Transformation media producing quasi-perfect isotropic emission

doi:10.1364/OE.19.020551

« Show journal navigation

Transformation media producing quasi-perfect isotropic emission

Paul-Henri Tichit, Shah Nawaz Burokur, and André de Lustrac »View Author Affiliations

Optics Express, Vol. 19, Issue 21, pp. 20551-20556 (2011)
http://dx.doi.org/10.1364/OE.19.020551

View Full Text Article

Acrobat PDF (1182 KB)

Journal Search
Article Lookup

Article Tools

Citations

Alert me when this article is cited
Export Citation/Save

Article Navigation

▸ Article Outline
– Abstract
1. Introduction
2. Transformation formulations
3. Simulation and results
4. Conclusion
– References and links

Article Tools
Full Text
Article Info
References
Cited By
Figures
Metrics
Download PDF

Viewing Equations in the
HTML Article

Optimal Viewing Recommendations

Full Text
Article Info
References (30)
Cited By
Figures (4)
Metrics

Abstract

Using the idea of wave manipulation via transformation optics, we propose a way to create a quasi-perfect isotropic emission from a directional one. The manipulation is enabled by composite metamaterials that correspond to a space stretching around the source. It is shown that the directive radiation of a plane source larger than the operating wavelength can be transformed into an isotropic one by modifying the electromagnetic properties of the space around it. A set of parameters allowing practical realization of the proposed device is defined. Numerical simulations using Finite Element Method (FEM) are performed to illustrate the proposed coordinate transformation. This idea, which consists in strongly reducing the apparent size of a radiating source, can find various applications in novel antenna design techniques.

1. Introduction

Transformation optics is a mathematical tool that consists in generating a new transformed space from an initial one where solutions of Maxwell’s equations are known. As in general relativity, the first step is to imagine a virtual space with the desired topological properties, which will contain the underlying physics. This approach has been revived when J. B. Pendry et al. [1

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006). [CrossRef] [PubMed]

] have proposed an interpretation where permeability and permittivity tensors components can be viewed as a material in the original space. It is as if the new material mimicks the defined topological space. Since this pioneering work of J. B. Pendry and that of U. Leonhardt et al. [2

U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006). [CrossRef] [PubMed]

], transformation optics is an emerging field where Maxwell’s equations are form invariant under a coordinate transformation. It offers an unconventional strategy to the design of novel class metamaterial devices. The most intriguing application conceived remains the invisibility cloak whose designs have been presented in microwave [3

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(5801), 977–980 (2006). [CrossRef] [PubMed]

–5

B. Kanté, D. Germain, and A. de Lustrac, “Experimental demonstration of non-magnetic metamaterial cloak at microwave frequencies,” Phys. Rev. B 80(20), 201104 (2009). [CrossRef]

] and optical regimes [6

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics 1(4), 224–227 (2007). [CrossRef]

–8

L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics 3(8), 461–463 (2009). [CrossRef]

]. Other interesting wave manipulation applications such as concentrators [9

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,” Photonics Nanostruct. Fundam. Appl. 6(1), 87–95 (2008). [CrossRef]

], rotators [10

H. Chen, B. Hou, S. Chen, X. Ao, W. Wen, and C. T. Chan, “Design and experimental realization of a broadband transformation media field rotator at microwave frequencies,” Phys. Rev. Lett. 102(18), 183903 (2009). [CrossRef] [PubMed]

], wormholes [11

A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Electromagnetic wormholes and virtual magnetic monopoles from metamaterials,” Phys. Rev. Lett. 99(18), 183901 (2007). [CrossRef] [PubMed]

], waveguide transitions and bends [12

M. Rahm, D. A. Roberts, J. B. Pendry, and D. R. Smith, “Transformation-optical design of adaptive beam bends and beam expanders,” Opt. Express 16(15), 11555–11567 (2008). [CrossRef] [PubMed]

–16

P.-H. Tichit, S. N. Burokur, and A. de Lustrac, “Waveguide taper engineering using coordinate transformation technology,” Opt. Express 18(2), 767–772 (2010). [CrossRef] [PubMed]

] have also been proposed. Concerning antenna applications, focusing lens antennas have been theoretically designed [17

