Recent studies on artificial materials demonstrate that substantial improvements in electromagnetic response can be attained by combining different materials subject to desired metrics. However, the perfect material combination is unique and extremely difficult to determine without automated synthesis schemes. In this paper, we develop a versatile approach to design the microstructure of periodic materials with prescribed dielectric and magnetic material tensors. The proposed framework is based on a robust material model and generalized inverse synthesis tool relying on topology optimization. The former is derived using homogenization theory and asymptotic expansion applied to Maxwell equations and can characterize the effects of anisotropy and loss of materials with periodic unit cells of arbitrary geometries and multi-phases much smaller than the wavelength. Resulting Partial Differential Equation (PDE) is solved numerically using Finite Element Analysis (FEA) and is validated with results in literature. The material model proves to be fast and numerically stable even with complex inclusions. The topology optimization problem is applied for the first time towards designing the unit cell topology of periodic electromagnetic materials from scratch with desired dielectric and magnetic tensors using off-the-shelf materials, i.e., readily available constituents obtained from isotropic ceramic powders. The proposed framework's capability is demonstrated with five design examples. Design with anisotropic permittivity is also fabricated. Results show that the framework is capable of designing, in an automated fashion, non-intuitive material compositions from scratch with desired electromagnetic properties.
2. Ferguson, B. and X. C. Zhang, "Materials for terahertz science and technology," Nature Materials, Vol. 1, 26-33, Sep. 2002.
3. Liu, N., H. C. Guo, L. W. Fu, S. Kaiser, H. Schweizer, and H. Giessen, "Three-dimensional photonic metamaterials at optical frequencies," Nature Materials, Vol. 7, 31-37, Jan. 2008.
4. Temelkuran, B., M. Bayindir, E. Ozbay, R. Biswas, M. M. Sigalas, G. Tuttle, and K. M. Ho, "Photonic crystal-based resonant antenna with a very high directivity," Journal of Applied Physics, Vol. 87, 603-605, Jan. 2000.
5. Ziolkowski, R. W. and E. Heyman, "Wave propagation in media having negative permittivity and permeability," Physical Review E, Vol. 64, 056625, 2001.
6. Baccarelli, P., P. Burghignoli, F. Frezza, A. Galli, P. Lampariello, G. Lovat, and S. Paulotto, "Fundamental modal properties of surface waves on metamaterial grounded slabs," IEEE Transactions on Microwave Theory and Techniques, Vol. 53, 1431-1442, 2005.
7. Ziolkowski, R. W. and A. D. Kipple, "Application of double negative materials to increase the power radiated by electrically small antennas," IEEE Transactions on Antennas and Propagation, Vol. 51, 2626-2640, Oct. 2003.
8. Tung, N. T., V. D. Lam, J. W. Park, M. H. Cho, J. Y. Rhee, W. H. Jang, and Y. P. Lee, "Single- and double-negative refractive indices of combined metamaterial structure," Journal of Applied Physics, Vol. 106, Sep. 2009.
9. Wu, B.-I., W.Wang, J. Pacheco, X. Chen, T. M. Grzegorczyk, and J. A. Kong, "A study of using metamaterials as antenna substrate to enhance gain," Progress In Electromagnetics Research, Vol. 51, 295-328, 2005.
10. Figotin, A. and I. Vitebskiy, "Electromagnetic unidirectionality in magnetic photonic crystals," Physical Review B, Vol. 67, 165210, 2003.
11. Perruisseau-Carrier, J. and A. K. Skrivervik, "Composite right/left-handed transmission line metamaterial phase shifters (mps) in mmic technology," IEEE Transactions on Microwave Theory and Techniques, Vol. 54, 1582-1589, 2006.
12. Ran, L., J. Huangfu, H. Chen, Y. Li, X. Zhang, K. Chen, and J. A. Kong, "Microwave solid-state left-handed material with a broad bandwidth and an ultralow loss," Physical Review B (Condensed Matter and Materials Physics), Vol. 70, 073102-3, 2004.
13. Sungjoon, L., C. Caloz, and T. Itoh, "Metamaterial-based electronically controlled transmission-line structure as a novel leaky-wave antenna with tunable radiation angle and beamwidth," IEEE Transactions on Microwave Theory and Techniques, Vol. 52, 2678-2690, 2004.
14. Sungjoon, L., C. Caloz, and T. Itoh, "Metamaterial-based electronically controlled transmission-line structure as a novel leaky-wave antenna with tunable radiation angle and beamwidth," IEEE Transactions on Microwave Theory and Techniques, Vol. 53, 161-173, 2005.
15. Fang, N., H. Lee, C. Sun, and X. Zhang, "Sub-diffraction-limited optical imaging with a silver superlens," Science, 534-537, Apr. 22, 2005.
