volume | PIER Journals

Abstract

Split-field finite-difference time-domain (SF-FDTD) meth-od can overcome the limitation of ordinary FDTD in analyzing periodic structures under oblique incidence. On the other hand, huge run times of 3D SF-FDTD, is practically a major burden in its usage for analysis and design of nanostructures, particularly when having dispersive media. Here, details of parallel implementation of 3D SF-FDTD method for dispersive media, combined with total-field/scattered-field (TF/SF) method for injecting oblique plane wave, are discussed. Graphics processing unit (GPU) has been used for this purpose, and very large speed up factors have been achieved. Also a previously reported formulation of SF-FDTD based on the Drude model for dispersive media, is extended to cover Drude-Lorentz model, which is usually needed for materials such as gold. The resulting reduction in the number of variables in this formulation, not only helps in reducing the computational time, but also makes it possible to be implemented in GPU, where its memory limitation is a major concern. As an example for demonstrating the importance of this method in optimization of nanophotonics structures, improvement in the performance of a refractive index sensor, made of an array of nanodisks, using suitable angle of incidence is reported. To the best of our knowledge this is the first report of GPU implementation of SF-FDTD method, capable of analyzing periodic dispersive media under oblique incidence.

1. Roden, J. A., S. D. Gedney, M. P. Kesler, J. G. Maloney, and P. H. Harms, "Time-domain analysis of periodic structures at oblique incidence: Orthogonal and nonorthogonal FDTD implementations," IEEE Trans. Microwave Theory Tech., Vol. 46, No. 4, 420-427, 1998.
doi:10.1109/22.664143

2. Maloney, J. G. and M. P. Kesler, "Analysis of antenna arrays using the split-field update FDTD method," Antennas and Propagation Society International Symposium, IEEE, 2036-2039, 1998.

3. Wu, B., E. Yang, J. A. Kong, J. A. Oswald, K. A. McIntosh, L. Mahoney, and S. Verghese, "Analysis of photonic crystal filters by the finite-difference time-domain technique," Microwave Opt. Technol. Lett., Vol. 27, No. 2, 81-87, 2000.
doi:10.1002/1098-2760(20001020)27:2<81::AID-MOP2>3.0.CO;2-S

4. Mosallaei, H. and Y. Rahmat-Samii, "Grand challenges in analyzing EM band-gap structures: An FDTD/Prony technique based on the split-field approach," Antennas and Propagation Society International Symposium, IEEE, 47-50, 2001.

5. Farahat, N. and R. Mittra, "Analysis of frequency selective surfaces using the finite difference time domain (FDTD) method," Antennas and Propagation Society International Symposium, IEEE, 568-571, 2002.

6. Amjadi, S. M. and M. Soleimani, "Design of band-pass waveguide filter using frequency selective surfaces loaded with surface mount capacitors based on split-field update FDTD method," Progress In Electromagnetics Research B, Vol. 3, 271-281, 2008.
doi:10.2528/PIERB07122402

7. Belkhir, A. and F. I. Baida, "Three-dimensional finite-difference time-domain algorithm for oblique incidence with adaptation of perfectly matched layers and nonuniform meshing: Application to the study of a radar dome ," Phys. Rev. E, Vol. 77, No. 5, 056701, 2008.
doi:10.1103/PhysRevE.77.056701

8. Oh, C. and M. J. Escuti, "Time-domain analysis of periodic anisotropic media at oblique incidence: An efficient FDTD implementation," Opt. Express, Vol. 14, No. 24, 11870-11884, 2006.
doi:10.1364/OE.14.011870

9. Baida, F. I. and A. Belkhir, "Split-field FDTD method for oblique incidence study of periodic dispersive metallic structures," Opt. Lett., Vol. 34, No. 16, 2453-2455, 2009.
doi:10.1364/OL.34.002453

10. Belkhir, A., O. Arar, S. S. Benabbes, O. Lamrous, and F. I. Baida, "Implementation of dispersion models in the split-field-finite-difference-time-domain algorithm for the study of metallic periodic structures at oblique incidence ," Phys. Rev. E, Vol. 81, 046705, 2010.
doi:10.1103/PhysRevE.81.046705

11. Shahmansouri, A. and B. Rashidian, "Comprehensive three-dimensional split-field finite difference time-domain method for analysis of periodic plasmonic nanostructures: Near- and far-field formulation," J. Opt. Soc. Am. B, Vol. 28, No. 11, 2690-2700, 2011.
doi:10.1364/JOSAB.28.002690

12. Taflove, A. and S. C. Hagness, Computational Electrodynamics: The Finite-Difference-Time-Domain Method, Artech House, 2005.

