Abstract

A new modeling system to determine the optical response function of a multilayer structure with imposed periodicity in the plane of the layers, a multilayer diffraction grating, is described. This new model has two essential ingredients. This model is based on the well-established coordinate transformation procedure developed by Chandezon et al. [ J. Opt. Soc. Am. 72, 839– 846 ( 1982)] in which a periodically modulated surface is transformed into a frame in which it is flat, permitting simpler use of Maxwell’s boundary conditions. Then, instead of using the conventional transfer-matrix method, we developed a scattering-matrix technique that permits the modeling of very thick (of the order of 1 μm or greater) multilayer systems with many field components without numerical instability. Model programs have been developed based on this new scattering-matrix approach and tested by comparison with other models and experimental data.

© 1995 Optical Society of America

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References

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  1. R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980).
    [CrossRef]
  2. M. Nevière, Thèse d’Etat (Centre National de la Recherche Scientifique, Marseille, France, 1975).
  3. M. Nevière, P. Vincent, “Differential theory of gratings: answer to an objection on its validity for TM polarization,” J. Opt. Soc. Am. A 5, 1522–1524 (1988).
    [CrossRef]
  4. E. Popov, L. Mashev, “Convergence of Rayleigh Fourier method and rigorous differential method for relief diffraction gratings,” Opt. Acta 33, 593–605 (1986).
    [CrossRef]
  5. D. M. Pai, K. A. Awada, “Analysis of dielectric gratings of arbitrary profiles and thicknesses,” J. Opt. Soc. Am. A 8, 755–762 (1991).
    [CrossRef]
  6. L. F. De Sandre, J. M. Elson, “Extinction theorem analysis of diffraction anomalies in overcoated gratings,” J. Opt. Soc. Am. A 8, 763–777 (1991).
    [CrossRef]
  7. L. Li, “Multilayer modal method for diffraction gratings of arbitrary profile, depth and conductivity,” J. Opt. Soc. Am. A 10, 2581–2591 (1993).
    [CrossRef]
  8. M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 71, 811–818 (1981).
    [CrossRef]
  9. M. G. Moharam, T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. 72, 1385–1392 (1982).
    [CrossRef]
  10. M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of metallic surface-relief gratings,” J. Opt. Soc. Am. 11, 1780–1787 (1986).
    [CrossRef]
  11. J. Chandezon, M. T. Dupuis, G. Cornet, D. Maystre, “Multicoated gratings: a differential formalism applicable in the entire optical region,” J. Opt. Soc. Am. 72, 839–846 (1982).
    [CrossRef]
  12. S. J. Elston, G. P. Bryan-Brown, T. W. Preist, J. R. Sambles, “Surface resonance polarization conversion mediated by broken surface symmetry,” Phys. Rev. B 44, 3483–3485 (1991).
    [CrossRef]
  13. S. J. Elston, G. P. Bryan-Brown, J. R. Sambles, “Polarization conversion from diffraction gratings,” Phys. Rev. B 44, 6393–6400 (1991).
    [CrossRef]
  14. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).
  15. P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).
  16. D. Y. K. Ko, J. R. Sambles, “Scattering matrix method for propagation of radiation in stratified media: attenuated total reflection studies of liquid crystals,” J. Opt. Soc. Am. A 5, 1863–1866 (1988).
    [CrossRef]
  17. E. Popov, L. Mashev, “Rigorous electromagnetic treatment of planar corrugated waveguides,” J. Opt. Commun. 7, 127–131 (1986).
  18. I. Pockrand, “Resonance anomalies in the light intensity reflected at silver gratings with dielectric coatings,” J. Phys. D 9, 2423–2432 (1976).
    [CrossRef]
  19. G. P. Bryan-Brown, “Optical excitation of electromagnetic modes using grating coupling,” Ph.D. dissertation (University of Exeter, Exeter, UK, 1991).

1993 (1)

1991 (4)

D. M. Pai, K. A. Awada, “Analysis of dielectric gratings of arbitrary profiles and thicknesses,” J. Opt. Soc. Am. A 8, 755–762 (1991).
[CrossRef]

L. F. De Sandre, J. M. Elson, “Extinction theorem analysis of diffraction anomalies in overcoated gratings,” J. Opt. Soc. Am. A 8, 763–777 (1991).
[CrossRef]

S. J. Elston, G. P. Bryan-Brown, T. W. Preist, J. R. Sambles, “Surface resonance polarization conversion mediated by broken surface symmetry,” Phys. Rev. B 44, 3483–3485 (1991).
[CrossRef]

S. J. Elston, G. P. Bryan-Brown, J. R. Sambles, “Polarization conversion from diffraction gratings,” Phys. Rev. B 44, 6393–6400 (1991).
[CrossRef]

1988 (2)

1986 (3)

E. Popov, L. Mashev, “Convergence of Rayleigh Fourier method and rigorous differential method for relief diffraction gratings,” Opt. Acta 33, 593–605 (1986).
[CrossRef]

M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of metallic surface-relief gratings,” J. Opt. Soc. Am. 11, 1780–1787 (1986).
[CrossRef]

E. Popov, L. Mashev, “Rigorous electromagnetic treatment of planar corrugated waveguides,” J. Opt. Commun. 7, 127–131 (1986).

