Abstract

A model is proposed that describes diffraction by a microrelief grating. A microrelief grating is a grating of large period with grooves containing a periodic microrelief with a period considerably smaller than the period of the grating. In the model the interaction of the incident wave with the fine structure is taken into account rigorously while features of the grating that are large compared with the wavelength are modeled as phase or amplitude objects.

© 2001 Optical Society of America

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References

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  1. P. Dirksen, C. A. Juffermans, A. Leeuwestein, C. Mutsaers, T. A. Nuijs, R. Pellens, R. Wolters, J. Gemen, “Effect of processing on the overlay performance of a wafer stepper,” in Metrology, Inspection, and Process Control for Microlithography XI, S. K. Jones, ed., Proc. SPIE3050, 102–113 (1997).
    [CrossRef]
  2. C. P. Kirk, “Theoretical models for the optical alignment of wafer steppers,” in Optical Microlithography VI, H. L. Stover, ed., Proc. SPIE772, 134–141 (1987).
    [CrossRef]
  3. C. P. Kirk, D. Nyyssonen, “Modeling the optical microscope images of thick layers for the purpose of linewidth measurement,” in Optical Microlithography IV, H. L. Stover, ed., Proc. SPIE538, 179–187 (1985).
    [CrossRef]
  4. M. G. Moharam, T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. A 72, 1385–1392 (1982).
    [CrossRef]
  5. L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A 14, 2758–2767 (1997).
    [CrossRef]
  6. D. Maystre, “Integral methods,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980).
  7. B. H. Kleemann, A. Mitreiter, F. Wyrowski, “Integral equation method with parametrization of grating profile: Theory and experiments,” J. Mod. Opt. 43, 1323–1349 (1996).
    [CrossRef]
  8. T. Delort, D. Maystre, “Finite element methods for gratings,” J. Opt. Soc. Am. A 10, 2592–2601 (1993).
    [CrossRef]
  9. T. Abboud, “Electromagnetic waves in periodic media,” in Mathematical and Numerical Aspects of Wave Propagation, R. Kleinman, T. Angell, D. Colton, F. Santosa, I. Stakgold, eds. (Society of Industrial and Applied Mathematics, Philadelphia, Pa., 1993).
  10. G. Bao, D. Dobson, J. A. Cox, “Mathematical studies of rigorous grating theory,” J. Opt. Soc. Am. A 12, 1029–1042 (1995).
    [CrossRef]
  11. H. P. Urbach, R. T. M. Merkx, “Finite element simulation of electromagnetic plane wave diffraction at gratings for arbitrary angles of incidence,” in Mathematical and Numerical Aspects of Wave Propagation Phenomena, G. Cohen, L. Halpern, P. Joly, eds. (Society of Industrial and Applied Mathematics, Philadelphia, 1991), pp. 89–99.
  12. A. G. Voronovich, Wave Scattering from Rough Surfaces (Springer-Verlag, Berlin, 1994).
  13. H. P. Urbach, “Convergence of the Galerkin method for two-dimensional electromagnetic problems,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 28, 697–710 (1991).
    [CrossRef]
  14. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1986).
  15. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

1997

1996

B. H. Kleemann, A. Mitreiter, F. Wyrowski, “Integral equation method with parametrization of grating profile: Theory and experiments,” J. Mod. Opt. 43, 1323–1349 (1996).
[CrossRef]

1995

1993

1991

H. P. Urbach, “Convergence of the Galerkin method for two-dimensional electromagnetic problems,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 28, 697–710 (1991).
[CrossRef]

1982

M. G. Moharam, T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. A 72, 1385–1392 (1982).
[CrossRef]

Abboud, T.

T. Abboud, “Electromagnetic waves in periodic media,” in Mathematical and Numerical Aspects of Wave Propagation, R. Kleinman, T. Angell, D. Colton, F. Santosa, I. Stakgold, eds. (Society of Industrial and Applied Mathematics, Philadelphia, Pa., 1993).

Bao, G.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1986).

Cox, J. A.

Delort, T.

Dirksen, P.

P. Dirksen, C. A. Juffermans, A. Leeuwestein, C. Mutsaers, T. A. Nuijs, R. Pellens, R. Wolters, J. Gemen, “Effect of processing on the overlay performance of a wafer stepper,” in Metrology, Inspection, and Process Control for Microlithography XI, S. K. Jones, ed., Proc. SPIE3050, 102–113 (1997).
[CrossRef]

Dobson, D.

Gaylord, T. K.

M. G. Moharam, T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. A 72, 1385–1392 (1982).
[CrossRef]

Gemen, J.

