Abstract

We propose an Adaptive Perfectly Matched Layer (APML) to be used in diffraction grating modeling. With a properly tailored co-ordinate stretching depending both on the incident field and on grating parameters, the APML may efficiently absorb diffracted orders near grazing angles (the so-called Wood’s anomalies). The new design is implemented in a finite element method (FEM) scheme and applied on a numerical example of a dielectric slit grating. Its performances are compared with classical PML with constant stretching coefficient.

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  1. J.-P. Bérenger, “A perfectly matched layer for the absorption of electromagnetic waves,” Journal of Computational Physics114, 185 – 200 (1994).
    [CrossRef]
  2. M. Lassas, J. Liukkonen, and E. Somersalo, “Complex riemannian metric and absorbing boundary conditions,” J. Math. Pure Appl.80, 739 – 768 (2001).
  3. M. Lassas and E. Somersalo, “Analysis of the PML equations in general convex geometry,” P. Roy. Soc. Edinb. A131, 1183–1207 (2001).
    [CrossRef]
  4. A. Nicolet, F. Zolla, Y. Ould Agha, and S. Guenneau, “Geometrical transformations and equivalent materials in computational electromagnetism,” Compel27, 806–819 (2008).
    [CrossRef]
  5. F. Zolla, G. Renversez, A. Nicolet, B. Kuhlmey, S. Guenneau, D. Felbacq, A. Argyros, and S. Leon-Saval, Foundations of Photonic Crystal Fibres (Imperial College Press, London, 2012), 2nd ed.
  6. R. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” P. Phys. Soc. Lond.18, 269–275 (1902).
    [CrossRef]
  7. Lord Rayleigh, “Note on the remarkable case of diffraction spectra described by prof. Wood,” Philos. Mag.14, 60–65 (1907).
    [CrossRef]
  8. Z. Chen and X. Liu, “An adaptive perfectly matched layer technique for time-harmonic scattering problems,” SIAM J. Numer. Anal.43, 645–671 (2005).
    [CrossRef]
  9. G. Bao, Z. Chen, and H. Wu, “Adaptive finite-element method for diffraction gratings,” J. Opt. Soc. Am. A22, 1106–1114 (2005).
    [CrossRef]
  10. A. Schädle, L. Zschiedrich, S. Burger, R. Klose, and F. Schmidt, “Domain decomposition method for Maxwell’s equations: scattering off periodic structures,” J. Comput. Phys.226, 477 – 493 (2007).
    [CrossRef]
  11. A. Sommerfeld, Partial Differential Equations in Physics (Academic Press, New York, 1949).
  12. Y. Ould Agha, A. Nicolet, F. Zolla, and S. Guenneau, “Leaky modes in twisted microstructured optical fibres,” Wave Random Complex17, 559–570 (2007).
    [CrossRef]
  13. G. Demésy, F. Zolla, A. Nicolet, M. Commandré, and C. Fossati, “The finite element method as applied to the diffraction by an anisotropic grating,” Opt. Express15, 18089–18102 (2007).
    [CrossRef] [PubMed]
  14. G. Demésy, F. Zolla, A. Nicolet, and M. Commandré, “Versatile full-vectorial finite element model for crossed gratings,” Opt. Lett.34, 2216–2218 (2009).
    [CrossRef] [PubMed]
  15. A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements (Springer, New York, 2004).
  16. L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A14, 2758–2767, (1997).
    [CrossRef]

2009 (1)

2008 (1)

A. Nicolet, F. Zolla, Y. Ould Agha, and S. Guenneau, “Geometrical transformations and equivalent materials in computational electromagnetism,” Compel27, 806–819 (2008).
[CrossRef]

2007 (3)

A. Schädle, L. Zschiedrich, S. Burger, R. Klose, and F. Schmidt, “Domain decomposition method for Maxwell’s equations: scattering off periodic structures,” J. Comput. Phys.226, 477 – 493 (2007).
[CrossRef]

Y. Ould Agha, A. Nicolet, F. Zolla, and S. Guenneau, “Leaky modes in twisted microstructured optical fibres,” Wave Random Complex17, 559–570 (2007).
[CrossRef]

G. Demésy, F. Zolla, A. Nicolet, M. Commandré, and C. Fossati, “The finite element method as applied to the diffraction by an anisotropic grating,” Opt. Express15, 18089–18102 (2007).
[CrossRef] [PubMed]

2005 (2)

Z. Chen and X. Liu, “An adaptive perfectly matched layer technique for time-harmonic scattering problems,” SIAM J. Numer. Anal.43, 645–671 (2005).
[CrossRef]

G. Bao, Z. Chen, and H. Wu, “Adaptive finite-element method for diffraction gratings,” J. Opt. Soc. Am. A22, 1106–1114 (2005).
[CrossRef]

2001 (2)

M. Lassas, J. Liukkonen, and E. Somersalo, “Complex riemannian metric and absorbing boundary conditions,” J. Math. Pure Appl.80, 739 – 768 (2001).

