## Conformal carpet and grating cloaks |

Optics Express, Vol. 18, Issue 23, pp. 24361-24367 (2010)

http://dx.doi.org/10.1364/OE.18.024361

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### Abstract

We introduce a class of conformal versions of the previously introduced quasi-conformal carpet cloak, and show how to construct such conformal cloaks for different cloak shapes. Our method provides exact refractive-index profiles in closed mathematical form for the usual carpet cloak as well as for other shapes. By analyzing their asymptotic behavior, we find that the performance of finite-size cloaks becomes much better for metal shapes with zero average value, *e.g.*, for gratings.

© 2010 Optical Society of America

## 1. Introduction

*finite-size*carpet cloaks can be systematically improved with respect to Refs. [9

9. B. Zhang, T. Chan, and B.-I. Wu, “Lateral shift makes a ground-plane cloak detectable,” Phys. Rev. Lett. **104**, 233903 (2010). [CrossRef] [PubMed]

## 2. Conformal mapping

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

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

*strictly conformal*transformations. For each transformation of this class, the shape of the bump results automatically,

*i.e.*, it can generally

*not*be chosen arbitrarily. However, in the special and rather important limit of shallow bumps (which

*all*of the aforementioned experiments have used [2

2. R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science **323**, 366–369 (2009). [CrossRef] [PubMed]

7. H. F. Ma and T. J. Cui, “Three-dimensional broadband ground-plane cloak made of metamaterials,” Nature Commun. **1**, 1–6 (2010). [CrossRef]

*u*(

*x*,

*0*),

*v*(

*x*,

*0*)). In general, it is difficult to obtain a closed expression for the parametric dependence of

*v*(

*x*, 0) on

*u*(

*x*, 0). However, for shallow bumps an explicit expression can be obtained. To see this, let us consider the example of a Gaussian for the coefficients

*c*in Eq. (1), The conformal map is then where erf is the error function (see the left panel of Fig. 1). For shallow bumps (

_{k}*h*≪

*w*in this example) we obtain

*u*(

*x*, 0) ≈

*x*, and derive the explicit form for the bump shape The parameter

*h*is therefore the height of the Gaussian bump, and

*w*is its width. In general, in this limit of shallow bumps, the

*c*=

_{k}*a*+ i

_{k}*b*are the Fourier coefficients of the bump shape since and hence, since

_{k}*u*(

*x*, 0) ≈

*x*, Thus, in this limit the coefficients

*c*for any desired conformal map (and refractive-index profile) can be obtained by Fourier transformation of the real-space bump shape

_{k}*v*(

*u*).

*z*↦

*f*(

*z*) refer to a different bump shape

*v*(

*u*) than in the shallow-bump limit. In the above example (4), at the critical ratio

*i.e.,*by setting

*n*= 1 outside of the finite-size cloak. This procedure delivers cloaking results which are similar to those obtained for the quasi-conformal carpet cloak [9

9. B. Zhang, T. Chan, and B.-I. Wu, “Lateral shift makes a ground-plane cloak detectable,” Phys. Rev. Lett. **104**, 233903 (2010). [CrossRef] [PubMed]

9. B. Zhang, T. Chan, and B.-I. Wu, “Lateral shift makes a ground-plane cloak detectable,” Phys. Rev. Lett. **104**, 233903 (2010). [CrossRef] [PubMed]

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

*n*= 1 far away from the bump. At large distances the refractive index due to Eq. (4) behaves like This decay is polynomial, which means that

*n*(

*u*,

*v*) approaches the vacuum limit

*n*= 1 rather slowly – necessitating undesirably large cloaking structures. This slow decay, which corresponds to small spatial-frequency components in

*f*(

*z*), is connected to the small spatial-frequency components of the bump shape

*v*(

*u*) itself.

