Chapter 6 Photonic Crystal Waveguides with Type-B Antiresonant Reflecting Optical
6.3 Characteristics of a PC Waveguide Based on an ARROW-B Platform
6.3.3 Butt Coupling Efficiency
To maintain the features of single-mode and PBGs, the core sizes of PC waveguides must be less than half of a micron. On the contrary, the core diameters of single-mode fibers operating at 1.55 μm are normally 6-10 μm. Due to mismatch of core sizes, the butt-coupling efficiencies between a conventional PC waveguide and a single-mode optical fiber are on the order of 20-30 dB. The core layer of a PC waveguide based on an ARROW-B platform is as thick as 8 μm; therefore, the coupling efficiency can be improved. An ARROW-B-based PC waveguide coupled directly with a single-mode optical fiber is shown in Fig. 6.7(a) and its top view is shown in Fig. 6.7(b). The refractive indices of fiber core and cladding are ncore = 1.451 and ncladding = 1.445, respectively, and the core diameter of fiber is 6 μm. Light source of fiber mode is launched from the optical fiber and the coupling efficiency is monitored. Coupling efficiency is defined as the ratio of power measured at the input of PC waveguide to that measured at the end of fiber. Although the interface changes abruptly from 6 μm to 0.6 μm as light propagates from a single-mode fiber to an ARROW-B-based PC waveguide, the coupling efficiencies calculated by 3D FDTD are 28.1% (-5.516 dB) for the TE mode and 34.4% (-4.634 dB) for the TM mode, respectively. The efficiency improvement is mainly a result of mode matching in the vertical direction due to the large core size of an ARROW-B-based PC waveguide.
The tapered coupler is the most popular device used in connecting PC waveguides and other optical devices. In order to further improve the coupling ability of our device, a PC tapered coupler which acts as a mode converter is placed between a single-mode optical fiber and an ARROW-B-based PC waveguide as shown in Fig. 6.7(c). The width of the PC tapered coupler decreases gradually from 6 μm to 0.6 μm and the length of
are 82.0% (-0.862 dB) and 82.5% (-0.834 dB) for TE and TM modes, respectively. The high coupling efficiencies are mainly the consequence of mode matching in the vertical direction by the ARROW-B structure and mode conversion in the lateral direction provided by the PC tapered coupler.
(a) (b)
(c) (d)
Fig. 6.7: (a) An ARROW-B-based PC waveguide coupled directly with a single-mode optical fiber. (b) The top view of (a). (c) An ARROW-B-based PC waveguide coupled with a single-mode optical fiber through a three-dimensional PC tapered coupler. (d) The top view of (c).
6.4 Summary
A photonic crystal waveguide based on an ARROW-B platform is proposed in this study. Optical confinement of this device is supported by PBGs in the lateral plane and by the antiresonant reflection in the vertical direction of the slab. Finite-thickness slabs can support modes with higher order. If the slab is too thick, the presence of these modes can result in closing of the bandgap [143]. Therefore, the thickness of the slab is a critical parameter in conventional structure and needs to be modeled. Even though the core size is as thick as 8 μm, this ARROW-B-based PC waveguide supports quasi-single mode. The ARROW-B-based PC waveguide has high transmission and low bending loss.
The propagation losses are 12.3 dB/mm and 19.3 dB/mm for TE and TM modes, respectively. The bending losses of 60° and 120° ARROW-B-based PC waveguide bends are 0.09 dB/bend and 0.76 dB/bend for the TE mode, and 0.40 dB/bend and 0.99 dB/bend for the TM mode. Furthermore, with the aid of a PC tapered structure, the butt coupling efficiencies between a single-mode optical fiber and a PC waveguide with an ARROW-B structure can be as high as 82.0% and 82.5% for TE and TM modes, respectively. Simulation results show that an ARROW-B structure can be served as a platform for PC devices.
Besides our group, we have noticed that other researchers studied the PC waveguides based on an antiresonant reflecting platform [144]-[147]. Litchinitser proposed a simple analytical theory for low-index core photonic bandgap optical waveguides based on an antiresonant reflecting guidance mechanism and discussed the implications of the results for photonic bandgap fibers [144]. Lavrinenko applied the antiresonant reflecting layers arrangement to silicon-on-insulator based photonic crystal waveguides [147]. They analyzed several layered structures with different combinations of materials. Simulation results reveal that PC waveguides with an ARROW or
ARROW-B structure have potential for competing with membrane-like photonic crystal waveguides.
