Abstract

We present a new technique for the design of transformation-optics devices based on large-scale optimization to achieve the optimal effective isotropic dielectric materials within prescribed index bounds, which is computationally cheap because transformation optics circumvents the need to solve Maxwell’s equations at each step. We apply this technique to the design of multimode waveguide bends (realized experimentally in a previous paper) and mode squeezers, in which all modes are transported equally without scattering. In addition to the optimization, a key point is the identification of the correct boundary conditions to ensure reflectionless coupling to untransformed regions while allowing maximum flexibility in the optimization. Many previous authors in transformation optics used a certain kind of quasiconformal map which overconstrained the problem by requiring that the entire boundary shape be specified a priori while at the same time underconstraining the problem by employing “slipping” boundary conditions that permit unwanted interface reflections.

© 2013 OSA

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. J. Ward and J. B. Pendry, “Refraction and geometry in Maxwell’s equations,” J. Mod. Opt.43(4):773–793 (1996).
    [CrossRef]
  2. U. Leonhardt, “Optical conformal mapping,” Science312:1777–1780 (2006).
    [CrossRef] [PubMed]
  3. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312:1780–1782 (2006).
    [CrossRef] [PubMed]
  4. U. Leonhardt and T. Philbin, Geometry and Light: The Science of Invisibility (Dover, 2010).
  5. H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater.9:387–396 (2010).
    [CrossRef] [PubMed]
  6. A. V. Kildishev and V. M. Shalaev, “Transformation optics and metamaterials,” Phys.-Ups.54(1):53–63 (2011).
  7. Y. Liu and X. Zhang, “Recent advances in transformation optics,” Nanoscale4(17):5277–5292 (2012).
    [CrossRef] [PubMed]
  8. 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]
  9. N. I. Landy and W. J. Padilla, “Guiding light with conformal transformations,” Opt. Express17(17):14872–14879 (2009).
    [CrossRef] [PubMed]
  10. M. Heiblum and J. H. Harris, “Analysis of curved optical waveguides by conformal transformation,” J. Quantum Electron.11(2):75–83 (1975).
    [CrossRef]
  11. Y. G. Ma, N. Wang, and C. K. Ong, “Application of inverse, strict conformal transformation to design waveguide devices,” J. Opt. Soc. Am. A27:968–972 (2010).
    [CrossRef]
  12. S. Han, Y. Xiong, D. Genov, Z. Liu, G. Bartel, and X. Zhang, “Ray optics at a deep-subwavelength scale: A transformation optics approach,” Nano Lett.8(12):4243–4247 (2008).
    [CrossRef]
  13. K. Yao and X. Jiang, “Designing feasible optical devices via conformal mapping,” J. Opt. Soc. Am. B28(5):1037–1042 (2011).
    [CrossRef]
  14. T. Han, C. Qiu, J. Dong, X. Tang, and S. Zouhdi, “Homogeneous and isotropic bends to tunnel waves through multiple different/equal waveguides along arbitrary directions,” Opt. Express19(14):13020–13030 (2011).
    [CrossRef] [PubMed]
  15. 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(251111) (2008).
    [CrossRef]
  16. J. Huangfu, S. Xi, F. Kong, J. Zhang, H. Chen, D. Wang, B. Wu, L. Ran, and J. A. Kong, “Application of coordinate transformation in bent waveguides,” J. Appl. Phys.104(014502) (2008).
    [CrossRef]
  17. H. Xu, B. Zhang, Y. Yu, G. Barbastathis, and H. Sun, “Dielectric waveguide bending adapter with ideal transmission,” Opt. Express29(6):1287–1290 (2012).
  18. Z. L. Mei and T. J. Cui, “Experimental realization of a broadband bend structure using gradient index metamaterials,” Opt. Express17(20):18354–18363 (2009).
    [CrossRef] [PubMed]
  19. Z. L. Mei and T. J. Cui, “Arbitrary bending of electromagnetic waves using isotropic materials,” J. Appl. Phys.105(104913) (2009).
    [CrossRef]
  20. 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]
  21. B. Vasić, G. Isić, R. Gajić, and K. Hingerl, “Coordinate transformation based design of confined metamaterial structures,” Phys. Rev. B79(085103) (2009).
    [CrossRef]
  22. C. García-Meca, M. M. Tung, J. V. Galán, R. Ortuño, F. J. Rodríguez-Fortuño, J. Martí, and A. Martínez, “Squeezing and expanding light without reflections via transformation optics,” Opt. Express19(4):3562–3757 (2011).
    [CrossRef] [PubMed]
  23. X. Zhang, H. Chen, X. Luo, and H. Ma, “Transformation media that turn a narrow slit into a large window,” Opt. Express16(16):11764–11768 (2008).
    [CrossRef] [PubMed]
  24. O. Ozgun and M. Kuzuoglu, “Utilization of anisotropic metamaterial layers in waveguide miniaturization and transitions,” IEEE Microw. Wirel. Compon. Lett.17(754) (2007).
    [CrossRef]
  25. B. Zhang, Y. Luo, X. Liu, and G. Barbastathis, “Macroscopic invisibility cloak for visible light,” Phys. Rev. Lett.106(033901) (2011).
    [CrossRef]
  26. Z. Liang, X. Jiang, F. Miao, S. Guenneau, and J. Li, “Transformation media with variable optical axes,” New J. Phys.14(103042) (2012).
    [CrossRef]
  27. Z. L. Mei, Y. S. Liu, F. Yang, and T. J. Cui, “A DC carpet cloak based on resistor networks,” Opt. Express20(23):25758–25764 (2012).
    [CrossRef] [PubMed]
  28. S. Wang and S. Liu, “Controlling electromagnetic scattering of a cavity by transformation media,” Opt. Express20(6):6777–6785 (2012).
    [CrossRef] [PubMed]
  29. 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]
  30. X. Chen, Y. Luo, J. Zhang, K. Jiang, J. B. Pendry, and S. Zhang, “Macroscopic invisibility cloaking of visible light,” Nat. Commun.2(176) (2011).
    [CrossRef]
  31. J. Mei, Q. Wu, and K. Zhang, “Multimultifunctional complementary cloak with homogeneous anisotropic material parameters,” J. Opt. Soc. Am. A29(10):2067–2073 (2012).
    [CrossRef]
  32. A. V. Novitsky, “Inverse problem in transformation optics,”J. Opt.13(035104) (2011).
    [CrossRef]
  33. J. Li and J. B. Pendry, “Hiding under the carpet: A new strategy for cloaking,” Phys. Rev. Lett101(203901) (2008).
    [CrossRef]
  34. Q. Wu, J. P. Turpin, and D. H. Werner, “Integrated photonic systems based on transformation optics enabled gradient index devices,” Light: Science and Applications1(e38) (2012).
  35. C. Garcia-Meca, A. Martinez, and U. Leonhardt, “Engineering antenna radiation patterns via quasi-conformal mappings,” Opt. Express19(24):23743–23750 (2011).
    [CrossRef] [PubMed]
  36. H. F. Ma and T. J. Cui, “Three-dimensional broadband and broad-angle transformation-optics lens,” Nat. Commun.1(124) (2010).
    [CrossRef] [PubMed]
  37. N. I. Landy, N. Kundtz, and D. R. Smith, “Designing three-dimensional transformation optical media using quasiconformal coordinate transformations,” Phys. Rev. Lett.105(193902) (2010).
    [CrossRef]
  38. Z. L. Mei, J. Bai, and T. J. Cui, “Illusion devices with quasi-conformal mapping,” J. Electromagn. Waves App.24(17):2561–2573 (2010).
    [CrossRef]
  39. N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater.9:129–132 (2010).
    [CrossRef]
  40. R. Liu, R. 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]
  41. J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater.8:568–571 (2009).
    [CrossRef] [PubMed]
  42. L. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics3:461–463 (2009).
    [CrossRef]
  43. T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science328(5976):337–339 (2010).
    [CrossRef] [PubMed]
  44. M. Yin, X. Y. Tian, H. X. Han, and D. C. Li, “Free-space carpet-cloak based on gradient index photonic crystals in metamaterial regime,” Appl. Phys. Lett.100(124101) (2012).
    [CrossRef]
  45. 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]
  46. P. Markov, J. G. Valentine, and S. M. Weiss, “Fiber-to-chip coupler designed using an optical transformation,” Opt. Express20(13):14705–14712 (2012).
    [CrossRef] [PubMed]
  47. L. H. Gabrielli and M. Lipson, “Transformation optics on a silicon platform,” J. Opt.13(024010) (2011).
    [CrossRef]
  48. Z. L. Mei, J. Bai, and T. J. Cui, “Experimental verification of a broadband planar focusing antenna based on transformation optics,” New J. Phys.13(063028) (2011).
    [CrossRef]
  49. L. Tang, J. Yin, G. Yuan, J. Du, H. Gao, X. Dong, Y. Lu, and C. Du, “General conformal transformation method based on Schwarz–Christoffel approach,” Opt. Express19(16):15119–15126 (2011).
    [CrossRef] [PubMed]
  50. J. P. Turpin, A. T. Massoud, Z. H. Jiang, P. L. Werner, and D. H. Wener, “Conformal mappings to achieve simple material parameters for transformation optics devices,” Opt. Express18(1):244–252 (2010).
    [CrossRef] [PubMed]
  51. D. R. Smith, Y. Urzhumov, N. B. Kundtz, and N. I. Landy, “Enhancing imaging systems using transformation optics,” Opt. Express18(20):21238–21251 (2010).
    [CrossRef] [PubMed]
  52. W. R. Frei, H. T. Johnson, and K. D. Choquette, “Optimization of a single defect photonic crystal laser cavity,” J. Appl. Phys.103(033102) (2008).
    [CrossRef]
  53. C. Y. Kao and F. Santosa, “Maximization of the quality factor of an optical resonator,” Wave motion45(4):412–427 (2008).
    [CrossRef]
  54. D. C. Dobson and F. Santosa, “Optimal localization of eigenfunctions in an inhomogeneous medium,” SIAM J. Appl. Math64(3):762–774 (2004).
    [CrossRef]
  55. J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, “Design of photonic crystal microcavities for cavity QED” Phys. Rev. E65(016608) (2002).
  56. J. Lu and J. Vuckovic, “Inverse design of nanophotonic structures using complementary convex optimization,” Opt. Express18(4):3793–3804 (2010).
    [CrossRef] [PubMed]
  57. J. S. Jensen and O. Sigmund, “Topology optimization for nano-photonics,” Laser Photon. Rev.5(2):308–321 (2011).
    [CrossRef]
  58. L. Frandsen, A. Harpøth, P. Borel, M. Kristensen, J. Jensen, and O. Sigmund, “Broadband photonic crystal waveguide 60° bend obtained utilizing topology optimization,” Opt. Express12(24):5916–5921 (2004).
    [CrossRef] [PubMed]
  59. W. R. Frei, H. T. Johnson, and D. A. Tortorelli, “Optimization of photonic nanostructures,” Comput. Method. Appl. M.197(41):3410–3416 (2008).
    [CrossRef]
  60. J. S. Jensen and O. Sigmund, “Systematic design of photonic crystal structures using topology optimization: low-loss waveguide bends,” Appl. Phys. Lett.84(12):2022–2024 (2004).
    [CrossRef]
  61. J. Andkjaer and O. Sigmund, “Topology optimized low-contrast all-dielectric optical cloak,” Appl. Phys. Lett.98(021112) (2011).
    [CrossRef]
  62. P. Borel, A. Harpøth, L. Frandsen, M. Kristensen, P. Shi, J. Jensen, and O. Sigmund, “Topology optimization and fabrication of photonic crystal structures,” Opt. Express12(9):1996 (2004).
    [CrossRef] [PubMed]
  63. S. J. Cox and D. C. Dobson, “Maximizing band gaps in two-dimensional photonic crystals,” SIAM J. Appl. Math59(6):2108–2120 (1999).
    [CrossRef]
  64. J. S. Jensen and O. Sigmund, “Topology optimization of photonic crystal structures: a high-bandwidth low-loss T-junction waveguide,” J. Opt. Soc. Am. B22(6):1191–1198 (2005).
    [CrossRef]
  65. J. Riishede and O. Sigmund, “Inverse design of dispersion compensating optical fiber using topology optimization,” J. Opt. Soc. Am. B25(1):88–97 (2008).
    [CrossRef]
  66. C. Y. Yao, S. Osher, and E. Yablonovitch, “Maximizing band gaps in two-dimensional photonic crystals by using level set methods,” Appl. Phys. B81(2):235–244 (2005).
    [CrossRef]
  67. W. R. Frei, D. A. Tortorelli, and H. T. Johnson, “Topology optimization of a photonic crystal waveguide termination to maximize directional emission,” Appl. Phys. Lett.86(111114) (2005).
    [CrossRef]
  68. Y. Tsuji and K. Hirayama, “Design of optical circuit devices using topology optimization method with function-expansionbased refractive index distribution,” IEEE Phot. Tech. Lett.20(12):982–984 (2008).
    [CrossRef]
  69. C. Y. Kao and S. Osher, “Incorporating topological derivatives into shape derivatives based level set methods,” J. Comp. Phys.225(1):891–909 (2007).
    [CrossRef]
  70. P. Seliger, M. Mahvash, C. Wang, and A. F. J. Levi, “Optimization of aperiodic dielectric structures,” J. Appl. Phys.100(034310) (2006).
    [CrossRef]
  71. Y. Watanabe, N. Ikeda, Y. Sugimoto, Y. Takata, Y. Kitagawa, A. Mizutani, N. Ozaki, and K. Asakawa, “Topology optimization of waveguide bends with wide, flat bandwidth in air–bridge-type photonic crystal slabs,” J. Appl. Phys.101(113108) (2007).
    [CrossRef]
  72. J. H. Lee, J. Blair, V. A. Tamma, Q. Wu, S. J. Rhee, C. J. Summers, and W. Park, “Direct visualization of optical frequency invisibility cloak based on silicon nanorod array,” Opt. Express17(15):12922–12928 (2009).
    [CrossRef] [PubMed]
  73. U. Leonhardt and T. Tyc, “Broadband invisibility by non-euclidian cloaking,” Science323(5910):110–112 (2009).
    [CrossRef]
  74. A. Greenleaf, M. Lassas, and G. Uhlmann, “Anisotropic conductivities that cannot be detected by EIT,” Physiol. Meas.24(413):413–419 (2003).
    [CrossRef] [PubMed]
  75. A. Greenleaf, M. Lassas, and G. Uhlmann, “On nonuniqueness for Calderón’s inverse problem,” Math. Res. Letters10(5):685–693 (2003).
  76. V. Liu and S. Fan, “Compact bends for multi-mode photonic crystal waveguides with high transmission and suppressed modal crosstalk,” Opt. Express21(7):8069–8075 (2013).
    [CrossRef] [PubMed]
  77. A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett.77:3787–3790 (1996).
    [CrossRef] [PubMed]
  78. C. Ma, Q. Zhang, and E. V. Keuren, “Right-angle slot waveguide bends with high bending efficiency,” Opt. Express16(19):14330 (2008).
    [CrossRef] [PubMed]
  79. V. Liu and S. Fan, “Compact bends for multi-mode photonic crystal waveguides with high transmission and suppressed modal crosstalk,” Opt. Express21(7):8069–8075 (2013).
    [CrossRef] [PubMed]
  80. A. Chutinan, M. Okano, and S. Noda, “Wider bandwidth with high transmission through waveguide bends in two-dimensional photonic crystal slabs,” Appl. Phys. Lett.80(10):1698–1699 (2002).
    [CrossRef]
  81. A. Chutinan and S. Noda, “Highly confined waveguides and waveguide bends in three-dimensional photonic crystal,” Appl. Phys. Lett.75(24):3739–3741 (1999).
    [CrossRef]
  82. B. Chen, T. Tang, and H. Chen, “Study on a compact flexible photonic crystal waveguide and its bends,” Opt. Express17(7):5033–5038 (2009).
    [CrossRef] [PubMed]
  83. Y. Zhang and B. Li, “Photonic crystal-based bending waveguides for optical interconnections,” Opt. Express14(12):5723–5732 (2006).
    [CrossRef] [PubMed]
  84. J. Smajic, C. Hafner, and D. Erni, “Design and optimization of an achromatic photonic crystal bend,” Opt. Express11(12):1378–1384 (2003).
    [CrossRef] [PubMed]
  85. M. Schmiele, V. S. Varma, C. Rockstuhl, and F. Lederer, “Designing optical elements from isotropic materials by using transformation optics,” Phys. Rev. A81(033837) (2010).
    [CrossRef]
  86. D. Schurig, J. B. Pendry, and D. R. Smith, “Transformation-designed optical elements,” Opt. Express15(22):14772–14782 (2007).
    [CrossRef] [PubMed]
  87. 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. Lett100(063903) (2008).
    [CrossRef] [PubMed]
  88. D. Smith, J. Mock, A. Starr, and D. Schurig, “Gradient index metamaterials,” Phys. Rev. E71(036609) (2005).
    [CrossRef]
  89. D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science305(5685):788–792 (2004).
    [CrossRef] [PubMed]
  90. F. Xu, R. C. Tyan, P. C. Sun, Y. Fainman, C. C. Cheng, and A. Scherer, “Fabrication, modeling, and characterization of form-birefringent nanostructures,” Opt. Lett.20(24):2457–2459 (1995).
    [CrossRef] [PubMed]
  91. H. Kurt and D. S. Citrin, “Graded index photonic crystals,” Opt. Express15(3):1240–1253 (2007).
    [CrossRef] [PubMed]
  92. B. Vasić, R. Gajić, and K. Hingerl, “Graded photonic crystals for implementation of gradient refractive index media,” J. Nanophotonics5(051806) (2011).
    [CrossRef]
  93. U. Levy, M. Abashin, K. Ikeda, A. Krishnamoorthy, J. Cunningham, and Y. Fainman, “Inhomogenous dielectric metamaterials with space-variant polarizability,” Phys. Rev. Lett.98(243901) (2007).
    [CrossRef]
  94. L. Gabrielli and M. Lipson, “Integrated luneburg lens via ultra-strong index gradient on silicon,” Opt. Express19(21):20122–20127 (2011).
    [CrossRef] [PubMed]
  95. Y. Wang, C. Sheng, H. Liu, Y.J. Zheng, C. Zhu, S. M. Wang, and S. N. Zhu, “Transformation bending device emulated by graded-index waveguide,” Opt. Express20(12):13006–13013 (2012).
    [CrossRef] [PubMed]
  96. R. Ulrich and R. J. Martin, “Geometric optics in thin film light guides,” Appl. Opt.10(9):2077–2085 (1971).
    [CrossRef] [PubMed]
  97. F. Zernike, “Luneburg lens for optical waveguide use,” Opt. Commun.12(4):379–381 (1974).
    [CrossRef]
  98. B. U. Chen, E. Marom, and A. Lee, “Geodesic lenses in single-mode LiNbO3waveguides,” Appl. Phys. Lett.31(4):263 (1977).
    [CrossRef]
  99. J. Brazas, G. Kohnke, and J. McMullen, “Mode-index waveguide lens with novel gradient boundaries developed for application to optical recording,” Appl. Opt.31(18):3420–3428 (1992).
    [CrossRef] [PubMed]
  100. W. Rudin, Real and Complex Analysis (McGraw–Hill, 1986).
  101. G. E. Shilov, Elementary Real and Complex Analysis (MIT, 1973).
  102. J. Li and J. B. Pendry, Private communication (2013).
  103. W. Yan, M. Yan, and M. Qiu, “Necessary and sufficient conditions for reflectionless transformation media in an isotropic and homogenous background,” arXiv:0806.3231 (2008).
  104. L. Bergamin, “Electromagnetic fields and boundary conditions at the interface of generalized transformation media,” Phys. Rev. A80(063835) (2009).
    [CrossRef]
  105. W. Yan, M. Yan, Z. Ruan, and M. Qiu, “Coordinate transformations make perfect invisibility cloaks with arbitrary shape,” New J. Phys.10(043040) (2008).
    [CrossRef]
  106. O. Weber, A. Myles, and D. Zorin, “Computing extremal quasiconformal maps,” Symp. Geom. Process.31(5):1679–1689 (2012).
  107. K. Astala, T. Iwaniec, and G. Martin, Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane (Princeton University, 2008).
  108. A. Papadopoulos, editor. Handbook of Teichmüller Theory, volume 1 (European Mathematical Society, 2007).
    [CrossRef]
  109. R. Kühnau, editor. Handbook of Complex Analysis: Geometric Function Theory, volume 2 (Elsevier B.V., 2005).
  110. L. Bers, “An extremal problem for quasi-conformal mappings and a problem of Thurston,” Acta Math.73–98 (1978).
    [CrossRef]
  111. L. V. Ahlfors, Lectures on quasiconformal mappings (American Mathematical Society, 1966).
  112. Z. Balogh, K. Fässler, and I. Platis, “Modulus of curve families and extremality of spiral-stretch maps,” J. Anal. Math.113(1):265–291 (2011).
    [CrossRef]
  113. K. Astala, T. Iwaniec, and G. Martin, “Deformations of annuli with smallest mean distortion,” Arch. Rational Mech. Anal.195:899–921 (2010).
    [CrossRef]
  114. Z. Balogh, K. Fässler, and I. Platis, “Modulus method and radial stretch map in the Heisenberg group,” Ann. Acad. Sci. Fenn.38(1):149–180 (2013).
    [CrossRef]
  115. J. P. Boyd, Chebyshev and Fourier Spectral Methods (Dover, 2001).
  116. S. Boyd and L. Vandenberghe, Convex Optimization (Cambridge University, 2004).
  117. L. Gabrielli, D. Liu, S. G. Johnson, and M. Lipson, “On-chip transformation optics for multimode waveguide bends,” Nat. Commun.3(1217) (2012).
    [CrossRef] [PubMed]
  118. U. Leonhardt and T. G. Philbin, “Transformation optics and the geometry of light,” Prog. Opt.53:69–152 (2009).
    [CrossRef]
  119. J. D. Jackson, Classical Electrodynamics (Wiley, 1998).
  120. J. Plebanski, “Electromagnetic waves in gravitational fields,” Phys. Rev.118(5):1396–1408 (1960).
    [CrossRef]
  121. U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” New J. Phys.8(10) (2006).
    [CrossRef]
  122. L. Ahlfors, “On quasiconformal mappings. J. Anal. Math.3(1):1–58 (1953).
    [CrossRef]
  123. W. Zeng, F. Luo, S. T. Yau, and X. D. Gu, “Surface quasi-conformal mapping by solving Beltrami equations,” Proc. Math. SurfacesXIII391–408 (2009).
    [CrossRef]
  124. J. F. Thompson, B. K. Soni, and N. P. Weatherill, Handbook of Grid Generation (CRC, 1999).
  125. P. Knupp and S. Steinberg, Fundamentals of Grid Generation (CRC, 1994).
  126. S. G. Johnson, M. L. Povinelli, M. Soljačić, A. Karalis, S. Jacobs, and J. D. Joannopoulos, “Roughness losses and volume-current methods in photonic-crystal waveguides,” Appl. Phys. B81(283–293) (2005).
    [CrossRef]
  127. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).
  128. W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE, 1995).
  129. M. J. D. Powell, “A direct search optimization method that models the objective and constraint functions by linear interpolation,” Adv. Optim. Numer. Anal. (1994).
    [CrossRef]
  130. M. J. D. Powell, “Direct search algorithms for optimization calculations,” Acta Numer.7:287–336 (1998).
    [CrossRef]
  131. S. G. Johnson, The NLopt nonlinear-optimization package ( http://ab-initio.mit.edu/nlopt ) (2007).
  132. A. Logg, K. A. Mardal, and G. N. Wells, Automated Solution of Differential Equations by the Finite Element Method (Springer, 2012).
    [CrossRef]
  133. A. Oskooi, A. Mutapcic, S. Noda, J. D. Joannopoulos, S. P. Boyd, and S. G. Johnson, “Robust optimization of adiabatic tapers for coupling to slow-light photonic-crystal waveguides,” Opt. Express20(19):21558–21575 (2012).
    [CrossRef] [PubMed]
  134. A. Mutapcica, S. Boyd, A. Farjadpour, S. G. Johnson, and Y. Avniel, “Robust design of slow-light tapers in periodic waveguides”. Eng. Optim.41(4):365–384 (2009).
    [CrossRef]
  135. T. F. Chan, J. Cong, T. Kong, and J. R. Shinnerl, “Multilevel optimization for large-scale circuit placement,” IEEE ICAD171–176 (2000).
  136. K. W. Chun and J. Ra, “Fast block-matching algorithm by successive refinement of matching criterion,” Proc. SPIE, Vis. Commun. Image Process., 1818:552–560 (1992).
  137. S. G. Johnson, P. Bienstman, M. A. Skorobogatiy, M. Ibanescu, E. Lidorikis, and J. D. Joannopoulos, “Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals,” Phys. Rev. E66(066608) (2002).
    [CrossRef]
  138. C. Manolatou, S. G. Johnson, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “High-density integrated optics,” J. Lightwave Technol.17(9):1682–1692 (1999).
    [CrossRef]

