volume | PIER Journals

Abstract

Spatial transform techniques like transformation optics and conformal mapping have arisen as the dominant techniques for designing metamaterial devices. However, these techniques only produce the electrical permittivity and permeability as a function of position. The manner in which these functions are converted into physical metamaterial lattices remains elusive, except in some simple or canonical configurations. Metamaterial lattices designed by spatial transforms are composed of elements of different sizes, orientations, and designs. The elements must be distributed and oriented in a manner that makes the final lattice smooth, continuous, have uniform density, be free of unintentional defects, and have minimal distortions to the elements. Any of these would weaken or destroy the electromagnetic properties of the lattice. This paper describes a general purpose method to generate such arbitrary metamaterial lattices. Inputs to the algorithm are the permittivity and permeability functions as well as the baseline metamaterials that can provide the necessary permittivity and permeability values. In prior research, we reported a simple finite-difference technique for calculating the permittivity and permeability functions for arbitrary shaped devices using transformation optics. The present work is illustrated by generating an electromagnetic cloak of arbitrary shape that was designed using the previously reported technique. The final metamaterial cloak is simulated using the finite-difference time-domain method and performance compared to other cloaks reported in the literature.

1. Pendry, J. B., D. Schurig, and D. R. Smith, "Controlling electromagnetic fields," Science, Vol. 312, 1780-1782, New York, N.Y., Jun. 2006.
doi:10.1126/science.1125907

2. Kwon, D.-H. and D. H. Werner, "Transformation electromagnetics: An overview of the theory and applications," IEEE Antennas and Propagation Magazine, Vol. 52, No. 1, 24-46, 2010.
doi:10.1109/MAP.2010.5466396

3. Leonhardt, U. and T. G. Philbin, "Transformation optics and the geometry of light," Progress in Optics, Vol. 53, No. 8, 69-152, 2009.
doi:10.1016/S0079-6638(08)00202-3

4. Valentine, J., J. Li, T. Zentgraf, G. Bartal, and X. Zhang, "An optical cloak made of dielectrics," Nature Materials, Vol. 8, No. 7, 568, 2009.
doi:10.1038/nmat2461

5. Ergin, T., N. Stenger, Brenner, J. B. Pendry, and M. Wegener, "Three-dimensional invisibility cloak at optical wavelengths," Science, 1186351, 2010.

6. Gabrielli, L. H., J. Cardenas, C. B. Poitras, and M. Lipson, "Silicon nanostructure cloak operating at optical frequencies," Nature Photonics, Vol. 3, No. 8, 461, 2009.
doi:10.1038/nphoton.2009.117

7. Liu, R., C. Ji, J. Mock, J. Chin, T. Cui, and D. Smith, "Broadband ground-plane cloak," Science, Vol. 323, No. 5912, 366-369, 2009.
doi:10.1126/science.1166949

8. Rahm, M., D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, "Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell's equations," Photonics and Nanostructures-fundamentals and Applications, Vol. 6, No. 1, 87-95, 2008.

9. Li, J. and J. B. Pendry, "Hiding under the carpet: a new strategy for cloaking," Physical Review Letters, Vol. 101, No. 20, 203901, 2008.
doi:10.1103/PhysRevLett.101.203901

10. Ma, H. F. and T. J. Cui, "Three-dimensional broadband and broad-angle transformation-optics lens," Nature communications, Vol. 1, 124, 2010.
doi:10.1038/ncomms1126

11. Roberts, D., N. Kundtz, and D. Smith, "Optical lens compression via transformation optics," Optics Express, Vol. 17, No. 19, 16535-16542, 2009.
doi:10.1364/OE.17.016535

12. Rahm, M., D. Roberts, J. Pendry, and D. Smith, "Transformation-optical design of adaptive beam bends and beam expanders," Optics Express, Vol. 16, No. 15, 11555-11567, 2008.
doi:10.1364/OE.16.011555

