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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 10 — May. 11, 2009
  • pp: 8614–8620
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Invisibility cloak with a twin cavity

Tungyang Chen and Chung-Ning Weng  »View Author Affiliations


Optics Express, Vol. 17, Issue 10, pp. 8614-8620 (2009)
http://dx.doi.org/10.1364/OE.17.008614


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Abstract

We study an invisibility cloak with a twin cavity, simulated by a plane algebraic curve- hippopede. The cloaked region, which looks like eight for some sets of geometric parameters, is expanded from one single point. Using a geometric transformation approach, we demonstrate that the material parameters of cloaking layer can be exactly determined. Numerical simulations show that the incoming rays pass in and out the cloaking region twice, and return to their original trajectory outside the curved cloak. A notable feature is that the cloaking region has two hollow regions in which two objects can be hidden at one time and that they could not perceive each other.

© 2009 Optical Society of America

1. Introduction

Pendry et al. [1

1. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006). [CrossRef] [PubMed]

] and Leonhardt [2

2. U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006). [CrossRef] [PubMed]

] found that by enclosing an object by a layer of tailored material, the electromagnetic field can be controlled, bent around an interior region and return to their original propagation direction without perturbing the exterior field. Numerous theoretical, numerical and experimental developments have been reported in the last few years. Among them, we mention the relevant works: rigorous proofs of form invariance upon transformation [3-5

3. E. J. Post, Formal Structure of Electromagnetics: General Covariance and Electromagnetics, (North-Holland, Amsterdam, 1962).

], experimental implementations [6

6. 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]

], various physical phenomena [7-11

7. A. Greenleaf, Y. Kurylev, M. Lassas, and G Uhlmann, “Electromagnetic wormholes and virtual magnetic monopoles from metamaterials,” Phys. Rev. Lett . 99, 183901 (2007). [CrossRef] [PubMed]

], inhomogeneous media [12-13

12. J. Zhang, J. Huangfu, Y. Luo, H. Chen, J. A. Kong, and B. I. Wu, “Cloak for multilayered and gradually changing media,” Phys. Rev. B 77, 035116 (2008). [CrossRef]

] and rotated electromagnetic waves [14

14. H. Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett . 90, 241105 (2007). [CrossRef]

]. We mention that analogous concepts of invisibility can be dated back to the works [15-16

15. R. V. Kohn and M. Vogelius, “Identification of an unknown conductivity by means of measurements at the boundary,” in Inverse problems, D. W. McLaughlin, ed., (American Mathematical Society, Providence, RI, 1984), pp. 113–123.

]. Also, earlier references of [17-19

17. A. Nicolet, J. F. Remade, B. Meys, A. Genon, and W. Legros, “Transformation methods in computational electromagnetics,” J. Appl. Phys . 75, 6036–6038 (1994). [CrossRef]

] suggested that arbitrary change of coordinates could be useful in electromagnetics, especially in numerical modeling. The latest progress includes that the proposition of anti-cloak [20

20. H. Chen, X. Luo, and H. Ma, “The anti-cloak,” Opt. Express 16, 14603–14608 (2008). [CrossRef] [PubMed]

] which annihilates some effects of invisibility cloak, cloaking under the carpet [21

21. J. Li and J. B. Pendry, “Hiding under the carpet: A new strategy for cloaking,” Phys. Rev. Lett . 101, 203901 (2008). [CrossRef] [PubMed]

], and cloaking objects at a distance outside the cloaking shell [22

22. Y. Lai, H. Chen, Z. Q. Zhang, and C. T. Chan, “Complementary media invisibility cloak that cloaks objects at a distance outside the cloaking shell,” Phys. Rev. Lett . 102, 093901 (2009). [CrossRef] [PubMed]

]. See also the review [23

23. U. Leonhardt and T. G. Philbin, “Transformation optics and the geometry of Light,” arXiv: 0805.4778; Prog. Optics (to appear).

] for a broader scope.

