Abstract

The rigorous coupled wave analysis (RCWA) is a widely used method for simulating diffraction from periodic structures. Since its recognized formulation by Moharam et al. [J. Opt. Soc. Am. A 12, 1068 and 1077 (1995)] , there still has been a discussion about convergence problems. Those problems are more or less solved for the diffraction from line gratings, but there remain different concurrent proposals about the convergence improvement for crossed gratings. We propose to combine Popov and Nevière's formulation of the differential method [Light Propagation in Periodic Media (Dekker, 2003) and J. Opt. Soc. Am. A 18, 2886 (2001)] with the classical RCWA. With a suitable choice of a normal vector field we obtain a better convergence than for the formulations that are known from the literature.

© 2007 Optical Society of America

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References

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  1. E. Popov and M. Nevière, Light Propagation in Periodic Media (Dekker, 2003).
  2. 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]
  3. M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, "Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings," J. Opt. Soc. Am. A 12, 1068-1076 (1995).
    [CrossRef]
  4. M. G. Moharam, D. A. Pommet, E. B. Grann, and T. K. Gaylord, "Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach," J. Opt. Soc. Am. A 12, 1077-1086 (1995).
    [CrossRef]
  5. M. Totzeck, "Numerical simulation of high-NA quantitative polarization microscopy and corresponding near-fields," Optik 112, 399-406 (2001).
    [CrossRef]
  6. P. Lalanne and G. M. Morris, "Highly improved convergence of the coupled-wave method for TM polarization," J. Opt. Soc. Am. A 13, 779-784 (1996).
    [CrossRef]
  7. L. Li, "Use of Fourier series in the analysis of discontinuous periodic structures," J. Opt. Soc. Am. A 13, 1870-1876 (1996).
    [CrossRef]
  8. E. Popov, B. Chernov, M. Nevière, and N. Bonod, "Differential theory: application to highly conducting gratings," J. Opt. Soc. Am. A 21, 199-206 (2004).
    [CrossRef]
  9. P. Lalanne, "Improved formulation of the coupled-wave method for two-dimensional gratings," J. Opt. Soc. Am. A 14, 1592-1598 (1997).
    [CrossRef]
  10. L. Li, "New formulation of the Fourier modal method for crossed surface-relief gratings," J. Opt. Soc. Am. A 14, 2758-2767 (1997).
    [CrossRef]
  11. E. Popov, M. Neviére, B. Gralak, and G. Tayeb, "Staircase approximation validity for arbitrary-shaped gratings," J. Opt. Soc. Am. A 19, 33-42 (2002).
    [CrossRef]
  12. T. Driscoll, "Schwarz-Christoffel toolbox for MATLAB," http://www.math.udel.edu/∼driscoll/software.
  13. I. Sneddon, Encylopaedic Dictionary of Mathematics for Engineers and Applied Scientists (Wheatons, 1976).

2004 (1)

2002 (1)

2001 (2)

1997 (2)

1996 (2)

1995 (2)

Bonod, N.

Chernov, B.

Driscoll, T.

T. Driscoll, "Schwarz-Christoffel toolbox for MATLAB," http://www.math.udel.edu/∼driscoll/software.

Gaylord, T. K.

Gralak, B.

Grann, E. B.

Lalanne, P.

Li, L.

Moharam, M. G.

Morris, G. M.

Neviére, M.

Nevière, M.

Pommet, D. A.

Popov, E.

Sneddon, I.

I. Sneddon, Encylopaedic Dictionary of Mathematics for Engineers and Applied Scientists (Wheatons, 1976).

Tayeb, G.

Totzeck, M.

M. Totzeck, "Numerical simulation of high-NA quantitative polarization microscopy and corresponding near-fields," Optik 112, 399-406 (2001).
[CrossRef]

J. Opt. Soc. Am. A (9)

P. Lalanne and G. M. Morris, "Highly improved convergence of the coupled-wave method for TM polarization," J. Opt. Soc. Am. A 13, 779-784 (1996).
[CrossRef]

L. Li, "Use of Fourier series in the analysis of discontinuous periodic structures," J. Opt. Soc. Am. A 13, 1870-1876 (1996).
[CrossRef]

E. Popov, B. Chernov, M. Nevière, and N. Bonod, "Differential theory: application to highly conducting gratings," J. Opt. Soc. Am. A 21, 199-206 (2004).
[CrossRef]

