Abstract

The article presents a new method for rigorous simulation of the light diffraction on one-dimensional gratings. The method is capable to solve metal-dielectric structures in linear time and consumed memory with respect to structure complexity. Exceptional performance and convergence for metal gratings are achieved by implementing a curvilinear coordinate transformation into the generalized source method previously developed for dielectric gratings.

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References

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  1. A. V. Tishchenko, “Generalized source method: new possibilities for waveguide and grating problems,” Opt. Quantum Electron.32(6/8), 971–980 (2000).
    [CrossRef]
  2. A. A. Shcherbakov and A. V. Tishchenko, “Fast numerical method for modeling one-dimensional diffraction gratings,” Quantum Electron.40(6), 538–544 (2010).
    [CrossRef]
  3. A. A. Shcherbakov and A. V. Tishchenko, “New fast and memory-sparing method for rigorous electromagnetic analysis of 2d periodic dielectric structures,” J. Quant. Spectrosc. Radiat. Transf.113(2), 158–171 (2012).
    [CrossRef]
  4. E. Popov, Gratings: Theory and Numerical Applications (Presses Universitaires de Provence, 2012).
  5. J. Chandezon, D. Maystre, and G. Raoult, “A new theoretical method for diffraction gratings and its numerical applications,” J. Optics (Paris)11(4), 235–241 (1980).
    [CrossRef]
  6. K. Edee, J. P. Plumey, and G. Granet, “On the Rayleigh-Fourier method and the Chandezon method: comparative study,” Opt. Commun.286, 34–41 (2013).
    [CrossRef]
  7. J. A. Schouten, Tensor Analysis for Physicists (Dover Publications, 1954).
  8. J. A. Stratton and L. J. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev.56(1), 99–107 (1939).
    [CrossRef]
  9. A. V. Tishchenko, “A generalized source method for wave propagation,” Pure Appl. Opt.7(6), 1425–1449 (1998).
    [CrossRef]
  10. G. Granet and B. Guizal, “Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization,” J. Opt. Soc. Am. A13(5), 1019–1023 (1996).
    [CrossRef]
  11. P. Lalanne and G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A13(4), 779–784 (1996).
    [CrossRef]
  12. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A13(9), 1870–1876 (1996).
    [CrossRef]
  13. Y. Saad, Iterative Methods for Sparse Linear Systems (SIAM, 2003).
  14. A. V. Tishchenko, “Numerical demonstration of the validity of the Rayleigh hypothesis,” Opt. Express17(19), 17102–17117 (2009).
    [CrossRef] [PubMed]
  15. I. Avrutsky, Y. Zhao, and V. Kochergin, “Surface-plasmon-assisted resonant tunneling of light through a periodically corrugated thin metal film,” Opt. Lett.25(9), 595–597 (2000).
    [CrossRef] [PubMed]
  16. Y. Jourlin, S. Tonchev, A. V. Tishchenko, C. Pedri, C. Veillas, O. Parriaux, A. Last, and Y. Lacroute, “Spatially and polarization resolved plasmon mediated transmission through continuous metal films,” Opt. Express17(14), 12155–12166 (2009).
    [CrossRef] [PubMed]

2013 (1)

K. Edee, J. P. Plumey, and G. Granet, “On the Rayleigh-Fourier method and the Chandezon method: comparative study,” Opt. Commun.286, 34–41 (2013).
[CrossRef]

2012 (1)

A. A. Shcherbakov and A. V. Tishchenko, “New fast and memory-sparing method for rigorous electromagnetic analysis of 2d periodic dielectric structures,” J. Quant. Spectrosc. Radiat. Transf.113(2), 158–171 (2012).
[CrossRef]

2010 (1)

A. A. Shcherbakov and A. V. Tishchenko, “Fast numerical method for modeling one-dimensional diffraction gratings,” Quantum Electron.40(6), 538–544 (2010).
[CrossRef]

2009 (2)

2000 (2)

I. Avrutsky, Y. Zhao, and V. Kochergin, “Surface-plasmon-assisted resonant tunneling of light through a periodically corrugated thin metal film,” Opt. Lett.25(9), 595–597 (2000).
[CrossRef] [PubMed]

A. V. Tishchenko, “Generalized source method: new possibilities for waveguide and grating problems,” Opt. Quantum Electron.32(6/8), 971–980 (2000).
[CrossRef]

1998 (1)

A. V. Tishchenko, “A generalized source method for wave propagation,” Pure Appl. Opt.7(6), 1425–1449 (1998).
[CrossRef]

1996 (3)

1980 (1)

J. Chandezon, D. Maystre, and G. Raoult, “A new theoretical method for diffraction gratings and its numerical applications,” J. Optics (Paris)11(4), 235–241 (1980).
[CrossRef]

1939 (1)

J. A. Stratton and L. J. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev.56(1), 99–107 (1939).
[CrossRef]

Avrutsky, I.

