Controlling electromagnetic scattering of a cavity by transformation media

doi:10.1364/OE.20.006777

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Controlling electromagnetic scattering of a cavity by transformation media

Shenyun Wang and Shaobin Liu »View Author Affiliations

Optics Express, Vol. 20, Issue 6, pp. 6777-6787 (2012)
http://dx.doi.org/10.1364/OE.20.006777

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▸ Article Outline
– Abstract
1. Introduction
2. Theory
3. Design and performance of transformation medium
4. Practical implementation of the transformation medium
4. Conclusions
– References and links

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Abstract

Based on the transformation media theory, we proposed a way to control the scattering of a cavity, or trench, located on a metallic plane. Specifically, we show how is possible to design transformation medium to fill up a cavity with arbitrary cross section, which is capable of enhancing the specularly reflection wave. As the inverse problem, we also address the design of transformation medium coating, which is laid on the metallic plane, to mimic the scattering of the cavity. Based on the effective medium theory, the transformation medium for the case of a polygonal cavity can be realized by oblique layered structures, and each layered structure is consisting of two kinds of isotropic dielectrics, thus leading an ease of practical fabrication.

1. Introduction

In the past few years, transformation optics has aroused a lot of scientific interest for its unprecedented flexibility in manipulating electromagnetic waves propagation. The first interesting application of this concept is the realization of invisibility cloaks [1

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

], in which electromagnetic waves are guided around a certain region of space rendering the interior of the region invisible for external detector. Exploiting the coordinate transformation and the invariance of Maxwell's equations, one can determine the permittivity and permeability of the transformation medium. Various approaches like ray tracing [4

D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express 14(21), 9794–9804 (2006). [CrossRef] [PubMed]

], analytical method [5

C. W. Qiu, A. Novitsky, H. Ma, and S. Qu, “Electromagnetic interaction of arbitrary radial-dependent anisotropic spheres and improved invisibility for nonlinear-transformation-based cloaks,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 80(1), 016604 (2009). [CrossRef] [PubMed]

] and numerical methods (both finite-element method [6

S. A. Cummer, B.-I. Popa, and D. R. Smith, “Full-wave simulation of electromagnetic cloaking structures,” Phys. Rev. Lett. 74, 036621 (2006).

] and the finite-difference time-domain method [7

C. Argyropoulos, Y. Zhao, and Y. Hao, “A radially-dependent dispersive finite-difference time-domain method for the evaluation of electromagnetic cloaks,” IEEE Trans. Antenn. Propag. 57(5), 1432–1441 (2009). [CrossRef]

]) have been used to simulate the electromagnetic properties of cloaks. However, this kind of cloak requires singular values on the inner boundary, and as a consequence it only works in a narrow frequency. Another kind of cloak called carpet cloak [8

J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. 101(20), 203901 (2008). [CrossRef] [PubMed]

], which can conceal an object on a plane ground by mimicking the specular reflection of a mirror ground, is more practical for real application. Since it can be constructed with non-resonant components, carpet cloak has been experimentally realized both in microwave and optical frequency recently [9

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]

–11

B. Zhang, Y. Luo, X. Liu, and G. Barbastathis, “Macroscopic invisibility cloak for visible light,” Phys. Rev. Lett. 106(3), 033901 (2011). [CrossRef] [PubMed]

After the concept has been applied on invisible cloaks, further efforts have been taken to utilize transformation optics to design various kinds of interesting devices, such as wave concentrator [12

W. X. Jiang, T. J. Cui, Q. Cheng, J. Y. Chin, X. M. Yang, R. Liu, and D. R. Smith, “Design of arbitrarily shaped concentrators based on conformally optical transformation of nonuniform rational B-spline surfaces,” Appl. Phys. Lett. 92(26), 264101 (2008). [CrossRef]

], rotator [13

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

], beam shifter [14

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]

], bends [15

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]

], focusing lenses [16

A. Di Falco, S. C. Kehr, and U. Leonhardt, “Luneburg lens in silicon photonics,” Opt. Express 19(6), 5156–5162 (2011). [CrossRef] [PubMed]

,17

D.-H. Kwon and D. H. Werner, “Flat focusing lens designs having minimized reflection based on coordinate transformation techniques,” Opt. Express 17(10), 7807–7817 (2009). [CrossRef] [PubMed]

], illusion devices [18

Y. Lai, J. Ng, H. Y. Chen, D. Z. Han, J. 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]

,19

W. X. Jiang, T. J. Cui, X. M. Yang, H. F. Ma, and Q. Cheng, “Design of arbitrarily shaped concentrators based on conformally optical transformation of nonuniform rational B-spline surfaces,” Appl. Phys. Lett. 98, 204101 (2011).

