Abstract

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.

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2012 (1)

2010 (3)

2009 (1)

2007 (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]

2005 (1)

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).

1997 (1)

1996 (3)

1995 (1)

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]

1994 (1)

1972 (2)

C. de Boor, “On calculating with B-splines,” J. Approx. Theory6, 50–62 (1972).
[CrossRef]

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

1955 (1)

R. Penrose, “A generalized inverse for matrices,” Math. Proc. Cambridge51, 406–413 (1955).
[CrossRef]

1920 (1)

E. H. Moore, “On the reciprocal of the general algebraic matrix,” Bull. Amer. Math. Soc.26, 394–395 (1920).

Anderson, E.

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).

Bai, Z.

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).

Bischof, C.

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).

Blackford, S.

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).

Bouchon, P.

Buffa, A.

A. Buffa, G. Sangalli, and R. Vazquez, “Isogeometric analysis in electromagnetics: B-splines approximation,” Comput. Method. Appl. M.199, 1143–1152 (2010).
[CrossRef]

Busch, K.

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. Theory6, 50–62 (1972).
[CrossRef]

C. de Boor, A Practical Guide to Splines(Springer, 2001).

Demmel, J.

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).

Dongarra, J.

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).

Du Croz, J.

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).

Edee, K.

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).

Essig, S.

Giessen, H.

Gippius, N. A.

Granet, G.

Greenbaum, A.

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).

Guizal, B.

Haïdar, R.

Hammarling, S.

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).

Kingsland, D.

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]

Küchenmeister, J.

Lalanne, P.

Lee, J.-F.

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]

Lee, R.

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]

Lehoucq, R. B.

R. B. Lehoucq, ARPACK users’ guide: solution of large-scale eigenvalue problems with implicitly restarted Arnoldi methods(SIAM, Philadelphia, 1998).

Li, L.

Linden, S.

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]

Love, J. D.

A. W. Snyder and J. D. Love, Optical Waveguide Theory(Chapman and Hall, 1983).

McKenney, A.

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).

Mingaleev, S. F.

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]

Moore, E. H.

E. H. Moore, “On the reciprocal of the general algebraic matrix,” Bull. Amer. Math. Soc.26, 394–395 (1920).

Morris, G. M.

Noponen, E.

Pardo, F.

Pelouard, J.-L.

Penrose, R.

R. Penrose, “A generalized inverse for matrices,” Math. Proc. Cambridge51, 406–413 (1955).
[CrossRef]

Sacks, Z.

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]

Sangalli, G.

A. Buffa, G. Sangalli, and R. Vazquez, “Isogeometric analysis in electromagnetics: B-splines approximation,” Comput. Method. Appl. M.199, 1143–1152 (2010).
[CrossRef]

Schiavone, P.

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).

Snyder, A. W.

A. W. Snyder and J. D. Love, Optical Waveguide Theory(Chapman and Hall, 1983).

Sorensen, D.

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).

Tikhodeev, S. G.

Tkeshelashvili, L.

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]

Turunen, J.

Vazquez, R.

A. Buffa, G. Sangalli, and R. Vazquez, “Isogeometric analysis in electromagnetics: B-splines approximation,” Comput. Method. Appl. M.199, 1143–1152 (2010).
[CrossRef]

von Freymann, G.

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]

Wegener, M.

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]

Weiss, T.

Zebrowski, T.

Bull. Amer. Math. Soc. (1)

E. H. Moore, “On the reciprocal of the general algebraic matrix,” Bull. Amer. Math. Soc.26, 394–395 (1920).

Comput. Method. Appl. M. (1)

A. Buffa, G. Sangalli, and R. Vazquez, “Isogeometric analysis in electromagnetics: B-splines approximation,” Comput. Method. Appl. M.199, 1143–1152 (2010).
[CrossRef]

IEEE T. Antenn. Propag. (1)

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]

IMA J. Appl. Math. (1)

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

J. Appl. Phys. (1)

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).

J. Approx. Theory (1)

C. de Boor, “On calculating with B-splines,” J. Approx. Theory6, 50–62 (1972).
[CrossRef]

J. Opt. Soc. Am. A (6)

Math. Proc. Cambridge (1)

R. Penrose, “A generalized inverse for matrices,” Math. Proc. Cambridge51, 406–413 (1955).
[CrossRef]

Opt. Express (3)

Phys. Rep. (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]

Other (4)

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).

R. B. Lehoucq, ARPACK users’ guide: solution of large-scale eigenvalue problems with implicitly restarted Arnoldi methods(SIAM, Philadelphia, 1998).

A. W. Snyder and J. D. Love, Optical Waveguide Theory(Chapman and Hall, 1983).

C. de Boor, A Practical Guide to Splines(Springer, 2001).

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