1982 (2)

1981 (1)

1976 (1)

I. Pockrand, “Resonance anomalies in the light intensity reflected at silver gratings with dielectric coatings,” J. Phys. D 9, 2423–2432 (1976).
[CrossRef]

Awada, K. A.

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

Bryan-Brown, G. P.

S. J. Elston, G. P. Bryan-Brown, T. W. Preist, J. R. Sambles, “Surface resonance polarization conversion mediated by broken surface symmetry,” Phys. Rev. B 44, 3483–3485 (1991).
[CrossRef]

S. J. Elston, G. P. Bryan-Brown, J. R. Sambles, “Polarization conversion from diffraction gratings,” Phys. Rev. B 44, 6393–6400 (1991).
[CrossRef]

G. P. Bryan-Brown, “Optical excitation of electromagnetic modes using grating coupling,” Ph.D. dissertation (University of Exeter, Exeter, UK, 1991).

Chandezon, J.

Cornet, G.

De Sandre, L. F.

Dupuis, M. T.

Elson, J. M.

Elston, S. J.

S. J. Elston, G. P. Bryan-Brown, T. W. Preist, J. R. Sambles, “Surface resonance polarization conversion mediated by broken surface symmetry,” Phys. Rev. B 44, 3483–3485 (1991).
[CrossRef]

S. J. Elston, G. P. Bryan-Brown, J. R. Sambles, “Polarization conversion from diffraction gratings,” Phys. Rev. B 44, 6393–6400 (1991).
[CrossRef]

Gaylord, T. K.

Ko, D. Y. K.

Li, L.

Mashev, L.

E. Popov, L. Mashev, “Convergence of Rayleigh Fourier method and rigorous differential method for relief diffraction gratings,” Opt. Acta 33, 593–605 (1986).
[CrossRef]

E. Popov, L. Mashev, “Rigorous electromagnetic treatment of planar corrugated waveguides,” J. Opt. Commun. 7, 127–131 (1986).

Maystre, D.

Moharam, M. G.

Nevière, M.

Pai, D. M.

Pockrand, I.

I. Pockrand, “Resonance anomalies in the light intensity reflected at silver gratings with dielectric coatings,” J. Phys. D 9, 2423–2432 (1976).
[CrossRef]

Popov, E.

E. Popov, L. Mashev, “Rigorous electromagnetic treatment of planar corrugated waveguides,” J. Opt. Commun. 7, 127–131 (1986).

E. Popov, L. Mashev, “Convergence of Rayleigh Fourier method and rigorous differential method for relief diffraction gratings,” Opt. Acta 33, 593–605 (1986).
[CrossRef]

Preist, T. W.

S. J. Elston, G. P. Bryan-Brown, T. W. Preist, J. R. Sambles, “Surface resonance polarization conversion mediated by broken surface symmetry,” Phys. Rev. B 44, 3483–3485 (1991).
[CrossRef]

Sambles, J. R.

S. J. Elston, G. P. Bryan-Brown, T. W. Preist, J. R. Sambles, “Surface resonance polarization conversion mediated by broken surface symmetry,” Phys. Rev. B 44, 3483–3485 (1991).
[CrossRef]

S. J. Elston, G. P. Bryan-Brown, J. R. Sambles, “Polarization conversion from diffraction gratings,” Phys. Rev. B 44, 6393–6400 (1991).
[CrossRef]

D. Y. K. Ko, J. R. Sambles, “Scattering matrix method for propagation of radiation in stratified media: attenuated total reflection studies of liquid crystals,” J. Opt. Soc. Am. A 5, 1863–1866 (1988).
[CrossRef]

Vincent, P.

Yeh, P.

P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).

J. Opt. Commun. (1)

E. Popov, L. Mashev, “Rigorous electromagnetic treatment of planar corrugated waveguides,” J. Opt. Commun. 7, 127–131 (1986).