], planar antenna [20

Z. L. Mei, J. Bai, and T. J. Cui, “Experimental verification of a broadband planar focusing antenna based on transformation optics,” New J. Phys. 13(6), 063028 (2011). [CrossRef]

], and waveguide connectors [21

K. Zhang, Q. Wu, F.-Y. Meng, and L. W. Li, “Arbitrary waveguide connector based on embedded optical transformation,” Opt. Express 18(16), 17273–17279 (2010). [CrossRef] [PubMed]

,22

C. García-Meca, M. M. Tung, J. V. Galán, R. Ortuño, F. J. Rodríguez-Fortuño, J. Martí, and A. Martínez, “Squeezing and expanding light without reflections via transformation optics,” Opt. Express 19(4), 3562–3575 (2011). [CrossRef] [PubMed]

]. In principle, the constructive transformation medium is inhomogeneous and anisotropic, which is usually realized by artificially structured metamaterials, such as split ring resonator [3

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]

,23

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000). [CrossRef] [PubMed]

], and some natural materials, like birefringent calcite [10

X. Chen, Y. Luo, J. Zhang, K. Jiang, J. B. Pendry, and S. Zhang, “Macroscopic invisibility cloaking of visible light,” Nat Commun 2, 176 (2011). [CrossRef] [PubMed]

,11

B. Zhang, Y. Luo, X. Liu, and G. Barbastathis, “Macroscopic invisibility cloak for visible light,” Phys. Rev. Lett. 106(3), 033901 (2011). [CrossRef] [PubMed]

]. However, the anisotropy of natural material is weak and cannot be engineered as desired. A different easy way to implement the transformation medium is employing thin alternating layers of normal isotropic materials based on the effective medium theory. For instances, the concentric layered structures of alternating homogeneous isotropic dielectrics have been constructed to realize cylindrical cloaks [24

Y. Huang, Y. Feng, and T. Jiang, “Electromagnetic cloaking by layered structure of homogeneous isotropic materials,” Opt. Express 15(18), 11133–11141 (2007). [CrossRef] [PubMed]

,25

M. Farhat, S. Guenneau, and S. Enoch, “Ultrabroadband elastic cloaking in thin plates,” Phys. Rev. Lett. 103(2), 024301 (2009). [CrossRef] [PubMed]

] and spherical cloaks [26

C. W. Qiu, L. Hu, X. Xu, and Y. Feng, “Spherical cloaking with homogeneous isotropic multilayered structures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 79(4), 047602 (2009). [CrossRef] [PubMed]

,27

C. W. Qiu, L. Hu, and S. Zouhdi, “Isotropic non-ideal cloaks providing improved invisibility yb adaptive segmentation and optimal refractive index profile from ordering isotropic materials,” Opt. Express 18, 14950–14959 (2010).

], while the oblique layered structures have been utilized to demonstrate carpet cloaks [28

X. Xu, Y. Feng, Y. Hao, J. Zhao, and T. Jiang, “Infrared carpet cloak designed with uniform silicon grating structure,” Appl. Phys. Lett. 95(18), 184102 (2009). [CrossRef]

–30

J. Zhang, L. Liu, Y. Luo, S. Zhang, and N. A. Mortensen, “Homogeneous optical cloak constructed with uniform layered structure,” Opt. Express 19(9), 8625–8631 (2011). [CrossRef]

], beam shifters [31

H. Chen and C. T. Chan, “Electromagnetic wave manipulation by layered systems using the transformation media concept,” Appl. Phys. B 78, 054204 (2008).

In free space, a planar reflector covered with transformed medium is presented for imitating the dihedral corner scattering response [32

I. Gallina, G. Castaldi, and V. Galdi, “Transformation thin planar retro directive reflector,” IEEE Microw. Wirel. Compon. Lett. 7, 603–605 (2008).

], and corner and wedge structures coated by special transformation medium can induce an electromagnetic scattering similar to that of a planar metallic sheet [33

I. Gallina, G. Castaldi, and V. Galdi, “Transformation optics-inspired metamaterial coatings for controlling the scattering response of wedge/corner-type structures,” Microw. Opt. Technol. Lett. 51, 2709–2712 (2009). [CrossRef]

]. Recently, the utilization of periodically layered structure is extended into the domain of transformation acoustics, and a bending layered structure has been cut into particular shapes for acoustic cloaking and cavity, or trough illusion [34

Z. Liang and J. Li, “Bending a periodically layered structure for transformation acoustics,” Appl. Phys. Lett. 98(24), 241914 (2011). [CrossRef]

]. Following this study, in this paper, we address the design of transformation medium to fill up a cavity under a metal plane which is capable of controlling the scattering by mimicking the specular reflection of flat metallic mirror. Such reflection of the cavity might be very useful in electromagnetic wave anti-detection by rendering a reduction of the overall visibility of the cavity. The inverse problem is also presented that a cavity scattering could be imitated by a transformation coating laid on the metallic plane surface. Since the electromagnetic field scattered by cavity is much irregular, such a transformation medium coating for the metallic plane can enhance the electromagnetic wave scattering, leading many desirable applications such as target identification and illusion. In most cases, the designed transformation medium requires continuously inhomogeneous and anisotropic material parameters and sometimes results in extreme or singular material properties, making the realization quite difficult.

In the following, we consider the case of simplified polygonal cavity, whose boundary is made up of planar metal walls. By employing a general way of transformation optics, we find that the cavity filling medium of ideal parameters can be divided into several regions and each region only requires homogeneous but anisotropic parameters. Based on the effective medium theory, the comprising region of homogeneous anisotropy can be realized by oblique alternating layers of isotropic dielectrics. We provide the validation of the mimicking performance of both our ideas by full-wave finite-element numerical simulation with both near field distributions and average power outflow patterns at an observation semicircle under a TM Gaussian beam illumination.

