## Achieving invisibility over a finite range of frequencies.

Optics Express, Vol. 16, Issue 8, pp. 5656-5661 (2008)

http://dx.doi.org/10.1364/OE.16.005656

Acrobat PDF (505 KB)

### Abstract

We analyze cloaking of transverse electric (TE) fields through homogenization of radially symmetric metallic structures. The two-dimensional circular cloak consists of concentric layers cut into a large number of small infinitely conducting sectors which is equivalent to a highly anisotropic permittivity. We find that a wave radiated by a magnetic line current source located a couple of wavelengths away from the cloak is almost unperturbed in magnitude but not in phase. Our structured cloak is shown to work for different wavelengths provided they are ten times larger than the outermost sectors.

© 2008 Optical Society of America

## 1. Introduction

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

*r*′,

*θ*′,

*x*′

_{3}are radially contracted cylindrical coordinates

*r*,

*θ*,

*x*

_{3}. The former transformation maps the disk

*R*

_{2}onto an annulus with outer radius

*R*

_{2}and inner radius

*R*

_{1}. In other words, if a source located outside the disk

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

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

6. R.C. McPhedran, N.A. Nicorovici, and G.W. Milton,“Optical and dielectric properties of partially resonant composites,” Phys. Rev. B **49**, 8479–8482 (1994). [CrossRef]

7. J.B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. **85**, 3966–3969 (2000). [CrossRef] [PubMed]

8. J.B. Pendry and S.A. Ramakrishna, “Focussing light using negative refraction,” J. Phys. Cond. Matter **15**, 6345–6364 (2003). [CrossRef]

9. D. Maystre and S. Enoch, “Perfect lenses with left-handed material: Alice’s mirror?,” J. Opt. Soc. Am. A **21**, 122 (2004). [CrossRef]

10. S.A. Ramakrishna, “Physics of negative refractive index materials,” Rep. Prog. Phys. **68**, 449–521 (2005). [CrossRef]

11. G.W. Milton and N.A. Nicorovici, “On the cloaking effects associated with anomalous localised resonance,” Proc. Roy. Lond. A **462**, 3027–3059 (2006). [CrossRef]

12. N.A.P. Nicorovici, G.W. Milton, R.C. McPhedran, and L.C. Botten, “Quasistatic cloaking of two-dimensional polarizable discrete systems by anomalous resonance,” Opt. Express **15**, 6314–6323 (2007). [CrossRef] [PubMed]

*x*

_{3}-axis only two entries of

*ε*and

_{r}*ε*). Moreover

_{θ}*µ*

_{3}is the only entry of

*ε*and

_{r}*ε*by

_{θ}*µ*

_{3}in (2) [13]:

*r*=

*R*

_{2}of the cloak. Importantly, we note that the value of

*ε*increases with decreasing values of

_{θ}*r*in the annulus until it eventually becomes infinite on the inner boundary

*r*=

*R*

_{1}of the cloak in (2). The cloak proposed by Cai

*et al.*does not exhibit a very large azimuthal anisotropy as it is clear that all parameters in (3) remain bounded (unlike the cloak proposed by Pendry

*et al.*). To meet the criterion of (3), Cai

*et al.*propose to use a locally resonant structure consisting of concentric layers of thin elongated ellipsoidal wires elongated along the radial direction. A derivation based on effective medium theory shows that such a cloak displays the prerequisite electromagnetic parameters around a given frequency (in the visible spectrum) at which the local resonators are excited by the TE wave.

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**977–980 (2006). [CrossRef] [PubMed]

## 2. Homogenization of the micro-structured cloak at fixed frequency

*λ*is large compared with the characteristic size

*d*of sectors

*i.e.*when

*η*=

*λ*/

*d*≪1, unlike for the proposals in [3

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**977–980 (2006). [CrossRef] [PubMed]

*η*~1/3. We notice that our micro-structured cloak which works through field averaging does not suffer from any blow-up of the electromagnetic field, unlike its locally resonant counterpart [14

14. A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Improvement of cylindrical cloaking with the SHS lining,” Opt. Express **15**, 12717–12734 (2007). [CrossRef] [PubMed]

