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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 24 — Nov. 21, 2011
  • pp: 23743–23750
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Engineering antenna radiation patterns via quasi-conformal mappings

Carlos García-Meca, Alejandro Martínez, and Ulf Leonhardt  »View Author Affiliations


Optics Express, Vol. 19, Issue 24, pp. 23743-23750 (2011)
http://dx.doi.org/10.1364/OE.19.023743


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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

Recently, the use of conformal transformations [16

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

] has been proposed for antenna engineering [17

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

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]

]. Such transformations have the advantage of requiring only isotropic media for their implementation and, for TE polarization (electric field pointing in the direction in which the problem is invariant), only non-magnetic media [7

7. U. Leonhardt and T. G. Philbin, Geometry and Light: The Science of Invisibility (Dover, 2010).

,17

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

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]

]. In [17

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]

], near-zero constitutive parameters arising from a conformal transformation are employed to transform an isotropic source into one, two or four directional beams. In a more general work, the Schwarz-Christoffel transformation was used to map the circle onto a regular polygon with 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]

]. The resulting device distributes equally the power of a point source located at the polygon center among 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

Infinitesimal balls are just scaled and rotated when transformed by a conformal mapping. This is the reason why conformal mappings give rise to isotropic transformation media. Quasi-conformal mappings transform infinitesimal balls to ellipsoids of bounded eccentricity. Thus, transformation media resulting from a quasi-conformal mapping have a bounded anisotropy that can be neglected if it is small enough. Our goal is to change the omnidirectional radiation pattern of a two-dimensional point source. For this purpose, we will consider the transformation of the unit circle (normalized units) in a flat virtual space to another shape in flat physical space. The functionality of the device will be defined solely by the transformation we apply to the boundary of the considered domain in virtual space (the circle in this case). Thus, we are interested in finding a map that simplifies its implementation. Clearly, a conformal mapping fulfils our requirement as it gives rise to isotropic media. However, except for some specific cases, we do not have at our disposal a conformal transformation mapping the unit circle onto the desired shape. To overcome this problem, we will employ quasi-conformal mappings. Since most algorithms used in the calculation of quasi-conformal mappings transform a rectangular region into other shape, first it is necessary to transform the circle into a rectangle. The best way to do this is by using simple conformal transformations with known analytical expressions. Thus, we will follow a two-step method instead of a direct transformation. First, we will transform the unit circle to a square (with side length 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):

q(w)=2i(11F(π/2|1/2)F(π2arcsin(wi)|12)).
(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 = w1 + iw2, with q = q1 + iq2. The refractive index that implements this transformation can be obtained as n1 = |dw/dq| [7

7. U. Leonhardt and T. G. Philbin, Geometry and Light: The Science of Invisibility (Dover, 2010).

]. As for the second step, we use a quasi-conformal mapping z(q1,q2) = x + iy 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(gij)-1/2 and an anisotropic permeability with in-plane components μT and μL in each of the two principal directions, where gij is the contravariant metric in the curved coordinates we want to implement [19

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]

]. To measure the degree of anisotropy, α = max((μT/μL)1/2,(μL/μT)1/2) is usually taken as the anisotropy factor. If α is close to one, μTμL ≈1 and the quasi-conformal mapping can be implemented with negligible error by using only a refractive index distribution n22 = ε = det(gij) -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]

]. In this case, the four sides of the square are mapped to four disjoint specified pieces of the transformed square boundary. The complete transformation refractive index is then given by n = n1n2 [7

7. U. Leonhardt and T. G. Philbin, Geometry and Light: The Science of Invisibility (Dover, 2010).

]. In Fig. 1
Fig. 1 Mapping of the unit circle to an arbitrary shape by using a conformal transformation q(w) followed by a quasi-conformal one z(q1,q2).
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.