2. Theory

As known, different coordinate transformations could be chosen to design the desirable transformation medium by such as space squeezing [1

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

J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. 101(20), 203901 (2008). [CrossRef] [PubMed]

], rotating [13

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

], and expanding [22

]. In this work, considering a general scenario, a bi-dimensional cavity (identical along z-axis), which has arbitrary inner perfect electrical conductor (PEC) boundary, is filled up with our interest physical medium (grid region) in real space

(x' y' z')

as illustrated in Fig. 1(a) . In order to induce the desirable specular reflection, we let the virtual space

(x' y' z')

be an infinite PEC plane as show in Fig. 1(b). A simple transformation between the two space [the dash lined region (ADC) in Fig. 1(b) and the grid region(A’D’C’B’) in Fig. 1(a)] is given by

{\begin{matrix} x' = x \\ y' = y \cdot \frac{y_{2} (x) - y_{1} (x)}{y_{2} (x)} + y_{1} (x) \\ z' = z \end{matrix}

(1)

where,

y_{1} (x)

represents the cavity inner boundary,

y_{2} (x)

stands for the physical outer boundary of the physical medium (which is invariant upon the coordinate transformation). The Jacobian transformation matrix can be gotten based on Eq. (1)

J = \frac{\partial (x', y', z')}{\partial (x, y, z)} = | \begin{matrix} 1 & 0 & 0 \\ K & P & 0 \\ 0 & 0 & 1 \end{matrix} |

(2)

where

K = \frac{\partial y'}{\partial x} = y \cdot \frac{d [(y_{2} (x) - y_{1} (x)) / y_{2} (x)]}{d x} + \frac{d y_{1} (x)}{d x}

and

P = \frac{\partial y'}{\partial y} = \frac{y_{2} (x) - y_{1} (x)}{y_{2} (x)}

Fig. 1 Geometry of problems. (a) A cavity filled up with transformation medium in real space; (b) A PEC plane in virtual space; (c) An cavity illusion coating laid on a PEC plane in real space; (d) A empty cavity under a PEC plane in virtual space.

Due to the concept of transformation optics [1

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

], the transformation of the coordinate can be mapped into the change of material properties in the real space. Both the relative permittivity and permeability tensors of the transformation medium can be expressed as

\bar{\bar{ε}} = \bar{\bar{μ}} = \frac{J \cdot J^{T}}{\det | J |} = | \begin{matrix} 1 / P & K / P & 0 \\ K / P & K^{2} / P + P & 0 \\ 0 & 0 & 1 / P \end{matrix} |

(3)

For the inverse problem, we imitate an arbitrary-shaped cavity scattering [Fig. 1(a)] by designing a transformation medium coating which is laid on the metallic plane in real space

(x' y' z')

as depicted in Fig. 1(c). For this case, the virtual space

(x' y' z')

should include an empty cavity under the PEC plane surface as illustrated in Fig. 1(d). The transformation coordinate could be obtained by mapping the virtual cavity inner boundary

y_{1} (x)

(ABC) into real plane surface boundary

y = 0

(A’C’), leaving the physical outer boundary

y_{2} (x)

invariant in transformation. The transformation functions between the two space [the dash lined region (ADCB) in Fig. 1(d) and the grid region (A’D’C’) in Fig. 1(c)] are expressed as

{\begin{matrix} x' = x \\ y' = y \cdot \frac{y_{2} (x)}{y_{2} (x) - y_{1} (x)} + \frac{y_{1} (x) y_{2} (x)}{y_{2} (x) - y_{1} (x)} \\ z' = z \end{matrix}

(4)

Similarly, we can get the Jacobian transformation matrix and relative permittivity and permeability tensor as same forms with Eqs. (2) and (3), respectively, but element parameters

K

and

P

are given as

K = \frac{\partial y'}{\partial x} = y \cdot \frac{d [\frac{y_{2} (x)}{y_{2} (x) - y_{1} (x)}]}{d x} - \frac{d [\frac{y_{1} (x) \cdot y_{2} (x)}{y_{2} (x) - y_{1} (x)}]}{d x} and P = \frac{\partial y'}{\partial y} = \frac{y_{2} (x) - y_{1} (x)}{y_{2} (x)}

Furthermore, the symmetrical permittivity and permeability tensor derived in Eq. (3) can be transformed into a diagonal matrix, which will be useful in the following demonstration of transformation medium realized by alternating layered structures. Next, both ideas proposed above will be verified by full-wave finite-element numerical simulation.

