Equivalent configurations of optical  transformation media

doi:10.1364/OE.17.021326

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Equivalent configurations of optical  transformation media

Tungyang Chen, Shen-Wen Cheng, and Chung-Ning Weng »View Author Affiliations

Optics Express, Vol. 17, Issue 23, pp. 21326-21335 (2009)
http://dx.doi.org/10.1364/OE.17.021326

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▸ Article Outline
– Abstract
1. Introduction
2. Two layers by linear coordinate transformations
3. Some simplifications and generalizations
4. Discussions and implications
– References and links

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Abstract

We demonstrate that a medium consisting of two adjoining distinct layers of transformation materials, corresponding respectively to two linear coordinate transformations, can behave effectively as that of the same region transformed by another linear transformation. The equivalence means that, irrespective of the direction of incident wave, the fields of the medium exterior to the transformed regions of the two configurations are exactly the same. This property can also apply to a domain that is transformed by a piecewise linear transformation function, and to a medium that is mapped by a general curved function. This proof is shown analytically based on a rigorous Fourier-Bessel analysis. The equivalence suggests that, for a given transformed domain, one can find an infinite number of complementary media that altogether can give a desired effective response of certain transformation path.

1. Introduction

The subject of transformation optics is fascinating with far-reaching implications. Transformation optics shows that the space for light can be controlled in an almost arbitrary way, in which the idea is to exploit coordinate transformations to yield an “exotic” optical path. The key feature to achieve this effect is that, with appropriately designed material parameters, the governing Maxwell’s equations remain the same under curvilinear coordinate transformation. Various concepts were proposed recently which offer new possibilities to design and engineer optical devices through the use of metamaterials. Among the latest developments, Pendry et al. [1

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

] and Leonhardt [2

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

] showed in 2006 the possibility of designing an invisibility cloak, which could bend light around an object and return to their original propagation direction. Among relevant developments, we mention the theoretical work [3

G. W. Milton, M. Briane, and J. R. Willis, “On cloaking for elasticity and physical equations with a transformation invariant form,” N. J. Phys. 8(10), 248 ( 2006). [CrossRef]

] which provided a rigorous proof on the possibility of form invariance of various physical phenomena, and the paper [4

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]

] for experimental demonstrations of cloak operating at microwave frequency. Other related works include the study of cloaks for different physical phenomena and various geometric shapes [5

S. A. Cummer and D. Schurig, “One path to acoustic cloaking,” N. J. Phys. 9(3), 45 ( 2007). [CrossRef]

–12

T. Chen, C. N. Weng, and J. S. Chen, “Cloak for curvilinearly anisotropic media in conduction,” Appl. Phys. Lett. 93(11), 114103 ( 2008). [CrossRef]

], see also the reviews [13

U. Leonhardt and T. G. Philbin, “Transformation optics and the geometry of light,” Prog. Opt. 53, 70–152 ( 2009).

,14

V. M. Shalaev, “Physics. Transforming light,” Science 322(5900), 384–386 ( 2008). [CrossRef] [PubMed]

] for a general scope. In addition to cloaking, which expands a point into a finite domain, we mention the work [15

H. Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett. 90(24), 241105 ( 2007). [CrossRef]

] which demonstrates how to manipulate the direction of electromagnetic waves by introducing a rotation mapping of coordinates. This was later referred to as field rotators. Another recent progress [16

Z. L. Mei and T. J. Cui, “Arbitrary bending of electromagnetic waves using isotropic materials,” J. Appl. Phys. 105(10), 104913 ( 2009). [CrossRef]

] is to employ the theory of geometrical optics in demonstrating bending of electromagnetic waves. They showed that the designed wave bending structures could be isotropic. In addition, we mention an important notion of folded geometry [17

G. W. Milton, N.-A. P. Nicorovici, R. C. McPhedran, K. Cherednichenko, and Z. Jacob, “Solutions in folded geometries, and associated cloaking due to anomalous resonance,” N. J. Phys. 10(11), 115021 ( 2008). [CrossRef]

,18

U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” N. J. Phys. 8(10), 247 ( 2006). [CrossRef]

], which will lead to materials with negatively refracting index [19

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. Usp. 10 509–14 (Engl. Transl.) (1968).

]. The use of negatively refracting materials enables applications in the design of perfect electric conductor (PEC) reshaper [20

H. Chen, X. Zhang, X. Luo, H. Ma, and C. T. Chan, “Reshaping the perfect electrical conductor cylinder arbitrarily,” N. J. Phys. 10(11), 113016 ( 2008). [CrossRef]

], superscatterers [21

T. Yang, H. Chen, X. Luo, and H. Ma, “Superscatterer: enhancement of scattering with complementary media,” Opt. Express 16(22), 18545–18550 ( 2008). [CrossRef] [PubMed]

] and light concentrators [22

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

,23

Y. Luo, H. Chen, J. Zhang, L. Ran, and J. A. Kong, “Design and analytical full-wave validation of the invisibility cloaks, concentrators, and field rotators created with a general class of transformations,” Phys. Rev. B 77(12), 125127 ( 2008). [CrossRef]

]. Anti-cloaks [24

H. Chen, X. Luo, H. Ma, and C. T. Chan, “The anti-cloak,” Opt. Express 16(19), 14603–14608 ( 2008). [CrossRef] [PubMed]

,25

G. Castaldi, I. Gallina, V. Galdi, A. Alù, and N. Engheta, “Cloak/anti-cloak interactions,” Opt. Express 17(5), 3101–3114 ( 2009). [CrossRef] [PubMed]

], which also invoke the concept of folded-geometries, can annihilate some effects of invisibility. Lai et al. [26

Y. Lai, J. Ng, H. Chen, D. Han, J. Xiao, Z. Q. Zhang, and C. T. Chan, “Illusion optics: The optical transformation of an object into another object,” Phys. Rev. Lett. 102(25), 253902 ( 2009). [CrossRef] [PubMed]

] recently outlined a mathematical formalism which showed that an object of arbitrary shape may appear to be like another object of other shape, see also Pendry [27

J. Pendry, “All smoke and metamaterials,” Nature 460(7255), 579–580 ( 2009). [CrossRef]

In this work, we show that two adjoining distinct pieces of metamaterials, transformed respectively from two linear coordinate transformations, can behave effectively as one single piece of metamaterial corresponding to another linear transformation. The equivalence means that, irrespective of the direction of incident wave, the electric and magnetic fields outside the transformed domains are exactly the same for the two configurations. The procedure, based on the method of transformation optics, is to first set up a linear functional relationship between the virtual space and the physical space of the two layers. This will give the prescription of the material parameters in the physical domain. As in invisibility cloaks, the constituent material parameters of the two regions are anisotropic and position varying. Next, we consider the same domain in the physical space mapped from the virtual space by one single linear transformation. In the analysis, we consider that the domains outside the transformed region are free space. We restrict our analysis to two-dimensional cylindrical configuration for simplicity, and employ the cylindrical wave expansion method to show analytically the equivalence of the two configurations. We mention that the proof can be extended to a domain that is constituted by multiple layers corresponding to a piecewise linear transformation function, and also to a medium that is mapped from a general curved function. In addition we show that the equivalence property remains valid when the core region is a PEC.

