Abstract

The Fourier modal method (FMM) has advanced greatly by using adaptive coordinates and adaptive spatial resolution. The convergence characteristics were shown to be improved significantly, a construction principle for suitable meshes was demonstrated and a guideline for the optimal choice of the coordinate transformation parameters was found. However, the construction guidelines published so far rely on a certain restriction that is overcome with the formulation presented in this paper. Moreover, a modularization principle is formulated that significantly eases the construction of coordinate transformations in unit cells with reappearing shapes and complex sub-structures.

© 2014 Optical Society of America

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References

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  1. K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, M. Wegener, “Periodic nanostructures for photonics,” Phys. Rep. 444, 101–202 (2007).
    [CrossRef]
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  6. 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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  7. G. Granet, J.-P. Plumey, “Parametric formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. A 4, S145–S149 (2002).
    [CrossRef]
  8. T. Vallius, M. Honkanen, “Reformulation of the Fourier modal method with adaptive spatial resolution: application to multilevel profiles,” Opt. Express 10, 24–34 (2002).
    [CrossRef] [PubMed]
  9. T. Weiss, G. Granet, N. A. Gippius, S. G. Tikhodeev, H. Giessen, “Matched coordinates and adaptive spatial resolution in the Fourier modal method,” Opt. Express 17, 8051–8061 (2009).
    [CrossRef] [PubMed]
  10. S. Essig, K. Busch, “Generation of adaptive coordinates and their use in the Fourier Modal Method,” Opt. Express 18, 23258–23274 (2010).
    [CrossRef] [PubMed]
  11. J. Küchenmeister, T. Zebrowski, K. Busch, “A construction guide to analytically generated meshes for the Fourier Modal Method,” Opt. Express 20, 17319–17347 (2012).
    [CrossRef] [PubMed]
  12. J. Küchenmeister, “Three-dimensional adaptive coordinate transformations for the Fourier modal method,” Opt. Express 22, 1342–1349 (2014).
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  13. L. Li, “Fourier modal method for crossed anisotropic gratings with arbitrary permittivity and permeability tensors,” J. Opt. A 5, 345–355 (2003).
    [CrossRef]

2014 (1)

2012 (1)

2010 (1)

2009 (1)

2007 (1)

K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, M. Wegener, “Periodic nanostructures for photonics,” Phys. Rep. 444, 101–202 (2007).
[CrossRef]

2003 (1)

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

2002 (2)

G. Granet, J.-P. Plumey, “Parametric formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. A 4, S145–S149 (2002).
[CrossRef]

T. Vallius, M. Honkanen, “Reformulation of the Fourier modal method with adaptive spatial resolution: application to multilevel profiles,” Opt. Express 10, 24–34 (2002).
[CrossRef] [PubMed]

1999 (1)

1996 (4)

Busch, K.

Essig, S.

Giessen, H.

Gippius, N. A.

Granet, G.

Guizal, B.

Honkanen, M.

Küchenmeister, J.

Lalanne, P.

Li, L.

Linden, S.

K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, M. Wegener, “Periodic nanostructures for photonics,” Phys. Rep. 444, 101–202 (2007).
[CrossRef]

Mingaleev, S. F.

K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, M. Wegener, “Periodic nanostructures for photonics,” Phys. Rep. 444, 101–202 (2007).
[CrossRef]

Morris, G. M.

Plumey, J.-P.

G. Granet, J.-P. Plumey, “Parametric formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. A 4, S145–S149 (2002).
[CrossRef]

Tikhodeev, S. G.

Tkeshelashvili, L.

K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, M. Wegener, “Periodic nanostructures for photonics,” Phys. Rep. 444, 101–202 (2007).
[CrossRef]

Vallius, T.

von Freymann, G.

K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, M. Wegener, “Periodic nanostructures for photonics,” Phys. Rep. 444, 101–202 (2007).
[CrossRef]

Wegener, M.

K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, M. Wegener, “Periodic nanostructures for photonics,” Phys. Rep. 444, 101–202 (2007).
[CrossRef]

Weiss, T.

Zebrowski, T.

J. Opt. A (2)

G. Granet, J.-P. Plumey, “Parametric formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. A 4, S145–S149 (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 (5)

Opt. Express (5)

Phys. Rep. (1)

K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, M. Wegener, “Periodic nanostructures for photonics,” Phys. Rep. 444, 101–202 (2007).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Systems like a layer of rotated gold crosses cannot be simulated reliably with classical FMM. For that, adaptive coordinates and adaptive spatial resolution have shown to be a powerful tool to improve the representation of the structure. The efficient mesh construction for this type of system is the scope of this paper.

Fig. 2
Fig. 2

The characteristic coordinate lines in panel (a) (red and blue) are mapped onto the surface of the structure, see panel (b). The rest of the mesh in panel (c) is a linear transition between the characteristic coordinate lines and the unit cell edge. This mesh construction scheme was presented in [11] where the mathematical formulation contains the constraint that the characteristic points P, Q, R, S need to be mapped onto themselves. This constraint will be overcome in the following. The figures are taken from [11] and are slightly modified.