M. Lassas and E. Somersalo, “Analysis of the PML equations in general convex geometry,” P. Roy. Soc. Edinb. A131, 1183–1207 (2001).
[CrossRef]

1997 (1)

1994 (1)

J.-P. Bérenger, “A perfectly matched layer for the absorption of electromagnetic waves,” Journal of Computational Physics114, 185 – 200 (1994).
[CrossRef]

1907 (1)

Lord Rayleigh, “Note on the remarkable case of diffraction spectra described by prof. Wood,” Philos. Mag.14, 60–65 (1907).
[CrossRef]

1902 (1)

R. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” P. Phys. Soc. Lond.18, 269–275 (1902).
[CrossRef]

Argyros, A.

F. Zolla, G. Renversez, A. Nicolet, B. Kuhlmey, S. Guenneau, D. Felbacq, A. Argyros, and S. Leon-Saval, Foundations of Photonic Crystal Fibres (Imperial College Press, London, 2012), 2nd ed.

Bao, G.

Bérenger, J.-P.

J.-P. Bérenger, “A perfectly matched layer for the absorption of electromagnetic waves,” Journal of Computational Physics114, 185 – 200 (1994).
[CrossRef]

Burger, S.

A. Schädle, L. Zschiedrich, S. Burger, R. Klose, and F. Schmidt, “Domain decomposition method for Maxwell’s equations: scattering off periodic structures,” J. Comput. Phys.226, 477 – 493 (2007).
[CrossRef]

Chen, Z.

G. Bao, Z. Chen, and H. Wu, “Adaptive finite-element method for diffraction gratings,” J. Opt. Soc. Am. A22, 1106–1114 (2005).
[CrossRef]

Z. Chen and X. Liu, “An adaptive perfectly matched layer technique for time-harmonic scattering problems,” SIAM J. Numer. Anal.43, 645–671 (2005).
[CrossRef]

Commandré, M.

Demésy, G.

Ern, A.

A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements (Springer, New York, 2004).

Felbacq, D.

F. Zolla, G. Renversez, A. Nicolet, B. Kuhlmey, S. Guenneau, D. Felbacq, A. Argyros, and S. Leon-Saval, Foundations of Photonic Crystal Fibres (Imperial College Press, London, 2012), 2nd ed.

Fossati, C.

Guenneau, S.

A. Nicolet, F. Zolla, Y. Ould Agha, and S. Guenneau, “Geometrical transformations and equivalent materials in computational electromagnetism,” Compel27, 806–819 (2008).
[CrossRef]

Y. Ould Agha, A. Nicolet, F. Zolla, and S. Guenneau, “Leaky modes in twisted microstructured optical fibres,” Wave Random Complex17, 559–570 (2007).
[CrossRef]

F. Zolla, G. Renversez, A. Nicolet, B. Kuhlmey, S. Guenneau, D. Felbacq, A. Argyros, and S. Leon-Saval, Foundations of Photonic Crystal Fibres (Imperial College Press, London, 2012), 2nd ed.

Guermond, J.-L.

A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements (Springer, New York, 2004).

Klose, R.

A. Schädle, L. Zschiedrich, S. Burger, R. Klose, and F. Schmidt, “Domain decomposition method for Maxwell’s equations: scattering off periodic structures,” J. Comput. Phys.226, 477 – 493 (2007).
[CrossRef]

Kuhlmey, B.

F. Zolla, G. Renversez, A. Nicolet, B. Kuhlmey, S. Guenneau, D. Felbacq, A. Argyros, and S. Leon-Saval, Foundations of Photonic Crystal Fibres (Imperial College Press, London, 2012), 2nd ed.

Lassas, M.

M. Lassas, J. Liukkonen, and E. Somersalo, “Complex riemannian metric and absorbing boundary conditions,” J. Math. Pure Appl.80, 739 – 768 (2001).

