Abstract

The material properties of toroidal invisibility cloaks are derived based on the coordinate transformation method. The permittivity and permeability tensors for toroidal cloaks are substantially different from those for spherical cloaks, but quite similar to those for 2D cylindrical cloaks because a singularity is involved at the inner boundary in both the cases. The cloaking effect is confirmed by the electric field distribution in the vicinity of toroidal cloaks simulated from the generalized discrete-dipole approximation (DDA) method. This study extends the concept of electromagnetic cloaking of arbitrarily-shaped objects to a complex geometry.

©2009 Optical Society of America

1. Introduction

Transformation optics [1, 2, 3, 4, 5, 6, 7] has recently become a powerful tool to control the propagation of electromagnetic waves by altering material properties as determined by coordinate transformations in a general coordinate system [1, 2, 3] or in Cartesian coordinates [4], or by optical conformal mapping [5, 6] in an arbitrary coordinate system. The new tool makes it possible to design optical devices with a variety of interesting functionalities. The invisibility cloak is one of the optical device designs using the coordinate transformation approach and has drawn a great deal of attention due to its potential applications. An invisibility cloak reroutes the incident electromagnetic radiation in the cloaking layer and inhibits it from penetrating into the cloaked region, while the electromagnetic fields outside of the cloaking region remain unchanged. Therefore, the cloaking layer and any object inside it are invisible. The material properties, i.e., the permittivity and permeability tensors, of an invisibility cloak derived from the coordinate transformation approach are anisotropic and inhomogeneous. These material properties were first given for spherical [2, 4] and 2D infinite cylindrical [3, 8] cloaks. Rigorous solutions to Maxwell’s equations have been reported in the spherical and cylindrical cases [9, 10] to confirm the cloaking effects for these geometries. Numerical simulations have been carried out to study the cloaking effects of invisibility cloaks of various shapes, including 2D cylinders [8, 11, 12, 13] and squares [14], and 3D ellipsoids, cylinders and cuboids [15]. Recently, designs of invisibility cloaks with arbitrary shapes have been studied in both 2D [16, 17, 18, 19] and 3D [20, 21, 22] cases. With recent advances in the metamaterial technology, simplified specifications of the required material properties have been practically implemented in the case of 2D cylinders. The cloaking effect has been observed at both microwave [23] and optical frequencies [24].

It has been discussed in Ref. [20] that the coordinate transformations utilized to design invisibility cloaks fall into two categories: point transformations, which transform a point into a cloaked region; and, line transformations, which transform a line into a cloaked region. Note that at the inner boundary of a line-transformed cloak, one tangential component of the material property tensors has a singularity. According to this classification scheme, all arbitrarily-shaped 3D cloaks reported in the literature are point-transformed cloaks, and all 2D cloaks are line-transformed cloaks.

These point-transformed “arbitrarily-shaped” 3D cloaks, however, do not represent all possible geometries in 3D space. From a topological perspective, they have a genus-zero surface, meaning that the surface possesses no “hole”. Surfaces exist with a genus greater than zero. The simplest genus one surface is a torus, which has one hole in the surface. The volume bounded by a torus, a toroid, is the simplest genus one object. To make a toroidal object invisible, we can simply put it into a genus-zero cloak, for example, a spherical cloak, but the volume of the genus-zero cloak may be unnecessarily large if the toroid has a long thin tube. In such situations, a toroidal cloak is preferred as it uses less cloaking material. Obviously, toroidal cloaks are line-transformed cloaks, since the toroidal cloaked region can only be transformed from a line. A toroidal cloak can be made by cutting out a segment from a line transformed 2D cylindrical cloak and gluing the two ends together. To date, invisibility cloaks for toroids have not been reported in the literature.

In this study, first we apply the coordinate transformation method to the toroidal geometry and derive the material properties required for toroidal cloaks. Second, we compare the material properties for toriodal cloaks with those for spherical cloaks and 2D cylindrical cloaks. Third, we show simulated electric field distributions in the vicinity of toroidal cloaks to confirm the cloaking effect.

2. Coordinate transformations for toroidal cloaks

Figure 1(a) shows a torus generated by revolving a circle about an axis in the same plane as itself in 3D space, as shown in Fig. 1(b), where it is assumed that the axis about which the circle revolves is the z-axis. Figure 1(b) also shows that Cartesian coordinates (x, y, z) of any point in a toroid can be expressed in terms of three parameters r, u and v in the form

x=(R+rsinu)cosv,
y=(R+rsinu)sinv,
z=rcosu,

where R is the distance from the center of the circle to the coplanar axis, r ∈ [0,b] is the distance from the point to the center of the circle with b the radius of the circle, and angles u, v ∈ [0,2π]. Unit vectors er, eu and ev span the toroidal coordinates. Angle u is the angle between er and the z-direction, and angle v is the angle between the er × eu plane and the x-direction. In Figs. 1(a) and 1(b), it is assumed that b = R/2.

To create a toroidal cloaked region, we consider the following coordinate transformation

r'=a+babr,u'=u,v'=v,

where b and a are the radii of the circles that form the outer and inner boundaries of the cloaking region, respectively. A toroidal cloak with b = R/2 and a = b/2 is illustrated in Fig. 1(c). Half of the outer boundary of the cloaking region (the dark green surface) is removed to expose the cloaked region. The transformation equation, Eq. (2), is identical to the coordinate transformations used for spherical cloaks [2, 4] and for 2D cylindrical cloaks [3, 8] in the sense that it involves a linear compression in the “radial” direction with the space in the angular directions remaining intact. However, the radius r is no longer the distance from a point in the space to the origin or to the z-axis, but the distance from the point to the center of the revolving circle, which itself forms a circle in the x-y plane when revolving about the z-axis.

