Abstract

A second-order finite-element adaptive strategy with error control for one-dimensional grating problems is developed. The unbounded computational domain is truncated to a bounded one by a perfectly-matched-layer (PML) technique. The PML parameters, such as the thickness of the layer and the medium properties, are determined through sharp a posteriori error estimates. The adaptive finite-element method is expected to increase significantly the accuracy and efficiency of the discretization as well as reduce the computation cost. Numerical experiments are included to illustrate the competitiveness of the proposed adaptive method.

© 2005 Optical Society of America

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  1. M. Nevière, G. Cerutti-Maori, M. Cadilhac, “Sur une nouvelle méthode de résolution du problème de la diffraction d’une onde plane par un réseau infiniment conducteur,” Opt. Commun. 3, 48–52 (1971).
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    [CrossRef]
  11. Z. Chen, H. Wu, “An adaptive finite element method with perfectly matched absorbing layers for the wave scattering by periodic structures,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 41, 799–826 (2003).
    [CrossRef]
  12. I. Babuška, C. Rheinboldt, “Error estimates for adaptive finite element computations,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 15, 736–754 (1978).
    [CrossRef]
  13. P. Morin, R. H. Nochetto, K. G. Siebert, “Data oscillation and convergence of adaptive FEM,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 38, 466–488 (2000).
    [CrossRef]
  14. Z. Chen, S. Dai, “On the efficiency of adaptive finite element methods for elliptic problems with discontinuous coefficients,” SIAM J. Sci. Comput. (USA) 24, 443–462 (2002).
    [CrossRef]
  15. R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh Refinement Techniques, (Teubner, Wiesbaden, Germany, 1996).
  16. P. Monk, “ A posteriori error indicators for Maxwell’s equations,” J. Comput. Appl. Math. 100, 173–190 (1998).
    [CrossRef]
  17. P. Monk, E. Süli, “The adaptive computation of far-field patterns by a posteriori error estimation of linear functionals,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 36, 251–274 (1998).
    [CrossRef]
  18. J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
    [CrossRef]
  19. E. Turkel, A. Yefet, “Absorbing PML boundary layers for wave-like equations,” Appl. Numer. Math. 27, 533–557 (1998).
    [CrossRef]
  20. G. Granet, “Reformulation of the lamellar grating problem through the concept of adaptive spatial resolution,” J. Opt. Soc. Am. A 16, 2510–2516 (1999).
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  23. E. Popov, M. Nevière, “Grating theory: new equations in Fourier space leading to fast converging results for TM polarization,” J. Opt. Soc. Am. A 17, 1773–1784 (2000).
    [CrossRef]
  24. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, P. A. Wolff, “Extraordinary optical transmission through subwavelength hole arrays,” Nature (London) 391, 667–669 (1998).
    [CrossRef]

2003 (1)

Z. Chen, H. Wu, “An adaptive finite element method with perfectly matched absorbing layers for the wave scattering by periodic structures,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 41, 799–826 (2003).
[CrossRef]

2002 (1)

Z. Chen, S. Dai, “On the efficiency of adaptive finite element methods for elliptic problems with discontinuous coefficients,” SIAM J. Sci. Comput. (USA) 24, 443–462 (2002).
[CrossRef]

2000 (3)

1999 (3)

1998 (4)

P. Monk, “ A posteriori error indicators for Maxwell’s equations,” J. Comput. Appl. Math. 100, 173–190 (1998).
[CrossRef]

P. Monk, E. Süli, “The adaptive computation of far-field patterns by a posteriori error estimation of linear functionals,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 36, 251–274 (1998).
[CrossRef]

E. Turkel, A. Yefet, “Absorbing PML boundary layers for wave-like equations,” Appl. Numer. Math. 27, 533–557 (1998).
[CrossRef]

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, P. A. Wolff, “Extraordinary optical transmission through subwavelength hole arrays,” Nature (London) 391, 667–669 (1998).
[CrossRef]

1997 (1)

1996 (1)

1995 (2)

1994 (1)

J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

1993 (1)

1982 (2)

1978 (1)

I. Babuška, C. Rheinboldt, “Error estimates for adaptive finite element computations,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 15, 736–754 (1978).
[CrossRef]

1971 (1)

M. Nevière, G. Cerutti-Maori, M. Cadilhac, “Sur une nouvelle méthode de résolution du problème de la diffraction d’une onde plane par un réseau infiniment conducteur,” Opt. Commun. 3, 48–52 (1971).
[CrossRef]

Babuška, I.