2013

2012

L. Gabrielli, D. Liu, S. G. Johnson, and M. Lipson, “On-chip transformation optics for multimode waveguide bends,” Nat. Commun.3(1217) (2012).
[CrossRef] [PubMed]

O. Weber, A. Myles, and D. Zorin, “Computing extremal quasiconformal maps,” Symp. Geom. Process.31(5):1679–1689 (2012).

Y. Wang, C. Sheng, H. Liu, Y.J. Zheng, C. Zhu, S. M. Wang, and S. N. Zhu, “Transformation bending device emulated by graded-index waveguide,” Opt. Express20(12):13006–13013 (2012).
[CrossRef] [PubMed]

A. Oskooi, A. Mutapcic, S. Noda, J. D. Joannopoulos, S. P. Boyd, and S. G. Johnson, “Robust optimization of adiabatic tapers for coupling to slow-light photonic-crystal waveguides,” Opt. Express20(19):21558–21575 (2012).
[CrossRef] [PubMed]

Y. Liu and X. Zhang, “Recent advances in transformation optics,” Nanoscale4(17):5277–5292 (2012).
[CrossRef] [PubMed]

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]

H. Xu, B. Zhang, Y. Yu, G. Barbastathis, and H. Sun, “Dielectric waveguide bending adapter with ideal transmission,” Opt. Express29(6):1287–1290 (2012).

Z. Liang, X. Jiang, F. Miao, S. Guenneau, and J. Li, “Transformation media with variable optical axes,” New J. Phys.14(103042) (2012).
[CrossRef]

Z. L. Mei, Y. S. Liu, F. Yang, and T. J. Cui, “A DC carpet cloak based on resistor networks,” Opt. Express20(23):25758–25764 (2012).
[CrossRef] [PubMed]

S. Wang and S. Liu, “Controlling electromagnetic scattering of a cavity by transformation media,” Opt. Express20(6):6777–6785 (2012).
[CrossRef] [PubMed]

Q. Wu, J. P. Turpin, and D. H. Werner, “Integrated photonic systems based on transformation optics enabled gradient index devices,” Light: Science and Applications1(e38) (2012).

J. Mei, Q. Wu, and K. Zhang, “Multimultifunctional complementary cloak with homogeneous anisotropic material parameters,” J. Opt. Soc. Am. A29(10):2067–2073 (2012).
[CrossRef]

M. Yin, X. Y. Tian, H. X. Han, and D. C. Li, “Free-space carpet-cloak based on gradient index photonic crystals in metamaterial regime,” Appl. Phys. Lett.100(124101) (2012).
[CrossRef]

P. Markov, J. G. Valentine, and S. M. Weiss, “Fiber-to-chip coupler designed using an optical transformation,” Opt. Express20(13):14705–14712 (2012).
[CrossRef] [PubMed]

2011

L. H. Gabrielli and M. Lipson, “Transformation optics on a silicon platform,” J. Opt.13(024010) (2011).
[CrossRef]

Z. L. Mei, J. Bai, and T. J. Cui, “Experimental verification of a broadband planar focusing antenna based on transformation optics,” New J. Phys.13(063028) (2011).
[CrossRef]

L. Tang, J. Yin, G. Yuan, J. Du, H. Gao, X. Dong, Y. Lu, and C. Du, “General conformal transformation method based on Schwarz–Christoffel approach,” Opt. Express19(16):15119–15126 (2011).
[CrossRef] [PubMed]

A. V. Novitsky, “Inverse problem in transformation optics,”J. Opt.13(035104) (2011).
[CrossRef]

C. García-Meca, M. M. Tung, J. V. Galán, R. Ortuño, F. J. Rodríguez-Fortuño, J. Martí, and A. Martínez, “Squeezing and expanding light without reflections via transformation optics,” Opt. Express19(4):3562–3757 (2011).
[CrossRef] [PubMed]

B. Zhang, Y. Luo, X. Liu, and G. Barbastathis, “Macroscopic invisibility cloak for visible light,” Phys. Rev. Lett.106(033901) (2011).
[CrossRef]

X. Chen, Y. Luo, J. Zhang, K. Jiang, J. B. Pendry, and S. Zhang, “Macroscopic invisibility cloaking of visible light,” Nat. Commun.2(176) (2011).
[CrossRef]

J. S. Jensen and O. Sigmund, “Topology optimization for nano-photonics,” Laser Photon. Rev.5(2):308–321 (2011).
[CrossRef]

J. Andkjaer and O. Sigmund, “Topology optimized low-contrast all-dielectric optical cloak,” Appl. Phys. Lett.98(021112) (2011).
[CrossRef]

C. Garcia-Meca, A. Martinez, and U. Leonhardt, “Engineering antenna radiation patterns via quasi-conformal mappings,” Opt. Express19(24):23743–23750 (2011).
[CrossRef] [PubMed]

A. V. Kildishev and V. M. Shalaev, “Transformation optics and metamaterials,” Phys.-Ups.54(1):53–63 (2011).

K. Yao and X. Jiang, “Designing feasible optical devices via conformal mapping,” J. Opt. Soc. Am. B28(5):1037–1042 (2011).
[CrossRef]

T. Han, C. Qiu, J. Dong, X. Tang, and S. Zouhdi, “Homogeneous and isotropic bends to tunnel waves through multiple different/equal waveguides along arbitrary directions,” Opt. Express19(14):13020–13030 (2011).
[CrossRef] [PubMed]

Z. Balogh, K. Fässler, and I. Platis, “Modulus of curve families and extremality of spiral-stretch maps,” J. Anal. Math.113(1):265–291 (2011).
[CrossRef]

B. Vasić, R. Gajić, and K. Hingerl, “Graded photonic crystals for implementation of gradient refractive index media,” J. Nanophotonics5(051806) (2011).
[CrossRef]

L. Gabrielli and M. Lipson, “Integrated luneburg lens via ultra-strong index gradient on silicon,” Opt. Express19(21):20122–20127 (2011).
[CrossRef] [PubMed]

2010

M. Schmiele, V. S. Varma, C. Rockstuhl, and F. Lederer, “Designing optical elements from isotropic materials by using transformation optics,” Phys. Rev. A81(033837) (2010).
[CrossRef]

K. Astala, T. Iwaniec, and G. Martin, “Deformations of annuli with smallest mean distortion,” Arch. Rational Mech. Anal.195:899–921 (2010).
[CrossRef]

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater.9:387–396 (2010).
[CrossRef] [PubMed]

Y. G. Ma, N. Wang, and C. K. Ong, “Application of inverse, strict conformal transformation to design waveguide devices,” J. Opt. Soc. Am. A27:968–972 (2010).
[CrossRef]

H. F. Ma and T. J. Cui, “Three-dimensional broadband and broad-angle transformation-optics lens,” Nat. Commun.1(124) (2010).
[CrossRef] [PubMed]

N. I. Landy, N. Kundtz, and D. R. Smith, “Designing three-dimensional transformation optical media using quasiconformal coordinate transformations,” Phys. Rev. Lett.105(193902) (2010).
[CrossRef]

Z. L. Mei, J. Bai, and T. J. Cui, “Illusion devices with quasi-conformal mapping,” J. Electromagn. Waves App.24(17):2561–2573 (2010).
[CrossRef]

N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater.9:129–132 (2010).
[CrossRef]

T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science328(5976):337–339 (2010).
[CrossRef] [PubMed]

J. Lu and J. Vuckovic, “Inverse design of nanophotonic structures using complementary convex optimization,” Opt. Express18(4):3793–3804 (2010).
[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. P. Turpin, A. T. Massoud, Z. H. Jiang, P. L. Werner, and D. H. Wener, “Conformal mappings to achieve simple material parameters for transformation optics devices,” Opt. Express18(1):244–252 (2010).
[CrossRef] [PubMed]

