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Broadband photonic crystal waveguide 60° bend obtained utilizing topology optimization L.H. Frandsen, A. Harpøth, P.I. Borel, M. Kristensen Research Center COM, Technical University of Denmark, DK-2800 Kgs. Lyngby, Denmark lhf@com.dtu.dk, harpoeth@com.dtu.dk, pib@com.dtu.dk, mk@com.dtu.dk http://www.com.dtu.dk/research/glass/pipe/index.html J.S. Jensen and O. Sigmund Department of Mechanical Engineering, Technical University of Denmark, DK-2800 Kgs. Lyngby, Denmark jsj@mek.dtu.dk, sigmund@mek.dtu.dk http://www.topopt.dtu.dk Abstract: Topology optimization has been used to design a 60° bend in a single-mode planar photonic crystal waveguide. The design has been realized in a silicon-on-insulator material and we demonstrate a recordbreaking 200nm transmission bandwidth with an average bend loss of 0.43±0.27 dB for the TE polarization. The experimental results agree well with 3D finite-difference-time-domain simulations. ©2004 Optical Society of America OCIS codes: (000.3860) Mathematical methods in physics; (000.4430) Numerical approximation and analysis; (130.2790) Guided waves; (130.3130) Integrated optics materials; (220.4830) Optical systems design; (230.5440) Polarization-sensitive devices; (230.7390) Waveguides, planar; (999.9999) Photonic crystals. References and links 1. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059-2062 (1987). 2. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486-2489 (1987). 3. T. F. Krauss, R. M. De La Rue, and S. Brand, “Two-dimensional photonic-bandgap structures operating at nearinfrared wavelengths,” Nature 383, 699-702 (1996). 4. M. Thorhauge, L. H. Frandsen and P. I. Borel, “Efficient Photonic Crystal Directional Couplers,” Opt. Lett. 28, 1525-1527 (2003). 5. Y. Sugimoto, Y. Tanaka, N. Ikeda, K. Kanamoto, Y. Nakamura, S. Ohkouchi, H. Nakamura, K. Inoue, H. Sasaki, Y. Watanabe, K. Ishida, H. Ishikawa, K. Asakawa, “Two Dimensional Semiconductor-Based Photonic Crystal Slab Waveguides for Ultra-Fast Optical Signal Processing Devices,” IEICE Trans. Electron. E87-C, 316-327 (2004). 6. L. H. Frandsen, P. I. Borel, Y. X. Zhuang, A. Harpøth, M. Thorhauge, M. Kristensen, W. Bogaerts, P. Dumon, R. Baets, V. Wiaux, J. Wouters, and S. Beckx, “Ultra-low-loss 3-dB Photonic Crystal Waveguide Splitter,” Opt. Lett. 29, 1623-1625 (2004). 7. T. Søndergaard, J. Arentoft, and M. Kristensen, ”Theoretical Analysis of Finite-Height Semiconductor-on- Insulator-Based Planar Photonic Crystal Waveguides,” J. Lightwave Technol. 20, 1619-1626 (2002) 8. C. Jamois, R. B. Wehrspohn, L.C. Andreani, C. Hermann, O. Hess, U. Gösele, “Silicon-based two-dimensional photonic crystal waveguides,” Photonics and Nanostructures – Fundamentals and Applications 1, 1-13 (2003). 9. P.I. Borel, L.H. Frandsen, A. Harpøth, J.B. Leon, H. Liu, M. Kristensen, W. Bogaerts, P. Dumon, R. Baets, W. Wiaux, J. Wouters, S. Beckx, “Bandwidth tuning of photonic crystal waveguide bends,” Electron. Lett. 40, 1263-1264 (2004). 10. A. Chutinan, M. Okano, S. Noda, “Wider bandwidth with high transmission through waveguide bends in twodimensional photonic crystal slabs,” Appl. Phys.Lett. 80, 1698-1700 (2002). 11. J. Smajic, C. Hafner, D. Erni, “Design and optimization of an achromatic photonic crystal bend,” Opt. Express 11, 1378-1384 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-12-1378. #5520 - $15.00 US Received 19 October 2004; revised 12 November 2004; accepted 15 November 2004 (C) 2004 OSA 29 November 2004 / Vol. 12, No. 24 / OPTICS EXPRESS 5916
12. P.I. Borel, A. Harpøth, L.H. Frandsen, M. Kristensen, P. Shi, J.S. Jensen and O. Sigmund, “Topology Optimization and fabrication of photonic crystal structures,” Opt. Express 12, 1996–2001 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-9-1996. 13. A. Lavrinenko, P.I. Borel, L.H. Frandsen, M. Thorhauge, A. Harpøth, M. Kristensen and T. Niemi, “Comprehensive FDTD Modelling of Photonic Crystal Waveguide Components,” Opt. Express 12, 234-248 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-2-234. 14. P.I. Borel, L. H. Frandsen, M. Thorhauge, A. Harpøth, Y. X. Zhuang, M. Kristensen, and H. M. H. Chong, “Efficient propagation of TM polarized light in photonic crystal components exhibiting band gaps for TE polarized light.” Opt. Express 11, 1757-1762 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-15-1757. 15. A. Talneau, L. Le Gouezigou, N. Bouadma, M. Kafesaki, C.M. Soukoulis, M. Agio, “Photonic-crystal ultrashort bends with improved transmission and low reflection at 1.55 µm,” Appl. Phys.Lett. 80, 547-549 (2002). 16. E. Chow, S. Y. Lin, J. R. Wendt, S. G. Johnson, J. D. Joannopoulos, “Quantitative analysis of bending efficiency in photonic-crystal waveguide bends at λ = 1.55 µm wavelengths,” Opt. Lett. 26, 286-288 (2001). 17. M. P. Bendsøe and O. Sigmund, Topology optimization — Theory, Methods and Applications (Springer-Verlag, 2003). 18. K. Svanberg, “The method of moving asymptotes: a new method for structural optimization,” Int. J. Numer. Meth. Engng. 24, 359-373 (1987). 19. J. S. Jensen and O. Sigmund, “Systematic design of photonic crystal structures using topology optimization: Low-loss waveguide bends,” Appl. Phys. Lett. 84, 2022-2024 (2004). 