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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 5 — Mar. 3, 2008
  • pp: 3161–3166
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Cloaking an object on a dielectric half-space

Pu Zhang, Yi Jin , and Sailing He  »View Author Affiliations


Optics Express, Vol. 16, Issue 5, pp. 3161-3166 (2008)
http://dx.doi.org/10.1364/OE.16.003161


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Abstract

Cloaking an object on a dielectric half-space for an observer in air is studied. The proposed cloaking configuration has two vertical matching strips of isotropic and homogeneous material under the two bottom surfaces of a semi-cylindrical cloaking cover. Simple expression for the material parameters of the matching strips is derived. The theoretical results of cloaking are verified numerically for the incidence of plane wave and line current source. The vertical matching strips can be terminated with a finite depth by introducing some loss to the matching material.

© 2008 Optical Society of America

1. Introduction

Recently, Pendry et al. have suggested a novel scheme (called cloaking [1

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

]) to hide objects from electromagnetic detection. In this scheme, objects are surrounded by a specially-designed inhomogeneous anisotropic shell (called the cloaking shell or cloak) whose permittivity and permeability tensors are obtained from the transformation theory [2

2. E. J. Post, Formal structure of electromagnetics (North-Holland, 1962).

,3

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

]. The cloak excludes the incident wave from the inner space surrounded by the cloak, and the field outside the cloak is not perturbed as if the cloak and the objects don’t exist. Later, both numerical simulation [4

4. S. A. Cummer, B. I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E 74, 036,621 (2006). [CrossRef]

] and experimental demonstration with metamaterials at microwave frequencies [5

5. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006). [CrossRef] [PubMed]

] confirmed this exciting theoretical prediction. Due to its vast significance and potential applications, the cloaking scheme has attracted much attention and various properties and improvements of cloaking have been discussed [6–9].

Fig. 1. Schematic diagram for a semi-cylindrical cloaking cover with vertical matching strips underneath interfaces AA’ and BB’.

2. Design of the cloaking cover and matching strips

In this paper, we fix at a single frequency and the time-harmonic factor is exp(-iωt). A two-dimensional case is considered and the electric field is polarized along the axis of the semi-cylindrical cloak (TE). (Similar formulas for the TM incidence can be easily obtained due to the duality of the electromagnetic theory.)

First we consider the following plane wave incident on the dielectric half-space

Ei,z=E0exp[ik0(x sinγycosγ)]exp(iωt),
(1)

where γ is the incident angle. If there is nothing on the dielectric half-space, the reflected wave is also a plane wave. When an object (without any cloak) exists on the dielectric half-space, however, the scattering of the object may distort the reflected wave significantly. The proposed semi-cylindrical cloaking cover should be able to restore the reflected wave to a plane wave (same as that reflected from a pure air-dielectric interface).

To utilize some properties of an ideal cylindrical cloak in the present analysis of the semicylindrical cloaking cover (exactly one half of the ideal cylindrical cloak), we envisage to make the semi-cylindrical cloak enclosed with a complemented one (with the same inner and outer radii as the semi-cylindrical cloak) and let a plane wave (cf. Eq. (1)) impinge on it. In the cloak design (according to Ref. [5

5. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006). [CrossRef] [PubMed]

]), one uses the following coordinate transformation to map a cylindrical region r′<b in a fictitious vacuum space to an annulus a<r<b in the real space,

r={r(ba)b+a,r<br      ,r>b, ϕ=ϕ,z=z·
(2)

Based on this transformation theory, the relative permittivity and permeability tensors of the cloak are (with cylindrical coordinate components)

ε̿r=μ̿r=diag((ra)r,r(ra),[b(ba)]2 (ra)r).
(3)

The material property of the cloaking shell is required to be anisotropic and inhomogeneous with the above functional dependence on the spatial coordiante. Note that the permittivity and permeability tensors are diagonal in terms of cylindrical coordinate components, but will become non-diagonal when expressed in terms of the Cartesian components, which are implemented in our numerical simulation. Using the same transformation, the electric field inside the cloak is given by

Ec,z=E0exp[ik0(xsinγycosγ)]exp(iωt).
(4)

The magnetic field inside the cloak can be obtained from the electric field according to Maxwell’s equations. Eq. (4) implies that E c,z has a constant amplitude and its wavefront, determined by x′sin γ-y′ cos γ=C (C is a constant), is curved in the real space due to the transformation Eq. (2). Eq. (4) gives a clear picture of wave propagation inside the cylindrical cloak, which has also been illustrated by numerical simulation in e.g. Ref. [4

4. S. A. Cummer, B. I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E 74, 036,621 (2006). [CrossRef]

]. After propagating through the outer boundary of the cylindrical cloak (without reflection), some electromagnetic wave is guided along the cloaking shell and there exists no electromagnetic wave in the interior air space surrounded by the cloaking shell.