W. X. Jiang, T. J. Cui, H. F. Ma, X. M. Yang, and Q. Cheng, “Layered high-gain lens antennas via discrete optical transformation,” Appl. Phys. Lett. 93(22), 221906 (2008). [CrossRef]

,18

Z. L. Mei, J. Bai, T. M. Niu, and T. J. Cui, “A planar focusing antenna design using quasi-conformal mapping,” PIER 13, 261–273 (2010). [CrossRef]

]. The performances of an omnidirectional retroreflector [19

Y. G. Ma, C. K. Ong, T. Tyc, and U. Leonhardt, “An omnidirectional retroreflector based on the transmutation of dielectric singularities,” Nat. Mater. 8(8), 639–642 (2009). [CrossRef] [PubMed]

] based on the transmutation of singularities [20

T. Tyc and U. Leonhardt, “Transmutation of singularities in optical instruments,” N. J. Phys. 10(11), 115038 (2008). [CrossRef]

] and Luneberg lenses [21

N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater. 9(2), 129–132 (2010). [CrossRef] [PubMed]

,22

H. F. Ma and T. J. Cui, “Three-dimensional broadband ground-plane cloak made of metamaterials,” Nat Commun 1(3), 1–6 (2010). [CrossRef] [PubMed]

] have been experimentally demonstrated. Recently, techniques of source transformation [23

Y. Luo, J. Zhang, L. Ran, H. Chen, and J. A. Kong, “Controlling the emission of electromagnetic source,” PIERS 4(7), 795–800 (2008). [CrossRef]

–28

P.-H. Tichit, S. N. Burokur, D. Germain, and A. de Lustrac, “Coordinate transformation based ultra-directive emission,” Electron. Lett. 47(10), 580–582 (2011). [CrossRef]

] have offered new opportunities for the design of active devices with source distribution included in the transformed space. Using this last approach, we have designed an ultra-directive emission by stretching a source into an extended coherent radiator [26

P.-H. Tichit, S. N. Burokur, and A. de Lustrac, “Ultradirective antenna via transformation optics,” J. Appl. Phys. 105(10), 104912 (2009). [CrossRef]

–28

P.-H. Tichit, S. N. Burokur, D. Germain, and A. de Lustrac, “Coordinate transformation based ultra-directive emission,” Electron. Lett. 47(10), 580–582 (2011). [CrossRef]

2. Transformation formulations

It is our aim in this present work to demonstrate a quasi-perfect isotropic emission from a plane source presenting a directive emission by manipulating the apparent size of the latter source. Such an antenna can be very important in communication systems since it allows designing a reference antenna for measurement setups. To do so, we define a transformation which stretches exponentially the area where the source is located. The wavelength will then be much larger in the vicinity of the source, which makes the latter appear much smaller from outside. To our knowledge, exponential transformation has not been developed in literature. The theoretical analysis devoted to this coordinate transformation is presented and numerical simulations are performed using finite element method to validate the proposed concept.

In this study, we will focus our attention on the transformation of radiating sources and our plan will be to show how coordinate transformation can be applied to transform directive emissions into isotropic ones. We will start from the basic transformation media approach. An intuitive schematic principle to illustrate the proposed method is presented in Fig. 1 . Let us consider a source radiating in a circular space as shown in Fig. 1(a) and a circular region bounded by the blue circle around this source limits the radiation zone. The “space stretching” coordinate transformation consists in stretching exponentially the central zone of this delimited circular region represented by the red circle as illustrated in Fig. 1(b). The expansion procedure is further followed by a compression of the annular region formed between the red and blue circles so as to secure a good impedance matching with free space. Figure 1(c) summarizes the exponential form of our coordinate transformation. The diameter of the transformed, which will be the generated metamaterial circular medium is noted D. Mathematically this transformation is expressed as:

{\begin{cases} r' = α (1 - e^{q r}) \\ θ' = θ \\ z' = z \end{cases} with α = \frac{D}{2} \frac{1}{1 - e^{\frac{qD}{2}}}