P. Dirksen, C. A. Juffermans, A. Leeuwestein, C. Mutsaers, T. A. Nuijs, R. Pellens, R. Wolters, J. Gemen, “Effect of processing on the overlay performance of a wafer stepper,” in Metrology, Inspection, and Process Control for Microlithography XI, S. K. Jones, ed., Proc. SPIE3050, 102–113 (1997).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Juffermans, C. A.

P. Dirksen, C. A. Juffermans, A. Leeuwestein, C. Mutsaers, T. A. Nuijs, R. Pellens, R. Wolters, J. Gemen, “Effect of processing on the overlay performance of a wafer stepper,” in Metrology, Inspection, and Process Control for Microlithography XI, S. K. Jones, ed., Proc. SPIE3050, 102–113 (1997).
[CrossRef]

Kirk, C. P.

C. P. Kirk, “Theoretical models for the optical alignment of wafer steppers,” in Optical Microlithography VI, H. L. Stover, ed., Proc. SPIE772, 134–141 (1987).
[CrossRef]

C. P. Kirk, D. Nyyssonen, “Modeling the optical microscope images of thick layers for the purpose of linewidth measurement,” in Optical Microlithography IV, H. L. Stover, ed., Proc. SPIE538, 179–187 (1985).
[CrossRef]

Kleemann, B. H.

B. H. Kleemann, A. Mitreiter, F. Wyrowski, “Integral equation method with parametrization of grating profile: Theory and experiments,” J. Mod. Opt. 43, 1323–1349 (1996).
[CrossRef]

Leeuwestein, A.

P. Dirksen, C. A. Juffermans, A. Leeuwestein, C. Mutsaers, T. A. Nuijs, R. Pellens, R. Wolters, J. Gemen, “Effect of processing on the overlay performance of a wafer stepper,” in Metrology, Inspection, and Process Control for Microlithography XI, S. K. Jones, ed., Proc. SPIE3050, 102–113 (1997).
[CrossRef]

Li, L.

Maystre, D.

T. Delort, D. Maystre, “Finite element methods for gratings,” J. Opt. Soc. Am. A 10, 2592–2601 (1993).
[CrossRef]

D. Maystre, “Integral methods,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980).

Merkx, R. T. M.

H. P. Urbach, R. T. M. Merkx, “Finite element simulation of electromagnetic plane wave diffraction at gratings for arbitrary angles of incidence,” in Mathematical and Numerical Aspects of Wave Propagation Phenomena, G. Cohen, L. Halpern, P. Joly, eds. (Society of Industrial and Applied Mathematics, Philadelphia, 1991), pp. 89–99.

Mitreiter, A.

B. H. Kleemann, A. Mitreiter, F. Wyrowski, “Integral equation method with parametrization of grating profile: Theory and experiments,” J. Mod. Opt. 43, 1323–1349 (1996).
[CrossRef]

Moharam, M. G.

M. G. Moharam, T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. A 72, 1385–1392 (1982).
[CrossRef]

Mutsaers, C.

P. Dirksen, C. A. Juffermans, A. Leeuwestein, C. Mutsaers, T. A. Nuijs, R. Pellens, R. Wolters, J. Gemen, “Effect of processing on the overlay performance of a wafer stepper,” in Metrology, Inspection, and Process Control for Microlithography XI, S. K. Jones, ed., Proc. SPIE3050, 102–113 (1997).
[CrossRef]

Nuijs, T. A.

P. Dirksen, C. A. Juffermans, A. Leeuwestein, C. Mutsaers, T. A. Nuijs, R. Pellens, R. Wolters, J. Gemen, “Effect of processing on the overlay performance of a wafer stepper,” in Metrology, Inspection, and Process Control for Microlithography XI, S. K. Jones, ed., Proc. SPIE3050, 102–113 (1997).
[CrossRef]

Nyyssonen, D.

C. P. Kirk, D. Nyyssonen, “Modeling the optical microscope images of thick layers for the purpose of linewidth measurement,” in Optical Microlithography IV, H. L. Stover, ed., Proc. SPIE538, 179–187 (1985).
[CrossRef]

Pellens, R.

P. Dirksen, C. A. Juffermans, A. Leeuwestein, C. Mutsaers, T. A. Nuijs, R. Pellens, R. Wolters, J. Gemen, “Effect of processing on the overlay performance of a wafer stepper,” in Metrology, Inspection, and Process Control for Microlithography XI, S. K. Jones, ed., Proc. SPIE3050, 102–113 (1997).
[CrossRef]

Urbach, H. P.

H. P. Urbach, “Convergence of the Galerkin method for two-dimensional electromagnetic problems,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 28, 697–710 (1991).
[CrossRef]

H. P. Urbach, R. T. M. Merkx, “Finite element simulation of electromagnetic plane wave diffraction at gratings for arbitrary angles of incidence,” in Mathematical and Numerical Aspects of Wave Propagation Phenomena, G. Cohen, L. Halpern, P. Joly, eds. (Society of Industrial and Applied Mathematics, Philadelphia, 1991), pp. 89–99.