*k*in Eq. (1), with

*c*=

*a*+ i

*b*. The resulting transformation is illustrated in the left panel of Fig. 2. Its refractive-index profile is shown in the right panel, where

*W*

_{0}is the Lambert function (the principal solution for

*w*in

*z*=

*we*). The refractive index approaches the vacuum limit

^{w}*n*= 1

*exponentially*fast with increasing vertical coordinate

*v*. The absence of zero and small spatial-frequency components means that the average value of the “carpet” shape

*v*(

*u*) is zero. This implies that the shape

*v*(

*u*) no longer only exhibits values

*above*the fictitious ground plane (maxima), but also values

*below*that ground plane (minima) – in sharp contrast to the usual carpet [1

**101**, 203901 (2008). [CrossRef] [PubMed]

*i.e.,*for a one-dimensional metal grating.

*κ*> 0 such that

*c*= 0 for all

_{k}*k*<

*κ*. In this case, the refractive index will approach the vacuum limit

*n*= 1 according to

*e*

^{−κv}for

*v*→ ∞. Hence, the metal surface

*v*(

*u*) can be almost perfectly cloaked using a finite-size refractive-index profile with an extent comparable to only 2

*π*/

*κ*.

*u*,

*v*) converge rapidly towards the Cartesian ones (

*x*,

*y*) when moving away from the mirror plane. In the right panel this finding translates into much smaller beam displacements than those discussed in Ref. [9

**104**, 233903 (2010). [CrossRef] [PubMed]

*exponentially,*as opposed to the polynomial decrease observed for

*κ*= 0. We have found the same behavior for rays impinging under different angles (not depicted).

*z*↦

*f*(

*z*) =

*z*– 1/(

*z*+ i

*δy*) and a mirror plane at fixed height

*δy*= 1.3; in our notation this cloak results from

*c*= iexp(−

_{k}*kδy*). Corresponding results for a finite-size cloaking structure with a height of 4.5 normalized units are shown in Fig. 4 – allowing for direct comparison with Figs. 1 and 3. The lateral beam displacement highlighted in Ref. [9

**104**, 233903 (2010). [CrossRef] [PubMed]

*c*at small spatial frequencies

_{k}*k*and thus a slowly decaying refractive-index profile:

*n <*1. Following Ref. [1

**101**, 203901 (2008). [CrossRef] [PubMed]

*n*= 1 can simply be multiplied with the so-called

*reference index*such that the global minimum refractive index becomes

*n*= 1. This multiplication leaves all of our results completely unaffected. Indeed, all carpet-cloak experiments [2

2. R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science **323**, 366–369 (2009). [CrossRef] [PubMed]

7. H. F. Ma and T. J. Cui, “Three-dimensional broadband ground-plane cloak made of metamaterials,” Nature Commun. **1**, 1–6 (2010). [CrossRef]

*v*(

*u*) can easily be obtained with our analytical approach. Textbook examples of conformal maps representing circular, rectangular, and triangular bumps, however, lead to infinities in the required refractive-index profiles. In contrast, as long as the shapes

*v*(

*u*) are smooth and do not exhibit any kinks, the resulting conformal maps and refractive-index profiles are smooth as well and represent realistic proposals for experimental cloaks.

## 3. Conclusion

*e.g.*, of gratings. Our analytical formulas can simply replace nontrivial numerical calculations along the lines of the quasi-conformal mapping. This step considerably eases working with these refractive-index profiles in practice. The analytical forms also allow us to study the asymptotic behavior of the refractive-index profiles (

*e.g.*, polynomial or exponential). This aspect is important for assessing and optimizing the performance of

*finite-size*cloaks as blueprints for experiments. In this regard, we obtain much smaller lateral beam displacements for certain metal profiles than previous quasi-conformal [8

8. I. I. Smolyaninov, V. N. Smolyaninova, A. V. Kildishev, and V. M. Shalaev, “Anisotropic metamaterials emulated by tapered waveguides: Application to optical cloaking,” Phys. Rev. Lett. **102**, 213901 (2009). [CrossRef] [PubMed]

