Abstract

The main goal of the method proposed in this paper is the numerical study of various kinds of anisotropic gratings deposited on isotropic substrates, without any constraint upon the diffractive pattern geometry or electromagnetic properties. To that end we propose a new FEM (Finite Element Method) formulation which rigorously deals with each infinite issue inherent to grating problems. As an example, 2D numerical experiments are presented in the cases of the diffraction of a plane wave by an anisotropic aragonite grating on silica substrate (for the two polarization cases and at normal or oblique incidence). We emphasize the interesting property that the diffracted field is non symmetric in a geometrically symmetric configuration.

© 2007 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  26. A. Nicolet, S. Guenneau, C. Geuzaine, and F. Zolla, "Modeling of electromagnetic waves in periodic media with finite elements," J. Comput. Appl. Math. 168 (1-2), 321-329 (2004).
    [CrossRef]
  27. Y. Ould Agha, F. Zolla, A. Nicolet, S. Guenneau, "On the use of PML for the computation of leaky modes: an application to Microstructured Optcal Fibres," The In.J. Comput. Math. Elect. Elect. Eng. 27, 95-109 (2008).
  28. A. Nicolet, F. Zolla, Y. Ould Agha, and S. Guenneau, "Leaky modes in twisted microstructured optical fibres," Waves Random Media,  17:4, 559-570 (2007).
  29. M. Lassas, J. Liukkonen and E. Somersalo, "Complex riemannian metric and absorbing boundary condition," J. Math Pures Appl. 80, 739-768 (2001).
  30. M. Lassas and E. Somersalo, "Analysis of the PML equations in general convex geometry," Proc. R. Soc. Edinburgh 131, 1183-1207 (2001).
    [CrossRef]
  31. G. Bao and H. Wu, "On the convergence of the solutions of PML equations for time harmonic Maxwell’s equations," SIAM (Soc. Ind. Appl. Math.)J. Numer. Anal. 43, 2121-2143 (2005).
    [CrossRef]
  32. P. Helluy, S. Maire, and P. Ravel, "Intégrations numériques d’ordre élevé de fonctions régulières ou singulières sur un intervalle," CR. Acad. Sci. Paris, Ser. I, Math,  327, 843-848 (1998).

2008

Y. Ould Agha, F. Zolla, A. Nicolet, S. Guenneau, "On the use of PML for the computation of leaky modes: an application to Microstructured Optcal Fibres," The In.J. Comput. Math. Elect. Elect. Eng. 27, 95-109 (2008).

2007

A. Nicolet, F. Zolla, Y. Ould Agha, and S. Guenneau, "Leaky modes in twisted microstructured optical fibres," Waves Random Media,  17:4, 559-570 (2007).

2005

2004

N. Kono and Y. Tsuji, "A novel finite-element method for nonreciprocal magneto-photonic crystal waveguides," J. Lightwave Technol. 22, 1741-1747 (2004).
[CrossRef]

A. Nicolet, S. Guenneau, C. Geuzaine, and F. Zolla, "Modeling of electromagnetic waves in periodic media with finite elements," J. Comput. Appl. Math. 168 (1-2), 321-329 (2004).
[CrossRef]

2003

H. Kikuta, H. Toyotai, and W. Yul, "Optical elements with subwavelength structured surfaces," Opt. Rev. 10, 63-73 (2003)
[CrossRef]

2002

2001

M. Lassas, J. Liukkonen and E. Somersalo, "Complex riemannian metric and absorbing boundary condition," J. Math Pures Appl. 80, 739-768 (2001).

M. Lassas and E. Somersalo, "Analysis of the PML equations in general convex geometry," Proc. R. Soc. Edinburgh 131, 1183-1207 (2001).
[CrossRef]

1999

1998

S. M. Schultz, E. N. Glytsis, and T. K. Gaylord, "Design of a high-efficiency volume grating coupler for line focusing," Appl. Opt. 37, 2278-2287 (1998).
[CrossRef]

P. Helluy, S. Maire, and P. Ravel, "Intégrations numériques d’ordre élevé de fonctions régulières ou singulières sur un intervalle," CR. Acad. Sci. Paris, Ser. I, Math,  327, 843-848 (1998).

1996

1994

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

1993

1992

1987

1983

1981

1980

J. Chandezon, D. Maystre, and G. Raoult, "A new theoretical method for diffraction gratings and its numerical application," J. Opt. (Paris) 11, 235241 (1980).
[CrossRef]

1978

1966

K. Yee, "Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media," IEEE Trans. Antennas Propag. AP-14, 302-307 (1966).