13. Zheng, F., Z. Chen, and J. Zhang, "A finite-difference time-domain method without the courant stability conditions," IEEE Microw. Guided Wave Lett., Vol. 9, No. 11, 441-443, 1999.
doi:10.1109/75.808026

14. Namiki, T., "A new FDTD algorithm based on alternating-direction implicit method ," IEEE Trans. Microwave Theory Tech., Vol. 47, No. 10, 2003-2007, 1999.
doi:10.1109/22.795075

15. Wang, S., F. L. Teixeira, and J. Chen, "An iterative ADI-FDTD with reduced splitting error," IEEE Microw. Wireless Compon. Lett., Vol. 15, No. 2, 92-94, 2005.
doi:10.1109/LMWC.2004.842835

16. Ahmed, I. and Z. Chen, "Error reduced ADI-FDTD methods," IEEE Antennas Wireless Propag. Lett., Vol. 4, 323-325, 2005.
doi:10.1109/LAWP.2005.855630

17. Shibayama, J., M. Muraki, J. Yamauchi, and H. Nakano, "Efficient implicit FDTD algorithm based on locally one-dimensional scheme," Electron. Lett., Vol. 41, No. 19, 1046-1047, 2005.
doi:10.1049/el:20052381

18. Li, E., I. Ahmed, and R. Vahldieck, "Numerical dispersion analysis with an improved LOD-FDTD method," IEEE Microw. Wireless Compon. Lett., Vol. 17, No. 5, 319-321, 2007.
doi:10.1109/LMWC.2007.895687

19. Tan, E. L., "Unconditionally stable LOD-FDTD method for 3-DMaxwel's equations," IEEE Microw. Wireless Compon. Lett., Vol. 17, No. 2, 85-87, 2007.
doi:10.1109/LMWC.2006.890166

20. Ahmed, I., E. K. Chua, E. P. Li, and Z. Chen, "Development of the three-dimensional unconditionally stable LOD-FDTD method," IEEE Trans. Antennas Propagat., Vol. 56, No. 11, 3596-3600, 2008.
doi:10.1109/TAP.2008.2005544

21. Shibayama, J., A. Nomura, R. Ando, J. Yamauchi, and H. Nakano, "A frequency-dependent LOD-FDTD method and its application to the analyses of plasmonic waveguide devices ," IEEE J. Quantum Electron., Vol. 46, No. 1, 40-49, 2010.
doi:10.1109/JQE.2009.2024328

22. Yu, W., R. Mittra, T. Su, Y. Liu, and X. Yang, Parallel Finite-Difference Time-Domain Method, Artech House, 2006.

23. Schneider, R. N., L. E. Turner, and M. M. Okoniewski, "Application of FPGA technology to accelerate the finite difference time-domain (FDTD) method," FPGA'02 Proc. of the ACM/SIGDA Tenth Int. Symp. on Field-programmable Gate Arrays, ACM, 2002.

24. Inman, M. J., A. Z. Elsherbeni, and C. E. Smith, "FDTD calculations using graphical processing units," IEEE/ACES Int. Conf. on Wireless Communications and Applied Computational Electromagnetics, 728-731, 2005.
doi:10.1109/WCACEM.2005.1469689

25. Adams, S., J. Payne, and R. Boppana, "Finite difference time domain (FDTD) simulations using graphics processors," DOD High Performance Computing Modernization Program Users Group Conf., IEEE, 334-338, 2007.
doi:10.1109/HPCMP-UGC.2007.34

26. Sypek, P., A. Dziekonski, and M. Mrozowski, "How to render FDTD computations more effective using a graphics accelerator," IEEE Trans. Magn., Vol. 45, No. 3, 1324-1327, 2009.
doi:10.1109/TMAG.2009.2012614

27. Nagaoka, T. and S. Watanabe, "A GPU-based calculation using the three-dimensional FDTD method for electromagnetic field analysis," Engineering in Medicine and Biology Society on Annu. Int. Conf. of the IEEE, 327-330, 2010.