Voronovich, A. G.

A. G. Voronovich, Wave Scattering from Rough Surfaces (Springer-Verlag, Berlin, 1994).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1986).

Wolters, R.

P. Dirksen, C. A. Juffermans, A. Leeuwestein, C. Mutsaers, T. A. Nuijs, R. Pellens, R. Wolters, J. Gemen, “Effect of processing on the overlay performance of a wafer stepper,” in Metrology, Inspection, and Process Control for Microlithography XI, S. K. Jones, ed., Proc. SPIE3050, 102–113 (1997).
[CrossRef]

Wyrowski, F.

B. H. Kleemann, A. Mitreiter, F. Wyrowski, “Integral equation method with parametrization of grating profile: Theory and experiments,” J. Mod. Opt. 43, 1323–1349 (1996).
[CrossRef]

J. Mod. Opt.

B. H. Kleemann, A. Mitreiter, F. Wyrowski, “Integral equation method with parametrization of grating profile: Theory and experiments,” J. Mod. Opt. 43, 1323–1349 (1996).
[CrossRef]

J. Opt. Soc. Am. A

SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal.

H. P. Urbach, “Convergence of the Galerkin method for two-dimensional electromagnetic problems,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 28, 697–710 (1991).
[CrossRef]

Other

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1986).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

D. Maystre, “Integral methods,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980).

P. Dirksen, C. A. Juffermans, A. Leeuwestein, C. Mutsaers, T. A. Nuijs, R. Pellens, R. Wolters, J. Gemen, “Effect of processing on the overlay performance of a wafer stepper,” in Metrology, Inspection, and Process Control for Microlithography XI, S. K. Jones, ed., Proc. SPIE3050, 102–113 (1997).
[CrossRef]

C. P. Kirk, “Theoretical models for the optical alignment of wafer steppers,” in Optical Microlithography VI, H. L. Stover, ed., Proc. SPIE772, 134–141 (1987).
[CrossRef]

C. P. Kirk, D. Nyyssonen, “Modeling the optical microscope images of thick layers for the purpose of linewidth measurement,” in Optical Microlithography IV, H. L. Stover, ed., Proc. SPIE538, 179–187 (1985).
[CrossRef]

H. P. Urbach, R. T. M. Merkx, “Finite element simulation of electromagnetic plane wave diffraction at gratings for arbitrary angles of incidence,” in Mathematical and Numerical Aspects of Wave Propagation Phenomena, G. Cohen, L. Halpern, P. Joly, eds. (Society of Industrial and Applied Mathematics, Philadelphia, 1991), pp. 89–99.

A. G. Voronovich, Wave Scattering from Rough Surfaces (Springer-Verlag, Berlin, 1994).

T. Abboud, “Electromagnetic waves in periodic media,” in Mathematical and Numerical Aspects of Wave Propagation, R. Kleinman, T. Angell, D. Colton, F. Santosa, I. Stakgold, eds. (Society of Industrial and Applied Mathematics, Philadelphia, Pa., 1993).

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Figures (8)

Fig. 1
Fig. 1

Hybrid grating consisting of P-periodic rectangular grooves where on the bottom of the grooves a periodic microrelief of period pP. Also shown is the Cartesian coordinate system (x1, x2, x3).

Fig. 2
Fig. 2

Amplitude of the reflected electric near field of the periodically extended fine grating (top) and the amplitude of the reflected electric near field of one period of the P-periodic hybrid grating (bottom). The pitch P=8 μm, the period of the microstructure is 0.8 μm, and the incident electric field is polarized along the x3 axis (TE polarization) and has an amplitude of 1.

Fig. 3
Fig. 3

Top, efficiencies of the reflected orders for the rectangular grating with P=16 μm and Q=P/2 without a microstructure. The solid and the dashed curves correspond to results obtained by the hybrid model and by full finite-element calculations, respectively. The incident plane wave is TE polarized. Bottom, enlargement for the lowest ten orders.

Fig. 4
Fig. 4

Reflected near field (obtained by a full finite-element computation) for an empty grating with pitch P=8 μm and Q=P/2. The incident field is TE polarized and has an amplitude of 1.

Fig. 5
Fig. 5

Efficiencies of the reflected orders computed with the hybrid model (solid curve) and with the finite-element method for TM and TE polarized incident fields (dashed curve), pitch P=8 μm, n=1.5, and for a fine grating consisting of ten grooves.

Fig. 6
Fig. 6

Efficiencies of the reflected orders computed with the hybrid model (solid curve) and the full finite-element method (dashed curve) for TM- and TE-polarized incident plane waves and for pitch P=8 μm, refractive index n=1.5, and for a microrelief with period p=0.8 μm.