(1)

where r’, θ’, and z’ are the coordinates in the transformed cylindrical space, and r, θ and z are those in the initial cylindrical space. In the initial space, we assume free space, with isotropic permittivity and permeability tensors ε₀ and μ_0. It can be noted that only parameter q appears in Eq. (1). q (in m⁻¹) must be negative in order to secure the impedance matching condition. This parameter is an expansion factor which can be physically viewed as to what extent space is expanded. A high (negative) value of q means a high expansion whereas a low (negative) value of q means a nearly zero expansion. The new material can then be described by permeability and permittivity tensors:

\bar{\bar{ε}} = \bar{\bar{ψ}} ε_{0} and \bar{\bar{μ}} = \bar{\bar{ψ}} μ_{0} with ψ^{i'j'} = \frac{J_{i}^{i'} J_{j}^{j'} δ^{ij}}{det(J)}

(2)

where

J_{α}^{α^{'}} = \frac{\partial {x^{'}}^{α}}{\partial x^{α}}

represents the Jacobian transformation matrix of Eq. (1) and

δ^{i j}

is the Kronecker symbol. Both electromagnetic parameters ε and μ have the same behavior, allowing an impedance matching with vacuum outside the transformed space. The inverse transformation is obtained from the initial transformation of Eq. (1) and derived by a substitution method, enabling the circular metamaterial design which leads to anisotropic permittivity and permeability tensors. More details on similar calculations can be found in [29

U. Leonhardt and T. G. Philbin, “Transformation optics and the geometry of light,” Prog. Opt. 53, 69–152 (2009). [CrossRef]

Fig. 1 Space stretching coordinate transformation: (a) the initial space (r, θ, z) is bounded by the blue circle; (b) the transformed space (r’, θ’, z’) is formed by an expansion of the central zone defined by the red circle and a compression of the annular region between the red and blue circles; (c) working principle of the transformation.

By substituting the new coordinate system in the tensor components, and after some simplifications, the material parameters are derived. Calculations lead to permeability and permittivity tensors given in the diagonal base by:

\bar{\bar{ψ}} = (\begin{matrix} ψ_{r r} & 0 & 0 \\ 0 & ψ_{θ θ} & 0 \\ 0 & 0 & ψ_{z z} \end{matrix}) = (\begin{matrix} \frac{q r (r' - α)}{r'} & 0 & 0 \\ 0 & \frac{r'}{q r (r' - α)} & 0 \\ 0 & 0 & \frac{r}{r' q (r' - α)} \end{matrix}) \begin{matrix} with & r = \frac{ln (1- \frac{r'}{α})}{q} \end{matrix}

(3)

The components in the Cartesian coordinate system are calculated and are as follows:

{\begin{cases} ψ_{x x} = ψ_{rr} \cos^{2} (θ)+ ψ_{θ θ} \sin^{2} (θ) \\ ψ_{x y} = ψ_{y x} = (ψ_{rr} - ψ_{θ θ}) \sin (θ) \cos (θ) \\ ψ_{y y} = ψ_{rr} \sin^{2} (θ) + ψ_{θ θ} \cos^{2} (θ) \end{cases}

(4)

The ε and μ tensors components present the same behavior as given in Eq. (4).

Figure 2 shows the variation of the permittivity and permeability tensor components in the new generated transformed space. The geometrical dimension D is chosen to be 20 cm and parameter q is fixed to −40 m⁻¹. It can be noted that components ψ_xx, ψ_yy and ψ_zz present variations and an extremum, that are simple to realize with commonly used metamaterials by reducing their inhomogeneous dependence [27

P.-H. Tichit, S. N. Burokur, D. Germain, and A. de Lustrac, “Design and experimental demonstration of a high-directive emission with transformation optics,” Phys. Rev. B 83(15), 155108 (2011). [CrossRef]

]. At the center of the transformed space, ε and µ present very low values (<< 1). Consequently, light velocity and the corresponding wavelength are much higher than in vacuum. The width of the plane source then appears very small compared to wavelength and the source can then be regarded as a radiating wire, which is in fact an isotropic source. The merit of this transformation depends effectively on the expansion factor q value and more generally, it can be applied to a wide range of electromagnetic objects, where the effective size can be reduced compared to a given wavelength.