 

Fig. 1. (a) A toroid with b = R/2; (b) The toroidal coordinates spanned by unit vectors er, eu and ev; (c) A toroidal cloak with b = R/2 and a = b/2.

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Figure 2 illustrates a toroidal cloak subject to an incident plane wave. The incident light rays (blue lines) and wave fronts (red lines) in the cloaking region are determined by the coordinate transformation Eq. (2). They are compressed into both inner and outer sides of the cloaked region. For simplicity, in this study we limit ourselves to toroidal cloaks with a conformal toroidal cloaked region. The conformal toroidal cloaks can be continuously deformed into cloaks with any genus one geometry in the same manner as spherical cloaks are generalized to arbitrarily-shaped genus zero cloaks, which has been extensively discussed in the literature.

3. Material properties for toroidal cloaks

To obtain the cloaking material property tensors ε and μ via the coordinate transformation approach, we follow the scheme used in Ref. [20] by working in the toroidal coordinates but use slightly different notations. Consider a coordinate transformation from coordinates (q 1,q 2,q 3) to coordinates (q1,q2,q3), the tensor representation of the transformation formula is given by [4]

εi'j'=μi'j'=det(Λi'i)1Λi'iΛj'j,

where we have used the fact that εij = μij = δij for the vacuum, and Λij = ∂qi/∂qj is the Jacobian matrix. The equivalent matrix representation of Eq. (3) is [20]

ε'=μ'=det(Λ)1ΛΛT.

A point in the original space is indicated as (x, y, z) in Cartesian coordinates and (r,u,v) in the toroidal coordinates, and the corresponding point in the transformed space is indicated as (x′,y′, z′) in Cartesian coordinates and (r′,u′,v′) in the toroidal coordinates. Furthermore, we consider the sequential coordinate transformations as follows:

(x,y,z)(r,u,v)(r',u',v')(x',y',z').
 

Fig. 2. 3-D view of the propagation of light rays (blue lines) and associated wave fronts (red lines) in a toroidal cloak.

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For toroidal cloaks, the transformations (x,y,z) → (r,u,v) and (r,u,v) → (r′,u′,v′) have been determined by Eqs. (1) and (2), respectively. The transformation (r′,u′,v′)→(x′,y′,z′) can be determined by Eq. (1) with (x,y, z) and (r,u,v) replaced by (x′,y′,z′) and (r′,u′,v′), respectively.

The Jacobian matrices involved in these coordinate transformations are

Λ1=(r,u,v)(x,y,v),Λ2=(r',u',v')(r,u,v),Λ3=(x',y',z')(r',u',v').

Since we have Eq. (1), it is more straightforward to calculate Λ-1 1 = ∂(x, y, z)/∂ (r,u,v) rather than Λ1. According to the chain rule, the Jacobian matrix associated with the coordinate transformation (x, y, z)→(x′,y′,z′) can be written as

Λ=(x',y',z')x,y,z=Λ3Λ2Λ1.

A combination of Eqs. (4) and (7) gives

ε'=μ'=det(Λ11)det(Λ3)det(Λ2)Λ3Λ2Q11Λ2TΛ3T,

where Q 1 = (Λ-1 1)TΛ-1 1. Eq. (8) is the matrix representation of the permittivity and permeability tensors in Cartesian coordinates, e.g., ε′ = ∑ij εij exi exj, where exi is the ith Cartesian unit vector. To write Eq. (8) as a matrix representation in the toroidal coordinates, we use the following relation

Truv=(DΛ31)Txyz(DΛ31)T,

where D = diag[h 1,h 2,h 3] with hi = (Λ2 3,1i + Λ2 3,2i + Λ2 3,3i)1/2 the scale factors of the toroidal coordinates and DΛ-1 3 is an orthogonal matrix (DΛ-1 3)T = (DΛ-1 3)-1. Eq. (8) now becomes

ε'ruv=μ'ruv=det(Λ11)det(Λ3)det(Λ2)DΛ2Q11Λ2TQ3D1,

where Q 3 = ΛT 3Λ3.

For the toroidal coordinates, it is straightforward from Eqs. (1) and (2) to show that

Q1=diag[1,r2,(R+rsinu)2],
Λ2=diag[bab,1,1],
Q3=diag[1,r'2,(R+r'sinu)2],
D=diag[1,r',(R+r'sinu)2].

Therefore, the non-vanishing diagonal components of the permittivity and permeability tensors in the toroidal cloaking region are given by

εrr=μrr=rar(ba)R+b(ra)sinu(ba)(R+rsinu),
εuu=μuu=rra(ba)R+b(ra)sinu(ba)(R+rsinu),
εvv=μvv=b2bararR+rsinu(ba)R+b(ra)sinu,

where we have eliminated primes on ε′, μ′ and r′. All off-diagonal components vanish.