Fig. 7
Fig. 7

Amplitudes of H3 of the first seven reflected orders drawn in the complex plane and computed by the hybrid model (top) and by the finite-element method (bottom). The incident field is TM polarized with H3 amplitude equal to 1, pitch P=8 μm, period of the microrelief p=0.8 μm, and refractive index n=1.5.

Fig. 8
Fig. 8

Amplitudes of E3 of the first seven reflected orders drawn in the complex plane and computed by the hybrid model (top) and by the finite-element method (bottom). The incident field is TE polarized with E3 amplitude equal to 1, pitch P=8 μm, period of the microrelief p=0.8 μm, and refractive index n=1.5.

Equations (29)

Equations on this page are rendered with MathJax. Learn more.

Ei(x1, x2, x3)=eiexp[i(α1x2-α2x2+α3x3)],
Hi(x1, x2, x3)=hiexp[i(α1x2-α2x2+α3x3)],
E(x1, x2, x3)=e(x1, x2)exp[i(α1x1+α3x3)],
H(x1, x2, x3)=h(x1, x2)exp[i(α1x1+α3x3)],
f(x1)
=ϕ(x1-mP)formPx1Q+mPHforQ+mP<x1<(m+1)P
forallintegerm,
Ur(x1, x2, x3)=ur(x1, x2)exp[i(α1x1+α3x3)],
Up,r(x1, x2, x3)=up,r(x1, x2)×exp[i(α1x1+α3x3)],
Ur(x1, x2, x3)=R(α1, α3)uiexp{i[α1x1+α2(1)(x2-2H)+α3x3]},
R(α1, α3)=1(a2-a1)2+ω2μ0(b1+b2)(1b1+2b2)×-(a2-a1)2+ω2μ0(b1+b2)(1b1-2b2)2ωμ0(a2-a1)b12ω(a2-a1)1b1-(a2-a1)2+ω2μ0(b1-b2)(1b1+2b2),
ai=α1α3α12+α2(i)2,bi=α2(i)α12+α2(i)2,
α2(i)=(ω2iμ0-α12-α32)1/2,i=1, 2.
ur(x1, h)=up,r(x1-mP, h)formP<x1<mP+QR(α1, α3)uiexp[iα2(1)(h-2H)]formP+Q<x1<(m+1)P
Ur(x1, h, x3)
=m=-1[0,Q](x1+mP)up,r(x1+mP, h)+m=-1[Q,P](x1+mP)Rui×exp[iα2(1)(h-2H)]×exp[i(α1x1+α3x3)]
={v(x1)+w(x1)}exp[i(α1x1+α3x3)],
F(Ur)(ξ1, h, ξ3)
={F1(v)(ξ1-α1/2π)+F1(w)(ξ1-α1/2π)}δ(ξ3-α3/2π),
F1(g)(ξ)=-exp(-2πixξ)g(x)dx.
gˆ(ν)=(1/P)0Pexp(-2πixν/P)g(x)dx.
F1(g)(ξ)=ν=-+gˆ(ν)δ(ξ-ν/P).
vˆ(ν)=1P0Pexp(-2πiνx1/P)v(x1)dx1=1P0Qexp(-2πiνx1/P)up,r(x1, h)dx1=1Pμ=-u^p,r(μ, h)0Qexp[2πi(μ/p-ν/P)x1]dx1=μ=-u^p,r(μ, h)exp[πi(μQ/p-νQ/P)]×sin[π(μQ/p-νQ/P)]π(μP/p-ν)ifμPνpQP,ifμP=νp,
u^p,r(μ, h)=1p0pexp(-2πiμx1)up,r(x1, h)dx1.
wˆ(ν)=1PQPexp(-2πiνx1/P)Rui×exp[iα2(1)(h-2H)]dx1=Ruiexp[iα2(1)(h-2H)](1-Q/P)if ν=0-Ruiexp[iα2(1)(h-2H)]exp(-πiνQ/P)×sin(πνQ/P)πνif ν0.
F(Ur)(ξ1, h, ξ3)
=ν=-{vˆ(ν)+wˆ(ν)}δ(ξ1-ν/P-α1/2π)×δ(ξ3-α3/2π)=ν=-Ruiexp[iα2(1)(h-2H)]×δν0-exp(-πiνQ/P) sin(πνQ/P)πν+μ=-u^p,r(μ, h)exp[πi(μQ/p-νQ/P)]×sin[π(μQ/p-νQ/P)]π(μP/p-ν)δ(ξ1-ν/P-α1/2π)δ(ξ3-α3/2π),
(α1+2πν/P)2+α32<k2n12.
S2=cos θ2 (ω1|e3r|2+ωμ0|h3r|2).

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