## Electromagnetic localization based on transformation optics

Optics Express, Vol. 18, Issue 11, pp. 11891-11897 (2010)

http://dx.doi.org/10.1364/OE.18.011891

Acrobat PDF (1351 KB)

### Abstract

Localization of an electromagnetic field can be achieved by transformation optics using metamaterials. A coordinate transformation structure different from traditional resonator is proposed. Wherein, arbitrary frequency of the whole band of electromagnetic wave can be localized without energy loss, i.e., the modes in this structure are continuous. Theoretical analysis and numerical simulation show that the material parameter variations at the outer boundary of the structure have little influence on the localization property. When realizable physical structure is considered, multi-layer approximation should be applied. The calculated results show that the estimated localization time is about 100 ns for an 8-layer inhomogeneous approximation, and it could reach several seconds for a 30-layer homogeneous approximation. The present work may present a new application of transformation optics.

© 2010 OSA

## 1. Introduction

1. A. Ward and J. Pendry, “Refraction and geometry in Maxwell’s equations,” J. Mod. Opt. **43**, 773–793 (1996). [CrossRef]

13. L. Gabrielli, J. Cardenas, C. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics **3**(8), 461–463 (2009). [CrossRef]

1. A. Ward and J. Pendry, “Refraction and geometry in Maxwell’s equations,” J. Mod. Opt. **43**, 773–793 (1996). [CrossRef]

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

3. U. Leonhardt, “Optical conformal mapping,” Science **312**(5781), 1777–1780 (2006). [CrossRef] [PubMed]

14. B. Zhang and B. I. Wu, “Electromagnetic detection of a perfect invisibility cloak,” Phys. Rev. Lett. **103**(24), 243901 (2009). [CrossRef]

14. B. Zhang and B. I. Wu, “Electromagnetic detection of a perfect invisibility cloak,” Phys. Rev. Lett. **103**(24), 243901 (2009). [CrossRef]

6. S. A. Cummer, B. I. Popa, D. Schurig, D. R. Smith, and J. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **74**, 036621 (2006). [CrossRef] [PubMed]

## 2. Theoretical considerations

_{2}, there is no field distribution within the region of R

_{1}, and the wave front and power flow of the electromagnetic wave are not disturbed after passing through the structure. This is just an invisibility cloak. Figure 1(c) is the complementary structure of Fig. 1(a) geometrically in which R

_{4}approaches infinity. The combination of Fig. 1(a) and Fig. 1(c) forms the whole plane. Figure 1(d) shows the coordinate transformation formed by compressing the boundary R

_{4}from infinity to R

_{3}, i.e. the structure shown in Fig. 1(d) is obtained by compressing a semi-infinity space r>R

_{2}(Fig. 1(c)) into an annular domain R

_{2}<r<R

_{3}. For this transformation, if the wave source is in the inner region of R

_{3}, the wave front reached at the outer boundary of R

_{3}(Fig. 1(d)) is equivalent to that reached at infinity before the transformation (Fig. 1(c)). So, there is no electromagnetic field distribution in the outer region of R

_{3}, and electromagnetic localization is achieved in the mechanical space.

15. L. Zhang, M. Yan, and M. Qiu, “The effect of transformation order on the invisibility performance of a practical cylindrical cloak,” J. Opt. A, Pure Appl. Opt. **10**(9), 095001 (2008). [CrossRef]

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

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**(5801), 977–980 (2006). [CrossRef] [PubMed]

15. L. Zhang, M. Yan, and M. Qiu, “The effect of transformation order on the invisibility performance of a practical cylindrical cloak,” J. Opt. A, Pure Appl. Opt. **10**(9), 095001 (2008). [CrossRef]

16. W. Cai, U. Chettiar, A. Kildishev, V. Shalaev, and G. Milton, “Nonmagnetic cloak with minimized scattering,” Appl. Phys. Lett. **91**(11), 111105 (2007). [CrossRef]

19. J. Xu and X. Zhang, “Cloaking radiation of moving electron beam and relativistic energy loss spectra,” Opt. Express **17**(6), 4758–4772 (2009). [CrossRef] [PubMed]

## 3. Numerical simulation and analysis

### 3.1 Calculations of eigenmodes

### 3.2 Calculations of response characteristics of a single frequency wave in the time domain

_{3}).

_{z}) distribution under various transformation orders of the structure. The frequency of E

_{z}is still 2 GHz. It can be seen that the amplitude of E

_{z}attenuated rapidly with the increase of transmission distance in the transformed region for different transformation orders, the wave front compressed rapidly and there was no field distribution near the outer boundary. It can also be seen that the lower the transformation order, the faster the attenuation of the field. For p = −0.4, the field approaches zero at 2/3 thickness of the transformation structure (0.3 m). It implies that the lower the transformation order, the higher the insensitivity of the localization effect to the variations of the material parameters at its outer boundary. It means that the material parameters within a certain thickness near the outer boundary would have no effect on the electromagnetic field localized in the structure. To demonstrate this property, we introduce a perturbation to the structure, i.e. the outer boundary is shifted inward for a distance (L in the insert of Fig. 7 ) toward the center as shown in Fig. 7, so that the new outer radius is located at

_{3}-R

_{2}).

