WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

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waveguides have inherently lower propagation losses than the photonic crystal waveguides, and may be alternatives/supplements to these in future photonic integrated circuits. Our topology optimization algorithm is based on a two-dimensional finite element model of the Helmholtz equation that models propagation of plane polarized light. The finite element model is used to compute the energy transmission through the waveguides and the optimization algorithm is used to redistribute air and dielectric in the trouble regions in order to maximize the transmission. A SIMP-like material model is used together with analytical sensitivity analysis and the mathematical programming software MMA. We use a penalization method based on artificial damping (PAMP<strong>IN</strong>G) to ensure a feasible binary design, and perform optimization for several frequencies by exploiting fast-frequencysweeps using Padé approximants. The photonic wire components are currently being analyzed using a 3D finite-difference-time-domain code and will soon be tested experimentally. Additionally, other components such as a wavelength splitters in photonic crystal and photonic wire waveguides are currently being developed as well as directional couplers. Future research include also optimization based on 3D finite element models. 7. References [1] Sugimoto Y, Tanaka Y, Ikeda N, Kanamoto K, Nakamura Y, Ohkouchi S, Nakamura H, Inoue K, Sasaki H, Watanabe Y, Ishida K, Ishikawa H and Asakawa K. Two dimensional semiconductorbased photonic crystal slab waveguides for ultra-fast optical signal processing devices, IEICE Trans. Electron., 2004, E87-C, 316-327. [2] Borel P I, Harpøth A, Frandsen L H, Kristensen M, Shi P, Jensen J S and Sigmund O. Topology optimization and fabrication of photonic crystal structures, Optics Express, 2004, 12(9), 1996-2001. [3] Vlasov Y A and McNab S J. Losses in single-mode silicon-on-insulator strip waveguides and bends, Optics Express, 2004, 12(8), 1622-1631. [4] Bendsøe M and Sigmund O. Topology Optimization - Theory, Methods and Applications. Springer Verlag, Berlin Heidelberg, 2003. [5] Svanberg K. The method of moving asymptotes - a new method for structural optimization, International Journal for Numerical Methods in Engineering, 1987, 24, 359-373. [6] Jensen J S and Sigmund O. Topology optimization of photonic crystal structures: A high-bandwidth low-loss T-junction waveguide, J. Opt. Soc. Am. B, 2005, 22(6), to appear. [7] Yablonovitch E. Inhibited spontaneous emission in solid-state physics and electronics, Physical Review Letters, 1987, 58, 2059-2062. [8] John S. Strong localization of photons in certain disordered dielectric superlattices, Physical Review Letters, 1987, 58, 2486-2489. [9] Joannopoulos J D, Meade R D and Winn J N. Photonic Crystals. Princeton University Press, New Jersey, 1995. [10] Ammari H. and Santosa F. Guided waves in a photonic bandgap structure with a line defect, SIAM J. Appl. Math., 2004, 64(6), 2018-2033. [11] Borel P I, Frandsen L H, Harpøth A, Leon J B, Liu H, Kristensen M, Bogaerts W, Dumon P, Baets R, Wiaux V, Wouters J and Beckx S. Bandwidth engineering of photonic crystal waveguide bends, Electron. Lett., 2004, 40, 1263-1264. [12] Frandsen L H, Harpøth A, Borel P I, Kristensen M, Jensen J S and Sigmund O. Broadband photonic crystal waveguide 60-degree bend obtained using topology optimization, Optics Express, 2004, 12(24), 5916-5921. 9
[13] Borel P I, Frandsen L H, Harpøth A, Kristensen M, Jensen J S and Sigmund O. Topology Optimised Broadband Photonic Crystal Y-Splitter, Electron. Lett., 2005, 41(2), 69-71. [14] Jensen J S, Sigmund O, Frandsen L H, Borel P I, Harpøth A and Kristensen M. Topology design and fabrication of an efficient double 90-degree photonic crystal waveguide bend, IEEE Photonics Technology Letters, 2005, 17(6), to appear. [15] Koshiba M, Tsuji Y and Sasaki S. High-performance absorbing boundary conditions for photonic crystal waveguide simulations, IEEE Microwave Wireless Components Lett., 2001, 11, 152-154. [16] Manolatou C, Johnson S G, Fan S, Villeneuve P R, Haus H A and Joannopoulos J D. High-density integrated optics, Journal of Lightwave Technology, 1999, 17(9), 1682-1692. [17] Sakai A, Fukazawa T and Baba T. Low loss ultra-small branches in a silicon photonic wire waveguide, IEICE Trans. Electron., 2002, E85-C, 1033-1038. 10
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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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972 In the following section, we sh
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974 Design variable, t Design varia
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976 Eigenvalue ratio, ω n+1 2 /ωn
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978 to a design parameter te are gi
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980 ARTICLE IN PRESS J.S. Jensen, N
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982 respect to the higher bound C1
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984 instead vary the value of b. Th
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986 References ARTICLE IN PRESS J.S
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a thick solid was considered. A few
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The above expressions provide insta
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In all the examples shown a density
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domain. To the very left the passiv
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Jensen JS, Sigmund O (2005) Topolog
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Page 207 and 208: Appl. Phys. Lett., Vol. 84, No. 12,
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Page 218 and 219: Topology optimization and fabricati
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Page 222 and 223: Fig. 4. Scanning electron micrograp
Page 224 and 225: Broadband photonic crystal waveguid
Page 226 and 227: 2. Design and fabrication Silicon-o
Page 228 and 229: Fig. 3. Steady-state magnetic field
Page 230 and 231: Topology optimised broadband photon
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Page 239 and 240: nanoimprinted patterns are transfer
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Page 245 and 246: 5. Photonic Wire Waveguides Figure
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Page 275 and 276: 1628 J. S. JENSEN Equation (A10) co
Page 277 and 278: 1630 J. S. JENSEN 7. Coyette J-P, L
Page 279 and 280: The number of masses in the chain t
Page 281 and 282: 5 Results The optimization algorith
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Page 286 and 287: Space-time topology optimization fo
Page 288 and 289: good for low to moderate stiffness
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Page 292 and 293: The parameters used in this example
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Page 296: at the Department of Mechanical Eng
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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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