*r*,

*θ*,

*x*

_{3}), thereby taking into account the axi-symmetric geometry of the structure. From a two-scale model of the problem, assuming that the cloak lies in vacuum, we found that the homogenized permittivity tensor of the proposed structure is given by:

*Y** denotes the elementary area around each scatterer

*B*and

*ϕ*represent corrective terms defined by:

_{ij}*Y**. Hence, thanks to the symmetry of the right matrix above (

*ϕ*=

_{ij}*ϕ*), the homogenized permittivity is given by the knowledge of three terms

_{ji}*ϕ*. One can note that

_{ij}*V*is the unique

_{j}*Y*-periodic solution with null mean of the following system, where derivatives are taken in the usual sense:

*B*denotes the boundary of

*B*, and

*n*,

_{j}*j*∈ {

*r*,

*θ*}, denotes the projection on the axis

**e**

*of a unit outward normal to ∂*

_{j}*B*. Although these results might at first glance look similar to those reported in [15], we emphasize that they differ since the earlier paper considered merely dielectric structures.

## 3. Numerical analysis of electromagnetic cloaking

### 3.1. The homogenized matrix

*𝓚*and

_{r}*𝓚*, for the geometry shown in Fig. 1 which provided us with two electrostatic potentials

_{θ}*V*and

_{r}*V*. Using (5) we found that

_{θ}*ϕ*and

_{rθ}*ϕ*vanish. From

_{θr}*ϕ*and

_{θθ}*ϕ*we deduced the following homogenized permittivity

_{rr}### 3.2. Structured infinitely conducting cloak

*e*(

*H*

_{3}) of the longitudinal component

*H*

_{3}of the magnetic field along the

*x*

_{2}-axis. The black curve represents ℜ

*e*(

*H*

_{3}) in free space; the red curve represents ℜ

*e*(

*H*

_{3}) with a F-shaped infinitely conducting obstacle; the blue curve represents ℜ

*e*(

*H*

_{3}) with a F-shaped infinitely conducting obstacle surrounded by the cloak. We notice that the black and blue curves have nearly the same amplitude outside the cloak, but experience a phase shift. This phase shift can be attributed to the change in the optical path followed by the waves around the cloak: the path is longer than that of rays going straightforwardly in free space.

*e*(

*H*

_{3}) of the longitudinal component

*H*

_{3}of the magnetic field when the F-shaped scatterer is coated (right) and uncoated (left) and for frequencies ν ranging from 3.5 to 5.5. Note that the modelling takes into account the actual structure of the infinitely conducting cloak. The diffraction is clearly reduced by the structure especially for the lowest frequencies. Moreover, the numerical results show that although not perfect the cloak allows us to reduce the scattering for frequencies in a relatively large domain thanks to the fact that the physical principle is not based on any resonant property of the structure. Of course, when the frequency increases the typical size of the sectors becomes significant when compared to the wavelength and thus the homogenized model is not longer valid.

16. D.P. Gaillot, C. Croenne, and D. Lippens, “An all-dielectric route for terahertz cloaking,” Opt. Express **16**, 3986–3992 (2008). [CrossRef] [PubMed]

*H*

_{3}(and hence the transverse electric field) is controlled by the cloak.

## 4. Conclusion

*d*to be of the order of magnitude of a millimeter, in which case results should hold for TE micro-waves propagating out-of-plane provided their propagation constant γ is such that γ

*d*≪1. This suggests potential applications lie in improved telecommunication lines, for instance to transfer secure information. Interestingly, our cloak works well for the near field, and it is broad band to certain extent. Our design might prove useful for instance in certain problems of electromagnetic noise insulation for antennas applications. The influence of the absorption remains to be evaluated if the visible spectrum is aimed at.