Bao, G.

G. Bao and H. Wu, "On the convergence of the solutions of PML equations for time harmonic Maxwell’s equations," SIAM (Soc. Ind. Appl. Math.)J. Numer. Anal. 43, 2121-2143 (2005).
[CrossRef]

Bao, Gang

Belyaev, V. V.

Berenger, J.-P.

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

Brucker, C. F.

Chandezon, J.

J. Chandezon, D. Maystre, and G. Raoult, "A new theoretical method for diffraction gratings and its numerical application," J. Opt. (Paris) 11, 235241 (1980).
[CrossRef]

Chen, Zhiming

Delort, T.

Erwin, J. K.

Gaylord, T. K.

Geuzaine, C.

A. Nicolet, S. Guenneau, C. Geuzaine, and F. Zolla, "Modeling of electromagnetic waves in periodic media with finite elements," J. Comput. Appl. Math. 168 (1-2), 321-329 (2004).
[CrossRef]

Glytsis, E. N.

Granet, G.

Guenneau, S.

Y. Ould Agha, F. Zolla, A. Nicolet, S. Guenneau, "On the use of PML for the computation of leaky modes: an application to Microstructured Optcal Fibres," The In.J. Comput. Math. Elect. Elect. Eng. 27, 95-109 (2008).

A. Nicolet, F. Zolla, Y. Ould Agha, and S. Guenneau, "Leaky modes in twisted microstructured optical fibres," Waves Random Media,  17:4, 559-570 (2007).

A. Nicolet, S. Guenneau, C. Geuzaine, and F. Zolla, "Modeling of electromagnetic waves in periodic media with finite elements," J. Comput. Appl. Math. 168 (1-2), 321-329 (2004).
[CrossRef]

Helluy, P.

P. Helluy, S. Maire, and P. Ravel, "Intégrations numériques d’ordre élevé de fonctions régulières ou singulières sur un intervalle," CR. Acad. Sci. Paris, Ser. I, Math,  327, 843-848 (1998).

Kikuta, H.

H. Kikuta, H. Toyotai, and W. Yul, "Optical elements with subwavelength structured surfaces," Opt. Rev. 10, 63-73 (2003)
[CrossRef]

Klyshkov, A. V.

Knop, K.

Kono, N.

Koshiba, M.

Kushnir, E. M.

Lassas, M.

M. Lassas, J. Liukkonen and E. Somersalo, "Complex riemannian metric and absorbing boundary condition," J. Math Pures Appl. 80, 739-768 (2001).

M. Lassas and E. Somersalo, "Analysis of the PML equations in general convex geometry," Proc. R. Soc. Edinburgh 131, 1183-1207 (2001).
[CrossRef]

Li, L.

Liukkonen, J.

M. Lassas, J. Liukkonen and E. Somersalo, "Complex riemannian metric and absorbing boundary condition," J. Math Pures Appl. 80, 739-768 (2001).

Maire, S.

P. Helluy, S. Maire, and P. Ravel, "Intégrations numériques d’ordre élevé de fonctions régulières ou singulières sur un intervalle," CR. Acad. Sci. Paris, Ser. I, Math,  327, 843-848 (1998).

Mansuripur, M.

Maystre, D.

Moharam, M. G.

Nicolet, A.

Y. Ould Agha, F. Zolla, A. Nicolet, S. Guenneau, "On the use of PML for the computation of leaky modes: an application to Microstructured Optcal Fibres," The In.J. Comput. Math. Elect. Elect. Eng. 27, 95-109 (2008).

A. Nicolet, F. Zolla, Y. Ould Agha, and S. Guenneau, "Leaky modes in twisted microstructured optical fibres," Waves Random Media,  17:4, 559-570 (2007).

A. Nicolet, S. Guenneau, C. Geuzaine, and F. Zolla, "Modeling of electromagnetic waves in periodic media with finite elements," J. Comput. Appl. Math. 168 (1-2), 321-329 (2004).
[CrossRef]

Ohkawa, Y.

Ould Agha, Y.

Y. Ould Agha, F. Zolla, A. Nicolet, S. Guenneau, "On the use of PML for the computation of leaky modes: an application to Microstructured Optcal Fibres," The In.J. Comput. Math. Elect. Elect. Eng. 27, 95-109 (2008).