16. David, O. S. M., J. B. Richard, and R. W. Conrad, "Submicron imaging with a planar silver lens," Applied Physics Letters, Vol. 84, 4403-4405, 2004.
17. Jensen, J. S., O. Sigmund, L. H. Frandsen, P. I. Borel, A. Harpoth, and M. Kristensen, "Topology design and fabrication of an efficient double 90 photonic crystal waveguide bend," IEEE Photonics Technology Letters, Vol. 17, 1202-1204, 2005.
18. Kiziltas, G., D. Psychoudakis, J. L. Volakis, and N. Kikuchi, "Topology design optimization of dielectric substrates for bandwidth improvement of a patch antenna," IEEE Transactions on Antennas and Propagation, Vol. 51, 2732-2743, 2003.
19. Wang, S. and J. Kang, "Topology optimization of nonlinear magnetostatics," IEEE Transactions on Magnetics, Vol. 38, 1029-1032, 2002.
20. Wongkasemand, N., A. Akyurtlu, and K. A. Marx, "Group theory based design of isotropic negative refractive index metamaterials," Progress In Electromagnetics Research, Vol. 63, 295-310, 2006.
21. Thomas, Z., T. Grzegorczyk, B.-I. Wu, X. Chen, and J. A. Kong, "Design and measurement of a four-port device using metamaterials," Optics Express, Vol. 13, 4737-4744, Jun. 2005.
22. Marqués, R., J. Martel, F. Mesa, and F. Medina, "A new 2d isotropic left-handed metamaterial design: Theory and experiment," Microwave and Optical Technology Letters, Vol. 35, 405-408, 2002.
23. Shin, J., A. Akyurtlu, and M. Deshpande, "Comments on design, fabrication, and testing of double negative metamaterials," IEEE Transactions on Antennas and Propagation, Vol. 53, 891, 2005.
24. Sounas, D. L., N. V. Kantartzis, and T. D. Tsiboukis, "Focusing efficiency analysis and performance optimization of arbitrarily sized dng metamaterial slabs with losses," IEEE Transactions on Microwave Theory and Techniques, Vol. 54, 4111-4121, 2006.
25. Bendsøe, M. and O. Sigmund, Topology optimization: Theory, Methods, and Applications, Springer Verlag, 2003.
26. Sanchez-Palencia, E., Non Homogeneous Media and Vibration Theory, Springer-Verlag, Berlin, 1980.
27. Shelukhin, V. V. and S. A. Terentev, "Frequency dispersion of dielectric permittivity and electric conductivity of rocks via two-scale homogenization of the Maxwell equations," Progress In Electromagnetics Research B, Vol. 14, 175-202, 2009.
28. Baena, J. D., L. Jelinek, R. Marques, and M. Silveirinha, "Unified homogenization theory for magnetoinductive and electromagnetic waves in split-ring metamaterials," Physical Review A (Atomic, Molecular, and Optical Physics), Vol. 78, 013842, 2008.
29. Silveirinha, M. G., "Metamaterial homogenization approach with application to the characterization of microstructured composites with negative parameters," Physical Review B, Vol. 75, 115104, 2007.
30. Engstrom, C. and D. Sjoberg, "On two numerical methods for homogenization of Maxwell's equations," Journal of Electromagnetic Waves and Applications, Vol. 21, No. 13, 1845-1856, 2007.
31. Smith, D. R. and J. B. Pendry, "Homogenization of metamaterials by field averaging (invited paper)," Journal of the Optical Society of America B-Optical Physics, Vol. 23, 391-403, Mar. 2006.
32. Ouchetto, O., C. W. Qiu, S. Zouhdi, L. W. Li, and A. Razek, "Homogenization of 3-d periodic bianisotropic metamaterials," IEEE Transactions on Microwave Theory and Techniques, Vol. 54, 3893-3898, 2006.
33. Silveirinha, M. G. and C. A. Fernandes, "Homogenization of 3-d-connected and nonconnected wire metamaterials," IEEE Transactions on Microwave Theory and Techniques, Vol. 53, 1418-1430, 2005.
34. Ouchetto, O., S. Zouhdi, A. Bossavit, G. Griso, and B. Miara, Homogenization of 3d structured composites of complex shaped inclusions, PIERS Proceedings, 22-26, Hangzhou, China, August 22--26, 2005.
35. Krokhin, A. A., P. Halevi, and J. Arriaga, "Long-wavelength limit (homogenization) for two-dimensional photonic crystals," Physical Review B, Vol. 65, 115208, 2002.
36. Caloz, C., A. Lai, and T. Itoh, "The challenge of homogenization in metamaterials," New Journal of Physics, Vol. 7, No. 15, Aug. 2005.