20. O. Sigmund and J. S. Jensen, “Systematic design of phononic band gap materials and structures by topology optimization,” Phil. Trans. R. Soc. Lond. A 361, 1001-1019 (2003). 21. J. S. Jensen and O. Sigmund, “Topology optimization of photonic crystal structures: A high bandwidth low loss T-junction waveguide,” J. Opt. Soc. Am. B, accepted (2004). 22. R. L. Espinola, R. U. Ahmad, F. Pizzuto, M. J. Steel, R. M. Osgood, Jr., ”A study of high-index contrast 90° waveguide bend structures,” Opt. Express 8, 517-528 (2001), http://www.opticsexpress.org/abstract.cfm? URI=OPEX-8-9-517. 1. Introduction Modulating the refractive index profile periodically in an optical material can create photonic bandgaps (PBGs) wherein no optical modes can exist [1-3]. Such structures are often referred to as photonic crystals (PhCs) and have attracted a lot of attention as they potentially allow ultra-compact photonic integrated circuits (PICs) to be realized [4-6]. Planar PhC structures are often defined as triangular arrangements of low dielectric pillars in a high dielectric material. This configuration gives rise to a large PBG for the transverse-electric (TE) polarization [7, 8]. Defects in the PhC can introduce modes in the PBG. In this way, photonic crystal waveguides (PhCWs) can be formed by locally breaking the periodicity along a specific direction of the PhC lattice. Due to the triangular lattice configuration, such PhCWs are naturally bent in steps of 60°, thus, making the 60° PhCW bend a key component in PhCbased PICs. In the field of planar photonic crystals, research has within the last decade mostly relied on an Edisonian design approach combining physical arguments and experimental/numerical verifications [9, 10]. Further optimizations have typically been done in an iterative trial-anderror procedure to improve a chosen performance measure of the PhC component. Such an approach is very time-consuming and does not guarantee optimal solutions. Recently, Smajic et al. [11] have shown that sensitivity analyses can be assistive in choosing the critical PhC rods/holes to be altered. A different approach was suggested in our previous work [12], in which we used an inverse design strategy called topology optimization to optimize the performance of a PhCW containing two consecutive 120° bends. This design method offers an effective and robust optimization of the photonic crystal structure irrespectively of the device under consideration. Here, we apply the topology optimization method to the much more important and commonly used PhCW 60° bend and demonstrate an experimental 1-dB transmission bandwidth of more than 200nm. Experimental transmission spectra are compared to spectra obtained from 3D finite-difference-time-domain (FDTD) calculations [13] and good agreement is found. #5520 - $15.00 US Received 19 October 2004; revised 12 November 2004; accepted 15 November 2004 (C) 2004 OSA 29 November 2004 / Vol. 12, No. 24 / OPTICS EXPRESS 5917
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WAVES AND VIBRATIONS IN INHOMOGENEO
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Denne afhandling er af Danmarks Tek
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List of Thesis Papers [1] J. S. Jen
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Contents Preface i List of Thesis P
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Chapter 1 Introduction This thesis
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eplacemen Phononic and photonic ban
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Optimal material distribution - top
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Chapter 2 The bandgap phenomenon In
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2.1 Band diagram and forced vibrati
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2.3 Experimental demonstration 11 F
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y 2.5 Nonlinearities 13 Response in
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Relations to recent work 15 Amplitu
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Chapter 3 Bandgap structures as opt
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3.1 Vibration-quenching structures
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3.2 Maximizing wave reflection 21 d
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3.4 Maximizing wave dissipation 23
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3.4 Maximizing wave dissipation 25
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3.5 Plate structures 27 a) b) Figur
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3.6 Acoustic design 29 design domai
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Chapter 4 Optimization of photonic
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4.1 Waveguide bends and junctions 3
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4.2 Photonic crystal building block
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4.2 Photonic crystal building block
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4.4 Ridge waveguides 39 w ε =1 ?
