Maria Bayard Dühring - Solid Mechanics

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performance will decrease and some details, as the layout of the top part, are crucial. 4 Conclusion A method to optimize the energy flow trough the core of a holey photonic-crystal fiber based on topology optimization has been presented. As the cladding material is lossy, the aim is to design the cross section of the fiber such that the mode shape is overlapping the cladding as little as possible. The optical model is based on the time-harmonic wave equation for the magnetic field. Matching values for the propagation constant and the optical wavelength are employed in order to solve the wave equation in a time-harmonic way where the wave is exited by a magnetic force in the center of the fiber. The topology optimization is based on linear interpolation functions between the refractive indices and the objective function is the sum of the squared absolute value of the magnetic fields in the plane. A close type morphology filter and a continuation method based on the damping are applied to obtain appropriate designs. The performance of the method is illustrated by an example where a 0-1 design is obtained, which is symmetric about a vertical axis in contrast to the 60 degree symmetry of the initial design. The objective function is increased 375% because the optimized design changes the distribution of the magnetic field such that its overlap with the lossy cladding material is decreased. It is shown, that by maximizing the objective function the time-average power flow in the propagation direction is increased 378% times, and its distribution in the initial and the optimized design is similar to the distribution of the energy measure form the objective function. Thus, the same improvement is obtained for power flow and the objective function. It is furthermore shown that simplified designs with the main features from the optimized design can be created, which are simpler to fabricate. The improvement is decreasing when the simplified designs deviate more from the optimized design. Further work encompasses the inclusion of a symmetry constraint around the vertical axis and a study of other types of filters to further eliminate small details and the gray transition zones at the air/cladding interfaces. Methods to optimize for tolerant designs are relevant to apply, see [32], as the drawing process tends to smooth out the details in the design. Optimization of wavelength intervals to obtain broadband solutions can be performed by the method described in [18]. Finally, the method can be extended to other types of photonic-crystal fibers, as fibers where the light is confined by index guiding, and to other objective functions where dispersion properties and nonlinear effects are optimized. 5 Acknowledgements This work received support from the Eurohorcs/ESF European Young Investigator Award (EURYI, www.esf.org/euryi) through the grant ”Synthesis and topology optimization of optomechanical systems” and from the Danish Center for Scientific Computing (DCSC). Also support from the EU Network of Excellence ePIXnet is gratefully acknowledged. The authors are thankful to Jakob S. Jensen from the Department of Mechanical Engineering, Technical University of Denmark, for helpful discussions related to the presented work. References [1] E. Yablonovitch, Inhibited spontaneous emission in solid-state physics and electronics, Phys. Rev. Lett., 58, 20, 2059-2062, 1987. [2] S. John, Strong localization of photons in certain disordered dielectric superlattices, Phys. Rev. Lett., 58, 23, 2486-2489, 1987. [3] T.F. Krauss, R.M. De La Rue and S. Brand, Two-dimensional photonic-bandgap structures operating at near-infrared wavelenghts, Nature, 383, 699-702, 1996. 10
[4] J.D. Joannopoulos, S.G. Johnson, J.N. Winn and R.D. Meade, Photonic crystals, molding the flow of light, 2nd edition, Princeton University Press, 2008. [5] T.A. Birks, P.J. Roberts, P.St.J. Russell, D.M. Atkin and T.J. Shepherd, Full 2-D photonic bandgaps in silica/air structures, Electron. Lett., 31, 22, 1941-1943, 1995. [6] S.E. Barkou, J. Broeng and A. Bjarklev, Silica-air photonic crystal fiber design that permits waveguiding by a true photonic bandgap effect, Opt. Lett., 24, 1, 46-48, 1999. [7] R. Syms and J. Cozens, Optical guided waves and devices, 1st ed., McGraw-Hill, 1992. [8] P. Russell, Photonic crystal fibers, Science, 299, 5605, 358-362, 2003. [9] D. Torres, O. Weisberg, G. Shapira, C. Anastassiou, B. Temelkuran, M. Shurgalin, S.A. Jacobs, R.U. Ahmad, T. Wang, U. Kolodny, S.M. Shapshay, Z. Wang, A.K. Devaiah, U.D. Upadhyay and J.A. Koufman, OmniGuide photonic bandgap fibers for flexible delivery of CO2 laser energy for laryngeal and airway surgery, Proc. of SPIE, 5686, 310-321, 2005. [10] H.K. Kim, J. Shin, S. Fan, M.J.F. Digonnet and G.S. Kino, Designing air-core photonic-bandgap fibers free of surface modes, IEEE J. Quantum Electron., 40, 5, 551-556, 2004. [11] P.J. Roberts, F. Couney, H. Sabert, B.J. Mangan, D.P. Williams, L. Farr, M.W. Mason, A. Tomlinson, T.A. Birks, J.C. Knight and P.St.J. Russell, Ultimate low loss of hollow-core photonic crystal fibres, Opt. Exp., 13, 1, 236-244, 2005. [12] J. Hu and C.R. Menyuk, Use of fingers in the core to reduce leakage loss in air-core photonic bandgap fibers, Proc. of OFC/NFOEC, 2007. [13] R. Amezcua-Correa, N.G.R. Broderick, M.N. Petrovich, F. Poletti and D.J. Richardson, Design of 7 and 19 cells core air-guiding photonic crystal fibers for low-loss, wide bandwidth and dispersion controlled operation, Opt. Exp., 15, 26, 17577-17586, 2007. [14] T. Murao, K. Saitoh and M. Koshiba, Structural optimization of air-guiding photonic bandgap fibers for realizing ultimate low loss waveguides, J. Lightwave tech., 26, 12, 1602-1612, 2008. [15] M. P. Bendsøe and N. Kikuchi, Generating optimal topologies in structural design using a homogenization method, Comp. Meth. Appl. Mech. Engrg., 71, 197-224, 1988. [16] M.P. Bendsøe and O. Sigmund, Topology optimization, theory, methods and applications, Springer, Berlin, 2003. [17] S.J. Cox and D.C. Dobson, Maximizing band gaps in two-dimensional photonic crystals, SIAM J. Appl. Math., 59, 6, 2108-2120, 1999. [18] 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 22, 1191-1198, 2005. [19] Y. Tsuji, K. Hirayama, T. Nomura, K. Sato, and S. Nishiwaki, Design of optical circuit devices based on topology optimization, IEEE photonics Tehnol. Lett., 18, 7, 850-852, 2006. [20] 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. Exp., 12, 9, 1996- 2001, 2004. [21] P.I. Borel, L.H. Frandsen, A. Harpøth, M. Kristensen, J.S. Jensen and O. Sigmund, Topology optimised broadband photonic crystal Y-splitter, Electron. Lett., 41, 2, 69-71, 2005. [22] R. Stainko and O. Sigmund, Tailoring dispersion properties of photonic crystal waveguuides by topology optimization, Waves in Random and Complex Media, 17, 4, 477-489, 2007. 11
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Optimization of acoustic, optical a
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Resumé (in Danish) Forskningsomr˚
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Publications The following publicat
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vi Contents 6 Design of acousto-opt
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2 Chapter 1 Introduction waves. The
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12 Chapter 3 Topology optimization
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Suzuki, A. Onoe, T. Adachi and K. F
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Maria Bayard Dühring - Solid Mechanics

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