volume | PIER Journals

Abstract

In this work, a matrix method is applied to study the propagation of electromagnetic waves inside a non-slanted reflection grating. The elements of the matrix which characterizes the periodic medium are obtained in terms of Mathieu functions and their derivatives, and the expressions of the efficiencies of reflected and transmitted orders are calculated in terms of the elements of the matrix. In addition the band structure of a general reflection grating is studied with the layer matrix of one single period. The results obtained by this matrix method are firstly compared to the results obtained by Kogelnik's expressions in index-matched media showing good agreement. The comparison is also made for a reflection grating embedded in two media with different refractive indexes, showing good agreement with an FDTD method, but slight differences with respect to Kogelnik's Coupled Wave Theory.

1. Kogelnik, H., "Coupled wave theory for thick hologram gratings," Bell Labs Tech. J., Vol. 48, No. 9, 2909-2947, 1969. Google Scholar

2. Moharam, M. G. and T. K. Gaylord, "Rigorous coupled-wave analysis of planar-grating diffraction," J. Opt. Soc. Am., Vol. 71, No. 7, 811-818, 1981.
doi:10.1364/JOSA.71.000811 Google Scholar

3. Moharam, M. G. and T. K. Gaylord, "Rigorous coupled-wave analysis of grating diffraction-e-mode polarization and losses," J. Opt. Soc. Am., Vol. 73, No. 4, 451-455, 1983.
doi:10.1364/JOSA.73.000451 Google Scholar

4. Moharam, M. G. and T. K. Gaylord, "Three-dimensional vector coupled-wave analysis of planar-grating diffraction," J. Opt. Soc. Am., Vol. 73, No. 9, 1105-1112, 1983.
doi:10.1364/JOSA.73.001105 Google Scholar

5. Moharam, M. G. and T. K. Gaylord, "Analysis and applications of optical diffraction by gratings," Proc. IEEE, Vol. 73, No. 5, 894-937, 1985.
doi:10.1109/PROC.1985.13220 Google Scholar

6. Moharam, M. G. and T. K. Gaylord, "Rigorous coupled-wave analysis of metallic surface-relief gratings," J. Opt. Soc. Am. A, Vol. 3, No. 11, 1780-1787, 1986.
doi:10.1364/JOSAA.3.001780 Google Scholar

7. Moharam, M. G., E. B. Grann, D. A. Pommet, and T. K. Gaylord, "Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings," J. Opt. Soc. Am. A, Vol. 12, No. 5, 1068-1076, 1995.
doi:10.1364/JOSAA.12.001068 Google Scholar

8. Kamiya, N., "Rigorous coupled-wave analysis for practical planar dielectric gratings: 1. Thickness-changed holograms and some characteristics of diffraction efficiency," Appl. Optics, Vol. 37, No. 25, 5843-5853, 1998.
doi:10.1364/AO.37.005843 Google Scholar

9. Neipp, C., A. Beléndez, S. Gallego, M. Ortuño, I. Pascual, and J. Sheridan, "Angular responses of the first and second diffracted orders in transmission diffraction grating recorded on photopolymer material," Opt. Express, Vol. 11, No. 16, 1835-1843, 2003.
doi:10.1364/OE.11.001835 Google Scholar

10. Dansas, P. and N. Paraire, "Fast modeling of photonic bandgap structures by use of a diffraction-grating approach," J. Opt. Soc. Am. A, Vol. 15, No. 6, 1586-1598, 1998.
doi:10.1364/JOSAA.15.001586 Google Scholar

11. Chang, N. Y. and C. J. Juo, "Algorithm based on rigorous coupled-wave analysis for diffractive optical element design," J. Opt. Soc. Am. A, Vol. 18, No. 10, 2491-2501, 2001.
doi:10.1364/JOSAA.18.002491 Google Scholar

12. Little, B. E. and W. P. Huang, "Coupled-mode theory for optical waveguides," Progress In Electromagnetics Research, Vol. 10, 217-270, 1995. Google Scholar

13. Jarem, J. M., "Rigorous coupled wave theory of anisotropic, azimuthally-inhomogeneous cylindrical systems," Progress In Electromagnetics Research, Vol. 19, 109-127, 1998.
doi:10.2528/PIER97103100 Google Scholar

14. Carretero, L., M. Pérez-Molina, P. Acebal, S. Blaya, and A. Fimia, "Matrix method for the study of wave propagation in one-dimensional general media," Opt. Express, Vol. 14, No. 23, 11385-11391, 2006.
doi:10.1364/OE.14.011385 Google Scholar

15. Lekner, J., Theory of Reflection of Electromagnetic and Particle Waves, Kluwer Academic Publishers Group, 1987.

16. Abramowitz, M. and I. A. Stegun, Handbook of Mathematical Functions, Ch. 20, 722-748, Dover Publications, Inc., New York, 1972.

