Advertisement

Quasi-Conformal Approaches for Two and Three-Dimensional Transformation Optical Media

  • Nathan Landy
  • Yaroslav Urzhumov
  • David R. Smith
Chapter

Abstract

Transformation optical design is generally complicated by the requirement for highly anisotropic and inhomogeneous constituent materials. Quasi-conformal mappings have appeared as an attractive subset of the general transformation optics method because they only require isotropic, dielectric-only materials. In this chapter, we examine the quasi-conformal method as it applies to transformation optics and show that while it does significantly ease the burden of material design and fabrication, it may also create severely aberrant behavior unless caution is taken. We also show how to extend the method to three dimensions, and examine the performance of an optic designed with the quasi-conformal method.

Keywords

Conformal Transformation Physical Domain Quasiconformal Mapping Conformal Module Optical Path Difference 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Pendry JB, Schurig D, Smith DR (2006) Controlling electromagnetic fields. Science 312:1780–1782. doi: 10.1126/science.1125907 MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Schurig D, Pendry JB, Smith DR (2007) Transformation-designed optical elements. Opt Express 15:14772–14782CrossRefGoogle Scholar
  3. 3.
    Liu R, Ji C, Mock JJ, Chin JY, Cui TJ, Smith DR (2009) Broadband ground-plane cloak. Science 323:366–369. doi: 10.1126/science.1166949 CrossRefGoogle Scholar
  4. 4.
    Kundtz N, Smith DR (2010) Extreme-angle broadband metamaterial lens. Nat Mater 9:129–132. doi: 10.1038/Nmat2610 CrossRefGoogle Scholar
  5. 5.
    Hunt J, Kundtz N, Landy N, Nguyen V, Perram T, Starr A, Smith DR (2011) Broadband wide angle lens implemented with dielectric metamaterials. Sensors-Basel 11:7982–7991. doi: 10.3390/S110807982 CrossRefGoogle Scholar
  6. 6.
    Hunt J, Jang G, Smith DR (2011) Perfect relay lens at microwave frequencies based on flattening a maxwell lens. J Opt Soc Am B 28:2025–2029CrossRefGoogle Scholar
  7. 7.
    Valentine J, Li JS, Zentgraf T, Bartal G, Zhang X (2009) An optical cloak made of dielectrics. Nat Mater 8:568–571. doi: 10.1038/Nmat2461 CrossRefGoogle Scholar
  8. 8.
    Gabrielli LH, Cardenas J, Poitras CB, Lipson M (2009) Silicon nanostructure cloak operating at optical frequencies. Nat Photonics 3:461–463. doi: 10.1038/Nphoton.2009.117 CrossRefGoogle Scholar
  9. 9.
    Gharghi M, Gladden C, Zentgraf T, Liu YM, Yin XB, Valentine J, Zhang X (2011) A carpet cloak for visible light. Nano Lett 11:2825–2828. doi: 10.1021/Nl201189z CrossRefGoogle Scholar
  10. 10.
    Hunt J, Tyler T, Dhar S, Tsai YJ, Bowen P, Larouche S, Jokerst NM, Smith DR (2012) Planar, flattened Luneburg lens at infrared wavelengths. Opt Express 20:1706–1713CrossRefGoogle Scholar
  11. 11.
    Ma HF, Cui TJ (2010) Three-dimensional broadband ground-plane cloak made of metamaterials. Nat Commun 1:21. doi: 10.1038/Ncomms1023 Google Scholar
  12. 12.
    Ma HF, Cui TJ (2010) Three-dimensional broadband and broad-angle transformation-optics lens. Nat Commun 1:124. doi: 10.1038/Ncomms1126 CrossRefGoogle Scholar
  13. 13.
    Fischer J, Ergin T, Wegener M (2011) Three-dimensional polarization-independent visible-frequency carpet invisibility cloak. Opt Lett 36:2059–2061CrossRefGoogle Scholar
  14. 14.
    Ergin T, Fischer J, Wegener M (2011) Three-dimensional invisibility carpet cloak at 700 nm wavelengths. In: 2011 Conference on lasers and electro-optics (Cleo)Google Scholar
  15. 15.
    Ergin T, Stenger N, Brenner P, Pendry JB, Wegener M (2010) Three-dimensional invisibility cloak at optical wavelengths. Science 328:337–339. doi: 10.1126/science.1186351 CrossRefGoogle Scholar
  16. 16.