Fig. 2 Variation in the permeability and permittivity tensor components of the transformed space for D = 20 cm and q = −40 m⁻¹.

By fixing the electric field directed along the z-axis and by adjusting the dispersion equation without changing propagation in the structure, the following reduced parameters can be obtained [3

{\begin{matrix} μ_{r r} = 1 \\ μ_{θ θ} = {(\frac{r'}{q r (r' - α)})}^{2} \\ ε_{z z} = {(\frac{r}{r'})}^{2} \end{matrix}

(5)

In Eq. (5), the parameters present positive values which can be easily achieved firstly by discretizing their continuous profile and secondly by using metamaterial resonators where magnetic and electric responses can be tailored and controlled. Such control has been lately performed in [27

] with a 200 MHz bandwidth at 10 GHz. A possible prototype of our proposed device is presented in Fig. 3 where permeability and permittivity metamaterial layers are used to assure both μ_θθ and ε_zz gradient.

Fig. 3 Schematic of a possible prototype with tailored permittivity and permeability values in a cylindrical configuration.

3. Simulation and results

The theoretical underlying physics of a “space stretching” coordinate transformation involved in this present study having been described, numerical verifications of the method using FEM based commercial solver Comsol Multiphysics [30

Comsol MULTIPHYSICS Modeling, (http://www.comsol.com).

] are performed to design and characterize the transformed quasi-isotropic emission. A plane current source is used as excitation to validate the concept.

Figure 4 presents simulations results of the source radiating in the initial circular space at an operating frequency of 4 GHz. The current direction of the source is supposed to be along the z-axis. Simulations are performed in a Transverse Electric (TE) mode with the electric field polarized along z-direction. The surface current source is considered to have a width of 10 cm, which is greater than the 7.5 cm wavelength at 4 GHz. Radiation boundary conditions are placed around the calculation domain in order to plot the radiation properties. Continuity of the field is assured in the interior boundaries. As shown in Fig. 4, different values for the expansion factor q are used in the simulations and as it can be observed, the emission becomes more and more isotropic as the q factor decreases from −5 m⁻¹ to −40 m⁻¹. As stated previously and verified from the different electric field distribution patterns, a high negative value of q leads to a quasi-perfect isotropic emission since the space expansion is higher. In Fig. 4(e), we can note the far-field isotropic pattern of our device compared to the directive source. Figure 4(f) gives an insight of the influence of the q parameter on the proposed coordinate transformation.

Fig. 4 Simulated electric field distribution for a TE wave polarization at 4 GHz: (a) current plane source used as excitation for the transformation. The current direction is perpendicular to the plane of the figure; [(b)-(d)] verification of the transformation for different values of expansion factor q; (e) Far field radiation pattern of the emission with (q=−40) and without transformation;(f) influence of the q parameter on the proposed coordinate transformation. The emitted radiation is more and more isotropic as q tends to high negative values.

4. Conclusion

In conclusion, using transformation optics, we have proposed a concept to manipulate electromagnetic waves and achieve a quasi-perfect isotropic emission from a directional source. The latter manipulation is enabled by using composite metamaterials corresponding to a space stretching around the source by coordinate transformation. Numerical verifications have been performed where the radiation of a plane source which is directive has been transformed into an isotropic emission by reducing the apparent size of this source. The proposed device can therefore be potentially useful in the reduction of an object’s Radar Cross Section (RCS).

