Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Fast Fourier factorization for differential method and RCWA: a powerful tool for the modeling of non-lamellar metallic diffraction gratings

  • 38 Accesses


The rigorous coupled-wave analysis (RCWA), also known as Fourier modal method, is one of the most popular methods for the modeling of diffraction gratings. It has been proven to be particularly effective for lamellar gratings. However, for non-lamellar metallic gratings, in TM polarization, the differential method (DM) or the RCWA need to be associated with the fast Fourier factorization (FFF) to provide more accurate solutions. In this study, we present the effectiveness of the FFF when it is associated with the DM and the RCWA for different grating’s profiles, and we show how to circumvent on the case where the FFF can diverge when the metal permittivity is close to a pure negative real value.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10


  1. Arnaud, L.: Diffraction et diffusion de la lumière: modélisation tridimensionnelle et application à la métrologie de la microélectronique et aux techniques d’imagerie sélective en milieu diffusant, PhD Dissertation, Aix-Marseille 3 (2008)

  2. Chandezon, M., Raoult, G., Maystre, D.: Reformulation of the eigenvalue a new theoretical method for diffraction gratings and its numerical application. J. Opt. 11, 235–241 (1980)

  3. Chiu, N.F., Nien, S.-Y., Yu, C., Lee, J.-H., Lin, C.-W.: Advanced metal nanostructure design for surface plasmon photonic bandgap biosensor device. In: 2006 International Conference of the IEEE Engineering in Medicine and Biology Society, pp. 6521–6524 (2006)

  4. Gonzalez-Alcalde, A., Salas-Montiel, R., Mohamad, H., Morand, A., Blaize, S., Macias, D.: Optimization ofall-dielectric structures for color generation. Appl. Opt. 57, 3959–3967 (2018)

  5. Goodno, G.D., Dadusc, G., Miller, R.J.D.: Ultrafast heterodyne-detected transient-grating spectroscopy using diffractive optics. JOSA B 15, 1791–1794 (1998)

  6. Gu, Y., Zhang, L., Yang, J.K.W., Yeo, S.P., Qiu, C.-W.: Color generation via subwavelength plasmonicnanostructures. Nanoscale 7, 6409–6419 (2015)

  7. Guizal, B., Yala, H., Felbacq, D.: Reformulation of the eigenvalue problem in the Fourier modal method with spatial adaptive resolution. Opt. Lett. 34, 2790–2792 (2009)

  8. Iqbal, T., Afsheen, S.: One dimensional plasmonic grating: high sensitive biosensor. Plasmonics 12, 19–25 (2017)

  9. Li, L.: Use of Fourier series in the analysis of discontinuous periodic structures. JOSA A. 13, 1870–1876 (1996)

  10. Li, L.: Field singularities at lossless metal-dielectric arbitrary-angle edges and their ramifications to the numerical modeling of gratings. JOSA A. 29, 593–604 (2012)

  11. Li, L., Granet, G.M.: Field singularities at lossless metal-dielectric right-angle edges and their ramifications to the numerical modeling of gratings. JOSA A. 28, 738–746 (2011)

  12. Lv, C., Jia, Z., Liu, Y., Mo, J., Li, P., Lv, X.: Angle-resolved diffraction grating biosensor based on porous silicon. J. Appl. Phys. 119(9), 094502 (2016)

  13. Lyndin, McGrating software,

  14. Lyndin, N.M., Parriaux, O., Tishchenko, A.V.: Modal analysis and suppression of the Fourier modal method instabilities in highly conductive gratings. JOSA A. 24, 3781–3788 (2007)

  15. Mei, Y., Liu, H., Zhong, Y.: Treatment of nonconvergence of Fourier modal method arising from irregular field singularities at lossless metal-dielectric right-angle edges. JOSA A. 31, 900–906 (2014)

  16. Neviere, M.: Sur la theorie du reseau conducteur et ses applications a l’optique. Nouv. Rev. Optique. 5, 65–77 (1974)

  17. Pai, D.M., Awada, K.A.: Analysis of dielectric gratings of arbitrary profiles and thicknesses. JOSA A. 8, 755–762 (1991)

  18. Popov, E., Neviere, M.: Grating, theory: new equations in Fourier space leading to fast converging results for TM polarization. JOSA A. 17, 1773–1784 (2000)

  19. Popov, E., Chernov, B., Nevière, M., Bonod, N.: Differential theory: application to highly conducting gratings. JOSA A. 21, 199–206 (2004)

  20. Sauvage-Vincent, J., Tonchev, S., Veillas, C., Reynaud, S., Jourlin, Y.: Optical security device for document protection using plasmon resonant transmission through a thin corrugated metallic film embedded in a plastic foil. J. Eur. Opt. Soc. Rapid Publ. 8 (2013)

  21. Tvingstedt, K., Persson, N-K., Inganäs, O., Rahachou, A., Zozoulenko, I.V.: Surface plasmon increase absorption in polymer photovoltaic cells. Appl. Phys. Lett. 91(11), 113514 (2007)

  22. Vallius, T., Honkanen, M.: Reformulation of the Fourier modal method with adaptive spatial resolution: application to multilevel profiles. Opt. Express 10, 24–34 (2002)

  23. Vincent, P.: New improvement of the differential formalism for high-modulated gratings. Int. Soc. Opt. Photonics 240, 147–154 (1981)

  24. Watanabe, K.: Study of the differential theory of lamellar gratings made of highly conducting materials. JOSA A. 23, 69–72 (2006)

  25. Zhu, J., Liu, H., Zhong, Y.: Treatment of nonconvergence of the Fourier modal method and C method arising from field hypersingularities at lossless metal-dielectric arbitrary-angle edges. JOSA A. 33, 845–853 (2016)

Download references


This work was supported by the ODISSEA project funded by the French National Agency of Research ‘ANR’ under Grant No 16-CE39-0016-01.

Author information

Correspondence to Habib Mohamad.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This article is part of the Topical Collection on Optical Wave and Waveguide Theory and Numerical Modelling.

Guest edited by Alejandro Ortega Moñux, Rafael Godoy Rubio and Jiri Ctyroky.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Mohamad, H., Essaidi, S., Blaize, S. et al. Fast Fourier factorization for differential method and RCWA: a powerful tool for the modeling of non-lamellar metallic diffraction gratings. Opt Quant Electron 52, 127 (2020).

Download citation


  • Computational electromagnetic
  • 1D diffraction grating
  • RCWA
  • Differential method
  • Fast Fourier factorization