## Reducing physical appearance of electromagnetic sources |

Optics Express, Vol. 21, Issue 4, pp. 5053-5062 (2013)

http://dx.doi.org/10.1364/OE.21.005053

Acrobat PDF (2262 KB)

### Abstract

We propose to use the concept of transformation optics for the design of novel radiating devices. By applying transformations that compress space, and then that match it to the surrounding environment, we show how the electromagnetic appearance of radiating elements can be tailored at will. Our efficient approach allows one to realize a large aperture emission from a small aperture one. We describe transformation of the metric space and the calculation of the material parameters. Full wave simulations are performed to validate the proposed approach on different space compression shapes, factors and impedance matching. The idea paves the way to interesting applications in various domains in microwave and optical regimes, but also in acoustics.

© 2013 OSA

## 1. Introduction

1. J. B. Pendry and S. A. Ramakrishna, “Focusing light using negative refraction,” J. Phys. Condens. Matter **15**(37), 6345–6364 (2003). [CrossRef]

5. D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express **14**(21), 9794–9804 (2006). [CrossRef] [PubMed]

6. U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” New J. Phys. **8**(10), 247 (2006). [CrossRef]

8. U. Leonhardt and T. G. Philbin, “Transformation optics and the geometry of light,” Prog. Opt. **53**, 69–152 (2009). [CrossRef]

1. J. B. Pendry and S. A. Ramakrishna, “Focusing light using negative refraction,” J. Phys. Condens. Matter **15**(37), 6345–6364 (2003). [CrossRef]

2. S. Guenneau, B. Gralak, and J. B. Pendry, “Perfect corner reflector,” Opt. Lett. **30**(10), 1204–1206 (2005). [CrossRef] [PubMed]

9. 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]

10. F. Zolla, S. Guenneau, A. Nicolet, and J. B. Pendry, “Electromagnetic analysis of cylindrical invisibility cloaks and the mirage effect,” Opt. Lett. **32**(9), 1069–1071 (2007). [CrossRef] [PubMed]

11. M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photon. Nanostruct.: Fundam. Appl. **6**(1), 87–95 (2008). [CrossRef]

12. H. Chen, B. Hou, S. Chen, X. Ao, W. Wen, and C. T. Chan, “Design and experimental realization of a broadband transformation media field rotator at microwave frequencies,” Phys. Rev. Lett. **102**(18), 183903 (2009). [CrossRef] [PubMed]

13. D.-H. Kwon and D. H. Werner, “Transformation optical designs for wave collimators, flat lenses and right-angle bends,” New J. Phys. **10**(11), 115023 (2008). [CrossRef]

16. D. A. Roberts, N. Kundtz, and D. R. Smith, “Optical lens compression via transformation optics,” Opt. Express **17**(19), 16535–16542 (2009). [CrossRef] [PubMed]

17. A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Electromagnetic wormholes and virtual magnetic monopoles from metamaterials,” Phys. Rev. Lett. **99**(18), 183901 (2007). [CrossRef] [PubMed]

18. A. Nicolet, F. Zolla, and S. Guenneau, “A finite element modelling for twisted electromagnetic waveguides,” Eur. J. Phys. Appl. Phys. **28**(2), 153–157 (2004). [CrossRef]

23. P.-H. Tichit, S. N. Burokur, and A. de Lustrac, “Waveguide taper engineering using coordinate transformation technology,” Opt. Express **18**(2), 767–772 (2010). [CrossRef] [PubMed]

24. V. Ginis, P. Tassin, C. M. Soukoulis, and I. Veretennicoff, “Confining light in deep subwavelength electromagnetic cavities,” Phys. Rev. B **82**(11), 113102 (2010). [CrossRef]

25. V. Ginis, P. Tassin, J. Danckaert, C. M. Soukoulis, and I. Veretennicoff, “Creating electromagnetic cavities using transformation optics,” New J. Phys. **14**(3), 033007 (2012). [CrossRef]

