Abstract

Microscopy of binary objects in partially coherent light is analyzed in terms of bilinear transfer. A transfer model using two apparent transfer functions is proposed. Its domain of validity is studied by numerical simulation.

© 1988 Optical Society of America

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References

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  1. See, for example, H. P. Baltes, Inverse Source Problems in Optics, Vol. 9 of Topics in Current Physics, H. P. Baltes, ed. (Springer-Verlag, New York, 1978), pp. 119–154.
    [CrossRef]
  2. W. M. Bullis, D. Nyyssonen, “Optical linewidth measurements on photomasks and wafers,”VLSI Electron. Microstructure Sci. 3, 301–346 (1982).
  3. H. H. Hopkins, “On the diffraction theory of optical images,” Proc. R. Soc. London Ser. A 217, 408–432 (1953).
    [CrossRef]
  4. B. Saleh, “Optical bilinear transformations: general properties,” Opt. Acta 26, 777–799 (1979).
    [CrossRef]
  5. R. Becherer, G. Parrent, “Nonlinearity in optical imaging systems,”J. Opt. Soc. Am. 57, 1479–1486 (1967).
    [CrossRef]
  6. M. E. Guillaume, N. Noailly, J. C. Reynaud, J. L. Buevoz, “Fourier transform method for optical linewidth measurement,” in Integrated Circuit Metrology II, D. Nyyssonen, ed., Proc. Soc. Photo-Opt. Instrum. Eng.480, 71–77 (1984).
    [CrossRef]
  7. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975), pp. 491–535.
  8. E. Kintner, “Method for the calculation of partially coherent imagery,” Appl. Opt. 17, 2747–2753 (1978).
    [CrossRef] [PubMed]
  9. D. Courjon, D. Charraut, P. Livrozet, “Bilinear transfer in microscopy,” Opt. Acta 34, 127–136 (1987).
  10. M. E. Guillaume, N. Noailly, M. Pichot, J. L. Buevoz, “Evaluation of a Fourier transform method for accurate critical dimension measurements,” Microcircuit Eng. 3, 211–218 (1985).
    [CrossRef]

1987

D. Courjon, D. Charraut, P. Livrozet, “Bilinear transfer in microscopy,” Opt. Acta 34, 127–136 (1987).

1985

M. E. Guillaume, N. Noailly, M. Pichot, J. L. Buevoz, “Evaluation of a Fourier transform method for accurate critical dimension measurements,” Microcircuit Eng. 3, 211–218 (1985).
[CrossRef]

1982

W. M. Bullis, D. Nyyssonen, “Optical linewidth measurements on photomasks and wafers,”VLSI Electron. Microstructure Sci. 3, 301–346 (1982).

1979

B. Saleh, “Optical bilinear transformations: general properties,” Opt. Acta 26, 777–799 (1979).
[CrossRef]

1978

1967

1953

H. H. Hopkins, “On the diffraction theory of optical images,” Proc. R. Soc. London Ser. A 217, 408–432 (1953).
[CrossRef]

Baltes, H. P.

See, for example, H. P. Baltes, Inverse Source Problems in Optics, Vol. 9 of Topics in Current Physics, H. P. Baltes, ed. (Springer-Verlag, New York, 1978), pp. 119–154.
[CrossRef]

Becherer, R.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975), pp. 491–535.

Buevoz, J. L.

M. E. Guillaume, N. Noailly, M. Pichot, J. L. Buevoz, “Evaluation of a Fourier transform method for accurate critical dimension measurements,” Microcircuit Eng. 3, 211–218 (1985).
[CrossRef]

M. E. Guillaume, N. Noailly, J. C. Reynaud, J. L. Buevoz, “Fourier transform method for optical linewidth measurement,” in Integrated Circuit Metrology II, D. Nyyssonen, ed., Proc. Soc. Photo-Opt. Instrum. Eng.480, 71–77 (1984).
[CrossRef]

Bullis, W. M.

W. M. Bullis, D. Nyyssonen, “Optical linewidth measurements on photomasks and wafers,”VLSI Electron. Microstructure Sci. 3, 301–346 (1982).

Charraut, D.

D. Courjon, D. Charraut, P. Livrozet, “Bilinear transfer in microscopy,” Opt. Acta 34, 127–136 (1987).

Courjon, D.

D. Courjon, D. Charraut, P. Livrozet, “Bilinear transfer in microscopy,” Opt. Acta 34, 127–136 (1987).

Guillaume, M. E.

M. E. Guillaume, N. Noailly, M. Pichot, J. L. Buevoz, “Evaluation of a Fourier transform method for accurate critical dimension measurements,” Microcircuit Eng. 3, 211–218 (1985).
[CrossRef]

M. E. Guillaume, N. Noailly, J. C. Reynaud, J. L. Buevoz, “Fourier transform method for optical linewidth measurement,” in Integrated Circuit Metrology II, D. Nyyssonen, ed., Proc. Soc. Photo-Opt. Instrum. Eng.480, 71–77 (1984).
[CrossRef]

Hopkins, H. H.

