A fast and simple design methodology for transformation optics (TO) is described for devices having completely arbitrary geometries. An intuitive approach to the design of arbitrary devices is presented which enables possibilities not available through analytical coordinate transformations. Laplace's equation is solved using the finite-difference method to generate the arbitrary spatial transforms. Simple techniques are presented for enforcing boundary conditions and for isolating the solution of Laplace's equation to just the device itself. It is then described how to calculate the permittivity and permeability functions via TO from the numerical spatial transforms. Last, a modification is made to the standard anisotropic finite-difference frequency-domain (AFDFD) method for much faster and more efficient simulations. Several examples are given at the end to benchmark and to demonstrate the versatility of the approach. This work provides the basis for a complete set of tools to design and simulate transformation electromagnetic devices of any shape and size.
2. Schurig, D., et al., "Metamaterial electromagnetic cloak at microwave frequencies," Science, Vol. 314, 977-980, 2006.
doi:10.1126/science.1133628
3. Hu, J., X. Zhou, and G. Hu, "Design method for electromagnetic cloak with arbitrary shapes based on Laplace's equation," Optics Express, Vol. 17, 1308-1320, 2009.
doi:10.1364/OE.17.001308
4. Chang, Z., X. Zhou, J. Hu, and G. Hu, "Design method for quasi-isotropic transformation materials based on inverse Laplace's equation with sliding boundaries," Optics Express, Vol. 18, 6089-6096, 2010.
doi:10.1364/OE.18.006089
5. 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 B, Vol. 61, 55-67, 2014.
doi:10.2528/PIERB14071606
6. Landy, N. I. and W. J. Padilla, "Guiding light with conformal transformations," Optics Express, Vol. 17, 14872-14879, 2009.
doi:10.1364/OE.17.014872
7. Ma, J.-J., X.-Y. Cao, K.-M. Yu, and T. Liu, "Determination the material parameters for arbitrary cloak based on Poisson's equation," Progress In Electromagnetics Research M, Vol. 9, 177-184, 2009.
doi:10.2528/PIERM09091405
8. Chen, X., Y. Fu, and N. Yuan, "Invisible cloak design with controlled constitutive parameters and arbitrary shaped boundaries through Helmholtz's equation," Optics Express, Vol. 17, 3581-3586, Mar. 2009.
doi:10.1364/OE.17.003581
9. Rumpf, R. C. and J. Pazos, "Synthesis of spatially variant lattices," Optics Express, Vol. 20, 15263-15274, 2012.
doi:10.1364/OE.20.015263
10. Smith, G. D., Numerical Solution of Partial Differential Equations: Finite Difference Methods, Oxford University Press, 1985.
11. LeVeque, R. J., "Finite difference methods for differential equations," Draft Version for Use in AMath, Vol. 585, 1998.
12. Golub, G. H. and C. F. van Loan, Matrix Computations, Vol. 3, JHU Press, 2012.
13. Johnson, H. and C. S. Burrus, "On the structure of e±cient DFT algorithms," IEEE Transactions on Acoustics, Speech and Signal Processing, Vol. 33, 248-254, 1985.
doi:10.1109/TASSP.1985.1164526
14. Iserles, A., A First Course in the Numerical Analysis of Differential Equations, Cambridge University Press, 2009.
15. Chapra, S. C. and R. P. Canale, Numerical Methods for Engineers, Vol. 2, McGraw-Hill, 2012.
16. Arfken, G. and H. Weber, Mathematical Methods for Physicists, 6th Ed., Academic, New York, 2005.
17. Morgan, M. A., Finite Element and Finite Difference Methods in Electromagnetic Scattering, Elsevier, 2013.
18. Luong, P., "A mathematical coastal ocean circulation system with breaking waves and numerical grid generation," Applied Mathematical Modelling, Vol. 21, 633-641, 1997.
doi:10.1016/S0307-904X(97)00076-0
19. Eiseman, P. R., "Grid generation for fluid mechanics computations," Annual Review of Fluid Mechanics, Vol. 17, 487-522, 1985.
doi:10.1146/annurev.fl.17.010185.002415
20. Thompson, J. F., Z. U. Warsi, and C. W. Mastin, Numerical Grid Generation: Foundations and Applications, Vol. 45, North-holland Amsterdam, 1985.
21. Sanmiguel-Rojas, E., J. Ortega-Casanova, C. del Pino, and R. Fernandez-Feria, "A Cartesian grid finite-difference method for 2D incompressible viscous flows in irregular geometries," Journal of Computational Physics, Vol. 204, 302-318, 2005.
doi:10.1016/j.jcp.2004.10.010
22. Akcelik, V., B. Jaramaz, and O. Ghattas, "Nearly orthogonal two-dimensional grid generation with aspect ratio control," Journal of Computational Physics, Vol. 171, 805-821, 2001.
doi:10.1006/jcph.2001.6811
23. Davis, T. A., "Algorithm 832: UMFPACK V4.3 --- An unsymmetric-pattern multifrontal method," ACM Transactions on Mathematical Software (TOMS), Vol. 30, 196-199, 2004.
doi:10.1145/992200.992206
24. Paige, C. C. and M. A. Saunders, "Solution of sparse indefinite systems of linear equations," SIAM Journal on Numerical Analysis, Vol. 12, 617-629, 1975.
doi:10.1137/0712047
25. Rumpf, R. C., "Simple implementation of arbitrarily shaped total-field/scattered-field regions in finite-difference frequency-domain," Progress In Electromagnetics Research B, Vol. 36, 221-248, 2012.
doi:10.2528/PIERB11092006
26. Schutz, B., A First Course in General Relativity, Cambridge University Press, 2009.
doi:10.1017/CBO9780511984181
27. Hobson, M. P., G. P. Efstathiou, and A. N. Lasenby, General Relativity: An Introduction for Physicists, Cambridge University Press, 2006.
doi:10.1017/CBO9780511790904
28. Liseikin, V., "Coordinate transformations," Grid Generation Methods, 31-66, Springer, Netherlands, 2010.
29. Kwon, D.-H. and D. H. Werner, "Transformation electromagnetics: An overview of the theory and applications," IEEE Antennas and Propagation Magazine, Vol. 52, 24-46, 2010.
doi:10.1109/MAP.2010.5466396
Submit
