volume | PIER Journals

Abstract

Two alternative approaches to the spatial spectral integral equation method are proposed. The first enhancement comprises a Hermite interpolation as the set of basis functions instead of the Gabor frame. The continuity, differentiability, equidistant spacing, and small support of these basis functions allows for an efficient and accurate numerical implementation. The second approach encompasses a method to transform between the spatial domain and the deformed path in the complexplane spectral domain. This method allows for more general path shapes, thereby removing the need to decompose the complex-plane spectral-domain path into distinct straight sections. Both enhancements are implemented for the case of TE polarization, and the results are validated against the finite element method and the rigorous coupled-wave analysis.

1. Dilz, R. J. and M. C. van Beurden, "An efficient complex spectral path formulation for simulating the 2D TE scattering problem in a layered medium using Gabor frames," Journal of Computational Physics, Vol. 345, 528-542, 2017.
doi:10.1016/j.jcp.2017.05.034

2. Dilz, R. J., M. G. G. M. van Kraaij, and M. C. van Beurden, "2D TM scattering problem for finite objects in a dielectric stratified medium employing Gabor frames in a domain integral equation," Journal of the Optical Society of America A, Vol. 34, No. 8, 1315-1321, 2017.
doi:10.1364/JOSAA.34.001315

3. Dilz, R. J., A spatial spectral domain integral equation solver for electromagnetic scattering in dielectric layered media, Ph.D. thesis, Chapter 8, Eindhoven University of Technology, 2017.

4. Dilz, R. J. and M. C. van Beurden, "Fast operations for a Gabor-frame based integral equation with equidistant sampling," IEEE Antennas and Wireless Propagation Letters, Vol. 12, No. 1, 82-85, 2018.
doi:10.1109/LAWP.2017.2775702

5. 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, 1757-1766, Sep. 1992.
doi:10.1109/22.156602

6. Diebold, A. C., Handbook of Silicon Semiconductor Metrology, CRC Press, 2001.
doi:10.1201/9780203904541

7. Dilz, R. J. and M. C. van Beurden, "The Gabor frame as a discretization for the 2D transverseelectric scattering-problem domain integral equation," Progress In Electromagnetics Research B, Vol. 69, 117-136, 2016.
doi:10.2528/PIERB16061406

8. Chew, W. C., Waves and Fields in Inhomogeneous Media, IEEE Press, 1995.

9. Felsen, L. B. and N. Marcuvitz, Radiation and Scattering of Waves, IEEE Press, 1973.

10. Kong, J. A., Theory of Electromagnetic Waves, John Wiley & Sons, Inc, 1975.

11. Wait, J. R., Electromagnetic Waves in Stratified Media, Pergamon Press, 1970.

12. Sommerfeld, A., "Uber der ausbreitung der wellen in der drahtlosen telegraphie," Annalen der Physik, Vol. 333, No. 4, 665-736, 1909.
doi:10.1002/andp.19093330402

13. Hochman, A. and Y. Leviatan, "A numerical methodology for efficient evaluation of 2D Sommerfield integral in the dielectric half-space problem," IEEE Transactions on Antennas and Propagation, Vol. 58, No. 2, 413-431, Feb. 2010.
doi:10.1109/TAP.2009.2037761

14. De Ruiter, H. M., "Limits on the propagation constants of planar optical waveguide modes," Applied Optics, Vol. 20, No. 5, 731-732, 1981.
doi:10.1364/AO.20.000731

15. Newman, E. H. and D. Forrai, "Scattering from a microstrip patch," IEEE Transactions on Antennas and Propagation, Vol. 35, No. 3, 245-251, Mar. 1987.
doi:10.1109/TAP.1987.1144084

16. Larson, M. G. and F. Bengzon, The Finite Element Method: Theory, Implementation, and Applications, Springer, 2013.
doi:10.1007/978-3-642-33287-6

17. Burger, S., L. Zschiedrich, J. Pomplun, and F. Schmidt, "Finite-element based electromagnetic field simulations: Benchmark results for isolated structures," Proc. SPIE 8880 Photomask Technology, Vol. 8880, 2013.

18. Moharam, M. G. and T. K. Gaylord, "Rigorous coupled-wave analysis of planar-grating diffraction," Journal of the Optical Society of America, Vol. 73, No. 4, 811-818, 1981.
doi:10.1364/JOSA.71.000811

19. 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

20. Pisarenco, M., J. Maubach, I. Setija, and R. Mattheij, "Aperiodic Fourier modal method in contrast-field formulation for simulation of scattering from finite structures," Journal of the Optical Society of America A, Vol. 27, No. 11, 2423-2431, 2010.
doi:10.1364/JOSAA.27.002423

21. Berezin, I. S. and N. P. Zhidkov, Computing Methods, Pergamon Press, 1965.

22. Dilz, R. J. and M. C. van Beurden, "Computational aspects of a spatial spectral domain integralequation for scattering by objects of large longitudinal extent," 2017 International Conference on Electromagnetics in Advanced Applications (ICEAA), Verona, Italy, Sep. 11–15, 2017.

23. Taillaert, D., F. Van Laere, M. Ayre, W. Bogaerts, D. Van Thourhout, P. Bienstman, and R. Baets, "Grating couplers for coupling between optical fibers and nanophotonic waveguides," Japanese Journal of Applied Physics, Vol. 45, No. 8a, 6071-6077, 2006.
doi:10.1143/JJAP.45.6071

24. Lawrence, G. N., K. E. Moore, and P. J. Cronkite, "Rotationally symmetric construction optics for a waveguide focusing grating," Appl. Opt., Vol. 29, No. 15, 2315-2319, May 1990.
doi:10.1364/AO.29.002315

25. Forouhar, S., R.-X. Lu, W. S. C. Chang, R. L. Davis, and S.-K. Yao, "Chirped grating lenses on nb2o5 transition waveguides," Appl. Opt., Vol. 22, No. 19, 3128-3132, Oct. 1983.
doi:10.1364/AO.22.003128

Mr. Roeland Johannes Dilz

Prof. Martijn Constant van Beurden