28. Zunoubi, M. R., J. Payne, and W. P. Roach, "CUDA implementation of TEz-FDTD solution of Maxwell's equations in dispersive media," IEEE Antennas Wireless Propag. Lett., Vol. 9, 756-759, 2010.
doi:10.1109/LAWP.2010.2060181

29. Tay, W. C., D. Y. Heh, and E. L. Ta, "GPU-accelerated fundamental ADI-FDTD with complex frequency shifted convolutional perfectly matched layer," Progress In Electromagnetics Research M, Vol. 14, 177-192, 2010.
doi:10.2528/PIERM10090605

30. Toivanen, J. I., T. P. Stefanski, N. Kuster, and N. Chavannes, "Comparison of CPML implementations for the GPU-accelerated FDTD solver," Progress In Electromagnetics Research M, Vol. 19, 61-75, 2011.
doi:10.2528/PIERM11061002

31. Lee, K. H., I. Ahmed, R. S. M. Goh, E. H. Khoo, E. P. Li, and T. G. G. Hung, "Implementation of the FDTD method based on lorentz-drude dispersive model on GPU for plasmonics applications," Progress In Electromagnetics Research, Vol. 116, 441-456, 2011.

32. Ma, L. C. and R. Mittra, "Parallel implementation of the periodic boundary condition (PBC) in the FDTD for the investigation of spatial filters," Antennas and Propagation Society International Symposium, IEEE, 1-4, 2008.

33. Mao, Y., B. Chen, B. Zhou, L. Cheng, and Q. Wu, "Parallel implementation of the split-field FDTD method for the analysis of periodic structure," 8’th International Symp. Antennas, Propagation and EM Theory, IEEE, 875-878, 2008.

34. Roden, J. A., J. P. Skinner, and S. L. Johns, "Shielding effectiveness of three dimensional gratings using the periodic FDTD technique and CPML absorbing boundary condition," IEEE/ACES Int. Conf. on Wireless Communications and Applied Computational Electromagnetics, 128-131, 2005.
doi:10.1109/WCACEM.2005.1469545

35. http://www.nvidia.com.

36. NVIDIA CUDA C Programming Guide, Version 3.2, NVIDIA, 2010.

37. Vial, A., A. Grimault, D. Mac´ıas, D. Barchiesi, and M. L. de la Chapelle, "Improved analytical fit of gold dispersion: Application to the modeling of extinction spectra with a finite-difference time-domain method ," Phys. Rev. B, Vol. 71, No. 8, 085416, 2005.
doi:10.1103/PhysRevB.71.085416

38. Born, M. and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, Pergamon, 1980.

39. Maier, S. A., Plasmonics: Fundamentals and Applications, Springer, New York, 2007.

40. Bohren, C. F. and D. R. Huffman, Absorption and Scattering of Light by Small Particles, Wiley Interscience, 1983.

41. Lee, K. S. and M. A. El-Sayed, "Gold and silver nanoparticles in sensing and imaging: Sensitivity of plasmon response to size, shape, and metal composition," J. Phys. Chem. B, Vol. 110, No. 39, 19220-19225, 2006.
doi:10.1021/jp062536y

42. Larsson, E. M., C. Langhammer, I. Zoric, and B. Kasemo, "Nanoplasmonic probes of catalytic reactions," Science, Vol. 326, 1091-1094, 2009.
doi:10.1126/science.1176593

43. Bera, M. and M. Ray, "Precise detection and signature of biological/chemical samples based on surface plasmon resonance (SPR)," J. Opt., Vol. 38, No. 4, 232-248, 2009.
doi:10.1007/s12596-009-0021-x

44. Shankaran, D. R., K. V. Gobi, and N. Miura, "Recent advancements in surface plasmon resonance immunosensors for detection of small molecules of biomedical, food and environmental interest," Sensors and Actuators B, Vol. 121, 158-177, 2007.
doi:10.1016/j.snb.2006.09.014

45. Ankerm, J. N., W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, "Biosensing with plasmonicnanosensors," Nature Materials, Vol. 7, 442-453, 2008.
doi:10.1038/nmat2162

46. Bingham, J. M., J. N. Anker, L. E. Kreno, and R. P. Van Duyne, "Gas sensing with high-resolution localized surface plasmon resonance spectroscopy," J. Am. Chem. Soc., Vol. 132, 17358-17359, 2010.
doi:10.1021/ja1074272

47. Nau, D., A. Seidel, R. B. Orzekowsky, S. H. Lee, S. Deb, and H. Giessen, "Hydrogen sensor based on metallic photonic crystal slabs," Opt. Lett., Vol. 35, No. 18, 3150-3152, 2010.
doi:10.1364/OL.35.003150

48. Homola, J., Surface Plasmon Resonance Based Sensors, Springer, Berlin, 2006.

49. Charles, D. E., M. Gara, D. Aherne, D. M. Ledwith, J. M. Kelly, W. J. Blau, and M. E. Brennan-Fournet, "Scaling of surface plasmon resonances in triangular silver nanoplate sols for enhanced refractive index sensing," Plasmonics, Vol. 6, 351-362, 2011.
doi:10.1007/s11468-011-9211-x

