Abstract

Several improvements have been introduced for the Fourier modal method in the last fifteen years. Among those, the formulation of the correct factorization rules and adaptive spatial resolution have been crucial steps towards a fast converging scheme, but an application to arbitrary two-dimensional shapes is quite complicated. We present a generalization of the scheme for non-trivial planar geometries using a covariant formulation of Maxwell’s equations and a matched coordinate system aligned along the interfaces of the structure that can be easily combined with adaptive spatial resolution. In addition, a symmetric application of Fourier factorization is discussed.

© 2009 Optical Society of America

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References

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  1. D. M. Whittaker and I. S. Culshaw, "Scattering Matrix treatment of patterned multilayer photonic structures," Phys. Rev. B 60, 2610-2618 (1999).
    [CrossRef]
  2. S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, "Quasiguided modes and optical properties of photonic crystal slabs," Phys. Rev. B66,  45,102-1-17 (2002).
    [CrossRef]
  3. L. Li, "Use of Fourier series in the analysis of discontinuous periodic structures," J. Opt. Soc. Am. A 13, 1870-1876 (1996).
    [CrossRef]
  4. G. Granet, "Reformulation of the lamellar grating problem through the concept of adaptive spatial resolution," J. Opt. Soc. Am. A 16, 2510-2516 (1999).
    [CrossRef]
  5. G. Granet and J. P. Plumey, "Parametric formulation of the Fourier modal method for crossed surface-relief gratings," J. Opt. A 4, 145-149 (2002).
    [CrossRef]
  6. P. Lalanne, "Improved formulation of the coupled-wave method for two-dimensional gratings," J. Opt. Soc. Am. A 14, 1592-1598 (1997).
    [CrossRef]
  7. E. Popov and M. Nevière, "Maxwell equations in Fourier space: fast-converging formulation for diffraction by arbitrary shaped, periodic, anisotropic media," J. Opt. Soc. Am. A 18, 2886-2894 (2001).
    [CrossRef]
  8. T. Schuster, J. Ruoff, N. Kerwien, S. Rafler, and W. Osten, "Normal vector method for convergence improvement using the RCWA for crossed gratings," J. Opt. Soc. Am. A 24, 2880-2890 (2007).
    [CrossRef]
  9. P. G¨otz, T. Schuster, K. Frenner, S. Rafler, and W. Osten, "Normal vector method for the RCWA with automated vector field generation," Opt. Ex. 16, 17,295-17,301 (2008).
  10. J. H. Heinbockel, Introduction to Tensor Calculus and Continuum Mechanics (Trafford Publishing, 2001).
  11. E. J. Post, Formal structure of Electromagnetism (North Holland, Amsterdam, 1962).
  12. L. Li, "New formulation of the Fourier modal method for crossed surface-relief gratings," J. Opt. Soc. Am. A 14, 2758-2767 (1997).
    [CrossRef]
  13. L. Li, "Fourier modal method for crossed anisotropic gratings with arbitrary permittivity and permeability tensors," J. Opt. A 5, 345-355 (2003).
    [CrossRef]
  14. J. D. Jackson, Classical Electrodynamics, 3rd ed. (John Wiley and Sons, 1999).

2008

P. G¨otz, T. Schuster, K. Frenner, S. Rafler, and W. Osten, "Normal vector method for the RCWA with automated vector field generation," Opt. Ex. 16, 17,295-17,301 (2008).

2007

2003

L. Li, "Fourier modal method for crossed anisotropic gratings with arbitrary permittivity and permeability tensors," J. Opt. A 5, 345-355 (2003).
[CrossRef]

2002

G. Granet and J. P. Plumey, "Parametric formulation of the Fourier modal method for crossed surface-relief gratings," J. Opt. A 4, 145-149 (2002).
[CrossRef]

2001

1999

D. M. Whittaker and I. S. Culshaw, "Scattering Matrix treatment of patterned multilayer photonic structures," Phys. Rev. B 60, 2610-2618 (1999).
[CrossRef]

G. Granet, "Reformulation of the lamellar grating problem through the concept of adaptive spatial resolution," J. Opt. Soc. Am. A 16, 2510-2516 (1999).
[CrossRef]

1997

1996

Culshaw, I. S.

D. M. Whittaker and I. S. Culshaw, "Scattering Matrix treatment of patterned multilayer photonic structures," Phys. Rev. B 60, 2610-2618 (1999).
[CrossRef]

Frenner, K.

P. G¨otz, T. Schuster, K. Frenner, S. Rafler, and W. Osten, "Normal vector method for the RCWA with automated vector field generation," Opt. Ex. 16, 17,295-17,301 (2008).

G¨otz, P.