Chapter 7 Conclusion
In this study, we have shown that a high efficient photonic crystal waveguide bend can be designed by mode matching technique. By shifting lattice points around the bend corner, the bound state in the waveguide bend and the guided mode in the straight waveguide can be matched. The transmissions of the 60° and 120° PC waveguide bend with mode matching are improved significantly. The bound state in a waveguide bend is similar to a cavity mode; therefore, the PC waveguide bend performs a narrow-band transmission. Frequency-shift of the spectra for a PC waveguide bend due to this lattice shifting is observed.
The multimode interference phenomena in multiple-line-defect PC waveguides have been studied. It is found that the optimal length of each section in a PC step tapered coupler is related to the imaging length of MMI. Therefore, a PC step tapered coupler with different section lengths is proposed. The optimal structure can provide mode matching between two adjacent sections and reduce the scattering loss occurring at the corners of abrupt steps. We also revealed the reason why in some cases the transmission of PC step tapered couplers decreases counterintuitively when the taper length is increased.
Based on the MMI effect, the compact PC polarization beam splitter is proposed.
This compact PC PBS is based on the difference of interference effect between TE and TM modes. Within the MMI coupler, the dependence of interference of modes on propagation distance is weak for a TE wave and strong for a TM wave; as a result, the length of the MMI section is only seven lattice constants. Simulation results show that this PBS has low insertion losses and high extinction ratios for TE and TM modes. The
multiplexing, and demultiplexing devices, etc.
We have shown that the ARROW-B structure is a good platform for PC devices.
Optical confinement of a PC slab waveguide based on an ARROW-B platform is supported by PBGs in the lateral plane and by the antiresonant reflection in the vertical direction of the slab. The core size of the ARROW-B-based PC waveguide is thicker than that of a conventional PC waveguide; however, it can support quasi-single mode and has high transmission for TE and TM modes. By mode matching technique, the 60° and 120° ARROW-B-based PC waveguide bends with low bending losses can be designed. Furthermore, with the aid of a PC step tapered structure, the butt coupling efficiency between a single-mode optical fiber and a PC waveguide with an ARROW-B structure is improved.
Though the test cases we studied in each chapter are different, the design methods we proposed are based on the mode matching technique and multimode interference those are basic physics; therefore, the methodologies can be applied to general cases. In the real world, the characteristics of a PC structure are three-dimensional problems to be solved. The finite heights and radiation losses are issues that need to be taken into account in practical 3D PC devices. The guided modes of a slab waveguide are not completely confined in the slab. The energy of the guided mode extends into the air;
therefore, the bands will be at higher frequencies than in the case of 2D. In addition, modes with frequencies above the light line will couple to the radiation modes and leak energy into the air, and thus represent the loss mechanism of the waveguide. However, we can investigate the MMI phenomena and find out the physics easily in a simplified two-dimensional model. While a 3D problem is converted to a 2D model, the dielectric constant of the host medium must be replaced by its effective index.
In the future, the fabrication and measurement of PC devices based on an ARROW-B platform will be carried out. The manufacturing of ARROW-B based PC
devices is going on at National Nano Device Laboratories (NDL). The transmission of a photonic crystal waveguide can be experimentally investigated using the end-fire coupling. Light from a diode laser that can be tuned over a broad wavelength range is transmitted through an optical fiber and focused onto the waveguide facet. The light emitted from the other end of the waveguide is imaged on a camera or detected by a photodiode. Further works will be required to set up these equipments and collect useful data.
Bibliography
[1] E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,“ Phys. Rev. Lett., vol. 58, no. 20, pp. 2059-2062, 1987.
[2] S. John, “Strong localization of photons in certain disordered dielectric superlattices,”
Phys. Rev. Lett., vol. 58, no. 23, pp. 2486-2489, 1987.
[3] V. P. Bykov, “Spontaneous emission in a periodic structure,” Sov. Phys., no. 35, pp.
269-273, 1972.
[4] V. P. Bykov, “Spontaneous emission from a medium with a band spectrum,” Sov. J.
Quant. Electron., no. 4, pp. 861-871, 1975.
[5] D. M. Pustai, A. Sharkawy, S. Shi, and D. W. Prather, “Tunable photonic crystal microcavities,” Appl. Opt., vol. 41, no. 20, pp. 5574-5579, 2002.
[6] G. Subramania, S. Y. Lin, J. R. Wendt, and J. M. Rivera, ”Tuning the microcavity resonant wavelength in a two-dimensional photonic crystal by modifying the cavity geometry,” Appl. Phys. Lett., vol. 83, no. 22, pp. 4491-4493, 2003.