37. Sihvola, A., Electromagnetic Mixing Formulas & Applications, Institution of Electrical Engineers, London, UK, 1999.
38. Smith, D. R., S. Schultz, P. Markos, and C. M. Soukoulis, "Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients," Physical Review B, Vol. 65, 195104, 2002.
39. Chen, X., T. M. Grzegorczyk, B.-I. Wu, J. J. Pacheco, and J. A. Kong, "Robust method to retrieve the constitutive effective parameters of metamaterials," Physical Review E (Statistical, Nonlinear, and Soft Matter Physics), Vol. 70, 016608-7, 2004.
40. Chen, X., B.-I. Wu, J. A. Kong, and T. M. Grzegorczyk, "Retrieval of the effective constitutive parameters of bianisotropic metamaterials," Physical Review E, Vol. 71, 046610, 2005.
41. Datta, S., C. T. Chan, K. M. Ho, and C. M. Soukoulis, "Effective dielectric constant of periodic composite structures," Physical Review B, Vol. 48, 14936, 1993.
42. Lamb, W., D. M. Wood, and N. W. Ashcroft, "Long wavelength electromagnetic propagation in heterogeneous media," Physical Review B, Vol. 21, 2248-2266, 1980.
43. Silveirinha, M. G., C. A. Fernandes, and J. R. Costa, "Electromagnetic characterization of textured surfaces formed by metallic pins," IEEE Transactions on Antennas and Propagation, Vol. 56, 405-415, Feb. 2008.
44. Pokrovsky, A. L. and A. L. Efros, "Electrodynamics of metallic photonic crystals and the problem of left-handed materials," Physical Review Letters, Vol. 89, 093901, 2002.
45. Bensoussan, A., J. L. Lions, and G. Papanicolaou, Asymptotic Analysis of Periodic Structures, Amsterdam, North-Holland, 1978.
46. Nguetseng, G., "A general convergence result for a functional related to the theory of homogenization," SIAM Journal on Mathematical Analysis, Vol. 20, 608-623, 1989.
47. Wellander, N. and S. Barbara, "Homogenization of the Maxwell equations: Case ii. Nonlinear conductivity," Applications of Mathematics, Vol. 47, 255-283, 2002.
48. Banks, H. T., V. A. Bokil, D. Cioranescu, N. L. Gibson, G. Griso, and B. Miara, "Homogenization of periodically varying coefficients in electromagnetic materials," Journal of Scientific Computing, Vol. 28, 191-221, 2006.
49. Kristensson, G., "Homogenization of corrugated interfaces in electromagnetics," Progress In Electromagnetics Research, Vol. 55, 1-31, 2005.
50. Hashin, Z. and S. Shtrikman, "A variational approach to the theory of the effective magnetic permeability of multiphase materials," Journal of Applied Physics, Vol. 33, 3125-3131, 1962.
51. Jackson, J. D., Classical Electrodynamics, 2 Ed., John Wiley & Sons, New York, 1975.
52. Kristensson, G., "Homogenization of the Maxwell equations in an anisotropic material," Radio Science, Vol. 38, 8018, Apr. 2003.
53. Sjoberg, D., "Homogenization of dispersive material parameters for Maxwell's equations using a singular value decomposition," Multiscale Modeling and Simulation, Vol. 4, 760-789, 2006.
54. Wellander, N., "Homogenization of the Maxwell equations: Case i. Linear theory," Applications of Mathematics, Vol. 46, 29-51, 2001.
55. Huang, K. and X. Yang, "A method for calculating the effective permittivity of a mixture solution during a chemical reaction by experimental results," Progress In Electromagnetics Research Letters, Vol. 5, 99-107, 2008.
56. Ouchetto, O., S. Zouhdi, A. Bossavit, G. Griso, and B. Miara, "Modeling of 3-d periodic multiphase composites by homogenization," IEEE Transactions on Microwave Theory and Techniques, Vol. 54, 2615-2619, 2006.
57. Wellander, N. and G. Kristensson, "Homogenization of the Maxwell equations at fixed frequency," SIAM Journal on Applied Mathematics, Vol. 64, 170-195, 2003.
58. Bossavit, A., "Effective penetration depth in spatially periodic grids: A novel approach to homogenization," EMC'94, International Symposium on Electromagnetic Compatibility, 859-864, Dept. Elec. Engng., University La Sapienza, Roma, Italy, 1994.
59. Bossavit, A., "Superconductivity modelling: Homogenization of bean's model in three dimensions, and the problem of transverse conductivity," IEEE Transactions on Magnetics, Vol. 31, 1769-1774, 1995.
60. Cherkaev, E., "Inverse homogenization for evaluation of effective properties of a mixture," Institute of Physics Publishing Inverse Problems, Vol. 17, 1203-1218, Jun. 7, 2001.