26. Y. Lai, J. Ng, H. Chen, D. Han, J. Xiao, Z.-Q. Zhang, and C. T. Chan, “Illusion optics: the optical transformation of an object into another object,” Phys. Rev. Lett. **102**(25), 253902 (2009). [CrossRef] [PubMed]

27. W. H. Wee and J. B. Pendry, “Shrinking optical devices,” New J. Phys. **11**(7), 073033 (2009). [CrossRef]

28. W. Lu, Z. Lin, H. Chen, and C. T. Chan, “Transformation media based super focusing antenna,” J. Phys. D Appl. Phys. **42**(21), 212002 (2009). [CrossRef]

35. P.-H. Tichit, S. N. Burokur, and A. de Lustrac, “Transformation media producing quasi-perfect isotropic emission,” Opt. Express **19**(21), 20551–20556 (2011). [CrossRef] [PubMed]

## 2. Transformation formulations

*R*

_{1}/

*q*

_{1}(with

*q*

_{1}< 1), delimited by the red circle in Fig. 1(a)) in a region of radius

*R*

_{1}. In the studied transformation, our space is described by polar coordinates and the angular part of these coordinates remains unchanged. The second part of the transformation consists in an impedance matching with the surrounding space through an annular expansion zone defined between circular regions with radius

*R*

_{1}and

*R*

_{2}, as illustrated in Fig. 1(b). This space expansion can be performed using three different transformations: a positive exponential transformation, a negative exponential transformation, and a linear one. We denote below and in the rest of the paper the two different zones by the index

*i*, where

*i*= 1 corresponds to the first zone and

*i*= 2 to the second zone. The final virtual space describing our device is represented in Fig. 1(c). Figure 1(d) summarizes the different transformations considered. To secure the impedance and metric matching, continuity of our transformations is assured at the boundary of the first region (point A in Fig. 1(d)) and at the outer boundary of the device (point B in Fig. 1(d)).

*f*

_{i,r}and

*f*

_{i,θ}represent the respective derivatives of

*f*

_{i}with respect to

*r*and

*θ*. To calculate permittivity and permeability tensors directly from the coordinate transformation in the cylindrical and orthogonal coordinates, we need to express the metric tensor in the initial and virtual spaces. The final Jacobian matrix needed for the permeability and permittivity tensors of our material is then given as:

*f*

_{i,θ}= 0 to simplify the calculations. To apply our proposed coordinate transformation, we consider a radial compression of the space in region 1. This leads to a material with high permittivity and permeability tensors. For the transformation, we choose

*q*

_{1}being a coefficient lower than 1. The physical meaning of the factor

*q*

_{1}is the compression factor applied in the central region. This factor has a transition value which can be defined as

*q*

_{1}<

*q*

_{0}the material presents a negative index and the final apparent size of the source can be larger than 2

*R*

_{2}. Now if this embedded source has a small aperture, much smaller than the wavelength, then after transformation this antenna will behave like one with a large aperture, typically

*q*

_{1}times larger and potentially much greater than the wavelength. A small aperture antenna is well known to radiate isotropically. The same antenna embedded in the material defined by Eq. (5) will present a directive radiation and therefore electrically appear as if its size is larger than the working wavelength. Moreover, we can obtain the radiation of a conventional array of antennas using much smaller dimensions for the latter array embedded in zone 1. To assure a good impedance matching for the radiated fields, a matching zone (region 2) is added around region 1. To design this zone, we consider three different possible transformations that match the space from

*R*

_{1}to

*R*

_{2}. The first studied transformation for this matching region is a linear one that takes the form

*r*from

*r*’ has an exponential form defined by:

*q*

_{2}that indicates the shape of the progressive metric matching to vacuum (

*q*

_{1}indicates a high compression of the space in the first region. To compensate this high compression, the transformation in the second region gives negative electromagnetic parameters due to the relative positions of points A and B (Fig. 1(d)), as presented in Fig. 2 . In such case, the wave propagates with a backward phase in this region. Figure 2 shows the variations of the different components of the permittivity and permeability tensors for the matching region 2. For the linear transformation, the minimum and maximum of the material parameters depend on the geometrical properties of the problem and thus they depend only on

*α*and

*γ*which are given by:where

*q*

_{1}is defined on ]0, 1].

