Abstract

The Legendre polynomial expansion method (LPEM), which has been successfully applied to homogenous and longitudinally inhomogeneous gratings [J. Opt. Soc. Am. B 24, 2676 (2007) ], is now generalized for the efficient analysis of arbitrary-shaped surface relief gratings. The modulated region is cut into a few sufficiently thin arbitrary-shaped subgratings of equal spatial period, where electromagnetic field dependence is now smooth enough to be approximated by keeping fewer Legendre basis functions. The R-matrix propagation algorithm is then employed to match the Legendre polynomial expansions of the transverse electric and magnetic fields across the upper and lower interfaces of every slice. The proposed strategy then enhances the overall computational efficiency, reduces the required memory size, and permits the efficient study of arbitrary-shaped gratings. Here the rigorous approach is followed, and analytical formulas of the involved matrices are given.

© 2008 Optical Society of America

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References

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  1. E. G. Loewen and E. Popov, Diffraction Gratings and Applications (Marcel Dekker, 1997).
  2. M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 71, 811-818 (1981).
    [CrossRef]
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    [CrossRef]
  4. 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]
  5. 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]
  6. Ph. 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. G. Garnet and B. Guizal, “Efficient implementation of the coupled wave method for metallic lamellar gratings in TM polarization,” J. Opt. Soc. Am. A 13, 1019-1023 (1996).
    [CrossRef]
  8. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870-1876 (1996).
    [CrossRef]
  9. E. Popov, M. Nevière, B. Gralak, and G. Tayeb, “Staircase approximation validity for arbitrary-shaped gratings,” J. Opt. Soc. Am. A 19, 33-42 (2002).
    [CrossRef]
  10. M. Nevière and E. Popov, Light Propagation in Periodic Media: Differential Theory and Design (Marcel Dekker, 2003).
  11. E. Popov and M. Nevière, “Grating theory: New equations in Fourier space leading to fast converging results for TM polarization,” J. Opt. Soc. Am. A 17, 1773-1784 (2000).
    [CrossRef]
  12. J. Chandezon, D. Maystre, and G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. (Paris) 11, 235-241 (1980).
    [CrossRef]
  13. L. Li, J. Chandezon, G. Granet, and J. P. Plumey, “Rigorous and efficient grating-analysis method made easy for optical engineers,” Appl. Opt. 38, 304-403 (1999).
    [CrossRef]
  14. L. Li and J. Chandezon, “Improvement of the coordinate transformation method for surface-relief gratings with sharp edges,” J. Opt. Soc. Am. A 13, 2247-2255 (1996).
    [CrossRef]
  15. G. Granet, J. Chandezon, J. P. Plumey, and K. Raniriharinosy, “Reformulation of the coordinate transformation method through the concept of adaptive spatial resolution. Application to trapezoidal gratings,” J. Opt. Soc. Am. A 18, 2102-2108 (2001).
    [CrossRef]
  16. S. D. Gedney and R. Mittra, “Analysis of the electromagnetic scattering by thick gratings using a combined FEM/MM solution,” IEEE Trans. Antennas Propag. 39, 1605-1614 (1991).
    [CrossRef]
  17. M. Chamanzar, K. Mehrany, and B. Rashidian, “Planar diffraction analysis of homogeneous and longitudinally inhomogeneous gratings based on Legendre expansion of electromagnetic fields,” IEEE Trans. Antennas Propag. 54, 3686-3694 (2006).
    [CrossRef]
  18. A. Khavasi, K. Mehrany, and B. Rashidian, “Three-dimensional diffraction analysis of gratings based on legendre expansion of electromagnetic fields,” J. Opt. Soc. Am. B 24, 2676-2685 (2007).
    [CrossRef]
  19. L. Li and C. W. Haggans, “Convergence of the coupled wave method for metallic lamellar diffraction gratings,” J. Opt. Soc. Am. A 10, 1184-1189 (1993).
    [CrossRef]
  20. P. Sarrafi, N. Zareian, and K. Mehrany, “Analytical extraction of leaky modes in circular slab waveguides with arbitrary refractive index profile,” Appl. Opt. 46, 8656-8667 (2007).
    [CrossRef] [PubMed]
  21. J. P. Boyd, Chebyshev and Fourier Spectral Methods, 2nd ed. (Dover, 2001).
  22. G. Granet, J. Chandezon, and O. Coudert, “Extension of the C method to nonhomogeneous media: Application to nonhomogeneous layers with parallel modulated faces and to inclined lamellar gratings,” J. Opt. Soc. Am. A 14, 1576-1582 (1997).
    [CrossRef]

