Abstract

We investigate the moiré fringe pattern of two concentric-circle gratings with the aim of measuring the relative two-dimensional planar displacement induced between the gratings. In the first stage we give a comprehensive mathematical approach for the analysis of fringe patterns. Then we suggest a fast computational measurement algorithm that uses a Fourier transformation technique. Finally, our experimental results prove that a measuring accuracy of 0.01 μm can be achieved with this method.

© 1994 Optical Society of America

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References

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  1. P. Theocaris, Moiré Fringes in Strain Analysis (Pergamon, Oxford, 1969).
  2. B. Fay, J. Trotel, A. Frichet, “Optical alignment system for submicron x-ray lithography,” J. Vac. Sci. Technol. 16, 1954–1958 (1979).
    [CrossRef]
  3. T. M. Lyszczarz, D. C. Flanders, N. P. Economou, P. D. DeGraff, “Experimental evaluation of interferometric alignment techniques for multiple mask registration,” J. Vac. Sci. Technol. 19, 1214–1218 (1981).
    [CrossRef]
  4. H. Kinoshita, A. Une, M. Iki, “A dual grating alignment technique for x-ray lithography,” J. Vac. Sci. Technol. B 1, 1276–1279 (1983).
    [CrossRef]
  5. M. C. King, D. H. Berry, “Photolithographic mask alignment using moiré techniques,” Appl. Opt. 11, 2455–2459 (1972).
    [CrossRef] [PubMed]
  6. B. Sandler, E. Keren, A. Livnat, O. Kafri, “Moiré patterns of skewed radial gratings,” Appl. Opt. 26, 772–773 (1987).
    [CrossRef] [PubMed]
  7. P. Szwaykowski, K. Patorski, “Moiré fringes by evolute gratings,” Appl. Opt. 28, 4679–4681 (1989).
    [CrossRef] [PubMed]
  8. Y. Vladimirsky, H. W. P. Koops, “Moiré method and zone plate pattern inaccuracies,” J. Vac. Sci. Technol. B 6, 2142–2146 (1988).
    [CrossRef]
  9. Z. Jaroszewicz, “A review of Fresnel zone plate moiré patterns obtained by translations,” Opt. Eng. 31, 458–464 (1992).
    [CrossRef]

1992 (1)

Z. Jaroszewicz, “A review of Fresnel zone plate moiré patterns obtained by translations,” Opt. Eng. 31, 458–464 (1992).
[CrossRef]

1989 (1)

1988 (1)

Y. Vladimirsky, H. W. P. Koops, “Moiré method and zone plate pattern inaccuracies,” J. Vac. Sci. Technol. B 6, 2142–2146 (1988).
[CrossRef]

1987 (1)

1983 (1)

H. Kinoshita, A. Une, M. Iki, “A dual grating alignment technique for x-ray lithography,” J. Vac. Sci. Technol. B 1, 1276–1279 (1983).
[CrossRef]

1981 (1)

T. M. Lyszczarz, D. C. Flanders, N. P. Economou, P. D. DeGraff, “Experimental evaluation of interferometric alignment techniques for multiple mask registration,” J. Vac. Sci. Technol. 19, 1214–1218 (1981).
[CrossRef]

1979 (1)

B. Fay, J. Trotel, A. Frichet, “Optical alignment system for submicron x-ray lithography,” J. Vac. Sci. Technol. 16, 1954–1958 (1979).
[CrossRef]

1972 (1)

Berry, D. H.

DeGraff, P. D.

T. M. Lyszczarz, D. C. Flanders, N. P. Economou, P. D. DeGraff, “Experimental evaluation of interferometric alignment techniques for multiple mask registration,” J. Vac. Sci. Technol. 19, 1214–1218 (1981).
[CrossRef]

Economou, N. P.