Chandezon, J.

J. Chandezon, D. Maystre, and G. Raoult, “A new theoretical method for diffraction gratings and its numerical applications,” J. Optics (Paris)11(4), 235–241 (1980).
[CrossRef]

Chu, L. J.

J. A. Stratton and L. J. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev.56(1), 99–107 (1939).
[CrossRef]

Edee, K.

K. Edee, J. P. Plumey, and G. Granet, “On the Rayleigh-Fourier method and the Chandezon method: comparative study,” Opt. Commun.286, 34–41 (2013).
[CrossRef]

Granet, G.

K. Edee, J. P. Plumey, and G. Granet, “On the Rayleigh-Fourier method and the Chandezon method: comparative study,” Opt. Commun.286, 34–41 (2013).
[CrossRef]

G. Granet and B. Guizal, “Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization,” J. Opt. Soc. Am. A13(5), 1019–1023 (1996).
[CrossRef]

Guizal, B.

Jourlin, Y.

Kochergin, V.

Lacroute, Y.

Lalanne, P.

Last, A.

Li, L.

Maystre, D.

J. Chandezon, D. Maystre, and G. Raoult, “A new theoretical method for diffraction gratings and its numerical applications,” J. Optics (Paris)11(4), 235–241 (1980).
[CrossRef]

Morris, G. M.

Parriaux, O.

Pedri, C.

Plumey, J. P.

K. Edee, J. P. Plumey, and G. Granet, “On the Rayleigh-Fourier method and the Chandezon method: comparative study,” Opt. Commun.286, 34–41 (2013).
[CrossRef]

Raoult, G.

J. Chandezon, D. Maystre, and G. Raoult, “A new theoretical method for diffraction gratings and its numerical applications,” J. Optics (Paris)11(4), 235–241 (1980).
[CrossRef]

Shcherbakov, A. A.

A. A. Shcherbakov and A. V. Tishchenko, “New fast and memory-sparing method for rigorous electromagnetic analysis of 2d periodic dielectric structures,” J. Quant. Spectrosc. Radiat. Transf.113(2), 158–171 (2012).
[CrossRef]

A. A. Shcherbakov and A. V. Tishchenko, “Fast numerical method for modeling one-dimensional diffraction gratings,” Quantum Electron.40(6), 538–544 (2010).
[CrossRef]

Stratton, J. A.

J. A. Stratton and L. J. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev.56(1), 99–107 (1939).
[CrossRef]

Tishchenko, A. V.

A. A. Shcherbakov and A. V. Tishchenko, “New fast and memory-sparing method for rigorous electromagnetic analysis of 2d periodic dielectric structures,” J. Quant. Spectrosc. Radiat. Transf.113(2), 158–171 (2012).
[CrossRef]

A. A. Shcherbakov and A. V. Tishchenko, “Fast numerical method for modeling one-dimensional diffraction gratings,” Quantum Electron.40(6), 538–544 (2010).
[CrossRef]

Y. Jourlin, S. Tonchev, A. V. Tishchenko, C. Pedri, C. Veillas, O. Parriaux, A. Last, and Y. Lacroute, “Spatially and polarization resolved plasmon mediated transmission through continuous metal films,” Opt. Express17(14), 12155–12166 (2009).
[CrossRef] [PubMed]

A. V. Tishchenko, “Numerical demonstration of the validity of the Rayleigh hypothesis,” Opt. Express17(19), 17102–17117 (2009).
[CrossRef] [PubMed]

A. V. Tishchenko, “Generalized source method: new possibilities for waveguide and grating problems,” Opt. Quantum Electron.32(6/8), 971–980 (2000).
[CrossRef]

A. V. Tishchenko, “A generalized source method for wave propagation,” Pure Appl. Opt.7(6), 1425–1449 (1998).
[CrossRef]

Tonchev, S.