2007 (2)

2006 (1)

M. Chamanzar, K. Mehrany, and B. Rashidian, “Planar diffraction analysis of homogeneous and longitudinally inhomogeneous gratings based on Legendre expansion of electromagnetic fields,” IEEE Trans. Antennas Propag. 54, 3686-3694 (2006).
[CrossRef]

2003 (1)

M. Nevière and E. Popov, Light Propagation in Periodic Media: Differential Theory and Design (Marcel Dekker, 2003).

2002 (1)

2001 (2)

2000 (1)

1999 (1)

1997 (2)

1996 (5)

1995 (1)

1993 (1)

1991 (1)

S. D. Gedney and R. Mittra, “Analysis of the electromagnetic scattering by thick gratings using a combined FEM/MM solution,” IEEE Trans. Antennas Propag. 39, 1605-1614 (1991).
[CrossRef]

1983 (1)

1981 (1)

1980 (1)

J. Chandezon, D. Maystre, and G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. (Paris) 11, 235-241 (1980).
[CrossRef]

Boyd, J. P.

J. P. Boyd, Chebyshev and Fourier Spectral Methods, 2nd ed. (Dover, 2001).

Chamanzar, M.

M. Chamanzar, K. Mehrany, and B. Rashidian, “Planar diffraction analysis of homogeneous and longitudinally inhomogeneous gratings based on Legendre expansion of electromagnetic fields,” IEEE Trans. Antennas Propag. 54, 3686-3694 (2006).
[CrossRef]

Chandezon, J.

Coudert, O.

Garnet, G.

Gaylord, T. K.

Gedney, S. D.

S. D. Gedney and R. Mittra, “Analysis of the electromagnetic scattering by thick gratings using a combined FEM/MM solution,” IEEE Trans. Antennas Propag. 39, 1605-1614 (1991).
[CrossRef]

Gralak, B.

Granet, G.

Grann, E. B.

Guizal, B.

Haggans, C. W.

Khavasi, A.

Lalanne, Ph.

Li, L.

Loewen, E. G.

E. G. Loewen and E. Popov, Diffraction Gratings and Applications (Marcel Dekker, 1997).

Maystre, D.

J. Chandezon, D. Maystre, and G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. (Paris) 11, 235-241 (1980).
[CrossRef]

Mehrany, K.

Mittra, R.

S. D. Gedney and R. Mittra, “Analysis of the electromagnetic scattering by thick gratings using a combined FEM/MM solution,” IEEE Trans. Antennas Propag. 39, 1605-1614 (1991).
[CrossRef]

Moharam, M. G.

Morris, G. M.

Nevière, M.

Plumey, J. P.

Pommet, D. A.

Popov, E.

Raniriharinosy, K.

Raoult, G.

J. Chandezon, D. Maystre, and G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. (Paris) 11, 235-241 (1980).
[CrossRef]

Rashidian, B.

A. Khavasi, K. Mehrany, and B. Rashidian, “Three-dimensional diffraction analysis of gratings based on legendre expansion of electromagnetic fields,” J. Opt. Soc. Am. B 24, 2676-2685 (2007).
[CrossRef]

M. Chamanzar, K. Mehrany, and B. Rashidian, “Planar diffraction analysis of homogeneous and longitudinally inhomogeneous gratings based on Legendre expansion of electromagnetic fields,” IEEE Trans. Antennas Propag. 54, 3686-3694 (2006).
[CrossRef]

Sarrafi, P.

Tayeb, G.

Zareian, N.