A. Nicolet, F. Zolla, Y. Ould Agha, and S. Guenneau, "Leaky modes in twisted microstructured optical fibres," Waves Random Media,  17:4, 559-570 (2007).

Petit, R.

Raoult, G.

J. Chandezon, D. Maystre, and G. Raoult, "A new theoretical method for diffraction gratings and its numerical application," J. Opt. (Paris) 11, 235241 (1980).
[CrossRef]

Ravel, P.

P. Helluy, S. Maire, and P. Ravel, "Intégrations numériques d’ordre élevé de fonctions régulières ou singulières sur un intervalle," CR. Acad. Sci. Paris, Ser. I, Math,  327, 843-848 (1998).

Rokushima, K.

Saj, W. M.

Schultz, S. M.

Somersalo, E.

M. Lassas and E. Somersalo, "Analysis of the PML equations in general convex geometry," Proc. R. Soc. Edinburgh 131, 1183-1207 (2001).
[CrossRef]

M. Lassas, J. Liukkonen and E. Somersalo, "Complex riemannian metric and absorbing boundary condition," J. Math Pures Appl. 80, 739-768 (2001).

Toyotai, H.

H. Kikuta, H. Toyotai, and W. Yul, "Optical elements with subwavelength structured surfaces," Opt. Rev. 10, 63-73 (2003)
[CrossRef]

Tsoi, V. I.

Tsuji, Y.

Watanabe, K.

Wu, H.

G. Bao and H. Wu, "On the convergence of the solutions of PML equations for time harmonic Maxwell’s equations," SIAM (Soc. Ind. Appl. Math.)J. Numer. Anal. 43, 2121-2143 (2005).
[CrossRef]

Wu, Haijun

Yamanaka, J.

Yee, K.

K. Yee, "Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media," IEEE Trans. Antennas Propag. AP-14, 302-307 (1966).

Yul, W.

H. Kikuta, H. Toyotai, and W. Yul, "Optical elements with subwavelength structured surfaces," Opt. Rev. 10, 63-73 (2003)
[CrossRef]

Zhou, A. F.

Zolla, F.

Y. Ould Agha, F. Zolla, A. Nicolet, S. Guenneau, "On the use of PML for the computation of leaky modes: an application to Microstructured Optcal Fibres," The In.J. Comput. Math. Elect. Elect. Eng. 27, 95-109 (2008).

A. Nicolet, F. Zolla, Y. Ould Agha, and S. Guenneau, "Leaky modes in twisted microstructured optical fibres," Waves Random Media,  17:4, 559-570 (2007).

A. Nicolet, S. Guenneau, C. Geuzaine, and F. Zolla, "Modeling of electromagnetic waves in periodic media with finite elements," J. Comput. Appl. Math. 168 (1-2), 321-329 (2004).
[CrossRef]

F. Zolla and R. Petit, "Method of fictitious sources as applied to the elctromagnetic diffraction of a plane wave by a grating in conical mounts," J. Opt. Soc. Am. A 13, 796-802 (1996).
[CrossRef]

Appl. Opt.

CR. Acad. Sci. Paris, S’er. I, Math

P. Helluy, S. Maire, and P. Ravel, "Intégrations numériques d’ordre élevé de fonctions régulières ou singulières sur un intervalle," CR. Acad. Sci. Paris, Ser. I, Math,  327, 843-848 (1998).

IEEE Trans. Antennas Propag.

K. Yee, "Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media," IEEE Trans. Antennas Propag. AP-14, 302-307 (1966).

J. Comput. Appl. Math.

A. Nicolet, S. Guenneau, C. Geuzaine, and F. Zolla, "Modeling of electromagnetic waves in periodic media with finite elements," J. Comput. Appl. Math. 168 (1-2), 321-329 (2004).
[CrossRef]

J. Comput. Math. Elect. Elect. Eng.

Y. Ould Agha, F. Zolla, A. Nicolet, S. Guenneau, "On the use of PML for the computation of leaky modes: an application to Microstructured Optcal Fibres," The In.J. Comput. Math. Elect. Elect. Eng. 27, 95-109 (2008).

J. Comput. Phys.

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

J. Lightwave Technol.

J. Math Pures Appl.

M. Lassas, J. Liukkonen and E. Somersalo, "Complex riemannian metric and absorbing boundary condition," J. Math Pures Appl. 80, 739-768 (2001).