P. G¨otz, T. Schuster, K. Frenner, S. Rafler, and W. Osten, "Normal vector method for the RCWA with automated vector field generation," Opt. Ex. 16, 17,295-17,301 (2008).

Granet, G.

G. Granet and J. P. Plumey, "Parametric formulation of the Fourier modal method for crossed surface-relief gratings," J. Opt. A 4, 145-149 (2002).
[CrossRef]

G. Granet, "Reformulation of the lamellar grating problem through the concept of adaptive spatial resolution," J. Opt. Soc. Am. A 16, 2510-2516 (1999).
[CrossRef]

Kerwien, N.

Lalanne, P.

Li, L.

Nevière, M.

Osten, W.

P. G¨otz, T. Schuster, K. Frenner, S. Rafler, and W. Osten, "Normal vector method for the RCWA with automated vector field generation," Opt. Ex. 16, 17,295-17,301 (2008).

T. Schuster, J. Ruoff, N. Kerwien, S. Rafler, and W. Osten, "Normal vector method for convergence improvement using the RCWA for crossed gratings," J. Opt. Soc. Am. A 24, 2880-2890 (2007).
[CrossRef]

Plumey, J. P.

G. Granet and J. P. Plumey, "Parametric formulation of the Fourier modal method for crossed surface-relief gratings," J. Opt. A 4, 145-149 (2002).
[CrossRef]

Popov, E.

Rafler, S.

P. G¨otz, T. Schuster, K. Frenner, S. Rafler, and W. Osten, "Normal vector method for the RCWA with automated vector field generation," Opt. Ex. 16, 17,295-17,301 (2008).

T. Schuster, J. Ruoff, N. Kerwien, S. Rafler, and W. Osten, "Normal vector method for convergence improvement using the RCWA for crossed gratings," J. Opt. Soc. Am. A 24, 2880-2890 (2007).
[CrossRef]

Ruoff, J.

Schuster, T.

P. G¨otz, T. Schuster, K. Frenner, S. Rafler, and W. Osten, "Normal vector method for the RCWA with automated vector field generation," Opt. Ex. 16, 17,295-17,301 (2008).

T. Schuster, J. Ruoff, N. Kerwien, S. Rafler, and W. Osten, "Normal vector method for convergence improvement using the RCWA for crossed gratings," J. Opt. Soc. Am. A 24, 2880-2890 (2007).
[CrossRef]

Whittaker, D. M.

D. M. Whittaker and I. S. Culshaw, "Scattering Matrix treatment of patterned multilayer photonic structures," Phys. Rev. B 60, 2610-2618 (1999).
[CrossRef]

J. Opt. A

G. Granet and J. P. Plumey, "Parametric formulation of the Fourier modal method for crossed surface-relief gratings," J. Opt. A 4, 145-149 (2002).
[CrossRef]

L. Li, "Fourier modal method for crossed anisotropic gratings with arbitrary permittivity and permeability tensors," J. Opt. A 5, 345-355 (2003).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Ex.

P. G¨otz, T. Schuster, K. Frenner, S. Rafler, and W. Osten, "Normal vector method for the RCWA with automated vector field generation," Opt. Ex. 16, 17,295-17,301 (2008).

Phys. Rev. B

D. M. Whittaker and I. S. Culshaw, "Scattering Matrix treatment of patterned multilayer photonic structures," Phys. Rev. B 60, 2610-2618 (1999).
[CrossRef]

Other

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, "Quasiguided modes and optical properties of photonic crystal slabs," Phys. Rev. B66,  45,102-1-17 (2002).
[CrossRef]

J. D. Jackson, Classical Electrodynamics, 3rd ed. (John Wiley and Sons, 1999).

J. H. Heinbockel, Introduction to Tensor Calculus and Continuum Mechanics (Trafford Publishing, 2001).

E. J. Post, Formal structure of Electromagnetism (North Holland, Amsterdam, 1962).

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Figures (5)

Fig. 1. (a)
Fig. 1. (a)

Schematic drawing of a standard structure and definition of a Cartesian coordinate system including the incidence parameters. (b) Sketch of lines with constant x̄1 and x̄2 in a plane normal to x̄3 = x 3 with the directions of the reciprocal bases.

Fig. 2.
Fig. 2.

(a) Matched coordinates including adaptive spatial resolution for a cylinder. (b) Comparison of the effective propagation constant k 3 with the calculated solution k calc of a dielectric cylinder. The scheme of Li is indicated by the small black markers.

Fig. 3.
Fig. 3.

(a) Absorption accuracy of a dielectric cylinder to show energy conservation using the symmetry-preserving formulas (white markers) and Li (black markers). (b) Differences in absorption accuracy A and the effective propagation constant k 3 between the scheme by Li and our formulation.

Fig. 4.
Fig. 4.

