Abstract

The coupled-wave method formulated by Moharam and Gaylord [ J. Opt. Soc. Am. 73, 451 ( 1983)] is known to be slowly converging, especially for TM polarization of metallic lamellar gratings. The slow convergence rate has been analyzed in detail by Li and Haggans [ J. Opt. Soc. Am. A 10, 1184 ( 1993)], who made clear that special care must be taken when coupled-wave methods are used for TM polarization. By reformulating the eigenproblem of the coupled-wave method, we provide numerical evidence and argue that highly improved convergence rates similar to the TE polarization case can be obtained. The discussion includes both nonconical and conical mountings.

© 1996 Optical Society of America

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References

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  1. L. Li, C. W. Haggans, “Convergence of the coupled-wave method for metallic lamellar diffraction gratings,” J. Opt. Soc. Am. A 10, 1184–1189 (1993).
    [CrossRef]
  2. M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of grating diffraction—E-mode polarization and losses,” J. Opt. Soc. Am. 73, 451–455 (1983).
    [CrossRef]
  3. M. G. Moharam, E. B. Grann, D. A. Pommet, T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, 1068–1086 (1995).
    [CrossRef]
  4. C. B. Burckhardt, “Diffraction of a plane wave at a sinusoidally stratified dielectric grating,” J. Opt. Soc. Am. 56, 1502–1509 (1966).
    [CrossRef]
  5. K. Knop, “Rigorous diffraction theory for transmission phase gratings with deep rectangular grooves,” J. Opt. Soc. Am. 68, 1206–1210 (1978).
    [CrossRef]
  6. S. Peng, G. M. Morris, “Efficient implementation of rigorous coupled-wave analysis for surface relief gratings,” J. Opt. Soc. Am. A 12, 1087–1096 (1995).
    [CrossRef]
  7. J. M. Miller, J. Turunen, E. Noponen, A. Vasara, M. R. Taghizadeh, “Rigorous modal theory for multiply grooved lamellar gratings,” Opt. Commun. 111, 526–535 (1994).
    [CrossRef]
  8. G. Bouchitte, R. Petit, “Homogenization techniques as applied in the electromagnetic theory of gratings,” Electromagnetics 5, 17–36 (1985).
    [CrossRef]
  9. Ph. Lalanne, D. Lemercier-Lalanne, “On the effective medium theory of subwavelength periodic structures,” submitted to J. Mod. Opt.

1995 (2)

1994 (1)

J. M. Miller, J. Turunen, E. Noponen, A. Vasara, M. R. Taghizadeh, “Rigorous modal theory for multiply grooved lamellar gratings,” Opt. Commun. 111, 526–535 (1994).
[CrossRef]

1993 (1)

1985 (1)

G. Bouchitte, R. Petit, “Homogenization techniques as applied in the electromagnetic theory of gratings,” Electromagnetics 5, 17–36 (1985).
[CrossRef]

1983 (1)

1978 (1)

1966 (1)

Bouchitte, G.

G. Bouchitte, R. Petit, “Homogenization techniques as applied in the electromagnetic theory of gratings,” Electromagnetics 5, 17–36 (1985).
[CrossRef]

Burckhardt, C. B.

Gaylord, T. K.

Grann, E. B.

Haggans, C. W.

Knop, K.

Li, L.

Miller, J. M.

J. M. Miller, J. Turunen, E. Noponen, A. Vasara, M. R. Taghizadeh, “Rigorous modal theory for multiply grooved lamellar gratings,” Opt. Commun. 111, 526–535 (1994).
[CrossRef]

Moharam, M. G.

Morris, G. M.

Noponen, E.

J. M. Miller, J. Turunen, E. Noponen, A. Vasara, M. R. Taghizadeh, “Rigorous modal theory for multiply grooved lamellar gratings,” Opt. Commun. 111, 526–535 (1994).
[CrossRef]

Peng, S.

Petit, R.

G. Bouchitte, R. Petit, “Homogenization techniques as applied in the electromagnetic theory of gratings,” Electromagnetics 5, 17–36 (1985).
[CrossRef]

Pommet, D. A.

Taghizadeh, M. R.

J. M. Miller, J. Turunen, E. Noponen, A. Vasara, M. R. Taghizadeh, “Rigorous modal theory for multiply grooved lamellar gratings,” Opt. Commun. 111, 526–535 (1994).
[CrossRef]

Turunen, J.

J. M. Miller, J. Turunen, E. Noponen, A. Vasara, M. R. Taghizadeh, “Rigorous modal theory for multiply grooved lamellar gratings,” Opt. Commun. 111, 526–535 (1994).
[CrossRef]

Vasara, A.

