Plane sets invisible in finitely many directions. (English) Zbl 1393.37049
Summary: We consider the problem of mirror invisibility for plane sets. Given a circle and a finite number of unit vectors (defining the directions of invisibility) such that the angles between them are commensurable with \(\pi\), for any \(\varepsilon >0\) there exists a set invisible in the chosen directions that contains the circle and is contained in its \(\varepsilon\)-neighborhood. This set is the disjoint union of infinitely many domains with piecewise smooth boundary.
MSC:
| 37D50 | Hyperbolic systems with singularities (billiards, etc.) (MSC2010) |
| 78A05 | Geometric optics |
References:
| [1] | Aleksenko, A.; Plakhov, A., Bodies of zero resistance and bodies invisible in one direction, Nonlinearity, 22, 1247-1258, (2009) · Zbl 1173.37035 · doi:10.1088/0951-7715/22/6/001 |
| [2] | Chen, X.; Luo, Y.; Zhang, J.; Jiang, K.; Pendry, J. B.; Zhang, S., Macroscopic invisibility cloaking of visible light, Nat. Commun., 2, 176, (2011) · doi:10.1038/ncomms1176 |
| [3] | Gharghi, M.; Gladden, C.; Zentgraf, T.; Liu, Y.; Yin, X.; Valentine, J.; Zhang, X., A carpet cloak for visible light, Nano Lett., 11, 2825-2828, (2011) · doi:10.1021/nl201189z |
| [4] | Ivrii, V. Y., The second term of the spectral asymptotics for a laplace-beltrami operator on manifolds with boundary, Func. Anal. Appl., 14, 98-106, (1980) · Zbl 0453.35068 · doi:10.1007/BF01086550 |
| [5] | Leonhardt, U., Optical conformal mapping, Science, 312, 1777-1780, (2006) · Zbl 1226.78001 · doi:10.1126/science.1126493 |
| [6] | Leonhardt, U., Notes on conformal invisibility devices, New J. Phys., 8, 118, (2006) · doi:10.1088/1367-2630/8/7/118 |
| [7] | Plakhov, A., Exterior Billiards. Systems with Impacts Outside Bounded Domains, (2012), New York: Springer, New York · Zbl 1257.37002 |
| [8] | Plakhov, A.; Roshchina, V., Invisibility in billiards, Nonlinearity, 24, 847-854, (2011) · Zbl 1215.37028 · doi:10.1088/0951-7715/24/3/007 |
| [9] | Plakhov, A.; Roshchina, V., Fractal bodies invisible in 2 and 3 directions, Discr. Contin. Dyn. A, 33, 1615-1631, (2013) · Zbl 1264.78005 · doi:10.3934/dcds.2013.33.1615 |
| [10] | Plakhov, A.; Roshchina, V., Bodies with mirror surface invisible from two points, Nonlinearity, 27, 1193-1203, (2014) · Zbl 1351.37162 · doi:10.1088/0951-7715/27/6/1193 |
| [11] | Plakhov, A.; Roshchina, V., Bodies invisible from one point |
| [12] | Pendry, J. B.; Schurig, D.; Smith, D. R., Controlling electromagnetic fields, Science, 312, 1780-1782, (2006) · Zbl 1226.78003 · doi:10.1126/science.1125907 |
| [13] | Schurig, D.; Mock, J. J.; Justice, B. J.; Cummer, S. A.; Pendry, J. B.; Starr, A. F.; Smith, D. R., Metamaterial electromagnetic cloak at microwave frequencies, Science, 314, 977-980, (2006) · doi:10.1126/science.1133628 |
| [14] | Zhang, B.; Luo, Y.; Liu, X.; Barbastathis, G., Macroscopic invisibility cloak for visible light, Phys. Rev. Lett., 106, (2011) · doi:10.1103/PhysRevLett.106.033901 |
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