M. Lassas and E. Somersalo, “Analysis of the PML equations in general convex geometry,” P. Roy. Soc. Edinb. A131, 1183–1207 (2001).
[CrossRef]

Leon-Saval, S.

F. Zolla, G. Renversez, A. Nicolet, B. Kuhlmey, S. Guenneau, D. Felbacq, A. Argyros, and S. Leon-Saval, Foundations of Photonic Crystal Fibres (Imperial College Press, London, 2012), 2nd ed.

Li, L.

Liu, X.

Z. Chen and X. Liu, “An adaptive perfectly matched layer technique for time-harmonic scattering problems,” SIAM J. Numer. Anal.43, 645–671 (2005).
[CrossRef]

Liukkonen, J.

M. Lassas, J. Liukkonen, and E. Somersalo, “Complex riemannian metric and absorbing boundary conditions,” J. Math. Pure Appl.80, 739 – 768 (2001).

Nicolet, A.

G. Demésy, F. Zolla, A. Nicolet, and M. Commandré, “Versatile full-vectorial finite element model for crossed gratings,” Opt. Lett.34, 2216–2218 (2009).
[CrossRef] [PubMed]

A. Nicolet, F. Zolla, Y. Ould Agha, and S. Guenneau, “Geometrical transformations and equivalent materials in computational electromagnetism,” Compel27, 806–819 (2008).
[CrossRef]

Y. Ould Agha, A. Nicolet, F. Zolla, and S. Guenneau, “Leaky modes in twisted microstructured optical fibres,” Wave Random Complex17, 559–570 (2007).
[CrossRef]

G. Demésy, F. Zolla, A. Nicolet, M. Commandré, and C. Fossati, “The finite element method as applied to the diffraction by an anisotropic grating,” Opt. Express15, 18089–18102 (2007).
[CrossRef] [PubMed]

F. Zolla, G. Renversez, A. Nicolet, B. Kuhlmey, S. Guenneau, D. Felbacq, A. Argyros, and S. Leon-Saval, Foundations of Photonic Crystal Fibres (Imperial College Press, London, 2012), 2nd ed.

Ould Agha, Y.

A. Nicolet, F. Zolla, Y. Ould Agha, and S. Guenneau, “Geometrical transformations and equivalent materials in computational electromagnetism,” Compel27, 806–819 (2008).
[CrossRef]

Y. Ould Agha, A. Nicolet, F. Zolla, and S. Guenneau, “Leaky modes in twisted microstructured optical fibres,” Wave Random Complex17, 559–570 (2007).
[CrossRef]

Rayleigh, Lord

Lord Rayleigh, “Note on the remarkable case of diffraction spectra described by prof. Wood,” Philos. Mag.14, 60–65 (1907).
[CrossRef]

Renversez, G.

F. Zolla, G. Renversez, A. Nicolet, B. Kuhlmey, S. Guenneau, D. Felbacq, A. Argyros, and S. Leon-Saval, Foundations of Photonic Crystal Fibres (Imperial College Press, London, 2012), 2nd ed.

Schädle, A.

A. Schädle, L. Zschiedrich, S. Burger, R. Klose, and F. Schmidt, “Domain decomposition method for Maxwell’s equations: scattering off periodic structures,” J. Comput. Phys.226, 477 – 493 (2007).
[CrossRef]

Schmidt, F.

A. Schädle, L. Zschiedrich, S. Burger, R. Klose, and F. Schmidt, “Domain decomposition method for Maxwell’s equations: scattering off periodic structures,” J. Comput. Phys.226, 477 – 493 (2007).
[CrossRef]

Somersalo, E.

M. Lassas and E. Somersalo, “Analysis of the PML equations in general convex geometry,” P. Roy. Soc. Edinb. A131, 1183–1207 (2001).
[CrossRef]

M. Lassas, J. Liukkonen, and E. Somersalo, “Complex riemannian metric and absorbing boundary conditions,” J. Math. Pure Appl.80, 739 – 768 (2001).

Sommerfeld, A.

A. Sommerfeld, Partial Differential Equations in Physics (Academic Press, New York, 1949).

Wood, R.

R. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” P. Phys. Soc. Lond.18, 269–275 (1902).
[CrossRef]

Wu, H.

Zolla, F.