*α*is therefore defined on ]-∞,1] and vanishes at

*q*

_{1}=

*q*

_{0}. Thus,

*γ*is a function of

*α*and is larger than 1 for

*q*

_{1}>

*q*

_{0}and is negative for

*q*

_{1}<

*q*

_{0}. In this last case such a medium is a left-handed material.

*R*

_{1},

*R*

_{2},

*q*

_{1}and

*q*

_{2}for the linear transformation. For the case of the exponential transformation in region 2, the parameters considered are

*q*

_{1}= 1/16,

*q*

_{2}= 15,

*R*

_{1}= 5 mm and

*R*

_{2}= 45 mm and as it can be observed, the calculated components ψ

_{xx}, ψ

_{yy}and ψ

_{zz}are always negative.

## 3. Numerical validation

*z*-axis). Different current sources perpendicular to the

*xy*plane are used as radiating elements in order to show that our transformation can be applied to any type of source embedded in the region 1. Continuity and matched conditions are applied respectively to the boundary of zone 1 and zone 2.

*R*

_{1}= 2 mm and

*R*

_{2}= 45 mm. The results obtained from linear transformations both in region 1 and 2, as defined by the continuous blue trace in Fig. 2(a), are presented in Fig. 3 . In Fig. 3(a), the electric field distribution of a current source radiating in free space is plotted. The source is supposed to have a width

*d*= 80 mm (2.7

*λ*at 10 GHz). For such a large size, the radiation is equivalent to that of an array of several elements and therefore, the radiated field is directive. Figure 3(b) shows a similar source but with a much smaller size

*d*= 2 mm (

*λ*/15 at 10 GHz) embedded in the metamaterial shell having a compression factor

*q*

_{1}= 1/40. In this scenario, a radiation pattern similar to the large aperture source is observed, demonstrating that small aperture antennas inserted in the proposed material shell present the same electromagnetic behavior as much larger aperture antennas in free space. However, this same miniature source will radiate in an isotropic manner in free space (Fig. 3(c)). The same observations can be made when replacing the linear current source by a crossed-type one, as illustrated in Figs. 3(d)-3(f).

*q*

_{2}= 15 in region 2 and the compression factor in region 1 is decreased to

*q*

_{1}= 1/40. The transformation is defined by the dashed blue trace in Fig. 2(a). The small size linear current source with

*d*=

*λ*/15 is embedded in the metamaterial shell defined by the proposed coordinate transformation. A directive emission is observed as in the previous case and as illustrated in the enlarged view of Fig. 4(b), we can clearly note the exponential form of the radiated field. We can also observe the perfect impedance matching between the regions 1 and 2 and between the region 2 and free space. This is clearly confirmed by the continuity of the electric field norm at the interface

*r*=

*R*

_{1}and by the absence of stationary waves in region 2 in Fig. 4(c).

*q*

_{1}= 1/16 and in region 2, the material is defined by an exponential transformation with

*q*

_{2}= 15. This transformation corresponds to the dashed red trace of Fig. 2(a) and the crossed-type source is embedded in the metamaterial shell. In this case also, a bidirectional directive beam can be observed even if the size of the source is very small compared to the working wavelength. In each case, the small aperture size of the radiating element has been transformed into a larger one: 40 times for the linear source and 16 times for the crossed-type source.

*r*=

*R*

_{1}.

*L*= 12.5 mm, spaced by a distance

*a*= 5 mm and with a 30° phase shift between each element. These sources radiate in vacuum and as illustrated in Fig. 6(a) , we observe a radiated beam pointing in an off-normal direction due to the phase shift applied between the different elements of the array. When the dimensions of these antennas are reduced by a factor of 25 (

*q*

_{1}= 1/25) the dimensions of the array become smaller compared to the wavelength and the radiated field becomes isotropic as shown in Fig. 6(b). By embedding the small sources in a material defined from the double linear transformation, we are able to recover the beam steering of the source array as shown in Fig. 6(c). This last example confirms the ability of our transformation to change the electromagnetic appearance of a group of radiators. Figures 6(d) and 6(e) show that in both cases, with and without transformed material, the impedance matching between the metamaterial shell and free space is perfect.

