Abstract

We derive the expression for the anisotropic heterogeneous matrices of permittivity and permeability associated with two-dimensional polygonal and star shaped cloaks. We numerically show using finite elements that the forward scattering worsens when we increase the number of sides in the latter cloaks, whereas it improves for the former ones. This antagonistic behavior is discussed using a rigorous asymptotic approach. We use a symmetry group theoretical approach to derive the cloaks design.

© 2009 Optical Society of America

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References

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  1. J. B. Pendry, D. Shurig, and D. R. Smith, "Controlling electromagnetic fields," Science 312, 1780-1782 (2006).
    [CrossRef] [PubMed]
  2. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial electromagnetic cloak at microwave frequencies," Science 314, 977-980 (2006).
    [CrossRef] [PubMed]
  3. A. Greenleaf, M. Lassas and G. Uhlmann, "On nonuniqueness for Calderons inverse problem," Math. Res. Lett. 10, 685-693 (2003).
  4. R. C. McPhedran, N. A. Nicorovici, and G. W. Milton,"Optical and dielectric properties of partially resonant composites," Phys. Rev. B 49, 8479-8482 (1994).
    [CrossRef]
  5. A. Nicolet, F. Zolla, S. Guenneau, "Electromagnetic analysis of cylindrical cloaks of an arbitrary cross section," Opt. Lett. 33, 1584-1586 (2008).
    [CrossRef] [PubMed]
  6. A. J. Ward and J. B. Pendry, "Refraction and geometry in Maxwell’s equations," J.Mod. Opt. 43, 773-793 (1996).
    [CrossRef]
  7. A. Nicolet, J. F. Remacle, B. Meys, A. Genon, and W. Legros, "Transformation methods in computational electromagnetism," J. Appl. Phys. 75, 6036-6038 (1994).
    [CrossRef]
  8. F. Zolla, S. Guenneau, A. Nicolet and J. B. Pendry, "Electromagnetic analysis of cylindrical invisibility cloaks and the mirage effect," Opt. Lett. 32, 1069-1071 (2007).
    [CrossRef] [PubMed]
  9. U. Leonhardt and T. G. Philbin, "General relativity in electrical engineering," New J. Phys. 8, 247 (2006).
    [CrossRef]
  10. P. H. Tichit, B. Kante, and A. de Lustrac, "Design of polygonal and elliptical cloaks," Proceedings of Nato ARW and META08, Marrakesh-Morocco,  7-10May (2008), 120-125.
  11. J. Zhang, Y. Luo, H. Chen, and B. Wu, "Cloak of arbitrary shape," J. Opt. Soc. Am. B 25, 1776-1779 (2008)
    [CrossRef]
  12. C. Li and F. Li, "Two-dimensional electromagnetic cloaks with arbitrary geometries," Opt. Express 16, 13414-13420 (2008)
    [CrossRef] [PubMed]
  13. M. Farhat, S. Guenneau, A. B. Movchan, and S. Enoch, "Achieving invisibility over a finite range of frequencies," Opt. Express 16, 5656-5661 (2008)
    [CrossRef] [PubMed]

2008 (5)

2007 (1)

2006 (3)

U. Leonhardt and T. G. Philbin, "General relativity in electrical engineering," New J. Phys. 8, 247 (2006).
[CrossRef]

J. B. Pendry, D. Shurig, and D. R. Smith, "Controlling electromagnetic fields," Science 312, 1780-1782 (2006).
[CrossRef] [PubMed]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial electromagnetic cloak at microwave frequencies," Science 314, 977-980 (2006).
[CrossRef] [PubMed]

2003 (1)

A. Greenleaf, M. Lassas and G. Uhlmann, "On nonuniqueness for Calderons inverse problem," Math. Res. Lett. 10, 685-693 (2003).

1996 (1)

A. J. Ward and J. B. Pendry, "Refraction and geometry in Maxwell’s equations," J.Mod. Opt. 43, 773-793 (1996).
[CrossRef]

1994 (2)

A. Nicolet, J. F. Remacle, B. Meys, A. Genon, and W. Legros, "Transformation methods in computational electromagnetism," J. Appl. Phys. 75, 6036-6038 (1994).
[CrossRef]

R. C. McPhedran, N. A. Nicorovici, and G. W. Milton,"Optical and dielectric properties of partially resonant composites," Phys. Rev. B 49, 8479-8482 (1994).
[CrossRef]

Chen, H.

Cummer, S. A.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial electromagnetic cloak at microwave frequencies," Science 314, 977-980 (2006).
[CrossRef] [PubMed]

de Lustrac, A.

P. H. Tichit, B. Kante, and A. de Lustrac, "Design of polygonal and elliptical cloaks," Proceedings of Nato ARW and META08, Marrakesh-Morocco,  7-10May (2008), 120-125.

