4 Mapeamento Quase-Conforme com Parametrização para Transformada
6.1 Trabalhos Futuros
Como trabalhos futuros, pode-se aplicar o m´etodo dos m´ınimos quadrados para se obter uma transforma¸c˜ao de coordenadas quase-conforme para a cria¸c˜ao
de meios de invisibilidade no espa¸co livre. Esse procedimento pode levar a um meio com baixa anisotropia e que funciona como uma capa de invisibilidade.
´
E poss´ıvel realizar uma otimiza¸c˜ao num´erica ao inv´es de utilizar regress˜ao ou interpola¸c˜ao para a redu¸c˜ao da anisotropia utilizando a modelagem das fun¸c˜oes de transforma¸c˜ao de coordenadas propostas. Este procedimento permite introduzir restri¸c˜oes nos limites do ´ındice de refra¸c˜ao ou ainda sobre a transforma¸c˜ao de coordenadas. Al´em disso, outras fun¸c˜oes de pertuba¸c˜ao, na forma de s´eries, podem ser utilizadas para a regress˜ao e interpola¸c˜ao.
As t´ecnicas de mapeamento quase-conforme apresentadas neste trabalho s˜ao t´ecnicas em 2D. No entanto essas t´ecnicas podem ser estendidas para 3D uma vez conhecidos quais condi¸c˜oes devem ser satisfeitas para obter transforma¸c˜oes de coordenadas conformes em 3D.
A TO pode ser estendida para aplica¸c˜ao em meios n˜ao lineares (PAUL; RAHM, 2012). Esta abordagem pode permitir novas funcionalidades como a cria¸c˜ao de portas l´ogicas ´opticas para o desenvolvimento da computa¸c˜ao ´optica.
Referências
AHLFORS, L. Lectures on Quasiconformal Mappings. USA: American Mathematical Society, 1966. (University lecture series). ISBN 9780821836446. AKAI, T. J. Applied Numerical Methods for Engineers. New York, NY: John Wiley & Sons, 1993.
ASTALA, K.; IWANIEC, T.; MARTIN, G. Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane (PMS-48). Princeton, NJ: Princeton University Press, 2008. (Princeton Mathematical Series). ISBN 9781400830114. ATKINSON, K. E. An Introduction to Numerical Analysis. New York, NY: John Wiley & Sons, 1988. ISBN 0471500232.
BARNES, W. L.; DEREUX, A.; EBBESEN, T. W. Surface plasmon subwavelength optics. Nature, v. 424, p. 824–830, ago. 2003.
BELTRAMI, D. R. et al. Fabrication and characterization of a planar gradient- index, plasma-enhanced chemical vapor deposition lens. Applied Optics, OSA, v. 36, n. 28, p. 7143–7145, Oct 1997.
BORN, M.; WOLF, E. Principles of Optics. 7. ed. Cambridge, UK: Cambridge University Press, 1999. ISBN 9780521642224.
BORN, M. et al. Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light. Cambridge, UK: Cambridge University Press, 1999. ISBN 9780521784498.
BOYD, R. W.; GAUTHIER, D. “Slow” and “Fast” light. New York: Elsevier Science, 2002.
BRETSCHER, O. Linear Algebra With Applications. Upper Saddle River, NJ: Prentice Hall, 1995.
CAI, W. et al. Nonmagnetic cloak with minimized scattering. Applied Physics Letters, v. 91, n. 11, 2007.
CHANG, Z. et al. Design method for quasi-isotropic transformation materials based on inverse Laplace’s equation with sliding boundaries. Optics Express, OSA, v. 18, n. 6, p. 6089–6096, Mar 2010.
CHEN, H.; CHAN, C. T. Acoustic cloaking and transformation acoustics. Journal of Physics D: Applied Physics, v. 43, n. 11, p. 113001, 2010.
CHEN, X.; FU, Y.; YUAN, N. Invisible cloak design with controlled constitutive parameters and arbitrary shaped boundaries through Helmholtz’s equation. Optics Express, OSA, v. 17, n. 5, p. 3581–3586, Mar 2009.
DACOROGNA, B. Introduction to the Calculus of Variations. 2. ed. Londres, UK: Imperial College Press, 2009. ISBN 9781848163331.
DAHLQUIST, G.; BJORCK, A. Numerical Methods. New York, NY: Dover Publications, 2003. (Dover Books on Mathematics). ISBN 9780486428079. ENGHETA, N.; ZIOLKOWSKI, R. W. Eletromagnetic Metamaterials: Physics and Engineering Explorations. New York, NY: John Wiley & Sons, 2006. ISBN 9780471784180.
FEDER, D. P. Automatic optical design. Applied Optics, OSA, v. 2, n. 12, p. 1209–1226, Dez 1963.