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (5)

J. Phys. D (1)

I. Pockrand, “Resonance anomalies in the light intensity reflected at silver gratings with dielectric coatings,” J. Phys. D 9, 2423–2432 (1976).
[CrossRef]

Opt. Acta (1)

E. Popov, L. Mashev, “Convergence of Rayleigh Fourier method and rigorous differential method for relief diffraction gratings,” Opt. Acta 33, 593–605 (1986).
[CrossRef]

Phys. Rev. B (2)

S. J. Elston, G. P. Bryan-Brown, T. W. Preist, J. R. Sambles, “Surface resonance polarization conversion mediated by broken surface symmetry,” Phys. Rev. B 44, 3483–3485 (1991).
[CrossRef]

S. J. Elston, G. P. Bryan-Brown, J. R. Sambles, “Polarization conversion from diffraction gratings,” Phys. Rev. B 44, 6393–6400 (1991).
[CrossRef]

Other (5)

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).

R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980).
[CrossRef]

M. Nevière, Thèse d’Etat (Centre National de la Recherche Scientifique, Marseille, France, 1975).

G. P. Bryan-Brown, “Optical excitation of electromagnetic modes using grating coupling,” Ph.D. dissertation (University of Exeter, Exeter, UK, 1991).

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Figures (4)

Fig. 1
Fig. 1

General system analyzed in this paper; there are Q layers of refractive index νj and thickness ej (j = 1, …, Q) with upper interface j and lower interface j − 1.

Fig. 2
Fig. 2

Comparison of transfer- and scattering-matrix reflectivity data for an air/dielectric/silver grating with a groove depth of 20 nm and a pitch of 800 nm at a wavelength of 632.8 nm. Circles indicate transfer-matrix theory, and the curve indicates scattering-matrix theory.

Fig. 3
Fig. 3

Comparisons between theory and experimental p-reflectivity data measured by Pockrand18 for grating structures of silica/photoresist/silver/air with different grating amplitudes. The gratings have a pitch of 891 nm, and the incident light has a wavelength of 550 nm. The curves are the theoretical fits, and the symbols indicate the experimental data for grating amplitudes and distortion parameters: (a) 24 nm, γ = 0.868, (b) 46.5 nm, γ = 1.038, (c) 82 nm, γ = 0.674.

Fig. 4
Fig. 4

Comparison between theory and experimental p-reflectivity data measured by Bryan-Brown19 using light of wavelength 632.8 nm for an 800.8-nm pitch silver grating coated with 325 nm of MgF. The data were fitted with a grating amplitude of 10.15 nm and a distortion factor of γ = 1.246. Circles indicate experiment, and the curve indicates theory.

Equations (59)

