WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

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40 Chapter 4 Optimization of photonic waveguides the optimized design for the case of H-polarized waves is more complicated with air holes inside the waveguide area. Relations to recent work The use of topology optimization for the design of photonic devices continues to be applied by different research groups. Different waveguide optimization studies have been performed recently e.g. by Frei et al. (2005) who optimized a photonic crystal waveguide termination for maximum directivity, by Tsuji et al. (2006), Hirayama et al. (2007) and Tsuji and Hirayama (2008) who considered various planar photonic waveguide optimization problems and by Nomura et al. (2009) who optimized the periodic microstructure of electromagnetic materials. Design of other photonic crystal components such as high-Q cavities have been considered e.g. by Dobsonand Santosa (2004) using a material distribution method and also by Frei et al. (2007) and Frei et al. (2008) using related geometry-projection/level set methods. Similar optimization problems have been analyzed out using different numerical optimization techniques such as simulated annealing (Kim and O’Brien, 2004) and genetic algorithms (Smajic et al., 2004). Somepublicationshaveelaborateddirectlyontheoptimizedstructurespresented in this thesis. This includes a study on the optimized 60-degree bend by Miao et al. (2004). Thestructuresoptimizedinpapers[12]–[15]havealsobeenfurtherexamined and subjected to additional experimental investigation by Tetu et al. (2005). Furthermore, a number of optimized structures have been created by other researchers using the software developed by the author and co-workers. This has been facilitated by an executable code and a graphical user interface (GUI) which have been made accessible to colleagues in Denmark and Japan. Fig. 4.8 shows a snapshot of the GUI. The structures created with the use of the software include a waveguide crossing, a waveguide taper and waveguide bends (Ikeda et al., 2006; Watanabe et al., 2006, 2007, 2008; Asakawa et al., 2006; Yang et al., 2007). The appearance of very small details in the optimized designs can be a major challenge with respect to fabrication and often these details cannot be reproduced exactly causing a deterioration of the performance. This problem is not addressed in this thesis. However, new work in this area (Sigmund, 2007, 2009) is expected to facilitate optimized structures that are robust to fabrication tolerances and uncertainties.
Relations to recent work 41 wave input design domains power evaluation Figure 4.8 Screenshot from the GUI interface to the developed topology optimization software for the design of photonic crystal waveguide components. The setup for the design of a 60-degree bend is shown with indications of wave input, design domains and power evaluation area.
Page 1 and 2: WAVES AND VIBRATIONS IN INHOMOGENEO
Page 3: Denne afhandling er af Danmarks Tek
Page 7 and 8: List of Thesis Papers [1] J. S. Jen
Page 9 and 10: Contents Preface i List of Thesis P
Page 11 and 12: Chapter 1 Introduction This thesis
Page 13 and 14: eplacemen Phononic and photonic ban
Page 15 and 16: Optimal material distribution - top
Page 17 and 18: Chapter 2 The bandgap phenomenon In
Page 19 and 20: 2.1 Band diagram and forced vibrati
Page 21 and 22: 2.3 Experimental demonstration 11 F
Page 23 and 24: y 2.5 Nonlinearities 13 Response in
Page 25 and 26: Relations to recent work 15 Amplitu
Page 27 and 28: Chapter 3 Bandgap structures as opt
Page 29 and 30: 3.1 Vibration-quenching structures
Page 31 and 32: 3.2 Maximizing wave reflection 21 d
Page 33 and 34: 3.4 Maximizing wave dissipation 23
Page 35 and 36: 3.4 Maximizing wave dissipation 25
Page 37 and 38: 3.5 Plate structures 27 a) b) Figur
Page 39 and 40: 3.6 Acoustic design 29 design domai
Page 41 and 42: Chapter 4 Optimization of photonic
Page 43 and 44: 4.1 Waveguide bends and junctions 3
Page 45 and 46: 4.2 Photonic crystal building block
Page 47 and 48: 4.2 Photonic crystal building block
Page 49: 4.4 Ridge waveguides 39 w ε =1 ?
Page 53 and 54: Chapter 5 Advanced optimization pro
Page 55 and 56: 5.1 Padé approximants 45 h 2h A f
Page 57 and 58: 5.3 Space-time topology optimizatio
Page 59 and 60: 5.3 Space-time topology optimizatio
Page 61 and 62: Relations to recent work 51 Figure
Page 63 and 64: Chapter 6 Conclusions In the first
Page 65 and 66: References K. Asakawa, Y. Sugimoto,
Page 67 and 68: References 57 W. R. Frei, D. A. Tor
Page 69 and 70: References 59 F. Maestre, A. Münch
Page 71 and 72: References 61 O. Sigmund, J. S. Jen
Page 73 and 74: References 63 X. M. Zhou and G. K.
Page 75 and 76: Dansk resumé 65 i strukturen. En r
Page 77 and 78: 1054 ARTICLE IN PRESS J.S. Jensen /
Page 81 and 82: 1058 fundamental eigenfrequency o
Page 83 and 84: 1060 ðo 2 y;j;k ARTICLE IN PRESS J
Page 85 and 86: 1062 local resonator and splits up
Page 87 and 88: 1064 3.1. Model equations A finite
Page 89 and 90: 1066 FRF (dB) 150 100 50 0 -50 -100
Page 91 and 92: 1068 3.3. Example 2: a 2-D waveguid
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1078 ARTICLE IN PRESS J.S. Jensen /
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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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paper extends on this work. For an
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R 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2
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objective 1 0.9 0.8 0.7 0.6 0.5 0.4
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The optimization algorithm employs
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Appl. Phys. Lett., Vol. 84, No. 12,
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J. S. Jensen and O. Sigmund Vol. 22
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Topology optimization and fabricati
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Normalized transmission 1.0 0.8 0.6
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Fig. 4. Scanning electron micrograp
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Broadband photonic crystal waveguid
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2. Design and fabrication Silicon-o
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Fig. 3. Steady-state magnetic field
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Topology optimised broadband photon
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1202 IEEE PHOTONICS TECHNOLOGY LETT
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1204 IEEE PHOTONICS TECHNOLOGY LETT
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11. P. Debackere, S. Scheerlinck, P
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nanoimprinted patterns are transfer
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emoved and the spectra changed dras
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Y-splitter W1 waveguide 60-degree b
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5. Photonic Wire Waveguides Figure
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Figure 5: Optimized designs and cor
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Figure 8: Optimized designs and cor
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[13] Borel P I, Frandsen L H, Harp
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1606 J. S. JENSEN To compute a freq
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1608 J. S. JENSEN Cramer’s rule [
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1610 J. S. JENSEN The following pro
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1612 J. S. JENSEN Table I. Coeffici
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1614 J. S. JENSEN h 2h Α fcos Ωt
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1616 J. S. JENSEN The derivative of
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1618 J. S. JENSEN which is solved f
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1620 J. S. JENSEN Based on the expr
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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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