## 4. Conclusion

## References and links

1. | J. B. Pendry and S. A. Ramakrishna, “Focusing light using negative refraction,” J. Phys. Condens. Matter |

2. | S. Guenneau, B. Gralak, and J. B. Pendry, “Perfect corner reflector,” Opt. Lett. |

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

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

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

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

7. | R. A. Crudo and J. G. O’Brien, “Metric approach to transformation optics,” Phys. Rev. A |

8. | U. Leonhardt and T. G. Philbin, “Transformation optics and the geometry of light,” Prog. Opt. |

9. | 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 |

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

11. | M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photon. Nanostruct.: Fundam. Appl. |

12. | H. Chen, B. Hou, S. Chen, X. Ao, W. Wen, and C. T. Chan, “Design and experimental realization of a broadband transformation media field rotator at microwave frequencies,” Phys. Rev. Lett. |

13. | D.-H. Kwon and D. H. Werner, “Transformation optical designs for wave collimators, flat lenses and right-angle bends,” New J. Phys. |

14. | M. Tsang and D. Psaltis, “Magnifying perfect lens and superlens design by coordinate transformation,” Phys. Rev. B |

15. | N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater. |

16. | D. A. Roberts, N. Kundtz, and D. R. Smith, “Optical lens compression via transformation optics,” Opt. Express |

17. | A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Electromagnetic wormholes and virtual magnetic monopoles from metamaterials,” Phys. Rev. Lett. |

18. | A. Nicolet, F. Zolla, and S. Guenneau, “A finite element modelling for twisted electromagnetic waveguides,” Eur. J. Phys. Appl. Phys. |

19. | M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, “Optical design of reflectionless complex media by finite embedded coordinate transformations,” Phys. Rev. Lett. |

20. | M. Rahm, D. A. Roberts, J. B. Pendry, and D. R. Smith, “Transformation-optical design of adaptive beam bends and beam expanders,” Opt. Express |

21. | J. Huangfu, S. Xi, F. Kong, J. Zhang, H. Chen, D. Wang, B.-I. Wu, L. Ran, and J. A. Kong, “Application of coordinate transformation in bent waveguide,” J. Appl. Phys. |

22. | D. A. Roberts, M. Rahm, J. B. Pendry, and D. R. Smith, “Transformation-optical design of sharp waveguide bends and corners,” Appl. Phys. Lett. |

23. | P.-H. Tichit, S. N. Burokur, and A. de Lustrac, “Waveguide taper engineering using coordinate transformation technology,” Opt. Express |

24. | V. Ginis, P. Tassin, C. M. Soukoulis, and I. Veretennicoff, “Confining light in deep subwavelength electromagnetic cavities,” Phys. Rev. B |

25. | V. Ginis, P. Tassin, J. Danckaert, C. M. Soukoulis, and I. Veretennicoff, “Creating electromagnetic cavities using transformation optics,” New J. Phys. |

26. | Y. Lai, J. Ng, H. Chen, D. Han, J. Xiao, Z.-Q. Zhang, and C. T. Chan, “Illusion optics: the optical transformation of an object into another object,” Phys. Rev. Lett. |

27. | W. H. Wee and J. B. Pendry, “Shrinking optical devices,” New J. Phys. |

28. | W. Lu, Z. Lin, H. Chen, and C. T. Chan, “Transformation media based super focusing antenna,” J. Phys. D Appl. Phys. |

29. | Y. Luo, J. Zhang, L. Ran, H. Chen, and J. A. Kong, “Controlling the emission of electromagnetic source,” PIERS Online |

30. | J. Allen, N. Kundtz, D. A. Roberts, S. A. Cummer, and D. R. Smith, “Electromagnetic source transformations using superellipse equations,” Appl. Phys. Lett. |