[7] K. Srinivasan, P. E. Barclay, and O. Painter, ”Fabrication-tolerant high quality factor photonic crystal microcavities,” Opt. Express, vol. 12, no. 7, pp. 1458-1463, 2004.
[8] Z. Zhang and M. Qiu, ”Small-volume waveguide-section high Q microcavities in 2D photonic crystal slabs,” Opt. Express, vol. 12, no. 17, pp. 3988-3995, 2004.
[9] D. Englund, I. Fushman, and J. Vuckovi, ”General recipe for designing photonic crystal cavities,” Opt. Express, vol. 12, no. 16, pp. 5961-5979, 2005.
[10] T. Asano, B. S. Song, and S. Noda, “Analysis of the experimental Q factors (~1 million) of photonic crystal nanocavities,” Opt. Express, vol. 14, no. 5, pp.
1996-2002, 2006.
[11] D. N. Chigrin, A. V. Lavrinenko, D. A. Yarotsky, and S. V. Gaponenko,
“All-dielectric one-dimensional periodic structures for total omnidirectional reflection and partial spontaneous emission control,” J. Lightwave Technol., vol. 17, no. 11, pp. 2018-2024, 1999.
[12] V. Tolmachev, T. Perova, and K. Berwick, “Design criteria and optical characteristics of one-dimensional photonic crystals based on periodically grooved silicon,” Appl. Opt., vol. 42, no. 28, pp. 5679-5683, 2003.
[13] A. S. Jugessur, P. Pottier, and R. M. De La Rue, “One-dimensional periodic photonic crystal microcavity filters with transition mode-matching features, embedded in ridge waveguides,” Electron. Lett., vol. 39, no. 4, pp. 367-369, 2003.
[14] J. T. Shen and S. Fan, “Coherent photon transport from spontaneous emission in one-dimensional waveguides,” Opt. Lett., vol. 30, no. 15, pp. 2001–2003, 2005.
[15] J. J. Li, Z. Y. Li, and D. Z. Zhang, “Second harmonic generation in one- dimensional nonlinear photonic crystals solved by the transfer matrix method,”
Phys. Rev. E, vol. 75, no. 5, pp. 056606-1~7, 2007.
[16] C. J. M. Smith, H. Benisty, D. Labilloy, U. Oesterle, R. Houdre, T. F. Krauss, R. M.
De La Rue and C. Weisbuch, “Near-infrared microcavities confined by two dimensional photonic bandgap crystals,” Electron. Lett., vol. 35, no. 3, pp. 228-230, 1999.
[17] E. Chow, S. Y. Lin, S. G. Johnson, P. R. Villeneuve, J. D. Joannopoulos, J. R.
Wendt, G. A. Vawter, W. Zubrzycki, H. Hou, and A. Alleman, “Three-dimensional control of light in a two-dimensional photonic crystal slab,” Nature, vol. 407, pp.
983- 86, 2000.
[18] Y. Akahane, T. Asano, H. Takano, B. S. Song, Y. Takana, and S. Noda,
“Two-dimensional photonic-crystal-slab channel drop filter with flat-top response,”
Opt. Express, vol. 13, no. 7, pp. 2512-2530, 2005.
square-lattice photonic-crystal lasers with TM-polarization,” Opt. Express, vol. 15, no. 7, pp. 3981-3990, 2007.
[20] C. Sell, C. Christensen, G. Tuttle, Z. Y. Li, and K. M. Ho, “Propagation loss in three-dimensional photonic crystal waveguides with imperfect confinement,” Phys.
Rev. B, vol. 68, no. 11, pp. 113106-1~4, 2003.
[21] L. Pierantoni, A. Massaro, and T. Rozzi, “Efficient modeling of 3-D photonic crystals for integrated optical devices,” IEEE Photon. Technol. Lett., vol. 18, no. 2, pp. 319-321, 2006.
[22] M. Che, Z. Y. Li, and R. J. Liu, “Tunable optical anisotropy in three-dimensional photonic crystals,” Phys. Rev. A, vol. 76, no. 2, pp. 023809-1~4, 2007.
[23] S. A. Rinne, F. G. Santamaria, and P. V. Braun, “Embedded cavities and waveguides in three-dimensional silicon photonic crystals,” Nature Photonics, vol.
2, pp. 52-56, 2008.