References and links

1.	J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006). [CrossRef] [PubMed]
2.	U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006). [CrossRef] [PubMed]
3.	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(5801), 977–980 (2006). [CrossRef] [PubMed]
4.	R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323(5912), 366–369 (2009). [CrossRef] [PubMed]
5.	B. Kanté, D. Germain, and A. de Lustrac, “Experimental demonstration of non-magnetic metamaterial cloak at microwave frequencies,” Phys. Rev. B 80(20), 201104 (2009). [CrossRef]
6.	W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics 1(4), 224–227 (2007). [CrossRef]
7.	J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8(7), 568–571 (2009). [CrossRef] [PubMed]
8.	L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics 3(8), 461–463 (2009). [CrossRef]
9.	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,” Photonics Nanostruct. Fundam. Appl. 6(1), 87–95 (2008). [CrossRef]
10.	H. Chen, B. Hou, S. Chen, X. Ao, W. Wen, and C. T. Chan, “Design and experimental realization of a broadband transformation media field rotator at microwave frequencies,” Phys. Rev. Lett. 102(18), 183903 (2009). [CrossRef] [PubMed]
11.	A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Electromagnetic wormholes and virtual magnetic monopoles from metamaterials,” Phys. Rev. Lett. 99(18), 183901 (2007). [CrossRef] [PubMed]
12.	M. Rahm, D. A. Roberts, J. B. Pendry, and D. R. Smith, “Transformation-optical design of adaptive beam bends and beam expanders,” Opt. Express 16(15), 11555–11567 (2008). [CrossRef] [PubMed]
13.	M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, “Optical design of reflectionless complex media by finite embedded coordinate transformations,” Phys. Rev. Lett. 100(6), 063903 (2008). [CrossRef] [PubMed]
14.	L. Lin, W. Wang, J. Cui, C. Du, and X. Luo, “Design of electromagnetic refractor and phase transformer using coordinate transformation theory,” Opt. Express 16(10), 6815–6821 (2008). [CrossRef] [PubMed]
15.	J. Huangfu, S. Xi, F. Kong, J. Zhang, H. Chen, D. Wang, B.-I. Wu, L. Ran, and J. A. Kong, “Application of coordinate transformation in bent waveguide,” J. Appl. Phys. 104(1), 014502 (2008). [CrossRef]
16.	P.-H. Tichit, S. N. Burokur, and A. de Lustrac, “Waveguide taper engineering using coordinate transformation technology,” Opt. Express 18(2), 767–772 (2010). [CrossRef] [PubMed]
17.	W. X. Jiang, T. J. Cui, H. F. Ma, X. M. Yang, and Q. Cheng, “Layered high-gain lens antennas via discrete optical transformation,” Appl. Phys. Lett. 93(22), 221906 (2008). [CrossRef]
18.	Z. L. Mei, J. Bai, T. M. Niu, and T. J. Cui, “A planar focusing antenna design using quasi-conformal mapping,” PIER 13, 261–273 (2010). [CrossRef]
19.	Y. G. Ma, C. K. Ong, T. Tyc, and U. Leonhardt, “An omnidirectional retroreflector based on the transmutation of dielectric singularities,” Nat. Mater. 8(8), 639–642 (2009). [CrossRef] [PubMed]
20.	T. Tyc and U. Leonhardt, “Transmutation of singularities in optical instruments,” N. J. Phys. 10(11), 115038 (2008). [CrossRef]
21.	N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater. 9(2), 129–132 (2010). [CrossRef] [PubMed]
22.	H. F. Ma and T. J. Cui, “Three-dimensional broadband ground-plane cloak made of metamaterials,” Nat Commun 1(3), 1–6 (2010). [CrossRef] [PubMed]
23.	Y. Luo, J. Zhang, L. Ran, H. Chen, and J. A. Kong, “Controlling the emission of electromagnetic source,” PIERS 4(7), 795–800 (2008). [CrossRef]
24.	J. Allen, N. Kundtz, D. A. Roberts, S. A. Cummer, and D. R. Smith, “Electromagnetic source transformations using superellipse equations,” Appl. Phys. Lett. 94(19), 194101 (2009). [CrossRef]
25.	B. I. Popa, J. Allen, and S. A. Cummer, “Conformal array design with transformation electromagnetics,” Appl. Phys. Lett. 94(24), 244102 (2009). [CrossRef]
26.	P.-H. Tichit, S. N. Burokur, and A. de Lustrac, “Ultradirective antenna via transformation optics,” J. Appl. Phys. 105(10), 104912 (2009). [CrossRef]
27.	P.-H. Tichit, S. N. Burokur, D. Germain, and A. de Lustrac, “Design and experimental demonstration of a high-directive emission with transformation optics,” Phys. Rev. B 83(15), 155108 (2011). [CrossRef]
28.	P.-H. Tichit, S. N. Burokur, D. Germain, and A. de Lustrac, “Coordinate transformation based ultra-directive emission,” Electron. Lett. 47(10), 580–582 (2011). [CrossRef]
29.	U. Leonhardt and T. G. Philbin, “Transformation optics and the geometry of light,” Prog. Opt. 53, 69–152 (2009). [CrossRef]
30.	Comsol MULTIPHYSICS Modeling, (http://www.comsol.com).