Enoch, S.

Farhat, M.

Genon, A.

A. Nicolet, J. F. Remacle, B. Meys, A. Genon, and W. Legros, "Transformation methods in computational electromagnetism," J. Appl. Phys. 75, 6036-6038 (1994).
[CrossRef]

Greenleaf, A.

A. Greenleaf, M. Lassas and G. Uhlmann, "On nonuniqueness for Calderons inverse problem," Math. Res. Lett. 10, 685-693 (2003).

Guenneau, S.

Justice, B. J.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial electromagnetic cloak at microwave frequencies," Science 314, 977-980 (2006).
[CrossRef] [PubMed]

Kante, B.

P. H. Tichit, B. Kante, and A. de Lustrac, "Design of polygonal and elliptical cloaks," Proceedings of Nato ARW and META08, Marrakesh-Morocco,  7-10May (2008), 120-125.

Lassas, M.

A. Greenleaf, M. Lassas and G. Uhlmann, "On nonuniqueness for Calderons inverse problem," Math. Res. Lett. 10, 685-693 (2003).

Legros, W.

A. Nicolet, J. F. Remacle, B. Meys, A. Genon, and W. Legros, "Transformation methods in computational electromagnetism," J. Appl. Phys. 75, 6036-6038 (1994).
[CrossRef]

Leonhardt, U.

U. Leonhardt and T. G. Philbin, "General relativity in electrical engineering," New J. Phys. 8, 247 (2006).
[CrossRef]

Li, C.

Li, F.

Luo, Y.

McPhedran, R. C.

R. C. McPhedran, N. A. Nicorovici, and G. W. Milton,"Optical and dielectric properties of partially resonant composites," Phys. Rev. B 49, 8479-8482 (1994).
[CrossRef]

Meys, B.

A. Nicolet, J. F. Remacle, B. Meys, A. Genon, and W. Legros, "Transformation methods in computational electromagnetism," J. Appl. Phys. 75, 6036-6038 (1994).
[CrossRef]

Milton, G. W.

R. C. McPhedran, N. A. Nicorovici, and G. W. Milton,"Optical and dielectric properties of partially resonant composites," Phys. Rev. B 49, 8479-8482 (1994).
[CrossRef]

Mock, J. J.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial electromagnetic cloak at microwave frequencies," Science 314, 977-980 (2006).
[CrossRef] [PubMed]

Movchan, A. B.

Nicolet, A.

Nicorovici, N. A.

R. C. McPhedran, N. A. Nicorovici, and G. W. Milton,"Optical and dielectric properties of partially resonant composites," Phys. Rev. B 49, 8479-8482 (1994).
[CrossRef]

Pendry, J. B.

F. Zolla, S. Guenneau, A. Nicolet and J. B. Pendry, "Electromagnetic analysis of cylindrical invisibility cloaks and the mirage effect," Opt. Lett. 32, 1069-1071 (2007).
[CrossRef] [PubMed]

J. B. Pendry, D. Shurig, and D. R. Smith, "Controlling electromagnetic fields," Science 312, 1780-1782 (2006).
[CrossRef] [PubMed]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial electromagnetic cloak at microwave frequencies," Science 314, 977-980 (2006).
[CrossRef] [PubMed]

A. J. Ward and J. B. Pendry, "Refraction and geometry in Maxwell’s equations," J.Mod. Opt. 43, 773-793 (1996).
[CrossRef]

Philbin, T. G.

U. Leonhardt and T. G. Philbin, "General relativity in electrical engineering," New J. Phys. 8, 247 (2006).
[CrossRef]

Remacle, J. F.

A. Nicolet, J. F. Remacle, B. Meys, A. Genon, and W. Legros, "Transformation methods in computational electromagnetism," J. Appl. Phys. 75, 6036-6038 (1994).
[CrossRef]

Schurig, D.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial electromagnetic cloak at microwave frequencies," Science 314, 977-980 (2006).
[CrossRef] [PubMed]

Shurig, D.

J. B. Pendry, D. Shurig, and D. R. Smith, "Controlling electromagnetic fields," Science 312, 1780-1782 (2006).
[CrossRef] [PubMed]

Smith, D. R.

J. B. Pendry, D. Shurig, and D. R. Smith, "Controlling electromagnetic fields," Science 312, 1780-1782 (2006).
[CrossRef] [PubMed]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial electromagnetic cloak at microwave frequencies," Science 314, 977-980 (2006).
[CrossRef] [PubMed]

Starr, A. F.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial electromagnetic cloak at microwave frequencies," Science 314, 977-980 (2006).
[CrossRef] [PubMed]

Tichit, P. H.

P. H. Tichit, B. Kante, and A. de Lustrac, "Design of polygonal and elliptical cloaks," Proceedings of Nato ARW and META08, Marrakesh-Morocco,  7-10May (2008), 120-125.