As predicted by Ref. [20], the tangential components εuu and μuu approach infinity at the inner boundary ra. This implies that, similar to perfect 2D cylindrical cloaks, perfect toroidal cloaks are difficult to realize using metamaterials. To fabricate toroidal cloaks, simplified material properties [23, 24, 25, 26, 27] may have to be used. It is interesting to compare Eqs. (12a)-(12c) with the material properties of 2D cylindrical cloaks given by, for example, Eqs. (1) and (2) in Ref. [8]

εr=μr=rR1r,
εϕ=μϕ=rrR1,
εz=μz=(R2R2R1)2rR1r,

where R 2 and R 1 are used to denote radii of the outer and inner boundaries, respectively. The correspondences between the r-, u- and v-components in the toroidal coordinates and the r-, ϕ - and z-components in the cylindrical coordinates are obvious. For toroidal cloaks, each diagonal component of the material property tensors has an extra term that depends on both r and u coordinates. This is reasonable since the geometry is no longer invariant with respect to the u coordinate. Specifically, Eqs. (12a)-(12c) reduce to Eqs. (13a)-(13c) when b/R approaches 0, or when sinu = 0, r = a or r = b if b/R is fixed.

Figure 3 shows the diagonal components of the permittivity and permeability tensors for a toroidal cloak with b = R/2 and a = b/2. Since the components depend on the u coordinate, we show them for various u values. The material properties for a toroidal cloak are the same as those for a 2D cylindrical cloak when u = 0. Additionally, the percentage changes (with respect to the 2D cylindrical results) introduced by the u-dependence are on the order of 10%, with the r- and u-components decreased and the v-component increased. The u-dependence seems negligible for angles where u > 60°.

For toroidal cloaks, the permittivity and permeability tensors also depend on the ratio b/R, as shown in Eqs. (12a). Figure 4 shows the diagonal components for a toroidal cloak with b = R/10. This is a toroid with a long thin tube. In this case, the changes introduced by the u-dependence are even smaller, with percentage changes of no more than 4%.

 

Fig. 3. Diagonal components of the permittivity and permeability tensors for a toroidal cloak with b = R/2 and a = b/2. The angle u varies from 0° to 90°.

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Fig. 4. Same as Fig. 3, except that b = R/10.

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4. Numerical simulations

To confirm the cloaking effect of toroidal invisibility cloaks, we simulated the distribution of the electric field in the vicinity of toroidal cloaks using the generalized discrete-dipole approximation (DDA) method [28, 29, 30, 31]. Since the u-components of the required permittivity and permeability tensors vary with a large gradient and become infinite at the inner boundary of the cloaking region, it is not likely possible to perfectly mimic the required u-components in a discretized numerical calculation. In contrast, the non-singular r- and v-components can be represented by the discrete DDA grids to the extent that the simulated electric field distributions show the cloaking effect [15, 31].

Figure 5 shows the simulated electric field distribution in the vicinity of two toroidal cloaks. Shown in the figure are the field contours observed in the x-y plane. The scales are in units of the wavelength (λ) of the incident plane wave propagating from left to right. The cross sections of the inner and outer parts of the cloaking region are separated (the thin annuli confined by black circles). The cross sections of the cloaked region are the thicker annuli between the two parts of the cloaking region. The cloaked regions are filled with isotropic and homogeneous impedance-matched material with ε =μ = 1.2. Figure 5(a) shows a toroidal cloak with a = 3/4b, b = R/2 and R = 0.45λ. The cloaked region is a relatively small toroid with a fat tube and a thin layer of cloaking material compared with the cloaked toroid. Figure 5(b) shows a cloak with a = 3/4b, b = 3/4R, and R = 1.196λ. The cloaked region is a larger and thinner toroid. The first phenomenon noticed from these figures is that, the electric field is highly compressed into the cloaking region, leading to a radiation-free cloaked region. As predicted by the coordinate transformation approach shown in Fig. 2, the electric field is compressed into both sides of the cloaked region. Outside of the cloaking region, the plane wave structure is well kept, as if no scatterer is present. There are slight deviations from the plane wave patterns at the far end of the cloaks, as the infinitely large u-components at the inner boundary of the cloaking region were not perfectly represented. The implementation of the material property specifications in this numerical simulation is practical for an experiment using metamaterials, yet the resultant cloaking effect is impressive.

 

Fig. 5. The DDA simulation of the electric field distributions in the vicinity of toroidal cloaks subject to incident plane waves. (a) a=3/4b, b=R/2 and R=0.45λ ; (b) a=3/4b, b = 3/4R, and R = 1.196λ. The scales are in units of λ.

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5. Conclusions

The invisibility cloak for the simplest genus-one geometry in three-dimensional space, the toroid, was defined on the basis of the coordinate transformation approach. This design broadens the extent of arbitrarily-shaped 3D cloaks. The permittivity and permeability tensors for toroidal cloaks are similar to those for 2D cylindrical cloaks rather than those for spherical cloaks. A tangential component is required to be infinite at the inner boundary of the cloaking region. Numerical calculations based on the generalized DDA method were carried out to simulate the electric field distributions in the vicinity of toroidal cloaks subject to the incident plane wave. The results suggest that the electric field outside of a cloak is almost unaltered as if neither the cloaking material nor the cloaked toroids were present.