FENG, Y.; XU, X.; YU, Z. Practical realization of transformation-optics designed invisibility cloak through layered structures. In: Antennas and Propagation (EUCAP), Proceedings of the 5th European Conference on. Roma, It: [s.n.], 2011. p. 3456–3460.
GABRIELLI, L. H. et al. Silicon nanostructure cloak operating at optical frequencies. Nature Photonics, v. 3, p. 461–463, ago. 2009.
GABRIELLI, L. H. et al. On-chip transformation optics for multimode waveguide bends. Nature Communications, Nature Publishing Group, v. 3, p. 1217, 2012. GARCIA, A.; STICHTENOTH, H. On chebyshev polynomials and maximal curves. Acta Arithmetica, Institute of Mathematics Polish Academy of Sciences, v. 90, n. 4, p. 301–311, 1999.
GARC´IA-MECA, C. et al. Negative refractive index metamaterials aided byextraordinary optical transmission. Optics Express, OSA, v. 17, n. 8, p. 6026–6031, Apr 2009.
GARCIA, N.; NIETO-VESPERINAS, M. Left-handed materials do not make a perfect lens. Physical Review Letters, v. 88, n. 20, p. 207403, may 2002.
GHATAK, A. Optics. 4. ed. New York, NY: McGraw-Hill, 2009. ISBN 978-0073380483.
GOLDBERG, D. Genetic Algorithms. [S.l.]: Pearson Education, 2006. ISBN 9788177588293.
HAMMING, R. W. Introduction to numerical analysis. New York, NY: Dover Publications, 1987. (Dover Books on Mathematics). ISBN 978-0486652412. H ¨ARDLE, W. Applied Nonparametric Regression. Cambridge, UK: Cambridge University Press, 1991. (Econometric Society Monographs). ISBN 9780521429504. HUNSPERGER, R. Integrated Optics: Theory and Technology. New York, USA: Springer, 2009. (Advanced texts in physics). ISBN 9780387897752.
JIN, J. The Finite Element Method in Electromagnetics. 3rd. ed. [S.l.]: Wiley-IEEE Press, 2014. ISBN 1118571363, 9781118571361.
KUNDTZ, N.; SMITH, D.; PENDRY, J. Electromagnetic design with
transformation optics. Proceedings of the IEEE, v. 99, n. 10, p. 1622–1633, Oct 2011. ISSN 0018-9219.
LANDY, N. I.; PADILLA, W. J. Guiding light with conformal transformations. Optics Express, OSA, v. 17, n. 17, p. 14872–14879, Aug 2009.
LAWDEN, D. F. Introduction to Tensor Calculus, Relativity and Cosmology. 3. ed. Nova York, NY: Dover Publications, 2003. (Dover Books on Physics). ISBN 9780486425405.
LEE, C. J.; LIN, J. F. Lens designs by ray tracing analyses for high-performance reflection optical modules. Journal of Optics, v. 12, n. 9, p. 095501, 2010.
LEONHARDT, U. Optical conformal mapping. Science, 312, n. 5781, p. 1777–80, 2006.
LEONHARDT, U. Optical metamaterials - invisibility cup. Nature Photonics, v. 1, p. 207 – 208, 2007.
LEONHARDT, U. Perfect imaging without negative refraction. New Journal of Physics, v. 11, n. 9, p. 093040, 2009.
LEONHARDT, U.; PHILBIN, T. Geometry and Light: The Science of Invisibility. Mineola, NY: Dover Publications, 2012. (Dover Books on Physics). ISBN 9780486476933.
LEONHARDT, U.; PHILBIN, T. G. General relativity in electrical engineering. New Journal of Physics, 8, n. 10, p. 247, 2006.
LEVINSON, H. J. Principles of Lithography. 3. ed. Bellingham, WA: SPIE Press, 2011. (Press Monographs). ISBN 9780819456601.
LI, J.; PENDRY, J. B. Hiding under the Carpet: A New Strategy for Cloaking. Physical Review Letters, v. 101, n. 20, p. 203901, nov. 2008.
LI, Z. et al. Composite chiral metamaterials with negative refractive index and high values of the figure of merit. Optics Express, OSA, v. 20, n. 6, p. 6146–6156, Mar 2012.
LIGHTHILL, J. An Introduction to Fourier Analysis and Generalised Functions. [S.l.]: Cambridge University Press, 1964. (Cambridge Monographs on Mechanics). ISBN 9780521091282.
LIU, D. et al. Transformation inverse design. Optics Express, OSA, v. 21, n. 12, p. 14223–14243, Jun 2013.
