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).
[CrossRef]

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).
[CrossRef]

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).
[CrossRef]

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).
[CrossRef]

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).
[CrossRef]

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).
[CrossRef]

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).
[CrossRef]

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).
[CrossRef]

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).
[CrossRef]

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).
[CrossRef]

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).
[CrossRef]

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).
[CrossRef]

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).
[CrossRef]

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

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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Figures (5)

Fig. 1
Fig. 1

B-splines �� i n of degree n = 2 defined over a knot sequence ti with a degenerate knot at x = t4 = t5. The blue B-splines are continuously differentiable everywhere, whereas the green B-splines, �� 2 2 and �� 4 2, have discontinuous derivatives at the degenerate knot. The red B-spline �� 3 2 is continuous but not differentiable at the (m = n)-fold knot (left picture) and if an additional knot 6 is inserted (right panel) splits into two discontinuous B-splines, �� ^ 3 2 and �� ^ 4 2, featuring a jump at the additionally inserted knot.

Fig. 2
Fig. 2

(a) A periodic three-layer system invariant in y-direction with a unit cell size in x-direction of a = 10μm. The unit cell of the second layer is divided into two regions, an air part with ε = 1 of 9μm width and a waveguide part with ε = 5 of 1μm width. Layers (1) and (3) denote homogeneous half spaces filled with air (ε = 1). (b) A knot sequence as used in the calculation. In general, we place n-fold knots at every material interface and distribute the remaining knots so that each region contains the same number of knots. For δ = 1 the knots are spaced equidistantly in each region. The equidistant spacing is called Δxi. For δ ≠ 1, the distance of the first knot next to the interface is divided by δ as compared to an equidistant spacing. The remaining knots are again spaced equidistantly but, of course, with a slightly larger spacing (slightly larger than Δxi).

Fig. 3
Fig. 3

Convergence plot of the errors of the propagation constant λ and the mode profiles. The convergence behaviour is fitted as Nr which is shown as a straight line in the double logarithmic plot. The filled markers refer to a B-spline knot sequence with δ = 1 whereas the open markers refer to δ = 10; see the text and Fig. 2(b). (a) Convergence rates: rBMM4 = 7.1, rBMM7 = 9.7, rBMM10 = 11.9, rFMM = 3.2. (b) Convergence rates: rBMM4 = 8.7, rBMM7 = 10.9, rBMM10 = 12.9, rFMM = 3.2.

Fig. 4
Fig. 4

Convergence plot of the reflectance calculation for the FMM and the BMM using B-splines of degree n = 10 with different knot sequences identified by the values of δ. (a) The black horizontal line at R = 0.04228344 is the fitted limiting value of the FMM calculation which we assume to converge to the correct value. (b) The relative error is calculated against the limit value of the FMM. Convergence rates: rBMM1 = 0.9, rBMM10 = 0.9, rFMM = 2.5.

Fig. 5
Fig. 5

(a) Comparison of the error of the propagation constants using an FMM and a BMM calculation using B-splines of degree n = 7. (b) An illustration of a non-Cartesian knot mesh with multiple knot lines (bold lines) directly at the material interfaces.

Equations (30)

