This paper reports an exact and explicit representation of the differential operators from Maxwell's equations. In order to solve these equations, the spline basis functions with compact support are used. We describe the electromagnetic analysis of the lamellar grating as an eigenvalues problem. We choose the second degree spline as basis functions. The basis functions are projected onto a set of test functions. We use and compare several test functions namely: Dirac, Pulse and Spline. We show that the choice of the basis and test functions has a great influence on the convergence speed. The outcomes are compared with those obtained by implementing the Finite-Difference Modal Method which is used as a reference. In order to improve the numerical results an adaptive spatial resolution is used. Compared to the reference method, we show a significantly improved convergence when using the spline expansion projected onto spline test functions.
2. Botten, L. C., M. C. Craig, R. C McPherdran, L. R. Adams, and J. R. Andrewartha, "The finitely conducting lamellar diffraction grating," Opt. Acta, Vol. 28, 1087-1102, 1981.
3. Moharam, M. G. and T. K. Gaylord, "Diffraction analysis of dielectric surface-relief gratings," J. Opt. Soc. Am. A, Vol. 72, 1385-1392, 1982.
doi:10.1364/JOSA.72.001385
4. Neviere, M. and E. Popov, Light Propagation in Periodic Media: Differential Theory and Design, Marcel Dekker, New York, 2003.
5. Morf, R. H., "Exponentially convergent and numerically efficient solution of Maxwell's equations for lamellar gratings," J. Opt. Soc. Am., Vol. 12, No. 5, 1043-1056, 1995.
doi:10.1364/JOSAA.12.001043
6. Lalanne, P. and J. P. Hugonin, "Numerical performance of finite-difference Modal Method for the electromagnetic analysis of one-dimensional grating," J. Opt. Soc. Am., Vol. 17, No. 6, 1033-1042, 2000.
doi:10.1364/JOSAA.17.001033
7. Modisette, J. P., P. Nordlander, J. L. Kinsey, and B. R. Johnson, "Wavelet based in eigenvalue problems in quantum mechanics," Chem. Phys. Letters, Vol. 250, 485-428, 1996.
doi:10.1016/0009-2614(96)00060-7
8. Beylkin, G., R. R. Coifman, and V. Rokhlin, "Fast wavelets transform and numerical algoritms I," Comm. Pure and Appl. Math., Vol. 44, 141-183, 1991, Yale University Technical Report YALEU/DCS/RR-696, August 1989.
doi:10.1002/cpa.3160440202
9. Wagner, R. L. and W. C. Chew, "A study of wavelets for the solution of electromagnetic intergal equations," IEEE Trans. Antennas Propagat., Vol. 43, 614-622, June 1995.
doi:10.1109/8.387178
10. Edee, K., P. Schiavone, and G. Granet, "Analysis of defect in extreme UV Lithography mask using a modal method based on nodal B-spline expansion," Japanese Journal of Applied Physics, Vol. 44, No. 9A, 6458-6462, 2005.
doi:10.1143/JJAP.44.6458
11. Armeanu, A., K. Edee, P. Schiavone, and G. Granet, "The lamellar diffraction grating problem: A spectral method based on spline expansion," Proceedings of ICMI 2 Conference, Vol. 19, No. 2, 37-46, 2009.
12. Jackson, J. D., Classical Electrodynamics, John Wiley and Sons, Inc., New York, 1962.
13. Harrington, R., Field computation by Moment Methods, The Macmillan, New York, 1968.
14. Harrington, R., "Matrix methods for field problem," Proceeding of the IEEE, Vol. 55, No. 2, 136-149, February 1967.
doi:10.1109/PROC.1967.5433
15. Guizal, B., H. Yala, and D. Felbacq, "Reformulation of the eigenvalue problem in the Fourier modal method with spatial adaptive resolution," Opt. Lett., Vol. 34, No. 18, 2790-2792, 2009.
doi:10.1364/OL.34.002790
16. Granet, G., "Reformulation of the lamellar grating problem through the concept of adaptive spatial resolution," J. Opt. Soc. Am., Vol. 16, No. 10, 2510-2516, 1999.
doi:10.1364/JOSAA.16.002510
Submit