OCIS Codes
(160.1190) Materials : Anisotropic optical materials
(260.2110) Physical optics : Electromagnetic optics
(260.2065) Physical optics : Effective medium theory
(160.3918) Materials : Metamaterials
(260.2710) Physical optics : Inhomogeneous optical media

ToC Category:
Physical Optics

History
Original Manuscript: August 1, 2011
Revised Manuscript: September 9, 2011
Manuscript Accepted: September 9, 2011
Published: October 3, 2011

Citation
Paul-Henri Tichit, Shah Nawaz Burokur, and André de Lustrac, "Transformation media producing quasi-perfect isotropic emission," Opt. Express 19, 20551-20556 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-21-20551

Sort: Author | Year | Journal | Reset

References

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312(5781), 1780–1782 (2006). [CrossRef] [PubMed]
U. Leonhardt, “Optical conformal mapping,” Science312(5781), 1777–1780 (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,” Science314(5801), 977–980 (2006). [CrossRef] [PubMed]
R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science323(5912), 366–369 (2009). [CrossRef] [PubMed]
B. Kanté, D. Germain, and A. de Lustrac, “Experimental demonstration of non-magnetic metamaterial cloak at microwave frequencies,” Phys. Rev. B80(20), 201104 (2009). [CrossRef]
W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics1(4), 224–227 (2007). [CrossRef]
J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater.8(7), 568–571 (2009). [CrossRef] [PubMed]
L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics3(8), 461–463 (2009). [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,” Photonics Nanostruct. Fundam. Appl.6(1), 87–95 (2008). [CrossRef]
H. Chen, B. Hou, S. Chen, X. Ao, W. Wen, and C. T. Chan, “Design and experimental realization of a broadband transformation media field rotator at microwave frequencies,” Phys. Rev. Lett.102(18), 183903 (2009). [CrossRef] [PubMed]
A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Electromagnetic wormholes and virtual magnetic monopoles from metamaterials,” Phys. Rev. Lett.99(18), 183901 (2007). [CrossRef] [PubMed]
M. Rahm, D. A. Roberts, J. B. Pendry, and D. R. Smith, “Transformation-optical design of adaptive beam bends and beam expanders,” Opt. Express16(15), 11555–11567 (2008). [CrossRef] [PubMed]
M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, “Optical design of reflectionless complex media by finite embedded coordinate transformations,” Phys. Rev. Lett.100(6), 063903 (2008). [CrossRef] [PubMed]
L. Lin, W. Wang, J. Cui, C. Du, and X. Luo, “Design of electromagnetic refractor and phase transformer using coordinate transformation theory,” Opt. Express16(10), 6815–6821 (2008). [CrossRef] [PubMed]
J. Huangfu, S. Xi, F. Kong, J. Zhang, H. Chen, D. Wang, B.-I. Wu, L. Ran, and J. A. Kong, “Application of coordinate transformation in bent waveguide,” J. Appl. Phys.104(1), 014502 (2008). [CrossRef]
P.-H. Tichit, S. N. Burokur, and A. de Lustrac, “Waveguide taper engineering using coordinate transformation technology,” Opt. Express18(2), 767–772 (2010). [CrossRef] [PubMed]
W. X. Jiang, T. J. Cui, H. F. Ma, X. M. Yang, and Q. Cheng, “Layered high-gain lens antennas via discrete optical transformation,” Appl. Phys. Lett.93(22), 221906 (2008). [CrossRef]
Z. L. Mei, J. Bai, T. M. Niu, and T. J. Cui, “A planar focusing antenna design using quasi-conformal mapping,” PIER13, 261–273 (2010). [CrossRef]
Y. G. Ma, C. K. Ong, T. Tyc, and U. Leonhardt, “An omnidirectional retroreflector based on the transmutation of dielectric singularities,” Nat. Mater.8(8), 639–642 (2009). [CrossRef] [PubMed]
T. Tyc and U. Leonhardt, “Transmutation of singularities in optical instruments,” N. J. Phys.10(11), 115038 (2008). [CrossRef]
N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater.9(2), 129–132 (2010). [CrossRef] [PubMed]
H. F. Ma and T. J. Cui, “Three-dimensional broadband ground-plane cloak made of metamaterials,” Nat Commun1(3), 1–6 (2010). [CrossRef] [PubMed]
Y. Luo, J. Zhang, L. Ran, H. Chen, and J. A. Kong, “Controlling the emission of electromagnetic source,” PIERS4(7), 795–800 (2008). [CrossRef]
J. Allen, N. Kundtz, D. A. Roberts, S. A. Cummer, and D. R. Smith, “Electromagnetic source transformations using superellipse equations,” Appl. Phys. Lett.94(19), 194101 (2009). [CrossRef]
B. I. Popa, J. Allen, and S. A. Cummer, “Conformal array design with transformation electromagnetics,” Appl. Phys. Lett.94(24), 244102 (2009). [CrossRef]
P.-H. Tichit, S. N. Burokur, and A. de Lustrac, “Ultradirective antenna via transformation optics,” J. Appl. Phys.105(10), 104912 (2009). [CrossRef]
P.-H. Tichit, S. N. Burokur, D. Germain, and A. de Lustrac, “Design and experimental demonstration of a high-directive emission with transformation optics,” Phys. Rev. B83(15), 155108 (2011). [CrossRef]
P.-H. Tichit, S. N. Burokur, D. Germain, and A. de Lustrac, “Coordinate transformation based ultra-directive emission,” Electron. Lett.47(10), 580–582 (2011). [CrossRef]
U. Leonhardt and T. G. Philbin, “Transformation optics and the geometry of light,” Prog. Opt.53, 69–152 (2009). [CrossRef]
Comsol MULTIPHYSICS Modeling, ( http://www.comsol.com ).