3. Design and performance of transformation medium

In this study, in order to validate ideas presented above, we suppose a semielliptical cavity under a standard metallic plane. It should be pointed out that the cavity boundary is also PEC material for absolute reflection of TM Gaussian incident wave (with the magnetic field along

z

axis), which is also utilized in all the following numerical simulations. As a representative example, we let the center of the semiellipse be the coordinate origin (0,0) and the major axis of the ellipse be 2.0 and minor axis be 1.0, and the cavity boundary function is given as

y' = y_{1} (x) = - 0.5 \sqrt{1 - x^{2}}

. Based on the imbedded transformation optics [21

K. Zhang, Q. Wu, F.-Y. Meng, and L. W. Li, “Arbitrary waveguide connector based on embedded optical transformation,” Opt. Express 18(16), 17273–17279 (2010). [CrossRef] [PubMed]

], for simplicity, we take the outer boundary of the transformed space as

y' = y_{2} (x) = - y_{1} (x)

, indicating that the transformed space takes a full elliptical shape. Inserting the boundary functions into Eqs. (2) and (3), the constitutive permittivity and permeability tensor of the transformed space could be obtained.

Figure 2(a) demonstrates the transverse magnetic field distribution of the semielliptical cavity filled up with transformation medium in a full ellipse shape. Here the incoming wave is properly selected to be a TM Gaussian beam of 2.0 GHz and its incident angle is set to be 45°, supposing the zero-angle along positive

x

axis in

x - y

plane. Similar to the carpet cloak, the cavity filled up with the designed transformation medium can reflect the incoming TM wave in the specular direction, mimicking that the incoming wave impinges onto a flat reflective surface (mirror) as depicted in Fig. 2(b). In Fig. 2(d), we plot the near magnetic field distribution of the empty cavity structure. As indicated, the incoming Gaussian wave could be confined around the left focus and then generate a broad and cured wavefronts. This scattering phenomenon could find interesting applications in target identification, enhancing radar-cross-section (RCS) scenarios. In order to imitate the diffuse scattering of this cavity, we can derive the constitutive properties of a transformation medium coating lain on a metallic plane. We let the same incident Gaussian beam illuminate the designed illusion coating, and the near field magnetic field is plotted in Fig. 2(c), from which it can be seen that the field outside the transformed space has a very excellent agreement with the case of empty cavity [Fig. 2(d)].

Fig. 2 Transverse magnetic field distribution of (a) a semielliptical cavity filled up with ideal transformation medium, (b) a PEC plane, (c) a cavity illusion coating laid on a PEC plane, (d) empty semiellipse, for an incident angle of 45°.

In addition, the performance of both cases is still verified by simulating total average scattering power outflow at an observation semicircle with its radius of 1.95m and origin at (0,0) [see dark lined semicircle in Fig. 2(b)]. In Fig. 3 , it is obvious that the total average power outflow patterns show tiny difference between cases of physical medium and the PEC plane or bare cavity.

Fig. 3 (a) Total average power outflow patterns of the semielliptical cavity filled up with ideal transformation medium and perfect metallic plane; (b) Total average power outflow patterns of a cavity illusion coating laid on a PEC plane and empty semielliptical cavity.

In most cases, the designed transformation medium based on the optical transformation requires continuously inhomogeneous and anisotropic material parameters and sometimes results in extreme or singular material properties, which is impossible for practical realization even with artificially structured metamaterials. For the above instances, a set of constitutive parameters of the two models above can be derived from Eq. (3), and the spatial variable parameters are simulated and shown in Fig. 4 . Both ends of the major axis of the ellipse or semiellipse require infinite values.

Fig. 4 Variable constitutive parameters. (a) (b) Permittivity elements of the cavity filling medium; (c) (d) Permittivity elements of the cavity illusion coating.

4. Practical implementation of the transformation medium

The above analysis indicates that the designed transformation medium derived from the embedded transformation optics is valid for our proposed ideas. Now we intend to simplify the cavity structure, which has constant constitutive parameters, thus leading an ease of practical fabrication. We let the constitutive parameters

K

and

P

be constant and the necessary condition is that both inner and outer boundary functions should be proportioned and linear functions. For the sake of simplicity, we still make inner and outer boundaries symmetrical along

x

axis. Now we can suppose that the cavity is composed of planar walls, then the cross section of the transformed space takes a symmetrical polygonal shape, which can be divided into several regions and each region only requires spatial invariant but anisotropic material parameters. As pointed in last paragraph in Section 2, the constitutive tensor is symmetrical, which can be obtained from a diagonal tensor through rotating the optical axis by angle

θ

\bar{\bar{ε}} = | \begin{matrix} ε_{11} & ε_{12} & 0 \\ ε_{21} & ε_{22} & 0 \\ 0 & 0 & ε_{33} \end{matrix} | = | \begin{matrix} 1 / P & K / P & 0 \\ K / P & K^{2} / P + P & 0 \\ 0 & 0 & 1 / P \end{matrix} | = | \begin{matrix} \cos θ & \sin θ & 0 \\ - \sin θ & \cos θ & 0 \\ 0 & 0 & 1 \end{matrix} | | \begin{matrix} ε'_{11} & 0 & 0 \\ 0 & ε'_{22} & 0 \\ 0 & 0 & ε'_{33} \end{matrix} | | \begin{matrix} \cos θ & - \sin θ & 0 \\ \sin θ & \cos θ & 0 \\ 0 & 0 & 1 \end{matrix} |

(5)

In subwavelength scale, the layered structure of alternating homogeneous isotropic materials could be effectively treated as single anisotropic medium with effective dielectric permittivity of