I. Babuška, C. Rheinboldt, “Error estimates for adaptive finite element computations,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 15, 736–754 (1978).
[CrossRef]

Bao, G.

Berenger, J.-P.

J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

Bozhkov, B.

Bruno, O. P.

Cadilhac, M.

M. Nevière, G. Cerutti-Maori, M. Cadilhac, “Sur une nouvelle méthode de résolution du problème de la diffraction d’une onde plane par un réseau infiniment conducteur,” Opt. Commun. 3, 48–52 (1971).
[CrossRef]

Cerutti-Maori, G.

M. Nevière, G. Cerutti-Maori, M. Cadilhac, “Sur une nouvelle méthode de résolution du problème de la diffraction d’une onde plane par un réseau infiniment conducteur,” Opt. Commun. 3, 48–52 (1971).
[CrossRef]

Chandezon, J.

Chen, Z.

Z. Chen, H. Wu, “An adaptive finite element method with perfectly matched absorbing layers for the wave scattering by periodic structures,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 41, 799–826 (2003).
[CrossRef]

Z. Chen, S. Dai, “On the efficiency of adaptive finite element methods for elliptic problems with discontinuous coefficients,” SIAM J. Sci. Comput. (USA) 24, 443–462 (2002).
[CrossRef]

Cornet, G.

Cox, J. A.

Dai, S.

Z. Chen, S. Dai, “On the efficiency of adaptive finite element methods for elliptic problems with discontinuous coefficients,” SIAM J. Sci. Comput. (USA) 24, 443–462 (2002).
[CrossRef]

Dobson, D. C.

Dupuis, M. T.

Ebbesen, T. W.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, P. A. Wolff, “Extraordinary optical transmission through subwavelength hole arrays,” Nature (London) 391, 667–669 (1998).
[CrossRef]

Gaylord, T. K.

Ghaemi, H. F.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, P. A. Wolff, “Extraordinary optical transmission through subwavelength hole arrays,” Nature (London) 391, 667–669 (1998).
[CrossRef]

Granet, G.

Hoose, J.

Hugonin, J.-P.

Lalanne, P.

Lezec, H. J.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, P. A. Wolff, “Extraordinary optical transmission through subwavelength hole arrays,” Nature (London) 391, 667–669 (1998).
[CrossRef]

Li, L.

Magnusson, R.

Mait, J. N.

Maystre, D.

Mirotznik, M. S.

Moharam, M. G.

Monk, P.

P. Monk, E. Süli, “The adaptive computation of far-field patterns by a posteriori error estimation of linear functionals,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 36, 251–274 (1998).
[CrossRef]

P. Monk, “ A posteriori error indicators for Maxwell’s equations,” J. Comput. Appl. Math. 100, 173–190 (1998).
[CrossRef]

Morin, P.

P. Morin, R. H. Nochetto, K. G. Siebert, “Data oscillation and convergence of adaptive FEM,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 38, 466–488 (2000).
[CrossRef]

Nevière, M.

E. Popov, M. Nevière, “Grating theory: new equations in Fourier space leading to fast converging results for TM polarization,” J. Opt. Soc. Am. A 17, 1773–1784 (2000).
[CrossRef]

M. Nevière, G. Cerutti-Maori, M. Cadilhac, “Sur une nouvelle méthode de résolution du problème de la diffraction d’une onde plane par un réseau infiniment conducteur,” Opt. Commun. 3, 48–52 (1971).
[CrossRef]

Nochetto, R. H.

P. Morin, R. H. Nochetto, K. G. Siebert, “Data oscillation and convergence of adaptive FEM,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 38, 466–488 (2000).
[CrossRef]

Popov, E.

Prather, D. W.

Reitich, F.

Rheinboldt, C.

I. Babuška, C. Rheinboldt, “Error estimates for adaptive finite element computations,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 15, 736–754 (1978).
[CrossRef]

Siebert, K. G.

P. Morin, R. H. Nochetto, K. G. Siebert, “Data oscillation and convergence of adaptive FEM,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 38, 466–488 (2000).
[CrossRef]

Süli, E.