Similar to 2D cylindrical cloaks, the toroidal cloaks may be difficult to realize due to the singularity of the material properties at the inner boundary. The present numerical simulations suggest that the cloaking effect can be impressive even if non-perfect material properties are used. Since the required material properties for a toroidal cloak are similar to those for an infinitely long cylindrical cloak, experimental realizations of toroidal cloaks are possible by using the metamaterial structures consisting of unit cells reported in the literature for cylindrical cloaks [23, 24, 25] and by bending the cylindrical structure into a toroid, except that the extra u-dependence needs to be included by varying the three components of the permittivity and permeability tensors of each individual unit cell along the u direction. However, this u-dependence may be neglected if both the cloaked toroid and the cloaking layer are extremely thin such that Rb.

Acknowledgments

This research was partially supported by the Office of Naval Research under contract N00014-06-1-0069. Ping Yang acknowledges support from the National Science Foundation (ATM-0803779).

References and links

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

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

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

4. D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express 14, 9794–9804 (2006). [CrossRef]   [PubMed]  

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

6. U. Leonhardt, “Notes on conformal invisibility devices,” New J. Phys. 8, 118 (2006). [CrossRef]  

7. D. Schurig, J. B. Pendry, and D. R. Smith, “Transformation-designed optical elements,” Opt. Express 15, 14772–14782 (2007). [CrossRef]   [PubMed]  

8. S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E 74, 036621 (2006). [CrossRef]  

9. H. Chen, B.-I. Wu, B. Zhang, and J. A. Kong, “Electromagnetic wave interaction with a metamaterial cloak,” Phys. Rev. Lett. 99, 063903 (2007). [CrossRef]   [PubMed]  

10. B. Zhang, H. Chen, B.-I. Wu, Y. Luo, L. Ran, and J. A. Kong, “Response of a cylindrical invisibility cloak to electromagnetic waves,” Phys. Rev. B 76, 121101 (2007). [CrossRef]  

11. H. Ma, S. Qu, Z. Xu, J. Zhang, B Chen, and J. Wang, “Material parameter equation for elliptical cylindrical cloaks,” Phys. Rev. A 77, 013825 (2008). [CrossRef]  

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

13. 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]  

14. 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 transformations of Maxwell’s equations,” Photon. Nanostruct.: Fundam. Applic. 6, 87–95 (2008). [CrossRef]  

15. 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]  

16. 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]  

17. 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]  

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

19. C. Li, K. Yao, and F. Li, “Two-dimensional electromagnetic cloaks with non-conformal inner and outer boundaries,” Opt. Express , 16, 19366–19374 (2008). [CrossRef]  

20. W. Yan, M. Yan, Z. Ruan, and M. Qiu, “Coordinate transformations make perfect invisibility cloaks with arbitrary shape”, New. J. Phys. 10, 043040 (2008). [CrossRef]  

21. 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]  

22. 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]  

23. 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]  

24. I. I. Smolyaninov, Y. J. Hung, and C. C. Davis, “Two-dimensional metamaterial structure exhibiting reduced visibility at 500 nm,” Opt. Lett. 33, 1342–1344 (2008). [CrossRef]   [PubMed]  

25. W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shavlaev, “Optical cloaking with metamaterials,” Nat. Photon. 1, 224–227 (2007). [CrossRef]  

26. P. Collins and J. McGuirk, “A novel methodology for deriving improved material parameter sets for simplified cylindrical cloaks,” J. Opt. A , 11, 015104 (2009). [CrossRef]  

27. S. Xi, H. Chen, B. Zhang, B.-I. Wu, and J. A. Kong, “A route to quasi-perfect invisibility cylindrical cloaks without extreme values in the parameters,” http://arxiv.org/abs/0812.1995 (2008).

28. E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973). [CrossRef]  

29. B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988). [CrossRef]  

30. B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11, 1491–1499 (1994). [CrossRef]  

31. Y. You, G. W. Kattawar, P.-W. Zhai, and P. Yang, “Zero-backscatter cloak for aspherical particles using a generalized DDA formalism,” Opt. Express 16, 2068–2079 (2008). [CrossRef]   [PubMed]  