This finding implies that, for a given path of linear transformed domain, there are infinite pairs of linear coordinate transformations that will give the same field outside the transformed region. One may consider that the two transformation media as one pair of regions in which one medium is complementary to the other. The fact of equivalence, offering us new insight of transformation optics, could be of theoretical interest, relating to neutral inclusions and effective media of heterogeneous and composite materials [28

G. W. Milton, The Theory of Composites (Cambridge University Press, Cambridge, 2002).

]. Also, identifications of equivalent paths of different geometric configurations may find practical applications, for example in establishing the effective response of interactions between cloak and anti-cloak and other novel optical devices.

2. Two layers by linear coordinate transformations

We consider two adjoining layers in the physical space

x^{'}

that are mapped from a virtual space x via coordinate transformations. Specifically, we consider one linear transformation [Fig. 1(a) ] that maps the circular annulus with the inner radius

r = a

onto

r^{'} = a^{'}

, and the outer radius

r = r_{0}

onto

r^{'} = r_{0}^{'}

; and also another linear transformation that maps the layer with inner radius

r = r_{0}

onto

r^{'} = r_{0}^{'}

, and outer radius

r = b

onto

r^{'} = b^{'} .

This mapping, illustrated in Fig. 1(a), can be written as

r^{'} = α_{1} r + β_{1} = \frac{r_{0}^{'} - a^{'}}{r_{0} - a} r + \frac{a^{'} r_{0} - a r_{0}^{'}}{r_{0} - a}, a \leq r \leq r_{0}, θ^{'} = θ, z^{'} = z,

(1)

r^{'} = α_{2} r + β_{2} = \frac{b^{'} - r_{0}^{'}}{b - r_{0}} r + \frac{b r_{0}^{'} - b^{'} r_{0}}{b - r_{0}}, r_{0} \leq r \leq b, θ^{'} = θ, z^{'} = z .

(2)

The mapping function associated with the vertical axis, the physical

r^{'}

space, needs to be single-valued and thus

a^{'} \leq r_{0}^{'} \leq b^{'} .

While that associated with the horizontal axis, the virtual space, could be multiple-valued (folded). In other words, albeit

b > a

, the value of

r_{0}

could be either less or greater than that of a. Specifically, the linear functions, (1) and (2), transforms the circular layer

a \leq r \leq b

in the virtual space onto the circular domain

a^{'} \leq r \leq b^{'}

in the physical space. Milton et al. [3

G. W. Milton, M. Briane, and J. R. Willis, “On cloaking for elasticity and physical equations with a transformation invariant form,” N. J. Phys. 8(10), 248 ( 2006). [CrossRef]

] and earlier work [29

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

] showed that Maxwell’s equations under coordinate transformation are invariant in their forms provided that the basic material parameters follow

ε^{'} = A ε A^{T} \cdot {| \det (A) |}^{- 1},

where

A = \partial (r^{'}, θ^{'}, z^{'}) / \partial (r, θ, z) .

The transformed electric permittivity in the physical space

x^{'}

can be derived as [3

G. W. Milton, M. Briane, and J. R. Willis, “On cloaking for elasticity and physical equations with a transformation invariant form,” N. J. Phys. 8(10), 248 ( 2006). [CrossRef]

]

ε_{1 r}^{'} = \frac{r^{'} - β_{1}}{r^{'}} ε_{0}, ε_{1 θ}^{'} = \frac{r^{'}}{r^{'} - β_{1}} ε_{0}, ε_{1 z}^{'} = {(\frac{1}{α_{1}})}^{2} \frac{r^{'} - β_{1}}{r^{'}} ε_{0}, for a^{'} \leq r^{'} \leq r_{0}^{'},

(3)

ε_{2 r}^{'} = \frac{r^{'} - β_{2}}{r^{'}} ε_{0}, ε_{2 θ}^{'} = \frac{r^{'}}{r^{'} - β_{2}} ε_{0}, ε_{2 z}^{'} = {(\frac{1}{α_{2}})}^{2} \frac{r^{'} - β_{2}}{r^{'}} ε_{0}, for r_{0}^{'} \leq r^{'} \leq b^{'} .

(4)

The transformed magnetic permeability takes the same expressions as in Eqs. (3) and (4) except that

ε_{0}

is now replaced by

μ_{0} .

Here

ε_{0}

and

μ_{0}

are vacuum permittivity and permeability. We mention that the transformed material parameters are anisotropic and position varying. Further one can verify that the impedance is perfectly matched at

r^{'} = r_{0}^{'} .

For convenience, in the following exposition, we shall denote the regions

r^{'} \leq a^{'}, a^{'} \leq r^{'} \leq r_{0}^{'}, r_{0}^{'} \leq r^{'} \leq b^{'}, r^{'} \geq b^{'},

respectively, as regions I, II, III and IV. We note that when the point P is inside the region ABCD [Fig. 1(a)], then the transformed material parameters of the two regions II and III are positive. In the case where the point P locates in the region EFBC, the transformed material parameters for the region II will be negative (equivalent to negatively refracting material), while those corresponding to the region III are positive. This can be seen geometrically from Fig. 1(a). We note from Eqs. (1) and (2) that the value

α_{1}

represents the slope of the line segment

\bar{C P},

while the value of

β_{1}

is the ordinate of line

\bar{C P}

joining the vertical axis. From Eq. (3), the sign of the transformed parameters in region II, in which

a^{'} \leq r^{'} \leq r_{0}^{'},

is the same as that of

(r^{'} - β_{1}) .

Thus, in the case of point P lying in the region EFBC, the transformed parameters in II corresponding to

\bar{P C}

are always negative. Similar observations hold for the line segment

\bar{A P} .

When the point P locates inside the region GADH, the transformed parameters associated with region III are negative. We conclude that the material parameters of the two transformed domains (regions II and III) can be composed either of two positively refracting materials, or of one positively and one negatively refracting materials.

Fig. 1 (a) An illustration of the coordinate transformation of Eqs. (1) and (2) in which

(r_{0}, r_{0}^{'}) = (0.12, 0.20),

a = 0.07, a^{'} = 0.1,

b = 0.25, b^{'} = 0.3.

(b) the value of

ε_{r}^{'} / ε_{0}

versus

r^{'}

. (c) the value of

ε_{θ}^{'} / ε_{0}

versus

r^{'} .

(d) the value of

ε_{z}^{'} / ε_{0}

versus

r^{'} .

We first consider that the regions inside

r^{'} \leq a^{'}

and outside

r^{'} \geq b^{'}

are free space and thus their corresponding material parameters are simply

ε_{0}

and

μ_{0} .