Allen, J.
Ao, X.
Bai, J.
Bartal, G.
Burokur, S. N.
Cai, W.
Cardenas, J.
Chan, C. T.
Chen, H.
Chen, S.
Cheng, Q.
Chettiar, U. K.
Chin, J. Y.
Cui, J.
Cui, T. J.
Cummer, S. A.
de Lustrac, A.
Du, C.
Gabrielli, L. H.
Germain, D.
Greenleaf, A.
Hou, B.
Huangfu, J.
Ji, C.
Jiang, W. X.
Justice, B. J.
Kanté, B.
Kildishev, A. V.
Kong, F.
Kong, J. A.
Kundtz, N.
Kurylev, Y.
Lassas, M.
Leonhardt, U.
Li, J.
Lin, L.
Lipson, M.
Liu, R.
Luo, X.
Luo, Y.
Ma, H. F.
Ma, Y. G.
Mei, Z. L.
Mock, J. J.
Niu, T. M.
Ong, C. K.
Pendry, J. B.
Philbin, T. G.
Poitras, C. B.
Popa, B. I.
Rahm, M.
Ran, L.
Roberts, D. A.
Schurig, D.
Shalaev, V. M.
Smith, D. R.
Starr, A. F.
Tichit, P.-H.
Tyc, T.
Uhlmann, G.
Valentine, J.
Wang, D.
Wang, W.
Wen, W.
Wu, B.-I.
Xi, S.
Yang, X. M.
Zentgraf, T.
Zhang, J.
Zhang, X.

Appl. Phys. Lett.

W. X. Jiang, T. J. Cui, H. F. Ma, X. M. Yang, and Q. Cheng, “Layered high-gain lens antennas via discrete optical transformation,” Appl. Phys. Lett.93(22), 221906 (2008). [CrossRef]
J. Allen, N. Kundtz, D. A. Roberts, S. A. Cummer, and D. R. Smith, “Electromagnetic source transformations using superellipse equations,” Appl. Phys. Lett.94(19), 194101 (2009). [CrossRef]
B. I. Popa, J. Allen, and S. A. Cummer, “Conformal array design with transformation electromagnetics,” Appl. Phys. Lett.94(24), 244102 (2009). [CrossRef]

Electron. Lett.