The key idea of cloaking is to expand one point or a line segment into a circular or spherical cloaking space. Apart from the aforementioned works, another category of studies has been directed toward the feasibility of design of cloaking devices other than circular or spherical shapes. Study of non-circular or non-spherical cloaks is not just of academic curiosity, it provides flexible potentials to fit for the geometries of hidden objects and also for some specific purposes. Relevant findings include slab cloaks [24

24. G. X. Yu, W. X. Jiang, and T. J. Cui, “Invisible slab cloaks via embedded optical transformation,” Appl. Phys. Lett . 94, 041904 (2009). [CrossRef]

], elliptic and eccentric elliptic cloaks [25-26

25. W. X. Jiang, T. J. Cui, G. X. Yu, X. Q. Lin, Q. Cheng, and J. Y. Chin, “Arbitrarily elliptical-cylindrical invisible cloaking,” J. Phys. D: Appl. Phys . 41, 085504 (2008). [CrossRef]

], rectangular cloaks with flat surfaces and sharp corners [27

27. M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transfromations of Maxwell’s equations,” Photonics Nanostruct. Fundam. Appl . 6, 87–95 (2008). [CrossRef]

], toroidal cloaks [28

28. Y. You, G. W. Kattawar, and P. Yang, “Invisibility cloaks for toroids,” Opt. Express 17, 6591–6599 (2009). [CrossRef] [PubMed]

], and polygonal and arbitrarily shaped cloaks [29-37

29. Y. You, G. W. Kattawar, P. W. Zhai, and P. Yang, “Invisibility cloaks for irregular particles using coordinate transformations,” Opt. Express 16, 6134–6145 (2008). [CrossRef] [PubMed]

]. Among the formulations, some utilized analytic approaches, others were simulated by numerical solutions. But, to our best knowledge, all the existing works on cloaking so far are confined to cloaks enclosed with a single hole. In this paper we propose a new type of invisibility cloak that contains two or more cavities. Imagine the situation of a train containing two or more cars. One potential application of a twin or multiple cavities is to hide two or more objects at one time, in which they could not be placed together inside one cavity or would not be detected by each other. We here mention that the twin cavity is indeed expanded from one single point to a simply-connected hollow region with indentations at top and bottom, in which the two indentation points coincide with one single point at the origin. Thus, the derivation of the material parameters of the cloaking layer follows the conventional route for design of cylindrical cloaks.

2. Coordinate transformations for hippopedal cloaks

To begin with, we simulate the cloaked region by a plane algebraic curve- hippopede. A hippopede is a plane curve obeying the equation in polar coordinates [38

38. J. D. Lawrence,. A Catalog of Special Plane Curves, (New York: Dover, 1972).

]

r2=4b(absin2θ),
(1)

where a and b are positive numbers. Simple algebra suggests that the contour follows the symmetry relations

r(θ)=r(θ)=r(πθ)=r(π+θ).
(2)
Fig. 1. The hippopedal curves, described by Eq. (1), with different sets of parameters (a) a > b (dashed lines), (b) a = b (solid line), (c) a < b (dotted lines).

Different values of a and b will correspond to different contours. For example, if a ≥ 2b, it is an oval, and if b < a ≤ 2b, it is an oval with indentations at top and bottom. If ab, the figure forms a figure eight (see Fig. 1 for an illustration). Here we are particularly interested in the latter situation in which the configuration is composed of two closed contours. Particularly, we note that when θ varies from 0 to 2π, each θ will at most correspond to one single value of r. For example, when ba, as θ increases from 0 to π/2, the value r will decrease from 2√ab to zero at θ = sin-1a/b. When sin-1a/bθπ - sin-1a/b , there is no real value of r-that is, r is an imaginary value. For the remaining parts of the locus π/2 ≤ θ ≤ 2π , the contour can be depicted following the symmetry condition Eq. (2). In Cartesian coordinates, one can write Eq. (1) as

(x2+y2)2+4b(bc)(x2+y2)4b2x2=0.
(3)

The form also suggests that the locus should be symmetric about the x-axis, y-axis as well as the coordinate center O. Particularly, when a = b, one can simplify Eq. (2) as

((xb)2+y2b2)((x+b)2+y2b2)=0.
(4)

This reflects that, in this particular case, the locus is composed of two isolated circles with the same radius b, centered respectively at (b, 0) and (−b, 0), and that the two circles intersects at the origin O. It is seen that, in this situation, each θ will correspond to one single value of r. When θ = π/2, we have r = 0. We mention that, for b = 2a , the hippopede corresponds to the lemniscate of Bernoulli [39

39. G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers, (New York: McGraw-Hill, 1961).

].