50. Steinbrück, A., O. Stranik, A. Csaki, and W. Fritzsche, "Sensoric potential of gold-silver core-shell nanoparticles," Anal. Bioanal. Chem., Vol. 401, 1241-1249, 2011.
doi:10.1007/s00216-011-5177-y

51. Svedendahl, M., S. Chen, A. Dmitriev, and M. Käll, "Refractometric sensing using propagating versus localized surface plasmons: A direct comparison," Nano Lett., Vol. 9, No. 12, 4428-4433, 2009.
doi:10.1021/nl902721z

52. Rodríguez-Cantó, P. J., M. Martínez-Marco, F. J. Rodríguez-Fortuño, B. Tomás-Navarro, R. Ortuño, S. Peransí-Llopis, and A. Martínez, "Demonstration of near infrared gas sensing using gold nanodisks on functionalized silicon," Opt. Express, Vol. 19, No. 8, 7664-7672, 2011.
doi:10.1364/OE.19.007664

53. Sosnova, M. V., N. L. Dmitruk, A. V. Korovin, S. V. Mamykin, V. I. Mynko, and O. S. Lytvyn, "Local plasmon excitations in one-dimensional array of metal nanowires for sensor applications," Appl. Phys. B, Vol. 99, 493-497, 2010.
doi:10.1007/s00340-009-3799-y

54. Ameling, R., L. Langguth, M. Hentschel, M. Mesch, P. V. Braun, and H. Giessen, "Cavity-enhanced localized plasmon resonance sensing," Appl. Phys. Lett., Vol. 97, 253116, 2010.
doi:10.1063/1.3530795

55. Ye, J. and P. Van Dorpe, "Improvement of figure of merit for gold nanobar array plasmonic sensors," Plasmonics, Vol. 6, 665-671, 2011.
doi:10.1007/s11468-011-9249-9

56. Jiang, H., J. Markowski, and J. Sabarinathan, "Near-infrared optical response of thin film pH-sensitive hydrogel coated on a gold nanocrescent array," Opt. Express, Vol. 17, No. 24, 21802-21807, 2009.
doi:10.1364/OE.17.021802

57. Rodríguez-Fortuño, F. J., M. Martínez-Marco, B. Tomás-Navarro, R. Ortuño, J. Martí, A. Martínez, and P. J. Rodríguez-Cantó, "Highly-sensitive chemical detection in the infrared regime using plasmonic gold nanocrosses," Appl. Phys. Lett., Vol. 98, 133118, 2011.
doi:10.1063/1.3558916

58. Kubo, W. and S. Fujikawa, "Au double nanopillars with nanogap for plasmonic sensor," Nano Lett., Vol. 11, 8-15, 2011.
doi:10.1021/nl100787b

59. Lamprecht, B., G. Schider, R. T. Lechner, H. Ditlbacher, J. R. Krenn, A. Leitner, and F. R. Aussenegg, "Metal nanoparticle gratings: Influence of dipolar particle interaction on the plasmon resonance," Phys. Rev. Lett., Vol. 84, No. 20, 4721-4724, 2000.
doi:10.1103/PhysRevLett.84.4721

60. Zou, S. and G. C. Schatz, "Narrow plasmonic/photonic extinction and scattering line shapes for one and two dimensional silver nanoparticle arrays," J. Chem. Phys., Vol. 121, No. 24, 2004.
doi:10.1063/1.1826036

61. Chu, Y., E. Schonbrun, T. Yang, and K. B. Crozier, "Experimental observation of narrow surface plasmon resonances in gold nanoparticle arrays," Appl. Phys. Lett., Vol. 93, 181108, 2008.
doi:10.1063/1.3012365

62. Offermans, P., M. C. Schaafsma, S. R. K. Rodriguez, Y. Zhang, M. Crego-Calama, S. H. Brongersma, and J. G. Rivas, "Universal scaling of the figure of merit of plasmonic sensors," ACS Nano, Vol. 5, No. 6, 5151-5157, 2011.
doi:10.1021/nn201227b

63. Kravets, V. G., F. Schedin, A. V. Kabashin, and A. N. Grigorenko, "Sensitivity of collective plasmon modes of gold nanoresonators to local environment," Opt. Lett., Vol. 35, No. 7, 956-958, 2010.
doi:10.1364/OL.35.000956

Dr. Afsaneh Shahmansouri

Dr. Bizhan Rashidian