J. M. Miller, J. Turunen, E. Noponen, A. Vasara, M. R. Taghizadeh, “Rigorous modal theory for multiply grooved lamellar gratings,” Opt. Commun. 111, 526–535 (1994).
[CrossRef]

Electromagnetics (1)

G. Bouchitte, R. Petit, “Homogenization techniques as applied in the electromagnetic theory of gratings,” Electromagnetics 5, 17–36 (1985).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (3)

Opt. Commun. (1)

J. M. Miller, J. Turunen, E. Noponen, A. Vasara, M. R. Taghizadeh, “Rigorous modal theory for multiply grooved lamellar gratings,” Opt. Commun. 111, 526–535 (1994).
[CrossRef]

Other (1)

Ph. Lalanne, D. Lemercier-Lalanne, “On the effective medium theory of subwavelength periodic structures,” submitted to J. Mod. Opt.

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Figures (5)

Fig. 1
Fig. 1

Geometry for the nonconical grating diffraction problem analyzed in Sections 2 and 3 for TM polarization.

Fig. 2
Fig. 2

Diffraction efficiency of the transmitted zeroth order of a metallic grating with TM polarized light. The solid curve is obtained by using the conventional eigenproblem formulation of Eq. (6b). The circles are provided by the new eigenproblem of Eq. (9). The grating parameters and the geometry problem are defined in Fig. 2 of Ref. 6.

Fig. 3
Fig. 3

Diffraction efficiencies of the reflected negative first and zeroth orders of a metallic grating with TM polarization. The circles are provided by the new eigenproblem method of Eq. (9). The grating parameters and the geometry problem are defined in Fig. 1 of Ref. 1. A direct comparison can be applied with Figs. 3(a) and 3(b) of Ref. 1, where simulation results obtained with the conventional eigenproblem and modal methods are presented.

Fig. 4
Fig. 4

Effect of the truncation on the accuracy of the conventional eigenproblem. The pluses and circles correspond to the imaginary and the real parts, respectively, of the error e = n - 1 / a 0. The results are obtained by solving the system of linear equations defined by Eqs. (10a) and (11).

Fig. 5
Fig. 5

Diffraction efficiencies of the reflected negative first and zeroth orders of a metallic grating for conical mount (θ = 30°, ϕ = 30°, and ψ = 45°). The grating parameters are defined in Fig. 1 of Ref. 1. The solid curves are obtained with the conventional eigenproblem formulation of Ref. 3. The dotted curves are obtained with the new formulation of Eq. (16).

Equations (27)

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E x = m S m ( z ) exp j ( K m + β ) x , E z = m f m ( z ) exp j ( K m + β ) x , H y = 0 μ 0 m U m ( z ) exp j ( K m + β ) x .
- E z x + E x x = - j ω μ 0 H y ,
H y z = - j ω E x ,
1 H y x = j ω E z .
- j ( m K + β ) f m + S m = - j k 0 U m ,
U m = - j k 0 p m - p S p ,
p ( p K + β ) a m - p U p = k 0 f m .
S m = - j k 0 U m + j ( m K / k 0 + β / k 0 ) p ( p K + β ) a m - p U p .
U m = - k 0 2 p m - p [ U p - ( p K / k 0 + β / k 0 ) × r ( r K / k 0 + β / k 0 ) a p - r U r ] .
k 0 - 2 [ U ] = [ E ( K x A K x - I ) ] [ U ] ,
k 0 - 2 [ U ] = [ E ( K x E - 1 K x - I ) ] [ U ] .
- p a m - p U p = j k 0 S m ,
( m K + β ) U m = k 0 p m - p f p .
( m K / k 0 + β / k 0 ) U m - p m - p p K / k 0 + β / k 0 U p = 1 k 0 2 l , p m - p a p - l p K / k 0 + β / k 0 U l .
k 0 - 2 [ U ] = [ A - 1 ( K x E - 1 K x - I ) [ U ] ,
n S 0 ( 0 ) = U 0 ( 0 ) ,
m 0 ,             p p a m - p U p ( 0 ) = 0 ,
m 0 ,             p 0 m - p S p ( 0 ) = - m S o ( 0 ) ,
n U 0 ( 0 ) = p 0 - p S p ( 0 ) + 0 S 0 ( 0 ) .
p 0 ( m 0 a - m m - p ) S p ( 0 ) = - m 0 a - m m S 0 ( 0 ) .
- a 0 * p 0 - p S p ( 0 ) = - ( 1 - a 0 # o ) S 0 ( 0 ) .
a 0 * ( n U 0 ( 0 ) - 0 S 0 ( 0 ) ) = ( 1 - a 0 # o ) S 0 ( 0 ) .
n 2 = 1 a 0 * + 0 ( 1 - a 0 # a 0 * ) .
m 0 ,             m K 2 k 0 2 U m - p 0 m - p p U p = p 0 , l m - p a p - l p k 0 2 U l ,
k 0 2 U 0 + p a - p U p = 0.
k 0 - 1 [ S y S x U y U x ] = [ 0 0 K y E - 1 K x I - K y E - 1 K y 0 0 K x E - 1 K x - K x E - 1 K y K x K y A - 1 - K y 2 0 0 K x 2 - E - K x K y 0 0 ] × [ S y S x U y U x ] .
k 0 - 2 [ U x ] = [ K y 2 + K x 2 - E ] [ U x ] k 0 - 2 [ S x ] = [ K x E - 1 K x A - 1 + K y 2 - A - 1 ] [ S x ] .

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