Abstract

We investigate the electromagnetic modeling of plane-wave diffraction by nonperiodic surfaces by using the curvilinear coordinate method (CCM). This method is often used with a Fourier basis expansion, which results in the periodization of both the geometry and the electromagnetic field. We write the CCM in a complex coordinate system in order to introduce the perfectly matched layer concept in a simple and efficient way. The results, presented for a perfectly conducting surface, show the efficiency of the model.

© 2007 Optical Society of America

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  1. J. Chandezon, D. Maystre, and G. Raoult, "A new theoretical method for diffraction gratings and its numerical application," J. Opt. 11, 235-241 (1980).
    [Crossref]
  2. R. Dusséaux and R. Oliveira, "Scattering of plane wave by 1-dimensional rough surface: study in a nonorthogonal coordinate system," Prog. Electromagn. Res. 34, 63-88 (2001).
    [Crossref]
  3. G. Granet, K. Edee, and D. Felbacq, "Scattering of a plane wave by rough surfaces: a new curvilinear coordinate system based approach," Prog. Electromagn. Res. 41, 235-250 (2003).
  4. K. Edee, G. Granet, R. Dusséaux, and C. Baudier, "A curvilinear coordinate based hybrid method for scattering of plane wave from rough surfaces," J. Electromagn. Waves Appl. 15, 1001-1015 (2004).
    [Crossref]
  5. C. Baudier, R. Dusséaux, K. Edee, and G. Granet, "Scattering of a plane wave by one-dimensional dielectric random rough surfaces: study with the curvilinear coordinate method," Waves Random Media 14, 61-74 (2004).
    [Crossref]
  6. J. Berenger, "A perfectly matched layer for the absorption of electromagnetic waves," J. Comput. Phys. 114, 185-200 (1994).
    [Crossref]
  7. W. C. Chew and W. H. Weedon, "A 3D perfectly matched medium from modified Maxwell's equations with stretched coordinates," Microwave Opt. Technol. Lett. 7, 599-604 (1994).
    [Crossref]
  8. W. C. Chew, J. M. Jin, and E. Michielssen, "Complex coordinate stretching as a generalized absorbing boundary condition," Microwave Opt. Technol. Lett. 15, 363-369 (1997).
    [Crossref]
  9. F. L. Teixeira and W. C. Chew, "Differential forms, metrics, and the reflectionless absorption of electromagnetic waves," J. Electromagn. Waves Appl. 13, 665-686 (1999).
    [Crossref]
  10. H. Derruder, F. Olyslager, and D. De Zutter, "An efficient series expansion for the 2-D Green's function of microstrip substrate using perfectly matched layers," IEEE Microw. Guid. Wave Lett. 9, 505-507 (1999).
    [Crossref]
  11. J. P. Plumey, K. Edee, and G. Granet, "Modal expansion for the 2D Green's function in a non-orthogonal coordinates system," Prog. Electromagn. Res. 59, 101-112 (2006).
    [Crossref]

2006 (1)

J. P. Plumey, K. Edee, and G. Granet, "Modal expansion for the 2D Green's function in a non-orthogonal coordinates system," Prog. Electromagn. Res. 59, 101-112 (2006).
[Crossref]

2004 (2)

K. Edee, G. Granet, R. Dusséaux, and C. Baudier, "A curvilinear coordinate based hybrid method for scattering of plane wave from rough surfaces," J. Electromagn. Waves Appl. 15, 1001-1015 (2004).
[Crossref]

C. Baudier, R. Dusséaux, K. Edee, and G. Granet, "Scattering of a plane wave by one-dimensional dielectric random rough surfaces: study with the curvilinear coordinate method," Waves Random Media 14, 61-74 (2004).
[Crossref]

2003 (1)

G. Granet, K. Edee, and D. Felbacq, "Scattering of a plane wave by rough surfaces: a new curvilinear coordinate system based approach," Prog. Electromagn. Res. 41, 235-250 (2003).

2001 (1)

R. Dusséaux and R. Oliveira, "Scattering of plane wave by 1-dimensional rough surface: study in a nonorthogonal coordinate system," Prog. Electromagn. Res. 34, 63-88 (2001).
[Crossref]

1999 (2)

F. L. Teixeira and W. C. Chew, "Differential forms, metrics, and the reflectionless absorption of electromagnetic waves," J. Electromagn. Waves Appl. 13, 665-686 (1999).
[Crossref]

H. Derruder, F. Olyslager, and D. De Zutter, "An efficient series expansion for the 2-D Green's function of microstrip substrate using perfectly matched layers," IEEE Microw. Guid. Wave Lett. 9, 505-507 (1999).
[Crossref]