Let us now consider two similar hippopedes Ωi and Ωo represented by

Ωi:ri=2abb2sin2θ,
Ωo:ro=2mabb2sin2θ,
(5)

x'=rrx=(α+rir)x,y=rry=(α+rir)y,z=z,
(6)

where α = (m - 1)/m. The new relative permittivity and permeability follow [4

4. G. W. Milton, M. Briane, and J. R. Willis, “On cloaking for elasticity and physical equations with a transformation invariant form,” New J. Phys . 8, 248 (2006). [CrossRef]

]

ε'=ATdetA,μ=ATdetA,
(7)

where ε and μ are the original properties of background material, which can be represented by an identity matrix in free space with rectangular coordinates, and A is the Jacobian matrix defined by

A=(x',y',z')(x,y,z)=(α+4by2((a+b)r22by2)r5ri4bxy((a+b)r22by2)r5ri04bxy(ar22by2)r5riα+4bx2(ar22by2)r5ri0001).
(8)

A referee brought to our attention that the exact connection of Eq. (7) was also stated in Nicolet et al. [40

40. A. Nicolet, S. Guenneau, and F. Zolla, “Modelling of twisted optical waveguides with edge elements,” Eur. Phys. J. Appl. Phys . 28153–157 (2004). [CrossRef]

]. Now working out the algebra in Eq. (7), we could express each component of transformed material properties as

εxx=μxx=rαr[(α+4by2((a+b)r22by2)r5ri)2+(4bxy((a+b)r22by2)r5ri)2],
εyy=μyy=rαr[α+(4bx2(ar22by2)r5ri)2+(4bxy(ar22by2)r5ri)2],
εzz=μzz=rαr,
εzz=μxy=rαr[(4bxy(4br2(ar22by2)(ar2+b(x2y2))αr5ri((2a+b)r24by2))r10ri2)].
(9)

In terms of polar coordinate coordinates, the transformed permittivity and permeability follow the form

ε'=μ'=(r'rir'+4b2sin22θri2r'(r'ri)2b2sin2θri(r'ri)0r'r'ri0sym1α2r'rir').
(10)

Equation (10) provides full design parameters for the hippopedal cloaks in the polar coordinates based on the center O. Clearly, the cloak is composed of inhomogeneous and anisotropic metamaterials. We note that, when we set θ = 0 in Eq. (1), the contours characterized in Eq. (5) become two concentric circles and, in this situation, the transformed material properties in Eq. (10) recover to those of the circular cloak [6

6. 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]

]. It is known that, for circular cloaks, singular material parameters are distributed at the inner boundary. Here, in Fig. 2, we plot the values of εxx′, εyy′, εxy′ and εzz′ along the portion of cloaking layer for the case of a = 1 and b = 1.

Fig. 2. The magnitudes of material properties εxx′, εyy′, εzz′, and εxy′ inside the hippopedal cloaking layer (a = 1, b = 1) .

We find that extreme material properties also prevail along the inner boundary of the cloak as in the circular cloaks. In particular, we note that at the origin, the material property is necessarily singular as, at that point, the inner and outer cloaking boundaries shrink to one point and the thickness of the cloaking layer becomes effectively zero. The presence of a singularity arises from the fact that the origin point O is mapped onto the origin in the new space, but its neighborhood is tearing apart to four distinct directions in the transformed (new) space.

3. Numerical simulations

Fig. 3. Snapshot of the electric field distribution. The plane waves are incident in the horizontal direction. (a) a = 0.12, b = 0.1, (b) a = 0.15, b = 0.15, (c) a = 0.11, b = 0.16.
Fig. 4. Snapshot of the electric field distribution (a), and stream lines (b). The plane waves are incident at an oblique angle of 30° with a = 0.15, b = 0.15.
Fig. 5. A schematic illustration of a few plane curves. (a) double folium, r = 4acosθsin2 θ (dashed line), (b) sinusoidal spirals, r = (acos3θ)1/3 (dotted line).

