WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

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Transmission 1.00 0.95 0.90 0.85 Structure 1 0.80 0.75 1480 1500 1520 1540 1560 Structure 2 Structure 3 Structure 4 1580 1600 1620 Wavelength (nm) Figure 7: Total energy transmitted through top and bottom output ports relative to the input wave energy for the 4 structures in Figures 5 and 6. the first two structures is basically kept but enlarged aft of the structure. As can been seen this allows for placing an extra reflector aft of the waveguide at a distance commensurable with the wavelength. By comparing the response for structure 4 and structure 2 in Figure 7, it is noticed that this difference of the otherwise quite similar structures has caused a considerable improvement of the 2D transmission. 5.4. T-splitter - TE polarization The second example is for TE-polarized waves. The basic system dimensions and material properties are taken from [3] in which propagation losses in straight waveguides with simple bends were studied experimentally. In this case w = 445nm and n = 3.5. The optimized structures in Figure 8 are both obtained by simulating dissipation in the air. Figure 8 (top) (structure 1) shows the optimized structure obtained by maximizing the transmission for a wavelength of 1.55µm. For TE waves this corresponds to a significantly shorter wavelength compared to the width of the waveguide than for the TM wave example. This difference is noticeable in the optimized structures which possess structural details at a corresponding smaller scale. It is noted that the transmission is almost 100% near the target frequency. The design shown in Figure 8 (bottom) (structure 2) was obtained by attempting to get a high transmission for a broader frequency range. This was done by modifying the objective in the optimization problem (Eq. (4)) to include the transmission at two distinct frequencies: max min Φ1(ω1), Φ1(ω2), Φ2(ω1), Φ2(ω2). (8) The two target frequencies ω1 and ω2 are not kept fixed during the optimization procedure, but are repeatedly updated (e.g. every 10-20 iterations) so that they correspond to the critical frequencies with minimum transmission. The critical frequencies are identifying by fast-frequency-sweeps using Padé approximants [6]. Naturally, the number of frequencies in Eq. (8) can be increased which is necessary if a larger frequency range is considered. Figure 9 shows a comparison of the performance of the two designs. As noted the bandwidth of structure 2 is slightly better than that of structure 1, but this has been obtained at the expense of the high transmission near 1.55µm. Including more frequencies in the optimization does not change this significantly. 6. Conclusions We have demonstrated the use of topology optimization to design low-loss photonic circuit components such as bends and splitters. Previously it has been shown that such components in photonic crystal waveguides can effectively be designed with topology optimization, leading to low losses in large frequency ranges. In this paper we have demonstrated design of similar low loss components also in photonic wire waveguides. These 7
Figure 8: Optimized designs and corresponding wave patterns for TE-polarized waves. Top: structure 1 obtained by maximizing transmission for a single frequency and bottom: structure 2 obtained by maximizing transmission for two distinct frequencies that are repeatedly updated during the optimization procedure. Transmission 1.00 0.95 0.90 0.85 0.80 0.75 Structure 1 Structure 2 1350 1400 1450 1500 1550 1600 1650 1700 1750 Wavelength (nm) Figure 9: Transmission for the two optimized structures in Figure 8. 8
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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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Page 207 and 208: Appl. Phys. Lett., Vol. 84, No. 12,
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Page 218 and 219: Topology optimization and fabricati
Page 220 and 221: Normalized transmission 1.0 0.8 0.6
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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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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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