Electromagnetic fields follow this transformation, so light rays emanating from a point source located at the center of the unit circle in original space will be normal to the transformed boundary in physical space as well. In order to shape the radiation diagram of the omnidirectional source, we have to orient each little piece of the transformed boundary so that it is perpendicular to the direction towards which we want to redirect the rays crossing that piece. This way we can engineer the angular distribution of the radiated power. This procedure is not exact because of the wave nature of light and the reflections appearing at the transformed circle boundary, since the transformation is not continuous at it. The other limitation is that we do not have full control of the density of rays crossing the transformed circle boundary. We can only decide where to map each fourth of the circle boundary so that we can distribute the radiated power among four desired sets of angular directions, but we cannot specify the angular distribution within each set. Despite the first limitation, the results achieved by this technique are quite accurate. In addition, the second limitation can be overcome to a certain extent as shown below, increasing the degree of control of the angular power distribution.

3. Examples

As a last example, we show that we can also engineer the device to have isotropic radiation within a certain angular range, and not only a set of directional beams. For instance, suppose that we want to have three main lobes radiating in the directions corresponding to 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)
Fig. 4 Generation of a radiation pattern combining high directivity in some directions and isotropic radiation in a desired angular range from an omnidirectional source. (a) Desired boundary of the transformed circle. (b) Resulting quasi-conformal mapping. (c) Refractive index. (d)–(e) Simulated directivity.
]. 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).

For the directional beams, the directivity achieves maximum values of 9.4 dB (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.

There appears a little ripple with amplitude Δ ≈0.7 dB owing to reflections at the boundary and the fact that the density of rays is higher at the center of the arc, as shown in Fig. 4(b). Nevertheless, the directivity is higher than 2 dB approximately between 70° and 110° with a beamwidth of around 52°. This last value is somewhat smaller than expected because of the reasons mentioned above and could be corrected by considering a higher angular region in the specifications or by optimizing the radius of curvature of the mapping boundary, which in general could provide a path for engineering a large variety of radiation patterns.

4. Frequency dependence and implementation

The performance of the proposed devices should be frequency-independent, provided that non-dispersive materials are employed for their implementation. However, there exists a limiting upper wavelength for which the performance of the device begins to deteriorate significantly. This is due to the fact that we used concepts of ray optics in the design of our device (see discussion above). Thus, its behavior should be closer to the desired one at shorter wavelengths. To analyze the frequency dependence of the proposed devices we focused on the example of Fig. 3. In Figs. 5(a)
Fig. 5 (a)–(f) Directivity of the device in Fig. 3 at different wavelengths. (g)–(h) Refractive index and directivity of a modified version of the device in Fig. 3.
5(f) we depict the simulated directivity of this device at different wavelengths. We used the size 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 = 6.6λ). As expected, we observe an improvement of all features at shorter wavelengths. Specifically, for sizes of d larger than 20λ the performance is optimal, reaching directivities around 13 dB and half-power beamwidths as small as 4°. On the other hand, for d = 4λ 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 5λ.

5. Conclusions

Acknowledgments

Financial support by the Spanish Ministerio de Ciencia e Innovación (contract CSD2008-00066 and FPU grant) is gratefully acknowledged.

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. 89(13), 131111 (2006). [CrossRef]

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. 92(5), 051105 (2008). [CrossRef]

3.

J. Li, A. Salandrino, and N. Engheta, “Optical spectrometer at the nanoscale using optical Yagi-Uda nanoantennas,” Phys. Rev. B 79(19), 195104 (2009). [CrossRef]

4.

Y. Chen, P. Lodahl, and A. F. Koenderink, “Dynamically reconfigurable directionality of plasmon-based single photon sources,” Phys. Rev. B 82(8), 081402 (2010). [CrossRef]

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 329(5994), 930–933 (2010). [CrossRef] [PubMed]

6.

D. Dregely, R. Taubert, J. Dorfmüller, R. Vogelgesang, K. Kern, and H. Giessen, “3D optical Yagi-Uda nanoantenna array,” Nat Commun 2, 267 (2011). [CrossRef] [PubMed]

7.

