Abstract

In a recent article [J. Opt. Soc. Am. A 27, 1694 (2010) ], we proposed a rectangular truncation method to mitigate the convergence problems arising from the boundary matching conditions of a binary metallic grating. The proposed method may underestimate the total power in the scattered field for certain grating parameters. In this article, we extend this method to preserve the total power by introducing appropriate constraints and solving the resulting problem as a constrained least squares minimization problem. We provide examples to show that the new method provides a convergent solution for both lossy and lossless binary metallic gratings while preserving the total power.

© 2010 Optical Society of America

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References

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  1. L. Li and C. Haggans, “Convergence of the coupled-wave method for metallic lamellar diffraction gratings,” J. Opt. Soc. Am. A 10, 1184–1189 (1993).
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  2. P. Lalanne and G. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13, 779–784 (1996).
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  3. G. Granet 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]
  4. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996).
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  5. 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).
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  6. A. Khavasi, K. Mehrany, and A. Jazayeri, “Study of the numerical artifacts in differential analysis of highly conducting gratings,” Opt. Lett. 33, 159–161 (2008).
    [CrossRef] [PubMed]
  7. N. Lyndin, O. Parriaux, and A. Tishchenko, “Modal analysis and suppression of the Fourier modal method instabilities in highly conductive gratings,” J. Opt. Soc. Am. A 24, 3781–3788 (2007).
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  8. K. Gundu and A. Mafi, “Reliable computation of scattering from metallic binary gratings using Fourier based modal methods,” J. Opt. Soc. Am. A 27, 1694–1700 (2010).
    [CrossRef]
  9. L. Botten, M. Craig, R. McPhedran, J. Adams, and J. Andrewartha, “Dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
    [CrossRef]
  10. J. Suratteau, M. Cadilhac, and R. Petit, “Sur la détermination numérique des efficacités de certains réseaux diélectriques profonds,” J. Opt. (Paris) 14, 273–288 (1983).
    [CrossRef]
  11. P. Johnson and R. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
    [CrossRef]

2010

2008

2007

2004

1996

1993

1983

J. Suratteau, M. Cadilhac, and R. Petit, “Sur la détermination numérique des efficacités de certains réseaux diélectriques profonds,” J. Opt. (Paris) 14, 273–288 (1983).
[CrossRef]

1981

L. Botten, M. Craig, R. McPhedran, J. Adams, and J. Andrewartha, “Dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

1972

P. Johnson and R. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

Adams, J.

L. Botten, M. Craig, R. McPhedran, J. Adams, and J. Andrewartha, “Dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

Andrewartha, J.

L. Botten, M. Craig, R. McPhedran, J. Adams, and J. Andrewartha, “Dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

Bonod, N.

Botten, L.

L. Botten, M. Craig, R. McPhedran, J. Adams, and J. Andrewartha, “Dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

Cadilhac, M.

J. Suratteau, M. Cadilhac, and R. Petit, “Sur la détermination numérique des efficacités de certains réseaux diélectriques profonds,” J. Opt. (Paris) 14, 273–288 (1983).
[CrossRef]

Chernov, B.

Christy, R.

P. Johnson and R. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

Craig, M.

L. Botten, M. Craig, R. McPhedran, J. Adams, and J. Andrewartha, “Dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

Granet, G.

Guizal, B.

Gundu, K.

Haggans, C.

Jazayeri, A.

Johnson, P.

P. Johnson and R. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

Khavasi, A.

Lalanne, P.

Li, L.

Lyndin, N.

Mafi, A.

McPhedran, R.

L. Botten, M. Craig, R. McPhedran, J. Adams, and J. Andrewartha, “Dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

Mehrany, K.

Morris, G.

Nevière, M.

Parriaux, O.

Petit, R.

J. Suratteau, M. Cadilhac, and R. Petit, “Sur la détermination numérique des efficacités de certains réseaux diélectriques profonds,” J. Opt. (Paris) 14, 273–288 (1983).
[CrossRef]

Popov, E.

Suratteau, J.

J. Suratteau, M. Cadilhac, and R. Petit, “Sur la détermination numérique des efficacités de certains réseaux diélectriques profonds,” J. Opt. (Paris) 14, 273–288 (1983).
[CrossRef]

Tishchenko, A.

J. Opt. (Paris)

J. Suratteau, M. Cadilhac, and R. Petit, “Sur la détermination numérique des efficacités de certains réseaux diélectriques profonds,” J. Opt. (Paris) 14, 273–288 (1983).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Acta

L. Botten, M. Craig, R. McPhedran, J. Adams, and J. Andrewartha, “Dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

Opt. Lett.

Phys. Rev. B

P. Johnson and R. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Scattering from a binary grating (only one period is shown).

Fig. 2
Fig. 2

Depicted in this picture are the propagating (straight solid and dashed lines), evanescent (solid curves), and growing (dashed curves) Rayleigh modes of the superstrate and the substrate.