A transverse-electric (TE) polarized EM incident field together with harmonic time dependence exp

(- i ω t)

is considered. The coordinate system is the cylindrical coordinate coinciding with the cylinder we consider. The wave equation of

E_{z}

in each region is governed by

\frac{1}{ε_{z}} \frac{1}{r} \frac{\partial}{\partial r} (\frac{r}{μ_{θ}} \frac{\partial E_{z}}{\partial r}) + \frac{1}{ε_{z}} \frac{1}{r^{2}} (\frac{1}{μ_{r}} \frac{\partial^{2} E_{z}}{\partial θ^{2}}) + k_{0}^{2} E_{z} = 0,

(5)

where

k_{0} = ω / c_{0} = ω \sqrt{μ_{0} ε_{0}}

is the wave number of the EM wave in vacuum. The electric fields in each domain can be expressed as [30

Z. Ruan, M. Yan, C. W. Neff, and M. Qiu, “Ideal cylindrical cloak: perfect but sensitive to tiny perturbations,” Phys. Rev. Lett. 99(11), 113903 ( 2007). [CrossRef] [PubMed]

]

E_{z}^{I} = \sum_{n} d_{n}^{I} J_{n} (k_{0} r^{'}) \exp (i n θ), for r^{'} \leq a^{'}

(6)

E_{z}^{I I} = \sum_{n} {d_{n}^{I I} J_{n} (k_{1} (r^{'} - β_{1})) + f_{n}^{I I} H_{n}^{(1)} (k_{1} (r^{'} - β_{1}))} \exp (i n θ), for a^{'} \leq r^{'} \leq r_{0}^{'},

(7)

E_{z}^{I I I} = \sum_{n} {d_{n}^{I I I} J_{n} (k_{2} (r^{'} - β_{2})) + f_{n}^{I I I} H_{n}^{(1)} (k_{2} (r^{'} - β_{2}))} \exp (i n θ), for r_{0}^{'} \leq r^{'} \leq b^{'},

(8)

E_{z}^{I V} = \sum_{n} {d_{n}^{i n} J_{n} (k_{0} r^{'}) + f_{n}^{s c} H_{n}^{(1)} (k_{0} r^{'})} \exp (i n θ), for r^{'} \geq b^{'}

(9)

where

k_{1} = k_{0} (r_{0} - a) / (r_{0}^{'} - a^{'})

and

k_{2} = k_{0} (b - r_{0}) / (b^{'} - r_{0}^{'}) .

J_{n}

and

H_{n}^{(1)}

are the n-order Bessel and Hankel functions of the 1^st kind. The coefficients

d_{n}^{i n}

correspond to the prescribed incident field and

d_{n}^{I},

d_{n}^{I I},

d_{n}^{I I I},

f_{n}^{I},

f_{n}^{I I},

f_{n}^{I I I}

and the scattering coefficient

f_{n}^{s c}

are unknown coefficients to be determined from the interface conditions at the interfaces. The interface conditions suggest that the tangential fields

E_{z}

and

H_{θ} = \frac{1}{i ω μ_{θ}} \frac{\partial E_{z}}{\partial r}

be continuous at

r^{'} = b^{'}, r^{'} = r_{0}^{'} and r^{'} = a^{'} .

By virtue of orthogonality of

\exp (i n θ),

we find that the matching conditions at

r^{'} = b

are

d_{n}^{i n} J_{n} (k_{0} b^{'}) + f_{n}^{s c} H_{n}^{(1)} (k_{0} b^{'}) = d_{n}^{I I I} J_{n} (k_{2} (b^{'} - β_{2})) + f_{n}^{I I I} H_{n}^{(1)} (k_{2} (b^{'} - β_{2})),

(10)

\begin{array}{l} k_{0} d_{n}^{i n} J_{n}^{'} (k_{0} b^{'}) + k_{0} f_{n}^{s c} H_{n}^{(1)}^{'} (k_{0} b^{'}) \\ = \frac{k_{2}}{μ_{2 θ}^{'} (b^{'})} d_{n}^{I I I} J_{n}^{'} (k_{2} (b^{'} - β_{2})) + \frac{k_{2}}{μ_{2 θ}^{'} (b^{'})} f_{n}^{I I I} H_{n}^{(1)}^{'} (k_{2} (b^{'} - β_{2})) . \end{array}

(11)

where the prime denotes differentiation with respect to the entire argument. Since

k_{2} (b^{'} - β_{2}) = \frac{b - r_{0}}{b^{'} - r_{0}^{'}} k_{0} \frac{b^{'} b - r_{0}^{'} b}{b - r_{0}} = k_{0} b,

(12)

\frac{k_{2}}{μ_{2 θ}^{'} (b^{'})} = \frac{b - r_{0}}{b^{'} - r_{0}^{'}} k_{0} \frac{b^{'} - β_{2}}{b^{'}} = \frac{b - r_{0}}{b^{'} - r_{0}^{'}} \frac{k_{0}}{b^{'}} \frac{b^{'} b - r_{0}^{'} b}{b - r_{0}} = \frac{k_{0}}{b^{'}} b,

(13)

Equations (10) and (11) simplify to

d_{n}^{i n} J_{n} (k_{0} b^{'}) + f_{n}^{s c} H_{n}^{(1)} (k_{0} b^{'}) = d_{n}^{I I I} J_{n} (k_{0} b) + f_{n}^{I I I} H_{n}^{(1)} (k_{0} b),

(14)

b^{'} d_{n}^{i n} J_{n}^{'} (k_{0} b^{'}) + b^{'} f_{n}^{s c} H_{n}^{(1)}^{'} (k_{0} b^{'}) = b d_{n}^{I I I} J_{n}^{'} (k_{0} b) + b f_{n}^{I I I} H_{n}^{(1)}^{'} (k_{0} b) .