Veillas, C.

Zhao, Y.

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

J. Optics (Paris) (1)

J. Chandezon, D. Maystre, and G. Raoult, “A new theoretical method for diffraction gratings and its numerical applications,” J. Optics (Paris)11(4), 235–241 (1980).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transf. (1)

A. A. Shcherbakov and A. V. Tishchenko, “New fast and memory-sparing method for rigorous electromagnetic analysis of 2d periodic dielectric structures,” J. Quant. Spectrosc. Radiat. Transf.113(2), 158–171 (2012).
[CrossRef]

Opt. Commun. (1)

K. Edee, J. P. Plumey, and G. Granet, “On the Rayleigh-Fourier method and the Chandezon method: comparative study,” Opt. Commun.286, 34–41 (2013).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Opt. Quantum Electron. (1)

A. V. Tishchenko, “Generalized source method: new possibilities for waveguide and grating problems,” Opt. Quantum Electron.32(6/8), 971–980 (2000).
[CrossRef]

Phys. Rev. (1)

J. A. Stratton and L. J. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev.56(1), 99–107 (1939).
[CrossRef]

Pure Appl. Opt. (1)

A. V. Tishchenko, “A generalized source method for wave propagation,” Pure Appl. Opt.7(6), 1425–1449 (1998).
[CrossRef]

Quantum Electron. (1)

A. A. Shcherbakov and A. V. Tishchenko, “Fast numerical method for modeling one-dimensional diffraction gratings,” Quantum Electron.40(6), 538–544 (2010).
[CrossRef]

Other (3)

E. Popov, Gratings: Theory and Numerical Applications (Presses Universitaires de Provence, 2012).

J. A. Schouten, Tensor Analysis for Physicists (Dover Publications, 1954).

Y. Saad, Iterative Methods for Sparse Linear Systems (SIAM, 2003).

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Figures (5)

Fig. 1
Fig. 1

Problem under consideration: a plane wave diffraction on a smooth-profiled 1D metal- dielectric grating.

Fig. 2
Fig. 2

Transformation of a grating corrugation profile. (a) Curvilinear coordinates introduced in a bounded region Ω = {–b ≤□x3 ≤□b | b ≥□c}. (b) Grating profile becomes a plane interface z3 = 0, and everywhere outside Ω the new coordinates coincide with the Cartesian ones.

Fig. 3
Fig. 3

(a) Gold sinusoidal grating example and coordinate lines of the introduced curvilinear coordinate transformation. (b) Thin sinusoidal gold film placed in the air.

Fig. 4
Fig. 4

Simulation results for the diffraction of a plane wave of wavelength λ = 0.6328 μm on a metal-dielectric sinusoidal grating with refractive index ng = 0.25 + 6.25i placed onto a substrate of index ns = 1.5, and covered by air (nc = 1) with c = 0.1 μm, b = 0.11 μm, and Λ = 1 μm. (a) Convergence of the GSMCC and comparison of the GSMCC with the Rayleigh method with better than 10−8 accuracy. (b) Convergence of the GSMCC with the increasing number of diffraction orders for large number of slices NS = 2048.

Fig. 5
Fig. 5

Reflection and transmission curves of a plane wave of wavelength λ = 0.6328 μm incident under 10° angle on a thin sinusoidal gold film (ng = 0.25 + 6.25i) of thickness 0.02 μm and corrugation depth 0.1 μm versus grating period.

Tables (1)

Tables Icon

Table 1 Comparison of diffraction efficiencies calculated by the GSMCC and the FMM for the diffraction of a plane wave on a sinusoidal corrugation interface separating air and a substrate of refractive index 2.5. Grating period is 1 μm, grating depth is 0.1 μm. A plane wave of wavelength 0.6328 μm is incident under 10° angle from the air side

Equations (22)