Fig. 3
Fig. 3

Shown in this figure is the convergence behavior of reflectivity with the CLS-FMM method when the set of infinite linear equations are truncated with a rectangular matrix of size 2 ( 2 M + 1 ) × 2 ( M + 1 ) . The computed normalized total power of the scattered field is shown as a dashed line. We observe that the computed reflectivity converges uniformly while preserving the power balance.

Fig. 4
Fig. 4

Shown in this figure is the logarithm of the norm of the residual plotted against the number of grating modes used in the computation. We see that the norm of the residual converges to zero.

Fig. 5
Fig. 5

Plotted in this figure is the transmittivity computed with FMM and CLS-FMM for parameters ϵ 1 = 2.9 , ϵ 2 = 3.1 , ϵ r = 2.25 , ϵ b = 2.25 , Λ = 0.285 μ m , h = 0.280 μ m , and ρ = 0.5 . We see that the transmittivity plots lie on top of each other for various values of reduction factor r = 1 (FMM) and r = 0.90 , 0.75 , 0.50 (CLS-FMM).

Fig. 6
Fig. 6

Plotted in this figure is the reflectivity computed with FMM and CLS-FMM for a lossy grating with parameters ϵ 1 = ϵ A u ( λ ) , ϵ 2 = 1.69 , ϵ r = 2.25 , ϵ b = 1.69 , Λ = 0.330 μ m , h = 0.180 μ m , and ρ = 0.77 . We see good agreement between the two methods for various values of reduction factor r = 1 (FMM) and r = 0.90 , 0.75 , 0.50 (CLS-FMM). The top curve corresponds to the normalized power of the scattered field and is less than unity due to losses.

Equations (34)

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H y ( x , z ) = { m ( r m e i k r m ( z h / 2 ) + r m + e i k r m ( z h / 2 ) ) e i 2 π m x / Λ , z > h / 2 n ( g n e i k g n ( z + h / 2 ) + g n + e i k g n ( z h / 2 ) ) H y n g ( x ) , | z | < h / 2 m ( t m + e i k b m ( z + h / 2 ) + t m e i k b m ( z + h / 2 ) ) e i 2 π m x / Λ , z < h / 2 , }
v ± = M ± g ,
M ± = 1 2 ( [ H y g 0 0 H y g ] [ I X X I ] ± ω ϵ 0 [ K r 1 E x g 0 0 K b 1 E x g ] [ I X X I ] ) .
v + = [ r + t + ] ,     v = [ r t ] ,     g = [ g + g ] .
H y ( x ) = ( c 1 e γ z + c 2 e γ z ) e i κ x ,
P z = i γ ω ϵ ( c 1 c 2 c 1 c 2 ) ,
Re ( ( v + v + ) K ( v v + ) ) = 2 L ( g ) ,
K = [ K r 0 0 K b ] .
2 L ( g ) = g ( M g + M g ) g ,
M g = [ I X X I ] [ N g 0 0 N g ] [ I X X I ] ,
N g = ω ϵ 0 H y g E x g .
Re ( ( v + v + ) K ( v v + ) ) = Re ( ( v + δ ) K ( v δ ) ) ,
Re ( ( v + δ ) K ( v δ ) ) = 2 L ( g ) .
v ( K + K ) v + g ( M g + M g ) g = δ ( K + K ) δ .
minimize ( M + g δ ) ( M + g δ )
subject   to   g ( M ( K + K ) M + ( M g + M g ) ) g = δ ( K + K ) δ .
2 M + ( M + g δ ) ,
2 ( M ( K + K ) M + ( M g + M g ) ) g .
ϵ A u ( λ ) = ( 13.107 + 24.1895 i ) λ 4 ( 72.628 + 114.406 i ) λ 3 + ( 85.1995 + 206.83 i ) λ 2 ( 104.134 + 155.931 i ) λ + ( 36.4411 + 42.2957 i ) .
r + r + = H y g ( X g + g + ) ,
K r ( r r + ) = ω ϵ 0 E x g ( X g g + ) .
t + + t = H y g ( g + X g + ) ,
K b ( t + t ) = ω ϵ 0 E x g ( g X g + ) .
v ± = M ± g ,
M ± = 1 2 ( [ H y g 0 0 H y g ] [ I X X I ] ± ω ϵ 0 [ K r 1 E x g 0 0 K b 1 E x g ] [ I X X I ] ) .
ω ϵ 0 H y g E x g = N g ,
2 P r = Re ( ( r + r + ) K r ( r r + ) )
= Re ( ( X g + g + ) N g ( X g g + ) ) .
2 P b = Re ( ( t + + t ) K b ( t + t ) )
= Re ( ( g + X g + ) N g ( g X g + ) ) .
2 L ( g ) = 2 P r 2 P b = g ( M g + M g ) g ,
M g = [ I X X I ] [ N g 0 0 N g ] [ I X X I ] .
Re ( ( v + v + ) K ( v v + ) ) = 2 L ( g ) ,
K = [ K r 0 0 K b ] .

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