Now consider the semi-cylindrical cloaking cover shown in Fig. 1. After entering through the outer boundary of the semi-cylindrical cloaking cover, the electromagnetic wave does not undergo any reflection except when it reaches interfaces AA’ and BB’. Thus, the incident electric field (represented by c,z) inside the semi-cylindrical cloaking cover excited by the plane wave incident on the outer boundary of the cloak can be expressed with Eq. (4). Since c,z and the cloak material are inhomogeneous, we can define a local reflection coefficient at interfaces AA’ and BB’. Expanding the phase factor in Eq. (4) into power series of x and y and omitting the high-order terms, c,z around an arbitrary point P(xP,0) on interface AA’ or BB’ can be approximated as

E˜c,z=E0exp(ik0{(xxp)sinγ[b(ba)]ycosγ(rprp)+xpsin γ})exp(iωt).
(5)

Eq. (5) implies that c,z near point P can be treated locally as a plane wave. When such a local plane wave is reflected by a vertical (matching) strip underneath AA’ or BB’, we can derive the following local reflection coefficient

Rc=μrmcosγεrmμrm[(ba)b]2sin2γμrmcosγ+εrmμrm[(ba)b]2sin2γ,
(6)

where εmr and µmr are the relative permittivity and permeability of the matching strip, respectively. Variable xP disappears in Eq. (6), which indicates that the local reflection coefficient Rc is invariant at different position P. This interesting result is critical for the realization of our proposed cloaking configuration. Outside the cloaking cover, we have the following reflection coefficient for the plane wave reflected by the dielectric half-space,

Rd=μrdcosγεrdμrdsin2γμrdcosγ+εrdμrdsin2γ.
(7)

To make our semi-cylindrical cloaking cover (and the object inside) invisible, the reflection coefficient at interfaces AA’ and BB’ should be the same as that from pure air-dielectric interface, i.e.,

Rc=Rd.
(8)

It thus follows from Eq. (8) that

(εrmμrm[(ba)b]2sin2γ)(μrm)2=(εrdμrdsin2γ)(μrd)2.
(9)

Solution (εmr, µmr) to the above Eq. (9) is not unique. Among all the possible solutions, the most significant solution is

εrm=[b(ba)]2εrd,μrm=μrd,
(10)

as this solution (εmr, µmr) for the material parameters of the vertical matching strips is independent of the incident angle γ (so that the matching material can be physical). Thus, if we put such an isotropic and homogeneous matching material underneath the two bottom surfaces of the semi-cylindrical cloaking cover (see Fig. 1), excellent cloaking of any object on the dielectric half-space can be achieved. Note that if the half-space is a perfect electric conductor (PEC), a semi-cylindrical cloaking cover suffices to cloak the object (as Eq. (10) gives a perfectly conducting material). The suggested method does not deal with the scattering effect of edges A, A’, B and B’ of the cloaking configuration. The edge effect is, however, very small as shown in the following numerical simulation, and can be omitted.

3. Numerical simulation and validation

To validate and visualize the proposed cloaking configuration, numerical simulation based on the finite element method is performed. In Fig. 1, the dimension of the semi-cylindrical cloaking cover is set to a=10λ 0/3 and b=20λ 0/3 (λ 0 is the wavelength in vacuum). Its permittivity and permeability tensors are obtained with Eq. (3). We assume the dielectric half-space is nonmagnetic (i.e., µdr=1) and we choose its dielectric constant εdr=10 in our simulation. According to Eq. (10), the matching material will also be nonmagnetic (i.e., µmr=1) and its relative permittivity takes a value of 40 (i.e., εmr=40). Note that here we choose εdr=10 because this is the typical dielectric constant of the earth at the microwave. [At an optical frequency one can choose εdr=(SiO2) and then we have εmr=(SiNx).] To test the cloaking effect, a PEC scatterer (a cylinder on a triangular wedge) is placed on the axis of the semi-cylindrical cloaking cover. Utilizing the symmetry property of the configuration, we can replace one half of the configuration by a perfectly magnetic conductor (PMC) boundary at the symmetry plane. The computational domain is thus closed by this PMC boundary and perfect matched layers (PMLs) in the other three directions. Two kinds of incident wave are introduced separately to observe the cloaking effect.

Fig. 2. Snapshots of the scattered electric field for a normally incident plane wave on (a) a bare scatterer, (b) a scatterer covered by a semi-cylindrical cloak without matching strips, and (c) a scatterer covered by a semi-cylindrical cloak with two vertical matching strips.

Fig. 3. Snapshots of the scattered electric field of an external line current source for the case of (a) a bare scatterer, (b) a scatterer covered by a semi-cylindrical cloak without matching strips, and (c) a scatterer covered by the proposed cloaking structure (a semi-cylindrical cloak with two vertical matching strips). The black dots indicate the position of the line current source.