H. H. Hopkins, “On the diffraction theory of optical images,” Proc. R. Soc. London Ser. A 217, 408–432 (1953).
[CrossRef]

Kintner, E.

Livrozet, P.

D. Courjon, D. Charraut, P. Livrozet, “Bilinear transfer in microscopy,” Opt. Acta 34, 127–136 (1987).

Noailly, N.

M. E. Guillaume, N. Noailly, M. Pichot, J. L. Buevoz, “Evaluation of a Fourier transform method for accurate critical dimension measurements,” Microcircuit Eng. 3, 211–218 (1985).
[CrossRef]

M. E. Guillaume, N. Noailly, J. C. Reynaud, J. L. Buevoz, “Fourier transform method for optical linewidth measurement,” in Integrated Circuit Metrology II, D. Nyyssonen, ed., Proc. Soc. Photo-Opt. Instrum. Eng.480, 71–77 (1984).
[CrossRef]

Nyyssonen, D.

W. M. Bullis, D. Nyyssonen, “Optical linewidth measurements on photomasks and wafers,”VLSI Electron. Microstructure Sci. 3, 301–346 (1982).

Parrent, G.

Pichot, M.

M. E. Guillaume, N. Noailly, M. Pichot, J. L. Buevoz, “Evaluation of a Fourier transform method for accurate critical dimension measurements,” Microcircuit Eng. 3, 211–218 (1985).
[CrossRef]

Reynaud, J. C.

M. E. Guillaume, N. Noailly, J. C. Reynaud, J. L. Buevoz, “Fourier transform method for optical linewidth measurement,” in Integrated Circuit Metrology II, D. Nyyssonen, ed., Proc. Soc. Photo-Opt. Instrum. Eng.480, 71–77 (1984).
[CrossRef]

Saleh, B.

B. Saleh, “Optical bilinear transformations: general properties,” Opt. Acta 26, 777–799 (1979).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975), pp. 491–535.

Appl. Opt.

J. Opt. Soc. Am.

Microcircuit Eng.

M. E. Guillaume, N. Noailly, M. Pichot, J. L. Buevoz, “Evaluation of a Fourier transform method for accurate critical dimension measurements,” Microcircuit Eng. 3, 211–218 (1985).
[CrossRef]

Opt. Acta

D. Courjon, D. Charraut, P. Livrozet, “Bilinear transfer in microscopy,” Opt. Acta 34, 127–136 (1987).

B. Saleh, “Optical bilinear transformations: general properties,” Opt. Acta 26, 777–799 (1979).
[CrossRef]

Proc. R. Soc. London Ser. A

H. H. Hopkins, “On the diffraction theory of optical images,” Proc. R. Soc. London Ser. A 217, 408–432 (1953).
[CrossRef]

VLSI Electron. Microstructure Sci.

W. M. Bullis, D. Nyyssonen, “Optical linewidth measurements on photomasks and wafers,”VLSI Electron. Microstructure Sci. 3, 301–346 (1982).

Other

See, for example, H. P. Baltes, Inverse Source Problems in Optics, Vol. 9 of Topics in Current Physics, H. P. Baltes, ed. (Springer-Verlag, New York, 1978), pp. 119–154.
[CrossRef]

M. E. Guillaume, N. Noailly, J. C. Reynaud, J. L. Buevoz, “Fourier transform method for optical linewidth measurement,” in Integrated Circuit Metrology II, D. Nyyssonen, ed., Proc. Soc. Photo-Opt. Instrum. Eng.480, 71–77 (1984).
[CrossRef]

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975), pp. 491–535.

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Figures (9)

Fig. 1
Fig. 1

Microscope in Köhler’s illumination. The image SS of the thermal source (S0) is projected in the pupil plane (P) of the objective (Ob).

Fig. 2
Fig. 2

(a) Schematic representation of the object; nS and nM are the indices of the two media constituting the object. Plane waves before (Σ0) and after (ΣS and ΣM) reflection over the object are shown. (b), (c) Variation of T0 and ϕ0 over the object.

Fig. 3
Fig. 3

Three-dimensional representation of the BTF in partially coherent light (a) for a circular objective aperture and (b) for a one-dimensional aperture (coherence coefficient, σ = 0.67).

Fig. 4
Fig. 4

Integral I1 plotted for various coherence coefficients and various object widths. (U is in arbitrary units.) Curves 1, σ = 1; curves 2, σ = 0.67; curves 3, σ = 0.50; curves 4, σ = 0.30; curves 5, σ = 0.01.