(15)

At the interface of

r^{'} = r_{0}^{'},

we have

\begin{array}{l} d_{n}^{I I I} J_{n} (k_{2} (r_{0}^{'} - β_{2})) + f_{n}^{I I I} H_{n}^{(1)} (k_{2} (r_{0}^{'} - β_{2})) \\ = d_{n}^{I I} J_{n} (k_{1} (r_{0}^{'} - β_{1})) + f_{n}^{I I} H_{n}^{(1)} (k_{1} (r_{0}^{'} - β_{1})), \end{array}

(16)

\begin{array}{l} \frac{k_{2}}{μ_{2 θ}^{'} (r_{0}^{'})} d_{n}^{I I I} J_{n}^{'} (k_{2} (r_{0}^{'} - β_{2})) + \frac{k_{2}}{μ_{2 θ}^{'} (r_{0}^{'})} f_{n}^{I I I} H_{n}^{(1)}^{'} (k_{2} (r_{0}^{'} - β_{2})) \\ = \frac{k_{1}}{μ_{1 θ}^{'} (r_{0}^{'})} d_{n}^{I I} J_{n}^{'} (k_{1} (r_{0}^{'} - β_{1})) + \frac{k_{1}}{μ_{1 θ}^{'} (r_{0}^{'})} f_{n}^{I I} H_{n}^{(1)}^{'} (k_{1} (r_{0}^{'} - β_{1})) . \end{array}

(17)

Again since

k_{2} (r_{0}^{'} - β_{2}) = \frac{b - r_{0}}{b^{'} - r_{0}^{'}} k_{0} \frac{b^{'} r_{0} - r_{0}^{'} r_{0}}{b - r_{0}} = k_{0} r_{0},

(18)

k_{1} (r_{0}^{'} - β_{1}) = \frac{r_{0} - a}{r_{0}^{'} - a^{'}} k_{0} \frac{r_{0}^{'} r_{0} - a^{'} r_{0}}{r_{0} - a} = k_{0} r_{0},

(19)

and also

\frac{k_{2}}{μ_{2 θ}^{'} (r_{0}^{'})} = \frac{b - r_{0}}{b^{'} - r_{0}^{'}} k_{0} \frac{r_{0}^{'} - β_{2}}{r_{0}^{'}} = \frac{b - r_{0}}{b^{'} - r_{0}^{'}} \frac{k_{0}}{r_{0}^{'}} \frac{b^{'} r_{0} - r_{0}^{'} r_{0}}{b - r_{0}} = \frac{k_{0} r_{0}}{r_{0}^{'}},

(20)

\frac{k_{1}}{μ_{1 θ}^{'} (r_{0}^{'})} = \frac{r_{0} - a}{r_{0}^{'} - a^{'}} k_{0} \frac{r_{0}^{'} - β_{1}}{r_{0}^{'}} = \frac{r_{0} - a}{r_{0}^{'} - a^{'}} \frac{k_{0}}{r_{0}^{'}} \frac{r_{0}^{'} r_{0} - a^{'} r_{0}}{r_{0} - a} = \frac{k_{0} r_{0}}{r_{0}^{'}},

(21)

Equations (16) and (17) are reduced to the simple forms

(d_{n}^{I I I} - d_{n}^{I I}) J_{n} (k_{0} r_{0}) + (f_{n}^{I I I} - f_{n}^{I I}) H_{n}^{(1)} (k_{0} r_{0}) = 0,

(22)

(d_{n}^{I I I} - d_{n}^{I I}) J_{n}^{'} (k_{0} r_{0}) + (f_{n}^{I I I} - f_{n}^{I I}) H_{n}^{(1)}^{'} (k_{0} r_{0}) = 0,

(23)

which suggest that

d_{n}^{I I} = d_{n}^{I I I}, f_{n}^{I I} = f_{n}^{I I I} .

(24)

Lastly, at the interface

r^{'} = a^{'},

we have

d_{n}^{I I} J_{n} (k_{0} a) + f_{n}^{I I} H_{n}^{(1)} (k_{0} a) = d_{n}^{I} J_{n} (k_{0} a^{'}),

(25)

a d_{n}^{I I} J_{n}^{'} (k_{0} a) + a f_{n}^{I I} H_{n}^{(1)}^{'} (k_{0} a) = a^{'} d_{n}^{I} J_{n}^{'} (k_{0} a^{'}) .

(26)

Now for a given incident field, Eqs. (14), (15), (24), (25), (26) altogether, allow us to determine the unknown coefficients

d_{n}^{I},

d_{n}^{I I},

d_{n}^{I I I},

f_{n}^{I I},

f_{n}^{I I I}

and

f_{n}^{s c},

and thus the electric field as well as the magnetic field in each region can be determined readily. We shall not proceed to solve the linear algebraic system. Rather we will demonstrate that a different configuration will give the same fields in the regions I and IV (Fig. 2 ).

Fig. 2 A schematic illustration of equivalence of two configurations obtained from linear coordinate transformations. The equivalence means the electromagnetic fields in regions I and IV are identical for both configurations.

Let us now consider another linear coordinate transformation in which the circular annulus

a \leq r \leq b

in the virtual space is mapped onto the region

a^{'} \leq r^{'} \leq b^{'} .

The domains in the regions I and IV are again considered as free space. This linear transformation is written as [red line of Fig. 1(a)]

r^{'} = α_{3} r + β_{3} = \frac{b^{'} - a^{'}}{b - a} r + \frac{a^{'} b - b^{'} a}{b - a}, θ^{'} = θ, z^{'} = z,

(27)

and the corresponding material parameters for this transformed domain are

ε_{3 r}^{'} = \frac{r^{'} - β_{3}}{r^{'}} ε_{0}, ε_{3 θ}^{'} = \frac{r^{'}}{r^{'} - β_{3}} ε_{0}, ε_{3 z}^{'} = {(\frac{1}{α_{3}})}^{2} \frac{r^{'} - β_{3}}{r^{'}} ε_{0} .

(28)

Here we use a subscript ‘3′ to distinguish the notations in Eqs. (3) and (4). Numerical illustrations of the material parameters for Eqs. (3), (4) and (28) for a selected point of

(r_{0}, r_{0}^{'})

are exemplified in Fig. 1(b)–1(d). Analogous to (6)-(9), the electric fields in each domain can be written as

{\hat{E}}_{z}^{I} = \sum_{n} {\hat{d}}_{n}^{I} J_{n} (k_{0} r^{'}) \exp (i n θ), for r^{'} \leq a^{'},

(29)

{\hat{E}}_{z}^{I I + I I I} = \sum_{n} {{\hat{d}}_{n}^{I I + I I I} J_{n} (k_{3} (r^{'} - β_{3})) + {\hat{f}}_{n}^{I I + I I I} H_{n}^{(1)} (k_{3} (r^{'} - β_{3}))} \exp (i n θ), for a^{'} \leq r^{'} \leq b^{'},

(30)

{\hat{E}}_{z}^{I V} = \sum_{n} {d_{n}^{i n} J_{n} (k_{0} r^{'}) + {\hat{f}}_{n}^{s c} H_{n}^{(1)} (k_{0} r^{'})} \exp (i n θ), for r^{'} \geq b^{'},

(31)

where

k_{3} = k_{0} (b - a) / (b^{'} - a^{'})

and the ‘hat’ quantities are used to distinguish the fields with those in Eqs. (6)-(9). The objective now is to demonstrate that, under the same prescribed incident field

d_{n}^{i n},

the electric field Eqs. (29)-(31), and also the magnetic field, inside the regions I and IV will be the same with those of the configuration outlined in Eqs. (6)-(9) (Fig. 2). To show this, we again demand the interface continuity conditions at

r^{'} = a^{'}

and

r^{'} = b^{'} .