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Relations to recent work 41 wave in
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Chapter 5 Advanced optimization pro
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5.1 Padé approximants 45 h 2h A f
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5.3 Space-time topology optimizatio
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5.3 Space-time topology optimizatio
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Relations to recent work 51 Figure
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Chapter 6 Conclusions In the first
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References K. Asakawa, Y. Sugimoto,
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References 57 W. R. Frei, D. A. Tor
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References 59 F. Maestre, A. Münch
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References 61 O. Sigmund, J. S. Jen
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References 63 X. M. Zhou and G. K.
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Dansk resumé 65 i strukturen. En r
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1054 ARTICLE IN PRESS J.S. Jensen /
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1058 fundamental eigenfrequency o
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1060 ðo 2 y;j;k ARTICLE IN PRESS J
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1062 local resonator and splits up
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1064 3.1. Model equations A finite
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1066 FRF (dB) 150 100 50 0 -50 -100
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1068 3.3. Example 2: a 2-D waveguid
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wave-guiding properties show promis
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(a) (b) Acceleration response (dB)
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B.S. Lazarov, J.S. Jensen / Interna
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ω/κ ω/κ 1.5 1.4 1.3 1.2 1.1 1 0
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[B 2000 ] [B 2000 ] [B 2000 ] 0.25
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[B 400 ] 0.25 0.2 0.15 0.1 0.05 B.S
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1002 (a) (c) (e) 0. Sigmund and J.
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1004 O. Sigmund and J. S. Jensen wh
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1006 (a) - - - - - - _ - - - - I I
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1008 O. Sigmund and J. S. Jensen we
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1010 O. Sigmund and J. S. Jensen Th
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1012 (a) (b) O. Sigmund and J. S. J
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1014 . ct . _ a) Q^ s^c "3
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1016 O. Sigmund and J. S. Jensen (a
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1018 O. Sigmund and J. S. Jensen 4.
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Z. Kristallogr. 220 (2005) 895-905
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Inverse design of phononic crystals
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320 Optimization of the dissipation
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322 The reflection of the wave is f
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324 which averaged over a wave peri
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326 where the modified stress compo
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328 optimization algorithm is enhan
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330 reflectance are also seen (Fig.
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332 It should be emphasized that ot
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334 b Dissipation, D c Dissipation,
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336 7.2. Three-phase design ARTICLE
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968 ARTICLE IN PRESS J.S. Jensen, N
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970 To solve the wave equation with
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1628 J. S. JENSEN Equation (A10) co
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1630 J. S. JENSEN 7. Coyette J-P, L
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The number of masses in the chain t
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5 Results The optimization algorith
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Figure 9. Response for optimized de
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Space-time topology optimization fo
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good for low to moderate stiffness
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V ¼ 0:3 at a given time instant fo
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The parameters used in this example
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Normalized amplitude 1 0.9 0.8 0.7
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at the Department of Mechanical Eng
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600 J.S. JENSEN techniques for obta
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602 J.S. JENSEN 1 and 2, where the
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604 J.S. JENSEN 4. Numerical analys
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606 J.S. JENSEN Energy, E a) Energy
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608 J.S. JENSEN relaxed somewhat in
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610 J.S. JENSEN when the contrast i
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612 J.S. JENSEN 6. Summary and conc
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614 J.S. JENSEN Sigmund, O. and Jen
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