Fig. 5
Fig. 5

Integral I3 for the same coherence coefficients as in Fig. 4.

Fig. 6
Fig. 6

Modulus of the image spectrum: solid line, plotted from Eq. (7), i.e., without approximation; dotted line, plotted after modeling. LE = 1.35 μm, λ = 0.5145 μm, N.A. = 0.95, and σ = 0.67.

Fig. 7
Fig. 7

Image profile obtained (a) without approximation (Fourier transform of the spectrum plotted in Fig. 6) and (b) after modeling.

Fig. 8
Fig. 8

(a) Partial image profile due to the first apparent transfer. (b) Partial image profile due to the second apparent transfer (derivative effect).

Fig. 9
Fig. 9

Examples of experimental profiles for different parameters t0 and ϕ0: (a) A − 2B > A (amplitude object LE ≃ 1.5 μm); (b) A − 2BA (phase and amplitude object LE ≃ 1.5 μm); (c) A − 2B < A (phase object LE = 2.5 μm). σ = 0.67, λ = 0.5145 μm, N.A. = 0.95.

Equations (35)

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I ˜ ( u λ p ) = T ( u + u , u ) O ˜ ( u + u λ p ) O ˜ * ( u λ p ) d 2 u .
ω = u / λ p , ω = u / λ p , p / p = g
T ( ω , ω ) T ( ω · λ p , ω · λ p ) ,
I ˜ ( ω ) I ˜ ( ω / g ) ,
I ˜ ( ω ) = T ( ω + ω , ω ) O ˜ ( ω + ω ) O ˜ * ( ω ) d 2 ω ,
T ( ω , ω ) = S ( α ) P ( ω + α ) P * ( ω + α ) d 2 α .
O ( x ) = O R ( x ) + j O J ( x ) , O R ( x ) = 1 ( 1 t 0 cos ϕ 0 ) rect ( x / L ) , O J ( x ) = t 0 sin ϕ 0 rect ( x / L ) ,
I ˜ ( ω ) = T ( ω + ω , ω ) O ˜ ( ω + ω ) O ˜ * ( ω ) d ω ,
I ˜ ( ω ) = A I ˜ ( ω ) 2 B I ˜ ( ω ) for ω 0 .
I ˜ ( ω ) = 1 π 2 T ( ω + ω , ω ) sin [ π ( ω + ω ) L ] ( ω + ω ) × sin ( π ω L ) ω d ω
I ˜ ( ω ) = 1 π T ( ω , 0 ) sin ( π ω L ) ω .
A = 1 + t 0 2 2 t 0 cos ϕ 0 , B = 1 t 0 cos ϕ 0 .
1 ω ( ω + ω ) = 1 ω ( 1 ω 1 ω + ω ) .
I ˜ ( ω ) = sinc ( π ω L ) I ˜ 1 ( ω ) cos ( π ω L ) [ I ˜ 2 ( ω ) + I ˜ 3 ( ω ) ] ,
I ˜ 1 ( ω ) = 2 L 2 T ( ω + ω , ω ) sinc ( 2 π ω L ) d ω ,
I ˜ 2 ( ω ) = 1 π 2 ω T ( ω + ω , ω ) ω cos ( 2 π ω L ) d ω ,
I ˜ 3 ( ω ) = 1 2 π 2 T ( ω + ω , ω ) ( ω + ω ) ω d ω .
I ˜ 3 ( ω ) = 1 π 2 ω T ( ω + ω , ω ) ω d ω
T ( ω , ω ) = T * ( ω , ω ) ,
T ( ω , ω ) = T * ( ω , ω ) ,
I ˜ 1 L T ( ω , 0 ) ,
I ˜ ( ω ) L ( A 2 B ) sinc ( π ω L ) T A 1 A cos ( π ω L ) T A 2 ,
T A 1 = T ( ω , 0 )
T A 2 = I 3 = 1 π 2 ω T ( ω + ω , ω ) ω d ω
S ( α ) = 1 if | α | r = 0 if | α | > r , P ( α ) = 1 if | α | R = 0 if | α | > R ,
σ = r / R ,
I ˜ 1 L T ( ω , 0 )
λ = 0.5145 μ m , N . A . = 0.95 , σ = 0.67 , L = 2.5 , L E = 1.35 μ m , ϕ 0 85 ° , t 0 2 = 1.2 ,
A 2 B = 0 t 0 = 1 ,
A = ( 1 t 0 ) 2 0 for t 0 1.
A = 1 + t 0 2 2 t 0 cos ϕ 0 = 0 ;
( 1 + t 0 2 ) 2 t 0 1 ,
t 0 = 1 ± , ϕ 0 = 0.
A 2 t 0 ( 1 cos ϕ 0 ) = 0
A 2 B ± 2 .

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