We present an improvement of the standard Fourier Modal Method (FMM) for the analysis of lamellar gratings that is based on the use of automatically generated adaptive coordinates for arbitrarily shaped material profiles in the lateral plane of periodicity. This allows for an accurate resolution of small geometric features and/or large material contrasts within the unit. For dielectric gratings, we obtain considerable convergence accelerations. Similarly, for metallic gratings, our approach allows efficient and accurate computations of transmittance and reflectance coefficients into various Bragg orders, the spectral positions of Rayleigh anomalies, and field enhancement values within the grating structures.

© 2010 Optical Society of America

1. K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, and M. Wegener, "Periodic nanostructures for photonics," Phys. Rep. 444, 101-202 (2007). [CrossRef]
2. F. J. Garcia de Abajo, "Light scattering by particle and hole arrays," Rev. Mod. Phys. 79, 1267-1290 (2007). [CrossRef]
3. M. E. Stewart, C. R. Anderton, L. B. Thompson, J. Maria, S. K. Gray, J. A. Rogers, and R. G. Nuzzo, "Nanostructured plasmonic sensors," Chem. Rev. 108, 494-521 (2008). [CrossRef] [PubMed]
4. M. G. Moharam, and T. K. Gaylord, "Diffraction analysis fo dielectric surface-relief gratings," J. Opt. Soc. Am. 72, 1385-1392 (1982). [CrossRef]
5. M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, "Formulation of stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings," J. Opt. Soc. Am. A 12, 1068-1076 (1995). [CrossRef]
6. P. Lalanne, and G. M. Morris, "Highly improved convergence of the coupled-wave method for TM polarization," J. Opt. Soc. Am. A 13, 779-784 (1996). [CrossRef]
7. G. Granet, and B. Guizal, "Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization," J. Opt. Soc. Am. A 13, 1019-1023 (1996). [CrossRef]
8. L. Li, "Use of Fourier series in the analysis of discontinuous periodic structures," J. Opt. Soc. Am. A 13, 1870-1876 (1996). [CrossRef]
9. L. Li, "Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings," J. Opt. Soc. Am. A 13, 1024-1034 (1996). [CrossRef]
10. L. Li, "New formulation of the Fourier modal method for crossed surface-relief gratings," J. Opt. Soc. Am. A 14, 2758-2767 (1997). [CrossRef]
11. M. Nevière, and E. Popov, Light propagation in periodic media: Differential theory and design, Marcel Dekker (New York), 2003.
12. T. Schuster, J. Ruoff, N. Kerwien, S. Rafler, and W. Osten, "Normal vector method for convergence improvement using the RCWA for crossed gratings," J. Opt. Soc. Am. A 24, 2880-2890 (2007). [CrossRef]
13. G. Granet, "Reformulation of the lamellar grating problem through the concept of adaptive spatial resolution," J. Opt. Soc. Am. A 16, 2510-2516 (1999). [CrossRef]
14. J. Chandezon, M. T. Dupuis, G. Gornet, and D. Maystre, "Multicoated gratings: a differential formalism applicable in the entire optical region," J. Opt. Soc. Am. 72, 839-846 (1982). [CrossRef]
15. T. Vallius, and M. Honkanen, "Reformulation of the Fourier modal method with adaptive spatial resolution: application to multilevel profiles," Opt. Express 10, 24-34 (2002). [PubMed]
16. G. Granet, and J.-P. Plumey, "Parametric formulation of the Fourier modal method for crossed surface-relief gratings," J. Opt. A 4, S145-S149 (2002).
17. T. Weiss, G. Granet, N. A. Gippius, S. G. Tikhodeev, and H. Giessen, "Matched coordinates and adaptive spatial resolution in the Fourier modal method," Opt. Express 17, 8051-8061 (2009). [CrossRef] [PubMed]
18. U. Leonhardt, and T. G. Philbin, "General relativity in electrical engineering," N. J. Phys. 8, 247 (2006). [CrossRef]
19. A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, "Cloaking devices, electromagnetic wormholes, and transformation optics," SIAM Rev. 51, 3-33 (2009). [CrossRef]
20. G. J. Pearce, T. D. Hedley, and D. M. Bird, "Adaptive curvilinear coordinates in a plane-wave solution of Maxwell’s equations in photonic crystals," Phys. Rev. B 71, 195108 (2005). [CrossRef]
21. F. Gygi, "Electronic-structure calculations in adaptive coordinates," Phys. Rev. B 48, 11692-11700 (1993). [CrossRef]
22. P. G¨otz, T. Schuster, K. Frenner, S. Rafler, and W. Osten, "Normal vector method for the RCWA with automated vector field generation," Opt. Express 16, 17295-17301 (2008). [CrossRef] [PubMed]
23. L. Li, "Fourier modal method for crossed anisotropic gratings with arbitrary permittivity and permeability tensors," J. Opt. A 5, 345-355 (2003).
24. L. Li, "Note on the S-matrix propagation algorithm," J. Opt. Soc. Am. A 20, 655-660 (2003). [CrossRef]
25. G. Granet, and B. Guizal, "Analysis of strip gratings using a parametric modal method by Fourier expansions," Opt. Commun. 255, 1-11 (2005). [CrossRef]
26. http://www.gnu.org.
27. A. Vial, A.-S. Grimault, D. Macías, D. Barchiesi, and M. Lamy de la Chapelle, "Improved analytical fit of gold dispersion: Application to the modeling of extinction spectra with a finite-difference time-domain method," Phys. Rev. B 71, 085416 (2005). [CrossRef]
28. B. Bai, and L. Li, "Group-theoretic approach to enhancing the Fourier modal method for crossed gratings with C4 symmetry," J. Opt. A 7, 783-789 (2005).
29. J. S. Shumaker-Parry, H. Rochholz, and M. Kreiter, "Fabrication of Crescent-Shaped Optical Antennas," Adv. Mater. 17, 2131-2134 (2005). [CrossRef]
30. H. Rochholz, N. Bocchio, and M. Kreiter, "Tuning resonances on crescent-shaped noble-metal nanoparticles," N. J. Phys. 9, 53 (2007). [CrossRef]
31. Y. Choi, S. Hong, and L. P. Lee, "Shadow Overlap Ion-beam Lithography for Nanoarchitectures," Nano Lett. 9, 3726-3731 (2009). [CrossRef] [PubMed]

