volume | PIER Journals

Abstract

The finite-difference frequency-domain (FDFD) method is a very simple and powerful approach for rigorous analysis of electromagnetic structures. It may be the simplest of all methods to implement and is excellent for field visualization and for developing new ways to model devices. This paper describes a simple method for incorporating anisotropic materials with arbitrary tensors for both permittivity and permeability into the FDFD method. The algorithm is benchmarked by comparing transmission and reflection results for an anisotropic guided-mode resonant filter simulated in HFSS and FDFD. The anisotropic FDFD method is then applied to a lens and cloak designed by transformation optics.

1. Luo, G. Q., W. Hong, Z.-C. Hao, B. Liu, W. D. Li, J. X. Chen, et al. "Theory and experiment of novel frequency selective surface based on substrate integrated waveguide technology," IEEE Transactions on Antennas and Propagation, Vol. 53, 4035-4043, 2005.
doi:10.1109/TAP.2005.860010 Google Scholar

2. Rumpf, R. C., "Design and optimization of nano-optical elements by coupling fabrication to optical behavior,", University of Central Florida Orlando, Florida, 2006. Google Scholar

3. Sun, W., K. Liu, and C. A. Balanis, "Analysis of singly and doubly periodic absorbers by frequency-domain finite-difference method," IEEE Transactions on Antennas and Propagation, Vol. 44, 798-805, 1996.
doi:10.1109/8.509883 Google Scholar

4. Wu, S.-D. and E. N. Glytsis, "Volume holographic grating couplers: Rigorous analysis by use of the finite-difference frequency-domain method," Applied Optics, Vol. 43, 1009-1023, 2004.
doi:10.1364/AO.43.001009 Google Scholar

5. Shin, W. and S. Fan, "Choice of the perfectly matched layer boundary condition for frequency-domain Maxwell's equations solvers," Journal of Computational Physics, Vol. 231, 3406-3431, 2012.
doi:10.1016/j.jcp.2012.01.013 Google Scholar

6. Shin, W. and S. Fan, "Accelerated solution of the frequency-domain Maxwell's equations by engineering the eigenvalue distribution of the operator," Optics Express, Vol. 21, 22578-22595, 2013.
doi:10.1364/OE.21.022578 Google Scholar

7. 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 Google Scholar

8. Takayama, O., L.-C. Crasovan, S. K. Johansen, D. Mihalache, D. Artigas, and L. Torner, "Dyakonov surface waves: A review," Electromagnetics, Vol. 28, 126-145, 2008.
doi:10.1080/02726340801921403 Google Scholar

9. Figotin, A. and I. Vitebskiy, "Slow-wave resonance in periodic stacks of anisotropic layers," Physical Review A, Vol. 76, 053839, 2007.
doi:10.1103/PhysRevA.76.053839 Google Scholar

10. Schurig, D., J. Mock, B. Justice, S. A. Cummer, J. Pendry, A. Starr, et al. "Metamaterial electromagnetic cloak at microwave frequencies," Science, Vol. 314, 977-980, 2006.
doi:10.1126/science.1133628 Google Scholar

11. Kwon, D.-H. and D. H. Werner, "Polarization splitter and polarization rotator designs based on transformation optics," Optics Express, Vol. 16, 18731-18738, 2008.
doi:10.1364/OE.16.018731 Google Scholar

12. Al-Barqawi, H., N. Dib, and M. Khodier, "A two-dimensional full-wave finite-difference frequency-domain analysis of ferrite loaded structures," International Journal of Infrared and Millimeter Waves, Vol. 29, 443-456, 2008.
doi:10.1007/s10762-008-9349-6 Google Scholar

13. Loke, V. L., T. A. Nieminen, S. J. Parkin, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "FDFD/T-matrix hybrid method," Journal of Quantitative Spectroscopy and Radiative Transfer, Vol. 106, 274-284, 2007.
doi:10.1016/j.jqsrt.2007.01.040 Google Scholar

14. Mielewski, J., A. Cwikla, and M. Mrozowski, "Analysis of shielded anisotropic dielectric resonators using FDFD and the Arnoldi method," 12th International Conference on Microwaves and Radar MIKON'98, 335-339, 1998.
doi:10.1109/MIKON.1998.740799 Google Scholar

15. Pinheiro, H. F., A. J. Giarola, and C. L. D. S. S. Sobrinho, "Dispersion characteristics of asymmetric coupled anisotropic dielectric waveguides using FDFD," International Journal of Infrared and Millimeter Waves, Vol. 16, 1965-1975, 1995.
doi:10.1007/BF02072551 Google Scholar

16. Rappaport, C. M. and B. J. McCartin, "FDFD analysis of electromagnetic scattering in anisotropic media using unconstrained triangular meshes," IEEE Transactions on Antennas and Propagation, Vol. 39, 345-349, 1991.
doi:10.1109/8.76332 Google Scholar

17. Rappaport, C. M. and E. Smith, "Anisotropic FDFD computed on conformal meshes," IEEE Transactions on Magnetics, Vol. 27, 3848-3851, 1991.
doi:10.1109/20.104941 Google Scholar

18. Zhao, Y.-J., K.-L. Wu, and K.-K. Cheng, "A compact 2-D full-wave finite-difference frequency-domain method for general guided wave structures," IEEE Transactions on Microwave Theory and Techniques, Vol. 50, 1844-1848, 2002.
doi:10.1109/TMTT.2002.800447 Google Scholar

19. Chen, M.-Y., S.-M. Hsu, and H.-C. Chang, "A finite-difference frequency-domain method for full-vectroial mode solutions of anisotropic optical waveguides with arbitrary permittivity tensor," Optics Express, Vol. 17, 5965-5979, 2009.
doi:10.1364/OE.17.005965 Google Scholar