13. Rahm, M., S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, "Optical design of reflectionless complex media by finite embedded coordinate transformations," Physical Review Letters, Vol. 100, No. 6, 063903, 2008.
doi:10.1103/PhysRevLett.100.063903

14. Kwon, D.-H. and D. H. Werner, "Polarization splitter and polarization rotator designs based on transformation optics," Optics Express, Vol. 16, No. 23, 18731-18738, 2008.
doi:10.1364/OE.16.018731

15. Jiang, W. X., T. J. Cui, Q. Cheng, J. Y. Chin, X. M. Yang, R. Liu, and D. R. Smith, "Design of arbitrarily shaped concentrators based on conformally optical transformation of nonuniform rational b-spline surfaces," Applied Physics Letters, Vol. 92, No. 26, 264101, 2008.
doi:10.1063/1.2951485

16. Yang, J., M. Huang, C. Yang, Z. Xiao, and J. Peng, "Metamaterial electromagnetic concentrators with arbitrary geometries," Optics Express, Vol. 17, No. 22, 19656-19661, 2009.
doi:10.1364/OE.17.019656

17. Schurig, D., J. Mock, B. Justice, S. A. Cummer, J. B. Pendry, A. Starr, and D. Smith, "Metamaterial electromagnetic cloak at microwave frequencies," Science, Vol. 314, No. 5801, 977-980, 2006.
doi:10.1126/science.1133628

18. Hu, J., X. Zhou, and G. Hu, "Design method for electromagnetic cloak with arbitrary shapes based on laplace's equation," Optics Express, Vol. 17, No. 3, 1308-1320, 2009.
doi:10.1364/OE.17.001308

19. Berry, E. A., J. J. Gutierrez, and R. C. Rumpf, "Design and simulation of arbitrarily-shaped transformation optic devices using a simple finite-difference method," Progress In Electromagnetics Research, Vol. 68, 1-16, 2016.

20. Chen, H., C. T. Chan, and Sheng, "Transformation optics and metamaterials," Nature Materials, Vol. 9, No. 5, 387, 2010.
doi:10.1038/nmat2743

21. Mei, Z.-L., J. Bai, T. M. Niu, and T.-J. Cui, "A planar focusing antenna design with the quasi-conformal mapping," Progress In Electromagnetics Research, Vol. 13, 261-273, 2010.
doi:10.2528/PIERM10053102

22. Pendry, J. B., A. Holden, W. Stewart, and I. Youngs, "Extremely low frequency plasmons in metallic mesostructures," Physical Review Letters, Vol. 76, No. 25, 4773, 1996.
doi:10.1103/PhysRevLett.76.4773

23. Pendry, J. B., A. J. Holden, D. J. Robbins, and W. Stewart, "Magnetism from conductors and enhanced nonlinear phenomena," IEEE Transactions on Microwave Theory and Techniques, Vol. 47, No. 11, 2075-2084, 1999.
doi:10.1109/22.798002

24. Pedrola, G. L., Beam Propagation Method for Design of Optical Waveguide Devices, John Wiley & Sons, 2015.
doi:10.1002/9781119083405

25. Basser, J., J. Mattiello, and D. LeBihan, "Mr diffusion tensor spectroscopy and imaging," Biophysical Journal, Vol. 66, No. 1, 259-267, 1994.
doi:10.1016/S0006-3495(94)80775-1

26. Nye, J. F., Physical Properties of Crystals: Their Representation by Tensors and Matrices, Oxford University Press, 1985.