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y j = i j + 1 Q e i + α ( x ) ,
F = E z , G = ( ω / c ) Z 0 H
F = Z 0 H z , G = ( ω / c ) ν 2 E
Z 0 = ( μ 0 / 0 ) 1 / 2
F ( u , υ ) = m F m ( u ) exp ( i α m υ ) ,
G ( u , υ ) = m G m ( u ) exp ( i α m υ ) ,
α m = υ Q + 1 ( ω / c ) sin θ + 2 π m / λ g ,
ξ j ( u ) = M j ϕ j ( u ) b j ,
ξ p j ( u ) = M p q j ϕ q j ( u ) b q j ,
[ ϕ j ( u ) ] p q = ϕ q j ( u ) δ p q ,
ϕ q j ( u ) = exp ( i r q j u )
ξ m ( u ) ( F m , F m + 1 , . . . , F m ; Ĝ m , Ĝ m + 1 , . . . , Ĝ m ) ,
u = d j = i = j + 1 Q e i .
ξ j ( u ) = M j ϕ j ( u d j 1 ) b j ,
M j ϕ j ( d j d j 1 ) b j = M j + 1 ϕ j + 1 ( d j d j ) b j + 1
b j = [ ϕ j ( e j ) ] 1 ( M j ) 1 M j + 1 b j + 1 I ( j + 1 ) b j + 1 ,
d j d j 1 = i = j + 1 Q e i + i = j Q e i = e j .
I ( j + 1 ) = [ ϕ j ( e j ) ] 1 ( M j ) 1 M j + 1
( b + j b 0 ) = S ( 0 , j ) ( b + 0 b j ) .
( b + j b j ) = I ( j + 1 ) ( b + j + 1 b j + 1 ) ,
[ X 11 X 12 X 21 X 22 ] .
( b + j + 1 b 0 ) = S ( 0 , j + 1 ) ( b + 0 b j + 1 ) ,
S 11 ( 0 , j + 1 ) = [ I 11 ( j + 1 ) S 12 ( 0 , j ) I 21 ( j + 1 ) ] 1 × S 11 ( 0 , j ) ,
S 12 ( 0 , j + 1 ) = [ I 11 ( j + 1 ) S 12 ( 0 , j ) I 21 ( j + 1 ) ] 1 × [ S 12 ( 0 , j ) I 22 ( j + 1 ) I 12 ( j + 1 ) ] ,
S 21 ( 0 , j + 1 ) = [ S 22 ( 0 , j ) I 21 ( j + 1 ) S 11 ( 0 , j + 1 ) + S 21 ( 0 , j ) ] ,
S 22 ( 0 , j + 1 ) = [ S 22 ( 0 , j ) I 21 ( j + 1 ) S 12 ( 0 , j + 1 ) + S 22 ( 0 , j ) I 22 ( j + 1 ) ] .
( b + Q + 1 b 0 ) = S ( 0 , Q + 1 ) ( b + 0 b Q + 1 ) .
( b + Q + 1 b 0 ) = S ( b + 0 b Q + 1 ) ,
S = [ S 11 S 12 S 21 S 22 ] ,
b + Q + 1 = S 12 b Q + 1 ,
b 0 = S 22 b Q + 1 ,
ξ Q + 1 = ( M Q + 1 ) ( b + Q + 1 b Q + 1 ) .
( l l ) .
( M Q + 1 ) ( 0 ( b ̂ + ) q 0 ) ,
( M M ) ( R n ) ,
( M Q + 1 ) ( b + Q + 1 b Q + 1 ) = ( M Q + 1 ) ( 0 ( b ̂ + ) q 0 ) + ( l l ) + ( M M ) ( R n )
= ( M ̂ Q + 1 ) ( R n ( b ̂ + ) q 0 ) + ( l l )
= ( M ̂ Q + 1 ) ( R ̂ ) + ( l l ) ,
M ̂ Q + 1 = [ M ̂ 11 0 M ̂ 21 0 ] ,
R ̂ = ( R n ( b ̂ + ) q 0 ) = ( R 0 ) ,
( M ̂ 11 ) k k = ( M ) k k , ( M ̂ 21 ) k k = ( M ) k k , k = 1 , . . . , P ,
( M ̂ 11 ) k k = ( M 11 ) k k , ( M ̂ 21 ) k k = ( M 21 ) k k , k = P + 1 , . . . , 2 N + 1 , k = 1 , . . . , 2 N + 1 .
R = [ M ̂ 11 ( M 11 S 12 + M 12 ) ( M 21 S 12 + M 22 ) 1 M ̂ 21 ] 1 × [ ( M 11 S 12 + M 12 ) ( M 21 S 12 + M 22 ) 1 l = l ] ,
b 0 = S 22 ( M 21 S 12 + M 22 ) 1 ( M ̂ 21 R + l ) X Q + 1 .
b 0 = X Q + 1 ,
( M 0 ) ( 0 b 0 ) = ( M 0 ) ( 0 X Q + 1 ) .
( M 0 ) ( 0 0 ( b ̂ 0 ) k ) + ( M 0 M 0 ) T n = ( M 0 ) ( 0 X Q + 1 ) ,
( M ̂ ) ( 0 T ) = ( M ) ( 0 X Q + 1 ) ,
M ̂ = [ M 11 M ̂ 12 M 21 M ̂ 22 ] ,
( M ̂ 12 ) k k = ( M ) k k , ( M ̂ 21 ) k k = ( M ) k k , k = 1 , . . . , P ,
( M ̂ 12 ) k k = ( M 11 ) k k , ( M ̂ 22 ) k k = ( M 21 ) k k , k = P + 1 , . . . , 2 N + 1 ,
( T n ( b ̂ ) k ) .
M 1 ( M ̂ ) ( 0 T ) = ( 0 X Q + 1 )
Z = M 1 M ̂ = [ I 0 0 Z 22 ] ,
T = Z 22 1 X Q + 1 .
a ( x ) = A { [ cos ( 2 π x / λ g ) + 1 2 ] γ 1 2 } ,
I ( j + 1 ) = [ D 1 N 11 D 1 N 12 D 2 N 21 D 2 N 22 ] ,
S 11 ( j + 1 ) = [ N 11 Q 12 ( j ) N 21 ] 1 D 1 1 S 11 ( j ) , S 12 ( j + 1 ) = [ N 11 Q 12 ( j ) N 21 ] 1 [ Q 12 ( j ) N 22 N 12 ] , S 21 ( j + 1 ) = S 22 ( j ) D 2 N 21 S 11 ( j + 1 ) + S 21 ( j ) , S 22 ( j + 1 ) = S 22 ( j ) D 2 N 21 S 12 ( j + 1 ) + S 22 ( j ) D 2 N 22 ,
Q 12 ( j ) = D 1 1 S 12 ( j ) D 2 .

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