Abstract
We consider shape optimization for objects illuminated by light. More precisely, we focus on time-harmonic solutions of the Maxwell system in curl-curl-form scattered by an arbitrary shaped rigid object. Given a class of cost functionals, including the scattered energy and the extinction cross section, we develop an adjoint-based shape optimization scheme which is then applied to two key applications.
Keywords
- Relative Permeability
- Shape Optimization
- Extinction Spectrum
- Perfectly Match Layer
- Illumination Direction
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
V. Akcelik, G. Biros, O. Ghattas, D. Keyes, K. Ko, L.-Q. Lee, E.G. Ng, Adjoint methods for electromagnetic shape optimization of the low-loss cavity for the international linear collider. J. Phys. Conf. Ser. 16, 435–445 (2005). ISSN 1742-6588. doi:10.1088/1742-6596/16/1/059
G. Allaire, Conception Optimale de Structures (Springer, Berlin, 2006)
L. Andersen, J. Volakis, Hierarchical tangential vector finite elements for tetrahedra. IEEE Microw. Guid. Wave Lett. 8(3), 127–129, (1998). ISSN 10518207.doi:10.1109/75.661137
A. Andryieuski, R. Malureanu, G. Biagi, T. Holmgaard, A. Lavrinenko, Compact dipole nanoantenna coupler to plasmonic slot waveguide. Opt. Lett. 37(6), 1124–1126 (2012). ISSN 1539-4794. doi:10.1364/OL.37.001124
E. Angerson, Z. Bai, J. Dongarra, A. Greenbaum, A. McKenney, J. Du Croz, S. Hammarling, J. Demmel, C. Bischof, D. Sorensen, LAPACK: a portable linear algebra library for high-performance computers, in Proceedings SUPERCOMPUTING’90 (IEEE Computer Society Press, 1990), pp. 2–11. ISBN 0-8186-2056-0. doi:10.1109/SUPERC.1990.129995
J.-P. Berenger, A perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys. 114(2), 185–200 (1994). ISSN 00219991. doi:10.1006/jcph.1994.1159
C. Bohren, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983)
J. Cagnol, M. Eller, Boundary regularity for Maxwell’s equations with applications to shape optimization. J. Differ. Equ. 250(2), 1114–1136 (2011). ISSN 00220396.doi:10.1016/j.jde.2010.08.004
M. Cessenat, Mathematical Methods in Electromagnetism: Linear Theory and Applications (World Scientific, Singapore, 1996)
M. Daimon, A. Masumura, Measurement of the refractive index of distilled water from the near-infrared region to the ultraviolet region. Appl. Opt. 46(18), 3811 (2007). ISSN 0003-6935.doi:10.1364/AO.46.003811
M. Delfour, Shapes and Geometries: Analysis, Differential Calculus, and Optimization (Society for Industrial and Applied Mathematics, Philadelphia, 2001)
J. Haslinger, Introduction to Shape Optimization: Theory, Approximation, and Computation (SIAM Society for Industrial and Applied Mathematics, Philadelphia, 2003)
M. Hintermüller, A. Laurain, I. Yousept, Shape sensitivities for an inverse problem in magnetic induction tomography based on the Eddy current model. uni-graz.at (2014)
P.B. Johnson, R.W. Christy, Optical constants of the noble metals. Phys. Rev. B 6(12), 4370–4379 (1972). ISSN 0556-2805. doi:10.1103/PhysRevB.6.4370
A. Kriesch, S.P. Burgos, D. Ploss, H. Pfeifer, H.A. Atwater, U. Peschel, Functional plasmonic nanocircuits with low insertion and propagation losses. Nano Lett. 13(9), 4539–45, 2013. ISSN 1530-6992. doi:10.1021/nl402580c
C.M. Lalau-Keraly, S. Bhargava, O.D. Miller, E. Yablonovitch, Adjoint shape optimization applied to electromagnetic design. Opt. Express 21(18), 21693–21701 (2013). ISSN 1094-4087. doi:10.1364/OE.21.021693
P. Li, An inverse cavity problem for Maxwell’s equations. J. Differ. Equ. 252(4), 3209–3225 (2012). ISSN 00220396. doi:10.1016/j.jde.2011.10.023
M. Mishchenko, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge University Press, Cambridge, 2002)
P. Monk, Finite Element Methods for Maxwell’s Equations (Clarendon Press, Oxford, 2003)
C. Müller, Foundations of the Mathematical Theory of Electromagnetic Waves (Springer, Berlin, 1969). ISBN 978-3-662-11775-0. doi:10.1007/978-3-662-11773-6
J. C. Nédélec, A new family of mixed finite elements in \(\mathbb{R}^3\). Numer. Math. 50(1), 57–81 (1986). ISSN 0029-599X. doi:10.1007/BF01389668
D.M. Nguyen, A. Evgrafov, J. Gravesen. Isogeometric shape optimization for electromagnetic scattering problems. Prog. Electromagn. Res. B. 45, 117–146 (2012). ISSN 1937-6472. doi:10.2528/PIERB12091308
T. Radhakrishnan. Further studies on the temperature variation of the refractive index of crystals. Proc. Indian Acad. Sci. Sect. A, 33(1), 22–34 (1951). ISSN 0370-0089.doi:10.1007/BF03172255
J. Schöberl, S. Zaglmayr, High order Nédélec elements with local complete sequence properties. COMPEL Int. J. Comput. Math. Electr. Electron. Eng. 24(2), 374–384 (2005). ISSN 0332-1649. doi:10.1108/03321640510586015
J. Schwinger, L.L.J. Deraad, K.A. Milton, Classical Electrodynamics (Westview Press, Boston, 1998). ISBN 0813346622
Science & Technology Facility Council. HSL, a collection of Fortran codes for large scale scientific computation (2013). www.hsl.rl.ac.uk
H. Si, TetGen: a quality tetrahedral mesh generator and 3D delaunay triangulator (2013). www.tetgen.org
J. Sokolowski, Introduction to Shape Optimization: Shape Sensitivity Analysis (Springer, Berlin, 1992). ISBN 9783540541776
J. Stratton, Electromagnetic Theory (McGraw-Hill Book Company Inc., New York, 1941). ISBN 9780070621503
The MathWorks Inc. MATLAB Release 2014a (2014). www.mathworks.com
P. Varga, P. Török, The Gaussian wave solution of Maxwell’s equations and the validity of scalar wave approximation. Opt. Commun. 152(1–3), 108–118 (1998). doi:10.1016/S0030-4018(98)00092-3
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Semmler, J., Pflug, L., Stingl, M., Leugering, G. (2015). Shape Optimization in Electromagnetic Applications. In: Pratelli, A., Leugering, G. (eds) New Trends in Shape Optimization. International Series of Numerical Mathematics, vol 166. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-17563-8_11
Download citation
DOI: https://doi.org/10.1007/978-3-319-17563-8_11
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-319-17562-1
Online ISBN: 978-3-319-17563-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)