J. Numer. Anal.

G. Bao and H. Wu, "On the convergence of the solutions of PML equations for time harmonic Maxwell’s equations," SIAM (Soc. Ind. Appl. Math.)J. Numer. Anal. 43, 2121-2143 (2005).
[CrossRef]

J. Opt. (Paris)

J. Chandezon, D. Maystre, and G. Raoult, "A new theoretical method for diffraction gratings and its numerical application," J. Opt. (Paris) 11, 235241 (1980).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Opt. Technol.

Opt. Express

Opt. Rev.

H. Kikuta, H. Toyotai, and W. Yul, "Optical elements with subwavelength structured surfaces," Opt. Rev. 10, 63-73 (2003)
[CrossRef]

Proc. R. Soc. Edinburgh

M. Lassas and E. Somersalo, "Analysis of the PML equations in general convex geometry," Proc. R. Soc. Edinburgh 131, 1183-1207 (2001).
[CrossRef]

Waves Random Media

A. Nicolet, F. Zolla, Y. Ould Agha, and S. Guenneau, "Leaky modes in twisted microstructured optical fibres," Waves Random Media,  17:4, 559-570 (2007).

Other

R. Petit, Electromagnetic Theory of Gratings, (Springer Verlag, 1980).

F. Zolla, G. Renversez, A. Nicolet, B. Khulmey, S. Guenneau, D. Felbacq, Foundations of Photonic Crystal Fibres (Imperial College Press, London, 2005).
[CrossRef]

G. Tayeb, "Contributionàlétude de la diffraction des ondes électromagn étiques par des r éseaux. R éflexions sur les méthodes existantes et sur leur extension aux milieux anisotropes,’Université Paul Cézanne, PhD Thesis’ 1990.

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Figures (5)

Fig. 1.
Fig. 1.

Schematics of the gratings studied in this paper.

Fig. 2.
Fig. 2.

PML adapted to substratum and superstratum (TM case). Note (Fig. 2d) that the radiated field on each extreme boundary of the PML is at least 10-8 weaker than in the region of interest.

Fig. 3.
Fig. 3.

Rectangular groove grating : This pattern is repeatedly set up with a period d=1 µm. This grating has been studied by [23] and is one of our points of reference.

Fig. 4.
Fig. 4.

Diffractive element pattern. This element is made of aragonite for which the dielectric tensor is given by Eq.(41) and is deposited on a silica substrate (d=600nm).

Fig. 5.
Fig. 5.

Real part of the total calculated field depending on θ0 (TM and TE cases).

Tables (2)

Tables Icon

Table 1. Reflected efficiencies versus mesh refinement. Note that the efficiencies are properly computed (two significant digits) even for a rather coarse mesh.

Tables Icon

Table 2. Transmitted and reflected efficiencies of propagative orders deduced from field maps shown Fig.5

Equations (41)

Equations on this page are rendered with MathJax. Learn more.