Fig. 3
Fig. 3

This figure demonstrates that a different choice of the characteristic points can be reasonable. Panels (a) and (b) show how to obtain a mesh for the structure in Fig. 1 where the characteristic points are not mapped onto themselves, i.e., P0Pφ. Using the formulation from Section 3.2, the mesh in panel (c) can be constructed. The sketches are taken from [12] where the construction principle presented in Section 3.2 was already applied.

Fig. 4
Fig. 4

Construction of the help functions. To map the zone 1 from Fig. 3 (shown in panel (a)) correctly, the help functions H1 and H2 need to be chosen such that they both combined yield the correct result. Especially, we have to make sure that the mapping procedure maps P0 on Pφ.

Fig. 5
Fig. 5

Similar to Fig. 4 the characteristic point is not mapped onto itself. In addition, the mapped coordinate lines are not linear any more. To construct the suitable mapping, the help functions need to be varied as discussed below. The resulting mesh for this zone is displayed in panel (b).

Fig. 6
Fig. 6

Generalized construction principle. The rectangular area in panel (a) is mapped in an arbitrary fashion in panel (b). The characteristic points are not mapped onto themselves and no further restriction is imposed. The connecting functions e, f, g, h are chosen freely as well. The mesh obtained with the generalized expression presented below is shown in panel (c).

Fig. 7
Fig. 7

The modularization principle can be used when a mesh for a structure with complex sub-structures needs to be found. To obtain it, the characteristic points and coordinate lines in panel (a) are chosen such that the area for the sub-structure is mapped onto itself, see area A in panel (b). The resulting preliminary mesh is depicted in panel (c). In the second step, the sub-structures are entered in this region, see Fig. 8.

Fig. 8
Fig. 8

In the second step for the modularity concept, the area A that was mapped onto itself, see Figs. 7(a) and 7(b), is replaced by a scaled transformation of a previously known structure, see panel (a). This method also allows for a number of sub-structures as shown in panel (b).

Equations (20)

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

x ¯ 1 = x ¯ 1 ( x 1 , x 2 ) ,
x ¯ 2 = x ¯ 2 ( x 1 , x 2 ) ,
x ¯ 3 = x 3 .
ξ ρ σ τ σ E τ = i k 0 g μ ρ σ H σ ,
ξ ρ σ τ σ H τ = i k 0 g ε ρ σ E σ .
g ρ σ = x ρ x ¯ τ x σ x ¯ τ ,
g ε ρ σ = g x ρ x ¯ τ x σ x ¯ χ ε ¯ τ χ .
LT ( c , c ¯ , d , d ¯ , x ) = d ¯ c ¯ d c x + c ¯ c d ¯ c ¯ d c
x ¯ 1 ( x 1 , x 2 ) = LT ( 0 , 0 , P 0 , x 1 , H 1 ( x 2 ) , x 1 ) , ( x 1 , x 2 )
with H 1 ( x 2 ) = LT ( 0 , P 0 , x 1 , P 0 , x 2 , P φ , x 1 , x 2 ) ,
x ¯ 2 ( x 1 , x 2 ) = LT ( 0 , 0 , P 0 , x 2 , H 2 ( x 1 ) , x 2 ) , ( x 1 , x 2 )
with H 2 ( x 1 ) = LT ( 0 , P 0 , x 2 , P 0 , x 1 , P φ , x 2 , x 1 ) .
H 1 ( x 2 ) = h ( x h 2 ) = h ( P φ , x 2 P 0 , x 2 x 2 ) ( x 1 , x 2 ) ,
H 2 ( x 1 ) = g ( x g 1 ) = g ( P φ , x 1 P 0 , x 1 x 1 ) ( x 1 , x 2 ) ,
x ¯ 1 ( x 1 , x 2 ) = LT ( P 1 , x 1 , e ( x e 2 ) , S 1 , x 1 , g ( x g 2 ) , x 1 ) ,
x ¯ 2 ( x 1 , x 2 ) = LT ( P 1 , x 2 , f ( x f 1 ) , Q 1 , x 2 , h ( x h 1 ) , x 2 ) .
h ( x h 1 ) = h ( LT ( Q 1 , x 1 , Q 2 , x 1 , R 1 , x 1 , R 2 , x 1 , x 1 ) ) ,
f ( x f 1 ) = f ( LT ( P 1 , x 1 , P 2 , x 1 , S 1 , x 1 , S 2 , x 1 , x 1 ) ) ,
e ( x e 2 ) = e ( LT ( P 1 , x 2 , P 2 , x 2 , Q 1 , x 2 , Q 2 , x 2 , x 2 ) ) ,
g ( x g 2 ) = g ( LT ( S 1 , x 2 , S 2 , x 2 , R 1 , x 2 , R 2 , x 2 , x 2 ) ) .

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