\bar{\bar{ε}} = [\begin{matrix} ε_{∥} & ε_{⊥} & ε_{⊥} \end{matrix}]

, where

ε_{∥} = (ε_{1} + κ ε_{2}) / (1 + κ)

ε_{⊥} = (1 + κ) ε_{1} ε_{2} / (κ ε_{1} + ε_{2})

and

κ = d_{2} / d_{1}

denotes the ratio of the two layer widths [35

B. Wood, J. B. Pendry, and D. Tsai, “Directed subwavelength imaging using a layered metal-dielectric system,” Phys. Rev. B 74(11), 115116 (2006). [CrossRef]

]. In fact, the element value of

ε_{33} = ε'_{33}

could be arbitrary for TM wave. So in order to map the desired constitutive tensor into the anisotropy of layered structure, we let

ε'_{11} = ε_{∥} = 2 ε_{1} ε_{2} / (ε_{1} + ε_{2})

and

ε'_{22} = ε'_{33} = ε_{⊥} = (ε_{1} + ε_{2}) / 2

, where we have made the thickness of the layers kept identical (

κ = 1.0

) and sufficiently thin compared with the wavelength. For a TM Gaussian wave incidence, the parameters are simplified as

ε_{1}

ε_{2}

and

μ_{33}

, which can be determined as

ε_{1} = ε_{∥} + \sqrt{ε_{∥}^{2} - ε_{∥} ε_{⊥}}, ε_{2} = ε_{∥} - \sqrt{ε_{∥}^{2} - ε_{∥} ε_{⊥}}, μ_{33} = 1 / P

(6)

In the following, we demonstrate the implementation of transformation medium with a tri-sided polygonal cavity example. The designed alternating layered system is consisting of three regions denoted with Roman numbers as depicted in Fig. 5(a) . The essential parameters including cavity inner boundary functions, permittivity of the two layers

ε_{1}

and

ε_{2}

, alignment angles and ratios of layer thickness to wavelength for all regions are summarized in Table 1 .

Fig. 5 (a) Schematic view of cavity filling medium realized by isotropic layers; Magnetic field distribution of a tri-sided polygonal cavity filled up with (b) ideal transformation medium and (c) isotropic layers for an incident angle of 45°; Magnetic field distribution of a tri-sided polygonal cavity filled up with (d) ideal transformation medium and (e) isotropic layers for an incident angle of 18°.

Table 1 Cavity Boundary Functions, Permittivity, Alignment Angles and Ratios of Layer Thickness to Wavelength for All Regions of Layered System for Cavity Filling Medium

Region	$y_{1} (x)$	$ε_{1}$	$ε_{1}$	$θ$ (degree)	$Δ / λ$
I	$0.5 x - 0.5$	4.0801	0.2451	+ 8.55	0.0494
II	$0.25 x - 0.5$	3.8211	0.2617	−4.64	0.0269
III	$0.75 x - 0.375$	4.4913	0.2227	−11.42	0.0658

Figures 5(b) and 5(c) shows the near magnetic field distribution of the tri-sided polygonal cavity structure filled up with ideal anisotropic transformation medium and isotropic layers, respectively, for an incident angle of 45° from

x

axis. We find that the cavity filled up with the designed multilayer medium of alternating homogenous dielectrics can confine the incoming Gaussian wave highly in the specular direction as expected, and it is difficult to observe the difference among cases of Figs. 5(b) and 5(c) and Fig. 2(b) in field distribution outside the outer boundary. Furthermore, we investigate the robust performance of designed cavity filling medium for a much lower incident angle of 18°. The magnetic field distribution could be compared between the cavity filled up with ideal anisotropic medium and isotropic layers shown in Figs. 5(d) and 5(e).

We also simulated the total average power outflow at an observation semicircle defined in Fig. 2(b). In Fig. 6(a) , we plot the average power outflow patterns of cases in Figs. 5 (b) and 5(c) and Fig. 2(b), which shows almost the same, for an incident angle of 45°. A similar plot is also simulated for the incident angle of 18° as shown in Fig. 6(b), from which a bit difference could be seen between cases of the isotropic layers and anisotropic parameters for angle from 120° to 160°. Figure 6 further confirms that the approach of the utilizing alternating layers to realize the anisotropic material is quite convincing with much less sensitive response to the angular variations of the excitation conditions.

Fig. 6 The total average power outflow patterns of different cases at the observation semicircle for an incident angle of (a) 45°, (b) 18°.

Next, as the inverse problem, we design a layered coating laid on the metallic flat to mimic the empty tri-sided polygonal cavity designed above. In Figs. 7(a) and 7(b), we plot the magnetic field distribution of the empty cavity for incident angles of 45° and 18°, respectively. As indicated, the magnetic field scattered by the empty cavity boundary is quite irregular, which might be very useful in target identification and communication. Now we construct an illusion coating laid on the metallic plane realized by alternating layers system which is also consisting of three regions denoted with Roman numbers as shown in Fig. 8(a) . The parameters for each constitutive layered structure are summarized in Table 2 . It shows that cavity boundary functions have been taken the same, leading same values of permittivity for the alternating layers but different alignment angles compared with Table 1. Figure 8 shows magnetic field distribution of the illusion coating constructed with anisotropic parameters and isotropic layers for incident angles of 45° and 18°. It is obviously seen that outer magnetic field distribution plotted in Fig. 8 has little difference with the reference distribution of empty cavity shown in Fig. 7, indicating the illusion coating is valid for imitating electromagnetic scattering of the empty cavity.