T. M. Lyszczarz, D. C. Flanders, N. P. Economou, P. D. DeGraff, “Experimental evaluation of interferometric alignment techniques for multiple mask registration,” J. Vac. Sci. Technol. 19, 1214–1218 (1981).
[CrossRef]

Fay, B.

B. Fay, J. Trotel, A. Frichet, “Optical alignment system for submicron x-ray lithography,” J. Vac. Sci. Technol. 16, 1954–1958 (1979).
[CrossRef]

Flanders, D. C.

T. M. Lyszczarz, D. C. Flanders, N. P. Economou, P. D. DeGraff, “Experimental evaluation of interferometric alignment techniques for multiple mask registration,” J. Vac. Sci. Technol. 19, 1214–1218 (1981).
[CrossRef]

Frichet, A.

B. Fay, J. Trotel, A. Frichet, “Optical alignment system for submicron x-ray lithography,” J. Vac. Sci. Technol. 16, 1954–1958 (1979).
[CrossRef]

Iki, M.

H. Kinoshita, A. Une, M. Iki, “A dual grating alignment technique for x-ray lithography,” J. Vac. Sci. Technol. B 1, 1276–1279 (1983).
[CrossRef]

Jaroszewicz, Z.

Z. Jaroszewicz, “A review of Fresnel zone plate moiré patterns obtained by translations,” Opt. Eng. 31, 458–464 (1992).
[CrossRef]

Kafri, O.

Keren, E.

King, M. C.

Kinoshita, H.

H. Kinoshita, A. Une, M. Iki, “A dual grating alignment technique for x-ray lithography,” J. Vac. Sci. Technol. B 1, 1276–1279 (1983).
[CrossRef]

Koops, H. W. P.

Y. Vladimirsky, H. W. P. Koops, “Moiré method and zone plate pattern inaccuracies,” J. Vac. Sci. Technol. B 6, 2142–2146 (1988).
[CrossRef]

Livnat, A.

Lyszczarz, T. M.

T. M. Lyszczarz, D. C. Flanders, N. P. Economou, P. D. DeGraff, “Experimental evaluation of interferometric alignment techniques for multiple mask registration,” J. Vac. Sci. Technol. 19, 1214–1218 (1981).
[CrossRef]

Patorski, K.

Sandler, B.

Szwaykowski, P.

Theocaris, P.

P. Theocaris, Moiré Fringes in Strain Analysis (Pergamon, Oxford, 1969).

Trotel, J.

B. Fay, J. Trotel, A. Frichet, “Optical alignment system for submicron x-ray lithography,” J. Vac. Sci. Technol. 16, 1954–1958 (1979).
[CrossRef]

Une, A.

H. Kinoshita, A. Une, M. Iki, “A dual grating alignment technique for x-ray lithography,” J. Vac. Sci. Technol. B 1, 1276–1279 (1983).
[CrossRef]

Vladimirsky, Y.

Y. Vladimirsky, H. W. P. Koops, “Moiré method and zone plate pattern inaccuracies,” J. Vac. Sci. Technol. B 6, 2142–2146 (1988).
[CrossRef]

Appl. Opt. (3)

J. Vac. Sci. Technol. (2)

B. Fay, J. Trotel, A. Frichet, “Optical alignment system for submicron x-ray lithography,” J. Vac. Sci. Technol. 16, 1954–1958 (1979).
[CrossRef]

T. M. Lyszczarz, D. C. Flanders, N. P. Economou, P. D. DeGraff, “Experimental evaluation of interferometric alignment techniques for multiple mask registration,” J. Vac. Sci. Technol. 19, 1214–1218 (1981).
[CrossRef]

J. Vac. Sci. Technol. B (2)

H. Kinoshita, A. Une, M. Iki, “A dual grating alignment technique for x-ray lithography,” J. Vac. Sci. Technol. B 1, 1276–1279 (1983).
[CrossRef]

Y. Vladimirsky, H. W. P. Koops, “Moiré method and zone plate pattern inaccuracies,” J. Vac. Sci. Technol. B 6, 2142–2146 (1988).
[CrossRef]

Opt. Eng. (1)

Z. Jaroszewicz, “A review of Fresnel zone plate moiré patterns obtained by translations,” Opt. Eng. 31, 458–464 (1992).
[CrossRef]

Other (1)

P. Theocaris, Moiré Fringes in Strain Analysis (Pergamon, Oxford, 1969).

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Figures (13)

Fig. 1
Fig. 1

Moiré principle of two concentric-circle gratings.