P. Lalanne, "Improved formulation of the coupled-wave method for two-dimensional gratings," J. Opt. Soc. Am. A 14, 1592-1598 (1997).
[CrossRef]

L. Li, "New formulation of the Fourier modal method for crossed surface-relief gratings," J. Opt. Soc. Am. A 14, 2758-2767 (1997).
[CrossRef]

E. Popov, M. Neviére, B. Gralak, and G. Tayeb, "Staircase approximation validity for arbitrary-shaped gratings," J. Opt. Soc. Am. A 19, 33-42 (2002).
[CrossRef]

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]

M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, "Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings," J. Opt. Soc. Am. A 12, 1068-1076 (1995).
[CrossRef]

M. G. Moharam, D. A. Pommet, E. B. Grann, and T. K. Gaylord, "Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach," J. Opt. Soc. Am. A 12, 1077-1086 (1995).
[CrossRef]

Optik (1)

M. Totzeck, "Numerical simulation of high-NA quantitative polarization microscopy and corresponding near-fields," Optik 112, 399-406 (2001).
[CrossRef]

Other (3)

E. Popov and M. Nevière, Light Propagation in Periodic Media (Dekker, 2003).

T. Driscoll, "Schwarz-Christoffel toolbox for MATLAB," http://www.math.udel.edu/∼driscoll/software.

I. Sneddon, Encylopaedic Dictionary of Mathematics for Engineers and Applied Scientists (Wheatons, 1976).

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Figures (12)

Fig. 1
Fig. 1

Relative orientation of Cartesian field components and boundaries for two example structures.

Fig. 2
Fig. 2

Tilted line grating modeled as a 2D structure.

Fig. 3
Fig. 3

Convergence curves for tilted line grating.

Fig. 4
Fig. 4

Different unit cells for a checkerboard grating and different ways to set up the NV field.

Fig. 5
Fig. 5

Convergence curves for checkerboard cell A.

Fig. 6
Fig. 6

Convergence curves for checkerboard cell B.

Fig. 7
Fig. 7

Most obvious NV fields for simple geometric objects in the unit cell.

Fig. 8
Fig. 8

Coordinate lines obtained by Schwarz–Christoffel transformation.

Fig. 9
Fig. 9

Normal vector fields set up using the electrostatic model algorithm for three example structures.

Fig. 10
Fig. 10

Convergence curves for an array of circles.

Fig. 11
Fig. 11

Convergence curves for an array of oblique ellipses.

Fig. 12
Fig. 12

Convergence curves for an array of squares.

Equations (19)

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c ( x ) = a ( x ) b ( x ) c j = k = N N a j k b k .
[ D ] = ϵ 0 ε [ E ] .
[ D ] = ϵ 0 1 ε 1 [ E ]
× H g = i ω D g ,
E g = j [ S x j ( z ) x + S y j ( z ) y + S z j ( z ) z ] exp [ i ( k x j x + k y j y ) ] ,
H g = i ( ϵ 0 μ 0 ) 1 2 j [ U x j ( z ) x + U y j ( z ) y + U z j ( z ) z ] exp [ i ( k x j x + k y j y ) ] .
[ U x z U y z ] = [ K x K y K x 2 E E K y 2 K x K y ] [ S x S y ] ,
[ D x ] = { ε Δ N x 2 } [ E x ] { Δ N x N y } [ E y ] ,
[ D y ] = { ε Δ N y 2 } [ E y ] { Δ N x N y } [ E x ] ,
N = [ sin α cos α ] .
[ N x ] k = sin α δ k 0 , [ N y ] k = cos α δ k 0 ,
G = [ K x K y Δ sin α cos α K x 2 ε sin 2 α 1 ε 1 cos 2 α ε cos 2 α + 1 ε 1 sin 2 α K y 2 K x K y + Δ sin α cos α ] .
G = [ K x K y K x 2 ε 1 ε 1 K y 2 K x K y ] ,
U x z = ( K x 2 ε ) S y ,
U y z = 1 ε 1 S x .
[ S T S N ] = [ cos α sin α sin α cos α ] [ S x S y ] .
G = [ K x K y K x 2 1 ε 1 ε K y 2 K x K y ] .
[ U T z U N z ] = [ K T K N K T 2 1 ε 1 ε K N 2 K T K N ] [ S T S N ] .
[ U x z U y z ] = [ cos α sin α sin α cos α ] [ K T K N K T 2 1 ε 1 ε K N 2 K T K N ] [ cos α sin α sin α cos α ] [ S x S y ] .

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