1997 (1)

W. C. Chew, J. M. Jin, and E. Michielssen, "Complex coordinate stretching as a generalized absorbing boundary condition," Microwave Opt. Technol. Lett. 15, 363-369 (1997).
[Crossref]

1994 (2)

J. Berenger, "A perfectly matched layer for the absorption of electromagnetic waves," J. Comput. Phys. 114, 185-200 (1994).
[Crossref]

W. C. Chew and W. H. Weedon, "A 3D perfectly matched medium from modified Maxwell's equations with stretched coordinates," Microwave Opt. Technol. Lett. 7, 599-604 (1994).
[Crossref]

1980 (1)

J. Chandezon, D. Maystre, and G. Raoult, "A new theoretical method for diffraction gratings and its numerical application," J. Opt. 11, 235-241 (1980).
[Crossref]

Baudier, C.

K. Edee, G. Granet, R. Dusséaux, and C. Baudier, "A curvilinear coordinate based hybrid method for scattering of plane wave from rough surfaces," J. Electromagn. Waves Appl. 15, 1001-1015 (2004).
[Crossref]

C. Baudier, R. Dusséaux, K. Edee, and G. Granet, "Scattering of a plane wave by one-dimensional dielectric random rough surfaces: study with the curvilinear coordinate method," Waves Random Media 14, 61-74 (2004).
[Crossref]

Berenger, J.

J. Berenger, "A perfectly matched layer for the absorption of electromagnetic waves," J. Comput. Phys. 114, 185-200 (1994).
[Crossref]

Chandezon, J.

J. Chandezon, D. Maystre, and G. Raoult, "A new theoretical method for diffraction gratings and its numerical application," J. Opt. 11, 235-241 (1980).
[Crossref]

Chew, W. C.

F. L. Teixeira and W. C. Chew, "Differential forms, metrics, and the reflectionless absorption of electromagnetic waves," J. Electromagn. Waves Appl. 13, 665-686 (1999).
[Crossref]

W. C. Chew, J. M. Jin, and E. Michielssen, "Complex coordinate stretching as a generalized absorbing boundary condition," Microwave Opt. Technol. Lett. 15, 363-369 (1997).
[Crossref]

W. C. Chew and W. H. Weedon, "A 3D perfectly matched medium from modified Maxwell's equations with stretched coordinates," Microwave Opt. Technol. Lett. 7, 599-604 (1994).
[Crossref]

De Zutter, D.

H. Derruder, F. Olyslager, and D. De Zutter, "An efficient series expansion for the 2-D Green's function of microstrip substrate using perfectly matched layers," IEEE Microw. Guid. Wave Lett. 9, 505-507 (1999).
[Crossref]

Derruder, H.

H. Derruder, F. Olyslager, and D. De Zutter, "An efficient series expansion for the 2-D Green's function of microstrip substrate using perfectly matched layers," IEEE Microw. Guid. Wave Lett. 9, 505-507 (1999).
[Crossref]

Dusséaux, R.

K. Edee, G. Granet, R. Dusséaux, and C. Baudier, "A curvilinear coordinate based hybrid method for scattering of plane wave from rough surfaces," J. Electromagn. Waves Appl. 15, 1001-1015 (2004).
[Crossref]

C. Baudier, R. Dusséaux, K. Edee, and G. Granet, "Scattering of a plane wave by one-dimensional dielectric random rough surfaces: study with the curvilinear coordinate method," Waves Random Media 14, 61-74 (2004).
[Crossref]

R. Dusséaux and R. Oliveira, "Scattering of plane wave by 1-dimensional rough surface: study in a nonorthogonal coordinate system," Prog. Electromagn. Res. 34, 63-88 (2001).
[Crossref]

Edee, K.

J. P. Plumey, K. Edee, and G. Granet, "Modal expansion for the 2D Green's function in a non-orthogonal coordinates system," Prog. Electromagn. Res. 59, 101-112 (2006).
[Crossref]

K. Edee, G. Granet, R. Dusséaux, and C. Baudier, "A curvilinear coordinate based hybrid method for scattering of plane wave from rough surfaces," J. Electromagn. Waves Appl. 15, 1001-1015 (2004).
[Crossref]

C. Baudier, R. Dusséaux, K. Edee, and G. Granet, "Scattering of a plane wave by one-dimensional dielectric random rough surfaces: study with the curvilinear coordinate method," Waves Random Media 14, 61-74 (2004).
[Crossref]

G. Granet, K. Edee, and D. Felbacq, "Scattering of a plane wave by rough surfaces: a new curvilinear coordinate system based approach," Prog. Electromagn. Res. 41, 235-250 (2003).