    Xu HY, Zhang BL, Yu TY, Barbastathis G, Sun HD (2012) Dielectric waveguide bending adapter with ideal transmission: practical design strategy of area-preserving affine transformation optics. J Opt Soc Am B 29:1287–1290CrossRefGoogle Scholar
  17. 17.
    Zhang BL, Luo, YA, Liu XG, Barbastathis, G (2011) Macroscopic invisibility cloak for visible light. Phys Rev Lett 106:Artn 033901. doi: 10.1103/Physrevlett.106.033901
  18. 18.
    Chen XZ, Luo Y, Zhang JJ, Jiang K, Pendry JB, Zhang SA (2011) Macroscopic invisibility cloaking of visible light. Nat Commun 2:176. doi: 10.1038/Ncomms1176 CrossRefGoogle Scholar
  19. 19.
    Urzhumov Y, Chen WC, Bingham C, Padilla W, Smith DR (2012) Magnetic levitation of metamaterial bodies enhanced with magnetostatic surface resonances. Phys Rev B 85:054430. doi: 10.1103/Physrevb.85.054430 CrossRefGoogle Scholar
  20. 20.
    Leonhardt U (2006) Optical conformal mapping. Science 312:1777–1780. doi: 10.1126/science.1126493 MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Turpin JP, Massoud AT, Jiang ZH, Werner PL, Werner DH (2010) Conformal mappings to achieve simple material parameters for transformation optics devices. Opt Express 18:244–252CrossRefGoogle Scholar
  22. 22.
    Urzhumov YA, Kundtz NB, Smith DR, Pendry JB (2011) Cross-section comparisons of cloaks designed by transformation optical and optical conformal mapping approaches. J Opt-Uk 13:024002. doi: 10.1088/2040-8978/13/2/024002 CrossRefGoogle Scholar
  23. 23.
    Rahm M, Cummer SA, Schurig D, Pendry JB, Smith DR (2008) Optical design of reflectionless complex media by finite embedded coordinate transformations. Phys Rev Lett 100:063903. doi: 10.1103/Physrevlett.100.063903 CrossRefGoogle Scholar
  24. 24.
    Thompson JF, Soni Bk, Weatherill NP (1999) Handbook of grid generation. CRC Press, Boca RatonzbMATHGoogle Scholar
  25. 25.
    Li JS, Pendry JB (2008) Hiding under the carpet: a new strategy for cloaking. Phys Rev Lett 101:203901. doi: 10.1103/Physrevlett.101.203901 CrossRefGoogle Scholar
  26. 26.
    Knupp P, Steinberg S (1994) Fundamentals of grid generation. CRC Press, Boca RatonzbMATHGoogle Scholar
  27. 27.
    Tang LL, Yin JC, Yuan GS, Du JL, Gao HT, Dong XC, Lu YG, Du CL (2011) General conformal transformation method based on Schwarz-Christoffel approach. Opt Express 19:15119–15126CrossRefGoogle Scholar
  28. 28.
    Chang Z, Zhou XM, Hu J, Hu GK (2010) Design method for quasi-isotropic transformation materials based on inverse Laplace’s equation with sliding boundaries. Opt Express 18:6089–6096CrossRefGoogle Scholar
  29. 29.
    Zhang BL, Chan T, Wu BI (2010) Lateral shift makes a ground-plane cloak detectable. Phys Rev Lett 104:233903. doi: 10.1103/Physrevlett.104.233903 CrossRefGoogle Scholar
  30. 30.
    Tang WX, Argyropoulos C, Kallos E, Song W, Hao Y (2010) Discrete coordinate transformation for designing all-dielectric flat antennas. Ieee T Antenn Propag 58:3795–3804. doi: 10.1109/Tap.2010.2078475 CrossRefGoogle Scholar
  31. 31.
    Kong FM, Wu BII, Kong JA, Huangfu JT, Xi S, Chen HS (2007) Planar focusing antenna design by using coordinate transformation technology. Appl Phys Lett 91:253509 10.1063/1.2826283CrossRefGoogle Scholar
  32. 32.
    Mei ZL, Bai J, Cui TJ (2011) Experimental verification of a broadband planar focusing antenna based on transformation optics. New J Phys 13:063028. doi: 10.1088/1367-2630/13/6/063028 CrossRefGoogle Scholar
  33. 33.
    Garcia-Meca C, Martinez A, Leonhardt U (2011) Engineering antenna radiation patterns via quasi-conformal mappings. Opt Express 19:23743–23750CrossRefGoogle Scholar
  34. 34.
    Roberts DA, Rahm M, Pendry JB, Smith DR (2008) Transformation-optical design of sharp waveguide bends and corners. Appl Phys Lett 93:251111. doi: 10.1063/1.3055604 CrossRefGoogle Scholar
  35. 35.
    Rahm M, Roberts DA, Pendry JB, Smith DR (2008) Transformation-optical design of adaptive beam bends and beam expanders. Opt Express 16:11555–11567CrossRefGoogle Scholar