12. H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nature Mater. **9**, 387–396 (2010). [CrossRef]

## Acknowledgments

## References and links

1. | J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. |

2. | R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science |

3. | J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nature Mater. |

4. | L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nature Photon. |

5. | J. H. Lee, J. Blair, V. A. Tamma, Q. Wu, S. J. Rhee, C. J. Summers, and W. Park, “Direct visualization of optical frequency invisibility cloak based on silicon nanorod array,” Opt. Express |

6. | T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science |

7. | H. F. Ma and T. J. Cui, “Three-dimensional broadband ground-plane cloak made of metamaterials,” Nature Commun. |

8. | I. I. Smolyaninov, V. N. Smolyaninova, A. V. Kildishev, and V. M. Shalaev, “Anisotropic metamaterials emulated by tapered waveguides: Application to optical cloaking,” Phys. Rev. Lett. |

9. | B. Zhang, T. Chan, and B.-I. Wu, “Lateral shift makes a ground-plane cloak detectable,” Phys. Rev. Lett. |

10. | N.-A. P. Nicorovici, R. C. McPhedran, S. Enoch, and G. Tayeb, “Finite wavelength cloaking by plasmonic resonance,” New J. Phys. |

11. | V. M. Shalaev, “Transforming light,” Science |

12. | H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nature Mater. |

13. | P. Zhang, M. Lobet, and S. He, “Carpet cloaking on a dielectric half-space,” Opt. Express |

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

**OCIS Codes**

(080.0080) Geometric optics : Geometric optics

(080.2710) Geometric optics : Inhomogeneous optical media

(160.3918) Materials : Metamaterials

(230.3205) Optical devices : Invisibility cloaks

**History**

Original Manuscript: September 13, 2010

Revised Manuscript: October 19, 2010

Manuscript Accepted: October 22, 2010

Published: November 5, 2010

**Citation**

Roman Schmied, Jad C. Halimeh, and Martin Wegener, "Conformal carpet and grating cloaks," Opt. Express **18**, 24361-24367 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-23-24361

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### References

- J. Li, and J. B. Pendry, “Hiding under the carpet: A new strategy for cloaking,” Phys. Rev. Lett. 101, 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,” Science 323, 366–369 (2009). [CrossRef] [PubMed]
- J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8, 568–571 (2009). [CrossRef]
- L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics 3, 461–463 (2009). [CrossRef]
- J. H. Lee, J. Blair, V. A. Tamma, Q. Wu, S. J. Rhee, C. J. Summers, and W. Park, “Direct visualization of optical frequency invisibility cloak based on silicon nanorod array,” Opt. Express 17, 12922–12928 (2009). [CrossRef] [PubMed]
- T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science 328, 337–339 (2010). [CrossRef] [PubMed]
- H. F. Ma, and T. J. Cui, “Three-dimensional broadband ground-plane cloak made of metamaterials,” Nature Commun. 1, 1–6 (2010). [CrossRef]
- I. I. Smolyaninov, V. N. Smolyaninova, A. V. Kildishev, and V. M. Shalaev, “Anisotropic metamaterials emulated by tapered waveguides: Application to optical cloaking,” Phys. Rev. Lett. 102, 213901 (2009). [CrossRef] [PubMed]
- B. Zhang, T. Chan, and B.-I. Wu, “Lateral shift makes a ground-plane cloak detectable,” Phys. Rev. Lett. 104, 233903 (2010). [CrossRef] [PubMed]
- N.-A. P. Nicorovici, R. C. McPhedran, S. Enoch, and G. Tayeb, “Finite wavelength cloaking by plasmonic resonance,” N. J. Phys. 10, 115020 (2008). [CrossRef]
- V. M. Shalaev, “Transforming light,” Science 322, 384–386 (2008). [CrossRef] [PubMed]
- H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9, 387–396 (2010). [CrossRef]
- P. Zhang, M. Lobet, and S. He, “Carpet cloaking on a dielectric half-space,” Opt. Express 18, 15158–18163 (2010).
- U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006). [CrossRef] [PubMed]

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