Felbacq, D.

G. Granet, K. Edee, and D. Felbacq, "Scattering of a plane wave by rough surfaces: a new curvilinear coordinate system based approach," Prog. Electromagn. Res. 41, 235-250 (2003).

Granet, G.

J. P. Plumey, K. Edee, and G. Granet, "Modal expansion for the 2D Green's function in a non-orthogonal coordinates system," Prog. Electromagn. Res. 59, 101-112 (2006).
[Crossref]

K. Edee, G. Granet, R. Dusséaux, and C. Baudier, "A curvilinear coordinate based hybrid method for scattering of plane wave from rough surfaces," J. Electromagn. Waves Appl. 15, 1001-1015 (2004).
[Crossref]

C. Baudier, R. Dusséaux, K. Edee, and G. Granet, "Scattering of a plane wave by one-dimensional dielectric random rough surfaces: study with the curvilinear coordinate method," Waves Random Media 14, 61-74 (2004).
[Crossref]

G. Granet, K. Edee, and D. Felbacq, "Scattering of a plane wave by rough surfaces: a new curvilinear coordinate system based approach," Prog. Electromagn. Res. 41, 235-250 (2003).

Jin, J. M.

W. C. Chew, J. M. Jin, and E. Michielssen, "Complex coordinate stretching as a generalized absorbing boundary condition," Microwave Opt. Technol. Lett. 15, 363-369 (1997).
[Crossref]

Maystre, D.

J. Chandezon, D. Maystre, and G. Raoult, "A new theoretical method for diffraction gratings and its numerical application," J. Opt. 11, 235-241 (1980).
[Crossref]

Michielssen, E.

W. C. Chew, J. M. Jin, and E. Michielssen, "Complex coordinate stretching as a generalized absorbing boundary condition," Microwave Opt. Technol. Lett. 15, 363-369 (1997).
[Crossref]

Oliveira, R.

R. Dusséaux and R. Oliveira, "Scattering of plane wave by 1-dimensional rough surface: study in a nonorthogonal coordinate system," Prog. Electromagn. Res. 34, 63-88 (2001).
[Crossref]

Olyslager, F.

H. Derruder, F. Olyslager, and D. De Zutter, "An efficient series expansion for the 2-D Green's function of microstrip substrate using perfectly matched layers," IEEE Microw. Guid. Wave Lett. 9, 505-507 (1999).
[Crossref]

Plumey, J. P.

J. P. Plumey, K. Edee, and G. Granet, "Modal expansion for the 2D Green's function in a non-orthogonal coordinates system," Prog. Electromagn. Res. 59, 101-112 (2006).
[Crossref]

Raoult, G.

J. Chandezon, D. Maystre, and G. Raoult, "A new theoretical method for diffraction gratings and its numerical application," J. Opt. 11, 235-241 (1980).
[Crossref]

Teixeira, F. L.

F. L. Teixeira and W. C. Chew, "Differential forms, metrics, and the reflectionless absorption of electromagnetic waves," J. Electromagn. Waves Appl. 13, 665-686 (1999).
[Crossref]

Weedon, W. H.

W. C. Chew and W. H. Weedon, "A 3D perfectly matched medium from modified Maxwell's equations with stretched coordinates," Microwave Opt. Technol. Lett. 7, 599-604 (1994).
[Crossref]

IEEE Microw. Guid. Wave Lett. (1)

H. Derruder, F. Olyslager, and D. De Zutter, "An efficient series expansion for the 2-D Green's function of microstrip substrate using perfectly matched layers," IEEE Microw. Guid. Wave Lett. 9, 505-507 (1999).
[Crossref]

J. Comput. Phys. (1)

J. Berenger, "A perfectly matched layer for the absorption of electromagnetic waves," J. Comput. Phys. 114, 185-200 (1994).
[Crossref]

J. Electromagn. Waves Appl. (2)

K. Edee, G. Granet, R. Dusséaux, and C. Baudier, "A curvilinear coordinate based hybrid method for scattering of plane wave from rough surfaces," J. Electromagn. Waves Appl. 15, 1001-1015 (2004).
[Crossref]

F. L. Teixeira and W. C. Chew, "Differential forms, metrics, and the reflectionless absorption of electromagnetic waves," J. Electromagn. Waves Appl. 13, 665-686 (1999).
[Crossref]

J. Opt. (1)

J. Chandezon, D. Maystre, and G. Raoult, "A new theoretical method for diffraction gratings and its numerical application," J. Opt. 11, 235-241 (1980).
[Crossref]

Microwave Opt. Technol. Lett. (2)

