, L. B., F. V. Veerman, M. K. Smit, B. H. Verbeek, A. H. Dubost, and
E. C. M. Pennings, “Planar monomode optical couplers based on multi-mode
interference,” J. Lightwave Technol., vol. 10, pp. 1843–1850, 1992.
[59] Soldano, L. B., M. K. Smith, A. H. de Vreede, J. W. M. van Uffelen, B. H.
Verbeek, P. van Bennekom, W. H. C. de Krom, and W. van Etten, “New allpassive
4×4 planar optical phase diversity network,” in Proc. European Conf.
Opt. Commun. (ECOC), 1991.
[60] Sudb , A. S., “Why are accurate computations of mode fields in rectangular
dielectric waveguide difficult?,” J. Lightwave Technol., vol. 10, pp. 418–419,
1992.
[61] Suzuki, K., H. Kubota, S. Kawanishi, M. Tanaka, and M. Fujita, “High-speed
bi-directional polarization division multiplexed optical transmission in ultra
low-loss (1.3 dB/km) polarization-maintaining photonic crystal fiber,” Electron.
Lett., vol. 37, pp. 1399–1401, 2001.
[62] Thyagarajan, K., R. K. Varshney, P. Palai, A. K. Ghatak, and I. C. Goyal, “A
novel design of a dispersion compensating fiber,” IEEE Photon. Technol. Lett.,
vol. 8, pp. 1510–1512, 1996.
[63] Ulrich, R., “Image formation by phase coincidences in optical waveguides,” Opt.
Commun., vol. 13, pp. 259–264, 1975.
[64] White, T. P., B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C.
Martijn de Sterke, and L. C. Botten, “Multipole method for microstructured
optical fibers. I. Formulation,” J. Opt. Soc. Am. B, vol. 19, pp. 2322–2326,
2002.
[65] Wu, T. L., and C. H. Chao, “A novel ultraflattened dispersion photonic crystal
fiber,” IEEE. Photon. Technol. Lett., vol. 17, pp. 67–69, 2005.[66] Xu, C. L., W. P. Huang, and S. K. Chaudhuri, “Efficient and accurate vector
mode calculations by beam propagation method,” J. Lightwave Techmol., vol.
11, pp. 1209–1215, 1993.
[67] Xu, C. L., W. P. Huang, and S. K. Chaudhuri, “Full vectorial mode calculations
by finite difference method”, Inst. Elec. Eng. Proc.-J., vol. 141, pp. 281–286,
1994.
[68] Yamauchi, J., G. Takahashi, and H. Nakano, “Full-vectorial beam-propagation
method based on Mckee-Mitchell scheme with improved finite-difference formula,”
J. Lightwave Technol., vol. 16, pp. 2458–2464, 1998.
[69] Yang, S., Y. Zhang, X. Peng, Y. Lu, S. Xie, J. Li, W. Chen, Z. Jiang, J. Peng,
and H. Li, “Theoretical study and experimental fabrication of high negative
dispersion photonic crystal fiber with large area mode field,” Opt. Express, vol.
14, pp. 3015–3023, 2006.
[70] Yee, K. S., “Numerical solution of initial boundary value problems involving
Maxwell’s equations on isotropic media,” IEEE Trans. Antenna Propagat., vol.
14, pp. 302–307, 1966.
[71] Yu, C.-P., and H.-C. Chang, “Applications of the finite difference mode solution
method to photonic crystal structures,” Opt. Quantum Electron., vol. 36, pp.
145–163, 2004.
[72] Yu, C.-P., and H.-C. Chang, “Yee-mesh-based finite difference eigenmode solver
with PML absorbing boundary conditions for optical waveguides and photonic
crystal fibers,” J. Opt. Soc. Amer., vol. 12, pp. 6165–6176, 2004.
[73] Yu, C.-P., and H.-C. Chang, “Compact finite-difference frequency-domain
method for the analysis of two-dimensional photonic crystal fibers,” Opt. Express,
vol. 12, pp 1379–1408, 2004.
[74] Zucker, J. E., K. L. Jones, T. H. Chiu, and K. Brown-Goebeler, “Strained
quantum wells for polarization-independent electropic waveguide switches,” J.
Lightwave Technol., vol. 10, pp. 1926–1930, 1992.
[75] Zhu, Z., and T. G. Brown, “Full-vectorial finite-difference analysis of microstructured
optical fibers,” Opt. Express, vol.10, pp. 853–864, 2002.