This will give

d_{n}^{i n} J_{n} (k_{0} b^{'}) + {\hat{f}}_{n}^{s c} H_{n}^{(1)} (k_{0} b^{'}) = {\hat{d}}_{n}^{I I + I I I} J_{n} (k_{0} b) + {\hat{f}}_{n}^{I I + I I I} H_{n}^{(1)} (k_{0} b),

(32)

b^{'} d_{n}^{i n} J_{n}^{'} (k_{0} b^{'}) + b^{'} {\hat{f}}_{n}^{s c} H_{n}^{(1)}^{'} (k_{0} b^{'}) = b {\hat{d}}_{n}^{I I + I I I} J_{n}^{'} (k_{0} b) + b {\hat{f}}_{n}^{I I + I I I} H_{n}^{(1)}^{'} (k_{0} b),

(33)

and

{\hat{d}}_{n}^{I I + I I I} J_{n} (k_{0} a) + {\hat{f}}_{n}^{I I + I I I} H_{n}^{(1)} (k_{0} a) = {\hat{d}}_{n}^{I} J_{n} (k_{0} a^{'}),

(34)

a {\hat{d}}_{n}^{I I + I I I} J_{n}^{'} (k_{0} a) + a {\hat{f}}_{n}^{I I + I I I I} H_{n}^{(1)}^{'} (k_{0} a) = a^{'} {\hat{d}}_{n}^{I} J_{n}^{'} (k_{0} a^{'}) .

(35)

Now in comparison of the interface conditions of the two configurations, that is Eqs. (14), (15), (24), (25), (26) and Eqs. (32)-(35), we find interestingly that

f_{n}^{s c} = {\hat{f}}_{n}^{s c}, d_{n}^{I} = {\hat{d}}_{n}^{I}, d_{n}^{I I} = d_{n}^{I I I} = {\hat{d}}_{n}^{I I + I I I}, f_{n}^{I I} = f_{n}^{I I I} = {\hat{f}}_{n}^{I I + I I I} .

(36)

This suggests that, in the innermost region I

(r^{'} \leq a^{'})

and also the outermost region IV

(r^{'} \geq b^{'})

, the electric and the magnetic fields of the two configurations are identical

E_{z}^{I} = {\hat{E}}_{z}^{I}, E_{z}^{I V} = {\hat{E}}_{z}^{I V}, H_{θ}^{I} = {\hat{H}}_{θ}^{I}, H_{θ}^{I V} = {\hat{H}}_{θ}^{I V} .

(37)

This observation means that the transformed media described by the two linear coordinate transformations [black lines, Fig. 1(a)] are effectively equivalent to that by one linear transformed domain [red line, Fig. 1(a)], irrespective of the direction of incident wave. The two adjoining transformed media (regions II and III) can be regarded as two-phase domains in which one medium is complementary to the other. The equivalence of both configurations is akin to the effect of neutral inclusion and effective medium of heterogeneous and composite media [28

G. W. Milton, The Theory of Composites (Cambridge University Press, Cambridge, 2002).

3. Some simplifications and generalizations

The proof of equivalence can be extended to transformed media that are described by a multiple piecewise linear transformation with continued end points [Fig. 3(a) ]. This can be perceived geometrically. First, one can replace any two linear transformations with one linear transformation. Repeating the process, the transformation medium can be eventually equivalent to that of a single linear transformation that links the end points. Note that in Fig. 3, the piecewise linear transformations could also be composed of a few line segments that may cross the equivalent line segment (red line). To see this, we can identify the effect of the first two line segments with that of the dashed line, and then the dashed line together with the third line segment will lead to the net effect of the red line. In fact, for a general transformation characterized by a curved function rather than a straight linear function (Fig. 3, right panel), the equivalence is still valid. Note that, however, the mapping function along with the

r^{'}

axis needs to be single-valued since

r^{'}

corresponds to the physical space. The effective transformation of a curved function is then equivalent to one linear transformation function linked by the two ends of the curve. The reason is simple. We can discretize the curve into many line segments with sufficiently small length and thus, in principle, it is possible to utilize one single linear transformation to mimic a transformed region corresponding to a curved one.

Fig. 3 (left panel) A schematic illustration of equivalence of a multiple piecewise linear transformation to a single linear coordinate transformation (red line). (right panel) equivalence of a general curved transformation function to a single linear coordinate transformation.

Suppose now that the region I of Fig. 2 is a PEC rather than free space. In this case, Eqs. (25) and (26) simplify to

d_{n}^{I I} J_{n} (k_{0} a) + f_{n}^{I I} H_{n}^{(1)} (k_{0} a) = 0.

(38)

Thus, Eqs. (14), (15), (24) and (38) can be used to determine the 5 unknown coefficients

d_{n}^{I I},

d_{n}^{I I I},

f_{n}^{I I},

f_{n}^{I I I}

and

f_{n}^{s c}

. Let us now consider the configuration of the right panel of Fig. 2. Again, the region I is a PEC. Instead of (34) and (35), we now have

{\hat{d}}_{n}^{I I + I I I} J_{n} (k_{0} a) + {\hat{f}}_{n}^{I I + I I I} H_{n}^{(1)} (k_{0} a) = 0.

(39)

Hence the electric field of this configuration can be resolved from (32), (33) and (39). Now in comparison of these two linear algebraic systems, we find again that

f_{n}^{s c} = {\hat{f}}_{n}^{s c}, d_{n}^{I I} = d_{n}^{I I I} = {\hat{d}}_{n}^{I I + I I I}, f_{n}^{I I} = f_{n}^{I I I} = {\hat{f}}_{n}^{I I + I I I} .

(40)

This suggests that when the region I is a PEC, the electromagnetic fields in regions I and IV in both configurations remains the same.

Lastly we consider the simple situation in which

a^{'} = a

and

b = b^{'} .

In this case, the interface conditions of Eqs. (14) and (15) simplify to

(d_{n}^{i n} - d_{n}^{I I I}) J_{n} (k_{0} b) + (f_{n}^{s c} - f_{n}^{I I I}) H_{n}^{(1)} (k_{0} b) = 0,

(41)

(d_{n}^{i n} - d_{n}^{I I I}) J_{n}^{'} (k_{0} b) + (f_{n}^{s c} - d_{n}^{I I I}) H_{n}^{(1)}^{'} (k_{0} b) = 0,

(42)

r^{'} = b^{'},

and those of Eqs. (25) and (26) to

(d_{n}^{I I} - d_{n}^{I}) J_{n} (k_{0} a) + (f_{n}^{I I}) H_{n}^{(1)} (k_{0} a) = 0,

(43)

(d_{n}^{I I} - d_{n}^{I}) J_{n}^{'} (k_{0} a) + (f_{n}^{I I}) H_{n}^{(1)}^{'} (k_{0} a) = 0,

(44)

r^{'} = a^{'} .

Note that Eq. (24) remains unchanged. All these together allow us to solve the unknown coefficients as

d_{n}^{i n} = d_{n}^{I I} = d_{n}^{I I I}, f_{n}^{s c} = f_{n}^{I I} = f_{n}^{I I I} = 0.