D. R. Smith, Y. Urzhumov, N. B. Kundtz, and N. I. Landy, “Enhancing imaging systems using transformation optics,” Opt. Express18(20):21238–21251 (2010).
[CrossRef] [PubMed]

2009

J. H. Lee, J. Blair, V. A. Tamma, Q. Wu, S. J. Rhee, C. J. Summers, and W. Park, “Direct visualization of optical frequency invisibility cloak based on silicon nanorod array,” Opt. Express17(15):12922–12928 (2009).
[CrossRef] [PubMed]

U. Leonhardt and T. Tyc, “Broadband invisibility by non-euclidian cloaking,” Science323(5910):110–112 (2009).
[CrossRef]

R. Liu, R. 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]

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater.8:568–571 (2009).
[CrossRef] [PubMed]

L. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics3:461–463 (2009).
[CrossRef]

B. Vasić, G. Isić, R. Gajić, and K. Hingerl, “Coordinate transformation based design of confined metamaterial structures,” Phys. Rev. B79(085103) (2009).
[CrossRef]

N. I. Landy and W. J. Padilla, “Guiding light with conformal transformations,” Opt. Express17(17):14872–14879 (2009).
[CrossRef] [PubMed]

Z. L. Mei and T. J. Cui, “Experimental realization of a broadband bend structure using gradient index metamaterials,” Opt. Express17(20):18354–18363 (2009).
[CrossRef] [PubMed]

Z. L. Mei and T. J. Cui, “Arbitrary bending of electromagnetic waves using isotropic materials,” J. Appl. Phys.105(104913) (2009).
[CrossRef]

A. Mutapcica, S. Boyd, A. Farjadpour, S. G. Johnson, and Y. Avniel, “Robust design of slow-light tapers in periodic waveguides”. Eng. Optim.41(4):365–384 (2009).
[CrossRef]

W. Zeng, F. Luo, S. T. Yau, and X. D. Gu, “Surface quasi-conformal mapping by solving Beltrami equations,” Proc. Math. SurfacesXIII391–408 (2009).
[CrossRef]

B. Chen, T. Tang, and H. Chen, “Study on a compact flexible photonic crystal waveguide and its bends,” Opt. Express17(7):5033–5038 (2009).
[CrossRef] [PubMed]

L. Bergamin, “Electromagnetic fields and boundary conditions at the interface of generalized transformation media,” Phys. Rev. A80(063835) (2009).
[CrossRef]

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

2008

W. Yan, M. Yan, Z. Ruan, and M. Qiu, “Coordinate transformations make perfect invisibility cloaks with arbitrary shape,” New J. Phys.10(043040) (2008).
[CrossRef]

C. Ma, Q. Zhang, and E. V. Keuren, “Right-angle slot waveguide bends with high bending efficiency,” Opt. Express16(19):14330 (2008).
[CrossRef] [PubMed]

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. Lett100(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]

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(251111) (2008).
[CrossRef]

J. Huangfu, S. Xi, F. Kong, J. Zhang, H. Chen, D. Wang, B. Wu, L. Ran, and J. A. Kong, “Application of coordinate transformation in bent waveguides,” J. Appl. Phys.104(014502) (2008).
[CrossRef]

S. Han, Y. Xiong, D. Genov, Z. Liu, G. Bartel, and X. Zhang, “Ray optics at a deep-subwavelength scale: A transformation optics approach,” Nano Lett.8(12):4243–4247 (2008).
[CrossRef]

J. Riishede and O. Sigmund, “Inverse design of dispersion compensating optical fiber using topology optimization,” J. Opt. Soc. Am. B25(1):88–97 (2008).
[CrossRef]

Y. Tsuji and K. Hirayama, “Design of optical circuit devices using topology optimization method with function-expansionbased refractive index distribution,” IEEE Phot. Tech. Lett.20(12):982–984 (2008).
[CrossRef]

W. R. Frei, H. T. Johnson, and D. A. Tortorelli, “Optimization of photonic nanostructures,” Comput. Method. Appl. M.197(41):3410–3416 (2008).
[CrossRef]

W. R. Frei, H. T. Johnson, and K. D. Choquette, “Optimization of a single defect photonic crystal laser cavity,” J. Appl. Phys.103(033102) (2008).
[CrossRef]

C. Y. Kao and F. Santosa, “Maximization of the quality factor of an optical resonator,” Wave motion45(4):412–427 (2008).
[CrossRef]

J. Li and J. B. Pendry, “Hiding under the carpet: A new strategy for cloaking,” Phys. Rev. Lett101(203901) (2008).
[CrossRef]

X. Zhang, H. Chen, X. Luo, and H. Ma, “Transformation media that turn a narrow slit into a large window,” Opt. Express16(16):11764–11768 (2008).
[CrossRef] [PubMed]

2007

O. Ozgun and M. Kuzuoglu, “Utilization of anisotropic metamaterial layers in waveguide miniaturization and transitions,” IEEE Microw. Wirel. Compon. Lett.17(754) (2007).
[CrossRef]

C. Y. Kao and S. Osher, “Incorporating topological derivatives into shape derivatives based level set methods,” J. Comp. Phys.225(1):891–909 (2007).
[CrossRef]

H. Kurt and D. S. Citrin, “Graded index photonic crystals,” Opt. Express15(3):1240–1253 (2007).
[CrossRef] [PubMed]

D. Schurig, J. B. Pendry, and D. R. Smith, “Transformation-designed optical elements,” Opt. Express15(22):14772–14782 (2007).
[CrossRef] [PubMed]

Y. Watanabe, N. Ikeda, Y. Sugimoto, Y. Takata, Y. Kitagawa, A. Mizutani, N. Ozaki, and K. Asakawa, “Topology optimization of waveguide bends with wide, flat bandwidth in air–bridge-type photonic crystal slabs,” J. Appl. Phys.101(113108) (2007).
[CrossRef]

U. Levy, M. Abashin, K. Ikeda, A. Krishnamoorthy, J. Cunningham, and Y. Fainman, “Inhomogenous dielectric metamaterials with space-variant polarizability,” Phys. Rev. Lett.98(243901) (2007).
[CrossRef]

2006

Y. Zhang and B. Li, “Photonic crystal-based bending waveguides for optical interconnections,” Opt. Express14(12):5723–5732 (2006).
[CrossRef] [PubMed]

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

P. Seliger, M. Mahvash, C. Wang, and A. F. J. Levi, “Optimization of aperiodic dielectric structures,” J. Appl. Phys.100(034310) (2006).
[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]

U. Leonhardt, “Optical conformal mapping,” Science312:1777–1780 (2006).
[CrossRef] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312:1780–1782 (2006).
[CrossRef] [PubMed]

2005

C. Y. Yao, S. Osher, and E. Yablonovitch, “Maximizing band gaps in two-dimensional photonic crystals by using level set methods,” Appl. Phys. B81(2):235–244 (2005).
[CrossRef]

W. R. Frei, D. A. Tortorelli, and H. T. Johnson, “Topology optimization of a photonic crystal waveguide termination to maximize directional emission,” Appl. Phys. Lett.86(111114) (2005).
[CrossRef]

S. G. Johnson, M. L. Povinelli, M. Soljačić, A. Karalis, S. Jacobs, and J. D. Joannopoulos, “Roughness losses and volume-current methods in photonic-crystal waveguides,” Appl. Phys. B81(283–293) (2005).
[CrossRef]

J. S. Jensen and O. Sigmund, “Topology optimization of photonic crystal structures: a high-bandwidth low-loss T-junction waveguide,” J. Opt. Soc. Am. B22(6):1191–1198 (2005).
[CrossRef]

D. Smith, J. Mock, A. Starr, and D. Schurig, “Gradient index metamaterials,” Phys. Rev. E71(036609) (2005).
[CrossRef]

2004

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science305(5685):788–792 (2004).
[CrossRef] [PubMed]

J. S. Jensen and O. Sigmund, “Systematic design of photonic crystal structures using topology optimization: low-loss waveguide bends,” Appl. Phys. Lett.84(12):2022–2024 (2004).
[CrossRef]

P. Borel, A. Harpøth, L. Frandsen, M. Kristensen, P. Shi, J. Jensen, and O. Sigmund, “Topology optimization and fabrication of photonic crystal structures,” Opt. Express12(9):1996 (2004).
[CrossRef] [PubMed]

L. Frandsen, A. Harpøth, P. Borel, M. Kristensen, J. Jensen, and O. Sigmund, “Broadband photonic crystal waveguide 60° bend obtained utilizing topology optimization,” Opt. Express12(24):5916–5921 (2004).
[CrossRef] [PubMed]

D. C. Dobson and F. Santosa, “Optimal localization of eigenfunctions in an inhomogeneous medium,” SIAM J. Appl. Math64(3):762–774 (2004).
[CrossRef]

2003

A. Greenleaf, M. Lassas, and G. Uhlmann, “Anisotropic conductivities that cannot be detected by EIT,” Physiol. Meas.24(413):413–419 (2003).
[CrossRef] [PubMed]

A. Greenleaf, M. Lassas, and G. Uhlmann, “On nonuniqueness for Calderón’s inverse problem,” Math. Res. Letters10(5):685–693 (2003).

J. Smajic, C. Hafner, and D. Erni, “Design and optimization of an achromatic photonic crystal bend,” Opt. Express11(12):1378–1384 (2003).
[CrossRef] [PubMed]

2002

A. Chutinan, M. Okano, and S. Noda, “Wider bandwidth with high transmission through waveguide bends in two-dimensional photonic crystal slabs,” Appl. Phys. Lett.80(10):1698–1699 (2002).
[CrossRef]

S. G. Johnson, P. Bienstman, M. A. Skorobogatiy, M. Ibanescu, E. Lidorikis, and J. D. Joannopoulos, “Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals,” Phys. Rev. E66(066608) (2002).
[CrossRef]

J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, “Design of photonic crystal microcavities for cavity QED” Phys. Rev. E65(016608) (2002).

2000

T. F. Chan, J. Cong, T. Kong, and J. R. Shinnerl, “Multilevel optimization for large-scale circuit placement,” IEEE ICAD171–176 (2000).

1999

C. Manolatou, S. G. Johnson, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “High-density integrated optics,” J. Lightwave Technol.17(9):1682–1692 (1999).
[CrossRef]

A. Chutinan and S. Noda, “Highly confined waveguides and waveguide bends in three-dimensional photonic crystal,” Appl. Phys. Lett.75(24):3739–3741 (1999).
[CrossRef]

S. J. Cox and D. C. Dobson, “Maximizing band gaps in two-dimensional photonic crystals,” SIAM J. Appl. Math59(6):2108–2120 (1999).
[CrossRef]

1998

M. J. D. Powell, “Direct search algorithms for optimization calculations,” Acta Numer.7:287–336 (1998).
[CrossRef]

1996

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett.77:3787–3790 (1996).
[CrossRef] [PubMed]

A. J. Ward and J. B. Pendry, “Refraction and geometry in Maxwell’s equations,” J. Mod. Opt.43(4):773–793 (1996).
[CrossRef]

1995

1994

M. J. D. Powell, “A direct search optimization method that models the objective and constraint functions by linear interpolation,” Adv. Optim. Numer. Anal. (1994).
[CrossRef]

1992

K. W. Chun and J. Ra, “Fast block-matching algorithm by successive refinement of matching criterion,” Proc. SPIE, Vis. Commun. Image Process., 1818:552–560 (1992).

J. Brazas, G. Kohnke, and J. McMullen, “Mode-index waveguide lens with novel gradient boundaries developed for application to optical recording,” Appl. Opt.31(18):3420–3428 (1992).
[CrossRef] [PubMed]

1978

L. Bers, “An extremal problem for quasi-conformal mappings and a problem of Thurston,” Acta Math.73–98 (1978).
[CrossRef]

1977

B. U. Chen, E. Marom, and A. Lee, “Geodesic lenses in single-mode LiNbO3waveguides,” Appl. Phys. Lett.31(4):263 (1977).
[CrossRef]

1975

M. Heiblum and J. H. Harris, “Analysis of curved optical waveguides by conformal transformation,” J. Quantum Electron.11(2):75–83 (1975).
[CrossRef]

1974

F. Zernike, “Luneburg lens for optical waveguide use,” Opt. Commun.12(4):379–381 (1974).
[CrossRef]

1971

1966

L. V. Ahlfors, Lectures on quasiconformal mappings (American Mathematical Society, 1966).

1960

J. Plebanski, “Electromagnetic waves in gravitational fields,” Phys. Rev.118(5):1396–1408 (1960).
[CrossRef]

1953

L. Ahlfors, “On quasiconformal mappings. J. Anal. Math.3(1):1–58 (1953).
[CrossRef]

Abashin, M.

U. Levy, M. Abashin, K. Ikeda, A. Krishnamoorthy, J. Cunningham, and Y. Fainman, “Inhomogenous dielectric metamaterials with space-variant polarizability,” Phys. Rev. Lett.98(243901) (2007).
[CrossRef]

Ahlfors, L.

L. Ahlfors, “On quasiconformal mappings. J. Anal. Math.3(1):1–58 (1953).
[CrossRef]

Ahlfors, L. V.

L. V. Ahlfors, Lectures on quasiconformal mappings (American Mathematical Society, 1966).

Andkjaer, J.

J. Andkjaer and O. Sigmund, “Topology optimized low-contrast all-dielectric optical cloak,” Appl. Phys. Lett.98(021112) (2011).
[CrossRef]

Asakawa, K.

Y. Watanabe, N. Ikeda, Y. Sugimoto, Y. Takata, Y. Kitagawa, A. Mizutani, N. Ozaki, and K. Asakawa, “Topology optimization of waveguide bends with wide, flat bandwidth in air–bridge-type photonic crystal slabs,” J. Appl. Phys.101(113108) (2007).
[CrossRef]

Astala, K.

K. Astala, T. Iwaniec, and G. Martin, “Deformations of annuli with smallest mean distortion,” Arch. Rational Mech. Anal.195:899–921 (2010).
[CrossRef]

K. Astala, T. Iwaniec, and G. Martin, Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane (Princeton University, 2008).

Aubry, A.

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]

Avniel, Y.

A. Mutapcica, S. Boyd, A. Farjadpour, S. G. Johnson, and Y. Avniel, “Robust design of slow-light tapers in periodic waveguides”. Eng. Optim.41(4):365–384 (2009).
[CrossRef]

Bai, J.

Z. L. Mei, J. Bai, and T. J. Cui, “Experimental verification of a broadband planar focusing antenna based on transformation optics,” New J. Phys.13(063028) (2011).
[CrossRef]

Z. L. Mei, J. Bai, and T. J. Cui, “Illusion devices with quasi-conformal mapping,” J. Electromagn. Waves App.24(17):2561–2573 (2010).
[CrossRef]

Balogh, Z.

Z. Balogh, K. Fässler, and I. Platis, “Modulus method and radial stretch map in the Heisenberg group,” Ann. Acad. Sci. Fenn.38(1):149–180 (2013).
[CrossRef]

Z. Balogh, K. Fässler, and I. Platis, “Modulus of curve families and extremality of spiral-stretch maps,” J. Anal. Math.113(1):265–291 (2011).
[CrossRef]

Barbastathis, G.

H. Xu, B. Zhang, Y. Yu, G. Barbastathis, and H. Sun, “Dielectric waveguide bending adapter with ideal transmission,” Opt. Express29(6):1287–1290 (2012).

B. Zhang, Y. Luo, X. Liu, and G. Barbastathis, “Macroscopic invisibility cloak for visible light,” Phys. Rev. Lett.106(033901) (2011).
[CrossRef]

Bartal, G.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater.8:568–571 (2009).
[CrossRef] [PubMed]

Bartel, G.

S. Han, Y. Xiong, D. Genov, Z. Liu, G. Bartel, and X. Zhang, “Ray optics at a deep-subwavelength scale: A transformation optics approach,” Nano Lett.8(12):4243–4247 (2008).
[CrossRef]

Bergamin, L.

L. Bergamin, “Electromagnetic fields and boundary conditions at the interface of generalized transformation media,” Phys. Rev. A80(063835) (2009).
[CrossRef]

Bers, L.

L. Bers, “An extremal problem for quasi-conformal mappings and a problem of Thurston,” Acta Math.73–98 (1978).
[CrossRef]

Bienstman, P.

