## Engineering antenna radiation patterns via quasi-conformal mappings |

Optics Express, Vol. 19, Issue 24, pp. 23743-23750 (2011)

http://dx.doi.org/10.1364/OE.19.023743

Acrobat PDF (2678 KB)

### Abstract

We use a combination of conformal and quasi-conformal mappings to engineer isotropic electromagnetic devices that modify the omnidirectional radiation pattern of a point source. For TE waves, the designed devices are also non-magnetic. The flexibility offered by the proposed method is much higher than that achieved with conformal mappings. As a result, it is shown that complex radiation patterns can be achieved, which can combine high directivity in a desired number of arbitrary directions and isotropic radiation in other specified angular ranges. In addition, this technique enables us to control the power radiated in each direction to a certain extent. The obtained results are valid for any part of the spectrum. The potential of this method is illustrated with some examples. Finally, we study the frequency dependence of the considered devices and propose a practical dielectric implementation.

© 2011 OSA

## 1. Introduction

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

17. J. P. Turpin, A. T. Massoud, Z. H. Jiang, P. L. Werner, and D. H. Werner, “Conformal mappings to achieve simple material parameters for transformation optics devices,” Opt. Express **18**(1), 244–252 (2010). [CrossRef] [PubMed]

18. M. Schmiele, V. S. Varma, C. Rockstuhl, and F. Lederer, “Designing optical elements from isotropic materials by using transformation optics,” Phys. Rev. A **81**(3), 033837 (2010). [CrossRef]

17. J. P. Turpin, A. T. Massoud, Z. H. Jiang, P. L. Werner, and D. H. Werner, “Conformal mappings to achieve simple material parameters for transformation optics devices,” Opt. Express **18**(1), 244–252 (2010). [CrossRef] [PubMed]

18. M. Schmiele, V. S. Varma, C. Rockstuhl, and F. Lederer, “Designing optical elements from isotropic materials by using transformation optics,” Phys. Rev. A **81**(3), 033837 (2010). [CrossRef]

17. J. P. Turpin, A. T. Massoud, Z. H. Jiang, P. L. Werner, and D. H. Werner, “Conformal mappings to achieve simple material parameters for transformation optics devices,” Opt. Express **18**(1), 244–252 (2010). [CrossRef] [PubMed]

*N*sides [18

18. M. Schmiele, V. S. Varma, C. Rockstuhl, and F. Lederer, “Designing optical elements from isotropic materials by using transformation optics,” Phys. Rev. A **81**(3), 033837 (2010). [CrossRef]

*N*directional beams perpendicular to each polygon side. Thus, these techniques are limited to the design of symmetric antennas radiating in

*N*discrete directions. In this work, we combine this kind of conformal transformations with quasi-conformal mappings to gain more flexibility in the design of radiation-pattern-shaping devices. Although the proposed devices were devised for the optical range, the results are valid for any part of the spectrum. Therefore, normalized distances (in terms of the free-space wavelength

*λ*) are used throughout the text for the sake of generality.

## 2. Quasi-conformal mappings for antennas

*d*= 2) by using a combination of a Möbius transformation mapping the circle to the half upper plane, followed by a Schwarz-Christoffel transformation mapping the half upper plane to the square. The complete transformation is given by Eq. (1):

*F*(

*φ*|

*m*) is the incomplete elliptic integral of the first kind, with amplitude

*φ*and parameter

*m*. We have expressed this two-dimensional transformation as a function of the complex variable

*w*=

*w*

_{1}+ i

*w*

_{2}, with

*q*=

*q*

_{1}+ i

*q*

_{2}. The refractive index that implements this transformation can be obtained as

*n*

_{1}= |

*dw*/

*dq*| [7]. As for the second step, we use a quasi-conformal mapping

*z*(

*q*

_{1},

*q*

_{2}) =

*x*+ i

*y*to transform the square to the desired final shape. The advantage of this kind of quasi-conformal transformations is that they always exist and that, they can be easily calculated numerically. We only need to be careful so that this transformation has a negligible associated anisotropy. For a TE two-dimensional problem, the implementation of the exact quasi-conformal mapping would require a permittivity

*ε*= det(

*g*)

^{ij}^{-1/2}and an anisotropic permeability with in-plane components

*μ*

_{T}and

*μ*

_{L}in each of the two principal directions, where

*g*is the contravariant metric in the curved coordinates we want to implement [19

^{ij}19. J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. **101**(20), 203901 (2008). [CrossRef] [PubMed]

*α*= max((

*μ*

_{T}/

*μ*

_{L})