Fig. 7 Magnetic field distribution of the empty tri-sided polygonal cavity for incident angle of (a) 45° and (b) 18°.

Fig. 8 (a) Schematic view of cavity illusion coating realized by isotropic layers; Magnetic field distribution of the tri-sided polygonal cavity illusion coating with (b) ideal transformation medium and (c) isotropic layers, for an incident angle of 45°; Magnetic field distribution of the tri-sided polygonal cavity illusion coating with (d) ideal transformation medium and (e) isotropic layers, for an incident angle of 18°.

Table 2 Cavity Boundary Functions, Permittivity, Alignment Angles and Ratios of Layer Thickness to Wavelength for All Regions of Layered System for Cavity Illusion Coating

Region	$y_{1} (x)$	$ε_{1}$	$ε_{1}$	$θ$ (degree)	$Δ / λ$
I	$0.5 x - 0.5$	4.0801	0.2451	−71.98	0.0155
II	$0.25 x - 0.5$	3.8211	0.2617	+ 80.60	0.0160
III	$0.75 x - 0.375$	4.4913	0.2227	+ 64.55	0.0147

At last, we have also calculated the average power outflow patterns at the observation semicircle defined above for incident angles of 45° and 18°. As clearly shown in Fig. 9 , when the illusion coating laid on the metallic plane is realized by isotropic layers, the average power outflow pattern still has very small error compared with the cases of the empty cavity and the ideal transformation medium coating for both incident angles, behaving a robust illusion performance with excitation from different angles.

Fig. 9 The total average power outflow patterns of different cases at the observation semicircle for an incident angle of (a) 45°, (b) 18°.

4. Conclusions

In conclusion, we have presented a general approach to design transformation medium to fill up an arbitrary-shaped cavity under a metallic plane for reduction of the electromagnetic scattering. The inverse problem is also investigated that the empty cavity scattering could be imitated by an illusion coating laid on the metallic plane. We have verified our both presented ideas by taking a semielliptical cavity as a representative example. According to the effective medium theory, we find alternating layers of normal isotropic dielectrics can realize the transformation medium as one desired in case that the cavity takes a polygonal shape. This scheme of constructing transformation medium simply by isotropic alternating layers behaves a good illusion performance of both PEC plane and empty cavity under the excitation for different angles.

Acknowledgments

The authors are grateful for the support partly from the Chinese Natural Science Foundation (grant No. 60971122), and partly from the Open Research Program in China’s State Key Laboratory of Millimeter Waves (No. K201103).

