In this paper, a generalization of cloaking is presented: instead of an empty
region of space, an inhomogeneous structure is transformed via Pendry's map in
order to give, to any object hidden in the central hole of the cloak, a
completely arbitrary appearance.Comment: 14 pages, 4 figure

Geuzaine, C.

Nicolet, A.

Zolla, F.

English

arXiv

In this paper, a generalization of cloaking is presented: instead of an empty region of space, an inhomogeneous structure is transformed via Pendry's map in order to give, to any object hidden in the central hole of the cloak, a completely arbitrary appearance

Nicolet, André

Zolla, Frederic

Geuzaine, Christophe

HAL AMU

Generalized Cloaking and Optical PolyjuiceAndre´ Nicolet, Frederic Zolla, Christophe GeuzaineTo cite this version:Andre´ Nicolet, Frederic Zolla, Christophe Geuzaine. Generalized Cloaking and OpticalPolyjuice. 2009. <hal-00413492>HAL Id: hal-00413492https://hal.archives-ouvertes.fr/hal-00413492Submitted on 4 Sep 2009HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.Generalized Cloaking and Optical PolyjuiceAndre´ Nicolet,1,∗ Fre´de´ric Zolla,1 and Christophe Geuzaine21Institut Fresnel UMR CNRS 6133, Aix-Marseille Universite´, Ecole Centrale deMarseille, Campus de Saint-Je´roˆme, F-13013 Marseille, France2University of Lie`ge, Dept. of Electrical Engineering and Computer Science (ACE),B-4000 Lie`ge, Belgium∗Corresponding author: andre.nicolet@fresnel.frIn this paper, a generalization of cloaking is presented: instead of anempty region of space, an inhomogeneous structure is transformed viaPendry’s map in order to give, to any object hidden in the central hole of thecloak, a completely arbitrary appearance. c© 2009 Optical Society of AmericaOCIS codes: 000.4895, 260.2110In 2006, it was suggested by Pendry et al. [1] that an object surrounded by a coat-ing consisting of an exotic material becomes invisible to electromagnetic waves. Thisdevice was named “invisibility cloak” in reference to Harry Potter, the popular char-acter of J.K. Rowling. Beside his famous cloak, the little wizard has other spells togo unnoticed. Among the most spectacular is the “polyjuice potion” that is able toturn somebody into anybody else’s appearance [2]. In this paper, we do not present1a potion but rather an optical device able to accomplish the same task, i.e. to givean arbitrary optical response chosen in advance to any other object placed inside thedevice. In fact, the principle is here very similar to the design of Pendry’s invisibilitycloak but, instead of geometrically transforming an empty domain, it is a region con-taining the object to be imitated that is transformed leading thus to a generalizationof cloaking.In this section, we present a generalization of cloaking able to arbitrarily transformthe electromagnetic appearance of an object. The basic principle is to obtain theconstitutive relations of the cloak by application of a space transformation to a non-empty region.A geometric transformation is given by a map ϕ from a space N to a space M . Forall our practical purposes, M and N will be here the whole or parts of R3. Given aCartesian coordinate system x on M and an arbitrary coordinate system x′ on N , ϕ :N →M is described by x(x′), i.e. x given as function of x′. All the useful information,i.e. necessary to transform differential forms (indeed the formalism of differentialgeometry is the most natural to write the Maxwell’s equation [3–6] specially whenquite general geometric transformations are involved) and other covariant tensors, iscontained in the Jacobian matrix field J(x′) = ∂x(x′)/∂x′.The basic principle of transformation optics is that, when you have an electromag-netic system described by the tensor fields ε(x) for the dielectric permittivity and2µ(x) for the magnetic permeability in the space M , if you replace your initial mate-rial properties by equivalent material properties given by the following rule [7–10]:ε′(x′) = J−1(x′)ε(x(x′))J−T (x′) det(J(x′)),µ′(x′) = J−1(x′)µ(x(x′))J−T (x′) det(J(x′)), (1)(J−T is the transposed of the inverse of J), you get an equivalent problem on N .Here, an equivalent problem means that the solution of the new problem on N , i.e.electromagnetic quantities described as differential forms, are the pulled back of thesolution [5] of the original problem on M and that the same Maxwell’s equations(i.e. as if we were in Cartesian coordinates or, more accurately, having the same formwritten with the exterior derivative) are still satisfied.In the case of the cylindrical Pendry’s map [1,11,12], described by the transforma-tion of the 2D cross section, the plane R2 minus a disk D1 of radius R1 is mappedon the whole plane R2 in such a way that a disk D2 of radius R2 > R1, concentricwith D1, is the image of the annulus D2\D1 by a radial transformation. In cylindricalcoordinates, this transformation is given by:r = (r′ −R1)R2/(R2 −R1) for R1 ≤ r′ ≤ R2,θ = θ′, z = z′.