31. | B. I. Popa, J. Allen, and S. A. Cummer, “Conformal array design with transformation electromagnetics,” Appl. Phys. Lett. |

32. | P.-H. Tichit, S. N. Burokur, D. Germain, and A. de Lustrac, “Design and experimental demonstration of a high-directive emission with transformation optics,” Phys. Rev. B |

33. | P.-H. Tichit, S. N. Burokur, D. Germain, and A. de Lustrac, “Coordinate transformation based ultra-directive emission,” Electron. Lett. |

34. | Z. H. Jiang, M. D. Gregory, and D. H. Werner, “Experimental demonstration of a broadband transformation optics lens for highly directive multibeam emission,” Phys. Rev. B |

35. | P.-H. Tichit, S. N. Burokur, and A. de Lustrac, “Transformation media producing quasi-perfect isotropic emission,” Opt. Express |

**OCIS Codes**

(160.1190) Materials : Anisotropic optical materials

(260.2110) Physical optics : Electromagnetic optics

(260.2065) Physical optics : Effective medium theory

(160.3918) Materials : Metamaterials

(260.2710) Physical optics : Inhomogeneous optical media

**ToC Category:**

Physical Optics

**History**

Original Manuscript: December 27, 2012

Revised Manuscript: February 8, 2013

Manuscript Accepted: February 11, 2013

Published: February 21, 2013

**Citation**

Paul-Henri Tichit, Shah Nawaz Burokur, and André de Lustrac, "Reducing physical appearance of electromagnetic sources," Opt. Express **21**, 5053-5062 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-4-5053