(45)

This implies that there is no scattering field and that the electric fields in regions I and IV are exactly identical. We mention, however, that inside the region II and III, the field solutions are not the same.

4. Discussions and implications

In summary, we have shown analytically in this work, by resolving the electromagnetic field solutions of the two different configurations, that a medium consisting of two adjoining distinct layers of transformation materials, corresponding respectively to two linear coordinate transformations, can behave effectively as that of the same region transformed by another linear transformation. We note that as the r axis could be folded, the point

P (r_{0}, r_{0}^{'})

may lie on any point inside the semi-infinite strip of EFGH. Figure 4 depicts a schematic illustration of possible equivalent transformation paths that give to the same effect. In fact, different transformation paths may correspond to different optical behavior. For example, the path indicated by the blue dashed line (Fig. 4) could simulate the effects of anti-cloaks [24

H. Chen, X. Luo, H. Ma, and C. T. Chan, “The anti-cloak,” Opt. Express 16(19), 14603–14608 ( 2008). [CrossRef] [PubMed]

], while others (green and black dashed lines) can be modeled as concentrators.

Fig. 4 Possible transformation paths that are equivalent to the linear one (red).

It should be noted that the transformed material properties (3) and (4) are derived based on cylindrical configurations, which in fact involve the geometric information of the adjoining layers. We conjecture that the equivalence of the configurations could also be derivable from Fermat’s principle based on extremal optical path [31

V. G. Veselago, “About the wording of Fermat’s principle for light propagation in media with negative refraction index,” (2002) http://arxiv.org/abs/cond-mat/0203451.

,32

M. Born, and E. Wolf, Principles of Optics, 7th (Expanded) Edition, (Cambridge University Press, New York, 2002).

]. In a separate study we have performed a numerical check on the equivalence property based on finite element calculations. The equivalence between two configurations offers a new insight at the fundamentals of transformation optics. For instance, given a transformed domain, one can always find an infinite number of complementary media that altogether can give an effective response of certain transformed path. Identifications of equivalent paths of different geometric configurations may find useful in some practical applications, for example in establishing the effective response of combined transformed domains. For potential applications, when a certain desired optical ray-path outcome would require a medium which is realistically unfeasible one could look for a series of adjacent media which are realistically feasible and lead to the same net effect. Particularly, with the recent advances of fabrication methods of metamaterials, one can design synthetic structures with a desired distribution of electric permittivity and magnetic permeability for practical implementations.

In a future study, as in the design of cloaks of arbitrary cross-sections [9

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

], one can explore whether the equivalence property applies to other geometrical configurations. To analytically prove the equivalence, one may first think of extending the proof to three-dimensional spherical regions and also to regions with elliptical geometry. For spherical regions, one can derive the full wave solution based on Mie scattering [33

H. Chen, B. I. Wu, B. Zhang, and J. A. Kong, “Electromagnetic wave interactions with a metamaterial cloak,” Phys. Rev. Lett. 99(6), 063903 ( 2007). [CrossRef] [PubMed]

]. Similar electromagnetic scattering of elliptical layers, in terms of series expansions of Mathieu functions [34

S. Caorsi, M. Pastorino, and M. Raffetto, “Electromagnetic scattering by a multilayer elliptic cylinder under transverse-magnetic illumination: series solution in terms of Mathieu functions,” IEEE Trans. Antenn. Propag. 45(6), 926–935 ( 1997). [CrossRef]

], can be envisaged based on elliptical coordinates. More generally, one may wonder if the equivalence property holds for any configurations that can be characterized by orthogonal coordinates [10

H. Chen, “Transformation optics in orthogonal coordinates,” J. Opt. A, Pure Appl. Opt. 11(7), 075102 ( 2009). [CrossRef]

], in which a full wave series solution could be analyzed in closed forms.