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.	D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express 14(21), 9794–9804 (2006). [CrossRef] [PubMed]
5.	C. W. Qiu, A. Novitsky, H. Ma, and S. Qu, “Electromagnetic interaction of arbitrary radial-dependent anisotropic spheres and improved invisibility for nonlinear-transformation-based cloaks,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 80(1), 016604 (2009). [CrossRef] [PubMed]
6.	S. A. Cummer, B.-I. Popa, and D. R. Smith, “Full-wave simulation of electromagnetic cloaking structures,” Phys. Rev. Lett. 74, 036621 (2006).
7.	C. Argyropoulos, Y. Zhao, and Y. Hao, “A radially-dependent dispersive finite-difference time-domain method for the evaluation of electromagnetic cloaks,” IEEE Trans. Antenn. Propag. 57(5), 1432–1441 (2009). [CrossRef]
8.	J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. 101(20), 203901 (2008). [CrossRef] [PubMed]
9.	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]
10.	X. Chen, Y. Luo, J. Zhang, K. Jiang, J. B. Pendry, and S. Zhang, “Macroscopic invisibility cloaking of visible light,” Nat Commun 2, 176 (2011). [CrossRef] [PubMed]
11.	B. Zhang, Y. Luo, X. Liu, and G. Barbastathis, “Macroscopic invisibility cloak for visible light,” Phys. Rev. Lett. 106(3), 033901 (2011). [CrossRef] [PubMed]
12.	W. X. Jiang, T. J. Cui, Q. Cheng, J. Y. Chin, X. M. Yang, R. Liu, and D. R. Smith, “Design of arbitrarily shaped concentrators based on conformally optical transformation of nonuniform rational B-spline surfaces,” Appl. Phys. Lett. 92(26), 264101 (2008). [CrossRef]
13.	H. Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett. 90(24), 241105 (2007). [CrossRef]
14.	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]
15.	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.	A. Di Falco, S. C. Kehr, and U. Leonhardt, “Luneburg lens in silicon photonics,” Opt. Express 19(6), 5156–5162 (2011). [CrossRef] [PubMed]
17.	D.-H. Kwon and D. H. Werner, “Flat focusing lens designs having minimized reflection based on coordinate transformation techniques,” Opt. Express 17(10), 7807–7817 (2009). [CrossRef] [PubMed]
18.	Y. Lai, J. Ng, H. Y. Chen, D. Z. Han, J. 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]
19.	W. X. Jiang, T. J. Cui, X. M. Yang, H. F. Ma, and Q. Cheng, “Design of arbitrarily shaped concentrators based on conformally optical transformation of nonuniform rational B-spline surfaces,” Appl. Phys. Lett. 98, 204101 (2011).
20.	Z. L. Mei, J. Bai, and T. J. Cui, “Experimental verification of a broadband planar focusing antenna based on transformation optics,” New J. Phys. 13(6), 063028 (2011). [CrossRef]
21.	K. Zhang, Q. Wu, F.-Y. Meng, and L. W. Li, “Arbitrary waveguide connector based on embedded optical transformation,” Opt. Express 18(16), 17273–17279 (2010). [CrossRef] [PubMed]
22.	C. García-Meca, M. M. Tung, J. V. Galán, R. Ortuño, F. J. Rodríguez-Fortuño, J. Martí, and A. Martínez, “Squeezing and expanding light without reflections via transformation optics,” Opt. Express 19(4), 3562–3575 (2011). [CrossRef] [PubMed]
23.	D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000). [CrossRef] [PubMed]
24.	Y. Huang, Y. Feng, and T. Jiang, “Electromagnetic cloaking by layered structure of homogeneous isotropic materials,” Opt. Express 15(18), 11133–11141 (2007). [CrossRef] [PubMed]
25.	M. Farhat, S. Guenneau, and S. Enoch, “Ultrabroadband elastic cloaking in thin plates,” Phys. Rev. Lett. 103(2), 024301 (2009). [CrossRef] [PubMed]
26.	C. W. Qiu, L. Hu, X. Xu, and Y. Feng, “Spherical cloaking with homogeneous isotropic multilayered structures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 79(4), 047602 (2009). [CrossRef] [PubMed]
27.	C. W. Qiu, L. Hu, and S. Zouhdi, “Isotropic non-ideal cloaks providing improved invisibility yb adaptive segmentation and optimal refractive index profile from ordering isotropic materials,” Opt. Express 18, 14950–14959 (2010).
28.	X. Xu, Y. Feng, Y. Hao, J. Zhao, and T. Jiang, “Infrared carpet cloak designed with uniform silicon grating structure,” Appl. Phys. Lett. 95(18), 184102 (2009). [CrossRef]
29.	X. Xu, Y. Feng, Z. Yu, T. Jiang, and J. Zhao, “Simplified ground plane invisibility cloak by multilayer dielectrics,” Opt. Express 18(24), 24477–24485 (2010). [CrossRef] [PubMed]
30.	J. Zhang, L. Liu, Y. Luo, S. Zhang, and N. A. Mortensen, “Homogeneous optical cloak constructed with uniform layered structure,” Opt. Express 19(9), 8625–8631 (2011). [CrossRef]
31.	H. Chen and C. T. Chan, “Electromagnetic wave manipulation by layered systems using the transformation media concept,” Appl. Phys. B 78, 054204 (2008).
32.	I. Gallina, G. Castaldi, and V. Galdi, “Transformation thin planar retro directive reflector,” IEEE Microw. Wirel. Compon. Lett. 7, 603–605 (2008).
33.	I. Gallina, G. Castaldi, and V. Galdi, “Transformation optics-inspired metamaterial coatings for controlling the scattering response of wedge/corner-type structures,” Microw. Opt. Technol. Lett. 51, 2709–2712 (2009). [CrossRef]
34.	Z. Liang and J. Li, “Bending a periodically layered structure for transformation acoustics,” Appl. Phys. Lett. 98(24), 241914 (2011). [CrossRef]
35.	B. Wood, J. B. Pendry, and D. Tsai, “Directed subwavelength imaging using a layered metal-dielectric system,” Phys. Rev. B 74(11), 115116 (2006). [CrossRef]

OCIS Codes
(160.1190) Materials : Anisotropic optical materials
(230.0230) Optical devices : Optical devices

ToC Category:
Physical Optics

History
Original Manuscript: December 22, 2011
Revised Manuscript: February 19, 2012
Manuscript Accepted: March 4, 2012
Published: March 8, 2012

Citation
Shenyun Wang and Shaobin Liu, "Controlling electromagnetic scattering of a cavity by transformation media," Opt. Express 20, 6777-6787 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-6-6777