Adv. Mater.

• J. S. Shumaker-Parry, H. Rochholz, and M. Kreiter, "Fabrication of Crescent-Shaped Optical Antennas," Adv. Mater. 17, 2131-2134 (2005). [CrossRef]

Chem. Rev.

• M. E. Stewart, C. R. Anderton, L. B. Thompson, J. Maria, S. K. Gray, J. A. Rogers, and R. G. Nuzzo, "Nanostructured plasmonic sensors," Chem. Rev. 108, 494-521 (2008). [CrossRef] [PubMed]

J. Opt. A

• G. Granet, and J.-P. Plumey, "Parametric formulation of the Fourier modal method for crossed surface-relief gratings," J. Opt. A 4, S145-S149 (2002).
• B. Bai, and L. Li, "Group-theoretic approach to enhancing the Fourier modal method for crossed gratings with C4 symmetry," J. Opt. A 7, 783-789 (2005).
• L. Li, "Fourier modal method for crossed anisotropic gratings with arbitrary permittivity and permeability tensors," J. Opt. A 5, 345-355 (2003).

J. Opt. Soc. Am.

• J. Chandezon, M. T. Dupuis, G. Gornet, and D. Maystre, "Multicoated gratings: a differential formalism applicable in the entire optical region," J. Opt. Soc. Am. 72, 839-846 (1982). [CrossRef]
• M. G. Moharam, and T. K. Gaylord, "Diffraction analysis fo dielectric surface-relief gratings," J. Opt. Soc. Am. 72, 1385-1392 (1982). [CrossRef]