W. C. Chew and W. H. Weedon, "A 3D perfectly matched medium from modified Maxwell's equations with stretched coordinates," Microwave Opt. Technol. Lett. 7, 599-604 (1994).
[Crossref]

W. C. Chew, J. M. Jin, and E. Michielssen, "Complex coordinate stretching as a generalized absorbing boundary condition," Microwave Opt. Technol. Lett. 15, 363-369 (1997).
[Crossref]

Prog. Electromagn. Res. (3)

J. P. Plumey, K. Edee, and G. Granet, "Modal expansion for the 2D Green's function in a non-orthogonal coordinates system," Prog. Electromagn. Res. 59, 101-112 (2006).
[Crossref]

R. Dusséaux and R. Oliveira, "Scattering of plane wave by 1-dimensional rough surface: study in a nonorthogonal coordinate system," Prog. Electromagn. Res. 34, 63-88 (2001).
[Crossref]

G. Granet, K. Edee, and D. Felbacq, "Scattering of a plane wave by rough surfaces: a new curvilinear coordinate system based approach," Prog. Electromagn. Res. 41, 235-250 (2003).

Waves Random Media (1)

C. Baudier, R. Dusséaux, K. Edee, and G. Granet, "Scattering of a plane wave by one-dimensional dielectric random rough surfaces: study with the curvilinear coordinate method," Waves Random Media 14, 61-74 (2004).
[Crossref]

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Figures (9)

Fig. 1
Fig. 1

Schematic of the profile.

Fig. 2
Fig. 2

Amplitude of magnetic field H z radiated at θ = 45 ° computed for different values of η. Numerical parameters: θ 0 = 0 ° , λ = 1 , h = 0.4 λ , l = 4 λ , b = 3 , 1 Δ α = 10.5 , e PML = λ .

Fig. 3
Fig. 3

Schematic of the profile for PML implementation.

Fig. 4
Fig. 4

Field map calculated without PML. Numerical parameters: θ 0 = 0 ° , λ = 1 , h = 0.4 λ , l = 4 λ , b = 3 , 1 Δ α = 10 , η = 0 .

Fig. 5
Fig. 5

Field map calculated with PML. Numerical parameters: θ 0 = 0 ° , λ = 1 , h = 0.4 λ , l = 4 λ , b = 3 , 1 Δ α = 10 , η = 2 , e PML = λ .

Fig. 6
Fig. 6

Angular repartition of the amplitude of H z at ρ = 4 λ and for different values of 1 Δ α . Numerical parameters: θ 0 = 0 ° , λ = 1 , h = λ , l = 2 λ , η = 0 .

Fig. 7
Fig. 7

Angular repartition of the amplitude of H z at ρ = 4 λ and for different values of 1 Δ α . Numerical parameters: θ 0 = 0 ° , λ = 1 , h = λ , l = 2 λ , η = 2 , e PML = λ .

Fig. 8
Fig. 8

Amplitude of magnetic field H z ( x 1 , x 2 ) at x 2 = λ , for different values of 1 Δ α . Numerical parameters: θ 0 = 0 ° , λ = 1 , h = λ , l = 2 λ , η = 2 , e PML = λ .

Fig. 9
Fig. 9

Amplitude of magnetic field H z ( x 1 , x 2 ) at x 2 = 10 λ , for different values of 1 Δ α . Numerical parameters: θ 0 = 0 ° , λ = 1 , h = λ , l = 2 λ , η = 2 , e PML = λ .

Equations (38)