(a) Spectral behavior (transmission T, reflection R, and absorption A) of an array of metallic cylinders (see the text for a detailed description of parameters) for 625 harmonics. (b) Convergence of transmission intensity at 370 THz with matched coordinates (squares) and additionally adaptive spatial resolution (diamonds).

Fig. 5.
Fig. 5.

(a) Difference in transmission between the scheme by Li and our formulation. (b) Spurious transmission asymmetry between two independent polarizations P and S for the totally symmetric cylinder at the resonance.

Equations (45)

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r(x1,x2,x3)=xm em , m=1,2,3 .
r(x̅1,x̅2,x̅3)=xm (x̅1,x̅2,x̅3) em .
e̅m=rx̅m,e̅m=x̅m.
A=Am e̅m =Ame̅m.
Am=(e̅m·e̅n)An=pxpx̅mxpx̅nAn=gmnAn,
Am=(e̅m·e̅n)An=px̅mxpx̅nxpAn=gmnAn.
ξmnpnEp=ik0μ̅mn Hn ,
ξmnpnEp=ik0ε̅mn En ,
ε̅mn=g Λpm Λqn εpq ,
μ̅mn=g Λpm Λqn μpq ,
Λnm=x̅mxn .
ε==(ε11ε120ε21ε22000ε33),μ==(μ11μ120μ21μ22000μ33),
k0232(E1E2)=𝓛EH 𝓛HE (E1E2) ,
𝓛EH=(1(ε33)12+k02μ211(ε33)11+k02μ222(ε33)12+k02μ112(ε33)11+k02μ12),
𝓛HE=(1(μ33)12+k02ε211(μ33)11+k02ε222(μ33)12+k02ε112(μ33)11+k02ε12).
k1=ksin(θ)cos(ϕ)+2πP1j1,k2=ksin(θ)sin(ϕ)+2πP2j2,j1,2,
fm=ngmnhn.
[f]=1g1 [h] .
[f]=1g1[h1]+1g111g1g2[h2].
[D3]12=ε3312[E3]12.
D1=ε11E1+ε12E2 ,
D2=ε21E1+ε22E2 ,
E1=Δε1 (ε22D1ε12D2) ,
E2=Δε1 (ε21D1+ε11D2) ,
[D2]2=1ε22211ε22ε212[E1]2+1ε2221[E2]2.
[D1]2=1ε22Δε2[E1]2+[1ε22ε12]2[D2]2.
[D1]2=ε112[E1]2+ε122[E2]2,
[D2]1=ε211[E1]1+ε221[E2]1,
ε112=1ε22Δε2+1ε22ε1221ε22211ε22ε212,
ε122=1ε22ε1221ε2221,
ε211=1ε11ε2111ε1111,
ε221=1ε11Δε1+1ε11ε2111ε11111ε11ε121.
ε1121=ε112111,
ε1221=ε112111ε1121ε1221,
ε2112=ε221121ε2211ε2112,
ε2212=ε221121.
x±m(x˜nm,x˜±m)={P2±R2(x˜nP2)2forx˜nx˜nx˜+nx˜±melsewhere
xm(x˜nm,x˜m)={x˜mx˜mxmfor0x˜m<x˜mx˜mx˜mx˜+mx˜mx+m(x˜n)+x˜mx˜+mx˜mx˜+mxm(x˜n)x˜mPx˜+mPx+m(x˜n)+x˜mx˜+mPx˜+mPforx˜+m<x˜mPforx˜mx˜mx˜+m.
ε̅==(x˜2x̅2(x˜1x̅1)1ε11ε120ε21x˜1x̅1(x˜2x̅2)1ε22000x˜1x1x˜2x̅2ε33).
ε̅1121ASR=x˜1x̅1(ε̅112ASR)111,
ε̅112ASR = x˜2x̅21ε22Δε2+x˜2x̅21ε22ε122 x˜2x̅21ε2221 x˜2x̅21ε22ε212
x˜m(x̅m)={a1(x̅m)2+ax̅mfor0x̅m<x̅ma2+a3x̅m+a4sin2π(x̅mx̅m)x̅+mx̅+mforx̅mx̅mx̅+m,a1(x̅m)2a+x̅m+a5forx̅+m<x̅mP
a±=2x˜m(x̅m)2x̅±m(1η)P±x̅+mx̅m,a1=x˜m(x̅m)21ηx̅m,
a2=x̅+mx˜mx̅mx˜+mx̅+mx̅m, a3=x˜+mx˜mx̅+mx̅m,
a4=(1η)(x̅+mx̅m)(x˜+mx˜m)2π , a5=x˜+m+x˜m(x̅m)2(x̅+m)2(1η) Px̅+mx̅m .

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