We apply the Gabor frame as a projection method to numerically solve a 2D transverse electric-polarized domain-integral equation for a homogeneous medium. Since the Gabor frame is spatially as well as spectrally very well convergent, it is convenient to use for solving a domain integral equation. The mixed spatial and spectral nature of the Gabor frame creates a natural and fast way to Fourier transform a function. In the spectral domain we employ a coordinate scaling to smoothen the branchcut found in the Green function. We have developed algorithms to perform multiplication and convolution efficiently, scaling as O(NlogN) on the number of Gabor coefficients, yielding an overall algorithm that also scales as O(NlogN).
2. Ribot, C., P. Lalanne, M.-S.-L. Lee, B. Loiseaux, and J.-P. Huignard, "Analysis of blazed diffractive optical elements formed with artificial dielectrics," Jounal of the Optical Society of America A, Vol. 24, No. 12, 3819-3826, 2007.
doi:10.1364/JOSAA.24.003819
3. Glaser, T., S. Schroter, H. Bartelt, H.-J. Fuchs, and E.-B. Kley, "Diffractive optical isolator made of high-efficiency dielectric gratings only," Applied Optics, Vol. 41, No. 18, 3558-3566, 2002.
doi:10.1364/AO.41.003558
4. Dzibrou, D. O., J. J. G. M. van der Tol, and M. K. Smit, "Tolerant polarization converter for ingaaspinp photonic integrated circuits," Optics Letters, Vol. 38, No. 18, 3482-3484, 2013.
doi:10.1364/OL.38.003482
5. Wang, L., Y. Wang, and X. Zhang, "Embedded metallic focus grating for silicon nitride waveguide with enhanced coupling and directive radiation," Optical Express, Vol. 20, No. 16, 2012.
6. Shlager, K. L. and J. B. Schneider, "A selective survey of the finite-difference time-domain literature," IEEE Antennas and Propagation Magazine, Vol. 37, No. 4, 1995.
doi:10.1109/74.414731
7. Bengzon, F. and M. G. Larson, The Finite Element Method: Theory, Implementation, and Applications, Springer, 2013.
doi:10.1007/978-3-642-33287-6
8. Clemens, M. and T. Weiland, "Discrete electromagnetism with the finite integration technique," Progress In Electromagnetics Research, Vol. 32, 65-87, 2001.
doi:10.2528/PIER00080103
9. Zwamborn, P. and P. M. van den Berg, "The three-dimensional weak form of the conjugate gradient FFT method for solving scattering problems," IEEE Trans. Microwave Theory Tech., Vol. 40, No. 9, 1992.
doi:10.1109/22.156602
10. Bleszynski, E., M. Bleszynski, and T. Jaroszewicz, "AIM: Adaptive integral method for solving large-scale electromagnetic scattering and radiation problems," Radio Science, Vol. 31, No. 5, 1225-1251, 1996.
doi:10.1029/96RS02504
11. Philips, J. R. and J. K. White, "Efficient capacitiance extraction of 3D structures using generalized precorrected FFT methods," IEEE Transactions on Microwave Theory and Techniques, 253-256, 1994.
12. Botten, I. C., M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, "The dielectric Lamellar diffraction grating," Optica Acta, Vol. 28, No. 3, 413-428, 1981.
doi:10.1080/713820571
13. Moharam, M. G. and T. K. Gaylord, "Rigorous coupled-wave analysis of planar-grating diffraction," Jounal of the Optical Society of America, Vol. 71, No. 7, 1981.
14. Chandezon, J., D. Maystre, and G. Raoult, "A new theoretical method for diffraction gratings and its numerical application," Journal of Optics, Vol. 11, 235-241, 1980.
doi:10.1088/0150-536X/11/4/005
15. Poyedinchuk, A. Y., Y. A. Tuchkin, N. P. Yashina, J. Chandezon, and G. Granet, "C-method: Several aspects of spectral theorry of gratings," Progress In Electromagnetics Research, Vol. 59, 113-149, 2006.
doi:10.2528/PIER05050901
16. Van Beurden, M. C., "A spectral volume integral equation method for arbitrary bi-periodic gratings with explicit Fourier factorization," Progress In Electromagnetics Research B, Vol. 36, 133-149, 2012.
doi:10.2528/PIERB11100307
17. Pisarenco, M., J. Maubach, I. Setija, and R. Mattheij, "Formulation for simulation of scattering from finite structures," Journal of The Optical Society of America A, 2010.
