Heider, Pascal; Berebichez, D.; Kohn, Robert V.; Weinstein, M. I. Optimization of scattering resonances. (English) Zbl 1273.74045 Struct. Multidiscip. Optim. 36, No. 5, 443-456 (2008). Summary: The increasing use of micro- and nano-scale components in optical, electrical, and mechanical systems makes the understanding of loss mechanisms and their quantification issues of fundamental importance. In many situations, performance-limiting loss is due to scattering and radiation of waves into the surrounding structure. In this paper, we study the problem of systematically improving a structure by altering its design so as to decrease the loss. We use sensitivity analysis and local gradient optimization, applied to the scattering resonance problem, to reduce the loss within the class of piecewise constant structures. For a class of optimization problems where the material parameters are constrained by upper and lower bounds, it is observed that an optimal structure is piecewise constant with values achieving the bounds. Cited in 10 Documents MSC: 74E15 Crystalline structure 78A45 Diffraction, scattering 74M25 Micromechanics of solids 49K40 Sensitivity, stability, well-posedness 35L20 Initial-boundary value problems for second-order hyperbolic equations 49K20 Optimality conditions for problems involving partial differential equations PDFBibTeX XMLCite \textit{P. Heider} et al., Struct. Multidiscip. Optim. 36, No. 5, 443--456 (2008; Zbl 1273.74045) Full Text: DOI References: [1] Burger M, Osher S, Yablonovitch E (2004) Inverse problem techniques for the design of photonic crystals. IEICE Trans Electron 87:258–265 [2] Cox SJ, Dobson DC (1999) Maximizing band gaps in two-dimensional photonic crystals. SIAM J Appl Math 59:2108–2120 · Zbl 1027.78521 · doi:10.1137/S0036139998338455 [3] Cuccagna S (2007) Dispersion for Schrödinger equation with periodic potential in 1D. arxiv:math/0611919v1 [math.AP] [4] Dobson DC, Santosa F (2004) Optimal localization of eigenfunctions in an inhomogeneous medium. SIAM J Appl Math 64:762–774 · Zbl 1060.65068 · doi:10.1137/S0036139903426162 [5] Figotin A, Klein A (1998) Midgap defect modes in dielectric and acoustic media. SIAM J Appl Math 58:1748–1773 · Zbl 0963.78005 · doi:10.1137/S0036139997320536 [6] Geremia JM, Williams J, Mabuchi H (2002) Inverse-problem approach to designing phononic crystals for cavity QED experiments. Phys Rev E 66:066606 [7] Joannopoulos JD, Meade RD, Winn JN (1995) Photonic crystal: molding the flow of light. Princeton Univ. Press, Princeton, NJ · Zbl 1035.78500 [8] Kao C-Y, Santosa F (2007) Maximization of the quality factor of an optical resonator. doi: 10.1016/j.wavemoti.2007.07.012 · Zbl 1231.78007 [9] Kao CY, Osher S, Yablonovich E (2005) Maximizing band gaps in two-dimensional photonic crystals by using level set methods. Appl Phys B Lasers Optics 81:235–244 · doi:10.1007/s00340-005-1877-3 [10] Korotyaev E (1997) The propagation of waves in periodic media at large time. Asymptot Anal 15:1–24 · Zbl 0951.34054 [11] Lipton RP, Shipman SP, Venakides S (2003) Optimization of resonances of photonic crystal slabs. In: Proceedings SPIE, vol 5184, pp 168–177 [12] Osher S, Santosa F (2001) Level set methods for optimization problems involving geometry and constraints–I. Frequencies of two-density inhomogeneous drum. J Comp Phys 171:272–288 · Zbl 1056.74061 · doi:10.1006/jcph.2001.6789 [13] Pironneau O (1984) Optimal shape design for elliptic systems. Springer · Zbl 0534.49001 [14] Ramdani K, Shipman S (2007) Transmission through a thick periodic slab. Math Models Methods Appl Sci · Zbl 1171.35317 [15] Sigmund O, Jensen JS (2003) Systematic design of phononic band-gap materials and structures by topology optimization. Phil Trans R Soc London A 361:1001–1019 · Zbl 1067.74053 · doi:10.1098/rsta.2003.1177 [16] Tang S-H, Zworski M (2000) Resonance expansions of scattered waves. Commun Pure Appl Math 53:1305–1334 · Zbl 1032.35148 · doi:10.1002/1097-0312(200010)53:10<1305::AID-CPA4>3.0.CO;2-# This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.