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.

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2013 (4)

2012 (14)

L. Gabrielli, D. Liu, S. G. Johnson, and M. Lipson, “On-chip transformation optics for multimode waveguide bends,” Nat. Commun. 3(1217) (2012).
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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. Express 20(12):13006–13013 (2012).
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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. Express 20(19):21558–21575 (2012).
[Crossref] [PubMed]

Y. Liu and X. Zhang, “Recent advances in transformation optics,” Nanoscale 4(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,” Science 337(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. Express 29(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. Express 20(23):25758–25764 (2012).
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S. Wang and S. Liu, “Controlling electromagnetic scattering of a cavity by transformation media,” Opt. Express 20(6):6777–6785 (2012).
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Q. Wu, J. P. Turpin, and D. H. Werner, “Integrated photonic systems based on transformation optics enabled gradient index devices,” Light: Science and Applications 1(e38) (2012).

J. Mei, Q. Wu, and K. Zhang, “Multimultifunctional complementary cloak with homogeneous anisotropic material parameters,” J. Opt. Soc. Am. A 29(10):2067–2073 (2012).
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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).
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P. Markov, J. G. Valentine, and S. M. Weiss, “Fiber-to-chip coupler designed using an optical transformation,” Opt. Express 20(13):14705–14712 (2012).
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2011 (16)

L. H. Gabrielli and M. Lipson, “Transformation optics on a silicon platform,” J. Opt. 13(024010) (2011).
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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. Express 19(16):15119–15126 (2011).
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A. V. Novitsky, “Inverse problem in transformation optics,”J. Opt. 13(035104) (2011).
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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. Express 19(4):3562–3757 (2011).
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B. Zhang, Y. Luo, X. Liu, and G. Barbastathis, “Macroscopic invisibility cloak for visible light,” Phys. Rev. Lett. 106(033901) (2011).
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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. Express 19(24):23743–23750 (2011).
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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. B 28(5):1037–1042 (2011).
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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. Express 19(14):13020–13030 (2011).
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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. Nanophotonics 5(051806) (2011).
[Crossref]

L. Gabrielli and M. Lipson, “Integrated luneburg lens via ultra-strong index gradient on silicon,” Opt. Express 19(21):20122–20127 (2011).
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2010 (13)

M. Schmiele, V. S. Varma, C. Rockstuhl, and F. Lederer, “Designing optical elements from isotropic materials by using transformation optics,” Phys. Rev. A 81(033837) (2010).
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K. Astala, T. Iwaniec, and G. Martin, “Deformations of annuli with smallest mean distortion,” Arch. Rational Mech. Anal. 195:899–921 (2010).
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H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9:387–396 (2010).
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Y. G. Ma, N. Wang, and C. K. Ong, “Application of inverse, strict conformal transformation to design waveguide devices,” J. Opt. Soc. Am. A 27:968–972 (2010).
[Crossref]

H. F. Ma and T. J. Cui, “Three-dimensional broadband and broad-angle transformation-optics lens,” Nat. Commun. 1(124) (2010).
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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).
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N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater. 9:129–132 (2010).
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T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science 328(5976):337–339 (2010).
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J. Lu and J. Vuckovic, “Inverse design of nanophotonic structures using complementary convex optimization,” Opt. Express 18(4):3793–3804 (2010).
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Z. Chang, X. Zhou, J. Hu, and G. Hu, “Design method for quasi-isotropic transformation materials based on inverse Laplace’s equation with sliding boundaries,” Opt. Express 18(6):6089–6096 (2010).
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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. Express 18(1):244–252 (2010).
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D. R. Smith, Y. Urzhumov, N. B. Kundtz, and N. I. Landy, “Enhancing imaging systems using transformation optics,” Opt. Express 18(20):21238–21251 (2010).
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2009 (14)

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. Express 17(15):12922–12928 (2009).
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U. Leonhardt and T. Tyc, “Broadband invisibility by non-euclidian cloaking,” Science 323(5910):110–112 (2009).
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R. Liu, R. C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323(5912):366–369 (2009).
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J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8:568–571 (2009).
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L. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics 3:461–463 (2009).
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B. Vasić, G. Isić, R. Gajić, and K. Hingerl, “Coordinate transformation based design of confined metamaterial structures,” Phys. Rev. B 79(085103) (2009).
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N. I. Landy and W. J. Padilla, “Guiding light with conformal transformations,” Opt. Express 17(17):14872–14879 (2009).
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Z. L. Mei and T. J. Cui, “Experimental realization of a broadband bend structure using gradient index metamaterials,” Opt. Express 17(20):18354–18363 (2009).
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Z. L. Mei and T. J. Cui, “Arbitrary bending of electromagnetic waves using isotropic materials,” J. Appl. Phys. 105(104913) (2009).
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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).
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W. Zeng, F. Luo, S. T. Yau, and X. D. Gu, “Surface quasi-conformal mapping by solving Beltrami equations,” Proc. Math. Surfaces XIII391–408 (2009).
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B. Chen, T. Tang, and H. Chen, “Study on a compact flexible photonic crystal waveguide and its bends,” Opt. Express 17(7):5033–5038 (2009).
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L. Bergamin, “Electromagnetic fields and boundary conditions at the interface of generalized transformation media,” Phys. Rev. A 80(063835) (2009).
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U. Leonhardt and T. G. Philbin, “Transformation optics and the geometry of light,” Prog. Opt. 53:69–152 (2009).
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2008 (14)

