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Topology optimization and fabrication of photonic crystal structures P. I. Borel, A. Harpøth, L. H. Frandsen, M. Kristensen Research Center COM, Technical University of Denmark, DK-2800 Kgs. Lyngby, Denmark pib@com.dtu.dk, harpoeth@com.dtu.dk, lhf@com.dtu.dk, mk@com.dtu.dk http://www.com.dtu.dk/research/glass/pipe/index.html P. Shi Department of Micro and Nanotechnology, Technical University of Denmark, DK-2800 Kgs. Lyngby, Denmark pxshi@mic.dtu.dk 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 is used to design a planar photonic crystal waveguide component resulting in significantly enhanced functionality. Exceptional transmission through a photonic crystal waveguide Z-bend is obtained using this inverse design strategy. The design has been realized in a silicon-on-insulator based photonic crystal waveguide. A large low loss bandwidth of more than 200 nm for the TE polarization is experimentally confirmed. ©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 near-infrared 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. 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. (to be published). 6. D. Taillaert, H. Chong, P.I. Borel, L.H. Frandsen, R.M. De La Rue, and R. Baets, “A Compact Twodimensional Grating Coupler used as a Polarization Splitter,” IEEE Photon. Technol. Lett. 15, 1249-1251 (2003). 7. M. P. Bendsøe and N. Kikuchi, “Generating optimal topologies in structural design using a homogenization method,” Comput. Meth. Appl. Mech. Eng. 71, 197-224 (1988). 8. M. P. Bendsøe and O. Sigmund, Topology optimization — Theory, Methods and Applications (Springer- Verlag, 2003). 9. T. P. Felici and D. F. G. Gallagher, “Improved waveguide structures derived from new rapid optimization techniques,” Proc. SPIE 4986, 375-385 (2003). 10. J. Smajic, C. Hafner and 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. #4140 - $15.00 US Received 30 March 2004; revised 23 April 2004; accepted 26 April 2004 (C) 2004 OSA 3 May 2004 / Vol. 12 No. 9 / OPTICS EXPRESS 1996
11. W. J. Kim and J. D. O’Brien, “Optimization of a two-dimensional photonic-crystal waveguide branch by simulated annealing and the finite element method,” J. Opt. Soc. Am. B 21, 289-295 (2004). 12. M. Tokushima, H. Kosaka, A. Tomita and H.Yamada, “Lightwave propagation through a 120° sharply bent single-line-defect photonic crystal waveguide,” Appl. Phys. Lett. 76, 952-954 (2000). 13. T. Uusitupa, K. Kärkkäinen and K. Nikoskinen, “Studying 120° PBG waveguide bend using FDTD,” Microwave Opt. Technol. Lett. 39, 326-333 (2003). 14. It should be emphasized that the method can readily be implemented in a 3D finite element model where the computational requirements naturally will be significantly higher. 15. K. Svanberg, “The method of moving asymptotes: a new method for structural optimization,” Int. J. Numer. Meth. Engng. 24, 359-373 (1987). 16. 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). 17. 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). 18. 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. 19. A. Lavrinenko, P. I. Borel, L. H. Frandsen, M. Thorhauge, A. Harpøth, M. Kristensen, T. Niemi, and H. M. H. Chong, “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. 1. Introduction The planar photonic crystal (PhC) is an optical nano-material with periodic modulation of the refractive index. The modulation is designed to forbid propagation of light in certain wavelength ranges, so-called photonic bandgaps (PBGs) [1-3]. Breaking the crystal symmetry by introducing line defects and other discontinuities allows control of the light on a subwavelength scale in the PhCs. Therefore, photonic devices based on the PBG effect may be up to one million times smaller than traditional integrated optical devices. PhC structures with 20-40 nm useful optical bandwidths have previously been demonstrated [4-6]. Until now, however, no bandgap-based PhC components have been demonstrated with satisfactory performance in a broad wavelength range. A major reason for this has been the lack of efficient inverse design tools that can be applied irrespectively of the device under consideration. Therefore, most PhC design structures today are obtained either by intuition or by varying one or two design parameters—typically the position or size of a PhC element— using the trial-and-error method. In this paper we show exceptional transmission through a Z-bend consisting of two successive 120° PhC waveguide bends. The design of the bends is obtained using an efficient inverse design strategy called topology optimization. The optimized design is experimentally realized in a silicon-based PhC. Measurements have confirmed a large low-loss bandwidth of more than 200 nm for TE polarized light. 2. Topology optimization The systematic design method based on topology optimization allows creation of improved PhC components with previously unseen low transmission losses and high operational bandwidths, or with wavelength selective functionalities. The method was originally developed for structural optimization problems [7], but has recently been extended to a range of other design problems [8]. The method is based on repeated finite element analyses where the distribution of material in a given design area is iteratively modified in order to improve a chosen performance measure. The resulting designs are inherently free from geometrical restrictions such as the number of holes, hole shapes etc., thereby allowing the large potentials of PhC components to be exploited to hitherto unseen levels. Previously reported optimization tools for such components have all been restricted to deal with circular holes [9-11]. #4140 - $15.00 US Received 30 March 2004; revised 23 April 2004; accepted 26 April 2004 (C) 2004 OSA 3 May 2004 / Vol. 12 No. 9 / OPTICS EXPRESS 1997
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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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1622 J. S. JENSEN Response, ln(u ti
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1624 J. S. JENSEN h 2h fcos Ωt Fi
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1626 J. S. JENSEN details and have
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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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