17. Abeles, F., Optics of Thin Films in Advanced Optical Techniques, North-Holland Publishing Co., Amsterdam, 1967.

18. Frenkel, D. and R. Portugal, "Algebraic methods to compute Mathieu functions," J. Phys. A: Math. Gen., Vol. 34, 3541-3351, 2001.
doi:10.1088/0305-4470/34/17/302 Google Scholar

19. Vahabi Sani, N., A. Mohammadi, A. Abdipour, and F. M. Ghannouchi, "Analysis of multiport receivers using FDTD technique," Journal of Electromagnetic Waves and Applications, Vol. 23, No. 5-6, 635-643, 2009.
doi:10.1163/156939309788019921 Google Scholar

20. Silva, A. O., R. Bertholdo, M. G. Chiavetto, B.-H. V. Borges, S. J. L. Ribeiro, Y. Messaddeq, and M. A. Romero, "Comparative analysis between experimental characterization results and numerical FDTD modeling of self-assembled photonic crystals," Progress In Electromagnetics Research B, Vol. 23, No. 19, 329-342, 2010.
doi:10.2528/PIERB10060404 Google Scholar

21. Sullivan, D. M., Electromagnetic Simulation Using the FDTD Method, IEEE Press Editorial Board, 2000.

22. Balanis, C. A., Advanced Engineering Electromagnetics, Wiley, New York, 1989.

23. Yee, K. S., "Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media," IEEE Trans. Antennas Propag., Vol. 14, No. 3, 302-307, 1996. Google Scholar

24. Taflove, A., Computational Electrodynamics: The Finite-di®erence Time-domain Method, Artech House Publishers, 1995.

25. Berenger, J. P., "A perfectly matched layer for the absorption of electromagnetic waves," J. Comput. Phys., Vol. 114, No. 2, 185-200, 1994.
doi:10.1006/jcph.1994.1159 Google Scholar

26. Sullivan, D. M., "A simplified PML for use with the FDTD method," Microwave and Guided Wave Letters IEEE, Vol. 6, No. 2, 97-99, 1996.
doi:10.1109/75.482001 Google Scholar

27. Kunz, K. S. and R. J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics, CRC Press, 1993.

28. Pérez-Ocón, F., J. R. Jiménez, and A. M. Pozo, "Exponential discretization of the perfectly matched layer (PML) absorbing boundary condition simulation in FDTD 3D ," Optik, Vol. 113, No. 8, 354-360, 2002.
doi:10.1078/0030-4026-00176 Google Scholar

29. Francés, J., C. Neipp, M. Pérez-Molina, A. Beléndez, and , "Rigorous interference and diffraction analysis of diffractive optic elements using the finite-difference time-domain method ," Comput. Phys. Commun., Vol. 181, No. 12, 1963-1973, 2010.
doi:10.1016/j.cpc.2010.09.005 Google Scholar

30. Zheng, G., A. A. Kishk, A. W. Glisson, and A. B. Yakovlev, "Implementation of Mur's absorbing boundaries with periodic structures to speed up the design process using finite-difference time-domain method," Progress In Electromagnetics Research, Vol. 58, 101-114, 2006.
doi:10.2528/PIER05062103 Google Scholar

31. Zheng, G., B.-Z.Wang, H. Li, X.-F. Liu, and S. Ding, "Analysis of finite periodic dielectric gratings by the finite-difference frequency-domain method with the sub-entire-domain basis functions and wavelets," Progress In Electromagnetics Research, Vol. 99, 453-463, 2009.
doi:10.2528/PIER09111502 Google Scholar

32. Suyama, T., Y. Okuno, and T. Matsuda, "Surface plasmon resonance absorption in a multilayered thin-film grating," Journal of Electromagnetic Waves and Applications, Vol. 23, No. 13, 1773-1783, 2009.
doi:10.1163/156939309789566914 Google Scholar