4. Conclusions

In summary, we have demonstrated that, by using the concept of coordinate transformation, it is possible to expand one point into a twin cavity so that two objects can be hidden at one time. Numerical simulations show that the wave can be bent around the cloaking region in and out twice with two hidden objects inside. Creating the desired properties for the cloaking shell with singular properties could be a challenging task, but reports of new findings of design of cloaking devices may encourage further developments and inspire potential applications of cloaking with a twin cavity. We mention that recent studies [41-42

41. U. Leonhardt and T. Tyc, “Broadband invisibility by non-Euclidean cloaking,” Science 323, 110 (2009). [CrossRef]

] have proposed some schemes that may allow an invisibility cloak to be designed without singularities. Also, it should be mentioned that hippopedes in plane algebraic geometry are not the only plane curve that may lead to multiply connected domains. There are other kinds of geometries, such as double folium or sinusoidal spirals (Fig. 5) [38

38. J. D. Lawrence,. A Catalog of Special Plane Curves, (New York: Dover, 1972).

], which may also lead to a configuration with multiple cavities. The design of cloaks of the latter geometries is indeed parallel to the formulation presented in this work.

Acknowledgment

This work was supported by the National Science Council, Taiwan, under contract NSC97-2211-E-006-117-MY3.

References and links

1.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006). [CrossRef] [PubMed]

2.

U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006). [CrossRef] [PubMed]

3.

E. J. Post, Formal Structure of Electromagnetics: General Covariance and Electromagnetics, (North-Holland, Amsterdam, 1962).

4.

G. W. Milton, M. Briane, and J. R. Willis, “On cloaking for elasticity and physical equations with a transformation invariant form,” New J. Phys . 8, 248 (2006). [CrossRef]

5.

G. W. Milton and J. R. Willis, “On modifications of Newton’s second law and linear continuum elastodynamics,“ Proc. R. Soc. London, Ser. A 463, 855–880 (2007). [CrossRef]

6.

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]

7.

A. Greenleaf, Y. Kurylev, M. Lassas, and G Uhlmann, “Electromagnetic wormholes and virtual magnetic monopoles from metamaterials,” Phys. Rev. Lett . 99, 183901 (2007). [CrossRef] [PubMed]

8.

A. Greenleaf, M. Lassas, and G. Uhlmann, “Anisotropic conductivities that cannot be detected by EIT,” Physiol. Meas . 24, 413–419 (2003). [CrossRef] [PubMed]

9.

S. A. Cummer and D. Schurig, “One path to acoustic cloaking,” New J. Phys . 9, 45 (2007). [CrossRef]

10.

H. Chen and C. T. Chan, “Acoustic cloaking in three dimensions using acoustic metamaterials,” Appl. Phys. Lett . 91, 183518 (2007). [CrossRef]

11.

A. N. Norris, “Acoustic cloaking theory,” Proc. R. Soc. London, Ser. A 464, 2411–2434 (2008). [CrossRef]

12.

J. Zhang, J. Huangfu, Y. Luo, H. Chen, J. A. Kong, and B. I. Wu, “Cloak for multilayered and gradually changing media,” Phys. Rev. B 77, 035116 (2008). [CrossRef]

13.

T. Chen, C. N. Weng, and J. S. Chen, “Cloak for curvilinearly anisotropic media in conduction,” Appl. Phys. Lett . 93, 114103(2008). [CrossRef]

14.

H. Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett . 90, 241105 (2007). [CrossRef]

15.

R. V. Kohn and M. Vogelius, “Identification of an unknown conductivity by means of measurements at the boundary,” in Inverse problems, D. W. McLaughlin, ed., (American Mathematical Society, Providence, RI, 1984), pp. 113–123.

16.

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

17.

A. Nicolet, J. F. Remade, B. Meys, A. Genon, and W. Legros, “Transformation methods in computational electromagnetics,” J. Appl. Phys . 75, 6036–6038 (1994). [CrossRef]

18.

J. F. Remade, A. Nicolet, A. Genon, and W. Legros, “Comparison of boundary elements and transformed finite elements for open magnetic problems,” in Boundary Element Method XVI, C. A. Brebbia, ed., (Computational Mechanics Publications, Southhampton, 1994), pp. 109–116.

19.

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

20.

H. Chen, X. Luo, and H. Ma, “The anti-cloak,” Opt. Express 16, 14603–14608 (2008). [CrossRef] [PubMed]

21.

J. Li and J. B. Pendry, “Hiding under the carpet: A new strategy for cloaking,” Phys. Rev. Lett . 101, 203901 (2008). [CrossRef] [PubMed]

22.