S. G. Johnson, P. Bienstman, M. A. Skorobogatiy, M. Ibanescu, E. Lidorikis, and J. D. Joannopoulos, “Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals,” Phys. Rev. E66(066608) (2002).
[CrossRef]

Blair, J.

Borel, P.

Boyd, J. P.

J. P. Boyd, Chebyshev and Fourier Spectral Methods (Dover, 2001).

Boyd, S.

A. Mutapcica, S. Boyd, A. Farjadpour, S. G. Johnson, and Y. Avniel, “Robust design of slow-light tapers in periodic waveguides”. Eng. Optim.41(4):365–384 (2009).
[CrossRef]

S. Boyd and L. Vandenberghe, Convex Optimization (Cambridge University, 2004).

Boyd, S. P.

Brazas, J.

Brenner, P.

T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science328(5976):337–339 (2010).
[CrossRef] [PubMed]

Cardenas, J.

L. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics3:461–463 (2009).
[CrossRef]

Chan, C. T.

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater.9:387–396 (2010).
[CrossRef] [PubMed]

Chan, T. F.

T. F. Chan, J. Cong, T. Kong, and J. R. Shinnerl, “Multilevel optimization for large-scale circuit placement,” IEEE ICAD171–176 (2000).

Chang, Z.

Chen, B.

Chen, B. U.

B. U. Chen, E. Marom, and A. Lee, “Geodesic lenses in single-mode LiNbO3waveguides,” Appl. Phys. Lett.31(4):263 (1977).
[CrossRef]

Chen, H.

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater.9:387–396 (2010).
[CrossRef] [PubMed]

B. Chen, T. Tang, and H. Chen, “Study on a compact flexible photonic crystal waveguide and its bends,” Opt. Express17(7):5033–5038 (2009).
[CrossRef] [PubMed]

X. Zhang, H. Chen, X. Luo, and H. Ma, “Transformation media that turn a narrow slit into a large window,” Opt. Express16(16):11764–11768 (2008).
[CrossRef] [PubMed]

J. Huangfu, S. Xi, F. Kong, J. Zhang, H. Chen, D. Wang, B. Wu, L. Ran, and J. A. Kong, “Application of coordinate transformation in bent waveguides,” J. Appl. Phys.104(014502) (2008).
[CrossRef]

Chen, J. C.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett.77:3787–3790 (1996).
[CrossRef] [PubMed]

Chen, X.

X. Chen, Y. Luo, J. Zhang, K. Jiang, J. B. Pendry, and S. Zhang, “Macroscopic invisibility cloaking of visible light,” Nat. Commun.2(176) (2011).
[CrossRef]

Cheng, C. C.

Chew, W. C.

W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE, 1995).

Chin, J. Y.

R. Liu, R. 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]

Choquette, K. D.

W. R. Frei, H. T. Johnson, and K. D. Choquette, “Optimization of a single defect photonic crystal laser cavity,” J. Appl. Phys.103(033102) (2008).
[CrossRef]

Chun, K. W.

K. W. Chun and J. Ra, “Fast block-matching algorithm by successive refinement of matching criterion,” Proc. SPIE, Vis. Commun. Image Process., 1818:552–560 (1992).

Chutinan, A.

A. Chutinan, M. Okano, and S. Noda, “Wider bandwidth with high transmission through waveguide bends in two-dimensional photonic crystal slabs,” Appl. Phys. Lett.80(10):1698–1699 (2002).
[CrossRef]

A. Chutinan and S. Noda, “Highly confined waveguides and waveguide bends in three-dimensional photonic crystal,” Appl. Phys. Lett.75(24):3739–3741 (1999).
[CrossRef]

Citrin, D. S.

Cong, J.

T. F. Chan, J. Cong, T. Kong, and J. R. Shinnerl, “Multilevel optimization for large-scale circuit placement,” IEEE ICAD171–176 (2000).

Cox, S. J.

S. J. Cox and D. C. Dobson, “Maximizing band gaps in two-dimensional photonic crystals,” SIAM J. Appl. Math59(6):2108–2120 (1999).
[CrossRef]

Cui, T. J.

Z. L. Mei, Y. S. Liu, F. Yang, and T. J. Cui, “A DC carpet cloak based on resistor networks,” Opt. Express20(23):25758–25764 (2012).
[CrossRef] [PubMed]

Z. L. Mei, J. Bai, and T. J. Cui, “Experimental verification of a broadband planar focusing antenna based on transformation optics,” New J. Phys.13(063028) (2011).
[CrossRef]

H. F. Ma and T. J. Cui, “Three-dimensional broadband and broad-angle transformation-optics lens,” Nat. Commun.1(124) (2010).
[CrossRef] [PubMed]

Z. L. Mei, J. Bai, and T. J. Cui, “Illusion devices with quasi-conformal mapping,” J. Electromagn. Waves App.24(17):2561–2573 (2010).
[CrossRef]

R. Liu, R. 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]

Z. L. Mei and T. J. Cui, “Arbitrary bending of electromagnetic waves using isotropic materials,” J. Appl. Phys.105(104913) (2009).
[CrossRef]

Z. L. Mei and T. J. Cui, “Experimental realization of a broadband bend structure using gradient index metamaterials,” Opt. Express17(20):18354–18363 (2009).
[CrossRef] [PubMed]

Cummer, S. A.

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. Lett100(063903) (2008).
[CrossRef] [PubMed]

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]

Cunningham, J.

U. Levy, M. Abashin, K. Ikeda, A. Krishnamoorthy, J. Cunningham, and Y. Fainman, “Inhomogenous dielectric metamaterials with space-variant polarizability,” Phys. Rev. Lett.98(243901) (2007).
[CrossRef]

Dobson, D. C.

D. C. Dobson and F. Santosa, “Optimal localization of eigenfunctions in an inhomogeneous medium,” SIAM J. Appl. Math64(3):762–774 (2004).
[CrossRef]

S. J. Cox and D. C. Dobson, “Maximizing band gaps in two-dimensional photonic crystals,” SIAM J. Appl. Math59(6):2108–2120 (1999).
[CrossRef]

Dong, J.

Dong, X.

Du, C.

Du, J.

Ergin, T.

T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science328(5976):337–339 (2010).
[CrossRef] [PubMed]

Erni, D.

Fainman, Y.

U. Levy, M. Abashin, K. Ikeda, A. Krishnamoorthy, J. Cunningham, and Y. Fainman, “Inhomogenous dielectric metamaterials with space-variant polarizability,” Phys. Rev. Lett.98(243901) (2007).
[CrossRef]

F. Xu, R. C. Tyan, P. C. Sun, Y. Fainman, C. C. Cheng, and A. Scherer, “Fabrication, modeling, and characterization of form-birefringent nanostructures,” Opt. Lett.20(24):2457–2459 (1995).
[CrossRef] [PubMed]

Fan, S.

Farjadpour, A.

A. Mutapcica, S. Boyd, A. Farjadpour, S. G. Johnson, and Y. Avniel, “Robust design of slow-light tapers in periodic waveguides”. Eng. Optim.41(4):365–384 (2009).
[CrossRef]

Fässler, K.

Z. Balogh, K. Fässler, and I. Platis, “Modulus method and radial stretch map in the Heisenberg group,” Ann. Acad. Sci. Fenn.38(1):149–180 (2013).
[CrossRef]

Z. Balogh, K. Fässler, and I. Platis, “Modulus of curve families and extremality of spiral-stretch maps,” J. Anal. Math.113(1):265–291 (2011).
[CrossRef]

Frandsen, L.

Frei, W. R.

W. R. Frei, H. T. Johnson, and D. A. Tortorelli, “Optimization of photonic nanostructures,” Comput. Method. Appl. M.197(41):3410–3416 (2008).
[CrossRef]

W. R. Frei, H. T. Johnson, and K. D. Choquette, “Optimization of a single defect photonic crystal laser cavity,” J. Appl. Phys.103(033102) (2008).
[CrossRef]

W. R. Frei, D. A. Tortorelli, and H. T. Johnson, “Topology optimization of a photonic crystal waveguide termination to maximize directional emission,” Appl. Phys. Lett.86(111114) (2005).
[CrossRef]

Gabrielli, L.

L. Gabrielli, D. Liu, S. G. Johnson, and M. Lipson, “On-chip transformation optics for multimode waveguide bends,” Nat. Commun.3(1217) (2012).
[CrossRef] [PubMed]

L. Gabrielli and M. Lipson, “Integrated luneburg lens via ultra-strong index gradient on silicon,” Opt. Express19(21):20122–20127 (2011).
[CrossRef] [PubMed]

L. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics3:461–463 (2009).
[CrossRef]

Gabrielli, L. H.

L. H. Gabrielli and M. Lipson, “Transformation optics on a silicon platform,” J. Opt.13(024010) (2011).
[CrossRef]

Gajic, R.

B. Vasić, R. Gajić, and K. Hingerl, “Graded photonic crystals for implementation of gradient refractive index media,” J. Nanophotonics5(051806) (2011).
[CrossRef]

B. Vasić, G. Isić, R. Gajić, and K. Hingerl, “Coordinate transformation based design of confined metamaterial structures,” Phys. Rev. B79(085103) (2009).
[CrossRef]

Galán, J. V.

Gao, H.

Garcia-Meca, C.

García-Meca, C.

Genov, D.

S. Han, Y. Xiong, D. Genov, Z. Liu, G. Bartel, and X. Zhang, “Ray optics at a deep-subwavelength scale: A transformation optics approach,” Nano Lett.8(12):4243–4247 (2008).
[CrossRef]

Greenleaf, A.

A. Greenleaf, M. Lassas, and G. Uhlmann, “Anisotropic conductivities that cannot be detected by EIT,” Physiol. Meas.24(413):413–419 (2003).
[CrossRef] [PubMed]

A. Greenleaf, M. Lassas, and G. Uhlmann, “On nonuniqueness for Calderón’s inverse problem,” Math. Res. Letters10(5):685–693 (2003).

Gu, X. D.

W. Zeng, F. Luo, S. T. Yau, and X. D. Gu, “Surface quasi-conformal mapping by solving Beltrami equations,” Proc. Math. SurfacesXIII391–408 (2009).
[CrossRef]

Guenneau, S.

Z. Liang, X. Jiang, F. Miao, S. Guenneau, and J. Li, “Transformation media with variable optical axes,” New J. Phys.14(103042) (2012).
[CrossRef]

Hafner, C.

Han, H. X.

M. Yin, X. Y. Tian, H. X. Han, and D. C. Li, “Free-space carpet-cloak based on gradient index photonic crystals in metamaterial regime,” Appl. Phys. Lett.100(124101) (2012).
[CrossRef]

Han, S.

S. Han, Y. Xiong, D. Genov, Z. Liu, G. Bartel, and X. Zhang, “Ray optics at a deep-subwavelength scale: A transformation optics approach,” Nano Lett.8(12):4243–4247 (2008).
[CrossRef]

Han, T.

Harpøth, A.

Harris, J. H.

M. Heiblum and J. H. Harris, “Analysis of curved optical waveguides by conformal transformation,” J. Quantum Electron.11(2):75–83 (1975).
[CrossRef]

Haus, H. A.

Heiblum, M.

M. Heiblum and J. H. Harris, “Analysis of curved optical waveguides by conformal transformation,” J. Quantum Electron.11(2):75–83 (1975).
[CrossRef]

Hingerl, K.

B. Vasić, R. Gajić, and K. Hingerl, “Graded photonic crystals for implementation of gradient refractive index media,” J. Nanophotonics5(051806) (2011).
[CrossRef]

B. Vasić, G. Isić, R. Gajić, and K. Hingerl, “Coordinate transformation based design of confined metamaterial structures,” Phys. Rev. B79(085103) (2009).
[CrossRef]

Hirayama, K.

Y. Tsuji and K. Hirayama, “Design of optical circuit devices using topology optimization method with function-expansionbased refractive index distribution,” IEEE Phot. Tech. Lett.20(12):982–984 (2008).
[CrossRef]

Hu, G.

Hu, J.

Huangfu, J.

J. Huangfu, S. Xi, F. Kong, J. Zhang, H. Chen, D. Wang, B. Wu, L. Ran, and J. A. Kong, “Application of coordinate transformation in bent waveguides,” J. Appl. Phys.104(014502) (2008).
[CrossRef]

Ibanescu, M.

S. G. Johnson, P. Bienstman, M. A. Skorobogatiy, M. Ibanescu, E. Lidorikis, and J. D. Joannopoulos, “Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals,” Phys. Rev. E66(066608) (2002).
[CrossRef]

Ikeda, K.

U. Levy, M. Abashin, K. Ikeda, A. Krishnamoorthy, J. Cunningham, and Y. Fainman, “Inhomogenous dielectric metamaterials with space-variant polarizability,” Phys. Rev. Lett.98(243901) (2007).
[CrossRef]

Ikeda, N.

Y. Watanabe, N. Ikeda, Y. Sugimoto, Y. Takata, Y. Kitagawa, A. Mizutani, N. Ozaki, and K. Asakawa, “Topology optimization of waveguide bends with wide, flat bandwidth in air–bridge-type photonic crystal slabs,” J. Appl. Phys.101(113108) (2007).
[CrossRef]

Isic, G.

B. Vasić, G. Isić, R. Gajić, and K. Hingerl, “Coordinate transformation based design of confined metamaterial structures,” Phys. Rev. B79(085103) (2009).
[CrossRef]

Iwaniec, T.

K. Astala, T. Iwaniec, and G. Martin, “Deformations of annuli with smallest mean distortion,” Arch. Rational Mech. Anal.195:899–921 (2010).
[CrossRef]

K. Astala, T. Iwaniec, and G. Martin, Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane (Princeton University, 2008).

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, 1998).

Jacobs, S.

S. G. Johnson, M. L. Povinelli, M. Soljačić, A. Karalis, S. Jacobs, and J. D. Joannopoulos, “Roughness losses and volume-current methods in photonic-crystal waveguides,” Appl. Phys. B81(283–293) (2005).
[CrossRef]

Jensen, J.

Jensen, J. S.

J. S. Jensen and O. Sigmund, “Topology optimization for nano-photonics,” Laser Photon. Rev.5(2):308–321 (2011).
[CrossRef]

J. S. Jensen and O. Sigmund, “Topology optimization of photonic crystal structures: a high-bandwidth low-loss T-junction waveguide,” J. Opt. Soc. Am. B22(6):1191–1198 (2005).
[CrossRef]

J. S. Jensen and O. Sigmund, “Systematic design of photonic crystal structures using topology optimization: low-loss waveguide bends,” Appl. Phys. Lett.84(12):2022–2024 (2004).
[CrossRef]

Ji, R. C.

R. Liu, R. 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]

Jiang, K.

X. Chen, Y. Luo, J. Zhang, K. Jiang, J. B. Pendry, and S. Zhang, “Macroscopic invisibility cloaking of visible light,” Nat. Commun.2(176) (2011).
[CrossRef]

Jiang, X.

Z. Liang, X. Jiang, F. Miao, S. Guenneau, and J. Li, “Transformation media with variable optical axes,” New J. Phys.14(103042) (2012).
[CrossRef]

K. Yao and X. Jiang, “Designing feasible optical devices via conformal mapping,” J. Opt. Soc. Am. B28(5):1037–1042 (2011).
[CrossRef]

Jiang, Z. H.

Joannopoulos, J. D.

A. Oskooi, A. Mutapcic, S. Noda, J. D. Joannopoulos, S. P. Boyd, and S. G. Johnson, “Robust optimization of adiabatic tapers for coupling to slow-light photonic-crystal waveguides,” Opt. Express20(19):21558–21575 (2012).
[CrossRef] [PubMed]

S. G. Johnson, M. L. Povinelli, M. Soljačić, A. Karalis, S. Jacobs, and J. D. Joannopoulos, “Roughness losses and volume-current methods in photonic-crystal waveguides,” Appl. Phys. B81(283–293) (2005).
[CrossRef]

S. G. Johnson, P. Bienstman, M. A. Skorobogatiy, M. Ibanescu, E. Lidorikis, and J. D. Joannopoulos, “Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals,” Phys. Rev. E66(066608) (2002).
[CrossRef]

C. Manolatou, S. G. Johnson, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “High-density integrated optics,” J. Lightwave Technol.17(9):1682–1692 (1999).
[CrossRef]

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett.77:3787–3790 (1996).
[CrossRef] [PubMed]

Johnson, H. T.