(2)As for the outside of the disk D2, the map between the two copies of R2\D2 is theidentity map.3The material properties given by rule (1) corresponding to this transformationprovide an ideal invisibility cloak: outside D2, everything behaves as if we were infree space, including the propagation of electromagnetic waves across the cloak, andis completely independent of the content of D1.Now, rule (1) may be applied to D2 containing objects with arbitrary electromag-netic properties so that a region cloaked by this device is still completely hidden buthas the appearance of the objects originally in D2. We may call this optical effectmasking [13] or “polyjuice” effect.Figs. 2 and 3 show the effect of masking on a scattering structure. On Fig. 2,a cylindrical TM wave emitted by a circular cylindrical antenna is scattered by aconducting triangular cylinder (the longest side of the cross section is 1.62λ andεr = 1 + 40i). The field map represents the longitudinal electric field Ez(x, y) andthe outer boundary of the cloak is shown to ease the comparison with the maskedcase. On Fig. 3, the same cylindrical TM wave is scattered by a masked triangularcylinder (but the diffracting object inside the cloak may be arbitrary as far as it issmall enough to fit inside the cloak). This triangular cylinder is the symmetric of theprevious one with respect to the horizontal plane containing the central fibre of thecylindrical antenna. This bare scatterer would therefore give the Fig. 2 image invertedupside-down but, here, this object is surrounded by a cloak in order to give the verysame scattering as before. Indeed, on both sides, the electric fields outside the cloaklimit are alike.4Fig. 4 highlights the different scattering patterns by displaying the value of <e(Ez)on a circle of radius 4λ located around the antenna-scatterer system in the threefollowing cases: the case of Fig. 2 (original) with the triangle alone, the case of Fig. 3(coated), and the triangle of Fig. 3 without the coating (reversed). It is obvious thatthe coating restores the field distribution independently of the object present in thecentral hole.The numerical computation is performed using the finite element method (via thefree GetDP [14] and Gmsh [15] software tools). The mesh is made of 148,000 secondorder triangles including the Perfectly Matched Layers used to truncate the compu-tation domain . The singularity of ε and µ requires a very fine mesh in the vicinity ofthe inner boundary of the cloak (see Fig. 1) and is also responsible for the small dis-crepancies between the numerical model and a perfect cloak (see Fig. 3) — includingthe non zero field in the hole of the cloak.Note that a small technical problem arises in practice when rule (1) is applied:the material properties are defined piecewise on various domains and it is very usefulto know explicitly the boundaries of these domains, e.g. to build the finite elementmesh (see Fig. 1). These boundaries are curves in the cross section and are thuscontravariant objects. Therefore, their transformation requires the inverse map ϕ−1from M to N . Fortunately, map (2) is very simple to invert.More explicitly, for a given curve x(t) of parameter t in the initial Cartesian coor-5dinates, its push forward by Pendry’s map is:x′(t) = ϕ−1(x(t)) = (R2 −R1R2+R1‖x(t)‖)x(t), (3)with the same variation of the parameter t. Note that the most common curves usedin the design of devices, i.e. line segments and arc of circles, are transformed to lessusual curves except for radial segments (with respect to the center of the cloak) andarc of circles concentric with the cloak.On Fig. 3, the image by ϕ−1 of the triangle of Fig. 2 is the curvilinear triangleinside the coating region of the cloak. In practice, this anamorphosis of the triangleis described by three splines interpolating each 40 points that are images of points ofthe segments by ϕ−1.Transformation optics do not offer only the possibility make optically disappearobjects in invisibility cloaks but also to completely tune their optical signature i.e.to give them an arbitrary appearance. The possibility to place an object inside thecoating of the cloak and that it will therefore appear different was already consideredin [11] where a point source was shifted creating a mirage effect. A more generalcase is considered here since both the shape and the position of the object placedin the coating are modified . Note that if the object used to create the illusion isperfectly conducting and surrounds the central point of the cloak, its anamorphosiswill surround the inner boundary of the cloak and it de facto suppresses the singularbehavior of the material properties. In this particular case, the present approach is6similar to the the hiding under the carpet idea of Li and Pendry [16] but here abounded object replaces the infinite reflecting plane. This technique can be naturallyextended to cloaks of arbitrary shapes [17].References1. J.B. Pendry, D. Shurig, D.R. Smith, ”Controlling electromagnetic fields”, Science312, 1780 (2006).2. J.K. Rowling, Harry Potter and the Chamber of Secrets, (Bloomsbury PublishingPLC, 1998).3. G. A. Deschamps,”Electromagnetics and differential forms”, Proc. IEEE, 69, 676(1981).4. W. L. Burke, Applied Differential Geometry, (Cambridge University Press, 1985).5. A. Nicolet, J. F. Remacle, B. Meys, A. Genon, W. Legros, ”Transformation meth-ods in computational electromagnetism”, J. Appl. Phys. 75, 6036 (1994).6. A. Bossavit, J. Japan Soc. Appl. Electromagn. & Mech., 6, ”On the geometryof electromagnetism. (1): Euclidean space”,17, ”On the geometry of electromag-netism. (2): Geometrical objects”,114, ”On the geometry of electromagnetism.