OSA's Digital Library

Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 12 — Jun. 17, 2013
  • pp: 14683–14697

B-spline modal method: A polynomial approach compared to the Fourier modal method

Michael Walz, Thomas Zebrowski, Jens Küchenmeister, and Kurt Busch  »View Author Affiliations

Optics Express, Vol. 21, Issue 12, pp. 14683-14697 (2013)

View Full Text Article

Enhanced HTML    Acrobat PDF (2173 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



A detailed analysis of the B-spline Modal Method (BMM) for one- and two-dimensional diffraction gratings and a comparison to the Fourier Modal Method (FMM) is presented. Owing to its intrinsic capability to accurately resolve discontinuities, BMM avoids the notorious problems of FMM that are associated with the Gibbs phenomenon. As a result, BMM facilitates significantly more efficient eigenmode computations. With regard to BMM-based transmission and reflection computations, it is demonstrated that a novel Galerkin approach (in conjunction with a scattering-matrix algorithm) allows for an improved field matching between different layers. This approach is superior relative to the traditional point-wise field matching. Moreover, only this novel Galerkin approach allows for an competitive extension of BMM to the case of two-dimensional diffraction gratings. These improvements will be very useful for high-accuracy grating computations in general and for the analysis of associated electromagnetic field profiles in particular.

© 2013 osa

OCIS Codes
(050.1970) Diffraction and gratings : Diffractive optics
(050.1755) Diffraction and gratings : Computational electromagnetic methods
(160.5298) Materials : Photonic crystals

ToC Category:
Diffraction and Gratings

Original Manuscript: March 25, 2013
Manuscript Accepted: May 15, 2013
Published: June 13, 2013

Michael Walz, Thomas Zebrowski, Jens Küchenmeister, and Kurt Busch, "B-spline modal method: A polynomial approach compared to the Fourier modal method," Opt. Express 21, 14683-14697 (2013)

Sort:  Author  |  Year  |  Journal  |  Reset  


  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. L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A14, 2758–2767 (1997). [CrossRef]
  3. L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A13, 1024–1035 (1996). [CrossRef]
  4. K. Edee, P. Schiavone, and G. Granet, “Analysis of defect in extreme UV lithography mask using a modal method based on nodal B-spline expansion,” Jpn. J. Appl. Phys.44, 6458–6462 (2005).
  5. P. Bouchon, F. Pardo, R. Haïdar, and J.-L. Pelouard, “Fast modal method for subwavelength gratings based on B-spline formulation,” J. Opt. Soc. Am. A27, 696–702 (2010). [CrossRef]
  6. A. Buffa, G. Sangalli, and R. Vazquez, “Isogeometric analysis in electromagnetics: B-splines approximation,” Comput. Method. Appl. M.199, 1143–1152 (2010). [CrossRef]
  7. C. de Boor, “On calculating with B-splines,” J. Approx. Theory6, 50–62 (1972). [CrossRef]
  8. M. G. Cox, “The numerical evaluation of B-Splines,” IMA J. Appl. Math.10, 134–149 (1972). [CrossRef]
  9. C. de Boor, A Practical Guide to Splines(Springer, 2001).
  10. P. Lalanne and G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A13, 779–784 (1996). [CrossRef]
  11. G. Granet and B. Guizal, “Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization,” J. Opt. Soc. Am. A13, 1019–1023 (1996). [CrossRef]
  12. Z. Sacks, D. Kingsland, R. Lee, and J.-F. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE T. Antenn. Propag.43, 1460–1463 (1995). [CrossRef]
  13. 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. Express17, 8051–8061 (2009). [CrossRef] [PubMed]
  14. S. Essig and K. Busch, “Generation of adaptive coordinates and their use in the Fourier Modal Method,” Opt. Express18, 23258 (2010). [CrossRef] [PubMed]
  15. J. Küchenmeister, T. Zebrowski, and K. Busch, “A construction guide to analytically generated meshes for the Fourier Modal Method,” Opt. Express20, 17319–17347 (2012). [CrossRef] [PubMed]
  16. E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users’ Guide(SIAM, Philadelphia, 1999). [CrossRef]
  17. R. B. Lehoucq, ARPACK users’ guide: solution of large-scale eigenvalue problems with implicitly restarted Arnoldi methods(SIAM, Philadelphia, 1998). [CrossRef]
  18. A. W. Snyder and J. D. Love, Optical Waveguide Theory(Chapman and Hall, 1983).
  19. E. H. Moore, “On the reciprocal of the general algebraic matrix,” Bull. Amer. Math. Soc.26, 394–395 (1920).
  20. R. Penrose, “A generalized inverse for matrices,” Math. Proc. Cambridge51, 406–413 (1955). [CrossRef]
  21. E. Noponen and J. Turunen, “Eigenmode method for electromagnetic synthesis of diffractive elements with three-dimensional profiles,” J. Opt. Soc. Am. A11, 2494–2502 (1994). [CrossRef]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.


Fig. 1 Fig. 2 Fig. 3
Fig. 4 Fig. 5

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited