Abstract

We present a new algorithm that enables the analysis of large two-dimensional optical gratings with very small feature sizes using the Fourier modal method (FMM). With the conventional algorithm such structures cannot be solved because of limitations in computer memory and calculation time. By dividing the grating into several smaller subgratings and solving them sequentially, both memory requirement and calculation time can be reduced dramatically. We have calculated a grating with 32×32 pixels for a different number of subgratings. We show that the increased performance is directly related to the size of the subgratings. The field-stitched calculations prove to be very accurate and agree well with the predictions from the standard FMM approach.

© 2009 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2009 (1)

2007 (1)

2005 (2)

B. Bai, and L. Li, “Group-theoretic approach to the enhancement of the Fourier modal method for crossed gratings: C2 symmetry case,” J. Opt. Soc. Am. A 22, 654-661 (2005).
[CrossRef]

B. Bai and L. Li, “Group-theoretic approach to enhancing the Fourier modal method for crossed gratings with one or two reflection symmetries,” J. Opt. A, Pure Appl. Opt. 7, 271-278 (2005).
[CrossRef]

2004 (2)

C. Zhou and L. Li, “Formulation of the Fourier modal method for symmetric crossed gratings in symmetric mountings,” J. Opt. A, Pure Appl. Opt. 6, 43-50 (2004).
[CrossRef]

B. Bai, and L. Li, “Reduction of computation time for crossed-grating problems: a group-theoretic approach,” J. Opt. Soc. Am. A 21, 1886-1894 (2004).
[CrossRef]

2002 (1)

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

2001 (1)

1999 (1)

1998 (1)

L. Li, “Reformulation of the Fourier modal method for surface-relief gratings made with anisotropic materials,” J. Mod. Opt. 45, 1313-1334 (1998).
[CrossRef]

1997 (3)

1996 (2)

Bai, B.

Bergey, J. S.

Granet, G.

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

Hudelist, F.

Kerwien, N.

Lalanne, P.

Layet, B.

Li, L.

B. Bai and L. Li, “Group-theoretic approach to enhancing the Fourier modal method for crossed gratings with one or two reflection symmetries,” J. Opt. A, Pure Appl. Opt. 7, 271-278 (2005).
[CrossRef]

B. Bai, and L. Li, “Group-theoretic approach to the enhancement of the Fourier modal method for crossed gratings: C2 symmetry case,” J. Opt. Soc. Am. A 22, 654-661 (2005).
[CrossRef]

B. Bai, and L. Li, “Reduction of computation time for crossed-grating problems: a group-theoretic approach,” J. Opt. Soc. Am. A 21, 1886-1894 (2004).
[CrossRef]

C. Zhou and L. Li, “Formulation of the Fourier modal method for symmetric crossed gratings in symmetric mountings,” J. Opt. A, Pure Appl. Opt. 6, 43-50 (2004).
[CrossRef]

L. Li, “Reformulation of the Fourier modal method for surface-relief gratings made with anisotropic materials,” J. Mod. Opt. 45, 1313-1334 (1998).
[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]

L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A 13, 1024-1035 (1996).
[CrossRef]

Nevière, M.

Osten, W.

Plumey, J.-P.

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

Popov, E.

Prather, D. W.

Rafler, S.

Ruoff, J.

Schuster, T.

Shi, S.

Taghizadeh, M. R.

Waddie, A. J.

Zhou, C.

C. Zhou and L. Li, “Formulation of the Fourier modal method for symmetric crossed gratings in symmetric mountings,” J. Opt. A, Pure Appl. Opt. 6, 43-50 (2004).
[CrossRef]

J. Mod. Opt. (1)

L. Li, “Reformulation of the Fourier modal method for surface-relief gratings made with anisotropic materials,” J. Mod. Opt. 45, 1313-1334 (1998).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (3)

B. Bai and L. Li, “Group-theoretic approach to enhancing the Fourier modal method for crossed gratings with one or two reflection symmetries,” J. Opt. A, Pure Appl. Opt. 7, 271-278 (2005).
[CrossRef]

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

C. Zhou and L. Li, “Formulation of the Fourier modal method for symmetric crossed gratings in symmetric mountings,” J. Opt. A, Pure Appl. Opt. 6, 43-50 (2004).
[CrossRef]

J. Opt. Soc. Am. A (8)

Opt. Lett. (3)

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Figures (7)

Fig. 1
Fig. 1

The grating is divided into adjacent rectangular subgratings numbered from 1 to N. They do not necessarily have to have the same size and shape.

Fig. 2
Fig. 2

Each subgrating is extended by a frame with dimension x s and y s . This ensures that the fields at the edges are calculated correctly.

Fig. 3
Fig. 3

Subgratings that can be transformed into each each other by reflection (left) or rotation (right) need to be solved only once. The transmission coefficients and reflection coefficients for all partners can easily be calculated.

Fig. 4
Fig. 4

Size of the S matrix as a function of the effective truncation order.

Fig. 5
Fig. 5

Diffractive phase grating designed as 10 × 10 beam splitter. The grating consists of 32 × 32  pixels and seven refractive index levels.

Fig. 6
Fig. 6

Convergence of the absolute value of three transmission orders. All three cases show the convergence for the standard FMM approach (triangles) and for the field stitching algorithm with 2 × 2 subgratings (circles) and 4 × 4 subgratings (stars).

Fig. 7
Fig. 7

Calculation time as a function of the effective truncation order. Curves with circles show the trend for the undivided structure, diamonds for a 2 × 2 division, and squares for a 4 × 4 division.

Equations (39)

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E σ 1 = m n ( I σ , m n + R σ , m n ) × exp [ i ( α m x + β n y + γ m n 1 ) ] ,
E σ 2 = m n T σ , m n exp [ i ( α m x + β n y + γ m n 1 ) ] ,
r n = ( x 1 n x s n , y 1 n y s n , 0 ) ,
d x n = ( x 2 n + x s n ) ( x 1 n x s n ) ,
d y n = ( y 2 n + y s n ) ( y 1 n y s n ) ,
k p q = ( α p , β q )
= ( α 0 + 2 π p d x 1 , β 0 + 2 π q d y 1 )
k p q n = ( α 0 n + 2 π p ( d x n ) 1 , β 0 n + 2 π q ( d y n ) 1 )
U n ( r ) = U ( r + r n ) ,
I p q n = ( d x n d y n ) 1 0 d x n 0 d y n f ( r ) exp ( i k p q n r ) d x d y = ( d x n d y n ) 1 st I st exp ( i k st r n ) × 0 d x n 0 d y n exp ( i ( k st k p q n ) r ) d x d y .
U ( r ) = U n ( r r n ) { x 1 n < x < x 2 n y 1 n < y < y 2 n } .
T st = ( d x d y ) 1 p q T p q n exp ( i k p q n r n ) × 0 d x 0 d y exp ( i ( k p q n k st ) r ) d x d y .
T st = ( d x d y ) 1 n p q T p q n exp ( i k p q n r n ) × x 1 n x 2 n y 1 n y 2 n exp ( i ( k p q n k st ) r ) d x d y
( R st + I st ) = ( d x d y ) 1 n p q ( R p q n + I p q n ) exp ( i k p q n r n ) × x 1 n x 2 n y 1 n y 2 n exp ( i ( k p q n k st ) r ) d x d y .
T st = ( d x d y ) 1 p q T p q exp ( i k p q r ) × x 1 x 2 y 1 y 2 exp ( i ( k p q k st ) r ) d x d y ,
X σ m n = X σ N 2 + m N + n ,
U ( 2 ) ( x , y ) = U ( 1 ) ( x , y ) ,
U ( 3 ) ( x , y ) = U ( 1 ) ( x , y ) ,
U ( 4 ) ( x , y ) = U ( 1 ) ( x , y ) .
U ( 2 ) ( x , y ) = U ( 1 ) ( y , x ) ,
U ( 3 ) ( x , y ) = U ( 1 ) ( x , y ) ,
U ( 4 ) ( x , y ) = U ( 1 ) ( y , x ) .
U ( 2 ) ( x , y ) = 1 d x d y p q X p q ( 2 ) exp [ i ( α p x + β q y ) ] = 1 d x d y p q X p q ( 1 ) exp { i [ α p x + β q ( y ) ] } = 1 d x d y p q X p , q ( 1 ) exp [ i ( α p x + β q y ) ] ,
p q ( X p q ( 2 ) X p , q ( 1 ) ) exp ( i k r ) = 0 .
X p , q ( 2 ) = X p , q ( 1 ) .
X p q ( 3 ) = X p , q ( 1 ) ,
X p q ( 4 ) = X p , q ( 1 ) .
X p q ( 2 ) = X q , p ( 1 ) ,
X p q ( 3 ) = X p , q ( 1 ) ,
X p q ( 4 ) = X q , p ( 1 ) .
C b , a ( X σ , m , n ) = X σ , n , m ,
C a , b ( X σ , m , n ) = X σ , m , n .
T ( n ) = C ( T ( 1 ) ) .
I ( n ) = C ( I ( 1 ) ) .
T = S I .
T ( n ) = C ( T ) .
T ( n ) = C ( S ( 1 ) C ( I ( n ) ) ) .
α max = 2 π m ( d x a ) = a α max .
N eff = a N .

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