  36. 36.
    Landy NI, Padilla WJ (2009) Guiding light with conformal transformations. Opt Express 17:14872–14879CrossRefGoogle Scholar
  37. 37.
    Ma YG, Wang N, Ong CK (2010) Application of inverse, strict conformal transformation to design waveguide devices. J Opt Soc Am A 27:968–972CrossRefGoogle Scholar
  38. 38.
    Schurig D (2008) An aberration-free lens with zero F-number. New J Phys 10:115034. doi: 10.1088/1367-2630/10/11/115034 CrossRefGoogle Scholar
  39. 39.
    Luneburg RK, Herzberger M, Brown University. Graduate School (1944) Mathematical theory of optics. Providence, R.IGoogle Scholar
  40. 40.
    Smith DR, Urzhumov Y, Kundtz NB, Landy NI (2010) Enhancing imaging systems using transformation optics. Opt Express 18:21238–21251CrossRefGoogle Scholar
  41. 41.
    Driscoll T, Lipworth G, Hunt J, Landy N, Kundtz N, Basov DN, Smith DR (2012) Performance of a three dimensional transformation-optical-flattened Luneburg lens. Opt Express 20:13262–13273CrossRefGoogle Scholar
  42. 42.
    Birchenhall C (1994) Numerical recipes in C—the art of scientific computing. Econ J 104:725–726CrossRefGoogle Scholar
  43. 43.
    Sluijter M, de Boer DKG, Braat JJM (2008) General polarized ray-tracing method for inhomogeneous uniaxially anisotropic media. J Opt Soc Am A 25:1260–1273CrossRefGoogle Scholar
  44. 44.
    Sluijter M, de Boer DKG, Urbach HP (2009) Ray-optics analysis of inhomogeneous biaxially anisotropic media. J Opt Soc Am A 26:317–329CrossRefGoogle Scholar
  45. 45.
    Damaskos NJ, Maffett AL, Uslenghi PLE (1982) Dispersion-relation for general anisotropic media. Ieee T Antenn Propag 30:991–993CrossRefGoogle Scholar
  46. 46.
    Landy NI, Kundtz N, Smith DR (2010) Designing three-dimensional transformation optical media using quasiconformal coordinate transformations. Phys Rev Lett 105:193902. doi: 10.1103/Physrevlett.105.193902 CrossRefGoogle Scholar
  47. 47.
    Magnus F, Wood B, Moore J, Morrison K, Perkins G, Fyson J, Wiltshire MCK, Caplin D, Cohen LF, Pendry JB (2008) A d.c. magnetic metamaterial. Nat Mater 7:295–297. doi: 10.1038/Nmat2126 CrossRefGoogle Scholar
  48. 48.
    Wood B, Pendry JB (2007) Metamaterials at zero frequency. J Phys-Condens Mat 19:Artn 076208. doi: 10.1088/0953-8984/19/7/076208
  49. 49.
    Gutman AS (1954) Modified luneberg lens. J Appl Phys 25:855–859zbMATHCrossRefGoogle Scholar
  50. 50.
    Morgan SP (1958) General solution of the luneberg lens problem. J Appl Phys 29:1358–1368MathSciNetzbMATHCrossRefGoogle Scholar
  51. 51.
    Schurig D, Pendry JB, Smith DR (2006) Calculation of material properties and ray tracing in transformation media. Opt Express 14:9794–9804CrossRefGoogle Scholar
  52. 52.
    Jacob Z, Alekseyev LV, Narimanov E (2007) Semiclassical theory of the Hyperlens. In: 2007 Conference on Lasers & electro-optics/quantum electronics and laser science (Cleo/Qels 2007), Vols 1–5, pp 2095–2096Google Scholar
  53. 53.
    Halimeh JC, Ergin T, Mueller J, Stenger N, Wegener M (2009) Photorealistic images of carpet cloaks. Opt Express 17:19328–19336CrossRefGoogle Scholar
  54. 54.
    Ergin T, Halimeh JC, Stenger N, Wegener M (2010) Optical microscopy of 3D carpet cloaks:ray-tracing calculations. Opt Express 18:20535–20545CrossRefGoogle Scholar
  55. 55.
    Urzhumov Y, Landy N, Smith DR (2012) Isotropic-medium three-dimensional cloaks for acoustic and electromagnetic waves. J Appl Phys 111:053105. doi: 10.1063/1.3691242 CrossRefGoogle Scholar
  56. 56.
    Balanis CA (1989) Advanced engineering electromagnetics. Wiley, New YorkGoogle Scholar
  57. 57.
    Andrease MG (1965) Scattering from bodies of revolution. Ieee T Antenn Propag 13:303–310CrossRefGoogle Scholar

Copyright information

© Springer-Verlag London 2014

Authors and Affiliations

  1. 1.Center for Metamaterials and Enhanced PlasmonicsDuke UniversityDurhamUSA

Personalised recommendations