References

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  1. A. J. Ward and J. B. Pendry, “Refraction and geometry in Maxwell’s equations,” J. Mod. Opt. 43, 773–793 (1996).
    [Crossref]
  2. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
    [Crossref] [PubMed]
  3. U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” New J. Phys. 8, 247 (2006).
    [Crossref]
  4. D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express 14, 9794–9804 (2006).
    [Crossref] [PubMed]
  5. U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
    [Crossref] [PubMed]
  6. U. Leonhardt, “Notes on conformal invisibility devices,” New J. Phys. 8, 118 (2006).
    [Crossref]
  7. D. Schurig, J. B. Pendry, and D. R. Smith, “Transformation-designed optical elements,” Opt. Express 15, 14772–14782 (2007).
    [Crossref] [PubMed]
  8. S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E 74, 036621 (2006).
    [Crossref]
  9. H. Chen, B.-I. Wu, B. Zhang, and J. A. Kong, “Electromagnetic wave interaction with a metamaterial cloak,” Phys. Rev. Lett. 99, 063903 (2007).
    [Crossref] [PubMed]
  10. B. Zhang, H. Chen, B.-I. Wu, Y. Luo, L. Ran, and J. A. Kong, “Response of a cylindrical invisibility cloak to electromagnetic waves,” Phys. Rev. B 76, 121101 (2007).
    [Crossref]
  11. H. Ma, S. Qu, Z. Xu, J. Zhang, B Chen, and J. Wang, “Material parameter equation for elliptical cylindrical cloaks,” Phys. Rev. A 77, 013825 (2008).
    [Crossref]
  12. D.-H. Kwon and D. H. Werner, “Two-dimensional eccentric elliptic electromagnetic cloaks,” Appl. Phys. Lett. 92, 013505 (2008).
    [Crossref]
  13. 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]
  14. 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 transformations of Maxwell’s equations,” Photon. Nanostruct.: Fundam. Applic. 6, 87–95 (2008).
    [Crossref]
  15. 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]
  16. 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]
  17. 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]
  18. C. Li and F. Li, “Two-dimensional electromagnetic cloaks with arbitrary geometries,” Opt. Express 16, 13414–13420 (2008).
    [Crossref] [PubMed]
  19. C. Li, K. Yao, and F. Li, “Two-dimensional electromagnetic cloaks with non-conformal inner and outer boundaries,” Opt. Express,  16, 19366–19374 (2008).
    [Crossref]
  20. W. Yan, M. Yan, Z. Ruan, and M. Qiu, “Coordinate transformations make perfect invisibility cloaks with arbitrary shape”, New. J. Phys. 10, 043040 (2008).
    [Crossref]
  21. 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]
  22. 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]
  23. 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]
  24. I. I. Smolyaninov, Y. J. Hung, and C. C. Davis, “Two-dimensional metamaterial structure exhibiting reduced visibility at 500 nm,” Opt. Lett. 33, 1342–1344 (2008).
    [Crossref] [PubMed]
  25. W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shavlaev, “Optical cloaking with metamaterials,” Nat. Photon. 1, 224–227 (2007).
    [Crossref]
  26. P. Collins and J. McGuirk, “A novel methodology for deriving improved material parameter sets for simplified cylindrical cloaks,” J. Opt. A,  11, 015104 (2009).
    [Crossref]
  27. S. Xi, H. Chen, B. Zhang, B.-I. Wu, and J. A. Kong, “A route to quasi-perfect invisibility cylindrical cloaks without extreme values in the parameters,” http://arxiv.org/abs/0812.1995 (2008).
  28. E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
    [Crossref]
  29. B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
    [Crossref]
  30. B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11, 1491–1499 (1994).
    [Crossref]
  31. Y. You, G. W. Kattawar, P.-W. Zhai, and P. Yang, “Zero-backscatter cloak for aspherical particles using a generalized DDA formalism,” Opt. Express 16, 2068–2079 (2008).
    [Crossref] [PubMed]

2009 (2)

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]

P. Collins and J. McGuirk, “A novel methodology for deriving improved material parameter sets for simplified cylindrical cloaks,” J. Opt. A,  11, 015104 (2009).
[Crossref]

2008 (13)

I. I. Smolyaninov, Y. J. Hung, and C. C. Davis, “Two-dimensional metamaterial structure exhibiting reduced visibility at 500 nm,” Opt. Lett. 33, 1342–1344 (2008).
[Crossref] [PubMed]

Y. You, G. W. Kattawar, P.-W. Zhai, and P. Yang, “Zero-backscatter cloak for aspherical particles using a generalized DDA formalism,” Opt. Express 16, 2068–2079 (2008).
[Crossref] [PubMed]

H. Ma, S. Qu, Z. Xu, J. Zhang, B Chen, and J. Wang, “Material parameter equation for elliptical cylindrical cloaks,” Phys. Rev. A 77, 013825 (2008).
[Crossref]

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

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]

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 transformations of Maxwell’s equations,” Photon. Nanostruct.: Fundam. Applic. 6, 87–95 (2008).
[Crossref]

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]

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]

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]

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

C. Li, K. Yao, and F. Li, “Two-dimensional electromagnetic cloaks with non-conformal inner and outer boundaries,” Opt. Express,  16, 19366–19374 (2008).
[Crossref]

W. Yan, M. Yan, Z. Ruan, and M. Qiu, “Coordinate transformations make perfect invisibility cloaks with arbitrary shape”, New. J. Phys. 10, 043040 (2008).
[Crossref]

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]

2007 (4)

D. Schurig, J. B. Pendry, and D. R. Smith, “Transformation-designed optical elements,” Opt. Express 15, 14772–14782 (2007).
[Crossref] [PubMed]

H. Chen, B.-I. Wu, B. Zhang, and J. A. Kong, “Electromagnetic wave interaction with a metamaterial cloak,” Phys. Rev. Lett. 99, 063903 (2007).
[Crossref] [PubMed]

B. Zhang, H. Chen, B.-I. Wu, Y. Luo, L. Ran, and J. A. Kong, “Response of a cylindrical invisibility cloak to electromagnetic waves,” Phys. Rev. B 76, 121101 (2007).
[Crossref]

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shavlaev, “Optical cloaking with metamaterials,” Nat. Photon. 1, 224–227 (2007).
[Crossref]

2006 (7)

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]

S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E 74, 036621 (2006).
[Crossref]

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

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

D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express 14, 9794–9804 (2006).
[Crossref] [PubMed]

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

U. Leonhardt, “Notes on conformal invisibility devices,” New J. Phys. 8, 118 (2006).
[Crossref]

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 (1)

1988 (1)

B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
[Crossref]

1973 (1)

E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[Crossref]

Cai, W.