33. Ni, J., B. Chen, S. L. Zheng, X.-M. Zhang, X.-F. Jin, and H. Chi, "Ultra-wideband bandpass filter with notched band based on electrooptic phase modulator and phase-shift fiber Bragg grating ," Journal of Electromagnetic Waves and Applications, Vol. 24, No. 5-6, 795-802, 2010.
doi:10.1163/156939310791036395 Google Scholar

34. Liau, J.-J., N.-H. Sun, S.-C. Lin, R.-Y. Ro, J.-S. Chiang, C.-L. Pan, and H.-W. Chang, "A new look at numerical analysis of uniform fiber Bragg gratings using coupled mode theory," Progress In Electromagnetics Research, Vol. 93, 385-401, 2009.
doi:10.2528/PIER09031102 Google Scholar

35. Sun, N.-H., J.-J. Liau, Y.-W. Kiang, S.-C. Lin, R.-Y. Ro, J.-S. Chiang, and H.-W. Chang, "Numerical analysis of apodized fiber Bragg gratings using coupled mode theory," Progress In Electromagnetics Research, Vol. 99, 289-306, 2009.
doi:10.2528/PIER09102704 Google Scholar

36. Swillam, M. A., R. H. Gohary, M. H. Bakr, and X. Li, "Efficient approach for sensitivity analysis of lossy and leaky structures using FDTD," Progress In Electromagnetics Research, Vol. 94, 197-212, 2009.
doi:10.2528/PIER09061708 Google Scholar

37. Faghihi, F. and H. Heydari, "Time domain physical optics for the higher-order FDTD modeling in electromagnetic scattering from 3-D complex and combined multiple materials objects," Progress In Electromagnetics Research, Vol. 95, 87-102, 2009.
doi:10.2528/PIER09040407 Google Scholar

38. Zhang, Y.-Q. and D.-B. Ge, "A unified FDTD approach for electromagnetic analysis of dispersive objects," Progress In Electromagnetics Research, Vol. 96, 155-172, 2009.
doi:10.2528/PIER09072603 Google Scholar

39. Yang, S., Y. Chen, and Z.-P. Nie, "Simulation of time modulated linear antenna arrays using the FDTD method," Progress In Electromagnetics Research, Vol. 98, 175-190, 2009.
doi:10.2528/PIER09092507 Google Scholar

40. Xiao, S.-Q., Z. Shao, and B.-Z. Wang, "Application of the improved matrix type FDTD method for active antenna analysis," Progress In Electromagnetics Research, Vol. 100, 245-263, 2010.
doi:10.2528/PIER09112204 Google Scholar

41. Kalaee, P. and J. Rashed-Mohassel, "Investigation of dipole radiation pattern above a chiral media using 3D BI-FDTD approach," Journal of Electromagnetic Waves and Applications, Vol. 23, No. 1, 75-86, 2009.
doi:10.1163/156939309787604706 Google Scholar

42. Tay, W. C. and E. L. Tan, "Implementations of PMC and PEC boundary conditions for efficient fundamental ADI- and LOD-FDTD," Journal of Electromagnetic Waves and Applications, Vol. 24, No. 4, 565-573, 2010. Google Scholar

43. Dai, S.-Y., C. Zhang, and Z.-S. Wu, "Electromagnetic scattering of objects above ground using MRTD/FDTD hybrid method," Journal of Electromagnetic Waves and Applications, Vol. 23, No. 16, 2187-2196, 2009.
doi:10.1163/156939309790109306 Google Scholar

44. Li, J., L.-X. Guo, and H. Zeng, "FDTD method investigation on the polarimetric scattering from 2-D rough surface," Progress In Electromagnetics Research, Vol. 101, 173-188, 2010.
doi:10.2528/PIER09120104 Google Scholar

45. Xu, K., Z. Fan, D.-Z. Ding, and R.-S. Chen, "Gpu accelerated unconditionally stable crank-nicolson FDTD method for the analysis of three-dimensional microwave circuits," Progress In Electromagnetics Research, Vol. 102, 381-395, 2010.
doi:10.2528/PIER10020606 Google Scholar

46. Izadi, M., M. Z. A. Ab Kadir, C. Gomes, and W. F. W. Ahmad, "An analytical second-FDTD method for evaluation of electric and magnetic fields at intermediate distances from lightning channel," Progress In Electromagnetics Research, Vol. 110, 329-352, 2010.
doi:10.2528/PIER10080801 Google Scholar

Dr. Jorge Frances Monllor

Dr. Cristian Neipp

Dr. Andres Marquez Ruiz

Dr. Augusto Belendez

Dr. Inmaculada Pascual