W. R. Frei, H. T. Johnson, and K. D. Choquette, “Optimization of a single defect photonic crystal laser cavity,” J. Appl. Phys.103(033102) (2008).
[CrossRef]

W. R. Frei, H. T. Johnson, and D. A. Tortorelli, “Optimization of photonic nanostructures,” Comput. Method. Appl. M.197(41):3410–3416 (2008).
[CrossRef]

W. R. Frei, D. A. Tortorelli, and H. T. Johnson, “Topology optimization of a photonic crystal waveguide termination to maximize directional emission,” Appl. Phys. Lett.86(111114) (2005).
[CrossRef]

Johnson, S. G.

A. Oskooi, A. Mutapcic, S. Noda, J. D. Joannopoulos, S. P. Boyd, and S. G. Johnson, “Robust optimization of adiabatic tapers for coupling to slow-light photonic-crystal waveguides,” Opt. Express20(19):21558–21575 (2012).
[CrossRef] [PubMed]

L. Gabrielli, D. Liu, S. G. Johnson, and M. Lipson, “On-chip transformation optics for multimode waveguide bends,” Nat. Commun.3(1217) (2012).
[CrossRef] [PubMed]

A. Mutapcica, S. Boyd, A. Farjadpour, S. G. Johnson, and Y. Avniel, “Robust design of slow-light tapers in periodic waveguides”. Eng. Optim.41(4):365–384 (2009).
[CrossRef]

S. G. Johnson, M. L. Povinelli, M. Soljačić, A. Karalis, S. Jacobs, and J. D. Joannopoulos, “Roughness losses and volume-current methods in photonic-crystal waveguides,” Appl. Phys. B81(283–293) (2005).
[CrossRef]

S. G. Johnson, P. Bienstman, M. A. Skorobogatiy, M. Ibanescu, E. Lidorikis, and J. D. Joannopoulos, “Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals,” Phys. Rev. E66(066608) (2002).
[CrossRef]

C. Manolatou, S. G. Johnson, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “High-density integrated optics,” J. Lightwave Technol.17(9):1682–1692 (1999).
[CrossRef]

Justice, B. J.

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]

Kao, C. Y.

C. Y. Kao and F. Santosa, “Maximization of the quality factor of an optical resonator,” Wave motion45(4):412–427 (2008).
[CrossRef]

C. Y. Kao and S. Osher, “Incorporating topological derivatives into shape derivatives based level set methods,” J. Comp. Phys.225(1):891–909 (2007).
[CrossRef]

Karalis, A.

S. G. Johnson, M. L. Povinelli, M. Soljačić, A. Karalis, S. Jacobs, and J. D. Joannopoulos, “Roughness losses and volume-current methods in photonic-crystal waveguides,” Appl. Phys. B81(283–293) (2005).
[CrossRef]

Keuren, E. V.

Kildishev, A. V.

A. V. Kildishev and V. M. Shalaev, “Transformation optics and metamaterials,” Phys.-Ups.54(1):53–63 (2011).

Kitagawa, Y.

Y. Watanabe, N. Ikeda, Y. Sugimoto, Y. Takata, Y. Kitagawa, A. Mizutani, N. Ozaki, and K. Asakawa, “Topology optimization of waveguide bends with wide, flat bandwidth in air–bridge-type photonic crystal slabs,” J. Appl. Phys.101(113108) (2007).
[CrossRef]

Knupp, P.

P. Knupp and S. Steinberg, Fundamentals of Grid Generation (CRC, 1994).

Kohnke, G.

Kong, F.

J. Huangfu, S. Xi, F. Kong, J. Zhang, H. Chen, D. Wang, B. Wu, L. Ran, and J. A. Kong, “Application of coordinate transformation in bent waveguides,” J. Appl. Phys.104(014502) (2008).
[CrossRef]

Kong, J. A.

J. Huangfu, S. Xi, F. Kong, J. Zhang, H. Chen, D. Wang, B. Wu, L. Ran, and J. A. Kong, “Application of coordinate transformation in bent waveguides,” J. Appl. Phys.104(014502) (2008).
[CrossRef]

Kong, T.

T. F. Chan, J. Cong, T. Kong, and J. R. Shinnerl, “Multilevel optimization for large-scale circuit placement,” IEEE ICAD171–176 (2000).

Krishnamoorthy, A.

U. Levy, M. Abashin, K. Ikeda, A. Krishnamoorthy, J. Cunningham, and Y. Fainman, “Inhomogenous dielectric metamaterials with space-variant polarizability,” Phys. Rev. Lett.98(243901) (2007).
[CrossRef]

Kristensen, M.

Kundtz, N.

N. I. Landy, N. Kundtz, and D. R. Smith, “Designing three-dimensional transformation optical media using quasiconformal coordinate transformations,” Phys. Rev. Lett.105(193902) (2010).
[CrossRef]

N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater.9:129–132 (2010).
[CrossRef]

Kundtz, N. B.

Kurland, I.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett.77:3787–3790 (1996).
[CrossRef] [PubMed]

Kurt, H.

Kuzuoglu, M.

O. Ozgun and M. Kuzuoglu, “Utilization of anisotropic metamaterial layers in waveguide miniaturization and transitions,” IEEE Microw. Wirel. Compon. Lett.17(754) (2007).
[CrossRef]

Landy, N. I.

Lassas, M.

A. Greenleaf, M. Lassas, and G. Uhlmann, “Anisotropic conductivities that cannot be detected by EIT,” Physiol. Meas.24(413):413–419 (2003).
[CrossRef] [PubMed]

A. Greenleaf, M. Lassas, and G. Uhlmann, “On nonuniqueness for Calderón’s inverse problem,” Math. Res. Letters10(5):685–693 (2003).

Lederer, F.

M. Schmiele, V. S. Varma, C. Rockstuhl, and F. Lederer, “Designing optical elements from isotropic materials by using transformation optics,” Phys. Rev. A81(033837) (2010).
[CrossRef]

Lee, A.

B. U. Chen, E. Marom, and A. Lee, “Geodesic lenses in single-mode LiNbO3waveguides,” Appl. Phys. Lett.31(4):263 (1977).
[CrossRef]

Lee, J. H.

Leonhardt, U.

C. Garcia-Meca, A. Martinez, and U. Leonhardt, “Engineering antenna radiation patterns via quasi-conformal mappings,” Opt. Express19(24):23743–23750 (2011).
[CrossRef] [PubMed]

U. Leonhardt and T. Tyc, “Broadband invisibility by non-euclidian cloaking,” Science323(5910):110–112 (2009).
[CrossRef]

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

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

U. Leonhardt, “Optical conformal mapping,” Science312:1777–1780 (2006).
[CrossRef] [PubMed]

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

Levi, A. F. J.

P. Seliger, M. Mahvash, C. Wang, and A. F. J. Levi, “Optimization of aperiodic dielectric structures,” J. Appl. Phys.100(034310) (2006).
[CrossRef]

Levy, U.

U. Levy, M. Abashin, K. Ikeda, A. Krishnamoorthy, J. Cunningham, and Y. Fainman, “Inhomogenous dielectric metamaterials with space-variant polarizability,” Phys. Rev. Lett.98(243901) (2007).
[CrossRef]

Li, B.

Li, D. C.

M. Yin, X. Y. Tian, H. X. Han, and D. C. Li, “Free-space carpet-cloak based on gradient index photonic crystals in metamaterial regime,” Appl. Phys. Lett.100(124101) (2012).
[CrossRef]

Li, J.

J. Li and J. B. Pendry, Private communication (2013).

Z. Liang, X. Jiang, F. Miao, S. Guenneau, and J. Li, “Transformation media with variable optical axes,” New J. Phys.14(103042) (2012).
[CrossRef]

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater.8:568–571 (2009).
[CrossRef] [PubMed]

J. Li and J. B. Pendry, “Hiding under the carpet: A new strategy for cloaking,” Phys. Rev. Lett101(203901) (2008).
[CrossRef]

Liang, Z.

Z. Liang, X. Jiang, F. Miao, S. Guenneau, and J. Li, “Transformation media with variable optical axes,” New J. Phys.14(103042) (2012).
[CrossRef]

Lidorikis, E.

S. G. Johnson, P. Bienstman, M. A. Skorobogatiy, M. Ibanescu, E. Lidorikis, and J. D. Joannopoulos, “Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals,” Phys. Rev. E66(066608) (2002).
[CrossRef]

Lipson, M.

L. Gabrielli, D. Liu, S. G. Johnson, and M. Lipson, “On-chip transformation optics for multimode waveguide bends,” Nat. Commun.3(1217) (2012).
[CrossRef] [PubMed]

L. H. Gabrielli and M. Lipson, “Transformation optics on a silicon platform,” J. Opt.13(024010) (2011).
[CrossRef]

L. Gabrielli and M. Lipson, “Integrated luneburg lens via ultra-strong index gradient on silicon,” Opt. Express19(21):20122–20127 (2011).
[CrossRef] [PubMed]

L. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics3:461–463 (2009).
[CrossRef]

Liu, D.

L. Gabrielli, D. Liu, S. G. Johnson, and M. Lipson, “On-chip transformation optics for multimode waveguide bends,” Nat. Commun.3(1217) (2012).
[CrossRef] [PubMed]

Liu, H.

Liu, R.

R. Liu, R. 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]

Liu, S.

Liu, V.

Liu, X.

B. Zhang, Y. Luo, X. Liu, and G. Barbastathis, “Macroscopic invisibility cloak for visible light,” Phys. Rev. Lett.106(033901) (2011).
[CrossRef]

Liu, Y.

Y. Liu and X. Zhang, “Recent advances in transformation optics,” Nanoscale4(17):5277–5292 (2012).
[CrossRef] [PubMed]

Liu, Y. S.

Liu, Z.

S. Han, Y. Xiong, D. Genov, Z. Liu, G. Bartel, and X. Zhang, “Ray optics at a deep-subwavelength scale: A transformation optics approach,” Nano Lett.8(12):4243–4247 (2008).
[CrossRef]

Logg, A.

A. Logg, K. A. Mardal, and G. N. Wells, Automated Solution of Differential Equations by the Finite Element Method (Springer, 2012).
[CrossRef]

Loncar, M.

J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, “Design of photonic crystal microcavities for cavity QED” Phys. Rev. E65(016608) (2002).

Love, J. D.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).

Lu, J.

Lu, Y.

Luo, F.

W. Zeng, F. Luo, S. T. Yau, and X. D. Gu, “Surface quasi-conformal mapping by solving Beltrami equations,” Proc. Math. SurfacesXIII391–408 (2009).
[CrossRef]

Luo, X.

Luo, Y.

B. Zhang, Y. Luo, X. Liu, and G. Barbastathis, “Macroscopic invisibility cloak for visible light,” Phys. Rev. Lett.106(033901) (2011).
[CrossRef]

X. Chen, Y. Luo, J. Zhang, K. Jiang, J. B. Pendry, and S. Zhang, “Macroscopic invisibility cloaking of visible light,” Nat. Commun.2(176) (2011).
[CrossRef]

Ma, C.

Ma, H.

Ma, H. F.

H. F. Ma and T. J. Cui, “Three-dimensional broadband and broad-angle transformation-optics lens,” Nat. Commun.1(124) (2010).
[CrossRef] [PubMed]

Ma, Y. G.

Mabuchi, H.

J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, “Design of photonic crystal microcavities for cavity QED” Phys. Rev. E65(016608) (2002).

Mahvash, M.

P. Seliger, M. Mahvash, C. Wang, and A. F. J. Levi, “Optimization of aperiodic dielectric structures,” J. Appl. Phys.100(034310) (2006).
[CrossRef]

Maier, S. A.

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]

Manolatou, C.

Mardal, K. A.

A. Logg, K. A. Mardal, and G. N. Wells, Automated Solution of Differential Equations by the Finite Element Method (Springer, 2012).
[CrossRef]

Markov, P.

Marom, E.

B. U. Chen, E. Marom, and A. Lee, “Geodesic lenses in single-mode LiNbO3waveguides,” Appl. Phys. Lett.31(4):263 (1977).
[CrossRef]

Martí, J.

Martin, G.

K. Astala, T. Iwaniec, and G. Martin, “Deformations of annuli with smallest mean distortion,” Arch. Rational Mech. Anal.195:899–921 (2010).
[CrossRef]

K. Astala, T. Iwaniec, and G. Martin, Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane (Princeton University, 2008).

Martin, R. J.

Martinez, A.

Martínez, A.

Massoud, A. T.

McMullen, J.

Mei, J.

Mei, Z. L.

Z. L. Mei, Y. S. Liu, F. Yang, and T. J. Cui, “A DC carpet cloak based on resistor networks,” Opt. Express20(23):25758–25764 (2012).
[CrossRef] [PubMed]

Z. L. Mei, J. Bai, and T. J. Cui, “Experimental verification of a broadband planar focusing antenna based on transformation optics,” New J. Phys.13(063028) (2011).
[CrossRef]

Z. L. Mei, J. Bai, and T. J. Cui, “Illusion devices with quasi-conformal mapping,” J. Electromagn. Waves App.24(17):2561–2573 (2010).
[CrossRef]

Z. L. Mei and T. J. Cui, “Arbitrary bending of electromagnetic waves using isotropic materials,” J. Appl. Phys.105(104913) (2009).
[CrossRef]

Z. L. Mei and T. J. Cui, “Experimental realization of a broadband bend structure using gradient index metamaterials,” Opt. Express17(20):18354–18363 (2009).
[CrossRef] [PubMed]

Mekis, A.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett.77:3787–3790 (1996).
[CrossRef] [PubMed]

Miao, F.

Z. Liang, X. Jiang, F. Miao, S. Guenneau, and J. Li, “Transformation media with variable optical axes,” New J. Phys.14(103042) (2012).
[CrossRef]

Mizutani, A.

Y. Watanabe, N. Ikeda, Y. Sugimoto, Y. Takata, Y. Kitagawa, A. Mizutani, N. Ozaki, and K. Asakawa, “Topology optimization of waveguide bends with wide, flat bandwidth in air–bridge-type photonic crystal slabs,” J. Appl. Phys.101(113108) (2007).
[CrossRef]

Mock, J.

D. Smith, J. Mock, A. Starr, and D. Schurig, “Gradient index metamaterials,” Phys. Rev. E71(036609) (2005).
[CrossRef]

Mock, J. J.

R. Liu, R. 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]

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]

Mutapcic, A.

Mutapcica, A.

A. Mutapcica, S. Boyd, A. Farjadpour, S. G. Johnson, and Y. Avniel, “Robust design of slow-light tapers in periodic waveguides”. Eng. Optim.41(4):365–384 (2009).
[CrossRef]

Myles, A.

O. Weber, A. Myles, and D. Zorin, “Computing extremal quasiconformal maps,” Symp. Geom. Process.31(5):1679–1689 (2012).

Noda, S.

A. Oskooi, A. Mutapcic, S. Noda, J. D. Joannopoulos, S. P. Boyd, and S. G. Johnson, “Robust optimization of adiabatic tapers for coupling to slow-light photonic-crystal waveguides,” Opt. Express20(19):21558–21575 (2012).
[CrossRef] [PubMed]

A. Chutinan, M. Okano, and S. Noda, “Wider bandwidth with high transmission through waveguide bends in two-dimensional photonic crystal slabs,” Appl. Phys. Lett.80(10):1698–1699 (2002).
[CrossRef]

A. Chutinan and S. Noda, “Highly confined waveguides and waveguide bends in three-dimensional photonic crystal,” Appl. Phys. Lett.75(24):3739–3741 (1999).
[CrossRef]

Novitsky, A. V.

A. V. Novitsky, “Inverse problem in transformation optics,”J. Opt.13(035104) (2011).
[CrossRef]

Okano, M.

A. Chutinan, M. Okano, and S. Noda, “Wider bandwidth with high transmission through waveguide bends in two-dimensional photonic crystal slabs,” Appl. Phys. Lett.80(10):1698–1699 (2002).
[CrossRef]

Ong, C. K.

Ortuño, R.

Osher, S.

C. Y. Kao and S. Osher, “Incorporating topological derivatives into shape derivatives based level set methods,” J. Comp. Phys.225(1):891–909 (2007).
[CrossRef]

C. Y. Yao, S. Osher, and E. Yablonovitch, “Maximizing band gaps in two-dimensional photonic crystals by using level set methods,” Appl. Phys. B81(2):235–244 (2005).
[CrossRef]

Oskooi, A.

Ozaki, N.

