## Cloaking an object on a dielectric half-space

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

*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]

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]

*ε*and permeability

^{d}_{r}*µ*; we choose

^{d}_{r}*ε*=10 and

^{d}_{r}*µ*=1 in our numerical simulation) from being sensed by any observer above the dielectric half-space, however, an enclosed cloaking shell can not be used [assuming it is not practical to replace the supporting medium under the object with some cloaking structure and the reflected field will also become different from that for a pure flat interface]. In this paper, we propose to use a semi-cylindrical cloaking cover (with inner and outer radii

^{d}_{r}*a*and

*b*) to hide an object on the dielectric half-space (see Fig. 1) so that the reflected field (for any form of incidence) should be the same as that from pure air-dielectric flat interface (as if the cloak and the object don’t exist). Below we show first theoretically that in order to achieve a satisfactory cloaking effect we need to introduce two vertical matching strips right under the bottom surfaces AA’ and BB’ of the cloaking cover (see Fig. 1).

## 2. Design of the cloaking cover and matching strips

*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.)

*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).

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]

*r*′<

*b*in a fictitious vacuum space to an annulus

*a*<

*r*<

*b*in the real space,

*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]

*Ẽ*

_{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

*Ẽ*

_{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,

*Ẽ*

_{c,z}around an arbitrary point P(

*x*,0) on interface AA’ or BB’ can be approximated as

_{P}*Ẽ*

_{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

*ε*and

^{m}_{r}*µ*are the relative permittivity and permeability of the matching strip, respectively. Variable

^{m}_{r}*x*disappears in Eq. (6), which indicates that the local reflection coefficient

_{P}*R*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,

_{c}*ε*,

^{m}_{r}*µ*) to the above Eq. (9) is not unique. Among all the possible solutions, the most significant solution is

^{m}_{r}*ε*,

^{m}_{r}*µ*) for the material parameters of the vertical matching strips is independent of the incident angle

^{m}_{r}*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

*a*=10

*λ*

_{0}/3 and

*b*=20

*λ*

_{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.,

*µ*=1) and we choose its dielectric constant

^{d}_{r}*ε*=10 in our simulation. According to Eq. (10), the matching material will also be nonmagnetic (i.e.,

^{d}_{r}*µ*=1) and its relative permittivity takes a value of 40 (i.e.,

^{m}_{r}*ε*=40). Note that here we choose

^{m}_{r}*ε*=10 because this is the typical dielectric constant of the earth at the microwave. [At an optical frequency one can choose

^{d}_{r}*ε*=(SiO

^{d}_{r}_{2}) and then we have

*ε*=(SiN

^{m}_{r}_{x}).] 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.

*ε*=40+4

^{m}_{r}*i*without much change in the wave impedance of the matching strips used in the previous numerical example (note that there exist dielectrics with rather high permittivity and small loss tangent at e.g. radio and microwave frequencies [11]). The depth of the strips is chosen to be one vacuum wavelength. The cloaking effect for a normally incident plane wave is shown in the right part of Fig. 4 (only the right half is shown due to the symmetry). For comparison, the left part of Fig. 4 shows the corresponding lossless case (only the left half is shown due to the symmetry). For the lossless case, standing wave appears (as a result of multiple reflections between the top and bottom surfaces of each finite-depth matching strip) and some horizontal nonuniformity is observed in the field intensity distribution above the semi-cylindrical cloaking cover. In contrast, for the lossy case, the refracted wave gets absorbed gradually in the matching strip, and thus the reflection at the bottom surface of the finite-depth matching strip is negligible. The reflected wave in air outside the semi-cylindrical cloaking cover behaves as a uniform plane wave. This shows that the vertical matching strips in our cloaking configuration can be terminated successfully by choosing some matching material with some loss.

## 4. Conclusion

## Acknowledgments

## References and links

1. | J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science |

2. | E. J. Post, |

3. | D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Exp. |

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 |

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 |

6. | F. Zolla, S. Guenneau, A. Nicolet, and J. B. Pendry, “Electromagnetic analysis of cylindrical invisibility cloaks and the mirage effect,” Opt. Lett. |

7. | W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photon. |

8. | Y. Huang, Y. Feng, and T. Jiang, “Electromagnetic cloaking by layered structure of homogeneous isotropic materials,” Opt. Exp. |

9. | W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Nonmagnetic cloak with minimized scattering,” Appl. Phys. Lett. |

10. | I. V. Lindell, |

11. | A. R. von Hippel, |

**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

- J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling electromagnetic fields," Science 312, 1780-1782 (2006). [CrossRef] [PubMed]
- E. J. Post, Formal Structure of Electromagnetics, (North-Holland, 1962).
- 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]
- 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]
- 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]
- 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]
- W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, "Optical cloaking with metamaterials," Nat. Photonics 1, 224-227 (2007). [CrossRef]
- Y. Huang, Y. Feng, and T. Jiang, "Electromagnetic cloaking by layered structure of homogeneous isotropic materials," Opt. Express 15, 11133-11141 (2007). [CrossRef]
- 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]
- I. V. Lindell, Methods for Electromagnetic Field Analysis (Oxford Univ. Press, 1995).
- A. R. von Hippel, Dielectric Materials and Applications (MIT Press, 1954).

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