## Fast modal method for subwavelength gratings based on B-spline formulation

JOSA A, Vol. 27, Issue 4, pp. 696-702 (2010)

http://dx.doi.org/10.1364/JOSAA.27.000696

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### Abstract

We present a B-spline modal method for analyzing a stack of complex structured layers. Thanks to a B-spline approximation of the field, we solve the Maxwell equations. Diffraction calculation is based on the scattering matrices algorithm. We prove a good convergence of this method. Moreover, B-spline approximation results in very sparse matrices, which are used to hasten the computation of eigenmodes. A method for cleaning the inverted sparse matrix is also presented.

© 2010 Optical Society of America

**OCIS Codes**

(050.1950) Diffraction and gratings : Diffraction gratings

(050.1755) Diffraction and gratings : Computational electromagnetic methods

(050.6624) Diffraction and gratings : Subwavelength structures

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: November 6, 2009

Revised Manuscript: January 26, 2010

Manuscript Accepted: January 28, 2010

Published: March 15, 2010

**Citation**

Patrick Bouchon, Fabrice Pardo, Riad Haïdar, and Jean-Luc Pelouard, "Fast modal method for subwavelength gratings based on B-spline formulation," J. Opt. Soc. Am. A **27**, 696-702 (2010)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-27-4-696

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### References

- H. Liu and P. Lalanne, “Microscopic theory of the extraordinary optical transmission,” Nature 452, 728-731(2008). [CrossRef] [PubMed]
- A. V. Kabashin, P. Evans, S. Pastkovsky, W. Hendren, G. A. Wurtz, R. Atkinson, R. Pollard, V. A. Podolskiy, and A. V. Zayats, “Plasmonic nanorod metamaterials for biosensing,” Nature Mater. 8, 867-871 (2009). [CrossRef]
- W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824-830 (2003). [CrossRef] [PubMed]
- G. Vincent, R. Haidar, S. Collin, N. Guérineau, J. Primot, E. Cambril, and J. L. Pelouard, “Realization of sinusoidal transmittance with subwavelength metallic structures,” J. Opt. Soc. Am. B 25, 834-840 (2008). [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]
- A. Taflove and K. R. Umashankar, “The finite-difference time-domain method for numerical modeling of electromagnetic wave interactions,” Electromagnetics 10, 105-126 (1990). [CrossRef]
- K. Yee, “Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302-307 (1966). [CrossRef]
- X. Wei, A. J. Wachters, and H. P. Urbach, “Finite-element model for three-dimensional optical scattering problems,” J. Opt. Soc. Am. A 24, 866-881 (2007). [CrossRef]
- P. Lalanne and J. P. Hugonin, “Numerical performance of finite-difference modal methods for the electromagnetic analysis of one-dimensional lamellar gratings,” J. Opt. Soc. Am. A 17, 1033-1042 (2000). [CrossRef]
- J. P. Berenger, “Three-dimensional perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 127, 363-379 (1996). [CrossRef]
- F. Pardo, Y. Gottesman, and J. L. Pelouard, are preparing a manuscript to be called “Exact formulation of Maxwell's equations for lamellar gratings analysis,” available from fabrice.pardo@lpn.cnrs.fr.
- I. J. Schoenberg, “Contributions to the problem of approximation of equidistant data by analytic functions. On the problem of smoothing or graduation. A first class of analytic approximation formulae,” Q. Appl. Math. 4, 45-99 (1946).
- I. J. Schoenberg, “Contributions to the problem of approximation of equidistant data by analytic functions. On the problem of osculatory interpolation. A second class of analytic approximation formulae,” Q. Appl. Math. 4, 112-141 (1946).
- M. G. Cox, “The numerical evaluation of B-splines*,” IMA J. Appl. Math. 10, 134-149 (1972). [CrossRef]
- C. De Boor, “On calculating with B-splines,” J. Approx. Theory 6, 50-62 (1972). [CrossRef]
- L. Li and C. W. Haggans, “Convergence of the coupled-wave method for metallic lamellar diffraction gratings,” J. Opt. Soc. Am. A 10, 1184-1189 (1993). [CrossRef]
- E. D. Palik and G. Ghosh, Handbook of Optical Constants of Solids, (Academic, 1985).
- D. Y. K. Ko and J. R. Sambles, “Scattering matrix method for propagation of radiation in stratified media: attenuated total reflection studies of liquid crystals,” J. Opt. Soc. Am. A 5, 1863-1866 (1988). [CrossRef]
- D. C. Skigin and R. A. Depine, “Transmission resonances of metallic compound gratings with subwavelength slits,” Phys. Rev. Lett. 95, 217402 (2005). [CrossRef] [PubMed]
- We solve the full eigenequation with the function eig of LAPACK for all the previous results.
- P. Bouchon, F. Pardo, R. Haïdar, and J. L. Pelouard, are preparing a manuscript to be called “Reduced scattering-matrix algorithm.”
- R. B. Lehoucq, D. C. Sorensen, and C. Yang, ARPACK Users' Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods (SIAM, 1998).

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