LIU, S. et al. Tunable ultrathin mantle cloak via varactor-diode-loaded metasurface. Optics Express, OSA, v. 22, n. 11, p. 13403–13417, Jun 2014. LUNEBURG, R.; HERZBERGER, M. Mathematical Theory of Optics. Oakland, CA: University of California Press, 1964. (Economic forces in American history). MA, H. F.; CUI, T. J. Three-dimensional broadband and broad-angle
transformation-optics lens. Nat Commun, v. 1, n. 8, p. 124, 2010. ISSN 2041-1723. MA, H. F. et al. Compact-sized and broadband carpet cloak and free-space cloak. Optics Express, OSA, v. 17, n. 22, p. 19947–19959, Oct 2009.
MACKAY, T. G.; LAKHTAKIA, A. Electromagnetic Anisotropy and
Bianisotropy: A Field Guide. New York: World Scientific Publishing Company, 2009.
MALING, D. Coordinate Systems and Map Projections. Cambridge, MA: Elsevier Science & Technology Books, 1992. ISBN 9780080372341.
MARSHALL, G. F. Optical Scanning. New York, NY: Marcel Dekker Inc., 1991. (Optical Engineering, v. 31). ISBN 0824784731.
MEI, Z. L.; CUI, T. J. Experimental realization of a broadband bend structure using gradient index metamaterials. Optics Express, OSA, v. 17, n. 20, p. 18354–18363, Sep 2009.
MERZBACHER, E. Quantum Mechanics. New York, NY: John Wiley & Sons, 1998. ISBN 9780471887027.
MILONNI, P. Fast Light, Slow Light and Left-Handed Light. [S.l.]: CRC Press, 2004. (Series in Optics and Optoelectronics). ISBN 9781420034332.
MONTGOMERY, G. C. R. D. C. Applied Statistics and Probability for Engineers. 5. ed. New York, NY: John Wiley & Sons, 2010. ISBN 0470053046.
MOORE, D. T. Gradient-index optics: a review. Applied Optics, OSA, v. 19, n. 7, p. 1035–1038, Abr 1980.
NETO, J. B. Mecˆanica Newtoniana, Lagrangiana e Hamiltoniana. S˜ao Paulo, SP: Editora Livraria da F´ısica, 2004. ISBN 9788588325265.
PAUL, O.; RAHM, M. Covariant description of transformation optics in nonlinear media. Opt. Express, OSA, v. 20, n. 8, p. 8982–8997, Apr 2012.
PENDRY, J. B. Negative refraction makes a perfect lens. Phys. Rev. Lett., American Physical Society, v. 85, p. 3966–3969, Oct 2000.
PENDRY, J. B.; SCHURIG, D.; SMITH, D. R. Controlling Electromagnetic Fields. Science, 312, n. 23, p. 1780–1782, 2006.
PFEIFFER, C.; GRBIC, A. A printed, broadband luneburg lens antenna. Antennas and Propagation, IEEE Transactions on, v. 58, n. 9, p. 3055–3059, Sept 2010. ISSN 0018-926X.
POWELL, M. A direct search optimization method that models the objective and constraint functions by linear interpolation. In: GOMEZ, S.; HENNART, J.-P. (Ed.). Advances in Optimization and Numerical Analysis. [S.l.]: Springer Netherlands, 1994, (Mathematics and Its Applications, v. 275). p. 51–67. ISBN 978-90-481-4358-0.
PRESSLEY, A. Elementary Differential Geometry. Londres, UK: Springer, 2001. (Springer undergraduate mathematics series). ISBN 9781848828902.
RAHM, M. et al. Optical Design of Reflectionless Complex Media by Finite Embedded Coordinate Transformations. Physical Review Letters, v. 100, n. 6, p. 063903, fev. 2008.
RAHM, M. et al. Transformation-optical design of adaptive beam bends and beam expanders. Optics Express, OSA, v. 16, n. 15, p. 11555–11567, Jul 2008.
RAMM, A. Does negative refraction make a perfect lens? Physics Letters A, v. 372, n. 43, p. 6518 – 6520, 2008. ISSN 0375-9601.
REIF, J.; TYGAR, J.; YOSHIDA, A. Computability and complexity of ray tracing. Discrete & Computational Geometry, Springer-Verlag, v. 11, n. 1, p. 265–288, 1994. ISSN 0179-5376.
REITZ, J. R.; MILFORD, F. J.; CHRISTY, R. W. Foundations of Electromagnetic Theory. 4. ed. Boston, MA: Addison-Wesley, 2008. ISBN 9780321581747.
RIBEIRO, J. Propaga¸c˜ao das ondas eletromagn´eticas: principios e aplica¸c˜oes. 2. ed. S˜ao Paulo, SP: ´Erica, 2008. ISBN 9788571949935.