Appl. Opt. (2)

IEEE Trans. Antennas Propag. (2)

S. D. Gedney and R. Mittra, “Analysis of the electromagnetic scattering by thick gratings using a combined FEM/MM solution,” IEEE Trans. Antennas Propag. 39, 1605-1614 (1991).
[CrossRef]

M. Chamanzar, K. Mehrany, and B. Rashidian, “Planar diffraction analysis of homogeneous and longitudinally inhomogeneous gratings based on Legendre expansion of electromagnetic fields,” IEEE Trans. Antennas Propag. 54, 3686-3694 (2006).
[CrossRef]

J. Opt. (Paris) (1)

J. Chandezon, D. Maystre, and G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. (Paris) 11, 235-241 (1980).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (11)

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]

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]

Ph. 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]

G. Garnet and B. Guizal, “Efficient implementation of the coupled wave method for metallic lamellar gratings in TM polarization,” J. Opt. Soc. Am. A 13, 1019-1023 (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, M. Nevière, B. Gralak, and G. Tayeb, “Staircase approximation validity for arbitrary-shaped gratings,” J. Opt. Soc. Am. A 19, 33-42 (2002).
[CrossRef]

L. Li and C. W. Haggans, “Convergence of the coupled wave method for metallic lamellar diffraction gratings,” J. Opt. Soc. Am. A 10, 1184-1189 (1993).
[CrossRef]

L. Li and J. Chandezon, “Improvement of the coordinate transformation method for surface-relief gratings with sharp edges,” J. Opt. Soc. Am. A 13, 2247-2255 (1996).
[CrossRef]

G. Granet, J. Chandezon, J. P. Plumey, and K. Raniriharinosy, “Reformulation of the coordinate transformation method through the concept of adaptive spatial resolution. Application to trapezoidal gratings,” J. Opt. Soc. Am. A 18, 2102-2108 (2001).
[CrossRef]

E. Popov and M. Nevière, “Grating theory: New equations in Fourier space leading to fast converging results for TM polarization,” J. Opt. Soc. Am. A 17, 1773-1784 (2000).
[CrossRef]

G. Granet, J. Chandezon, and O. Coudert, “Extension of the C method to nonhomogeneous media: Application to nonhomogeneous layers with parallel modulated faces and to inclined lamellar gratings,” J. Opt. Soc. Am. A 14, 1576-1582 (1997).
[CrossRef]

J. Opt. Soc. Am. B (1)

Other (3)

E. G. Loewen and E. Popov, Diffraction Gratings and Applications (Marcel Dekker, 1997).

M. Nevière and E. Popov, Light Propagation in Periodic Media: Differential Theory and Design (Marcel Dekker, 2003).

J. P. Boyd, Chebyshev and Fourier Spectral Methods, 2nd ed. (Dover, 2001).

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Figures (7)

Fig. 1
Fig. 1

Geometry of a typical surface relief grating represented as a stack of L arbitrary-shaped subgratings.

Fig. 2
Fig. 2

Computational efficiency of the DM (circles), the RCWA (triangles), and the LPEM (diamonds).

Fig. 3
Fig. 3

TE polarized 1 st reflected order of the metallic surface relief grating in case B versus L and M in applying the LPEM with N = 15 .

Fig. 4
Fig. 4

TE-polarized 1 st reflected order of the metallic surface relief grating in case B (a) versus the number of integration steps in applying the DM with N = 15 (b) versus the number of slices in the staircase approximation based on the RCWA with N = 15 .

Fig. 5
Fig. 5

TM-polarized zeroth reflected order of the metallic surface relief grating in case B versus L and M in applying the LPEM with N = 15 .

Fig. 6
Fig. 6

TM-polarized zeroth reflected order of the metallic surface relief grating in case B versus the number of integration steps in applying the DM with N = 15 .

Fig. 7
Fig. 7

Resonance anomaly in the TM reflectivity of a sinusoidal aluminum grating: LPEM with N = 20 , M = 3 , L = 150 (solid curve), and the C-method (dashed curve).