Y. Watanabe, N. Ikeda, Y. Sugimoto, Y. Takata, Y. Kitagawa, A. Mizutani, N. Ozaki, and K. Asakawa, “Topology optimization of waveguide bends with wide, flat bandwidth in air–bridge-type photonic crystal slabs,” J. Appl. Phys.101(113108) (2007).
[CrossRef]

Ozgun, O.

O. Ozgun and M. Kuzuoglu, “Utilization of anisotropic metamaterial layers in waveguide miniaturization and transitions,” IEEE Microw. Wirel. Compon. Lett.17(754) (2007).
[CrossRef]

Padilla, W. J.

Park, W.

Pendry, J. B.

J. Li and J. B. Pendry, Private communication (2013).

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]

X. Chen, Y. Luo, J. Zhang, K. Jiang, J. B. Pendry, and S. Zhang, “Macroscopic invisibility cloaking of visible light,” Nat. Commun.2(176) (2011).
[CrossRef]

T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science328(5976):337–339 (2010).
[CrossRef] [PubMed]

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(251111) (2008).
[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. Lett100(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. Li and J. B. Pendry, “Hiding under the carpet: A new strategy for cloaking,” Phys. Rev. Lett101(203901) (2008).
[CrossRef]

D. Schurig, J. B. Pendry, and D. R. Smith, “Transformation-designed optical elements,” Opt. Express15(22):14772–14782 (2007).
[CrossRef] [PubMed]

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]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312:1780–1782 (2006).
[CrossRef] [PubMed]

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science305(5685):788–792 (2004).
[CrossRef] [PubMed]

A. J. Ward and J. B. Pendry, “Refraction and geometry in Maxwell’s equations,” J. Mod. Opt.43(4):773–793 (1996).
[CrossRef]

Philbin, T.

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

Philbin, T. G.

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

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

Platis, I.

Z. Balogh, K. Fässler, and I. Platis, “Modulus method and radial stretch map in the Heisenberg group,” Ann. Acad. Sci. Fenn.38(1):149–180 (2013).
[CrossRef]

Z. Balogh, K. Fässler, and I. Platis, “Modulus of curve families and extremality of spiral-stretch maps,” J. Anal. Math.113(1):265–291 (2011).
[CrossRef]

Plebanski, J.

J. Plebanski, “Electromagnetic waves in gravitational fields,” Phys. Rev.118(5):1396–1408 (1960).
[CrossRef]

Poitras, C. B.

L. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics3:461–463 (2009).
[CrossRef]

Povinelli, M. L.

S. G. Johnson, M. L. Povinelli, M. Soljačić, A. Karalis, S. Jacobs, and J. D. Joannopoulos, “Roughness losses and volume-current methods in photonic-crystal waveguides,” Appl. Phys. B81(283–293) (2005).
[CrossRef]

Powell, M. J. D.

M. J. D. Powell, “Direct search algorithms for optimization calculations,” Acta Numer.7:287–336 (1998).
[CrossRef]

M. J. D. Powell, “A direct search optimization method that models the objective and constraint functions by linear interpolation,” Adv. Optim. Numer. Anal. (1994).
[CrossRef]

Qiu, C.

Qiu, M.

W. Yan, M. Yan, Z. Ruan, and M. Qiu, “Coordinate transformations make perfect invisibility cloaks with arbitrary shape,” New J. Phys.10(043040) (2008).
[CrossRef]

W. Yan, M. Yan, and M. Qiu, “Necessary and sufficient conditions for reflectionless transformation media in an isotropic and homogenous background,” arXiv:0806.3231 (2008).

Ra, J.

K. W. Chun and J. Ra, “Fast block-matching algorithm by successive refinement of matching criterion,” Proc. SPIE, Vis. Commun. Image Process., 1818:552–560 (1992).

Rahm, M.

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(251111) (2008).
[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. Lett100(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]

Ran, L.

J. Huangfu, S. Xi, F. Kong, J. Zhang, H. Chen, D. Wang, B. Wu, L. Ran, and J. A. Kong, “Application of coordinate transformation in bent waveguides,” J. Appl. Phys.104(014502) (2008).
[CrossRef]

Rhee, S. J.

Riishede, J.

Roberts, D. A.

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(251111) (2008).
[CrossRef]

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]

Rockstuhl, C.

M. Schmiele, V. S. Varma, C. Rockstuhl, and F. Lederer, “Designing optical elements from isotropic materials by using transformation optics,” Phys. Rev. A81(033837) (2010).
[CrossRef]

Rodríguez-Fortuño, F. J.

Ruan, Z.

W. Yan, M. Yan, Z. Ruan, and M. Qiu, “Coordinate transformations make perfect invisibility cloaks with arbitrary shape,” New J. Phys.10(043040) (2008).
[CrossRef]

Rudin, W.

W. Rudin, Real and Complex Analysis (McGraw–Hill, 1986).

Santosa, F.

C. Y. Kao and F. Santosa, “Maximization of the quality factor of an optical resonator,” Wave motion45(4):412–427 (2008).
[CrossRef]

D. C. Dobson and F. Santosa, “Optimal localization of eigenfunctions in an inhomogeneous medium,” SIAM J. Appl. Math64(3):762–774 (2004).
[CrossRef]

Scherer, A.

J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, “Design of photonic crystal microcavities for cavity QED” Phys. Rev. E65(016608) (2002).

F. Xu, R. C. Tyan, P. C. Sun, Y. Fainman, C. C. Cheng, and A. Scherer, “Fabrication, modeling, and characterization of form-birefringent nanostructures,” Opt. Lett.20(24):2457–2459 (1995).
[CrossRef] [PubMed]

Schmiele, M.

M. Schmiele, V. S. Varma, C. Rockstuhl, and F. Lederer, “Designing optical elements from isotropic materials by using transformation optics,” Phys. Rev. A81(033837) (2010).
[CrossRef]

Schurig, D.

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. Lett100(063903) (2008).
[CrossRef] [PubMed]

D. Schurig, J. B. Pendry, and D. R. Smith, “Transformation-designed optical elements,” Opt. Express15(22):14772–14782 (2007).
[CrossRef] [PubMed]

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]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312:1780–1782 (2006).
[CrossRef] [PubMed]

D. Smith, J. Mock, A. Starr, and D. Schurig, “Gradient index metamaterials,” Phys. Rev. E71(036609) (2005).
[CrossRef]

Seliger, P.

P. Seliger, M. Mahvash, C. Wang, and A. F. J. Levi, “Optimization of aperiodic dielectric structures,” J. Appl. Phys.100(034310) (2006).
[CrossRef]

Shalaev, V. M.

A. V. Kildishev and V. M. Shalaev, “Transformation optics and metamaterials,” Phys.-Ups.54(1):53–63 (2011).

Sheng, C.

Sheng, P.

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater.9:387–396 (2010).
[CrossRef] [PubMed]

Shi, P.

Shilov, G. E.

G. E. Shilov, Elementary Real and Complex Analysis (MIT, 1973).

Shinnerl, J. R.

T. F. Chan, J. Cong, T. Kong, and J. R. Shinnerl, “Multilevel optimization for large-scale circuit placement,” IEEE ICAD171–176 (2000).

Sigmund, O.

Skorobogatiy, M. A.

S. G. Johnson, P. Bienstman, M. A. Skorobogatiy, M. Ibanescu, E. Lidorikis, and J. D. Joannopoulos, “Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals,” Phys. Rev. E66(066608) (2002).
[CrossRef]

Smajic, J.

Smith, D.

D. Smith, J. Mock, A. Starr, and D. Schurig, “Gradient index metamaterials,” Phys. Rev. E71(036609) (2005).
[CrossRef]

Smith, D. R.

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]

D. R. Smith, Y. Urzhumov, N. B. Kundtz, and N. I. Landy, “Enhancing imaging systems using transformation optics,” Opt. Express18(20):21238–21251 (2010).
[CrossRef] [PubMed]

N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater.9:129–132 (2010).
[CrossRef]

N. I. Landy, N. Kundtz, and D. R. Smith, “Designing three-dimensional transformation optical media using quasiconformal coordinate transformations,” Phys. Rev. Lett.105(193902) (2010).
[CrossRef]

R. Liu, R. 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]

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. Lett100(063903) (2008).
[CrossRef] [PubMed]

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(251111) (2008).
[CrossRef]

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]

D. Schurig, J. B. Pendry, and D. R. Smith, “Transformation-designed optical elements,” Opt. Express15(22):14772–14782 (2007).
[CrossRef] [PubMed]

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]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312:1780–1782 (2006).
[CrossRef] [PubMed]

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science305(5685):788–792 (2004).
[CrossRef] [PubMed]

Snyder, A. W.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).

Soljacic, M.

S. G. Johnson, M. L. Povinelli, M. Soljačić, A. Karalis, S. Jacobs, and J. D. Joannopoulos, “Roughness losses and volume-current methods in photonic-crystal waveguides,” Appl. Phys. B81(283–293) (2005).
[CrossRef]

Soni, B. K.

J. F. Thompson, B. K. Soni, and N. P. Weatherill, Handbook of Grid Generation (CRC, 1999).

Starr, A.

D. Smith, J. Mock, A. Starr, and D. Schurig, “Gradient index metamaterials,” Phys. Rev. E71(036609) (2005).
[CrossRef]

Starr, A. F.

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]

Steinberg, S.

P. Knupp and S. Steinberg, Fundamentals of Grid Generation (CRC, 1994).

Stenger, N.

T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science328(5976):337–339 (2010).
[CrossRef] [PubMed]

Sugimoto, Y.

Y. Watanabe, N. Ikeda, Y. Sugimoto, Y. Takata, Y. Kitagawa, A. Mizutani, N. Ozaki, and K. Asakawa, “Topology optimization of waveguide bends with wide, flat bandwidth in air–bridge-type photonic crystal slabs,” J. Appl. Phys.101(113108) (2007).
[CrossRef]

Summers, C. J.

Sun, H.

H. Xu, B. Zhang, Y. Yu, G. Barbastathis, and H. Sun, “Dielectric waveguide bending adapter with ideal transmission,” Opt. Express29(6):1287–1290 (2012).

Sun, P. C.

Takata, Y.

Y. Watanabe, N. Ikeda, Y. Sugimoto, Y. Takata, Y. Kitagawa, A. Mizutani, N. Ozaki, and K. Asakawa, “Topology optimization of waveguide bends with wide, flat bandwidth in air–bridge-type photonic crystal slabs,” J. Appl. Phys.101(113108) (2007).
[CrossRef]

Tamma, V. A.

Tang, L.

Tang, T.

Tang, X.

Thompson, J. F.

J. F. Thompson, B. K. Soni, and N. P. Weatherill, Handbook of Grid Generation (CRC, 1999).

Tian, X. Y.

M. Yin, X. Y. Tian, H. X. Han, and D. C. Li, “Free-space carpet-cloak based on gradient index photonic crystals in metamaterial regime,” Appl. Phys. Lett.100(124101) (2012).
[CrossRef]

Tortorelli, D. A.

W. R. Frei, H. T. Johnson, and D. A. Tortorelli, “Optimization of photonic nanostructures,” Comput. Method. Appl. M.197(41):3410–3416 (2008).
[CrossRef]

W. R. Frei, D. A. Tortorelli, and H. T. Johnson, “Topology optimization of a photonic crystal waveguide termination to maximize directional emission,” Appl. Phys. Lett.86(111114) (2005).
[CrossRef]

Tsuji, Y.

Y. Tsuji and K. Hirayama, “Design of optical circuit devices using topology optimization method with function-expansionbased refractive index distribution,” IEEE Phot. Tech. Lett.20(12):982–984 (2008).
[CrossRef]

Tung, M. M.

Turpin, J. P.

Q. Wu, J. P. Turpin, and D. H. Werner, “Integrated photonic systems based on transformation optics enabled gradient index devices,” Light: Science and Applications1(e38) (2012).

J. P. Turpin, A. T. Massoud, Z. H. Jiang, P. L. Werner, and D. H. Wener, “Conformal mappings to achieve simple material parameters for transformation optics devices,” Opt. Express18(1):244–252 (2010).
[CrossRef] [PubMed]

Tyan, R. C.

Tyc, T.

U. Leonhardt and T. Tyc, “Broadband invisibility by non-euclidian cloaking,” Science323(5910):110–112 (2009).
[CrossRef]

Uhlmann, G.

A. Greenleaf, M. Lassas, and G. Uhlmann, “On nonuniqueness for Calderón’s inverse problem,” Math. Res. Letters10(5):685–693 (2003).

A. Greenleaf, M. Lassas, and G. Uhlmann, “Anisotropic conductivities that cannot be detected by EIT,” Physiol. Meas.24(413):413–419 (2003).
[CrossRef] [PubMed]

Ulrich, R.

Urzhumov, Y.

Valentine, J.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater.8:568–571 (2009).
[CrossRef] [PubMed]

Valentine, J. G.

Vandenberghe, L.

S. Boyd and L. Vandenberghe, Convex Optimization (Cambridge University, 2004).

Varma, V. S.

M. Schmiele, V. S. Varma, C. Rockstuhl, and F. Lederer, “Designing optical elements from isotropic materials by using transformation optics,” Phys. Rev. A81(033837) (2010).
[CrossRef]

Vasic, B.

B. Vasić, R. Gajić, and K. Hingerl, “Graded photonic crystals for implementation of gradient refractive index media,” J. Nanophotonics5(051806) (2011).
[CrossRef]

B. Vasić, G. Isić, R. Gajić, and K. Hingerl, “Coordinate transformation based design of confined metamaterial structures,” Phys. Rev. B79(085103) (2009).
[CrossRef]

Villeneuve, P. R.

C. Manolatou, S. G. Johnson, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “High-density integrated optics,” J. Lightwave Technol.17(9):1682–1692 (1999).
[CrossRef]

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett.77:3787–3790 (1996).
[CrossRef] [PubMed]

Vuckovic, J.

J. Lu and J. Vuckovic, “Inverse design of nanophotonic structures using complementary convex optimization,” Opt. Express18(4):3793–3804 (2010).
[CrossRef] [PubMed]

J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, “Design of photonic crystal microcavities for cavity QED” Phys. Rev. E65(016608) (2002).

Wang, C.

P. Seliger, M. Mahvash, C. Wang, and A. F. J. Levi, “Optimization of aperiodic dielectric structures,” J. Appl. Phys.100(034310) (2006).
[CrossRef]

Wang, D.

J. Huangfu, S. Xi, F. Kong, J. Zhang, H. Chen, D. Wang, B. Wu, L. Ran, and J. A. Kong, “Application of coordinate transformation in bent waveguides,” J. Appl. Phys.104(014502) (2008).
[CrossRef]

Wang, N.

Wang, S.

Wang, S. M.

Wang, Y.

Ward, A. J.

A. J. Ward and J. B. Pendry, “Refraction and geometry in Maxwell’s equations,” J. Mod. Opt.43(4):773–793 (1996).
[CrossRef]

Watanabe, Y.

Y. Watanabe, N. Ikeda, Y. Sugimoto, Y. Takata, Y. Kitagawa, A. Mizutani, N. Ozaki, and K. Asakawa, “Topology optimization of waveguide bends with wide, flat bandwidth in air–bridge-type photonic crystal slabs,” J. Appl. Phys.101(113108) (2007).
[CrossRef]

Weatherill, N. P.

J. F. Thompson, B. K. Soni, and N. P. Weatherill, Handbook of Grid Generation (CRC, 1999).

Weber, O.

O. Weber, A. Myles, and D. Zorin, “Computing extremal quasiconformal maps,” Symp. Geom. Process.31(5):1679–1689 (2012).

Wegener, M.

T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science328(5976):337–339 (2010).
[CrossRef] [PubMed]

Weiss, S. M.

Wells, G. N.

A. Logg, K. A. Mardal, and G. N. Wells, Automated Solution of Differential Equations by the Finite Element Method (Springer, 2012).
[CrossRef]

Wener, D. H.

Werner, D. H.

Q. Wu, J. P. Turpin, and D. H. Werner, “Integrated photonic systems based on transformation optics enabled gradient index devices,” Light: Science and Applications1(e38) (2012).

Werner, P. L.

Wiltshire, M. C. K.

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science305(5685):788–792 (2004).
[CrossRef] [PubMed]

Wu, B.

J. Huangfu, S. Xi, F. Kong, J. Zhang, H. Chen, D. Wang, B. Wu, L. Ran, and J. A. Kong, “Application of coordinate transformation in bent waveguides,” J. Appl. Phys.104(014502) (2008).
[CrossRef]

Wu, Q.

Xi, S.