[24] S. Y. Lin and J. G. Fleming, “A three-dimensional optical photonic crystal,” J.
Lightwave Technol., vol. 17, no. 11, pp. 1944-1947, 1999.
[25] S. Kawashima, L. H. Lee, M. Okano, M. Imada, and S. Noda, “Design of donor-type line-defect waveguides in three-dimensional photonic crystals,” Opt.
Express, vol. 13, no. 24, pp. 9774-9781, 2005.
[26] A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos,
“High transmission through sharp bends in photonic crystal waveguides,” Phys.
Rev. Lett., vol. 77, no. 18, pp. 3787-3790, 1996.
[27] B. Temelkuran and E. Ozbay, “Experimental demonstration of photonic crystal based waveguides,” Appl. Phys. Lett., vol. 74, no. 4, pp. 486 - 488, 1999.
[28] M. Loncar, D. Nedeljkovic, T. Doll, J. Vuckovic, A. Scherer, and T. P. Pearsall,
“Waveguiding in planar photonic crystals,” Appl. Phys. Lett., vol. 77, no. 13, pp.
1937-1939, 2000.
[29] S. Boscolo, C. Conti, M. Midrio, and C. G. Someda, “Numerical analysis of propagation and impedance matching in 2-D photonic crystal waveguides with finite length,” J. Lightwave Technol., vol. 20, no. 2, pp. 304-310, 2002.
[30] N. Susa, “Towards perfect vertical photonic band gap confinement in a photonic crystal slab,” Jpn J. Appl. Phys., vol. 42, part 1, no. 11, pp. 7157–7162, 2003.
[31] J. Witzens, T. B. Jones, M. Hochberg, M. Loncar, and A. Scherer, “Photonic crystal waveguide-mode orthogonality conditions and computation of intrinsic waveguide losses,” J. Opt. Soc. Am. A, vol. 20, no. 10, pp. 1963-1968, 2003.
[32] 湯家榮, “A study of photonic band gap materials by finite-difference time-domain method,” M. S. Thesis, National Taiwan University, 2001.
[33] 汪天仁, “二維光子晶體之帶隙分析,” M. S. Thesis, National Taiwan University, 2001.
[34] 巫金樺, “Potential applications of photonic crystals by using finite-difference time-domain method,” M. S. Thesis, Tatung University, 2001.
[35] 許委斌, “The study of finite-difference time-domain method and its applications on photonic crystals,” M. S. Thesis, Tatung University, 2001.
[36] 蔡雅芝, “Formation and properties of band gaps in photonic crystals,” M. S.
Thesis, National Tsing Hua University, 1997.
[37] 王 志 明 , “Terahertz photonic crystals,” M. S. Thesis, National Tsing Hua University, 1998.
[38] 王海蒂, “Two-dimensional photonic crystal in THz range - Using etched silicon wafer stack,” M. S. Thesis, National Tsing Hua University, 2000.
[39] 賴瑋治, “The study of two dimensional terahertz photonic crystal,” M. S. Thesis, National Tsing Hua University, 2001.
[40] 林鳳瑜, “Fabrication and measurement of two-dimensional and three-dimensional
[41] 闕欣男, “Localized defect modes calculation of photonic crystal,” M. S. Thesis, National Chiao Tung University, 2001.
[42] 侯鴻龍, “The basic study of different 2-D photonic crystal structures,” M. S.
Thesis, National Chiao Tung University, 2000.
[43] 林永倫, “Negative index of refraction phenomenon in photonic crystal,” M. S.
Thesis, National Chiao Tung University, 2002.
[44] T. W. Lu, P. T. Lin, K. U. Sio, and P. T. Lee, “Square lattice photonic crystal point-shifted nanocavity with lowest-order whispering-gallery mode” Opt.
Express, vol. 18, no. 3, pp. 2566-2572, 2010.
[45] L. W. Chung and S. L. Lee, ”Multimode-interference-based broad-band demultiplexers with internal photonic crystals,” Opt. Express, vol. 14, no. 11, pp.
4923-4927, 2006.
[46] C. C. Lin and S. L. Tsao, ”Multimode interference-based heterostructure photonic crystal waveguide power splitter,” Proceedings of SPIE, vol. 5907, 9 pages, 2005.
[47] T. Y. Tsai, Z. C. Lee, J. R. Chen, C. C. Chen, Y. C. Fang, and M. H. Cha, “A novel ultra compact two-mode-interference wavelength division multiplexer for 1.5-μm operation,” Opt. Lett., vol. 41, no. 5, pp. 741-746, 2005.