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shavlaev, “Optical cloaking with metamaterials,” Nat. Photon. 1, 224–227 (2007).
[Crossref]

Chen, B

H. Ma, S. Qu, Z. Xu, J. Zhang, B Chen, and J. Wang, “Material parameter equation for elliptical cylindrical cloaks,” Phys. Rev. A 77, 013825 (2008).
[Crossref]

Chen, H.

H. Chen, B.-I. Wu, B. Zhang, and J. A. Kong, “Electromagnetic wave interaction with a metamaterial cloak,” Phys. Rev. Lett. 99, 063903 (2007).
[Crossref] [PubMed]

B. Zhang, H. Chen, B.-I. Wu, Y. Luo, L. Ran, and J. A. Kong, “Response of a cylindrical invisibility cloak to electromagnetic waves,” Phys. Rev. B 76, 121101 (2007).
[Crossref]

S. Xi, H. Chen, B. Zhang, B.-I. Wu, and J. A. Kong, “A route to quasi-perfect invisibility cylindrical cloaks without extreme values in the parameters,” http://arxiv.org/abs/0812.1995 (2008).

Cheng, Q.

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]

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]

Chettiar, U. K.

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shavlaev, “Optical cloaking with metamaterials,” Nat. Photon. 1, 224–227 (2007).
[Crossref]

Chin, J. Y.

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]

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]

Collins, P.

P. Collins and J. McGuirk, “A novel methodology for deriving improved material parameter sets for simplified cylindrical cloaks,” J. Opt. A,  11, 015104 (2009).
[Crossref]

Cui, T. J.

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]

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]

Cummer, S. A.

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 transformations of Maxwell’s equations,” Photon. Nanostruct.: Fundam. Applic. 6, 87–95 (2008).
[Crossref]

S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E 74, 036621 (2006).
[Crossref]

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]

Davis, C. C.

Draine, B. T.

B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11, 1491–1499 (1994).
[Crossref]

B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
[Crossref]

Flatau, P. J.

Guenneau, S.

Hu, G.

Hu, J.

Hung, Y. J.

Jiang, W. X.

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]

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]

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]

Kattawar, G. W.

Kildishev, A. V.

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shavlaev, “Optical cloaking with metamaterials,” Nat. Photon. 1, 224–227 (2007).
[Crossref]

Kong, J. A.

H. Chen, B.-I. Wu, B. Zhang, and J. A. Kong, “Electromagnetic wave interaction with a metamaterial cloak,” Phys. Rev. Lett. 99, 063903 (2007).
[Crossref] [PubMed]

B. Zhang, H. Chen, B.-I. Wu, Y. Luo, L. Ran, and J. A. Kong, “Response of a cylindrical invisibility cloak to electromagnetic waves,” Phys. Rev. B 76, 121101 (2007).
[Crossref]

S. Xi, H. Chen, B. Zhang, B.-I. Wu, and J. A. Kong, “A route to quasi-perfect invisibility cylindrical cloaks without extreme values in the parameters,” http://arxiv.org/abs/0812.1995 (2008).

Kwon, D.-H.

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

Leonhardt, U.

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

U. Leonhardt, “Notes on conformal invisibility devices,” New J. Phys. 8, 118 (2006).
[Crossref]

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

Li, C.

Li, F.

Li, Z.

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]

Lin, X. Q.

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]

Liu, R.

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]

Luo, Y.

B. Zhang, H. Chen, B.-I. Wu, Y. Luo, L. Ran, and J. A. Kong, “Response of a cylindrical invisibility cloak to electromagnetic waves,” Phys. Rev. B 76, 121101 (2007).
[Crossref]

Ma, H.

H. Ma, S. Qu, Z. Xu, J. Zhang, B Chen, and J. Wang, “Material parameter equation for elliptical cylindrical cloaks,” Phys. Rev. A 77, 013825 (2008).
[Crossref]

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]

McGuirk, J.

P. Collins and J. McGuirk, “A novel methodology for deriving improved material parameter sets for simplified cylindrical cloaks,” J. Opt. A,  11, 015104 (2009).
[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]

Nicolet, A.

Pendry, J. B.

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 transformations of Maxwell’s equations,” Photon. Nanostruct.: Fundam. Applic. 6, 87–95 (2008).
[Crossref]

D. Schurig, J. B. Pendry, and D. R. Smith, “Transformation-designed optical elements,” Opt. Express 15, 14772–14782 (2007).
[Crossref] [PubMed]

D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express 14, 9794–9804 (2006).
[Crossref] [PubMed]

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

S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E 74, 036621 (2006).
[Crossref]

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]

Pennypacker, C. R.

E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[Crossref]

Philbin, T. G.

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

Popa, B.-I.

S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E 74, 036621 (2006).
[Crossref]

Purcell, E. M.

E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[Crossref]

Qiu, M.

W. Yan, M. Yan, Z. Ruan, and M. Qiu, “Coordinate transformations make perfect invisibility cloaks with arbitrary shape”, New. J. Phys. 10, 043040 (2008).
[Crossref]

Qu, S.

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]

H. Ma, S. Qu, Z. Xu, J. Zhang, B Chen, and J. Wang, “Material parameter equation for elliptical cylindrical cloaks,” Phys. Rev. A 77, 013825 (2008).
[Crossref]

Rahm, M.

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 transformations of Maxwell’s equations,” Photon. Nanostruct.: Fundam. Applic. 6, 87–95 (2008).
[Crossref]

Ran, L.