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×E=iω μ 0 H, ×H=iωεE.
ξ αβγ z β E γ =iω μ 0 g g αβ H β , ξ αβγ z β H γ =iωε g g αβ E β ,
ξ αβγ z β E γ = M α +iω μ b δ αβ H β , ξ αβγ z β H γ = J α iω ε b δ αβ E β .
J=j( z 3 )exp( i k 1 z 1 +i k 2 z 2 ), M=m( z 3 )exp( i k 1 z 1 +i k 2 z 2 ).
( E α ( z ) H α ( z ) )=( E 0α ( z ) H 0α ( z ) )+exp( i k 1 z 1 +i k 2 z 2 )×{ δ α3 ( j 3 ( z 3 ) iω ε b m 3 ( z 3 ) iω μ b ) + Y αβ + z 3 ( j β ( ζ ) m β ( ζ ) )exp[ i k 3 ( z 3 ζ ) ]dζ + Y αβ z 3 ( j β ( ζ ) m β ( ζ ) )exp[ i k 3 ( z 3 ζ ) ]dζ }
Y αβ ± =( k α ± k β ± δ αβ k b 2 2ω ε b k 3 ξ αγβ k γ 2 k 3 ξ αγβ k γ 2 k 3 k α ± k β ± δ αβ k b 2 2ω μ b k 3 ).
( J α M α )=iω V αβ ( E β H β ),
V αβ =( ε ^ αβ ε b δ αβ 0 0 μ ^ αβ μ b δ αβ ).
E ˜ 3 ( z )= E 3 ( z ) δ 33 J 3 ( z ) iω ε b , H ˜ 3 ( z )= H 3 ( z ) δ 33 M 3 ( z ) iω μ b .
E ˜ 3 = E 3 + δ 33 ( ε ^ 3β ε b δ 3β ) E β ε b = δ 33 ε ^ 3β ε b E β , H ˜ 3 = H 3 + δ 33 ( μ ^ 3β μ b δ 3β ) H β μ b = δ 33 μ ^ 3β μ b H β .
f( z )=exp( i k 2 inc z 2 ) n= f n ( z 3 )exp( i k 1n z 1 )
( E αm ( z 3 ) H αm ( z 3 ) )=( E 0α ( z 3 ) H 0α ( z 3 ) ) δ m0 iω{ z 3 exp[ i k 3 ( z 3 ζ ) ] Y αβ + V βγ ( E γ ( ζ ) H γ ( ζ ) )dζ + z 3 exp[ i k 3 ( z 3 ζ ) ] Y αβ V βγ ( E γ ( ζ ) H γ ( ζ ) )dζ }.
( E ˜ 1m E ˜ 2m E ˜ 3m H ˜ 1m H ˜ 2m H ˜ 3m )= Q m ( a em + a em a hm + a hm ),
Q m =( k 2 γ m k 2 γ k 1m k 3m ω ε b γ m k 1m k 3m ω ε b γ m k 1m γ m k 1m γ m k 2 k 3m ω ε b γ m k 2 k 3m ω ε b γ m 0 0 γ m ω ε b γ m ω ε b k 1m k 3m ω μ b γ m k 1m k 3m ω μ b γ m k 2 γ m k 2 γ m k 2 k 3m ω μ b γ m k 2 k 3m ω μ b γ m k 1m γ m k 1m γ m γ m ω μ b γ m ω μ b 0 0 ),
( a em + a em a hm + a hm )= P m ( J 1m J 2m J 3m M 1m M 2m M 3m )
P m = 1 2 ( ω μ b k 2 inc γ m k 3m ω μ b k 1m γ m k 3m 0 k 1m γ m k 2 inc γ γ m k 3m ω μ b k 2 inc γ k 3m ω μ b k 1m γ m k 3m 0 k 1m γ m k 2 inc γ m γ m k 3m k 1m γ m k 2 inc γ m γ m k z ω ε b k 2 inc γ m k 3m ω ε b k 1m γ m k 3m 0 k 1m γ m k 2 inc γ m γ m k z ω ε b k 2 inc γ m k 3m ω ε b k 1m γ m k 3m 0 ).
R mpq ± = θ pq ± Δhexp[ ±i k 3m ( z p 3 z q 3 ) ],
θ s ± = 1 2 +{ ± 1 2 ,s>0 0,s=0 1 2 ,s<0 .
a= ( IRPVQ ) 1 a inc
a out = a inc +TPVQ ( IRPVQ ) 1 a inc .
T mp + | z 3 =b =exp[ i k 3m ( b z p 3 ) ], T mp | z 3 =b =exp[ i k 3m ( b+ z p 3 ) ], T mp + | z 3 =b = T mp | z 3 =b =0.
x 1 z 1 , x 2 z 2 , x 3 ={ c( 1 | z 3 | /b )sin( K x 2 )+ z 3 , | z 3 |b, z 3 , | z 3 |>b.

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