RIBEIRO, J. Comunica¸coes ´opticas. 4. ed. S˜ao Paulo, SP: ´Erica, 2009. ISBN 9788571949652.
RICCI, M.; LEVI-CIVITA, T. M´ethodes de calcul diff´erentiel absolu et leurs applications. Mathematische Annalen, v. 54, p. 125–201, 1901.
RUDIN, W. Principles of Mathematical Analysis. New York, NY: McGraw-Hill, 1976. ISBN 9780070542358.
SANCHEZ, E. Tensores. 1. ed. Rio de Janeiro, RJ: Interciˆencia, 2007. ISBN 9788571931787.
SANCHEZ, E. C´alculo Tensorial. 1. ed. Rio de Janeiro, RJ: Interciˆencia, 2011. ISBN 9788571932517.
SCHRADER, K. N. et al. Ray tracing in a finite-element domain using nodal basis functions. Applied Optics, OSA, v. 53, n. 24, p. F10–F20, Aug 2014.
SCHURIG, D. et al. Metamaterial electromagnetic cloak at microwave frequencies. Science, v. 314, n. 5801, p. 977–980, Nov 2006.
SCHUTZ, B. A First Course in General Relativity. Cambridge, UK: Cambridge University Press, 1985. (Series in physics). ISBN 9780521277037.
SHILOV, G. Elementary Real and Complex Analysis. 2. ed. New York: Dover Publications, 1996. (Dover books on mathematics, v. 2). ISBN 9780486689227. SHINGYOUCHI, K.; KONISHI, S. Gradient-index doped silica rod lenses produced by a solgel method. Applied Optics, OSA, v. 29, n. 28, p. 4061–4063, Oct 1990.
SMITH, J. Mathematics of the Discrete Fourier Transform. 2. ed. Stanford, CA: W3K Publishing, 2007. ISBN 9780974560748.
SORIC, J. C. et al. Demonstration of an ultralow profile cloak for scattering suppression of a finite-length rod in free space. New Journal of Physics, v. 15, n. 3, p. 033037, 2013.
SPADOTI, D. H. et al. Focusing light in a curved-space. Optics Express, OSA, v. 18, n. 3, p. 3181–3186, Feb 2010.
SPENCER, G. H.; MURTY, M. V. R. K. General ray-tracing procedure. J. Optics Soc. Am., OSA, v. 52, n. 6, p. 672–676, Jun 1962.
STEFFENSEN, J. F. Interpolation. 2. ed. New York, NY: Dover Publications, 2006. (Dover Books on Mathematics).
STEINBERG, S. Fundamentals of Grid Generation. Abingdon, OX: Taylor & Francis, 1993. (The Fundamentals of Grid Generation). ISBN 9780849389870. SVAITER, N.; MENEZES, G.; ALCALDE, M. T´opicos em teoria quˆantica dos campos. Rio de Janeiro, RJ: Centro Brasileiro de Pesquisas F´ısicas, 2006. SWAINSON, C. A.; MAXWELL, J. C. Problems. Cambridge Dublin Math, p. 188–189, 1854.
TAO, H. et al. Recent progress in electromagnetic metamaterial devices for terahertz applications. Selected Topics in Quantum Electronics, IEEE Journal of, v. 17, n. 1, p. 92–101, Jan 2011. ISSN 1077-260X.
VENKATARAMAN, P. Applied Optimization with MATLAB Programming. New Jersey: Wiley, 2002. (A Wiley-Interscience publication). ISBN 9780471349587. WU, Q.; TURPIN, J.; WERNER, D. Quasi-conformal transformation
electromagnetics enabled flat collimating lenses. In: Antennas and Propagation Society International Symposium (APSURSI), 2012 IEEE. [S.l.: s.n.], 2012. p. 1–2. ISSN 1522-3965.
XUE, L.; FUSCO, V. 24 GHz automotive radar planar Luneburg lens.
Microwaves, Antennas Propagation, IET, v. 1, n. 3, p. 624–628, June 2007. ISSN 1751-8725.
YAN, W.; YAN, M.; QIU, M. Necessary and sufficient conditions for reflectionless transformation media in an isotropic and homogenous background. ArXiv e-prints, jun 2008.
ZHANG, P.; JIN, Y.; HE, S. Inverse transformation optics and reflection analysis for two-dimensional finite-embedded coordinate transformation. Selected Topics in Quantum Electronics, IEEE Journal of, v. 16, n. 2, p. 427–432, March 2010. ISSN 1077-260X.
ZHANG, X.; LIU, Z. Superlenses to overcome the diffraction limit. Nat. Mater., v. 7, n. 6, p. 435–441, 2008.
ZYGMUND, A. Trigonometric interpolation. Chicago, IL: University of Chicago, 1950. (Trigonometric Interpolation, v. 1-2).