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

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  1. H. Liu and P. Lalanne, “Microscopic theory of the extraordinary optical transmission,” Nature 452, 728-731(2008).
    [CrossRef] [PubMed]
  2. 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]
  3. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824-830 (2003).
    [CrossRef] [PubMed]
  4. 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]
  5. 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]
  6. 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]
  7. 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]
  8. 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]
  9. 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]
  10. J. P. Berenger, “Three-dimensional perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 127, 363-379 (1996).
    [CrossRef]
  11. 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.
  12. 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).
  13. 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).
  14. M. G. Cox, “The numerical evaluation of B-splines*,” IMA J. Appl. Math. 10, 134-149 (1972).
    [CrossRef]
  15. C. De Boor, “On calculating with B-splines,” J. Approx. Theory 6, 50-62 (1972).
    [CrossRef]
  16. 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]
  17. E. D. Palik and G. Ghosh, Handbook of Optical Constants of Solids, (Academic, 1985).
  18. 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]
  19. D. C. Skigin and R. A. Depine, “Transmission resonances of metallic compound gratings with subwavelength slits,” Phys. Rev. Lett. 95, 217402 (2005).
    [CrossRef] [PubMed]
  20. We solve the full eigenequation with the function eig of LAPACK for all the previous results.
  21. P. Bouchon, F. Pardo, R. Haïdar, and J. L. Pelouard, are preparing a manuscript to be called “Reduced scattering-matrix algorithm.”
  22. 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).

2009 (1)

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]

2008 (2)

2007 (1)

2005 (1)

D. C. Skigin and R. A. Depine, “Transmission resonances of metallic compound gratings with subwavelength slits,” Phys. Rev. Lett. 95, 217402 (2005).
[CrossRef] [PubMed]

2003 (1)

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824-830 (2003).
[CrossRef] [PubMed]

2000 (1)

1996 (2)

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]

J. P. Berenger, “Three-dimensional perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 127, 363-379 (1996).
[CrossRef]

1993 (1)

1990 (1)

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]

1988 (1)

1972 (2)

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]

1966 (1)

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]

1946 (2)

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).

Atkinson, R.

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]

Barnes, W. L.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824-830 (2003).
[CrossRef] [PubMed]

Berenger, J. P.

J. P. Berenger, “Three-dimensional perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 127, 363-379 (1996).
[CrossRef]

Bouchon, P.

P. Bouchon, F. Pardo, R. Haïdar, and J. L. Pelouard, are preparing a manuscript to be called “Reduced scattering-matrix algorithm.”

Cambril, E.

Collin, S.

Cox, M. G.

M. G. Cox, “The numerical evaluation of B-splines*,” IMA J. Appl. Math. 10, 134-149 (1972).
[CrossRef]

De Boor, C.

C. De Boor, “On calculating with B-splines,” J. Approx. Theory 6, 50-62 (1972).
[CrossRef]

Depine, R. A.

D. C. Skigin and R. A. Depine, “Transmission resonances of metallic compound gratings with subwavelength slits,” Phys. Rev. Lett. 95, 217402 (2005).
[CrossRef] [PubMed]

Dereux, A.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824-830 (2003).
[CrossRef] [PubMed]

Ebbesen, T. W.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824-830 (2003).
[CrossRef] [PubMed]

Evans, P.

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]

Ghosh, G.

E. D. Palik and G. Ghosh, Handbook of Optical Constants of Solids, (Academic, 1985).

Gottesman, Y.

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.

Guérineau, N.

Haggans, C. W.

Haidar, R.

Haïdar, R.

P. Bouchon, F. Pardo, R. Haïdar, and J. L. Pelouard, are preparing a manuscript to be called “Reduced scattering-matrix algorithm.”

Hendren, W.

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]

Hugonin, J. P.

Kabashin, A. V.

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]

Ko, D. Y. K.

Lalanne, P.

Lehoucq, R. B.

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).

Li, L.

Liu, H.

H. Liu and P. Lalanne, “Microscopic theory of the extraordinary optical transmission,” Nature 452, 728-731(2008).
[CrossRef] [PubMed]

Morris, G. M.

Palik, E. D.

E. D. Palik and G. Ghosh, Handbook of Optical Constants of Solids, (Academic, 1985).

Pardo, F.

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.

P. Bouchon, F. Pardo, R. Haïdar, and J. L. Pelouard, are preparing a manuscript to be called “Reduced scattering-matrix algorithm.”

Pastkovsky, S.

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]

Pelouard, J. L.

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]

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.

P. Bouchon, F. Pardo, R. Haïdar, and J. L. Pelouard, are preparing a manuscript to be called “Reduced scattering-matrix algorithm.”

Podolskiy, V. A.

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]

Pollard, R.

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]

Primot, J.

Sambles, J. R.

Schoenberg, I. J.

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).

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).

Skigin, D. C.

D. C. Skigin and R. A. Depine, “Transmission resonances of metallic compound gratings with subwavelength slits,” Phys. Rev. Lett. 95, 217402 (2005).
[CrossRef] [PubMed]

Sorensen, D. C.

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).

Taflove, A.

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]

Umashankar, K. R.

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]

Urbach, H. P.

Vincent, G.

Wachters, A. J.

Wei, X.

Wurtz, G. A.

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]

Yang, C.

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).

Yee, K.

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]

Zayats, A. V.