^{1/2},(

*μ*

_{L}/

*μ*

_{T})

^{1/2}) is usually taken as the anisotropy factor. If

*α*is close to one,

*μ*

_{T}≈

*μ*

_{L}≈1 and the quasi-conformal mapping can be implemented with negligible error by using only a refractive index distribution

*n*

_{2}

^{2}=

*ε*= det(

*g*)

^{ij}^{-1/2}, as if our mapping were conformal. We will use a simple way of computing such quasi-conformal mappings, which is based on the solution of the inverse Laplace equation supplemented with sliding boundary conditions [20

20. Z. Chang, X. Zhou, J. Hu, and G. Hu, “Design method for quasi-isotropic transformation materials based on inverse Laplace’s equation with sliding boundaries,” Opt. Express **18**(6), 6089–6096 (2010). [CrossRef] [PubMed]

*n*=

*n*

_{1}

*n*

_{2}[7]. In Fig. 1 we illustrate the two steps of this transformation with an example. Conformal maps preserve angles,

*i.e.*, two curves meeting at a certain angle in virtual space are mapped to curves in physical space that meet at the same angle. Lines perpendicular to the unit circle boundary will be perpendicular to its transformed counterpart in physical space.

## 3. Examples

*θ*= 90°,

*θ*= 180°,

*θ*= −20°, and

*θ*= −100°. In Fig. 2(a) we have depicted a possible choice for the boundary of the final device. Note that we have made use of the flexibility allowed by the quasi-conformal mapping technique in order to avoid steep vertices, which have been rounded. This way, the required refractive index

*n*is always greater than zero. We will also apply this kind of smoothing to the next examples. In Fig. 2(b) we show how the calculated mapping transforms the grid in the

*w*-plane depicted in Fig. 1 to the

*z*-plane. The refractive index that implements such mapping is included in Fig. 2(c). In this case,

*n*ranges from 0.1 to 1.75, and

*α*is approximately 1.04 so the anisotropy can be neglected. This also applies for the other examples analyzed below, for which similar values of

*α*are obtained. To verify the behavior of the designed devices, we have performed numerical calculations with the commercially available software COMSOL Multiphysics, based on the finite element technique. Isotropic dielectric media have been used in all simulations (anisotropy neglected). In Fig. 2(d) we render the power flow distribution of a point source located in the transformed center of the circle. In addition, we have calculated the far-field distribution

*E*

_{far}(

*θ*) radiated by the system. This enables us to evaluate the directivity, which can be defined as

*D*= |

*E*

_{far}(

*θ*)/

*E*

_{omni}|

^{2}for a two-dimensional TE problem, where

*E*

_{omni}is the electric far field radiated by a two-dimensional point source in any direction. In Fig. 2(e) we have depicted

*D*in polar coordinates for this first example. We can observe that the radiation pattern consists of four well-defined narrow beams in the desired directions (with a maximum angle deviation of 0.1°). The directivity in each of these directions is very similar and is higher than 6 (7.8 dB), with a half-power beamwidth BW between 10° and 13°. For comparison purposes, we simulated the exact implementation of the device (anisotropy not neglected). No appreciable differences were observed, as corresponds to small values of

*α.*

*θ*= 0°,

*θ*= 90°,

*θ*= 150°,

*θ*= −90°, and

*θ*= −150°. Moreover, we want the beams associated with the directions

*θ*= 150° and

*θ*= −150° to have a smaller directivity than the other ones. These specifications can be accomplished by assigning the left side of the square in the

*q-*plane to two segments in the

*z*-plane, one of them perpendicular to the direction

*θ*= 150° and the other one to the direction

*θ*= −150°, while leaving the other three square sides unchanged [Fig. 3(a) ]. Since the device is symmetric with respect to the horizontal axis, we know that the beams exiting each of the smaller segments will carry the same power, approximately a quarter of the power carried by each of the other three beams. As in the previous example, the resulting mapping, required refractive index, power flow distribution and directivity are shown in Figs. 3(b)–3(e). The maximum directivity is 3.62 dB for the two secondary lobes and it is between 9.03 dB and 9.2 dB for the main lobes. The refractive index ranges from 0.2 to 1.3.

*θ*= 0°,

*θ*= 180°, and

*θ*= −90°, and that we want an isotropic radiation in a 60° angular region defined by the interval

*θ*ϵ [60°, 120°]. To achieve this, we can transform the upper side of the square in the

*q*-plane to a circular boundary, while leaving the other three square sides with the same orientation [see Fig. 4(a) ]. This circular boundary must be an arc subtending an angle of 60° in order to distribute the power uniformly in the desired range. According to Fig. 4(a), the radius

*r*of the circle should be

*r*= 1/sin(30) = 2. The mapping, refractive index (varying between 0.2 and 1.32) and directivity are shown in Figs. 4(b)–4(d).