P.-H. Tichit, S. N. Burokur, D. Germain, and A. de Lustrac, “Coordinate transformation based ultra-directive emission,” Electron. Lett.47(10), 580–582 (2011). [CrossRef]

J. Appl. Phys.

P.-H. Tichit, S. N. Burokur, and A. de Lustrac, “Ultradirective antenna via transformation optics,” J. Appl. Phys.105(10), 104912 (2009). [CrossRef]
J. Huangfu, S. Xi, F. Kong, J. Zhang, H. Chen, D. Wang, B.-I. Wu, L. Ran, and J. A. Kong, “Application of coordinate transformation in bent waveguide,” J. Appl. Phys.104(1), 014502 (2008). [CrossRef]

N. J. Phys.

T. Tyc and U. Leonhardt, “Transmutation of singularities in optical instruments,” N. J. Phys.10(11), 115038 (2008). [CrossRef]

Nat Commun

H. F. Ma and T. J. Cui, “Three-dimensional broadband ground-plane cloak made of metamaterials,” Nat Commun1(3), 1–6 (2010). [CrossRef] [PubMed]

Nat. Mater.

N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater.9(2), 129–132 (2010). [CrossRef] [PubMed]
Y. G. Ma, C. K. Ong, T. Tyc, and U. Leonhardt, “An omnidirectional retroreflector based on the transmutation of dielectric singularities,” Nat. Mater.8(8), 639–642 (2009). [CrossRef] [PubMed]
J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater.8(7), 568–571 (2009). [CrossRef] [PubMed]

Nat. Photonics

L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics3(8), 461–463 (2009). [CrossRef]
W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics1(4), 224–227 (2007). [CrossRef]

Opt. Express

Photonics Nanostruct. Fundam. Appl.

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,” Photonics Nanostruct. Fundam. Appl.6(1), 87–95 (2008). [CrossRef]

Phys. Rev. B

B. Kanté, D. Germain, and A. de Lustrac, “Experimental demonstration of non-magnetic metamaterial cloak at microwave frequencies,” Phys. Rev. B80(20), 201104 (2009). [CrossRef]
P.-H. Tichit, S. N. Burokur, D. Germain, and A. de Lustrac, “Design and experimental demonstration of a high-directive emission with transformation optics,” Phys. Rev. B83(15), 155108 (2011). [CrossRef]

Phys. Rev. Lett.

M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, “Optical design of reflectionless complex media by finite embedded coordinate transformations,” Phys. Rev. Lett.100(6), 063903 (2008). [CrossRef] [PubMed]
H. Chen, B. Hou, S. Chen, X. Ao, W. Wen, and C. T. Chan, “Design and experimental realization of a broadband transformation media field rotator at microwave frequencies,” Phys. Rev. Lett.102(18), 183903 (2009). [CrossRef] [PubMed]
A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Electromagnetic wormholes and virtual magnetic monopoles from metamaterials,” Phys. Rev. Lett.99(18), 183901 (2007). [CrossRef] [PubMed]

PIER

Z. L. Mei, J. Bai, T. M. Niu, and T. J. Cui, “A planar focusing antenna design using quasi-conformal mapping,” PIER13, 261–273 (2010). [CrossRef]

PIERS

Y. Luo, J. Zhang, L. Ran, H. Chen, and J. A. Kong, “Controlling the emission of electromagnetic source,” PIERS4(7), 795–800 (2008). [CrossRef]

Prog. Opt.

U. Leonhardt and T. G. Philbin, “Transformation optics and the geometry of light,” Prog. Opt.53, 69–152 (2009). [CrossRef]

Science

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312(5781), 1780–1782 (2006). [CrossRef] [PubMed]
U. Leonhardt, “Optical conformal mapping,” Science312(5781), 1777–1780 (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,” Science314(5801), 977–980 (2006). [CrossRef] [PubMed]
R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science323(5912), 366–369 (2009). [CrossRef] [PubMed]

Other

Comsol MULTIPHYSICS Modeling, ( http://www.comsol.com ).

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

Click here to see a list of articles that cite this paper