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]

Electromagnetics (1)

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]

IEEE Trans. Antennas Propag. (1)

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]

IMA J. Appl. Math. (1)

M. G. Cox, “The numerical evaluation of B-splines*,” IMA J. Appl. Math. 10, 134-149 (1972).
[CrossRef]

J. Approx. Theory (1)

C. De Boor, “On calculating with B-splines,” J. Approx. Theory 6, 50-62 (1972).
[CrossRef]

J. Comput. Phys. (1)

J. P. Berenger, “Three-dimensional perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 127, 363-379 (1996).
[CrossRef]

J. Opt. Soc. Am. A (5)

J. Opt. Soc. Am. B (1)

Nature (2)

H. Liu and P. Lalanne, “Microscopic theory of the extraordinary optical transmission,” Nature 452, 728-731(2008).
[CrossRef] [PubMed]

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824-830 (2003).
[CrossRef] [PubMed]

Nature Mater. (1)

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]

Phys. Rev. Lett. (1)

D. C. Skigin and R. A. Depine, “Transmission resonances of metallic compound gratings with subwavelength slits,” Phys. Rev. Lett. 95, 217402 (2005).
[CrossRef] [PubMed]

Q. Appl. Math. (2)

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).

Other (5)

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.

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).

E. D. Palik and G. Ghosh, Handbook of Optical Constants of Solids, (Academic, 1985).

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Figures (8)

Fig. 1
Fig. 1

Grating system: Region B is a metallic grating. Regions A and C are considered to be semi-infinite and are homogeneous media.

Fig. 2
Fig. 2

Uniform B-splines of degree 0 to 4 beginning on the same knot. The mesh of knots t 1 t 6 is uniform but could be defined as non-uniform.

Fig. 3
Fig. 3

B-splines of degree 2 defined on a non-uniform mesh with a 2-degenerated knot.

Fig. 4
Fig. 4

Choice of B-splines knots and interpolation points in homogeneous regions and near an interface. We define a 5-degenerated knot on each interface so there are no interpolation points on it. The four interpolation points are set uniformly on each side of each interface and its neighboring knots. There is no interpolation point set on an interface.

Fig. 5
Fig. 5

Convergence of first mode in a thick silver grating illuminated with a TM incident wave at a wavelength λ = 1 μ m with 30 ° incidence as a function of the number of knots in a period for various B-spline degrees. Grating period is 1 μ m and the fill factor is 50%. Silver refaction index is n = 0.22 + 6.71 i .

Fig. 6
Fig. 6

Specular reflection by a 1 μ m -thick silver grating illuminated with a TM incident wave at a wavelength λ = 1 μ m with 30 ° incidence as a function of the number of knots in a period for various B-spline degrees. Grating period is 1 μ m and the fill factor is 50%.

Fig. 7
Fig. 7

Zeroth-order transmittance of the 3-slit compound metallic grating for a TM-polarized, normally incident wave. The metallic grating is schematized in the inset for only one period with d = 1 μ m , a = c = 0.08 μ m and h = 1.14 μ m . The refraction index of the metal is n = 0.15 + 24.9 i .

Fig. 8
Fig. 8

Density of the inverted matrix B 1 after the procedure of cleaning (crosses) is plotted as a function of the roundoff criterion. Matrix B is computed to solve the eigenproblem of the silver grating previously described with a refined mesh of 1000 knots. To evaluate the effect of the cleaning procedure, relative error on the first mode in the silver grating after roundoff is also plotted (circles) as a function of the roundoff criterion.

Equations (29)

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t B + × E = 0 ,
t D + × H = j ( = 0 ) .
i ω B y x E z + i k z E x = 0 ,
D x = k z ω H y ,
D z = i ω x H y .
B = μ 0 μ H ,
D = ε 0 ε E .
μ y H y + 1 k 0 2 x ( 1 ε z x H y ) = k z N 2 H y ε x ,
k z N = k z k 0 .
ε y E y + 1 k 0 2 x ( 1 μ z x E y ) = k z N 2 E y μ x .
N i 0 ( t ) = { 1 if t [ t j     t j + 1 ] 0 otherwise ,
N i p ( t ) = t t j t j + n t j N i p 1 ( t ) + t j + n + 1 t t j + n + 1 t j + 1 N i + 1 p 1 ( t ) .
H y ( x ) = N i p ( x ) P i ,
i N i ( x int + ) P i ε z ( x int + ) i N i ( x int ) P i ε z ( x int ) = 0.
M cont P = 0.
P = Γ Q .
μ y i N i P i + 1 k 0 2 1 ε z i x 2 N i P i + 1 k 0 2 ( x 1 ε z ) i x N i P i = k z N 2 ε x i N i P i .
( μ y N + 1 k 0 2 1 ε z x 2 N + 1 k 0 2 x 1 ε z x N ) P = k z N 2 1 ε x N P .
( μ y N + 1 k 0 2 1 ε z x 2 N ) P = k z N 2 1 ε x N P .
( μ y N + 1 k 0 2 1 ε z x 2 N ) Γ Q = k z N 2 1 ε x M Γ Q .
H A + H A S A A = H B S B A ,
E A + E A S A A = E B S B A .
S A A = H A 1 ( Z A + Z B ) 1 ( Z A Z B ) H A ,
S B A = H B 1 ( Z A + Z B ) 1 2 Z A H A ,
Z i = E i H i 1 .
S A A grating = S A A α + S A B α ( 1 P S B B β P S B B α ) 1 P S B B β P S B A α ,
S C A grating = S C B β P ( 1 S B B α P S B B β P ) 1 S B A α .
P ( R 1 B ) Q = L U ,
B 1 = Q U 1 L 1 P R 1 .

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