*θ*= 0° and

*θ*= 180°) and 10.1 dB (

*θ*= −90°), with beamwidths of 9.8°. We have included in Fig. 4(e) a detail of the directivity in the region where we desire an isotropic radiation.

## 4. Frequency dependence and implementation

*d*of the square side (Fig. 1) as the reference length, since the lateral size of the device is of the order of

*d*(the results in Fig. 3 correspond to

*d*= 6.6

*λ*). As expected, we observe an improvement of all features at shorter wavelengths. Specifically, for sizes of

*d*larger than 20

*λ*the performance is optimal, reaching directivities around 13 dB and half-power beamwidths as small as 4°. On the other hand, for

*d*= 4

*λ*the secondary lobes do not point at the desired direction and the directivity is quite low. As an approximate rule, we found that the behavior of the device is acceptable for sizes down to 5

*λ*.

## 5. Conclusions

*λ*. Finally, we have proposed a feasible implementation that only requires isotropic dielectric media with refractive index values above unity.

## Acknowledgments

## References and links

1. | A. Martínez, M. A. Piqueras, and J. Martí, “Generation of highly directional beam by k-space filtering using a metamaterial flat slab with a small negative index of refraction,” Appl. Phys. Lett. |

2. | A. Martínez, R. García, A. Hakansson, M. A. Piqueras, and J. Sánchez-Dehesa, “Electromagnetic beaming from omnidirectional sources by inverse design,” Appl. Phys. Lett. |

3. | J. Li, A. Salandrino, and N. Engheta, “Optical spectrometer at the nanoscale using optical Yagi-Uda nanoantennas,” Phys. Rev. B |

4. | Y. Chen, P. Lodahl, and A. F. Koenderink, “Dynamically reconfigurable directionality of plasmon-based single photon sources,” Phys. Rev. B |

5. | A. G. Curto, G. Volpe, T. H. Taminiau, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “Unidirectional emission of a quantum dot coupled to a nanoantenna,” Science |

6. | D. Dregely, R. Taubert, J. Dorfmüller, R. Vogelgesang, K. Kern, and H. Giessen, “3D optical Yagi-Uda nanoantenna array,” Nat Commun |

7. | U. Leonhardt and T. G. Philbin, |

8. | F. Kong, B.-I. Wu, J. A. Kong, J. Huangfu, S. Xi, and H. Chen, “Planar focusing antenna design by using coordinate transformation technology,” Appl. Phys. Lett. |

9. | W. X. Jiang, T. J. Cui, H. F. Ma, X. Y. Zhou, and Q. Cheng, “Cylindrical-to-plane-wave conversion via embedded optical transformation,” Appl. Phys. Lett. |

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

11. | N. Kundtz, D. A. Roberts, J. Allen, S. Cummer, and D. R. Smith, “Optical source transformations,” Opt. Express |

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

13. | P.-H. Tichit, S. N. Burokur, and A. de Lustrac, “Ultradirective antenna via transformation optics,” J. Appl. Phys. |

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

15. | U. Leonhardt and T. Tyc, “Superantenna made of transformation media,” New J. Phys. |

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

17. | J. P. Turpin, A. T. Massoud, Z. H. Jiang, P. L. Werner, and D. H. Werner, “Conformal mappings to achieve simple material parameters for transformation optics devices,” Opt. Express |

18. | M. Schmiele, V. S. Varma, C. Rockstuhl, and F. Lederer, “Designing optical elements from isotropic materials by using transformation optics,” Phys. Rev. A |

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

20. | Z. Chang, X. Zhou, J. Hu, and G. Hu, “Design method for quasi-isotropic transformation materials based on inverse Laplace’s equation with sliding boundaries,” Opt. Express |

21. | J. Li, S. Han, S. Zhang, G. Bartal, and X. Zhang, “Designing the Fourier space with transformation optics,” Opt. Lett. |

22. | D. Korobkin, Y. Urzhumov, and G. Shvets, “Enhanced near-field resolution in midinfrared using metamaterials,” J. Opt. Soc. Am. B |