Fig. 4. Snapshots of the electric field distribution for a normally incident plane wave when the depth of the two vertical matching strips is finite. The matching strips are lossy in the right part and lossless in the left part (to show more clearly the perturbation outside the cloaking cover the electric field outside the cloak is for the reflected wave while that in the cloak, strips and dielectric half-space is for the total field).

4. Conclusion

In conclusion, we have proposed for the first time (to the best of our knowledge) a scheme to cloak objects on a dielectric half-space, which has many potential applications. The cloaking configuration consists of a semi-cylindrical cloaking cover and two vertical matching strips (of isotropic and homogeneous material) underneath. A uniform local reflection coefficient has been defined unambiguously at the interfaces between the cloaking cover and matching strips. By matching this local reflection coefficient to the air-dielectric reflection coefficient, we have derived simple expression Eq. (10) for the material parameters (independent of the angle of the incident wave) of the matching strips. Numerical simulation has shown satisfactory cloaking effects for the incidence of e.g. plane wave and line current source. The vertical matching strips can be terminated easily to a depth order of vacuum wavelength by letting the matching material have some loss. This enables us to cloak objects on a dielectric half-space effectively for any spatial waveform of incidence at the frequency considered. The present cloaking scheme may be generalized to a three-dimensional case.

Acknowledgments

This work is supported by the National Basic Research Program (973) of China (under Project No. 2004CB719800), the National Natural Science Foundation of China (60688401) and a Swedish Research Council (VR) grant on metamaterials.

References and links

1.

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

2.

E. J. Post, Formal structure of electromagnetics (North-Holland, 1962).

3.

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

4.

S. A. Cummer, B. I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E 74, 036,621 (2006). [CrossRef]

5.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006). [CrossRef] [PubMed]

6.

F. Zolla, S. Guenneau, A. Nicolet, and J. B. Pendry, “Electromagnetic analysis of cylindrical invisibility cloaks and the mirage effect,” Opt. Lett. 32, 1069–1071 (2007). [CrossRef] [PubMed]

7.

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

8.

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

9.

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Nonmagnetic cloak with minimized scattering,” Appl. Phys. Lett. 91, 111, 105 (2007). [CrossRef]

10.

I. V. Lindell, Methods for Electromagnetic Field Analysis (Oxford Univ. Press, 1995).

11.

A. R. von Hippel, Dielectric Materials and Applications (MIT Press, 1954).

OCIS Codes
(120.5700) Instrumentation, measurement, and metrology : Reflection
(160.1190) Materials : Anisotropic optical materials
(260.2110) Physical optics : Electromagnetic optics

ToC Category:
Physical Optics

History
Original Manuscript: December 14, 2007
Revised Manuscript: February 5, 2008
Manuscript Accepted: February 6, 2008
Published: February 21, 2008

Citation
Pu Zhang, Yi Jin, and Sailing He, "Cloaking an object on a dielectric half-space," Opt. Express 16, 3161-3166 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-5-3161


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References

  1. J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling electromagnetic fields," Science 312, 1780-1782 (2006). [CrossRef] [PubMed]
  2. E. J. Post, Formal Structure of Electromagnetics, (North-Holland, 1962).
  3. D. Schurig, J. B. Pendry, and D. R. Smith, "Calculation of material properties and ray tracing in transformation media," Opt. Express 14, 9794-9804 (2006). [CrossRef]
  4. S. A. Cummer, B. I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, "Full-wave simulations of electromagnetic cloaking structures," Phys. Rev. E 74, 036,621 (2006). [CrossRef]
  5. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial electromagnetic cloak at microwave frequencies," Science 314, 977-980 (2006). [CrossRef] [PubMed]
  6. F. Zolla, S. Guenneau, A. Nicolet, and J. B. Pendry, "Electromagnetic analysis of cylindrical invisibility cloaks and the mirage effect," Opt. Lett. 32, 1069-1071 (2007). [CrossRef] [PubMed]
  7. W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, "Optical cloaking with metamaterials," Nat. Photonics 1, 224-227 (2007). [CrossRef]
  8. Y. Huang, Y. Feng, and T. Jiang, "Electromagnetic cloaking by layered structure of homogeneous isotropic materials," Opt. Express 15, 11133-11141 (2007). [CrossRef]
  9. W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, "Nonmagnetic cloak with minimized scattering," Appl. Phys. Lett. 91, 111,105 (2007). [CrossRef]
  10. I. V. Lindell, Methods for Electromagnetic Field Analysis (Oxford Univ. Press, 1995).
  11. A. R. von Hippel, Dielectric Materials and Applications (MIT Press, 1954).

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