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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 26 — Dec. 30, 2013
  • pp: 32313–32326

Transmutation of singularities and zeros in graded index optical instruments: a methodology for designing practical devices

I. R. Hooper and T. G. Philbin  »View Author Affiliations

Optics Express, Vol. 21, Issue 26, pp. 32313-32326 (2013)

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We describe a design methodology for modifying the refractive index profile of graded-index optical instruments that incorporate singularities or zeros in their refractive index. The process maintains the device performance whilst resulting in graded profiles that are all-dielectric, do not require materials with unrealistic values, and that are impedance matched to the bounding medium. This is achieved by transmuting the singularities (or zeros) using the formalism of transformation optics, but with an additional boundary condition requiring the gradient of the co-ordinate transformation be continuous. This additional boundary condition ensures that the device is impedance matched to the bounding medium when the spatially varying permittivity and permeability profiles are scaled to realizable values. We demonstrate the method in some detail for an Eaton lens, before describing the profiles for an “invisible disc” and “multipole” lenses.

© 2013 Optical Society of America

OCIS Codes
(110.2760) Imaging systems : Gradient-index lenses
(120.4570) Instrumentation, measurement, and metrology : Optical design of instruments
(220.3630) Optical design and fabrication : Lenses
(160.3918) Materials : Metamaterials

ToC Category:
Imaging Systems

Original Manuscript: October 15, 2013
Revised Manuscript: November 23, 2013
Manuscript Accepted: November 25, 2013
Published: December 19, 2013

I. R. Hooper and T. G. Philbin, "Transmutation of singularities and zeros in graded index optical instruments: a methodology for designing practical devices," Opt. Express 21, 32313-32326 (2013)