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

**OCIS Codes**

(230.0230) Optical devices : Optical devices

(160.3918) Materials : Metamaterials

**ToC Category:**

Physical Optics

**History**

Original Manuscript: August 1, 2011

Revised Manuscript: September 12, 2011

Manuscript Accepted: September 16, 2011

Published: November 8, 2011

**Citation**

Carlos García-Meca, Alejandro Martínez, and Ulf Leonhardt, "Engineering antenna radiation patterns via quasi-conformal mappings," Opt. Express **19**, 23743-23750 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-24-23743

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### References

- A. Martínez, M. A. Piqueras, and J. Martí, “Generation of highly directional beam by k-space filtering using a metamaterial flat slab with a small negative index of refraction,” Appl. Phys. Lett.89(13), 131111 (2006). [CrossRef]
- A. Martínez, R. García, A. Hakansson, M. A. Piqueras, and J. Sánchez-Dehesa, “Electromagnetic beaming from omnidirectional sources by inverse design,” Appl. Phys. Lett.92(5), 051105 (2008). [CrossRef]
- J. Li, A. Salandrino, and N. Engheta, “Optical spectrometer at the nanoscale using optical Yagi-Uda nanoantennas,” Phys. Rev. B79(19), 195104 (2009). [CrossRef]
- Y. Chen, P. Lodahl, and A. F. Koenderink, “Dynamically reconfigurable directionality of plasmon-based single photon sources,” Phys. Rev. B82(8), 081402 (2010). [CrossRef]
- A. G. Curto, G. Volpe, T. H. Taminiau, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “Unidirectional emission of a quantum dot coupled to a nanoantenna,” Science329(5994), 930–933 (2010). [CrossRef] [PubMed]
- D. Dregely, R. Taubert, J. Dorfmüller, R. Vogelgesang, K. Kern, and H. Giessen, “3D optical Yagi-Uda nanoantenna array,” Nat Commun2, 267 (2011). [CrossRef] [PubMed]
- U. Leonhardt and T. G. Philbin, Geometry and Light: The Science of Invisibility (Dover, 2010).
- F. Kong, B.-I. Wu, J. A. Kong, J. Huangfu, S. Xi, and H. Chen, “Planar focusing antenna design by using coordinate transformation technology,” Appl. Phys. Lett.91(25), 253509 (2007). [CrossRef]
- W. X. Jiang, T. J. Cui, H. F. Ma, X. Y. Zhou, and Q. Cheng, “Cylindrical-to-plane-wave conversion via embedded optical transformation,” Appl. Phys. Lett.92(26), 261903 (2008). [CrossRef]
- Y. Luo, J. Zhang, L. Ran, H. Chen, and J. A. Kong, “Controlling the emission of electromagnetic source,” PIERS4(7), 795–800 (2008). [CrossRef]
- N. Kundtz, D. A. Roberts, J. Allen, S. Cummer, and D. R. Smith, “Optical source transformations,” Opt. Express16(26), 21215–21222 (2008). [CrossRef] [PubMed]
- 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, and A. de Lustrac, “Ultradirective antenna via transformation optics,” J. Appl. Phys.105(10), 104912 (2009). [CrossRef]
- P.-H. Tichit, S. 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]
- U. Leonhardt and T. Tyc, “Superantenna made of transformation media,” New J. Phys.10(11), 115026 (2008). [CrossRef]
- U. Leonhardt, “Optical conformal mapping,” Science312(5781), 1777–1780 (2006). [CrossRef] [PubMed]
- J. P. Turpin, A. T. Massoud, Z. H. Jiang, P. L. Werner, and D. H. Werner, “Conformal mappings to achieve simple material parameters for transformation optics devices,” Opt. Express18(1), 244–252 (2010). [CrossRef] [PubMed]
- M. Schmiele, V. S. Varma, C. Rockstuhl, and F. Lederer, “Designing optical elements from isotropic materials by using transformation optics,” Phys. Rev. A81(3), 033837 (2010). [CrossRef]
- J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett.101(20), 203901 (2008). [CrossRef] [PubMed]
- Z. Chang, X. Zhou, J. Hu, and G. Hu, “Design method for quasi-isotropic transformation materials based on inverse Laplace’s equation with sliding boundaries,” Opt. Express18(6), 6089–6096 (2010). [CrossRef] [PubMed]
- J. Li, S. Han, S. Zhang, G. Bartal, and X. Zhang, “Designing the Fourier space with transformation optics,” Opt. Lett.34(20), 3128–3130 (2009). [CrossRef] [PubMed]
- D. Korobkin, Y. Urzhumov, and G. Shvets, “Enhanced near-field resolution in midinfrared using metamaterials,” J. Opt. Soc. Am. B23(3), 468–478 (2006). [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]

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