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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]
D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express14(21), 9794–9804 (2006). [CrossRef] [PubMed]
C. W. Qiu, A. Novitsky, H. Ma, and S. Qu, “Electromagnetic interaction of arbitrary radial-dependent anisotropic spheres and improved invisibility for nonlinear-transformation-based cloaks,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.80(1), 016604 (2009). [CrossRef] [PubMed]
S. A. Cummer, B.-I. Popa, and D. R. Smith, “Full-wave simulation of electromagnetic cloaking structures,” Phys. Rev. Lett.74, 036621 (2006).
C. Argyropoulos, Y. Zhao, and Y. Hao, “A radially-dependent dispersive finite-difference time-domain method for the evaluation of electromagnetic cloaks,” IEEE Trans. Antenn. Propag.57(5), 1432–1441 (2009). [CrossRef]
J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett.101(20), 203901 (2008). [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]
X. Chen, Y. Luo, J. Zhang, K. Jiang, J. B. Pendry, and S. Zhang, “Macroscopic invisibility cloaking of visible light,” Nat Commun2, 176 (2011). [CrossRef] [PubMed]
B. Zhang, Y. Luo, X. Liu, and G. Barbastathis, “Macroscopic invisibility cloak for visible light,” Phys. Rev. Lett.106(3), 033901 (2011). [CrossRef] [PubMed]
W. X. Jiang, T. J. Cui, Q. Cheng, J. Y. Chin, X. M. Yang, R. Liu, and D. R. Smith, “Design of arbitrarily shaped concentrators based on conformally optical transformation of nonuniform rational B-spline surfaces,” Appl. Phys. Lett.92(26), 264101 (2008). [CrossRef]
H. Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett.90(24), 241105 (2007). [CrossRef]
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]
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]
A. Di Falco, S. C. Kehr, and U. Leonhardt, “Luneburg lens in silicon photonics,” Opt. Express19(6), 5156–5162 (2011). [CrossRef] [PubMed]
D.-H. Kwon and D. H. Werner, “Flat focusing lens designs having minimized reflection based on coordinate transformation techniques,” Opt. Express17(10), 7807–7817 (2009). [CrossRef] [PubMed]
Y. Lai, J. Ng, H. Y. Chen, D. Z. Han, J. 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]
W. X. Jiang, T. J. Cui, X. M. Yang, H. F. Ma, and Q. Cheng, “Design of arbitrarily shaped concentrators based on conformally optical transformation of nonuniform rational B-spline surfaces,” Appl. Phys. Lett.98, 204101 (2011).
Z. L. Mei, J. Bai, and T. J. Cui, “Experimental verification of a broadband planar focusing antenna based on transformation optics,” New J. Phys.13(6), 063028 (2011). [CrossRef]
K. Zhang, Q. Wu, F.-Y. Meng, and L. W. Li, “Arbitrary waveguide connector based on embedded optical transformation,” Opt. Express18(16), 17273–17279 (2010). [CrossRef] [PubMed]
C. García-Meca, M. M. Tung, J. V. Galán, R. Ortuño, F. J. Rodríguez-Fortuño, J. Martí, and A. Martínez, “Squeezing and expanding light without reflections via transformation optics,” Opt. Express19(4), 3562–3575 (2011). [CrossRef] [PubMed]
D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett.84(18), 4184–4187 (2000). [CrossRef] [PubMed]
Y. Huang, Y. Feng, and T. Jiang, “Electromagnetic cloaking by layered structure of homogeneous isotropic materials,” Opt. Express15(18), 11133–11141 (2007). [CrossRef] [PubMed]
M. Farhat, S. Guenneau, and S. Enoch, “Ultrabroadband elastic cloaking in thin plates,” Phys. Rev. Lett.103(2), 024301 (2009). [CrossRef] [PubMed]
C. W. Qiu, L. Hu, X. Xu, and Y. Feng, “Spherical cloaking with homogeneous isotropic multilayered structures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.79(4), 047602 (2009). [CrossRef] [PubMed]
C. W. Qiu, L. Hu, and S. Zouhdi, “Isotropic non-ideal cloaks providing improved invisibility yb adaptive segmentation and optimal refractive index profile from ordering isotropic materials,” Opt. Express18, 14950–14959 (2010).
X. Xu, Y. Feng, Y. Hao, J. Zhao, and T. Jiang, “Infrared carpet cloak designed with uniform silicon grating structure,” Appl. Phys. Lett.95(18), 184102 (2009). [CrossRef]
X. Xu, Y. Feng, Z. Yu, T. Jiang, and J. Zhao, “Simplified ground plane invisibility cloak by multilayer dielectrics,” Opt. Express18(24), 24477–24485 (2010). [CrossRef] [PubMed]
J. Zhang, L. Liu, Y. Luo, S. Zhang, and N. A. Mortensen, “Homogeneous optical cloak constructed with uniform layered structure,” Opt. Express19(9), 8625–8631 (2011). [CrossRef]
H. Chen and C. T. Chan, “Electromagnetic wave manipulation by layered systems using the transformation media concept,” Appl. Phys. B78, 054204 (2008).
I. Gallina, G. Castaldi, and V. Galdi, “Transformation thin planar retro directive reflector,” IEEE Microw. Wirel. Compon. Lett.7, 603–605 (2008).
I. Gallina, G. Castaldi, and V. Galdi, “Transformation optics-inspired metamaterial coatings for controlling the scattering response of wedge/corner-type structures,” Microw. Opt. Technol. Lett.51, 2709–2712 (2009). [CrossRef]
Z. Liang and J. Li, “Bending a periodically layered structure for transformation acoustics,” Appl. Phys. Lett.98(24), 241914 (2011). [CrossRef]
B. Wood, J. B. Pendry, and D. Tsai, “Directed subwavelength imaging using a layered metal-dielectric system,” Phys. Rev. B74(11), 115116 (2006). [CrossRef]

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