Fig. 2
Fig. 2

Analysis of the fringe patterns of two concentric-circle gratings.

Fig. 3
Fig. 3

Representative patterns of moiré fringes of two concentric-circle gratings.

Fig. 4
Fig. 4

Schematic diagram of the measuring system.

Fig. 5
Fig. 5

Grating G1.

Fig. 6
Fig. 6

Patterns of moiré fringes captured by a CCD camera.

Fig. 7
Fig. 7

Determination of the radial phase of moiré fringes: (a) raw image of moiré fringes, (b) distribution of wrapped radial phase, (c) unwrapped radial phase.

Fig. 8
Fig. 8

Determination of eccentricity magnitude and direction from the radial phase distribution.

Fig. 9
Fig. 9

Reconstructed image of pure moiré fringes.

Fig. 10
Fig. 10

Schematic diagram of the calibration system: GPIB, general-purpose interface bond.

Fig. 11
Fig. 11

Linearity for measurement of the eccentricity magnitude.

Fig. 12
Fig. 12

Deviation error of the measured position.

Fig. 13
Fig. 13

Repeatability of the measurement.

Equations (17)

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S 1 ( r , θ ) = 1 2 ( 1 + cos 2 π ( N - 1 ) R r ) .
S 2 ( r , θ ) = 1 2 ( 1 + cos 2 π N R r s ) ,
r s 2 = r 2 + 2 - 2 r cos ( θ - ϕ )
r s 2 = [ r - cos ( θ - ϕ ) ] 2 + 2 sin 2 ( ρ - ϕ ) .
r s r - cos ( θ - ϕ ) .
I ( r , θ ) = I 0 S r S s .
I ( r , θ ) = I 0 4 + I 0 4 cos { 2 π N [ r - e cos ( θ - ϕ ) ] } + I 0 4 cos [ 2 π ( N - 1 ) r ] + I 0 8 cos { 2 π [ ( 2 N - 1 ) r - N e cos ( θ - ϕ ) ] } + I 0 8 cos { 2 π [ r - N e cos ( θ - ϕ ) ] } .
I m ( r , θ ) = cos { 2 π [ r - N e cos ( θ - ϕ ) ] } .
I m ( r , θ ) = - 1
2 π [ r - N e cos ( θ - ϕ ) ] = ( 2 m + 1 ) π ,             m = 0 , ± 1 , ± 2 , ± 3 , .
r = ½ + m + N e cos ( θ - ϕ ) .
I m ( r , θ ) = cos [ 2 π r - ψ ( θ ) ] .
ψ ( θ ) = 2 π N e cos ( θ - ϕ ) .
ψ W ( θ ) = tan - 1 i = 0 n - 1 I ( R i n , θ ) sin 2 π i n i = 0 n - 1 I ( R i n , θ ) cos 2 π i n .
ψ ( θ ) = 2 π m + ψ W ( θ ) , m = 0 , ± 1 , ± 2 , ± 3 , .
ϕ = tan - 1 i = 0 n - 1 ψ ( 2 π i n ) sin 2 π i n i = 0 n - 1 ψ ( 2 π i n ) cos 2 π i n ,
= R π N n { [ i = 0 n - 1 ψ ( 2 π i n ) cos 2 π i n ] 2 + [ i = 0 n - 1 ψ ( 2 π i n ) sin 2 π i n ] 2 } 1 / 2 .

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