G. Demésy, F. Zolla, A. Nicolet, and M. Commandré, “Versatile full-vectorial finite element model for crossed gratings,” Opt. Lett.34, 2216–2218 (2009).
[CrossRef] [PubMed]

A. Nicolet, F. Zolla, Y. Ould Agha, and S. Guenneau, “Geometrical transformations and equivalent materials in computational electromagnetism,” Compel27, 806–819 (2008).
[CrossRef]

Y. Ould Agha, A. Nicolet, F. Zolla, and S. Guenneau, “Leaky modes in twisted microstructured optical fibres,” Wave Random Complex17, 559–570 (2007).
[CrossRef]

G. Demésy, F. Zolla, A. Nicolet, M. Commandré, and C. Fossati, “The finite element method as applied to the diffraction by an anisotropic grating,” Opt. Express15, 18089–18102 (2007).
[CrossRef] [PubMed]

F. Zolla, G. Renversez, A. Nicolet, B. Kuhlmey, S. Guenneau, D. Felbacq, A. Argyros, and S. Leon-Saval, Foundations of Photonic Crystal Fibres (Imperial College Press, London, 2012), 2nd ed.

Zschiedrich, L.

A. Schädle, L. Zschiedrich, S. Burger, R. Klose, and F. Schmidt, “Domain decomposition method for Maxwell’s equations: scattering off periodic structures,” J. Comput. Phys.226, 477 – 493 (2007).
[CrossRef]

Compel (1)

A. Nicolet, F. Zolla, Y. Ould Agha, and S. Guenneau, “Geometrical transformations and equivalent materials in computational electromagnetism,” Compel27, 806–819 (2008).
[CrossRef]

J. Comput. Phys. (1)

A. Schädle, L. Zschiedrich, S. Burger, R. Klose, and F. Schmidt, “Domain decomposition method for Maxwell’s equations: scattering off periodic structures,” J. Comput. Phys.226, 477 – 493 (2007).
[CrossRef]

J. Math. Pure Appl. (1)

M. Lassas, J. Liukkonen, and E. Somersalo, “Complex riemannian metric and absorbing boundary conditions,” J. Math. Pure Appl.80, 739 – 768 (2001).

J. Opt. Soc. Am. A (2)

Journal of Computational Physics (1)

J.-P. Bérenger, “A perfectly matched layer for the absorption of electromagnetic waves,” Journal of Computational Physics114, 185 – 200 (1994).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

P. Phys. Soc. Lond. (1)

R. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” P. Phys. Soc. Lond.18, 269–275 (1902).
[CrossRef]

P. Roy. Soc. Edinb. A (1)

M. Lassas and E. Somersalo, “Analysis of the PML equations in general convex geometry,” P. Roy. Soc. Edinb. A131, 1183–1207 (2001).
[CrossRef]

Philos. Mag. (1)

Lord Rayleigh, “Note on the remarkable case of diffraction spectra described by prof. Wood,” Philos. Mag.14, 60–65 (1907).
[CrossRef]

SIAM J. Numer. Anal. (1)

Z. Chen and X. Liu, “An adaptive perfectly matched layer technique for time-harmonic scattering problems,” SIAM J. Numer. Anal.43, 645–671 (2005).
[CrossRef]

Wave Random Complex (1)

Y. Ould Agha, A. Nicolet, F. Zolla, and S. Guenneau, “Leaky modes in twisted microstructured optical fibres,” Wave Random Complex17, 559–570 (2007).
[CrossRef]

Other (3)

A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements (Springer, New York, 2004).

A. Sommerfeld, Partial Differential Equations in Physics (Academic Press, New York, 1949).

F. Zolla, G. Renversez, A. Nicolet, B. Kuhlmey, S. Guenneau, D. Felbacq, A. Argyros, and S. Leon-Saval, Foundations of Photonic Crystal Fibres (Imperial College Press, London, 2012), 2nd ed.

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Figures (7)

Fig. 1
Fig. 1

Set up of the problem and notations

Fig. 2
Fig. 2

The basic cell used for the FEM computation of the diffracted field u 2 d.

Fig. 3
Fig. 3

Zeroth transmitted order by a grating with a rectangular cross section (see parameters in text, part 2.4) for different values of ζ″,− : blue line, ζ″,− = 1, correct damping; green line, ζ″,− = 0.1, underdamping; red line, ζ″,− = 20, overdamping.