Tables (5)

Tables Icon

Table 1 Required Run Times of the DM, RCWA, and LPEM to Achieve an Accuracy of 1% in the Largest Diffraction Efficiency of Both Major Polarizations

Tables Icon

Table 2 Transmitted Diffraction Efficiencies of the 1 st , 0th, and + 1 st Orders, Together with the Total Diffracted Energy of the Dielectric Surface Relief Grating in Case A, Are All Calculated Using the LPEM with N = 5 and M = 6 and for Different Values of L

Tables Icon

Table 3 Required Run Times of the C-Method and LPEM to Achieve an Accuracy of 1% in the Largest Diffraction Efficiency of Both Major Polarization in Cases A and B

Tables Icon

Table 4 Required Run Rimes of the C-Method and LPEM to Achieve and Accuracy of 1% in the Largest Diffraction Efficiency of Both Major Polarizations in Cases C and D

Tables Icon

Table 5 Incurred Reciprocity Error in Applying the LPEM for the Analysis of Different Surface Relief Gratings in Cases A–D

Equations (61)

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f ( x + Λ G ) = f ( x ) ,
E 1 = u ̂ e j k 10 r + i = + R i e j k 1 i r
E 3 = i = + T i e j k 3 i ( r d z ̂ )
k x i = k 1 sin α i 2 π Λ G ,
k z l i = k l 2 k x i 2 , k l = 2 π λ n l
u ̂ = { y ̂ TE cos α x ̂ sin α z ̂ TM .
E = i [ S y i ( z ) y ̂ ] exp ( j k x i x ) ,
H = ϵ 0 μ 0 i [ U x i ( z ) x ̂ + U z i ( z ) z ̂ ] exp ( j k x i x ) .
d S y i ( z ) d z = j k 0 U x i ( z ) ,
d U x i ( z ) d z = j k 0 [ ( k x i k 0 ) 2 S y i ( z ) + p ϵ i p ( z ) S y p ( z ) ] ,
d [ S y ( z ) ] d z = j k 0 I [ U x ( z ) ] ,
d [ U x ( z ) ] d z = j k 0 ( [ [ ϵ ] ] ( K x k 0 ) 2 ) [ S y ( z ) ] ,
S y i ( z ) = m = 0 + h m i P m ( ξ ) ,
U x i ( z ) = m = 0 + t m i P m ( ξ ) .
( 2 d l ) m h m i P m ( ξ ) j k 0 m t m i P m ( ξ ) = 0 ,
( 2 d l ) m t m i P m ( ξ ) + j k 0 m P m ( ξ ) [ ( k x i k 0 ) 2 h m i p ϵ i p ( ξ ) h m p ] = 0 ,
( 2 d l ) m h m i P n , P m j k 0 m t m i P n , P m = 0 ,
( 2 d l ) m t m i P n , P m + j k x i 2 k 0 m ( h m i P n , P m j k 0 p h m p P n , ϵ i p P m ) = 0 ,
P n , P m = { 0 m n 2 2 m + 1 m = n ,
P n , P m = { 2 m n = odd , m n > 0 , 0 oth
Q T E [ [ h ¯ m i ] [ t ¯ m i ] ] = 0 ,
E = i [ S x i ( z ) x ̂ + S z i ( z ) z ̂ ] exp ( j k x i x ) ,
H = ϵ 0 μ 0 i [ U y i ( z ) y ̂ ] exp ( j k x i x ) .
d d z [ [ S x ] [ U y ] ] = [ M T M 11 M T M 12 M T M 21 M T M 22 ] [ [ S x ] [ U y ] ] ,
M T M 11 = K x GB ,
M T M 12 = K x GK x k 0 + j k 0 I ,
M T M 21 = j k 0 ( A + [ [ 1 ϵ ] ] 1 j BGB ) ,