P. Monk, E. Süli, “The adaptive computation of far-field patterns by a posteriori error estimation of linear functionals,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 36, 251–274 (1998).
[CrossRef]

Thio, T.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, P. A. Wolff, “Extraordinary optical transmission through subwavelength hole arrays,” Nature (London) 391, 667–669 (1998).
[CrossRef]

Turkel, E.

E. Turkel, A. Yefet, “Absorbing PML boundary layers for wave-like equations,” Appl. Numer. Math. 27, 533–557 (1998).
[CrossRef]

Verfürth, R.

R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh Refinement Techniques, (Teubner, Wiesbaden, Germany, 1996).

Wang, S. S.

Wolff, P. A.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, P. A. Wolff, “Extraordinary optical transmission through subwavelength hole arrays,” Nature (London) 391, 667–669 (1998).
[CrossRef]

Wu, H.

Z. Chen, H. Wu, “An adaptive finite element method with perfectly matched absorbing layers for the wave scattering by periodic structures,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 41, 799–826 (2003).
[CrossRef]

Yefet, A.

E. Turkel, A. Yefet, “Absorbing PML boundary layers for wave-like equations,” Appl. Numer. Math. 27, 533–557 (1998).
[CrossRef]

Appl. Numer. Math. (1)

E. Turkel, A. Yefet, “Absorbing PML boundary layers for wave-like equations,” Appl. Numer. Math. 27, 533–557 (1998).
[CrossRef]

Appl. Opt. (2)

J. Comput. Appl. Math. (1)

P. Monk, “ A posteriori error indicators for Maxwell’s equations,” J. Comput. Appl. Math. 100, 173–190 (1998).
[CrossRef]

J. Comput. Phys. (1)

J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (8)

Nature (London) (1)

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, P. A. Wolff, “Extraordinary optical transmission through subwavelength hole arrays,” Nature (London) 391, 667–669 (1998).
[CrossRef]

Opt. Commun. (1)

M. Nevière, G. Cerutti-Maori, M. Cadilhac, “Sur une nouvelle méthode de résolution du problème de la diffraction d’une onde plane par un réseau infiniment conducteur,” Opt. Commun. 3, 48–52 (1971).
[CrossRef]

SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. (4)

P. Monk, E. Süli, “The adaptive computation of far-field patterns by a posteriori error estimation of linear functionals,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 36, 251–274 (1998).
[CrossRef]

Z. Chen, H. Wu, “An adaptive finite element method with perfectly matched absorbing layers for the wave scattering by periodic structures,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 41, 799–826 (2003).
[CrossRef]

I. Babuška, C. Rheinboldt, “Error estimates for adaptive finite element computations,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 15, 736–754 (1978).
[CrossRef]

P. Morin, R. H. Nochetto, K. G. Siebert, “Data oscillation and convergence of adaptive FEM,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 38, 466–488 (2000).
[CrossRef]

SIAM J. Sci. Comput. (USA) (1)

Z. Chen, S. Dai, “On the efficiency of adaptive finite element methods for elliptic problems with discontinuous coefficients,” SIAM J. Sci. Comput. (USA) 24, 443–462 (2002).
[CrossRef]

Other (2)

R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh Refinement Techniques, (Teubner, Wiesbaden, Germany, 1996).

G. Bao, L. Cowsar, and W. Masters, eds., Mathematical Modeling in Optical Sciences, SIAM Frontiers in Applied Mathematics (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 2001).
[CrossRef]

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Figures (12)

Fig. 1
Fig. 1

Geometry for the PML problem.

Fig. 2
Fig. 2

Problem geometry for Example 1.

Fig. 3
Fig. 3

Problem geometry for Example 2. The period is 0.3 μ m . The thicknesses of the layers corresponding to n 2 , n 3 , and n 4 are 0.188, 0.063, and 0.068 μ m , respectively.

Fig. 4
Fig. 4

TE spectral response of the triple-layer, waveguide-grating filter of Example 2.

Fig. 5
Fig. 5

Grating efficiency of Example 3 as a function of DoFs.

Fig. 6
Fig. 6

Mesh and surface plots of the real part of the associated solution of Example 3 after 44 adaptive iterations; TM case.

Fig. 7
Fig. 7

Problem geometry for Example 4. The period is 0.9 μ m , the groove height is 0.2 μ m , and the groove width is 0.02 μ m . The silver rods lie on a glass substrate of refractive index 1.44.

