volume | PIER Journals

Abstract

The Modal Method by Gegenbauer polynomials Expansion (MMGE) has been recently introduced for lamellar gratings by Edee [J. Opt. Soc. Am. 28, (2011)]. This method shows a promising potential of outstanding convergence but still suffers from instabilities when the number of polynomials is increased. In this work, we identify the origin of these instabilities and propose a way to remove them.

1. 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, 413-428, 1981.
doi:10.1080/713820571

2. Botten, I. C., M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, "The finitely conducting lamellar diffraction grating," Optica Acta, Vol. 28, 1087-1102, 1981.
doi:10.1080/713820680

3. Knop, K., "Rigorous diffraction theory for transmission phase gratings with deep rectangular grooves ," J. Opt. Soc. Am., Vol. 68, 1206-1210, 1978.
doi:10.1364/JOSA.68.001206

4. Li, L., "New formulation of the fourier modal method for crossed surface-relief gratings," J. Opt. Soc. Am., Vol. 14, 2758-2767, 1997.
doi:10.1364/JOSAA.14.002758

5. Morf, R. H., "Exponentially convergent and numerically efficient solution of Maxwell's equations for lamellar gratings," J. Opt. Soc. Am. A, Vol. 12, 1043-1056, 1995.
doi:10.1364/JOSAA.12.001043

6. Plumey, J. P., B. Guizal, and J. Chandezon, "Coordinate transformation method as applied to asymmetric gratings with vertical facets," J. Opt. Soc. Am. A, Vol. 14, 610-617, 1997.
doi:10.1364/JOSAA.14.000610

7. Edee, K., P. Schiavone, and G. Granet, "Analysis of defect in E.U.V. lithography mask using a modal method by nodal B-spline expansion," Japanese Journal of Applied Physics, Vol. 44, No. 9A, 6458-6462, 2005.
doi:10.1143/JJAP.44.6458

8. Armeanu, A. M., M. K. Edee, G. Granet, and P. Schiavone, "Modal method based on spline expansion for the electromagnetic analysis of the lamellar grating," Progress In Electromagnetics Research, Vol. 106, 243-261, 2010.
doi:10.2528/PIER10021902

9. Harrington, R. F., Field Computation by Moment Methods, The Macmillan Company, New York, 1968, reprinted by IEEE Press, New York, 1993.

10. Edee, K., "Modal method based on subsectional Gegenbauer polynomial expansion for lamellar gratings," J. Opt. Soc. Am., Vol. 28, 2006-2013, 2011.
doi:10.1364/JOSAA.28.002006

11. Hochstrasser, U. W., "Orthogonal polynomials," Handbook of Mathematical Functions, 771-802, M. Abramowitz and I. A. Stegun, eds., Dover, 1965.

12. Yala, H., B. Guizal, and D. Felbacq, "Fourier modal method with spatial adaptive resolution for structures comprising homogeneous layers," J. Opt. Soc. Am. A, Vol. 26, 2567-2570, 2009.
doi:10.1364/JOSAA.26.002567

13. Popov, E., B. Chernov, M. Neviµere, and N. Bonod, "Differential theory: Application to highly conducting gratings," J. Opt. Soc. Am. A, Vol. 21, 199-206, 2004.
doi:10.1364/JOSAA.21.000199

14. Lyndin, N. M., O. Parriaux, and A. V. Tishchenko, "Modal analysis and suppression of the Fourier modal method instabilities in highly conductive gratings," J. Opt. Soc. Am. A, Vol. 24, 3781-3788, 2007.
doi:10.1364/JOSAA.24.003781

15. Guizal, B., H. Yala, and D. Felbacq, "Reformulation of the eigenvalue problem in the Fourier modal method with spatial adaptive resolution," Opt. Lett. A, Vol. 34, 2790-2792, 2009.
doi:10.1364/OL.34.002790

Dr. M. Kofi Edee

Dr. Ismail Fenniche

Prof. Gerard Granet

Prof. Brahim Guizal