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(5781), 1780–1782 ( 2006). [CrossRef] [PubMed]
2.	U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 ( 2006). [CrossRef] [PubMed]
3.	G. W. Milton, M. Briane, and J. R. Willis, “On cloaking for elasticity and physical equations with a transformation invariant form,” N. J. Phys. 8(10), 248 ( 2006). [CrossRef]
4.	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.	S. A. Cummer and D. Schurig, “One path to acoustic cloaking,” N. J. Phys. 9(3), 45 ( 2007). [CrossRef]
6.	H. Chen and C. T. Chan, “Acoustic cloaking in three dimensions using acoustic metamaterials,” Appl. Phys. Lett. 91(18), 183518 ( 2007). [CrossRef]
7.	G. X. Yu, W. X. Jiang, and T. J. Cui, “Invisible slab cloaks via embedded optical transformation,” Appl. Phys. Lett. 94(4), 041904 ( 2009). [CrossRef]
8.	J. Hu, X. Zhou, and G. Hu, “Design method for electromagnetic cloak with arbitrary shapes based on Laplace’s equation,” Opt. Express 17(3), 1308–1320 ( 2009). [CrossRef] [PubMed]
9.	A. Nicolet, F. Zolla, and S. Guenneau, “Electromagnetic analysis of cylindrical cloaks of an arbitrary cross section,” Opt. Express 33, 1584–1586 ( 2008).
10.	H. Chen, “Transformation optics in orthogonal coordinates,” J. Opt. A, Pure Appl. Opt. 11(7), 075102 ( 2009). [CrossRef]
11.	N. Kundtz, D. A. Roberts, J. Allen, S. Cummer, and D. R. Smith, “Optical source transformations,” Opt. Express 16(26), 21215–21222 ( 2008). [CrossRef] [PubMed]
12.	T. Chen, C. N. Weng, and J. S. Chen, “Cloak for curvilinearly anisotropic media in conduction,” Appl. Phys. Lett. 93(11), 114103 ( 2008). [CrossRef]
13.	U. Leonhardt and T. G. Philbin, “Transformation optics and the geometry of light,” Prog. Opt. 53, 70–152 ( 2009).
14.	V. M. Shalaev, “Physics. Transforming light,” Science 322(5900), 384–386 ( 2008). [CrossRef] [PubMed]
15.	H. Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett. 90(24), 241105 ( 2007). [CrossRef]
16.	Z. L. Mei and T. J. Cui, “Arbitrary bending of electromagnetic waves using isotropic materials,” J. Appl. Phys. 105(10), 104913 ( 2009). [CrossRef]
17.	G. W. Milton, N.-A. P. Nicorovici, R. C. McPhedran, K. Cherednichenko, and Z. Jacob, “Solutions in folded geometries, and associated cloaking due to anomalous resonance,” N. J. Phys. 10(11), 115021 ( 2008). [CrossRef]
18.	U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” N. J. Phys. 8(10), 247 ( 2006). [CrossRef]
19.	V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. Usp. 10 509–14 (Engl. Transl.) (1968).
20.	H. Chen, X. Zhang, X. Luo, H. Ma, and C. T. Chan, “Reshaping the perfect electrical conductor cylinder arbitrarily,” N. J. Phys. 10(11), 113016 ( 2008). [CrossRef]
21.	T. Yang, H. Chen, X. Luo, and H. Ma, “Superscatterer: enhancement of scattering with complementary media,” Opt. Express 16(22), 18545–18550 ( 2008). [CrossRef] [PubMed]
22.	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,” Photon. Nanostruct.: Fundam. Appl. 6(1), 87–95 ( 2008). [CrossRef]
23.	Y. Luo, H. Chen, J. Zhang, L. Ran, and J. A. Kong, “Design and analytical full-wave validation of the invisibility cloaks, concentrators, and field rotators created with a general class of transformations,” Phys. Rev. B 77(12), 125127 ( 2008). [CrossRef]
24.	H. Chen, X. Luo, H. Ma, and C. T. Chan, “The anti-cloak,” Opt. Express 16(19), 14603–14608 ( 2008). [CrossRef] [PubMed]
25.	G. Castaldi, I. Gallina, V. Galdi, A. Alù, and N. Engheta, “Cloak/anti-cloak interactions,” Opt. Express 17(5), 3101–3114 ( 2009). [CrossRef] [PubMed]
26.	Y. Lai, J. Ng, H. Chen, D. Han, J. Xiao, Z. Q. Zhang, and C. T. Chan, “Illusion optics: The optical transformation of an object into another object,” Phys. Rev. Lett. 102(25), 253902 ( 2009). [CrossRef] [PubMed]
27.	J. Pendry, “All smoke and metamaterials,” Nature 460(7255), 579–580 ( 2009). [CrossRef]
28.	G. W. Milton, The Theory of Composites (Cambridge University Press, Cambridge, 2002).
29.	E. J. Post, Formal Structure of Electromagnetics: General Covariance and Electromagnetics, (North-Holland, Amsterdam, 1962).
30.	Z. Ruan, M. Yan, C. W. Neff, and M. Qiu, “Ideal cylindrical cloak: perfect but sensitive to tiny perturbations,” Phys. Rev. Lett. 99(11), 113903 ( 2007). [CrossRef] [PubMed]
31.	V. G. Veselago, “About the wording of Fermat’s principle for light propagation in media with negative refraction index,” (2002) http://arxiv.org/abs/cond-mat/0203451.
32.	M. Born, and E. Wolf, Principles of Optics, 7th (Expanded) Edition, (Cambridge University Press, New York, 2002).
33.	H. Chen, B. I. Wu, B. Zhang, and J. A. Kong, “Electromagnetic wave interactions with a metamaterial cloak,” Phys. Rev. Lett. 99(6), 063903 ( 2007). [CrossRef] [PubMed]
34.	S. Caorsi, M. Pastorino, and M. Raffetto, “Electromagnetic scattering by a multilayer elliptic cylinder under transverse-magnetic illumination: series solution in terms of Mathieu functions,” IEEE Trans. Antenn. Propag. 45(6), 926–935 ( 1997). [CrossRef]

OCIS Codes
(160.1190) Materials : Anisotropic optical materials
(260.2110) Physical optics : Electromagnetic optics
(160.2710) Materials : Inhomogeneous optical media

ToC Category:
Physical Optics

History
Original Manuscript: August 25, 2009
Revised Manuscript: October 21, 2009
Manuscript Accepted: November 2, 2009
Published: November 6, 2009

Citation
Tungyang Chen, Shen-Wen Cheng, and Chung-Ning Weng, "Equivalent configurations of optical  transformation media," Opt. Express 17, 21326-21335 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-23-21326

Sort: Author | Year | Journal | Reset

References

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006). [CrossRef] [PubMed]
U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006). [CrossRef] [PubMed]
G. W. Milton, M. Briane, and J. R. Willis, “On cloaking for elasticity and physical equations with a transformation invariant form,” N. J. Phys. 8(10), 248 (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(5801), 977–980 (2006). [CrossRef] [PubMed]
S. A. Cummer and D. Schurig, “One path to acoustic cloaking,” N. J. Phys. 9(3), 45 (2007). [CrossRef]
H. Chen and C. T. Chan, “Acoustic cloaking in three dimensions using acoustic metamaterials,” Appl. Phys. Lett. 91(18), 183518 (2007). [CrossRef]
G. X. Yu, W. X. Jiang, and T. J. Cui, “Invisible slab cloaks via embedded optical transformation,” Appl. Phys. Lett. 94(4), 041904 (2009). [CrossRef]
J. Hu, X. Zhou, and G. Hu, “Design method for electromagnetic cloak with arbitrary shapes based on Laplace’s equation,” Opt. Express 17(3), 1308–1320 (2009). [CrossRef] [PubMed]
A. Nicolet, F. Zolla, and S. Guenneau, “Electromagnetic analysis of cylindrical cloaks of an arbitrary cross section,” Opt. Express 33, 1584–1586 (2008).
H. Chen, “Transformation optics in orthogonal coordinates,” J. Opt. A, Pure Appl. Opt. 11(7), 075102 (2009). [CrossRef]
N. Kundtz, D. A. Roberts, J. Allen, S. Cummer, and D. R. Smith, “Optical source transformations,” Opt. Express 16(26), 21215–21222 (2008). [CrossRef] [PubMed]
T. Chen, C. N. Weng, and J. S. Chen, “Cloak for curvilinearly anisotropic media in conduction,” Appl. Phys. Lett. 93(11), 114103 (2008). [CrossRef]
U. Leonhardt and T. G. Philbin, “Transformation optics and the geometry of light,” Prog. Opt. 53, 70–152 (2009).
V. M. Shalaev, “Physics. Transforming light,” Science 322(5900), 384–386 (2008). [CrossRef] [PubMed]
H. Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett. 90(24), 241105 (2007). [CrossRef]
Z. L. Mei and T. J. Cui, “Arbitrary bending of electromagnetic waves using isotropic materials,” J. Appl. Phys. 105(10), 104913 (2009). [CrossRef]
G. W. Milton, N.-A. P. Nicorovici, R. C. McPhedran, K. Cherednichenko, and Z. Jacob, “Solutions in folded geometries, and associated cloaking due to anomalous resonance,” N. J. Phys. 10(11), 115021 (2008). [CrossRef]
U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” N. J. Phys. 8(10), 247 (2006). [CrossRef]
V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. Usp. 10 509–14 (Engl. Transl.) (1968).
H. Chen, X. Zhang, X. Luo, H. Ma, and C. T. Chan, “Reshaping the perfect electrical conductor cylinder arbitrarily,” N. J. Phys. 10(11), 113016 (2008). [CrossRef]
T. Yang, H. Chen, X. Luo, and H. Ma, “Superscatterer: enhancement of scattering with complementary media,” Opt. Express 16(22), 18545–18550 (2008). [CrossRef] [PubMed]
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,” Photon. Nanostruct.: Fundam. Appl. 6(1), 87–95 (2008). [CrossRef]
Y. Luo, H. Chen, J. Zhang, L. Ran, and J. A. Kong, “Design and analytical full-wave validation of the invisibility cloaks, concentrators, and field rotators created with a general class of transformations,” Phys. Rev. B 77(12), 125127 (2008). [CrossRef]
H. Chen, X. Luo, H. Ma, and C. T. Chan, “The anti-cloak,” Opt. Express 16(19), 14603–14608 (2008). [CrossRef] [PubMed]
G. Castaldi, I. Gallina, V. Galdi, A. Alù, and N. Engheta, “Cloak/anti-cloak interactions,” Opt. Express 17(5), 3101–3114 (2009). [CrossRef] [PubMed]
Y. Lai, J. Ng, H. Chen, D. Han, J. Xiao, Z. Q. Zhang, and C. T. Chan, “Illusion optics: The optical transformation of an object into another object,” Phys. Rev. Lett. 102(25), 253902 (2009). [CrossRef] [PubMed]
J. Pendry, “All smoke and metamaterials,” Nature 460(7255), 579–580 (2009). [CrossRef]
G. W. Milton, The Theory of Composites (Cambridge University Press, Cambridge, 2002).
E. J. Post, Formal Structure of Electromagnetics: General Covariance and Electromagnetics, (North-Holland, Amsterdam, 1962).
Z. Ruan, M. Yan, C. W. Neff, and M. Qiu, “Ideal cylindrical cloak: perfect but sensitive to tiny perturbations,” Phys. Rev. Lett. 99(11), 113903 (2007). [CrossRef] [PubMed]
V. G. Veselago, “About the wording of Fermat’s principle for light propagation in media with negative refraction index,” (2002) http://arxiv.org/abs/cond-mat/0203451 .
M. Born, and E. Wolf, Principles of Optics, 7th (Expanded) Edition, (Cambridge University Press, New York, 2002).
H. Chen, B. I. Wu, B. Zhang, and J. A. Kong, “Electromagnetic wave interactions with a metamaterial cloak,” Phys. Rev. Lett. 99(6), 063903 (2007). [CrossRef] [PubMed]
S. Caorsi, M. Pastorino, and M. Raffetto, “Electromagnetic scattering by a multilayer elliptic cylinder under transverse-magnetic illumination: series solution in terms of Mathieu functions,” IEEE Trans. Antenn. Propag. 45(6), 926–935 (1997). [CrossRef]