J. Huangfu, S. Xi, F. Kong, J. Zhang, H. Chen, D. Wang, B. Wu, L. Ran, and J. A. Kong, “Application of coordinate transformation in bent waveguides,” J. Appl. Phys.104(014502) (2008).
[CrossRef]

Xiong, Y.

S. Han, Y. Xiong, D. Genov, Z. Liu, G. Bartel, and X. Zhang, “Ray optics at a deep-subwavelength scale: A transformation optics approach,” Nano Lett.8(12):4243–4247 (2008).
[CrossRef]

Xu, F.

Xu, H.

H. Xu, B. Zhang, Y. Yu, G. Barbastathis, and H. Sun, “Dielectric waveguide bending adapter with ideal transmission,” Opt. Express29(6):1287–1290 (2012).

Yablonovitch, E.

C. Y. Yao, S. Osher, and E. Yablonovitch, “Maximizing band gaps in two-dimensional photonic crystals by using level set methods,” Appl. Phys. B81(2):235–244 (2005).
[CrossRef]

Yan, M.

W. Yan, M. Yan, Z. Ruan, and M. Qiu, “Coordinate transformations make perfect invisibility cloaks with arbitrary shape,” New J. Phys.10(043040) (2008).
[CrossRef]

W. Yan, M. Yan, and M. Qiu, “Necessary and sufficient conditions for reflectionless transformation media in an isotropic and homogenous background,” arXiv:0806.3231 (2008).

Yan, W.

W. Yan, M. Yan, Z. Ruan, and M. Qiu, “Coordinate transformations make perfect invisibility cloaks with arbitrary shape,” New J. Phys.10(043040) (2008).
[CrossRef]

W. Yan, M. Yan, and M. Qiu, “Necessary and sufficient conditions for reflectionless transformation media in an isotropic and homogenous background,” arXiv:0806.3231 (2008).

Yang, F.

Yao, C. Y.

C. Y. Yao, S. Osher, and E. Yablonovitch, “Maximizing band gaps in two-dimensional photonic crystals by using level set methods,” Appl. Phys. B81(2):235–244 (2005).
[CrossRef]

Yao, K.

Yau, S. T.

W. Zeng, F. Luo, S. T. Yau, and X. D. Gu, “Surface quasi-conformal mapping by solving Beltrami equations,” Proc. Math. SurfacesXIII391–408 (2009).
[CrossRef]

Yin, J.

Yin, M.

M. Yin, X. Y. Tian, H. X. Han, and D. C. Li, “Free-space carpet-cloak based on gradient index photonic crystals in metamaterial regime,” Appl. Phys. Lett.100(124101) (2012).
[CrossRef]

Yu, Y.

H. Xu, B. Zhang, Y. Yu, G. Barbastathis, and H. Sun, “Dielectric waveguide bending adapter with ideal transmission,” Opt. Express29(6):1287–1290 (2012).

Yuan, G.

Zeng, W.

W. Zeng, F. Luo, S. T. Yau, and X. D. Gu, “Surface quasi-conformal mapping by solving Beltrami equations,” Proc. Math. SurfacesXIII391–408 (2009).
[CrossRef]

Zentgraf, T.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater.8:568–571 (2009).
[CrossRef] [PubMed]

Zernike, F.

F. Zernike, “Luneburg lens for optical waveguide use,” Opt. Commun.12(4):379–381 (1974).
[CrossRef]

Zhang, B.

H. Xu, B. Zhang, Y. Yu, G. Barbastathis, and H. Sun, “Dielectric waveguide bending adapter with ideal transmission,” Opt. Express29(6):1287–1290 (2012).

B. Zhang, Y. Luo, X. Liu, and G. Barbastathis, “Macroscopic invisibility cloak for visible light,” Phys. Rev. Lett.106(033901) (2011).
[CrossRef]

Zhang, J.

X. Chen, Y. Luo, J. Zhang, K. Jiang, J. B. Pendry, and S. Zhang, “Macroscopic invisibility cloaking of visible light,” Nat. Commun.2(176) (2011).
[CrossRef]

J. Huangfu, S. Xi, F. Kong, J. Zhang, H. Chen, D. Wang, B. Wu, L. Ran, and J. A. Kong, “Application of coordinate transformation in bent waveguides,” J. Appl. Phys.104(014502) (2008).
[CrossRef]

Zhang, K.

Zhang, Q.

Zhang, S.

X. Chen, Y. Luo, J. Zhang, K. Jiang, J. B. Pendry, and S. Zhang, “Macroscopic invisibility cloaking of visible light,” Nat. Commun.2(176) (2011).
[CrossRef]

Zhang, X.

Y. Liu and X. Zhang, “Recent advances in transformation optics,” Nanoscale4(17):5277–5292 (2012).
[CrossRef] [PubMed]

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater.8:568–571 (2009).
[CrossRef] [PubMed]

S. Han, Y. Xiong, D. Genov, Z. Liu, G. Bartel, and X. Zhang, “Ray optics at a deep-subwavelength scale: A transformation optics approach,” Nano Lett.8(12):4243–4247 (2008).
[CrossRef]

X. Zhang, H. Chen, X. Luo, and H. Ma, “Transformation media that turn a narrow slit into a large window,” Opt. Express16(16):11764–11768 (2008).
[CrossRef] [PubMed]

Zhang, Y.

Zheng, Y.J.

Zhou, X.

Zhu, C.

Zhu, S. N.

Zorin, D.

O. Weber, A. Myles, and D. Zorin, “Computing extremal quasiconformal maps,” Symp. Geom. Process.31(5):1679–1689 (2012).

Zouhdi, S.

Acta Math.

L. Bers, “An extremal problem for quasi-conformal mappings and a problem of Thurston,” Acta Math.73–98 (1978).
[CrossRef]

Acta Numer.

M. J. D. Powell, “Direct search algorithms for optimization calculations,” Acta Numer.7:287–336 (1998).
[CrossRef]

Adv. Optim. Numer. Anal.

M. J. D. Powell, “A direct search optimization method that models the objective and constraint functions by linear interpolation,” Adv. Optim. Numer. Anal. (1994).
[CrossRef]

Ann. Acad. Sci. Fenn.

Z. Balogh, K. Fässler, and I. Platis, “Modulus method and radial stretch map in the Heisenberg group,” Ann. Acad. Sci. Fenn.38(1):149–180 (2013).
[CrossRef]

Appl. Opt.

Appl. Phys. B

S. G. Johnson, M. L. Povinelli, M. Soljačić, A. Karalis, S. Jacobs, and J. D. Joannopoulos, “Roughness losses and volume-current methods in photonic-crystal waveguides,” Appl. Phys. B81(283–293) (2005).
[CrossRef]

C. Y. Yao, S. Osher, and E. Yablonovitch, “Maximizing band gaps in two-dimensional photonic crystals by using level set methods,” Appl. Phys. B81(2):235–244 (2005).
[CrossRef]

Appl. Phys. Lett.

W. R. Frei, D. A. Tortorelli, and H. T. Johnson, “Topology optimization of a photonic crystal waveguide termination to maximize directional emission,” Appl. Phys. Lett.86(111114) (2005).
[CrossRef]

J. S. Jensen and O. Sigmund, “Systematic design of photonic crystal structures using topology optimization: low-loss waveguide bends,” Appl. Phys. Lett.84(12):2022–2024 (2004).
[CrossRef]

J. Andkjaer and O. Sigmund, “Topology optimized low-contrast all-dielectric optical cloak,” Appl. Phys. Lett.98(021112) (2011).
[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(251111) (2008).
[CrossRef]

M. Yin, X. Y. Tian, H. X. Han, and D. C. Li, “Free-space carpet-cloak based on gradient index photonic crystals in metamaterial regime,” Appl. Phys. Lett.100(124101) (2012).
[CrossRef]

A. Chutinan, M. Okano, and S. Noda, “Wider bandwidth with high transmission through waveguide bends in two-dimensional photonic crystal slabs,” Appl. Phys. Lett.80(10):1698–1699 (2002).
[CrossRef]

A. Chutinan and S. Noda, “Highly confined waveguides and waveguide bends in three-dimensional photonic crystal,” Appl. Phys. Lett.75(24):3739–3741 (1999).
[CrossRef]

B. U. Chen, E. Marom, and A. Lee, “Geodesic lenses in single-mode LiNbO3waveguides,” Appl. Phys. Lett.31(4):263 (1977).
[CrossRef]

Arch. Rational Mech. Anal.

K. Astala, T. Iwaniec, and G. Martin, “Deformations of annuli with smallest mean distortion,” Arch. Rational Mech. Anal.195:899–921 (2010).
[CrossRef]

Comput. Method. Appl. M.

W. R. Frei, H. T. Johnson, and D. A. Tortorelli, “Optimization of photonic nanostructures,” Comput. Method. Appl. M.197(41):3410–3416 (2008).
[CrossRef]

Eng. Optim.

A. Mutapcica, S. Boyd, A. Farjadpour, S. G. Johnson, and Y. Avniel, “Robust design of slow-light tapers in periodic waveguides”. Eng. Optim.41(4):365–384 (2009).
[CrossRef]

IEEE ICAD

T. F. Chan, J. Cong, T. Kong, and J. R. Shinnerl, “Multilevel optimization for large-scale circuit placement,” IEEE ICAD171–176 (2000).

IEEE Microw. Wirel. Compon. Lett.

O. Ozgun and M. Kuzuoglu, “Utilization of anisotropic metamaterial layers in waveguide miniaturization and transitions,” IEEE Microw. Wirel. Compon. Lett.17(754) (2007).
[CrossRef]

IEEE Phot. Tech. Lett.

Y. Tsuji and K. Hirayama, “Design of optical circuit devices using topology optimization method with function-expansionbased refractive index distribution,” IEEE Phot. Tech. Lett.20(12):982–984 (2008).
[CrossRef]

J. Anal. Math.

Z. Balogh, K. Fässler, and I. Platis, “Modulus of curve families and extremality of spiral-stretch maps,” J. Anal. Math.113(1):265–291 (2011).
[CrossRef]

L. Ahlfors, “On quasiconformal mappings. J. Anal. Math.3(1):1–58 (1953).
[CrossRef]

J. Appl. Phys.

P. Seliger, M. Mahvash, C. Wang, and A. F. J. Levi, “Optimization of aperiodic dielectric structures,” J. Appl. Phys.100(034310) (2006).
[CrossRef]

Y. Watanabe, N. Ikeda, Y. Sugimoto, Y. Takata, Y. Kitagawa, A. Mizutani, N. Ozaki, and K. Asakawa, “Topology optimization of waveguide bends with wide, flat bandwidth in air–bridge-type photonic crystal slabs,” J. Appl. Phys.101(113108) (2007).
[CrossRef]

Z. L. Mei and T. J. Cui, “Arbitrary bending of electromagnetic waves using isotropic materials,” J. Appl. Phys.105(104913) (2009).
[CrossRef]

J. Huangfu, S. Xi, F. Kong, J. Zhang, H. Chen, D. Wang, B. Wu, L. Ran, and J. A. Kong, “Application of coordinate transformation in bent waveguides,” J. Appl. Phys.104(014502) (2008).
[CrossRef]

W. R. Frei, H. T. Johnson, and K. D. Choquette, “Optimization of a single defect photonic crystal laser cavity,” J. Appl. Phys.103(033102) (2008).
[CrossRef]

J. Comp. Phys.

C. Y. Kao and S. Osher, “Incorporating topological derivatives into shape derivatives based level set methods,” J. Comp. Phys.225(1):891–909 (2007).
[CrossRef]

J. Electromagn. Waves App.

Z. L. Mei, J. Bai, and T. J. Cui, “Illusion devices with quasi-conformal mapping,” J. Electromagn. Waves App.24(17):2561–2573 (2010).
[CrossRef]

J. Lightwave Technol.

J. Mod. Opt.

A. J. Ward and J. B. Pendry, “Refraction and geometry in Maxwell’s equations,” J. Mod. Opt.43(4):773–793 (1996).
[CrossRef]

J. Nanophotonics

B. Vasić, R. Gajić, and K. Hingerl, “Graded photonic crystals for implementation of gradient refractive index media,” J. Nanophotonics5(051806) (2011).
[CrossRef]

J. Opt.

A. V. Novitsky, “Inverse problem in transformation optics,”J. Opt.13(035104) (2011).
[CrossRef]

L. H. Gabrielli and M. Lipson, “Transformation optics on a silicon platform,” J. Opt.13(024010) (2011).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

J. Quantum Electron.

M. Heiblum and J. H. Harris, “Analysis of curved optical waveguides by conformal transformation,” J. Quantum Electron.11(2):75–83 (1975).
[CrossRef]

Laser Photon. Rev.

J. S. Jensen and O. Sigmund, “Topology optimization for nano-photonics,” Laser Photon. Rev.5(2):308–321 (2011).
[CrossRef]

Lectures on quasiconformal mappings

L. V. Ahlfors, Lectures on quasiconformal mappings (American Mathematical Society, 1966).

Light: Science and Applications

Q. Wu, J. P. Turpin, and D. H. Werner, “Integrated photonic systems based on transformation optics enabled gradient index devices,” Light: Science and Applications1(e38) (2012).

Math. Res. Letters

A. Greenleaf, M. Lassas, and G. Uhlmann, “On nonuniqueness for Calderón’s inverse problem,” Math. Res. Letters10(5):685–693 (2003).

Nano Lett.

S. Han, Y. Xiong, D. Genov, Z. Liu, G. Bartel, and X. Zhang, “Ray optics at a deep-subwavelength scale: A transformation optics approach,” Nano Lett.8(12):4243–4247 (2008).
[CrossRef]

Nanoscale

Y. Liu and X. Zhang, “Recent advances in transformation optics,” Nanoscale4(17):5277–5292 (2012).
[CrossRef] [PubMed]

Nat. Commun.

H. F. Ma and T. J. Cui, “Three-dimensional broadband and broad-angle transformation-optics lens,” Nat. Commun.1(124) (2010).
[CrossRef] [PubMed]

X. Chen, Y. Luo, J. Zhang, K. Jiang, J. B. Pendry, and S. Zhang, “Macroscopic invisibility cloaking of visible light,” Nat. Commun.2(176) (2011).
[CrossRef]

L. Gabrielli, D. Liu, S. G. Johnson, and M. Lipson, “On-chip transformation optics for multimode waveguide bends,” Nat. Commun.3(1217) (2012).
[CrossRef] [PubMed]

Nat. Mater.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater.8:568–571 (2009).
[CrossRef] [PubMed]

N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater.9:129–132 (2010).
[CrossRef]

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater.9:387–396 (2010).
[CrossRef] [PubMed]

Nat. Photonics

L. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics3:461–463 (2009).
[CrossRef]

New J. Phys.

Z. L. Mei, J. Bai, and T. J. Cui, “Experimental verification of a broadband planar focusing antenna based on transformation optics,” New J. Phys.13(063028) (2011).
[CrossRef]

Z. Liang, X. Jiang, F. Miao, S. Guenneau, and J. Li, “Transformation media with variable optical axes,” New J. Phys.14(103042) (2012).
[CrossRef]

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

W. Yan, M. Yan, Z. Ruan, and M. Qiu, “Coordinate transformations make perfect invisibility cloaks with arbitrary shape,” New J. Phys.10(043040) (2008).
[CrossRef]

Opt. Commun.