Fig. 8
Fig. 8

Zeroth-order transmitted efficiency of a silver rectangular-rod grating lying on a glass substrate as a function of the wavelength. The extraordinary transmission through subwavelength hole arrays is found when the wavelength is near 0.984 μ m .

Fig. 9
Fig. 9

Mesh and surface plots of the real part of the associated solution of Example 4 after 14 adaptive iterations; λ = 0.984 μ m , TM case.

Fig. 10
Fig. 10

The 47 th -order efficiency as a function of DoFs for the echelle grating of 316   grooves mm , blaze angle 63.4° with coating of Mg F 2 of thickness 25 nm ; TM case with λ = 120 nm .

Fig. 11
Fig. 11

Mesh and surface plots of the real part of the associated solution of Example 5 near the apex after 14 adaptive iterations; TM case.

Fig. 12
Fig. 12

Total efficiency and 47 th -order efficiency versus wavelength for the echelle grating of 316 gr mm , blaze angle 63.4° with coating of Mg F 2 , thickness 25 nm for TM case. Mode, 47 th order.

Tables (1)

Tables Icon

Table 1 Zeroth-Order Efficiencies and Computation Time for Example 1

Equations (51)

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Δ u + k 2 ( x ) u = 0 ,
div [ 1 k 2 ( x ) u ] + u = 0 .
ϵ ( x 1 , x 3 ) = ϵ 1 in Ω 1 = { ( x 1 , x 3 ) : x 3 b 1 } ,
ϵ ( x 1 , x 3 ) = ϵ 2 in Ω 2 = { ( x 1 , x 3 ) : x 3 b 2 } .
Ω = { ( x 1 , x 3 ) : 0 < x 1 < L and b 2 < x 3 < b 1 } .
s 1 , s 2 C ( R ) , s 1 1 , s 2 0 ,
s ( x 3 ) = 1 for b 2 x 3 b 1 .
Ω 1 PML = { ( x 1 , x 3 ) : 0 < x 1 < L , b 1 < x 3 < b 1 + δ 1 } ,
Ω 2 PML = { ( x 1 , x 3 ) : 0 < x 1 < L , b 2 δ 2 < x 3 < b 2 }
L { x 1 [ s ( x 3 ) x 1 ] + x 3 [ 1 s ( x 3 ) x 3 ] + k 2 ( x ) s ( x 3 ) for TE polarization , x 1 [ s ( x 3 ) k 2 ( x ) x 1 ] + x 3 [ 1 s ( x 3 ) k 2 ( x ) x 3 ] + s ( x 3 ) for TM polarization } .
Γ 1 PML = { ( x 1 , x 3 ) : 0 < x 1 < L , x 3 = b 1 + δ 1 } ,
Γ 2 PML = { ( x 1 , x 3 ) : 0 < x 1 < L , x 3 = b 2 δ 2 } ,
D = { ( x 1 , x 3 ) : 0 < x 1 < L , b 2 δ 2 < x 3 < b 1 + δ 1 } .
L u ̂ = g in D ,
g = { L u I in Ω 1 PML , 0 elsewhere } .
a D ( u ̂ , ψ ) = D g ψ ¯ d x , ψ X ̊ ( D ) ,
a D ( ϕ , ψ ) = D [ A ( x ) ϕ ψ ¯ B ( x ) ϕ ψ ¯ ] d x .
A ( x ) = [ A 11 0 0 A 22 ] ,
A 11 = s ( x 3 ) , A 22 = 1 s ( x 3 ) ,
B ( x ) = k 2 ( x ) s ( x 3 ) in the TE case ;
A 11 = s ( x 3 ) k 2 ( x ) , A 22 = 1 s ( x 3 ) k 2 ( x ) ,