18. Bastiaans, M. J., "A sampling theorem for the complex spectrogram, and Gabor's expansion of a signal in Gaussian elementary signals," Optical Engineering, Vol. 20, No. 4, 594-598, 1981.
doi:10.1117/12.7972768
19. Bastiaans, M. J., Gabor's expansion and the Zak transform for continuous-time and discrete-time signals: Critical sampling and rational oversampling, Eindhoven University of Technology, Eindhoven, 1995.
20. Feichtinger, H. G. and T. Strohmer, Gabor Analysis and Algorithms: Theory and Applications, Birkhauser, 1998.
doi:10.1007/978-1-4612-2016-9
21. Battle, G., "Heisenberg proof of the Balian-Low theorem," Letters in Mathematical Physics, Vol. 15, 175-177, 1988.
doi:10.1007/BF00397840
22. Benedetto, J. J., C. Heil, and D. F. Walnut, "Differentiation and the Balian-Low theorem," The Journal of Fourier Analysis and Applications, Vol. 1, No. 4, 355-402, 1995.
doi:10.1007/s00041-001-4016-5
23. Sondergaard, P. L., Finite discrete Gabor analysis, PhD Thesis, Institut for Matematik, DTU, 2007.
24. Maciel, J. J. and L. B. Felsen, "Discretized Gabor-based beam algorithm for time-harmonic radiation from two-dimensional truncated planar aperture distributions --- I: Formulation and solution," IEEE Transactions on Antennas and Propagation, Vol. 50, No. 12, 1751-1759, 2002.
doi:10.1109/TAP.2002.807419
25. Maciel, J. J. and L. B. Felsen, "ctime-harmonic radiation from two-dimensional truncated planar aperture distributions --- II: Asymptotics and numerical tests," IEEE Transactions on Antennas and Propagation, Vol. 50, No. 12, 1760-1768, 2002.
doi:10.1109/TAP.2002.807381
26. Einziger, P. D., S. Raz, and M. Saphira, "Gabor representation and aperture theory," Journal of Journal of the Optical Society of America, Vol. 3, No. 4, 508-522, 1986.
doi:10.1364/JOSAA.3.000508
27. Maciel, J. J. and L. B. Felsen, "Systematic study of fields due to extended apertures by Gaussian discretization," IEEE Transactions on Antennas and Propagation, Vol. 37, No. 7, 884-892, 1989.
doi:10.1109/8.29383
28. Daubechies, I., S. Jaffard, and J.-L. Journe, "A simple Wilson orthonormal basis with exponential decay," SIAM Journal on Mathematical Analysis, Vol. 22, No. 2, 554-572, 1991.
doi:10.1137/0522035
29. Floris, S. J. and B. P. de Hon, "Electromagnetic field expansion in a Wilson basis," Proceedings of the 42nd European Microwave Conference (EuMC), Amsterdam, NL, Oct. 29-Nov. 1, 2012.
30. Lugara, D. and C. Letrou, "Printed antennas analysis by a Gabor frame-based method of moments," IEEE Transactions on Antennas and Propagation, Vol. 50, No. 11, 1588-1597, 2002.
doi:10.1109/TAP.2002.804023
31. Jackson, J. D., "Classical Electrodynamics," Wiley, 2007.
32. Szego, G., "Orthogonal polynomials," Royal American Mathematical Society, 1975.
33. Balanis, C. A., Advanced Engineering Electromagnetics, John Wiley & Sons, Inc., 1989.
34. Burger, S., L. Zschiedrich, J. Pomplun, and F. Schmidt, "Finite-element based electromagnetic field simulations: Benchmark results for isolated structures," Proc. SPIE, Vol. 8880, 2013.
Submit