6. S. A. Cummer, B. I. Popa, D. Schurig, D. R. Smith, and J. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **74**, 036621 (2006). [CrossRef] [PubMed]

_{z}) of the proposed structure. It can be seen that the wave energy inside the ELS leaked out when an eight layer (homogeneous) approximation was used. It means that for a practical ELS with multilayer approximation the localization time will be finite.

*Q*will be infinite. It means that electromagnetic field can be localized inside the equivalent cavity without energy loss and with infinite time. However, for an ELS with multilayer approximation, the

*Q*factor will be a finite value. It means that electromagnetic field can still be localized inside the equivalent cavity, but with energy loss and finite time. The corresponding

*Q*factor can be estimated by the formula

*E*is the total energy inside the cavity,

*P*is the rate of energy loss,

*ν*is the frequency of the electromagnetic field. This

*Q*factor can be used to estimate the localization time, and the estimation is equivalent to the above simulation.

## 3. Conclusion

## Acknowledgments

## References and links

1. | A. Ward and J. Pendry, “Refraction and geometry in Maxwell’s equations,” J. Mod. Opt. |

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

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

4. | U. Leonhardt, “Notes on conformal invisibility devices,” N. J. Phys. |

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. | S. A. Cummer, B. I. Popa, D. Schurig, D. R. Smith, and J. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

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

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

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

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

11. | U. Leonhardt and T. Tyc, “Broadband invisibility by non-Euclidean cloaking,” Science |

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

13. | L. Gabrielli, J. Cardenas, C. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics |

14. | B. Zhang and B. I. Wu, “Electromagnetic detection of a perfect invisibility cloak,” Phys. Rev. Lett. |

15. | L. Zhang, M. Yan, and M. Qiu, “The effect of transformation order on the invisibility performance of a practical cylindrical cloak,” J. Opt. A, Pure Appl. Opt. |

16. | W. Cai, U. Chettiar, A. Kildishev, V. Shalaev, and G. Milton, “Nonmagnetic cloak with minimized scattering,” Appl. Phys. Lett. |

17. | W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Designs for optical cloaking with high-order transformations,” Opt. Express |

18. | A. Hendi, J. Henn, and U. Leonhardt, “Ambiguities in the scattering tomography for central potentials,” Phys. Rev. Lett. |

19. | J. Xu and X. Zhang, “Cloaking radiation of moving electron beam and relativistic energy loss spectra,” Opt. Express |

**OCIS Codes**

(120.4570) Instrumentation, measurement, and metrology : Optical design of instruments

(160.4760) Materials : Optical properties

(160.3918) Materials : Metamaterials

**ToC Category:**

Physical Optics

**History**

Original Manuscript: April 5, 2010

Revised Manuscript: May 16, 2010

Manuscript Accepted: May 16, 2010

Published: May 20, 2010

**Citation**

Tianrui Zhai, Ying Zhou, Jinwei Shi, Zhaona Wang, Dahe Liu, and Jing Zhou, "Electromagnetic localization based on transformation optics," Opt. Express **18**, 11891-11897 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-11-11891

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### References

- A. Ward and J. Pendry, “Refraction and geometry in Maxwell’s equations,” J. Mod. Opt. 43, 773–793 (1996). [CrossRef]
- J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006). [CrossRef] [PubMed]
- U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006). [CrossRef] [PubMed]
- U. Leonhardt, “Notes on conformal invisibility devices,” N. J. Phys. 8(7), 118 (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(5801), 977–980 (2006). [CrossRef] [PubMed]
- S. A. Cummer, B. I. Popa, D. Schurig, D. R. Smith, and J. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74, 036621 (2006). [CrossRef] [PubMed]
- W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics 1(4), 224–227 (2007). [CrossRef]
- J. Lee, J. Blair, V. Tamma, Q. Wu, S. Rhee, C. Summers, and W. Park, “Direct visualization of optical frequency invisibility cloak based on silicon nanorod array,” Opt. Express 17(15), 12922–12928 (2009). [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]
- 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]
- U. Leonhardt and T. Tyc, “Broadband invisibility by non-Euclidean cloaking,” Science 323(5910), 110–112 (2009). [CrossRef]
- J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8(7), 568–571 (2009). [CrossRef] [PubMed]
- L. Gabrielli, J. Cardenas, C. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics 3(8), 461–463 (2009). [CrossRef]
- B. Zhang and B. I. Wu, “Electromagnetic detection of a perfect invisibility cloak,” Phys. Rev. Lett. 103(24), 243901 (2009). [CrossRef]
- L. Zhang, M. Yan, and M. Qiu, “The effect of transformation order on the invisibility performance of a practical cylindrical cloak,” J. Opt. A, Pure Appl. Opt. 10(9), 095001 (2008). [CrossRef]
- W. Cai, U. Chettiar, A. Kildishev, V. Shalaev, and G. Milton, “Nonmagnetic cloak with minimized scattering,” Appl. Phys. Lett. 91(11), 111105 (2007). [CrossRef]
- W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Designs for optical cloaking with high-order transformations,” Opt. Express 16(8), 5444–5452 (2008). [CrossRef] [PubMed]
- A. Hendi, J. Henn, and U. Leonhardt, “Ambiguities in the scattering tomography for central potentials,” Phys. Rev. Lett. 97(7), 073902 (2006). [CrossRef] [PubMed]
- J. Xu and X. Zhang, “Cloaking radiation of moving electron beam and relativistic energy loss spectra,” Opt. Express 17(6), 4758–4772 (2009). [CrossRef] [PubMed]

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