B ( x ) = s ( x 3 ) in the TE case .
σ 1 = b 1 b 1 + δ 1 s ( τ ) d τ , σ 2 = b 2 δ 2 b 2 s ( τ ) d τ .
( β j n ) 2 = k j 2 ( 2 π n L + α ) 2 , Im β j n 0 .
β j n = β j n ( α ) = { [ k j 2 ( 2 π n L + α ) 2 ] 1 2 if k j 2 ( 2 π n L + α ) 2 i [ ( 2 π n L + α ) 2 k j 2 ] 1 2 if k j 2 < ( 2 π n L + α ) 2 } .
Δ j = min { Re ( β j n ) : Re ( β j n ) > 0 } ,
Δ j + = min { Im ( β j n ) : Im ( β j n ) > 0 } .
M 1 = max [ 2 Δ 1 exp ( 2 σ 1 I Δ 1 ) 1 , 2 Δ 1 + exp ( 2 σ 1 R Δ 1 + ) 1 ] ,
M 2 = { max [ 2 Δ 2 exp ( 2 σ 2 I Δ 2 ) 1 , 2 Δ 2 + exp ( 2 σ 2 R Δ 2 + ) 1 ] if Im ϵ 2 = 0 ; 2 k 2 exp ( 2 σ 2 R Im k 2 ) 1 if Im ϵ 2 > 0 . }
u u ̂ H 1 ( Ω ) C M 1 u ̂ u I L 2 ( Γ 1 ) + C M 2 u ̂ L 2 ( Γ 2 ) ,
u u ̂ H 1 ( Ω ) 2 = u u ̂ L 2 ( Ω ) 2 + ( u u ̂ ) L 2 ( Ω ) 2
s ( x 3 ) = { 1 + σ 1 m ( x 3 b 1 δ 1 ) m if x 3 b 1 1 + σ 2 m ( b 2 x 3 δ 2 ) m if x 3 b 2 } , m 1 .
σ j R = ( 1 + Re σ j m m + 1 ) δ j , σ j I = Im σ j m m + 1 δ j .
a D ( u ̂ h , ψ h ) = D g ψ ¯ h d x , ψ h V ̊ h ( D ) .
R T L u ̂ h T + g T .
J e = ( A u ̂ h T 1 A u ̂ h T 2 ) : ν e ,
Γ left = { ( x 1 , x 3 ) : x 1 = 0 , b 2 δ 2 < x 3 < b 1 + δ 1 } ,
Γ right = { ( x 1 , x 3 ) : x 1 = L , b 2 δ 2 < x 3 < b 1 + δ 1 } .
J e = A 11 [ x 1 ( u ̂ h T ) exp ( i α L ) x 1 ( u ̂ h T ) ] ,
J e = A 11 [ exp ( i α L ) x 1 ( u ̂ h T ) x 1 ( u ̂ h T ) ] .
η T = max x T ̃ ρ ( x 3 ) [ h T R T L 2 ( T ) + ( 1 2 e T h e J e L 2 ( e ) 2 ) 1 2 ] ,
ρ ( x 3 ) = { s ( x 3 ) exp [ R j ( x 3 ) ] if x Ω j PML ¯ 1 if x Ω } ,
R 1 ( x 3 ) = min [ Δ 1 b 1 x 3 s 2 ( τ ) d τ , Δ 1 + b 1 x 3 s 1 ( τ ) d τ ] ,
R 2 ( x 3 ) = { min ( Δ 2 x 3 b 2 s 2 ( τ ) d τ , Δ 2 + x 3 b 2 s 1 ( τ ) d τ ) if Im ϵ 2 = 0 ; Im ( k 2 ) x 3 b 2 s 1 ( τ ) d τ if Im ϵ 2 > 0 . }
u u ̂ h H 1 ( Ω ) C M 1 u ̂ h u I L 2 ( Γ 1 ) + C M 2 u ̂ h L 2 ( Γ 2 ) + C M 3 I h u I u I L 2 ( Γ 1 PML ) + C ( T M h η T 2 ) 1 2 ,
M 3 = max [ 2 Δ 1 exp ( Δ 1 σ 1 I ) 1 exp ( 2 Δ 1 σ 1 I ) , 2 Δ 1 + exp ( Δ 1 + σ 1 R ) 1 exp ( 2 Δ 1 + σ 1 R ) ] .
E PML = M 1 u ̂ h u I L 2 ( Γ 1 ) + M 2 u ̂ h L 2 ( Γ 2 ) ,
E FEM = M 3 u ̂ h u I L 2 ( Γ 1 PML ) + ( T M h η T 2 ) 1 2 .
η ̃ T = η T + M 3 I h u I u I L 2 ( Γ 1 PML T ) .
Solution Estimation Refinement .
( T M ̂ h η ̃ T 2 ) 1 2 > 0.7 ( T M h η ̃ T 2 ) 1 2 ;

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