Y. Lai, H. Chen, Z. Q. Zhang, and C. T. Chan, “Complementary media invisibility cloak that cloaks objects at a distance outside the cloaking shell,” Phys. Rev. Lett . 102, 093901 (2009). [CrossRef] [PubMed]

23.

U. Leonhardt and T. G. Philbin, “Transformation optics and the geometry of Light,” arXiv: 0805.4778; Prog. Optics (to appear).

24.

G. X. Yu, W. X. Jiang, and T. J. Cui, “Invisible slab cloaks via embedded optical transformation,” Appl. Phys. Lett . 94, 041904 (2009). [CrossRef]

25.

W. X. Jiang, T. J. Cui, G. X. Yu, X. Q. Lin, Q. Cheng, and J. Y. Chin, “Arbitrarily elliptical-cylindrical invisible cloaking,” J. Phys. D: Appl. Phys . 41, 085504 (2008). [CrossRef]

26.

D. H. Kwon and D. H. Werner, “Two-dimensional eccentric elliptic electromagnetic cloaks,” Appl. Phys. Lett . 92, 013505 (2008). [CrossRef]

27.

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transfromations of Maxwell’s equations,” Photonics Nanostruct. Fundam. Appl . 6, 87–95 (2008). [CrossRef]

28.

Y. You, G. W. Kattawar, and P. Yang, “Invisibility cloaks for toroids,” Opt. Express 17, 6591–6599 (2009). [CrossRef] [PubMed]

29.

Y. You, G. W. Kattawar, P. W. Zhai, and P. Yang, “Invisibility cloaks for irregular particles using coordinate transformations,” Opt. Express 16, 6134–6145 (2008). [CrossRef] [PubMed]

30.

X. Chen, Y. Fu, and N. Yuan, “Invisible cloak design with controlled constitutive parameters and arbitrary shaped boundaries through Helmholtz’s equation,” Opt. Express 17, 3581–3586 (2009). [CrossRef] [PubMed]

31.

J. Hu, X. Zhou, and G. Hu, “Design method for electromagnetic cloak with arbitrary shapes based on Laplace’s equation,” Opt. Express 17, 1308–1320 (2009). [CrossRef] [PubMed]

32.

Q. Wu, K. Zhang, F. Y. Meng, and L. W. Li, “Material parameters characterization for arbitrary N-sided regular polygonal invisible cloak,” J. Phys. D: Appl. Phys . 42, 035408 (2009). [CrossRef]

33.

J. Zhang, Y. Luo, H. Chen, and B. I. Wu, “Cloak of arbitrary shape,” J. Opt. Soc. Am. B 25, 1776–1779 (2008). [CrossRef]

34.

A. Nicolet, F. Zolla, and S. Guenneau, “Electromagnetic analysis of cylindrical cloaks of an arbitrary cross section,” Opt. Lett . 33, 1584–1586 (2008). [CrossRef] [PubMed]

35.

W. X. Jiang, J. Y. Chin, Z. Li, Q. Cheng, R. Liu, and T. J. Cui, “Analytical design of conformally invisible cloaks for arbitrarily shaped objects,” Phys. Rev. E 77, 066607 (2008). [CrossRef]

36.

C. Li and F. Li, “Two-dimensional electromagnetic cloaks with arbitrary geometries,” Opt. Express 16, 13414–13420 (2008). [CrossRef] [PubMed]

37.

H. Ma, S. Qu, Z. Xu, and J. Wang, “Approximation approach of designing practical cloaks with arbitrary shapes,” Opt. Express 16, 15449–15454 (2008). [CrossRef] [PubMed]

38.

J. D. Lawrence,. A Catalog of Special Plane Curves, (New York: Dover, 1972).

39.

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers, (New York: McGraw-Hill, 1961).

40.

A. Nicolet, S. Guenneau, and F. Zolla, “Modelling of twisted optical waveguides with edge elements,” Eur. Phys. J. Appl. Phys . 28153–157 (2004). [CrossRef]

41.

U. Leonhardt and T. Tyc, “Broadband invisibility by non-Euclidean cloaking,” Science 323, 110 (2009). [CrossRef]

42.