J. Opt. Soc. Am. A

• M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, "Formulation of stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings," J. Opt. Soc. Am. A 12, 1068-1076 (1995). [CrossRef]
• P. Lalanne, and G. M. Morris, "Highly improved convergence of the coupled-wave method for TM polarization," J. Opt. Soc. Am. A 13, 779-784 (1996). [CrossRef]
• G. Granet, and B. Guizal, "Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization," J. Opt. Soc. Am. A 13, 1019-1023 (1996). [CrossRef]
• L. Li, "Use of Fourier series in the analysis of discontinuous periodic structures," J. Opt. Soc. Am. A 13, 1870-1876 (1996). [CrossRef]
• L. Li, "Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings," J. Opt. Soc. Am. A 13, 1024-1034 (1996). [CrossRef]
• L. Li, "New formulation of the Fourier modal method for crossed surface-relief gratings," J. Opt. Soc. Am. A 14, 2758-2767 (1997). [CrossRef]
• T. Schuster, J. Ruoff, N. Kerwien, S. Rafler, and W. Osten, "Normal vector method for convergence improvement using the RCWA for crossed gratings," J. Opt. Soc. Am. A 24, 2880-2890 (2007). [CrossRef]
• G. Granet, "Reformulation of the lamellar grating problem through the concept of adaptive spatial resolution," J. Opt. Soc. Am. A 16, 2510-2516 (1999). [CrossRef]
• L. Li, "Note on the S-matrix propagation algorithm," J. Opt. Soc. Am. A 20, 655-660 (2003). [CrossRef]

N. J. Phys.

• U. Leonhardt, and T. G. Philbin, "General relativity in electrical engineering," N. J. Phys. 8, 247 (2006). [CrossRef]
• H. Rochholz, N. Bocchio, and M. Kreiter, "Tuning resonances on crescent-shaped noble-metal nanoparticles," N. J. Phys. 9, 53 (2007). [CrossRef]

Nano Lett.

• Y. Choi, S. Hong, and L. P. Lee, "Shadow Overlap Ion-beam Lithography for Nanoarchitectures," Nano Lett. 9, 3726-3731 (2009). [CrossRef] [PubMed]

Opt. Commun.

• G. Granet, and B. Guizal, "Analysis of strip gratings using a parametric modal method by Fourier expansions," Opt. Commun. 255, 1-11 (2005). [CrossRef]

Opt. Express

• P. G¨otz, T. Schuster, K. Frenner, S. Rafler, and W. Osten, "Normal vector method for the RCWA with automated vector field generation," Opt. Express 16, 17295-17301 (2008). [CrossRef] [PubMed]
• T. Vallius, and M. Honkanen, "Reformulation of the Fourier modal method with adaptive spatial resolution: application to multilevel profiles," Opt. Express 10, 24-34 (2002). [PubMed]
• T. Weiss, G. Granet, N. A. Gippius, S. G. Tikhodeev, and H. Giessen, "Matched coordinates and adaptive spatial resolution in the Fourier modal method," Opt. Express 17, 8051-8061 (2009). [CrossRef] [PubMed]

Phys. Rep.

• K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, and M. Wegener, "Periodic nanostructures for photonics," Phys. Rep. 444, 101-202 (2007). [CrossRef]

Phys. Rev. B

• A. Vial, A.-S. Grimault, D. Macías, D. Barchiesi, and M. Lamy de la Chapelle, "Improved analytical fit of gold dispersion: Application to the modeling of extinction spectra with a finite-difference time-domain method," Phys. Rev. B 71, 085416 (2005). [CrossRef]
• G. J. Pearce, T. D. Hedley, and D. M. Bird, "Adaptive curvilinear coordinates in a plane-wave solution of Maxwell’s equations in photonic crystals," Phys. Rev. B 71, 195108 (2005). [CrossRef]
• F. Gygi, "Electronic-structure calculations in adaptive coordinates," Phys. Rev. B 48, 11692-11700 (1993). [CrossRef]

Rev. Mod. Phys.

• F. J. Garcia de Abajo, "Light scattering by particle and hole arrays," Rev. Mod. Phys. 79, 1267-1290 (2007). [CrossRef]

SIAM Rev.

• A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, "Cloaking devices, electromagnetic wormholes, and transformation optics," SIAM Rev. 51, 3-33 (2009). [CrossRef]

Other

• http://www.gnu.org.
• M. Nevière, and E. Popov, Light propagation in periodic media: Differential theory and design, Marcel Dekker (New York), 2003.

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