27. Kuprel, B. and A. Grbic, "Anisotropic inhomogeneous metamaterials using nonuniform transmission-line grids aligned with the principal axes," IEEE Antennas and Wireless Propagation Letters, Vol. 11, 358-361, 2012.
doi:10.1109/LAWP.2012.2191257

28. Lam, T. A., D. C. Vier, J. A. Nielsen, C. G. Parazzoli, and M. H. Tanielian, "Steering phased array antenna beams to the horizon using a buckyball nim lens," Proceedings of the IEEE, Vol. 99, No. 10, 1755-1767, 2011.
doi:10.1109/JPROC.2011.2128290

29. Ansys, H., "v15," ANSYS Corporation Software, Pittsburgh, PA, USA, 2014.

30. Smith, D. R., S. Schultz, Markoš, and C. Soukoulis, "Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients," Physical Review B, Vol. 65, No. 19, 195104, 2002.
doi:10.1103/PhysRevB.65.195104

31. Chen, X., T. M. Grzegorczyk, B.-I. Wu, J. Pacheco, Jr, and J. A. Kong, "Robust method to retrieve the constitutive effective parameters of metamaterials," Physical Review E, Vol. 70, No. 1, 016608, 2004.
doi:10.1103/PhysRevE.70.016608

32. Liu, R., T. J. Cui, D. Huang, B. Zhao, and D. R. Smith, "Description and explanation of electromagnetic behaviors in artificial metamaterials based on effective medium theory," Physical Review E, Vol. 76, No. 2, 026606, 2007.
doi:10.1103/PhysRevE.76.026606

33. Smith, D., D. Vier, T. Koschny, and C. Soukoulis, "Electromagnetic parameter retrieval from inhomogeneous metamaterials," Physical Review E, Vol. 71, No. 3, 036617, 2005.
doi:10.1103/PhysRevE.71.036617

34. Barton, J. H., C. R. Garcia, E. A. Berry, R. Salas, and R. C. Rumpf, "3-d printed all-dielectric frequency selective surface with large bandwidth and field of view," IEEE Transactions on Antennas and Propagation, Vol. 63, 1032-1039, March 2015.
doi:10.1109/TAP.2015.2388541

35. Fraser, A. S., "Simulation of genetic systems by automatic digital computers vi. epistasis," Australian Journal of Biological Sciences, Vol. 13, No. 2, 150-162, 1960.
doi:10.1071/BI9600150

36. Clerc, M., Particle Swarm Optimization, Vol. 93, John Wiley & Sons, 2010.

37. Rumpf, R. C., C. R. Garcia, E. A. Berry, and J. H. Barton, "Finite-difference frequency-domain algorithm for modeling electromagnetic scattering from general anisotropic objects," Progress In Electromagnetics Research, Vol. 61, 55-67, 2014.
doi:10.2528/PIERB14071606

38. Rumpf, R. C. and J. Pazos, "Synthesis of spatially variant lattices," Optics Express, Vol. 20, No. 14, 15263-15274, 2012.
doi:10.1364/OE.20.015263

39. Rumpf, R. C., "Engineering the dispersion and anisotropy of periodic electromagnetic structures," Solid State Physics, Vol. 66, 213-300, Elsevier, 2015.

40. Rumpf, R. C., J. Pazos, C. R. Garcia, L. Ochoa, and R. Wicker, "3d printed lattices with spatially variant self-collimation," Progress In Electromagnetics Research, Vol. 139, 1-15, 2013.
doi:10.2528/PIER13030507

41. Rumpf, R. C., J. J. Pazos, J. L. Digaum, and S. M. Kuebler, "Spatially variant periodic structures in electromagnetics," Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 373, No. 2049, 20140359, 2015.
doi:10.1098/rsta.2014.0359

42. Greville, T., "Some applications of the pseudoinverse of a matrix," SIAM Review, Vol. 2, No. 1, 15-22, 1960.
doi:10.1137/1002004

43. Noble, B. and J. W. Daniel, Applied Linear Algebra, 3rd Ed., Prentice Hall, 1988.

44. Sacks, Z. S., D. M. Kingsland, R. Lee, and J.-F. Lee, "A perfectly matched anisotropic absorber for use as an absorbing boundary condition," IEEE Transactions on Antennas and Propagation, Vol. 43, No. 12, 1460-1463, 1995.
doi:10.1109/8.477075

Dr. Eric A. Berry

Professor Raymond C. Rumpf