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### References

- J. B. Pendry and S. A. Ramakrishna, “Focusing light using negative refraction,” J. Phys. Condens. Matter15(37), 6345–6364 (2003). [CrossRef]
- S. Guenneau, B. Gralak, and J. B. Pendry, “Perfect corner reflector,” Opt. Lett.30(10), 1204–1206 (2005). [CrossRef] [PubMed]
- U. Leonhardt, “Optical conformal mapping,” Science312(5781), 1777–1780 (2006). [CrossRef] [PubMed]
- J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312(5781), 1780–1782 (2006). [CrossRef] [PubMed]
- D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express14(21), 9794–9804 (2006). [CrossRef] [PubMed]
- U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” New J. Phys.8(10), 247 (2006). [CrossRef]
- R. A. Crudo and J. G. O’Brien, “Metric approach to transformation optics,” Phys. Rev. A80(3), 033824 (2009). [CrossRef]
- U. Leonhardt and T. G. Philbin, “Transformation optics and the geometry of light,” Prog. Opt.53, 69–152 (2009). [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,” Science314(5801), 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(9), 1069–1071 (2007). [CrossRef] [PubMed]
- M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photon. Nanostruct.: Fundam. Appl.6(1), 87–95 (2008). [CrossRef]
- H. Chen, B. Hou, S. Chen, X. Ao, W. Wen, and C. T. Chan, “Design and experimental realization of a broadband transformation media field rotator at microwave frequencies,” Phys. Rev. Lett.102(18), 183903 (2009). [CrossRef] [PubMed]
- D.-H. Kwon and D. H. Werner, “Transformation optical designs for wave collimators, flat lenses and right-angle bends,” New J. Phys.10(11), 115023 (2008). [CrossRef]
- M. Tsang and D. Psaltis, “Magnifying perfect lens and superlens design by coordinate transformation,” Phys. Rev. B77(3), 035122 (2008). [CrossRef]
- N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater.9(2), 129–132 (2010). [CrossRef] [PubMed]
- D. A. Roberts, N. Kundtz, and D. R. Smith, “Optical lens compression via transformation optics,” Opt. Express17(19), 16535–16542 (2009). [CrossRef] [PubMed]
- A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Electromagnetic wormholes and virtual magnetic monopoles from metamaterials,” Phys. Rev. Lett.99(18), 183901 (2007). [CrossRef] [PubMed]
- A. Nicolet, F. Zolla, and S. Guenneau, “A finite element modelling for twisted electromagnetic waveguides,” Eur. J. Phys. Appl. Phys.28(2), 153–157 (2004). [CrossRef]
- M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, “Optical design of reflectionless complex media by finite embedded coordinate transformations,” Phys. Rev. Lett.100(6), 063903 (2008). [CrossRef] [PubMed]
- M. Rahm, D. A. Roberts, J. B. Pendry, and D. R. Smith, “Transformation-optical design of adaptive beam bends and beam expanders,” Opt. Express16(15), 11555–11567 (2008). [CrossRef] [PubMed]
- J. Huangfu, S. Xi, F. Kong, J. Zhang, H. Chen, D. Wang, B.-I. Wu, L. Ran, and J. A. Kong, “Application of coordinate transformation in bent waveguide,” J. Appl. Phys.104(1), 014502 (2008). [CrossRef]
- D. A. Roberts, M. Rahm, J. B. Pendry, and D. R. Smith, “Transformation-optical design of sharp waveguide bends and corners,” Appl. Phys. Lett.93(25), 251111 (2008). [CrossRef]
- P.-H. Tichit, S. N. Burokur, and A. de Lustrac, “Waveguide taper engineering using coordinate transformation technology,” Opt. Express18(2), 767–772 (2010). [CrossRef] [PubMed]
- V. Ginis, P. Tassin, C. M. Soukoulis, and I. Veretennicoff, “Confining light in deep subwavelength electromagnetic cavities,” Phys. Rev. B82(11), 113102 (2010). [CrossRef]
- V. Ginis, P. Tassin, J. Danckaert, C. M. Soukoulis, and I. Veretennicoff, “Creating electromagnetic cavities using transformation optics,” New J. Phys.14(3), 033007 (2012). [CrossRef]
- Y. Lai, J. Ng, H. Chen, D. Han, J. Xiao, Z.-Q. Zhang, and C. T. Chan, “Illusion optics: the optical transformation of an object into another object,” Phys. Rev. Lett.102(25), 253902 (2009). [CrossRef] [PubMed]
- W. H. Wee and J. B. Pendry, “Shrinking optical devices,” New J. Phys.11(7), 073033 (2009). [CrossRef]
- W. Lu, Z. Lin, H. Chen, and C. T. Chan, “Transformation media based super focusing antenna,” J. Phys. D Appl. Phys.42(21), 212002 (2009). [CrossRef]
- Y. Luo, J. Zhang, L. Ran, H. Chen, and J. A. Kong, “Controlling the emission of electromagnetic source,” PIERS Online4(7), 795–800 (2008). [CrossRef]
- J. Allen, N. Kundtz, D. A. Roberts, S. A. Cummer, and D. R. Smith, “Electromagnetic source transformations using superellipse equations,” Appl. Phys. Lett.94(19), 194101 (2009). [CrossRef]
- B. I. Popa, J. Allen, and S. A. Cummer, “Conformal array design with transformation electromagnetics,” Appl. Phys. Lett.94(24), 244102 (2009). [CrossRef]
- P.-H. Tichit, S. N. Burokur, D. Germain, and A. de Lustrac, “Design and experimental demonstration of a high-directive emission with transformation optics,” Phys. Rev. B83(15), 155108 (2011). [CrossRef]
- P.-H. Tichit, S. N. Burokur, D. Germain, and A. de Lustrac, “Coordinate transformation based ultra-directive emission,” Electron. Lett.47(10), 580–582 (2011). [CrossRef]
- Z. H. Jiang, M. D. Gregory, and D. H. Werner, “Experimental demonstration of a broadband transformation optics lens for highly directive multibeam emission,” Phys. Rev. B84(16), 165111 (2011). [CrossRef]
- P.-H. Tichit, S. N. Burokur, and A. de Lustrac, “Transformation media producing quasi-perfect isotropic emission,” Opt. Express19(21), 20551–20556 (2011). [CrossRef] [PubMed]

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