W. Yan, M. Yan, Z. Ruan, and M. Qiu, “Coordinate transformations make perfect invisibility cloaks with arbitrary shape,” New J. Phys. 10(043040) (2008).
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C. Ma, Q. Zhang, and E. V. Keuren, “Right-angle slot waveguide bends with high bending efficiency,” Opt. Express 16(19):14330 (2008).
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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. Lett 100(063903) (2008).
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M. Rahm, D. A. Roberts, J. B. Pendry, and D. R. Smith, “Transformation-optical design of adaptive beam bends and beam expanders,” Opt. Express 16(15):11555–11567 (2008).
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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).
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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).
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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).
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J. Riishede and O. Sigmund, “Inverse design of dispersion compensating optical fiber using topology optimization,” J. Opt. Soc. Am. B 25(1):88–97 (2008).
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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).
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W. R. Frei, H. T. Johnson, and D. A. Tortorelli, “Optimization of photonic nanostructures,” Comput. Method. Appl. M. 197(41):3410–3416 (2008).
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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).
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C. Y. Kao and F. Santosa, “Maximization of the quality factor of an optical resonator,” Wave motion 45(4):412–427 (2008).
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J. Li and J. B. Pendry, “Hiding under the carpet: A new strategy for cloaking,” Phys. Rev. Lett 101(203901) (2008).
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X. Zhang, H. Chen, X. Luo, and H. Ma, “Transformation media that turn a narrow slit into a large window,” Opt. Express 16(16):11764–11768 (2008).
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2007 (6)

O. Ozgun and M. Kuzuoglu, “Utilization of anisotropic metamaterial layers in waveguide miniaturization and transitions,” IEEE Microw. Wirel. Compon. Lett. 17(754) (2007).
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C. Y. Kao and S. Osher, “Incorporating topological derivatives into shape derivatives based level set methods,” J. Comp. Phys. 225(1):891–909 (2007).
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H. Kurt and D. S. Citrin, “Graded index photonic crystals,” Opt. Express 15(3):1240–1253 (2007).
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D. Schurig, J. B. Pendry, and D. R. Smith, “Transformation-designed optical elements,” Opt. Express 15(22):14772–14782 (2007).
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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).
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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).
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2006 (6)

Y. Zhang and B. Li, “Photonic crystal-based bending waveguides for optical interconnections,” Opt. Express 14(12):5723–5732 (2006).
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U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” New J. Phys. 8(10) (2006).
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P. Seliger, M. Mahvash, C. Wang, and A. F. J. Levi, “Optimization of aperiodic dielectric structures,” J. Appl. Phys. 100(034310) (2006).
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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,” Science 314(5801):977–980 (2006).
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U. Leonhardt, “Optical conformal mapping,” Science 312:1777–1780 (2006).
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J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312:1780–1782 (2006).
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2005 (5)

C. Y. Yao, S. Osher, and E. Yablonovitch, “Maximizing band gaps in two-dimensional photonic crystals by using level set methods,” Appl. Phys. B 81(2):235–244 (2005).
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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).
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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. B 81(283–293) (2005).
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J. S. Jensen and O. Sigmund, “Topology optimization of photonic crystal structures: a high-bandwidth low-loss T-junction waveguide,” J. Opt. Soc. Am. B 22(6):1191–1198 (2005).
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D. Smith, J. Mock, A. Starr, and D. Schurig, “Gradient index metamaterials,” Phys. Rev. E 71(036609) (2005).
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2004 (5)

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305(5685):788–792 (2004).
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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).
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P. Borel, A. Harpøth, L. Frandsen, M. Kristensen, P. Shi, J. Jensen, and O. Sigmund, “Topology optimization and fabrication of photonic crystal structures,” Opt. Express 12(9):1996 (2004).
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L. Frandsen, A. Harpøth, P. Borel, M. Kristensen, J. Jensen, and O. Sigmund, “Broadband photonic crystal waveguide 60° bend obtained utilizing topology optimization,” Opt. Express 12(24):5916–5921 (2004).
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D. C. Dobson and F. Santosa, “Optimal localization of eigenfunctions in an inhomogeneous medium,” SIAM J. Appl. Math 64(3):762–774 (2004).
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2003 (3)

A. Greenleaf, M. Lassas, and G. Uhlmann, “Anisotropic conductivities that cannot be detected by EIT,” Physiol. Meas. 24(413):413–419 (2003).
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J. Smajic, C. Hafner, and D. Erni, “Design and optimization of an achromatic photonic crystal bend,” Opt. Express 11(12):1378–1384 (2003).
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2002 (3)

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).
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J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, “Design of photonic crystal microcavities for cavity QED” Phys. Rev. E 65(016608) (2002).

2000 (1)

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

1999 (3)

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).
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Andkjaer, J.

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

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

J. B. Pendry, A. Aubry, D. R. Smith, and S. A. Maier, “Transformation optics and subwavelength control of light,” Science 337(6094):549–552 (2012).
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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).
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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).
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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).
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Barbastathis, G.

H. Xu, B. Zhang, Y. Yu, G. Barbastathis, and H. Sun, “Dielectric waveguide bending adapter with ideal transmission,” Opt. Express 29(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).
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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)

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× 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 ) .

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