[48] T. H. Chang, S. H. Chen, C. C. Lee, and H. L. Chen, ”Fabrication of autocloned photonic crystals using electron-beam guns with ion-assisted deposition,” Thin Solid Films, vol. 516, pp.1051-1055, 2008.
[49] T. H. Chang, S. H. Chen, C. H. Chan, Y. W. Yeh, C. C. Lee and C. C. Chen,
“Fabrication of three-dimensional photonic crystals using autocloning layers on the self-assembled microspheres,” Opt. Eng., vol. 48, no. 7, pp. 073401-1~5, 2009.
[50] M. A. Duguay, Y. Kokubun, T. L. Koch, and L. Pfeiffer, “Antiresonant reflecting optical waveguides in SiO2-Si multilayer structures,” Appl. Phys. Letters, vol. 49, no. 1, pp. 13-15, 1986.
[51] T. Baba, Y. Kokubun, T. Sakaki, and K. Iga, “Loss reduction of an ARROW waveguide in shorter wavelength and its stack configuration,” J. Lightwave Technol.
vol. 6, no. 9, pp. 1440-1445, 1988.
[52] W. Huang, R. M. Shubair, A. Nathan, and Y. L. Chow, “The model characteristics of ARROW structures,” J. Lightwave Technol., vol. 10, no. 8, pp. 1015-1022, 1992.
[53] J. M. Kubica, “A rigorous design method for antiresonant reflecting optical waveguides,” IEEE Photon. Technol. Lett., vol. 6, no. 12, pp. 1460-1462, 1994.
[54] S. H. Hsu and Y. T. Huang, “A novel Mach-Zehnder interferometer based on dual-ARROW structures for sensing applications,” J. Lightwave Technol., vol. 23, no. 12, pp. 4200-4206, 2005.
[55] T. Baba and Y. Kokubum, “New polarization-insensitive antiresonant reflecting optical waveguides (ARROW-B),” IEEE Photon. Technol. Lett., vol. 1, no. 8, pp.
232-234, 1989.
[56] T. Baba and Y. Kokubum, “Dispersion and radiation loss characteristics of antiresonant reflecting optical waveguides-numerical results and analytical expressions,” IEEE J. Quantum Electron., vol. 28, no. 7, pp. 1689-1700, 1992.
[57] S. Asakawa and Y. Kokubun, “ARROW-B type polarizer utilizing birefringence in multilayer stripe lateral confinement,” IEEE Photon. Technol. Lett., vol. 7, no. 1, pp. 38-40, 1995.
[58] K. Sakoda, Optical properties of photonic crystal, Springer Press, 2001.
[59] S. G. Johnson and J. D. Joannopoulos, Photonic crystal: The road from theory to practice, Kluwer academic publishers group, 4th ed., 2004.
[60] J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic crystal:
Molding the flow of light, Princeton University Press, 2nd ed., 2008.
[61] J. M. Lourtioz, H. Benisty, V. Berger, J. M. Gerard, D. Maystre, and A. Tchelnokov,
[62] D. W. Prather, S. Shi, A. Sharkawy, J. Murakowski, and G. J. Schneider, Photonic crystal: Theory, Applications, and Fabrication, John Wiley & Sons Inc., 2009.
[63] K. Aygün, B. Shanker, A. A. Ergin, and E. Michielssen, “A two-level plane wave
time-domain algorithm for fast analysis of EMC/EMI problems,” IEEE Trans.
Electromagnetic Comp.,vol. 44, no. 1, pp. 152-164, 2002.
[64] J. D. Pursel and P. M. Goggans, “A finite-difference time-domain method for solving electromagnetic problems with bandpass-limited sources,” IEEE Trans.
Antennas and Prop., vol. 47, no. 1, pp. 9-15, 1999.
[65] M. Yokoyama and S. Noda, “Finite-difference time-domain simulation of two dimensional photonic crystal surface-emitting laser,” Opt. Express, vol. 12, no. 8, pp.2869-2880, 2004.
[66] K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propagat., AP-14, 302, 1966.
[67] H. C. Chang and C. P. Yu, “Yee-mesh-based finite difference eigenmode analysis algorithms for optical waveguides and photonic crystals,” OSA IPR, IFE4, 2004.
[68] A. Mekis, S. Fan, and J. D. Joannopoulos, “Absorbing boundary conditions for FDTD simulations of photonic crystal waveguides,” IEEE Microwave and Guided Wave Letters, vol. 9, no. 12, pp. 502-504, 1999.