## Acknowledgments

## References and links

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

2. | U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” New J. Phys . |

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 |

4. | U. Leonhardt, “Optical conformal mapping,” Science |

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

6. | R.C. McPhedran, N.A. Nicorovici, and G.W. Milton,“Optical and dielectric properties of partially resonant composites,” Phys. Rev. B |

7. | J.B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. |

8. | J.B. Pendry and S.A. Ramakrishna, “Focussing light using negative refraction,” J. Phys. Cond. Matter |

9. | D. Maystre and S. Enoch, “Perfect lenses with left-handed material: Alice’s mirror?,” J. Opt. Soc. Am. A |

10. | S.A. Ramakrishna, “Physics of negative refractive index materials,” Rep. Prog. Phys. |

11. | G.W. Milton and N.A. Nicorovici, “On the cloaking effects associated with anomalous localised resonance,” Proc. Roy. Lond. A |

12. | N.A.P. Nicorovici, G.W. Milton, R.C. McPhedran, and L.C. Botten, “Quasistatic cloaking of two-dimensional polarizable discrete systems by anomalous resonance,” Opt. Express |

13. | W. Cai, U.K. Chettiar, A.V. Kildiev, and V.M. Shalaev, “Optical Cloaking with metamaterials,” Nature |

14. | A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Improvement of cylindrical cloaking with the SHS lining,” Opt. Express |

15. | S. Guenneau and F. Zolla, “Homogenization of three-dimensional finite photonic crystals,” JEWA14, 529–530 (2000) & Progress In Electromagnetics Research 27, 91–127 (2000). |

16. | D.P. Gaillot, C. Croenne, and D. Lippens, “An all-dielectric route for terahertz cloaking,” Opt. Express |

**OCIS Codes**

(230.3990) Optical devices : Micro-optical devices

(260.2110) Physical optics : Electromagnetic optics

**ToC Category:**

Metamaterials

**History**

Original Manuscript: February 26, 2008

Revised Manuscript: April 3, 2008

Manuscript Accepted: April 3, 2008

Published: April 7, 2008

**Citation**

M. Farhat, S. Guenneau, A. B. Movchan, and S. Enoch, "Achieving invisibility over a finite range of frequencies," Opt. Express **16**, 5656-5661 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-8-5656

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### References

- J. B. Pendry, D. Shurig, D. R. Smith, "Controlling electromagnetic fields," Science 312, 1780-1782 (2006). [CrossRef] [PubMed]
- U. Leonhardt and T. G. Philbin, "General relativity in electrical engineering," New J. Phys. 8, 247 (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]
- U. Leonhardt, "Optical conformal mapping," Science 312, 1777-1780 (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]
- R. C. McPhedran, N. A. Nicorovici, and G. W. Milton,"Optical and dielectric properties of partially resonant composites," Phys. Rev. B 49, 8479-8482 (1994). [CrossRef]
- J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-3969 (2000). [CrossRef] [PubMed]
- J. B. Pendry and S. A. Ramakrishna, "Focussing light using negative refraction," J. Phys. Cond. Matter 15, 6345-6364 (2003). [CrossRef]
- D. Maystre and S. Enoch, "Perfect lenses with left-handed material: Alice’s mirror?," J. Opt. Soc. Am. A 21, 122 (2004). [CrossRef]
- S. A. Ramakrishna, "Physics of negative refractive index materials," Rep. Prog. Phys. 68, 449-521 (2005). [CrossRef]
- G. W. Milton and N. A. Nicorovici, "On the cloaking effects associated with anomalous localised resonance," Proc. Roy. Lond. A 462, 3027-3059 (2006). [CrossRef]
- N. A. P. Nicorovici, G. W. Milton, R. C. McPhedran, and L. C. Botten, "Quasistatic cloaking of two-dimensional polarizable discrete systems by anomalous resonance," Opt. Express 15, 6314-6323 (2007). [CrossRef] [PubMed]
- W. Cai, U. K. Chettiar, A. V. Kildiev and V. M. Shalaev, "Optical Cloaking with metamaterials," Nature 1, 224-227 (2007).
- A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, "Improvement of cylindrical cloaking with the SHS lining," Opt. Express 15, 12717-12734 (2007). [CrossRef] [PubMed]
- S. Guenneau and F. Zolla, "Homogenization of three-dimensional finite photonic crystals," JEWA 14, 529-530 (2000) &Progress In Electromagnetics Research 27, 91-127 (2000).
- D. P. Gaillot, C. Croenne and D. Lippens, "An all-dielectric route for terahertz cloaking," Opt. Express 16, 3986-3992 (2008). [CrossRef] [PubMed]

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