U. Leonhardt and T. G. Philbin, Geometry and Light: The Science of Invisibility (Dover, 2010).

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. 91(25), 253509 (2007). [CrossRef]

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. 92(26), 261903 (2008). [CrossRef]

10.

Y. Luo, J. Zhang, L. Ran, H. Chen, and J. A. Kong, “Controlling the emission of electromagnetic source,” PIERS 4(7), 795–800 (2008). [CrossRef]

11.

N. Kundtz, D. A. Roberts, J. Allen, S. Cummer, and D. R. Smith, “Optical source transformations,” Opt. Express 16(26), 21215–21222 (2008). [CrossRef] [PubMed]

12.

B.-I. Popa, J. Allen, and S. A. Cummer, “Conformal array design with transformation electromagnetics,” Appl. Phys. Lett. 94(24), 244102 (2009). [CrossRef]

13.

P.-H. Tichit, S. N. Burokur, and A. de Lustrac, “Ultradirective antenna via transformation optics,” J. Appl. Phys. 105(10), 104912 (2009). [CrossRef]

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 83(15), 155108 (2011). [CrossRef]

15.

U. Leonhardt and T. Tyc, “Superantenna made of transformation media,” New J. Phys. 10(11), 115026 (2008). [CrossRef]

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]

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]

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]

21.

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]

22.

D. Korobkin, Y. Urzhumov, and G. Shvets, “Enhanced near-field resolution in midinfrared using metamaterials,” J. Opt. Soc. Am. B 23(3), 468–478 (2006). [CrossRef]

23.

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]

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

  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.89(13), 131111 (2006). [CrossRef]
  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.92(5), 051105 (2008). [CrossRef]
  3. J. Li, A. Salandrino, and N. Engheta, “Optical spectrometer at the nanoscale using optical Yagi-Uda nanoantennas,” Phys. Rev. B79(19), 195104 (2009). [CrossRef]
  4. Y. Chen, P. Lodahl, and A. F. Koenderink, “Dynamically reconfigurable directionality of plasmon-based single photon sources,” Phys. Rev. B82(8), 081402 (2010). [CrossRef]
  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,” Science329(5994), 930–933 (2010). [CrossRef] [PubMed]
  6. 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]
  7. U. Leonhardt and T. G. Philbin, Geometry and Light: The Science of Invisibility (Dover, 2010).
  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.91(25), 253509 (2007). [CrossRef]
  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.92(26), 261903 (2008). [CrossRef]
  10. Y. Luo, J. Zhang, L. Ran, H. Chen, and J. A. Kong, “Controlling the emission of electromagnetic source,” PIERS4(7), 795–800 (2008). [CrossRef]
  11. N. Kundtz, D. A. Roberts, J. Allen, S. Cummer, and D. R. Smith, “Optical source transformations,” Opt. Express16(26), 21215–21222 (2008). [CrossRef] [PubMed]
  12. B.-I. Popa, J. Allen, and S. A. Cummer, “Conformal array design with transformation electromagnetics,” Appl. Phys. Lett.94(24), 244102 (2009). [CrossRef]
  13. P.-H. Tichit, S. N. Burokur, and A. de Lustrac, “Ultradirective antenna via transformation optics,” J. Appl. Phys.105(10), 104912 (2009). [CrossRef]
  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. B83(15), 155108 (2011). [CrossRef]
  15. U. Leonhardt and T. Tyc, “Superantenna made of transformation media,” New J. Phys.10(11), 115026 (2008). [CrossRef]
  16. U. Leonhardt, “Optical conformal mapping,” Science312(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. Express18(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. A81(3), 033837 (2010). [CrossRef]
  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]
  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. Express18(6), 6089–6096 (2010). [CrossRef] [PubMed]
  21. 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]
  22. 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]
  23. 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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