[69] M. Koshiba, Y. Tsuji, and S. Sasaki, “High-performance absorbing boundary conditions for photonic crystal waveguide simulations,” IEEE Microwave and Guided Wave Letters, vol. 11, no. 4, pp. 152-154, 2001.
[70] T. Tamir, Guided-wave optoelectronics, Springer-Verlag, 1990.
[71] http://ab-initio.mit.edu/wiki/index.php/MIT_Photonic_Bands
[72] B. P. Hiett, J. M. Generowicz, S. J. Cox, M. Molinari, D. H. Beckett, G. J. Parker, and K. S. Thomas, “Finite element modelling of photonic crystals,” PREP, 2001.
[73] R. C. McPhedran, L. C. Botten, and N. A. Nicorovici, Advances in the Rayleigh Multipole Method for Problems in Photonics and Phononics, Springer-Verlag, 2006.
[74] BandSOLVE 3.0 User Guide, Rsoft Design Group, Inc.
[75] E. Chow, S. Y. Lin, J. R. Wendt, S. G. Johnson, and J. D. Joannopoulos,
“Quantitative analysis of bending efficiency in photonic- crystal waveguide bends at λ = 1.55 μm wavelengths,” Opt. Lett., vol. 26, no. 5, pp. 286-288, 2001.
[76] A. Talneau, Ph. Lalanne, M. Agio, and C. M. Soukoulis, “Low-reflection photonic-crystal taper for efficient coupling between guide sections of arbitrary widths,” Opt. Lett., vol. 27, no. 17, pp. 1522-1524, 2002.
[77] N. Moll and G. L. Bona, “Bend design for the low-group-velocity mode in photonic crystal-slab waveguides,” Appl. Phys. Lett., vol. 85, no. 19, pp.
4322-4324, 2004.
[78] S. Olivier, H. Benisty, M. Rattier, C. Weisbuch, M. Qiu, A. Karlsson, C. J. M.
Smith, R. Houdre, and U. Oesterle, “Resonant and nonresonant transmission through waveguide bends in a planar photonic crystal,” Appl. Phys. Lett., vol. 79, no. 16, pp. 2514-2516, 2001.
[79] B. Miao, C. Chen, S. Shi, J. Murakowski, and D. W. Prather, “High-efficiency broad-band transmission through a double-60° bend in a planar photonic crystal single-line defect waveguide,” IEEE Photon. Technol. Lett., vol. 16, no. 11, pp.
2469-2471, 2004.
[80] A. Chutinan, M. Okano, and S. Noda, “Wider bandwidth with high transmission through waveguide bends in two-dimensional photonic crystal slabs,” Appl. Phys.
Lett., vol. 80, no. 10, pp. 1698-1700, 2002.
[81] J. S. Jensen, O. Sigmund, L. H. Frandsen, P. I. Borel, A. Harpøth, and M.
crystal waveguide bend,” IEEE Photon. Technol. Lett., vol. 17, no. 6, pp.
1202-1204, 2002.
[82] A. Têtu, M. Kristensen, L. H. Frandsen, A. Harpøth, P. I. Borel, J. S. Jensen, and O.
Sigmund, “Broadband topology-optimized photonic crystal components for both TE and TM polarizations,” Opt. Express, vol. 13, no. 21, pp. 8606-8611, 2005.
[83] A. Mekis, S. Fan, and J. D. Joannopoulos, “Bound states in photonic crystal waveguides and waveguide bends,” Phys. Rev. B, vol. 58, no. 8, pp. 4809-4817, 1998.
[84] S. Blair and J. Goeckeritz, “Effect of vertical mode matching on defect resonances in one-dimensional photonic crystal slabs,” J. Lightwave Technol., vol. 24, no. 3, pp. 1456-1461, 2006.
[85] Z. Zhang and M. Qiu, “Small-volume waveguide-section high Q microcavities in 2D photonic crystal slabs,” Opt. Express, vol. 12, no. 17, pp. 3988-3995, 2004.
[86] K. Nozaki, S. Kita, and T. Baba, “Room temperature continuous wave operation and controlled spontaneous emission in ultrasmall photonic crystal nanolaser,”
Opt. Express, vol. 15, no. 12, pp. 7506-7514, 2007.
[87] M. F. Lu and Y. T. Huang, “Design of a photonic crystal taper coupler with
[87] M. F. Lu and Y. T. Huang, “Design of a photonic crystal taper coupler with