Appl. Phys. Lett.

H. Chen and C. T. Chan, “Acoustic cloaking in three dimensions using acoustic metamaterials,” Appl. Phys. Lett. 91(18), 183518 (2007). [CrossRef]
G. X. Yu, W. X. Jiang, and T. J. Cui, “Invisible slab cloaks via embedded optical transformation,” Appl. Phys. Lett. 94(4), 041904 (2009). [CrossRef]
T. Chen, C. N. Weng, and J. S. Chen, “Cloak for curvilinearly anisotropic media in conduction,” Appl. Phys. Lett. 93(11), 114103 (2008). [CrossRef]
H. Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett. 90(24), 241105 (2007). [CrossRef]

IEEE Trans. Antenn. Propag.

S. Caorsi, M. Pastorino, and M. Raffetto, “Electromagnetic scattering by a multilayer elliptic cylinder under transverse-magnetic illumination: series solution in terms of Mathieu functions,” IEEE Trans. Antenn. Propag. 45(6), 926–935 (1997). [CrossRef]

J. Appl. Phys.

Z. L. Mei and T. J. Cui, “Arbitrary bending of electromagnetic waves using isotropic materials,” J. Appl. Phys. 105(10), 104913 (2009). [CrossRef]

J. Opt. A, Pure Appl. Opt.

H. Chen, “Transformation optics in orthogonal coordinates,” J. Opt. A, Pure Appl. Opt. 11(7), 075102 (2009). [CrossRef]

N. J. Phys.

G. W. Milton, N.-A. P. Nicorovici, R. C. McPhedran, K. Cherednichenko, and Z. Jacob, “Solutions in folded geometries, and associated cloaking due to anomalous resonance,” N. J. Phys. 10(11), 115021 (2008). [CrossRef]
U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” N. J. Phys. 8(10), 247 (2006). [CrossRef]
G. W. Milton, M. Briane, and J. R. Willis, “On cloaking for elasticity and physical equations with a transformation invariant form,” N. J. Phys. 8(10), 248 (2006). [CrossRef]
S. A. Cummer and D. Schurig, “One path to acoustic cloaking,” N. J. Phys. 9(3), 45 (2007). [CrossRef]
H. Chen, X. Zhang, X. Luo, H. Ma, and C. T. Chan, “Reshaping the perfect electrical conductor cylinder arbitrarily,” N. J. Phys. 10(11), 113016 (2008). [CrossRef]

Nature

J. Pendry, “All smoke and metamaterials,” Nature 460(7255), 579–580 (2009). [CrossRef]

Opt. Express

H. Chen, X. Luo, H. Ma, and C. T. Chan, “The anti-cloak,” Opt. Express 16(19), 14603–14608 (2008). [CrossRef] [PubMed]
G. Castaldi, I. Gallina, V. Galdi, A. Alù, and N. Engheta, “Cloak/anti-cloak interactions,” Opt. Express 17(5), 3101–3114 (2009). [CrossRef] [PubMed]
T. Yang, H. Chen, X. Luo, and H. Ma, “Superscatterer: enhancement of scattering with complementary media,” Opt. Express 16(22), 18545–18550 (2008). [CrossRef] [PubMed]
J. Hu, X. Zhou, and G. Hu, “Design method for electromagnetic cloak with arbitrary shapes based on Laplace’s equation,” Opt. Express 17(3), 1308–1320 (2009). [CrossRef] [PubMed]
A. Nicolet, F. Zolla, and S. Guenneau, “Electromagnetic analysis of cylindrical cloaks of an arbitrary cross section,” Opt. Express 33, 1584–1586 (2008).
N. Kundtz, D. A. Roberts, J. Allen, S. Cummer, and D. R. Smith, “Optical source transformations,” Opt. Express 16(26), 21215–21222 (2008). [CrossRef] [PubMed]

Photon. 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 transfromations of Maxwell’s equations,” Photon. Nanostruct.: Fundam. Appl. 6(1), 87–95 (2008). [CrossRef]

Phys. Rev. B

Y. Luo, H. Chen, J. Zhang, L. Ran, and J. A. Kong, “Design and analytical full-wave validation of the invisibility cloaks, concentrators, and field rotators created with a general class of transformations,” Phys. Rev. B 77(12), 125127 (2008). [CrossRef]

Phys. Rev. Lett.

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