(3): Faraday’s law”,233, ”On the geometry of electromagnetism. (4): ’Maxwell’shouse’ ”,318 (1998), 7, ”Computational electromagnetism and geometry: Build-ing a finitedimensional ’Maxwell’s house’. (1): Network equations”,150, ”Com-putational electromagnetism and geometry. (2): Network constitutive laws”,294,7”Computational electromagnetism and geometry. (3): Convergence”,401 (1999),8, ”Computational electromagnetism and geometry. (4): From degrees of free-dom to fields”,102, ”Computational electromagnetism and geometry. (5): The’Galerkin hodge’”,203, ”Computational electromagnetism and geometry. (6):Some questions and answers”, 372 (2000).7. F. Zolla, G. Renversez, A. Nicolet, B. Kuhlmey, S. Guenneau, D. Felbacq, Foun-dations of Photonic Crystal Fibres, (Imperial College Press, 2005).8. G. W. Milton, M. Briane, J. R. Willis, ”On cloaking for elasticity and physicalequations with a transformation invariant form”, New J. of Phys. 8, 248 (2006).9. A. Nicolet, F. Zolla, S. Guenneau, ”Modelling of twisted optical waveguides withedge elements”, Eur. Phys. J. - Appl. Phys. 28, 153 (2004).10. A. Nicolet, F. Zolla, Y. Ould Agha, S. Guenneau, ”Leaky modes in twisted mi-crostructured optical fibers”, Waves in Random and Complex Media 17, 559(2007).11. F. Zolla, S. Guenneau, A. Nicolet, J. B. Pendry, ”Electromagnetic analysis ofcylindrical invisibility cloaks and the mirage effect”, Opt. Lett. 32, 1069 (2007).12. A. Nicolet, F. Zolla, Y. Ould Agha, S. Guenneau, ”On the use of PML for thecomputation of leaky modes : an application to gradient index MOF”, COMPEL27, 806 (2008).13. F. L. Teixeira, ”Differential form approach to the analysis of electromagnetic8cloaking and masking”, Microwave Opt. Technol. Lett. 49, 2051 (2007).14. P. Dular, C. Geuzaine, F. Henrotte, W. Legros, ”A general environment for thetreatment of discrete problems and its application to the finite element method”,IEEE Trans. Mag. 34, 3395 (1998). (see also http://www.geuz.org/getdp/)15. C. Geuzaine, J.-F. Remacle, ”Gmsh: a three-dimensional finite element mesh gen-erator with built-in pre- and post-processing facilities”, Int. J. Numer. MethodsEng. 79, 1309 (2009).16. J.S. Li, J.B. Pendry, ”Hiding under the Carpet: A New Strategy for Cloaking”,Phys. Rev. Lett. 101, 203901 (2008)17. A. Nicolet, F. Zolla, S. Guenneau, ”Electromagnetic analysis of cylindrical cloaksof an arbitrary cross section”, Opt. Lett. 33, 1584 (2008).9List of Figures1 This figure shows a part of the triangular mesh used for the finiteelement modeling of the scattering problem of Fig. 3. The singularbehavior of the permittivity and of the permeability requires a veryfine mesh along the inner boundary of the cloak in order to achieve asatisfactory accuracy with the numerical model. . . . . . . . . . . . . 112 A conducting triangular cylinder is scattering cylindrical waves. . . . 123 A triangular cylinder different from the one on Fig. 2 is surroundedby a cloak designed to reproduce the scattering pattern of the Fig.2 triangular cylinder in spite of the change of diffracting object. Ofcourse, the diffracting object inside the cloak may be arbitrary as faras it is small enough to fit inside the cloak . . . . . . . . . . . . . . . 134 The value of the electric field (the real part of Ez) on a circle of radius4λ concentric with the cloak is represented as a function of the positionangle θ (increasing counterclockwise and with θ = 0 corresponding tothe point the most on the right). The three configurations consideredhere are the ones of Fig. 3 (coated), Fig. 2 (original), and the triangleof Fig. 3 without the coating (reversed). . . . . . . . . . . . . . . . . 1410Fig. 1. This figure shows a part of the triangular mesh used for the finiteelement modeling of the scattering problem of Fig. 3. The singular behaviorof the permittivity and of the permeability requires a very fine mesh along theinner boundary of the cloak in order to achieve a satisfactory accuracy withthe numerical model.11Fig. 2. A conducting triangular cylinder is scattering cylindrical waves.12Fig. 3. A triangular cylinder different from the one on Fig. 2 is surrounded bya cloak designed to reproduce the scattering pattern of the Fig. 2 triangularcylinder in spite of the change of diffracting object. Of course, the diffractingobject inside the cloak may be arbitrary as far as it is small enough to fit insidethe cloak131 2 3 4 5 6-0.2-0.10.10.2reversedoriginalcoatedReHEzLHVmL vs. ΘHradLFig. 4. The value of the electric field (the real part of Ez) on a circle of radius4λ concentric with the cloak is represented as a function of the position angleθ (increasing counterclockwise and with θ = 0 corresponding to the point themost on the right). The three configurations considered here are the ones ofFig. 3 (coated), Fig. 2 (original), and the triangle of Fig. 3 without the coating(reversed).14

Generalized Cloaking and Optical Polyjuice

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Generalized Cloaking and Optical Polyjuice

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