W. X. Jiang, T. J. Cui, X. M. Yang, Q. Cheng, R. Liu, and D. R. Smith, “Invisibility cloak without singularity,” Appl. Phys. Lett . 93, 194102 (2008). [CrossRef]

OCIS Codes
(160.1190) Materials : Anisotropic optical materials
(260.2110) Physical optics : Electromagnetic optics
(160.2710) Materials : Inhomogeneous optical media
(230.3205) Optical devices : Invisibility cloaks

ToC Category:
Physical Optics

History
Original Manuscript: March 27, 2009
Revised Manuscript: April 26, 2009
Manuscript Accepted: May 1, 2009
Published: May 6, 2009

Citation
Tungyang Chen and Chung-Ning Weng, "Invisibility cloak with a twin cavity," Opt. Express 17, 8614-8620 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-10-8614


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References

  1. J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling electromagnetic fields," Science 312, 1780-1782 (2006). [CrossRef] [PubMed]
  2. U. Leonhardt, "Optical conformal mapping," Science 312, 1777-1780 (2006). [CrossRef] [PubMed]
  3. E. J. Post, Formal Structure of Electromagnetics: General Covariance and Electromagnetics (North-Holland, Amsterdam, 1962).
  4. G. W. Milton, M. Briane, and J. R. Willis, "On cloaking for elasticity and physical equations with a transformation invariant form," New J. Phys. 8, 248 (2006). [CrossRef]
  5. G. W. Milton and J. R. Willis, "On modifications of Newton’s second law and linear continuum elastodynamics," Proc. R. Soc. London, Ser. A 463, 855-880 (2007). [CrossRef]
  6. 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]
  7. A. Greenleaf, Y. Kurylev, M. Lassas and G. Uhlmann, "Electromagnetic wormholes and virtual magnetic monopoles from metamaterials," Phys. Rev. Lett. 99, 183901 (2007). [CrossRef] [PubMed]
  8. A. Greenleaf, M. Lassas and G. Uhlmann, "Anisotropic conductivities that cannot be detected by EIT," Physiol. Meas. 24, 413-419 (2003). [CrossRef] [PubMed]
  9. S. A. Cummer and D. Schurig, "One path to acoustic cloaking," New J. Phys. 9, 45 (2007). [CrossRef]
  10. H. Chen and C. T. Chan, "Acoustic cloaking in three dimensions using acoustic metamaterials," Appl. Phys. Lett. 91, 183518 (2007). [CrossRef]
  11. A. N. Norris, "Acoustic cloaking theory," Proc. R. Soc. London, Ser. A 464, 2411-2434 (2008). [CrossRef]
  12. J. Zhang, and J. Huangfu, Y. Luo, H. Chen, J. A. Kong, and B. I. Wu, "Cloak for multilayered and gradually changing media," Phys. Rev. B 77, 035116 (2008). [CrossRef]
  13. T. Chen, C. N. Weng and J. S. Chen, "Cloak for curvilinearly anisotropic media in conduction," Appl. Phys. Lett. 93, 114103 (2008). [CrossRef]
  14. H. Chen and C. T. Chan, "Transformation media that rotate electromagnetic fields," Appl. Phys. Lett. 90, 241105 (2007). [CrossRef]
  15. R. V. Kohn and M. Vogelius, "Identification of an unknown conductivity by means of measurements at the boundary," in Inverse problems, D.W. McLaughlin, ed., (American Mathematical Society, Providence, RI, 1984), pp. 113-123.
  16. N. A. Nicorovici, R. C. McPhedran, and G. W. Milton, "Optical and dielectric properties of partially resonant composites," Phys. Rev. B 49, 8479-8482 (1994). [CrossRef]
  17. A. Nicolet, J. F. Remacle, B. Meys, A. Genon, and W. Legros, "Transformation methods in computational electromagnetics," J. Appl. Phys. 75, 6036-6038 (1994). [CrossRef]
  18. J. F. Remacle, A. Nicolet, A. Genon, and W. Legros, "Comparison of boundary elements and transformed finite elements for open magnetic problems," in Boundary Element Method XVI, C. A. Brebbia, ed., (Computational Mechanics Publications, Southhampton, 1994), pp. 109-116.
  19. A. J. Ward and J. B. Pendry, "Refraction and geometry in Maxwell's equations," J. Mod. Opt. 43, 773-793. (1996). [CrossRef]
  20. H. Chen, X. Luo, and H. Ma, "The anti-cloak," Opt. Express 16, 14603-14608 (2008). [CrossRef] [PubMed]
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