B. Zhang, H. Chen, B.-I. Wu, Y. Luo, L. Ran, and J. A. Kong, “Response of a cylindrical invisibility cloak to electromagnetic waves,” Phys. Rev. B 76, 121101 (2007).
[Crossref]

Roberts, D. A.

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 transformations of Maxwell’s equations,” Photon. Nanostruct.: Fundam. Applic. 6, 87–95 (2008).
[Crossref]

Ruan, Z.

W. Yan, M. Yan, Z. Ruan, and M. Qiu, “Coordinate transformations make perfect invisibility cloaks with arbitrary shape”, New. J. Phys. 10, 043040 (2008).
[Crossref]

Schurig, D.

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 transformations of Maxwell’s equations,” Photon. Nanostruct.: Fundam. Applic. 6, 87–95 (2008).
[Crossref]

D. Schurig, J. B. Pendry, and D. R. Smith, “Transformation-designed optical elements,” Opt. Express 15, 14772–14782 (2007).
[Crossref] [PubMed]

S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E 74, 036621 (2006).
[Crossref]

D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express 14, 9794–9804 (2006).
[Crossref] [PubMed]

J. B. Pendry, D. Schurig, 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]

Shavlaev, V. M.

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shavlaev, “Optical cloaking with metamaterials,” Nat. Photon. 1, 224–227 (2007).
[Crossref]

Smith, D. R.

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 transformations of Maxwell’s equations,” Photon. Nanostruct.: Fundam. Applic. 6, 87–95 (2008).
[Crossref]

D. Schurig, J. B. Pendry, and D. R. Smith, “Transformation-designed optical elements,” Opt. Express 15, 14772–14782 (2007).
[Crossref] [PubMed]

S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E 74, 036621 (2006).
[Crossref]

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

D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express 14, 9794–9804 (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]

Smolyaninov, I. I.

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]

Wang, J.

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]

H. Ma, S. Qu, Z. Xu, J. Zhang, B Chen, and J. Wang, “Material parameter equation for elliptical cylindrical cloaks,” Phys. Rev. A 77, 013825 (2008).
[Crossref]

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]

Werner, D. H.

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

Wu, B.-I.

B. Zhang, H. Chen, B.-I. Wu, Y. Luo, L. Ran, and J. A. Kong, “Response of a cylindrical invisibility cloak to electromagnetic waves,” Phys. Rev. B 76, 121101 (2007).
[Crossref]

H. Chen, B.-I. Wu, B. Zhang, and J. A. Kong, “Electromagnetic wave interaction with a metamaterial cloak,” Phys. Rev. Lett. 99, 063903 (2007).
[Crossref] [PubMed]

S. Xi, H. Chen, B. Zhang, B.-I. Wu, and J. A. Kong, “A route to quasi-perfect invisibility cylindrical cloaks without extreme values in the parameters,” http://arxiv.org/abs/0812.1995 (2008).

Xi, S.

S. Xi, H. Chen, B. Zhang, B.-I. Wu, and J. A. Kong, “A route to quasi-perfect invisibility cylindrical cloaks without extreme values in the parameters,” http://arxiv.org/abs/0812.1995 (2008).

Xu, Z.

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]

H. Ma, S. Qu, Z. Xu, J. Zhang, B Chen, and J. Wang, “Material parameter equation for elliptical cylindrical cloaks,” Phys. Rev. A 77, 013825 (2008).
[Crossref]

Yan, M.

W. Yan, M. Yan, Z. Ruan, and M. Qiu, “Coordinate transformations make perfect invisibility cloaks with arbitrary shape”, New. J. Phys. 10, 043040 (2008).
[Crossref]

Yan, W.

W. Yan, M. Yan, Z. Ruan, and M. Qiu, “Coordinate transformations make perfect invisibility cloaks with arbitrary shape”, New. J. Phys. 10, 043040 (2008).
[Crossref]

Yang, P.

Yao, K.

You, Y.

Yu, G. X.

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]

Zhai, P.-W.

Zhang, B.

H. Chen, B.-I. Wu, B. Zhang, and J. A. Kong, “Electromagnetic wave interaction with a metamaterial cloak,” Phys. Rev. Lett. 99, 063903 (2007).
[Crossref] [PubMed]

B. Zhang, H. Chen, B.-I. Wu, Y. Luo, L. Ran, and J. A. Kong, “Response of a cylindrical invisibility cloak to electromagnetic waves,” Phys. Rev. B 76, 121101 (2007).
[Crossref]

S. Xi, H. Chen, B. Zhang, B.-I. Wu, and J. A. Kong, “A route to quasi-perfect invisibility cylindrical cloaks without extreme values in the parameters,” http://arxiv.org/abs/0812.1995 (2008).

Zhang, J.

H. Ma, S. Qu, Z. Xu, J. Zhang, B Chen, and J. Wang, “Material parameter equation for elliptical cylindrical cloaks,” Phys. Rev. A 77, 013825 (2008).
[Crossref]

Zhou, X.

Zolla, F.