Fig. 4
Fig. 4

Example of a co-ordinate mapping (yd) used for the APML (black solid line). The graph of yd(y) (blue solid line) is continued by a straight line t0(yd) tangent at y d 0 (red dashed line) to avoid the singular behaviour at y d = y d .

Fig. 5
Fig. 5

Field maps of the logarithm of the norm of Hz, Ex and Ey for the dielectric slit grating at λ 0 = 0.999 y d , + (same parameters as in part 2.4). (a) : classical PML with inefficient damping of Hz in the bottom PML. (b) : APML where the Hz field is correctly damped in the bottom PML. For both cases the thickness of the PML is h ^ = 1.1 y d , + .

Fig. 6
Fig. 6

Modulus of the un for the three propagating orders with adapted (dashed lines) and classical PMLs (solid lines). Note that the classical PMLs are efficient for all orders except for the grazing one (n = 1) as expected. This drawback is bypassed when using the adaptive PML.

Fig. 7
Fig. 7

Mean value of the norm of Hz along the outer boundary of the bottom PML γ = 〈|Hz(−ĥ)|〉x, for λ0 approaching the Wood’s anomaly y d , + by inferior values ( λ 0 = ( 1 10 n ) y d , + , red squares) and by superior value ( λ 0 = ( 1 + 10 n ) y d , + , blue circles) as a function of n.

Tables (1)

Tables Icon

Table 1 Diffraction efficiencies R0, T−1, T0 and T+1 of the four propagating orders, and energy balance B = R0 + T−1 + T0 + T+1, computed by three methods : RCWA (line 1), FEM formulation with APML (line 2), FEM formulation with classical PML (line 3).

Equations (27)

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ε = = ( ε x x ε a ¯ 0 ε a ε y y 0 0 0 ε z z ) and μ = = ( μ x x μ a ¯ 0 μ a μ y y 0 0 0 μ z z ) ,
E 0 = A e 0 e ( i k + r ) z ( resp . H 0 = A m 0 e ( i k + r ) z )
{ curl E = i ω μ 0 μ = H curl H = i ω ε 0 ε = E ,
ε ˜ = = ( ε x x ε a ¯ ε a ε y y ) and μ ˜ = = ( μ x x μ a ¯ ¯ μ a μ y y ) ,
ξ = , χ ( u ) : = div ( ξ = grad u ) + k 0 2 χ u = 0 ,
u = e , ξ = = μ ˜ = T / det ( μ ˜ = ) , χ = ε z z ,
u = h , ξ = = ε ˜ = T / det ( ε ˜ = ) , χ = μ z z ,
ε = s = J 1 ε = J T det ( J ) and μ = s = J 1 μ = J T det ( J ) ,
y s ( y ) = 0 y s y ( y ) d y .
ε = s = ( s y ε x x ε a ¯ 0 ε a s y 1 ε y y 0 0 0 s y ε z z ) and μ = s = ( s y μ x x μ a ¯ 0 μ a s y 1 μ y y 0 0 0 s y μ z z ) .
s y ( y ) = { ζ if y < y b 1 if y b < y < y t ζ + if y > y t
y ( y c ) = { y b + ζ ( y c y b ) if y c < y b y c if y b < y c < y t , y t + ζ + ( y c y t ) if y c > y t
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 d x , with α n = α + 2 π d n .
β n ± 2 = k ± 2 α n ± 2
u n d ( y ) = { u n + ( y ) = r n e i β n + y for y > h g u n ( y ) = t n e i β n y for y < 0 .
u n , s ( y c ) = u n ( y ( y c ) ) = t n e i β n [ y t + ζ ( y c y t ) ] .
U n ( y ) = t n exp ( ( β n , ζ , + β n , ζ , ) y c ) ,
U n ( y l n ) = U n ( y ) e .
l = max n l n .
α ( y d ) = α 0 + 2 π d y d λ 0 ,
exp [ i β ( y d ) y ( y d ) ] = exp ( i k 0 y d )
y ( y d ) = k 0 y d β ( y d ) = y d ε ( sin θ 0 + y d / d ) 2
{ n + / D + = min n + | y d , + n + λ 0 | n / D = min n | y d , + n λ 0 | ,
n / D = min n { n + , n } ( D + , D ) ,
y ˜ ( y d ) = { y ( y d ) for y d y d 0 t 0 ( y d ) for y d > y d 0 .
s y ( y d ) = ζ y ˜ y d ( y d ) .

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