61. Sigmund, O. and S. Torquato, "Design of materials with extreme thermal expansion using a three-phase topology optimization method," Journal of the Mechanics and Physics of Solids, Vol. 45, 1037-1067, 1997.
62. Sigmund, O., "A new class of extremal composites," Journal of the Mechanics and Physics of Solids, Vol. 48, 397-428, 2000.
63. Bloembaum, B., The use of structural optimization in automotive design-state of the art and vision, presented at the WCSMO-3, Buffalo, 1999.
64. Bendsoe, M. P. and N. Kikuchi, "Generating optimal topologies in structural design using a homogenization method," Computer Methods in Applied Mechanics and Engineering, Vol. 71, 197-224, 1988.
65. Bendsøe, M., "Optimal shape design as a material distribution problem," Structural and Multidisciplinary Optimization, Vol. 1, 193-202, 1989.
66. Hassani, H. and E. Hinton, Homogenization and Structural Topology Optimization Theory, Practice and Software, Springer, Berlin, 1998.
67. Rozvany, G., "Aims, scope, methods, history and unified terminology of computer-aided topology optimization in structural mechanics," Structural and Multidisciplinary Optimization, Vol. 21, 90-108, 2001.
66. Dyck, D. N. and D. A. Lowther, "Automated design of magnetic devices by optimizing material distribution," IEEE Transactions on Magnetics, Vol. 32, 1188-1193, May 1996.
69. Jensen, J. S. and O. Sigmund, "Systematic design of photonic crystal structures using topology optimization: Low-loss waveguide bends," Applied Physics Letters, Vol. 84, 2022-2024, 2004.
70. Borel, P., A. Harpøth, L. Frandsen, M. Kristensen, P. Shi, J. Jensen, and O. Sigmund, "Topology optimization and fabrication of photonic crystal structures," Opt. Express, Vol. 12, 1996-2001, 2004.
71. Nomura, T., K. Sato, K. Taguchi, and T. Kashiwa, "Topology optimization method for antenna design using FDTD method," IEICE Transactions on Electronics, Vol. 12, 2196-2205, 2006.
72. Fuchi, K., A. R. Diaz, E. Rothwell, R. Ouedraogo, and A. Temme, "Topology optimization of periodic layouts of dielectric materials," Structural and Multidisciplinary Optimization, Vol. 42, 483-493, Jun. 2010.
73. Nomura, T., S. Nishiwaki, K. Sato, and K. Hirayama, "Topology optimization for the design of periodic microstructures composed of electromagnetic materials," Finite Elements in Analysis and Design, Vol. 45, 210-226, February 2009.
74. Sigmund, O., "Materials with prescribed constitutive parameters: An inverse homogenization problem," International Journal of Solids and Structures, Vol. 31, 2313-2329, 1994.
75. Sigmund, O., Systematic design of metamaterials by topology optimization, Vol. 13, 151-159 Iutam Symposium on Modelling Nanomaterials and Nanosystems, Netherlands, R. Pyrz and J. C. Rauhe (eds.), Springer, Netherlands, 2008.
76. Smith, D. R., D. C. Vier, N. Kroll, and S. Schultz, "Direct calculation of permeability and permittivity for a left-handed metamaterial," Applied Physics Letters, Vol. 77, 2246-2248, 2000.
77. Kiziltas, G. and Z. N. Wing, Automated fabrication of three dimensional dielectric composites for RF applications, IEEE Antennas and Propagation Society International Symposium and USNC/URSI National Radio Science, Albuquerque, USA, 2006.
78. Li, Z., Y. Erdemli, J. Volakis, and P. Papalambros, "Design optimization of conformal antennas by integrating stochastic algorithms with the hybrid finite-element method," IEEE Transactions on Antennas and Propagation, Vol. 50, 676-684, 2002.
79. Sigmund, O., "Design of multi-physics actuators using topology optimization- part i: One material structures," Computer Methods in Applied Mechanics and Engineering, Vol. 190, 6577-6604, 2001.
80. Kiziltas, G., N. Kikuchi, J. Volakis, and J. Halloran, "Topology optimization of dielectric substrates for filters and antennas using simp," Archives of Computational Methods in Engineering, Vol. 11, 355-388, 2004.
81. Olesen, L. H., F. Okkels, and H. Bruus, "A high-level programming-language implementation of topology optimization applied to steady-state Navier-Stokes flow," International Journal for Numerical Methods in Engineering, Vol. 65, 975-1001, 2006.
82. Wing, Z. and J. Halloran, "Dry powder deposition and compaction for functionally graded ceramics," Journal of the American Ceramic Society, Vol. 89, 3406-3412, 2006.