M T M 22 = BGK x ,
Δ = [ [ ϵ ] ] [ [ 1 ϵ ] ] 1 ,
A = Δ [ [ c 2 ] ] ,
B = Δ [ [ c s ] ] ,
G = j ( [ [ ϵ ] ] A ) 1 .
S x i ( z ) = m = 0 + q m i P m ( ξ ) ,
U y i ( z ) = m = 0 + l m i P m ( ξ ) ,
m ( q m i 2 d l P n , P m + p q m p P n , M i p T M 11 P m + p l m p P n , M i p T M 12 P m ) = 0 ,
m ( l m i 2 d l P n , P m + p l m p P n , M i p T M 22 P m + p q m p P n , M i p T M 21 P m ) = 0 .
Q T M [ [ q ¯ m i ] [ l ¯ m i ] ] = 0 ,
[ [ S x ( z l ) ] [ S x ( z l 1 ) ] ] = r ̃ ( l ) T M [ [ U y ( z l ) ] [ U y ( z l 1 ) ] ] .
χ [ [ q ¯ m i ] [ l ¯ m i ] ] = [ [ U y ( z l ) ] [ U y ( z l 1 ) ] ] ,
ψ [ [ q ¯ m i ] [ l ¯ m i ] ] = [ [ S x ( z l ) ] [ S x ( z l 1 ) ] ] ,
r ̃ ( l ) T M = ψ [ Q ( 2 ( 2 N + 1 ) ( M 1 ) ) × ( 2 ( 2 N + 1 ) M ) T M χ ( 2 ( 2 N + 1 ) ) × ( 2 ( 2 N + 1 ) M ) ] 1 [ 0 ( 2 ( 2 N + 1 ) ( M 1 ) ) × ( 2 ( 2 N + 1 ) ) I ( 2 ( 2 N + 1 ) ) × ( 2 ( 2 N + 1 ) ) ] .
R 11 ( l ) = r ̃ 11 ( l ) r ̃ 12 ( l ) Z ( l ) r ̃ 21 ( l ) ,
R 12 ( l ) = r ̃ 12 ( l ) Z ( l ) R 12 ( l 1 ) ,
R 21 ( l ) = R 21 ( l 1 ) Z ( l ) r ̃ 21 ( l ) ,
R 22 ( l ) = R 22 ( l 1 ) + R 21 ( l 1 ) Z ( l ) R 12 ( l 1 ) ,
Z ( l ) = ( r ̃ 22 ( l ) R 11 ( l 1 ) ) 1 ,
e = D E 0 ( θ ) D E 0 ( θ ) D E 0 ( θ ) ,
Q T E ( n + ( i + N ) ( M 1 ) + 1 , m + ( i + N ) M + ( 2 N + 1 ) M + 1 ) = j k 0 P n , P m ,
Q T E ( n + ( i + N ) ( M 1 ) + ( 2 N + 1 ) ( M 1 ) + 1 , m + ( p + N ) M + 1 ) = j k 0 P n , ( [ [ ϵ ] ] ( K x k 0 ) 2 ) i p P m ,
Q T E ( n + ( i + N ) ( M 1 ) + 1 , m + ( i + N ) M + 1 ) = 2 d l P n , P m ,
Q T E ( n + ( i + N ) ( M 1 ) + ( 2 N + 1 ) ( M 1 ) + 1 , m + ( i + N ) M + ( 2 N + 1 ) M + 1 ) = 2 d l P n , P m ,
Q T M ( n + ( i + N ) ( M 1 ) + 1 , m + ( p + N ) M + 1 ) = P n , M i p T M 11 P m ,
Q T M ( n + ( i + N ) ( M 1 ) + 1 , m + ( p + N ) M + ( 2 N + 1 ) M + 1 ) = P n , M i p T M 12 P m ,
Q T M ( n + ( i + N ) ( M 1 ) + ( 2 N + 1 ) ( M 1 ) + 1 , m + ( p + N ) M + 1 ) = P n , M i p T M 21 P m ,
Q T M ( n + ( i + N ) ( M 1 ) + ( 2 N + 1 ) ( M 1 ) + 1 , m + ( p + N ) M + ( 2 N + 1 ) M + 1 ) = P n , M i p T M 22 P m ,
Q T M ( n + ( i + N ) ( M 1 ) + 1 , m + ( i + N ) M + 1 ) = 2 d l P n , P m ,
Q T M ( n + ( i + N ) ( M 1 ) + ( 2 N + 1 ) ( M 1 ) + 1 , m + ( i + N ) M + ( 2 N + 1 ) M + 1 ) = 2 d l P n , P m ,
χ = [ 0 χ 12 0 χ 22 ] ,
ψ = [ χ 12 0 χ 22 0 ] ,
χ 12 ( i + N + 1 , m + ( i + N ) M + 1 ) = 1 ,
χ 22 ( i + N + 1 , m + ( i + N ) M + 1 ) = ( 1 ) m .

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