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  1. U. Leonhardt, “Optical conformal mapping,” Science312(5781), 1777–1780 (2006). [CrossRef] [PubMed]
  2. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312(5781), 1780–1782 (2006). [CrossRef] [PubMed]
  3. U. Leonhardt and T. G. Philbin, Geometry and Light: The Science of Invisibility (Dover, 2010).
  4. K. B. Nathan, D. R. Smith, and J. B. Pendry, “Electromagnetic design with transformation optics,” Proc. IEEE99(10), 1622–1633 (2011). [CrossRef]
  5. H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater.9(5), 387–396 (2010). [CrossRef] [PubMed]
  6. W. Cai, U. K. Chettiar, A. V. Kildishev, V. M. Shalaev, and G. W. Milton, “Nonmagnetic cloak with minimized scattering,” Appl. Phys. Lett.91(11), 111105 (2007). [CrossRef]
  7. U. Leonhardt and T. Tyc, “Broadband invisibility by non-Euclidean cloaking,” Science323(5910), 110–112 (2009). [CrossRef] [PubMed]
  8. J. Li and J. B. Pendry, “Hiding under the carpet: A new strategy for cloaking,” Phys. Rev. Lett.101(20), 203901 (2008). [CrossRef] [PubMed]
  9. R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science323(5912), 366–369 (2009). [CrossRef] [PubMed]
  10. J. Perczel, T. Tyc, and U. Leonhardt, “Invisibility cloaking without superluminal propagation,” New J. Phys.13(8), 083007 (2011). [CrossRef]
  11. J. B. Pendry, A. Aubry, D. R. Smith, and S. A. Maier, “Transformation optics and subwavelength control of light,” Science337(6094), 549–552 (2012). [CrossRef] [PubMed]
  12. P.-H. Tichit, S. N. Burokur, and A. De Lustrac, “Ultradirective antenna via transformation optics,” J. Appl. Phys.105(10), 104912 (2009). [CrossRef]
  13. Y. Luo, J. Zhang, H. Chen, J. Huangfu, and L. Ran, “High-directivity antenna with small antenna aperture,” Appl. Phys. Lett.95(19), 193506 (2009). [CrossRef]
  14. 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. Nanostructures6(1), 87–95 (2008). [CrossRef]
  15. R. K. Luneburg, Mathematical Theory of Optics (Univ. of California Press, 1964).
  16. J. E. Eaton, “On spherically symmetric lenses,” Trans. IRE Antennas Propag.4, 66–71 (1952).
  17. J. C. Maxwell, “Problems,” Cambridge Dublin Math. J.8, 188–189 (1854).
  18. N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater.9(2), 129–132 (2010). [CrossRef] [PubMed]
  19. T. Driscoll, G. Lipworth, J. Hunt, N. Landy, N. Kundtz, D. N. Basov, and D. R. Smith, “Performance of a three dimensional transformation-optical-flattened Lüneburg lens,” Opt. Express20(12), 13262–13273 (2012). [CrossRef] [PubMed]
  20. A. Demetriadou and Y. Hao, “Slim Luneburg Lens for Antenna Applications,” Opt. Express19(21), 19925–19934 (2011). [CrossRef] [PubMed]
  21. R. Yang, W. Tang, and Y. Hao, “A broadband zone plate lens from transformation optics,” Opt. Express19(13), 12348–12355 (2011). [CrossRef] [PubMed]
  22. N. Engheta and R. W. Ziolkowski, Metamaterials: Physics and Engineering Explorations (Wiley & Sons, 2006).
  23. C. M. Soukoulis and M. Wegener, “Materials science. Optical metamaterials--more bulky and less lossy,” Science330(6011), 1633–1634 (2010). [CrossRef] [PubMed]
  24. D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science305(5685), 788–792 (2004). [CrossRef] [PubMed]
  25. A. N. Lagarkov and K. N. Rozanov, “High-frequency behavior of magnetic composites,” J. Magn. Magn. Mater.321(14), 2082–2092 (2009). [CrossRef]
  26. T. Tsutaoka, “Frequency dispersion of complex permeability in Mn–Zn and Ni–Zn spinel ferrites and their composite materials,” J. Appl. Phys.93(5), 2789–2796 (2003). [CrossRef]
  27. O. Acher, “Copper vs. iron: Microwave magnetism in the metamaterial age,” J. Magn. Magn. Mater.321(14), 2093–2101 (2009). [CrossRef]
  28. D. Bao, K. Z. Rajab, Y. Hao, E. Kallos, W. Tang, C. Argyropoulos, Y. Piao, and S. Yang, “All-dielectric invisibility cloaks made of BaTiO 3 -loaded polyurethane foam,” New J. Phys.13(10), 103023 (2011). [CrossRef]
  29. N. I. Landy, N. Kundtz, and D. R. Smith, “Designing three-dimensional transformation optical media using quasiconformal coordinate transformations,” Phys. Rev. Lett.105(19), 193902 (2010). [CrossRef] [PubMed]
  30. A. J. Danner, T. Tyc, and U. Leonhardt, “Controlling birefringence in dielectrics,” Nat. Photonics5(6), 357–359 (2011). [CrossRef]
  31. 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]
  32. Y. G. Ma, C. K. Ong, T. Tyc, and U. Leonhardt, “An omnidirectional retroreflector based on the transmutation of dielectric singularities,” Nat. Mater.8(8), 639–642 (2009). [CrossRef] [PubMed]
  33. T. Tyc and U. Leonhardt, “Transmutation of singularities in optical instruments,” New J. Phys.10(11), 115038 (2008). [CrossRef]
  34. J. Perczel, C. Garci-Meca, and U. Leonhardt, “Partial transmutation of singularities in optical instruments,” J. Opt.13(7), 075103 (2011). [CrossRef]
  35. D. F. Sievenpiper, E. Yablonovitch, J. N. Winn, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “3D metallo-dielectric photonic crystals with strong capacitative coupling,” Phys. Rev. Lett.80, 2829–2832 (1998). [CrossRef]
  36. J. T. Shen, P. B. Catrysse, and S. Fan, “Mechanism for designing metallic metamaterials with a high index of refraction,” Phys. Rev. Lett.94(19), 197401 (2005). [CrossRef] [PubMed]
  37. B. Wood and J. B. Pendry, “Metamaterials at zero frequency,” J. Phys. Condens. Matter19(7), 076208 (2007). [CrossRef] [PubMed]
  38. Y. N. Demkov and V. N. Ostrovsky, “Internal symmetry of the Maxwell “fish-eye” problem and the Fock group for the Hydrogen atom,” Sov. Phys. JETP33, 1083 (1971).
  39. T. Tyc, L. Herzánová, M. Šarbot, and K. Bering, “Absolute instruments and perfect imaging in geometrical optics,” New J. Phys.13(11), 115004 (2011). [CrossRef]
  40. P. Benítez, J. C. Miñano, and J. C. González, “Perfect focusing of scalar wave fields in three dimensions,” Opt. Express18(8), 7650–7663 (2010). [CrossRef] [PubMed]

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