20. Lavranos, C., G. Kyriacou, and J. Sahalos, "A 2-D finite difference frequency domain (FDFD) eigenvalue method for orthogonal curvilinear coordinates," PIERS Proceedings, 397-400, PISA, Italy, Mar. 28-31, 2004. Google Scholar

21. Pereda, J. A., A. Vegas, and A. Prieto, "An improved compact 2D full-wave FDFD method for general guided wave structures," Microwave and Optical Technology Letters, Vol. 38, 331-335, 2003.
doi:10.1002/mop.11052 Google Scholar

22. Zainud-Deen, S. H., A. Z. Botros, and M. S. Ibrahim, "Scattering from bodies coated with metamaterial using FDFD method," Progress In Electromagnetics Research B, Vol. 2, 279-290, 2008.
doi:10.2528/PIERB07112803 Google Scholar

23. Taflove, A. and S. C. Hagness, Computational Electrodynamics, Vol. 160, Artech House Boston, 2000.

24. Umashankar, K. and A. Taflove, "A novel method to analyze electromagnetic scattering of complex objects," IEEE Transactions on Electromagnetic Compatibility, 397-405, 1982.
doi:10.1109/TEMC.1982.304054 Google Scholar

25. Berenger, J.-P., "A perfectly matched layer for the absorption of electromagnetic waves," Journal of Computational Physics, Vol. 114, 185-200, 1994.
doi:10.1006/jcph.1994.1159 Google Scholar

26. Margengo, E., C. M. Rappaport, and E. L. Miller, "Optimum PML ABC conductivity profile in FDFD," IEEE Transactions on Magnetics, Vol. 35, 1506-1509, 1999.
doi:10.1109/20.767253 Google Scholar

27. 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, 1460-1463, 1995.
doi:10.1109/8.477075 Google Scholar

28. Yee, K. S., "Numerical solution of initial boundary value problems involving Maxwell's equations," IEEE Transactions on Antennas and Propagation, Vol. 14, 302-307, 1966.
doi:10.1109/TAP.1966.1138693 Google Scholar

29. Jordán, K., Calculus of Finite Differences, American Mathematical Soc., 1965.

30. Rumpf, R. C., C. R. Garcia, H. H. Tsang, J. E. Padilla, and M. D. Irwin, "Electromagnetic isolation of a microstrip by embedding in a spatially variant anisotropic metamaterial," Progress In Electromagnetics Research, Vol. 142, 243-260, 2013.
doi:10.2528/PIER13070308 Google Scholar

31. Horn, A., "Doubly stochastic matrices and the diagonal of a rotation matrix," Amer. J. Math, Vol. 76, 620-630, 1954.
doi:10.2307/2372705 Google Scholar

32. Magnusson, R. and S. Wang, "New principle for optical filters," Applied Physics Letters, Vol. 61, 1022-1024, 1992.
doi:10.1063/1.107703 Google Scholar

33. Barton, J. H., R. C. Rumpf, R. W. Smith, C. L. Kozikowski, and P. A. Zellner, "All-dielectric frequency selective surfaces with few number of periods," Progress In Electromagnetics Research B, Vol. 41, 269-283, 2012.
doi:10.2528/PIERB12042404 Google Scholar

34. Boonruang, S., A. Greenwell, and M. Moharam, "Multiline two-dimensional guided-mode resonant filters," Applied Optics, Vol. 45, 5740-5747, 2006.
doi:10.1364/AO.45.005740 Google Scholar

35. Pung, A. J., S. R. Carl, I. R. Srimathi, and E. G. Johnson, "Method of fabrication for encapsulated polarizing resonant gratings," IEEE Photonics Technology Letters, Vol. 25, 1432-1434, 2013.
doi:10.1109/LPT.2013.2266575 Google Scholar

36. Tibuleac, S. and R. Magnusson, "Re°ection and transmission guided-mode resonance filters," JOSA A, Vol. 14, 1617-1626, 1997.
doi:10.1364/JOSAA.14.001617 Google Scholar

37. Garcia, C. R., J. Correa, D. Espalin, J. H. Barton, R. C. Rumpf, R. Wicker, and V. Gonzalez, "3D printing of anisotropic metamaterials," Progress In Electromagnetics Research Letters, Vol. 34, 75-82, 2012.
doi:10.2528/PIERL12070311 Google Scholar

38. Leonhardt, U., "Optical conformal mapping," Science, Vol. 312, 1777-1780, 2006.
doi:10.1126/science.1126493 Google Scholar

39. Pendry, J. B., D. Schurig, and D. R. Smith, "Controlling electromagnetic fields," Science, Vol. 312, 1780-1782, 2006.
doi:10.1126/science.1125907 Google Scholar

40. Ward, A. and J. Pendry, "Refraction and geometry in Maxwell's equations," Journal of Modern Optics, Vol. 43, 773-793, 1996.
doi:10.1080/09500349608232782 Google Scholar

41. Landau, L. D. and E. M. Lifshits, The Classical Theory of Fields, Vol. 2, Butterworth-Heinemann, 1975.

42. Post, E. J., Formal Structure of Electromagnetics: General Covariance and Electromagnetics, Courier Dover Publications, 1997.

43. Kwon, D.-H. and D. H. Werner, "Transformation optical designs for wave collimators, flat lenses and right-angle bends ," New Journal of Physics, Vol. 10, 115023, 2008.
doi:10.1088/1367-2630/10/11/115023 Google Scholar

44. Cai, W., U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, "Optical cloaking with metamaterials," Nature Photonics, Vol. 1, 224-227, 2007.
doi:10.1038/nphoton.2007.28 Google Scholar

Dr. Raymond C. Rumpf

Dr. Cesar R. Garcia

Dr. Eric A. Berry

Dr. Jay H. Barton