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f ( x ) = i = 1 N c i 𝒩 i n ( x ) = i = 1 N + 1 c ^ i 𝒩 ^ i n ( x ) .
c ^ i = { c i for i k , c i 1 for i k + 1.
× E = i ω μ _ H , × H = i ω ε _ E .
[ μ y ( x ) + x ω 1 ε z ( x ) + x ω ] = : 𝒜 H y ( x ) = λ 2 ω 2 [ 1 ε x ( x ) ] = : H y ( x ) .
E x = λ ω ε x H y , and E z = i ω ε z x H y .
A c = λ 2 ω 2 B c ,
A j i = 𝒩 j n * μ y 𝒩 i n d x [ x ω 𝒩 j n * ] 1 ε z [ x ω 𝒩 i n ] d x , and B j i = 𝒩 j n * 1 ε x 𝒩 i n d x .
H y , m ( p ) ( x ) = i = 1 N H im ( p ) 𝒩 i n ( x ) .
error = 1 2 d x | H y ( x ) H y ( x ) | 2 = 1 Re d x H y * ( x ) H y ( x ) .
E x , m ( p ) ( x ) = i = 1 N ^ E im ( p ) 𝒩 ^ i n ( x ) .
E x , m ( p ) ( x ) ( 5 ) λ m ( p ) ε x ( p ) ( x ) H y , m ( p ) ( x ) = ( 8 ) i = 1 N λ m ( p ) ε x ( p ) ( x ) H im ( p ) 𝒩 i ( x ) .
F ( p ) ( x , z ) : = ( H y ( p ) ( x , z ) E x ( p ) ( x , z ) ) = ( 𝒩 1 ( x ) 𝒩 N ( x ) 0 0 𝒩 ^ 1 ( x ) 𝒩 ^ N ^ ( x ) ) = : N ( x ) ( H ( p ) H ( p ) E ( p ) E ( p ) ) = : W ( p ) ( u ( p ) ( z ) d ( p ) ( z ) ) = N ( x ) W ( p ) ( u ( p ) ( z ) d ( p ) ( z ) ) .
F ( p + 1 ) ( x , z p + 0 ) = ! F ( p ) ( x , z p 0 ) W ( p + 1 ) ( u ( p + 1 ) ( z p + 0 ) d ( p + 1 ) ( z p + 0 ) ) = ! W ( p ) ( u ( p ) ( z p 0 ) d ( p ) ( z p 0 ) )
t ( p ) = [ W ( p + 1 ) ] 1 W ( p ) .
S = ( A x b ) w ( A x b ) ,
[ A ] ls 1 : = ( A wA ) 1 A w .
t ( p ) = [ W ( p + 1 ) ] ls 1 W ( p ) .
w = N ( x ) N ( x ) d x .
S = | F ( p + 1 ) ( x ) F ( p ) ( x ) | 2 d x = | N ( x ) [ W ( p + 1 ) ( u ( p + 1 ) d ( p + 1 ) ) W ( p ) ( u ( p ) d ( p ) ) ] | 2 d x = ( W ( p + 1 ) A ( u ( p + 1 ) d ( p + 1 ) ) x W ( p ) ( u ( p ) d ( p ) ) b ) N ( x ) N ( x ) d x w ( W ( p + 1 ) A ( u ( p + 1 ) d ( p + 1 ) ) x W ( p ) ( u ( p ) d ( p ) ) b ) = ( A x b ) w ( A x b ) = S .
ε _ = ( ε x x ε x y 0 ε y x ε y y 0 0 0 ε z z ) and μ _ = ( μ x x μ x y 0 μ y x μ y y 0 0 0 μ z z ) .
( 0 0 x ω 1 ε z z y ω + μ y x + x ω 1 ε z z x ω + μ y y 0 0 y ω 1 ε z z y ω μ x x + y ω + 1 ε z z x ω μ x y + x ω 1 μ z z y ω ε y x x ω 1 μ z z x ω ε y y 0 0 + y ω 1 μ z z y ω + ε x x y ω 1 μ z z x ω + ε x y 0 0 ) = : ( 0 𝒢 0 ) ( E x E y H x H y ) = λ ω ( E x E y H x H y ) .
𝒢 ( E x E y ) = λ 2 ω 2 ( E x E y ) ,
( 𝒢 ) 11 = x ω 1 ε z z [ x ω ε x x + y ω ε y x ] + μ y x [ x ω + 1 μ z z y ω ε y x ] + μ y y [ y ω 1 μ z z y ω + ε x x ] ,
( 𝒢 ) 12 = x ω 1 ε z z [ x ω ε x y + y ω ε y y ] μ y x [ x ω 1 μ z z x ω + ε y y ] + μ y y [ y ω 1 μ z z x ω + ε x y ] ,
( 𝒢 ) 21 = y ω 1 ε z z [ x ω ε x x + y ω ε y x ] + μ x x [ x ω 1 μ z z y ω + ε y x ] μ x y [ y ω 1 μ z z y ω + ε x x ] ,
( 𝒢 ) 22 = y ω 1 ε z z [ x ω ε x y + y ω ε y y ] + μ x x [ x ω 1 μ z z x ω + ε y y ] + μ x y [ y ω 1 μ z z x ω ε x y ] .
𝒩 i , j n ( x , y ) : = 𝒩 i n ( x ) 𝒩 j n ( y ) .
1 ε i ε = i ln ( | ε | ) + ( 1 ln ( | ε | ) ) i ,
( 𝒢 ) 11 = μ ε + x 2 ω 2 ln ( | ε | ) + x ω ( 1 ln ( | ε | ) ) x ω + ln ( | μ | ) y 2 ω 2 + y ω ( 1 ln ( | μ | ) ) y ω .
ε a b ( x , y ) = i j Θ ( x x i ) Θ ( y y j ) ε ˜ a b i j ( x , y ) , μ a b analoguous .

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