ε ¯ ¯ = ( ε xx ε ¯ a 0 ε a ε yy 0 0 0 ε zz ) and μ ¯ ¯ = ( μ xx μ ¯ a 0 μ a μ yy 0 0 0 μ zz ) ,
E e 0 = A e 0 exp ( i k + · r ) z ( resp . H m 0 = A m 0 exp ( i k + · r ) z ) ,
{ curl E = i ω μ 0 μ ¯ ¯ H curl H = i ω ε 0 ε ¯ ¯ E
δ ˜ ¯ ¯ = ( δ xx δ ¯ a δ a δ y ) .
curl ( δ ¯ ¯ 1 curl ( u z ) ) = div ( δ ˜ ¯ ¯ T det ( δ ˜ ¯ ¯ ) u ) z ,
L ξ ¯ ¯ , χ ( u ) div ( ξ ¯ ¯ u ) + k 0 2 χ u = 0
u = e , ξ ¯ ¯ = μ ˜ ¯ ¯ T det ( μ ˜ ¯ ¯ ) , χ = ε zz ,
u = h , ξ ¯ ¯ = ε ˜ ¯ ¯ T det ( ε ˜ ¯ ¯ ) , χ = μ zz ,
ξ ¯ ¯ + = 1 μ + Id 2 and χ + = ε + in TE case
ξ ¯ ¯ + = 1 ε + Id 2 and χ + = μ + in TM case ,
ξ ¯ ¯ ( x , y ) { ξ ¯ ¯ + for y > H ξ ¯ ¯ g ( x , y ) for H > y > 0 ξ ¯ ¯ for y < 0 , χ ( x , y ) { χ + for y > H χ g ( x , y ) for H > y > 0 χ for y < 0 .
ξ ¯ ¯ ¯ 1 ( x , y ) { ξ ¯ ¯ + for y > 0 ξ ¯ ¯ for y < 0 , χ 1 ( x , y ) { χ + for y > 0 χ for y < 0 ,
u 0 ( x , y ) { u inc for y > H 0 for y < H
L ξ ¯ ¯ , χ ( u ) = 0 such that u d u u 0 satisfies an O.W.C.
L ξ ¯ ¯ 1 , χ 1 ( u 1 ) = 0 such that u 1 d u 1 u 0 satisfies an O.W.C.
u 2 d 𢉔 u u 1 = u d u 1 d .
L ξ ¯ ¯ , χ ( u 2 d ) = L ξ ¯ ¯ , χ ( u 1 ) ,
S 1 L ξ ¯ ¯ , χ ( u 1 ) = L ξ ¯ ¯ , χ ( u 1 ) L ξ ¯ ¯ 1 , χ 1 ( u 1 ) = 0 = L ξ ¯ ¯ ξ ¯ ¯ 1 , χ χ 1 ( u 1 ) .
S 1 = S 1 0 + S 1 d ,
S 1 0 = L ξ ¯ ¯ ξ ¯ ¯ 1 , χ χ 1 ( u 0 )
S 1 d = L ξ ¯ ¯ ξ ¯ ¯ 1 , χ χ 1 ( u 1 d ) .
S 1 0 = { i div [ ( ξ ¯ ¯ + ξ ¯ ¯ ) k + exp ( i k + · r ) ] + k 0 2 ( χ + χ ) exp ( i k + · r ) } .
S 1 d = ρ { i div [ ( ξ ¯ ¯ + ξ ¯ ¯ ) k exp ( i k · r ) ] + k 0 2 ( χ + χ ) exp ( i k · r ) } ,
ρ = p + p p + + p with p ± = { β ± in the TM case β ± ε ± in the TE case
R ξ ¯ ¯ , χ ( u , u ) = Ω ( ξ ¯ ¯ u ) · u ¯ + k 0 2 χ u u ¯ d Ω + Ω u ¯ ( ξ ¯ ¯ u ) · n d S
R ξ ¯ ¯ , χ ( u , u ) = 0 u L 2 ( curl , d , k ) .
δ s ¯ ¯ J s 1 δ ¯ ¯ J s T det ( J s ) for δ = { ε , μ } ,
δ s ¯ ¯ = ( s y δ xx δ d ¯ 0 δ d s y 1 δ yy 0 0 0 s y δ zz ) .
u n sc ( x c , y c ) u n ( x ( x c ) , y ( y c ) ) = e i α x c e i β + , n ( Y * + ζ ( y c Y * ) )
R ξ ¯ ¯ s + , χ s + ( u 2 d , u ) = 0 ,
R ξ ¯ ¯ + , χ + ( u 2 d , u ) = 0 ,
R ξ ¯ ¯ g , χ g ( u 2 d , u ' ) = R ξ ¯ ¯ g , χ g ( S 1 , u ' ) ,
R ξ ¯ ¯ , χ ( u 2 d , u ) = 0 ,
R ξ ¯ ¯ s , χ s ( u 2 d , u ) = 0 ,
for y < 0 and y > H , u d ( x , y ) = n u n d ( y ) e i α n x
u n d ( y ) = 1 d d 2 d 2 u d ( x , y ) e i α n x dx with α n = α + 2 π d n
u n d ( y ) = { s n e i β n + y + r n e i β n + y for y > H u n e i β n y + t n e i β n y for y < 0 with β n ± 2 = k ± 2 α n 2
{ r n = 1 d d 2 d 2 u d ( x , y 0 ) e i ( α n x + β n + y 0 ) dx for y 0 > H t n = 1 d d 2 d 2 u d ( x , y 0 ) e i ( α n x β n y 0 ) dx for y 0 < 0
{ R n r n r n ¯ β n + β + for y 0 > H T n t n t n ¯ β n β γ + γ for y 0 < 0 with γ ± = { 1 in the TM case ε ± in the TE case
ε ¯ ¯ Ca CO 3 = ( 2.843 0 0 0 2.341 0 0 0 2.829 ) and μ ¯ ¯ Ca CO 3 = ( μ 0 0 0 0 μ 0 0 0 0 μ 0 )
ε ¯ ¯ Ca CO 3 + 45 ° = ( 2.592 0.251 0 0.251 2.592 0 0 0 2.829 )

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