Appl. Phys. Lett. (1)

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

Astrophys. J. (2)

E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[Crossref]

B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
[Crossref]

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]

J. Opt. A (1)

P. Collins and J. McGuirk, “A novel methodology for deriving improved material parameter sets for simplified cylindrical cloaks,” J. Opt. A,  11, 015104 (2009).
[Crossref]

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

J. Phys. D: Appl. Phys. (1)

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]

Nat. Photon. (1)

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shavlaev, “Optical cloaking with metamaterials,” Nat. Photon. 1, 224–227 (2007).
[Crossref]

New J. Phys. (2)

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

U. Leonhardt, “Notes on conformal invisibility devices,” New J. Phys. 8, 118 (2006).
[Crossref]

New. J. Phys. (1)

W. Yan, M. Yan, Z. Ruan, and M. Qiu, “Coordinate transformations make perfect invisibility cloaks with arbitrary shape”, New. J. Phys. 10, 043040 (2008).
[Crossref]

Opt. Express (8)

Opt. Lett. (2)

Photon. Nanostruct.: Fundam. Applic. (1)

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 transformations of Maxwell’s equations,” Photon. Nanostruct.: Fundam. Applic. 6, 87–95 (2008).
[Crossref]

Phys. Rev. A (1)

H. Ma, S. Qu, Z. Xu, J. Zhang, B Chen, and J. Wang, “Material parameter equation for elliptical cylindrical cloaks,” Phys. Rev. A 77, 013825 (2008).
[Crossref]

Phys. Rev. B (1)

B. Zhang, H. Chen, B.-I. Wu, Y. Luo, L. Ran, and J. A. Kong, “Response of a cylindrical invisibility cloak to electromagnetic waves,” Phys. Rev. B 76, 121101 (2007).
[Crossref]

Phys. Rev. E (2)

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]

S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E 74, 036621 (2006).
[Crossref]

Phys. Rev. Lett. (1)

H. Chen, B.-I. Wu, B. Zhang, and J. A. Kong, “Electromagnetic wave interaction with a metamaterial cloak,” Phys. Rev. Lett. 99, 063903 (2007).
[Crossref] [PubMed]

Science (3)

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

J. B. Pendry, D. Schurig, 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]

Other (1)

S. Xi, H. Chen, B. Zhang, B.-I. Wu, and J. A. Kong, “A route to quasi-perfect invisibility cylindrical cloaks without extreme values in the parameters,” http://arxiv.org/abs/0812.1995 (2008).

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Figures (5)

Fig. 1.
Fig. 1. (a) A toroid with b = R/2; (b) The toroidal coordinates spanned by unit vectors er , eu and ev ; (c) A toroidal cloak with b = R/2 and a = b/2.
Fig. 2.
Fig. 2. 3-D view of the propagation of light rays (blue lines) and associated wave fronts (red lines) in a toroidal cloak.
Fig. 3.
Fig. 3. Diagonal components of the permittivity and permeability tensors for a toroidal cloak with b = R/2 and a = b/2. The angle u varies from 0° to 90°.
Fig. 4.
Fig. 4. Same as Fig. 3, except that b = R/10.
Fig. 5.
Fig. 5. The DDA simulation of the electric field distributions in the vicinity of toroidal cloaks subject to incident plane waves. (a) a=3/4b, b=R/2 and R=0.45λ ; (b) a=3/4b, b = 3/4R, and R = 1.196λ. The scales are in units of λ.

Equations (22)

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x = ( R + r sin u ) cos v ,
y = ( R + r sin u ) sin v ,
z = r cos u ,
r ' = a + b a b r , u ' = u , v ' = v ,
ε i ' j ' = μ i ' j ' = det ( Λ i ' i ) 1 Λ i ' i Λ j ' j ,
ε ' = μ ' = det ( Λ ) 1 ΛΛ T .
( x , y , z ) ( r , u , v ) ( r ' , u ' , v ' ) ( x ' , y ' , z ' ) .
Λ 1 = ( r , u , v ) ( x , y , v ) , Λ 2 = ( r ' , u ' , v ' ) ( r , u , v ) , Λ 3 = ( x ' , y ' , z ' ) ( r ' , u ' , v ' ) .
Λ = ( x ' , y ' , z ' ) x , y , z = Λ 3 Λ 2 Λ 1 .
ε ' = μ ' = det ( Λ 1 1 ) det ( Λ 3 ) det ( Λ 2 ) Λ 3 Λ 2 Q 1 1 Λ 2 T Λ 3 T ,
T ruv = ( D Λ 3 1 ) T xyz ( D Λ 3 1 ) T ,
ε ' ruv = μ ' ruv = det ( Λ 1 1 ) det ( Λ 3 ) det ( Λ 2 ) D Λ 2 Q 1 1 Λ 2 T Q 3 D 1 ,
Q 1 = diag [ 1 , r 2 , ( R + r sin u ) 2 ] ,
Λ 2 = diag [ b a b , 1,1 ] ,
Q 3 = diag [ 1 , r ' 2 , ( R + r ' sin u ) 2 ] ,
D = diag [ 1 , r ' , ( R + r ' sin u ) 2 ] .
ε r r = μ r r = r a r ( b a ) R + b ( r a ) sin u ( b a ) ( R + r sin u ) ,
ε u u = μ u u = r r a ( b a ) R + b ( r a ) sin u ( b a ) ( R + r sin u ) ,
ε v v = μ v v = b 2 b a r a r R + r sin u ( b a ) R + b ( r a ) sin u ,
ε r = μ r = r R 1 r ,
ε ϕ = μ ϕ = r r R 1 ,
ε z = μ z = ( R 2 R 2 R 1 ) 2 r R 1 r ,

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