Uhlmann, G.

A. Greenleaf, M. Lassas and G. Uhlmann, "On nonuniqueness for Calderons inverse problem," Math. Res. Lett. 10, 685-693 (2003).

Ward, A. J.

A. J. Ward and J. B. Pendry, "Refraction and geometry in Maxwell’s equations," J.Mod. Opt. 43, 773-793 (1996).
[CrossRef]

Wu, B.

Zhang, J.

Zolla, F.

J. Appl. Phys. (1)

A. Nicolet, J. F. Remacle, B. Meys, A. Genon, and W. Legros, "Transformation methods in computational electromagnetism," J. Appl. Phys. 75, 6036-6038 (1994).
[CrossRef]

J. Opt. Soc. Am. B (1)

J.Mod. Opt. (1)

A. J. Ward and J. B. Pendry, "Refraction and geometry in Maxwell’s equations," J.Mod. Opt. 43, 773-793 (1996).
[CrossRef]

Marrakesh-Morocco (1)

P. H. Tichit, B. Kante, and A. de Lustrac, "Design of polygonal and elliptical cloaks," Proceedings of Nato ARW and META08, Marrakesh-Morocco,  7-10May (2008), 120-125.

Math. Res. Lett. (1)

A. Greenleaf, M. Lassas and G. Uhlmann, "On nonuniqueness for Calderons inverse problem," Math. Res. Lett. 10, 685-693 (2003).

New J. Phys. (1)

U. Leonhardt and T. G. Philbin, "General relativity in electrical engineering," New J. Phys. 8, 247 (2006).
[CrossRef]

Opt. Express (2)

Opt. Lett. (2)

Phys. Rev. B (1)

R. C. McPhedran, N. A. Nicorovici, and G. W. Milton,"Optical and dielectric properties of partially resonant composites," Phys. Rev. B 49, 8479-8482 (1994).
[CrossRef]

Science (2)

J. B. Pendry, D. Shurig, and D. R. Smith, "Controlling electromagnetic fields," Science 312, 1780-1782 (2006).
[CrossRef] [PubMed]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial electromagnetic cloak at microwave frequencies," Science 314, 977-980 (2006).
[CrossRef] [PubMed]

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Figures (3)

Fig. 1.
Fig. 1.

(a) Triangle of vertices (0,0), (bi ,θi ) and (b i+1,θ i+1) used in the triangulation of a polygon (see (b)) or a polygram (see (c)) mapped under Eq. (1) onto the trapezoid of vertices (ai ,θi ), (bi ,θi ), (b i+1,θ i+1) and (a i+1,θ i+1). The angles θi and the segments d i, j are as in Eq. (2); (b) Dodecagon and first step in its four tessellations {12/2}–{12/5}; (c) Tessellation {12/5} and representative samples of its triangulation.

Fig. 2.
Fig. 2.

Real part of the longitudinal component E 3 of the electric field scattered by a (a) pentagon {5}, (b) hexagon {6}, (c) heptagon {7}, (d) octagon {8}, (e) nonagon {9}, (f) decagon {10}, (g) hendecagon {11} (h) hexadecagon {16} polygonal cloaks. Both forward and backward scattering for a plane wave propagating from the left are nearly vanishing.

Fig. 3.
Fig. 3.

Real part of the longitudinal component E 3 of the electric field scattered by a (a) pentagram {5,2}, (b) hexagram {6,2}, (c) heptagram {7/3}, (d) octagram {8/3}, (e) nonagram {9/4}, (f) decagram {10/4}, (g) hendecagram {11/5} (h) hexadecagram {16/7} star-shaped cloaks. Unlike for computations reported in Figure 2, scattering worsens from panels (a) to (h), and most specifically forward scattering.

Equations (8)

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{r=R1(θ)+r(R2(θ)R1(θ))R2(θ),0rR2(θ),θ=θ,0<θ2π,x3=x3,x3IR,
{Ri,j=di,jcos(θθ0),θiθθi+1,j=1,2di,j=ci,jci+1,jsin(θi+1θi)ci,j2+ci+1,j22ci,jci+1,jcos(θi+1θi),j=1,2θ0=θi+arccos(di,jci,j)=θi+1+arccos(di,jci+1,j)ci,1=ai,ci,2=bi
domainΩwithdomainΩwithcoordinatesxixi(yj)coordinatesyi
differentialdxipullbackdifferentialΣjxi(yj)yjdyj
ε͇=ε T1 , and μ͇=μ T1 where T=JTJdet(J).
T1=(e122+fr2e11frre12fr0e12fre11rfr000e11frr),
T1 =Diag(rR1r',rrR1,(R2R2R1)2rR1r).
×(µ¯¯'1×El)k2ε¯¯'El=0

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