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Ψ i ( x , y ) = exp ( i k α 0 x ) exp ( i k β 0 y ) ,
Ψ = Ψ t ( Ψ i + Ψ p ) ,
Ψ p ( x , y ) = R TM , TE exp ( i k α 0 x ) exp ( i k β 0 y ) ,
Ψ ( x , y ) = + R ( α ) exp ( i k β ( α ) y ) exp ( i k α x ) d α .
Ψ ( ρ , θ ) = k 1 2 π R ( θ ) cos ( θ ) exp ( i k ρ ) k ρ exp ( i π 4 ) .
ξ i j k j H k = i ω D i , ξ i j k j E k = i ω B i , i , j , k = 1 , 2 , 3 ,
D i = ϵ g g i j E j , B i = μ g g i j H j ,
d s 2 = g i j d x i d x j .
x = x 1 , y = x 2 + a ( x 1 ) , z = x 3 ,
[ g i j ( x 1 , x 2 , x 3 ) ] = [ 1 + a ̇ ( x 1 ) a ̇ ( x 1 ) a ̇ ( x 1 ) 0 a ̇ ( x 1 ) 1 0 0 0 1 ] ,
x 1 x ̃ 1 0 x 1 s ( ξ ) d ξ ,
s ( x 1 ) = { ( 1 i η ) x 1 if x 1 [ 0.5 d 0.5 d + e PML ] [ 0.5 d e PML 0.5 d ] x 1 otherwise } ,
d s ̃ 2 = g i j d x ̃ i d x ̃ j .
g i j ( x i ) = x ̃ k x i x ̃ l x j g k l ( x ̃ i ) .
g i j ( x 1 , x 2 , x 3 ) = [ s ( x 1 ) s ( x 1 ) + a ̇ ( x ̃ 1 ( x 1 ) ) a ̇ ( x ̃ 1 ( x 1 ) ) a ̇ ( x ̃ 1 ( x 1 ) ) 0 a ̇ ( x ̃ 1 ( x 1 ) ) 1 0 0 0 1 ] .
g g i j ( x 1 , x 2 , x 3 ) = [ 1 s ( x 1 ) a ̇ ( x ̃ 1 ( x 1 ) ) 0 a ̇ ( x ̃ 1 ( x 1 ) ) s ( x 1 ) [ 1 + a ̇ ( x ̃ 1 ( x 1 ) ) a ̇ ( x ̃ 1 ( x 1 ) ) ] 0 0 0 s ( x 1 ) ] ,
[ 1 s 1 1 s 1 ( 1 s 1 a ̇ + a ̇ 1 s 1 ) 2 + ( 1 + a ̇ a ̇ ) 2 2 + ω 2 ε μ ] Ψ ( x 1 , x 2 ) = 0 .
[ 1 s 1 1 s 1 + k 2 I 0 0 I ] ( Ψ 2 Ψ ) = 2 [ 1 s 1 a ̇ + a ̇ 1 s 1 ( 1 + a ̇ a ̇ ) I 0 ] ( Ψ 2 Ψ ) ,
Ψ ( x 1 , x 2 ) = exp ( i k r x 2 ) ψ ( x 1 ) ,
2 Ψ ( x 1 , x 2 ) = exp ( i k r x 2 ) ψ ̇ ( x 1 ) .
[ 1 s 1 1 s 1 + k 2 I 0 0 I ] ( ψ ψ ̇ ) = i k r [ 1 s 1 a ̇ + a ̇ 1 s 1 ( 1 + a ̇ a ̇ ) I 0 ] ( ψ ψ ̇ ) ,
ψ ̂ ( α ) = + ψ ( x 1 ) exp ( i k α x 1 ) d x 1 ,
a ̇ ̂ ( α ) = + a ̇ ( x 1 ) exp ( i k α x 1 ) d x 1 .
ψ ̂ ( α ) = n = + ψ n b ̂ n ( α ) ,
ψ ̇ ̂ ( α ) = n = + ψ ̇ n b ̂ n ( α ) ,
a ̇ ̂ ( α ) = n = + a ̇ n b ̂ n ( α ) ,
Ψ ̂ ( α , x 2 ) = q = 1 2 M + 1 R q Ψ ̂ q ( α , x 2 ) ,
Ψ ̂ q ( α , x 2 ) = exp ( i k r q x 2 ) n = M M ψ n q b ̂ n ( α ) ,
Ψ ( x 1 , x 2 ) = Δ α Π Δ α ( x 1 ) q R q exp ( i k r q x 2 ) n = M M ψ n q exp ( i k α n x 1 ) ,
Π Δ α ( x 1 ) = { 1 if x 1 [ λ 2 Δ α λ 2 Δ α ] 0 otherwise } .
Φ ( x 1 , x 2 ) = τ [ 2 a ̇ ( x 1 ) ( 1 a ̇ ( x 1 ) 1 ) ] Ψ ( x 1 , x 2 ) ,
a ( x ) = { h exp ( b b l 2 l 2 4 x 2 ) if x [ l 2 , l 2 ] 0 otherwise } .
Ψ ( x 1 , 0 ) = ( Ψ i ( x 1 , 0 ) + Ψ p ( x 1 , 0 ) ) , for TM polarization ,
Φ ( x 1 , 0 ) = ( Φ i ( x 1 , 0 ) + Φ p ( x 1 , 0 ) ) , for TE polarization .
[ ψ ] [ R ] = [ F ] + [ F + ] , for TM polarization ,
[ Φ ] [ R ] = [ G ] [ G + ] , for TE polarization .
F n ± = λ 2 Δ α λ 2 Δ α exp [ ± i k β 0 a ( x ) ] exp ( i k n Δ α x ) d x .
a ( x ) = { h [ cos ( 2 π l x ) + 1 ] if x [ l 2 , l 2 ] 0 otherwise } .

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