F. Zernike, “Luneburg lens for optical waveguide use,” Opt. Commun.12(4):379–381 (1974).
[CrossRef]

Opt. Express

Y. Zhang and B. Li, “Photonic crystal-based bending waveguides for optical interconnections,” Opt. Express14(12):5723–5732 (2006).
[CrossRef] [PubMed]

H. Kurt and D. S. Citrin, “Graded index photonic crystals,” Opt. Express15(3):1240–1253 (2007).
[CrossRef] [PubMed]

D. Schurig, J. B. Pendry, and D. R. Smith, “Transformation-designed optical elements,” Opt. Express15(22):14772–14782 (2007).
[CrossRef] [PubMed]

J. Smajic, C. Hafner, and D. Erni, “Design and optimization of an achromatic photonic crystal bend,” Opt. Express11(12):1378–1384 (2003).
[CrossRef] [PubMed]

P. Borel, A. Harpøth, L. Frandsen, M. Kristensen, P. Shi, J. Jensen, and O. Sigmund, “Topology optimization and fabrication of photonic crystal structures,” Opt. Express12(9):1996 (2004).
[CrossRef] [PubMed]

L. Frandsen, A. Harpøth, P. Borel, M. Kristensen, J. Jensen, and O. Sigmund, “Broadband photonic crystal waveguide 60° bend obtained utilizing topology optimization,” Opt. Express12(24):5916–5921 (2004).
[CrossRef] [PubMed]

T. Han, C. Qiu, J. Dong, X. Tang, and S. Zouhdi, “Homogeneous and isotropic bends to tunnel waves through multiple different/equal waveguides along arbitrary directions,” Opt. Express19(14):13020–13030 (2011).
[CrossRef] [PubMed]

L. Tang, J. Yin, G. Yuan, J. Du, H. Gao, X. Dong, Y. Lu, and C. Du, “General conformal transformation method based on Schwarz–Christoffel approach,” Opt. Express19(16):15119–15126 (2011).
[CrossRef] [PubMed]

L. Gabrielli and M. Lipson, “Integrated luneburg lens via ultra-strong index gradient on silicon,” Opt. Express19(21):20122–20127 (2011).
[CrossRef] [PubMed]

C. Garcia-Meca, A. Martinez, and U. Leonhardt, “Engineering antenna radiation patterns via quasi-conformal mappings,” Opt. Express19(24):23743–23750 (2011).
[CrossRef] [PubMed]

S. Wang and S. Liu, “Controlling electromagnetic scattering of a cavity by transformation media,” Opt. Express20(6):6777–6785 (2012).
[CrossRef] [PubMed]

Y. Wang, C. Sheng, H. Liu, Y.J. Zheng, C. Zhu, S. M. Wang, and S. N. Zhu, “Transformation bending device emulated by graded-index waveguide,” Opt. Express20(12):13006–13013 (2012).
[CrossRef] [PubMed]

P. Markov, J. G. Valentine, and S. M. Weiss, “Fiber-to-chip coupler designed using an optical transformation,” Opt. Express20(13):14705–14712 (2012).
[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]

X. Zhang, H. Chen, X. Luo, and H. Ma, “Transformation media that turn a narrow slit into a large window,” Opt. Express16(16):11764–11768 (2008).
[CrossRef] [PubMed]

C. Ma, Q. Zhang, and E. V. Keuren, “Right-angle slot waveguide bends with high bending efficiency,” Opt. Express16(19):14330 (2008).
[CrossRef] [PubMed]

B. Chen, T. Tang, and H. Chen, “Study on a compact flexible photonic crystal waveguide and its bends,” Opt. Express17(7):5033–5038 (2009).
[CrossRef] [PubMed]

J. H. Lee, J. Blair, V. A. Tamma, Q. Wu, S. J. Rhee, C. J. Summers, and W. Park, “Direct visualization of optical frequency invisibility cloak based on silicon nanorod array,” Opt. Express17(15):12922–12928 (2009).
[CrossRef] [PubMed]

N. I. Landy and W. J. Padilla, “Guiding light with conformal transformations,” Opt. Express17(17):14872–14879 (2009).
[CrossRef] [PubMed]

Z. L. Mei and T. J. Cui, “Experimental realization of a broadband bend structure using gradient index metamaterials,” Opt. Express17(20):18354–18363 (2009).
[CrossRef] [PubMed]

J. P. Turpin, A. T. Massoud, Z. H. Jiang, P. L. Werner, and D. H. Wener, “Conformal mappings to achieve simple material parameters for transformation optics devices,” Opt. Express18(1):244–252 (2010).
[CrossRef] [PubMed]

J. Lu and J. Vuckovic, “Inverse design of nanophotonic structures using complementary convex optimization,” Opt. Express18(4):3793–3804 (2010).
[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]

D. R. Smith, Y. Urzhumov, N. B. Kundtz, and N. I. Landy, “Enhancing imaging systems using transformation optics,” Opt. Express18(20):21238–21251 (2010).
[CrossRef] [PubMed]

C. García-Meca, M. M. Tung, J. V. Galán, R. Ortuño, F. J. Rodríguez-Fortuño, J. Martí, and A. Martínez, “Squeezing and expanding light without reflections via transformation optics,” Opt. Express19(4):3562–3757 (2011).
[CrossRef] [PubMed]

A. Oskooi, A. Mutapcic, S. Noda, J. D. Joannopoulos, S. P. Boyd, and S. G. Johnson, “Robust optimization of adiabatic tapers for coupling to slow-light photonic-crystal waveguides,” Opt. Express20(19):21558–21575 (2012).
[CrossRef] [PubMed]

Z. L. Mei, Y. S. Liu, F. Yang, and T. J. Cui, “A DC carpet cloak based on resistor networks,” Opt. Express20(23):25758–25764 (2012).
[CrossRef] [PubMed]

V. Liu and S. Fan, “Compact bends for multi-mode photonic crystal waveguides with high transmission and suppressed modal crosstalk,” Opt. Express21(7):8069–8075 (2013).
[CrossRef] [PubMed]

V. Liu and S. Fan, “Compact bends for multi-mode photonic crystal waveguides with high transmission and suppressed modal crosstalk,” Opt. Express21(7):8069–8075 (2013).
[CrossRef] [PubMed]

H. Xu, B. Zhang, Y. Yu, G. Barbastathis, and H. Sun, “Dielectric waveguide bending adapter with ideal transmission,” Opt. Express29(6):1287–1290 (2012).

Opt. Lett.

Phys. Rev.

J. Plebanski, “Electromagnetic waves in gravitational fields,” Phys. Rev.118(5):1396–1408 (1960).
[CrossRef]

Phys. Rev. A

L. Bergamin, “Electromagnetic fields and boundary conditions at the interface of generalized transformation media,” Phys. Rev. A80(063835) (2009).
[CrossRef]

M. Schmiele, V. S. Varma, C. Rockstuhl, and F. Lederer, “Designing optical elements from isotropic materials by using transformation optics,” Phys. Rev. A81(033837) (2010).
[CrossRef]

Phys. Rev. B

B. Vasić, G. Isić, R. Gajić, and K. Hingerl, “Coordinate transformation based design of confined metamaterial structures,” Phys. Rev. B79(085103) (2009).
[CrossRef]

Phys. Rev. E

J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, “Design of photonic crystal microcavities for cavity QED” Phys. Rev. E65(016608) (2002).

D. Smith, J. Mock, A. Starr, and D. Schurig, “Gradient index metamaterials,” Phys. Rev. E71(036609) (2005).
[CrossRef]

S. G. Johnson, P. Bienstman, M. A. Skorobogatiy, M. Ibanescu, E. Lidorikis, and J. D. Joannopoulos, “Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals,” Phys. Rev. E66(066608) (2002).
[CrossRef]

Phys. Rev. Lett

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. Lett100(063903) (2008).
[CrossRef] [PubMed]

J. Li and J. B. Pendry, “Hiding under the carpet: A new strategy for cloaking,” Phys. Rev. Lett101(203901) (2008).
[CrossRef]

Phys. Rev. Lett.

N. I. Landy, N. Kundtz, and D. R. Smith, “Designing three-dimensional transformation optical media using quasiconformal coordinate transformations,” Phys. Rev. Lett.105(193902) (2010).
[CrossRef]

B. Zhang, Y. Luo, X. Liu, and G. Barbastathis, “Macroscopic invisibility cloak for visible light,” Phys. Rev. Lett.106(033901) (2011).
[CrossRef]

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett.77:3787–3790 (1996).
[CrossRef] [PubMed]

U. Levy, M. Abashin, K. Ikeda, A. Krishnamoorthy, J. Cunningham, and Y. Fainman, “Inhomogenous dielectric metamaterials with space-variant polarizability,” Phys. Rev. Lett.98(243901) (2007).
[CrossRef]

Phys.-Ups.

A. V. Kildishev and V. M. Shalaev, “Transformation optics and metamaterials,” Phys.-Ups.54(1):53–63 (2011).

Physiol. Meas.

A. Greenleaf, M. Lassas, and G. Uhlmann, “Anisotropic conductivities that cannot be detected by EIT,” Physiol. Meas.24(413):413–419 (2003).
[CrossRef] [PubMed]

Private communication

J. Li and J. B. Pendry, Private communication (2013).

Proc. Math. Surfaces

W. Zeng, F. Luo, S. T. Yau, and X. D. Gu, “Surface quasi-conformal mapping by solving Beltrami equations,” Proc. Math. SurfacesXIII391–408 (2009).
[CrossRef]

Proc. SPIE, Vis. Commun. Image Process.

K. W. Chun and J. Ra, “Fast block-matching algorithm by successive refinement of matching criterion,” Proc. SPIE, Vis. Commun. Image Process., 1818:552–560 (1992).

Prog. Opt.

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

Science

U. Leonhardt and T. Tyc, “Broadband invisibility by non-euclidian cloaking,” Science323(5910):110–112 (2009).
[CrossRef]

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]

U. Leonhardt, “Optical conformal mapping,” Science312:1777–1780 (2006).
[CrossRef] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312:1780–1782 (2006).
[CrossRef] [PubMed]

R. Liu, R. 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]

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]

T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science328(5976):337–339 (2010).
[CrossRef] [PubMed]

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science305(5685):788–792 (2004).
[CrossRef] [PubMed]

SIAM J. Appl. Math

D. C. Dobson and F. Santosa, “Optimal localization of eigenfunctions in an inhomogeneous medium,” SIAM J. Appl. Math64(3):762–774 (2004).
[CrossRef]

S. J. Cox and D. C. Dobson, “Maximizing band gaps in two-dimensional photonic crystals,” SIAM J. Appl. Math59(6):2108–2120 (1999).
[CrossRef]

Symp. Geom. Process.

O. Weber, A. Myles, and D. Zorin, “Computing extremal quasiconformal maps,” Symp. Geom. Process.31(5):1679–1689 (2012).

Wave motion

C. Y. Kao and F. Santosa, “Maximization of the quality factor of an optical resonator,” Wave motion45(4):412–427 (2008).
[CrossRef]

Other

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

K. Astala, T. Iwaniec, and G. Martin, Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane (Princeton University, 2008).

A. Papadopoulos, editor. Handbook of Teichmüller Theory, volume 1 (European Mathematical Society, 2007).
[CrossRef]

R. Kühnau, editor. Handbook of Complex Analysis: Geometric Function Theory, volume 2 (Elsevier B.V., 2005).

J. D. Jackson, Classical Electrodynamics (Wiley, 1998).

J. P. Boyd, Chebyshev and Fourier Spectral Methods (Dover, 2001).

S. Boyd and L. Vandenberghe, Convex Optimization (Cambridge University, 2004).

W. Yan, M. Yan, and M. Qiu, “Necessary and sufficient conditions for reflectionless transformation media in an isotropic and homogenous background,” arXiv:0806.3231 (2008).

W. Rudin, Real and Complex Analysis (McGraw–Hill, 1986).

G. E. Shilov, Elementary Real and Complex Analysis (MIT, 1973).

J. F. Thompson, B. K. Soni, and N. P. Weatherill, Handbook of Grid Generation (CRC, 1999).

P. Knupp and S. Steinberg, Fundamentals of Grid Generation (CRC, 1994).

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).

W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE, 1995).

S. G. Johnson, The NLopt nonlinear-optimization package ( http://ab-initio.mit.edu/nlopt ) (2007).

A. Logg, K. A. Mardal, and G. N. Wells, Automated Solution of Differential Equations by the Finite Element Method (Springer, 2012).
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1

Three possible applications of transformation optics for multimode waveguides: squeezer, expander, and bend. Dark areas indicate higher refractive index.

Fig. 2
Fig. 2

The interface between the transformed and untransformed region must have x′ continuous in order for there not to be any interface reflections.

Fig. 3
Fig. 3

In the transformation process, the untransformed straight waveguide is bent, perturbed, and optimized. Darker regions indicate higher refractive index

Fig. 4
Fig. 4

Optimization decreases anisotropy by a factor of 10−4, while dramatically improving the scattered-power matrix.

Fig. 5
Fig. 5

FEM field profiles show heavy scattering in the conventional non-TO and scalarized circular bends, but very little scattering in the optimized bend.

Fig. 6
Fig. 6

Anisotropy profile and scattered-power matrices for optimized designs that minimize the mean and the peak, with R = 2.5, N = 3, and Nm = 6.

Fig. 7
Fig. 7

Successive optimization with N = 5, Nm = 8 results in a power law decaying trade-off maxx �� − 1 ∼ R−4 at low R and an exponentially decaying tradeoff at higher R. For comparison, the unoptimized anisotropy for the circular TO bend is shown above.

Fig. 8
Fig. 8

Optimized squeezer outperforms gaussian taper and stretched optimized squeezers in finite element simulations.

Equations (33)

Equations on this page are rendered with MathJax. Learn more.

× H = i ω ε ( x ) E × E = i ω μ 0 H .
× H = i ω ε E × E = i ω μ H ,
μ = μ 0 𝒥 T 𝒥 det 𝒥 ε = ε ( x ) 𝒥 T 𝒥 det 𝒥 .
| x | 2 | y | 2 = 0 x y = 0 .
𝒥 T 𝒥 det 𝒥 = ( 1 1 1 det 𝒥 ) .
× H = i ω ε ( x ) det 𝒥 E
× E = i ω μ 0 H .
× × E = ω 2 ε 0 μ 0 ( det 𝒥 ) E + 𝒪 ( det 𝒥 ) ,
μ B ( w , w ¯ ) ( w w ¯ ) ( w w ) 1 .
K 1 + | μ B | 2 1 | μ B | 2 = λ 1 λ 2 1 .
𝕂 ( x , y ) 1 2 ( K + 1 K ) = tr 𝒥 T 𝒥 2 det 𝒥 1 .
μ = λ 1 + λ 2 2 λ 1 λ 2 = tr 𝒥 T 𝒥 2 det 𝒥
min x ( x ) 𝕂 ( x ) subject to { x , 𝒥 continuous at input / output interfaces n min n ( x ) n max ,
x = r cos θ y = r sin θ z = z ,
tr 𝒥 T 𝒥 = | r | 2 + | r θ | 2
det 𝒥 = | r × r θ | .
r ( x , y ) = r ( x , y ) θ ( x , y ) = θ ( x , y ) .
r = x θ = ± π 4 r y = 0 θ y = 1 x .
n ( x ) = ε μ = n 0 p ( x ) tr 𝒥 T 𝒥 2 ( det 𝒥 ) 2
min r ( x ) , θ ( x ) , n 0 , L , t t subject to : { continuity conditions 17 , 18 n min n p p ( x ) tr 𝒥 T 𝒥 2 ( det 𝒥 ) 2 n max for x G R = R 0 𝕂 ( x ) t for x G .
min r ( x ) , θ ( x ) , n 0 , L 𝕂 x subject to : { continuity conditions 17 , 18 n min n 0 p ( x ) tr 𝒥 T 𝒥 2 ( det 𝒥 ) 2 n max for x G R = R 0 .
r ( x , y ) = x + , m N , N m C m r T ( 2 x 2 R 1 ) cos 2 m π y L θ ( x , y ) = π y 2 L + 1 x , m N , N m C m θ T ( 2 x 2 R 1 ) sin 2 m π y L ,
m N m C m r ( 1 ) m = 0 m N m C m θ ( 1 ) m m = { L 8 π R 4 1 8 , = 0 1 8 , = 1 0 , 2 .
min { C m r , θ } , n 0 , L , t t subject to : { constraint 23 n min n 0 p ( x ) tr 𝒥 T 𝒥 2 ( det 𝒥 ) 2 n max for x G R = R 0 𝕂 ( x ) t for x G .
T i j = | R R + 1 d x θ ^ ( E j 0 × H i ) | y = L 2 2 ,
W = 1 1 d x ( x x ) 2 + ( y x ) 2 | y = L 2 .
x ( x , y ) = x + , m N , N m C m x T ( x ) sin ( 2 m + 1 ) π y L
y ( x , y ) = y + , m N , N m C m y T ( x ) sin ( 2 m + 1 ) π y L ,
𝒥 = ( 1 0 δ 1 + Δ ) ,
𝕂 1 = 1 + δ 2 + ( 1 + Δ ) 2 2 ( 1 + Δ ) 1 = 1 2 ( δ 2 + Δ 2 ) + 𝒪 ( δ 2 Δ + Δ 3 ) .
det 𝒥 1 = Δ
= 2 ( 𝕂 1 ) δ 2 + 𝒪 ( δ 2 Δ + Δ 3 )
= 𝒪 ( 𝕂 1 ) .

Metrics