Abstract

Considering the necessity of alignment in practical applications of photolithography, distribution of complex amplitude of moiré fringe patterns that are produced in superposition of two gratings is analyzed in the viewpoint of Fourier Optics and the relationship between fringes and properties of these two gratings is concluded by means of an analysis model. The rule of one-dimensional gratings (1D-gratings) is extended to other form of the gratings which have quasi-periodic repetitive structures. Especially, moiré fringes generated by the two superposed 1D-gratings (used in alignment of lithography) can be expressed by an arithmetical operation of two vectors which include enough information about these 1D-gratings. Numerical analyses regarding the moiré model and its application in the alignment process of lithography are carried out. Our computational analyses results show that the moiré fringes of the two extended gratings can be refined as a transformed fringe pattern of two standard 1D-gratings. Finally, the results also make it out that the fringes which have magnified periods versus that of two 1D-gratings are highly sensitive to relative shift of two gratings thus might be applicable in alignment of lithography or correlated fields.

© 2008 Optical Society of America

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References

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  1. L. Raleigh, "On the manufacture and theory of diffraction gratings," Philos. Mag. 4, 81-93 (1874).
  2. J. Guid, The Interference Systems of Crossed Diffraction Gratings (Oxford University Press, London, 1956).
  3. J. Guid, Diffraction Gratings as Measuring Scales, (Oxford University Press, London,1960).
  4. I. G. O. Kafri, The Physics of Moire Metrology, (Wiley, New York,1989).
  5. G. Oster, M. Wasserman, and C. Zwerling, "Theoretical Interpretation of Moiré Patterns," J. Opt. Soc. Am. A 54, 169-175 (1964).
    [CrossRef]
  6. Y. Nishijima and G. Oster, "Moiré Patterns: Their Application to Refractive Index and Refractive Index Gradient Measurements," J. Opt. Soc. Am. A 54, 1-5 (1964).
    [CrossRef]
  7. O. Bryngdahl, "Moiré: Formation and interpretation," J. Opt. Soc. Am. A 64, 1287-1294 (1974).
    [CrossRef]
  8. O. Bryngdahl, "Moiré and higher grating harmonics," J. Opt. Soc. Am. A 65, 685-694 (1975).
    [CrossRef]
  9. I. Amidror and R. D. Hersch, "Fourier-based analysis of phase shifts in the superposition of periodic layers and their moire effects," J. Opt. Soc. Am. A 13, 974-987 (1996).
    [CrossRef]
  10. I. Amidror and R. D. Hersch, "Fourier-based analysis and synthesis of moirés in the superposition of geometrically transformed periodic structures," J. Opt. Soc. Am. A 15, 1100-1113 (1998).
    [CrossRef]
  11. M. C. King and D. H. Berry, "Photolithographic mask alignment using moire techniques," J. Vac. Sci. Technol. B 11, 2455-2458 (1972).
  12. K. Hara, Y. Uchida, T. Nomura, S. Kimura, D. Sugimoto, A. Yoshida, H. Miyake, T. Iida, and S. Hattori, "An alignment technique using diffracted moire signals," J. Vac. Sci. Technol. B 7, 1977-1979 (1989).
    [CrossRef]
  13. A. Moel, E. E. Moon, R. D. Frankel, and H. I. Smith, "Novel on-axis interferometric alignment method with sub-10 nm precision," J. Vac. Sci. Technol. B 11, 2191-2194 (1993).
    [CrossRef]
  14. J. H. Lee, C. H. Kim, Y.-S. Kim, K. M. Ho, K. Constant, W. Leung, and C. H. Oh, "Diffracted moire fringes as analysis and alignment tools for multilayer fabrication in soft lithography," Appl. Phys. Lett. 86, 204101-204101 -- 204101-204103 (2005).[DOI: 10. 1063/1.1927268]
  15. N. H. Li, W. Wu, and S. Y. Chou, "Sub-20-nm alignment in nanoimprint lithography using Moire fringe," Nano Lett. 6, 2626-2629 (2006).
    [CrossRef] [PubMed]
  16. M. Muhlberger, I. Bergmair, W. Schwinger, M. Gmainer, R. Schoftner, T. Glinsner, C. Hasenfuss, K. Hingerl, M. Vogler, H. Schmidt, and E. B. Kley, "A Moire method for high accuracy alignment in nanoimprint lithography," Microelectron. Eng. 84, 925-927 (2007).
    [CrossRef]
  17. L. Q. T, Physical Optics, (China Machine Press, Beijing, 1986).

2007 (1)

M. Muhlberger, I. Bergmair, W. Schwinger, M. Gmainer, R. Schoftner, T. Glinsner, C. Hasenfuss, K. Hingerl, M. Vogler, H. Schmidt, and E. B. Kley, "A Moire method for high accuracy alignment in nanoimprint lithography," Microelectron. Eng. 84, 925-927 (2007).
[CrossRef]

2006 (1)

N. H. Li, W. Wu, and S. Y. Chou, "Sub-20-nm alignment in nanoimprint lithography using Moire fringe," Nano Lett. 6, 2626-2629 (2006).
[CrossRef] [PubMed]

1998 (1)

1996 (1)

1993 (1)

A. Moel, E. E. Moon, R. D. Frankel, and H. I. Smith, "Novel on-axis interferometric alignment method with sub-10 nm precision," J. Vac. Sci. Technol. B 11, 2191-2194 (1993).
[CrossRef]

1989 (1)

K. Hara, Y. Uchida, T. Nomura, S. Kimura, D. Sugimoto, A. Yoshida, H. Miyake, T. Iida, and S. Hattori, "An alignment technique using diffracted moire signals," J. Vac. Sci. Technol. B 7, 1977-1979 (1989).
[CrossRef]

1975 (1)

O. Bryngdahl, "Moiré and higher grating harmonics," J. Opt. Soc. Am. A 65, 685-694 (1975).
[CrossRef]

1974 (1)

O. Bryngdahl, "Moiré: Formation and interpretation," J. Opt. Soc. Am. A 64, 1287-1294 (1974).
[CrossRef]

1972 (1)

M. C. King and D. H. Berry, "Photolithographic mask alignment using moire techniques," J. Vac. Sci. Technol. B 11, 2455-2458 (1972).

1964 (2)

G. Oster, M. Wasserman, and C. Zwerling, "Theoretical Interpretation of Moiré Patterns," J. Opt. Soc. Am. A 54, 169-175 (1964).
[CrossRef]

Y. Nishijima and G. Oster, "Moiré Patterns: Their Application to Refractive Index and Refractive Index Gradient Measurements," J. Opt. Soc. Am. A 54, 1-5 (1964).
[CrossRef]

1874 (1)

L. Raleigh, "On the manufacture and theory of diffraction gratings," Philos. Mag. 4, 81-93 (1874).

Amidror, I.

Bergmair, I.

M. Muhlberger, I. Bergmair, W. Schwinger, M. Gmainer, R. Schoftner, T. Glinsner, C. Hasenfuss, K. Hingerl, M. Vogler, H. Schmidt, and E. B. Kley, "A Moire method for high accuracy alignment in nanoimprint lithography," Microelectron. Eng. 84, 925-927 (2007).
[CrossRef]

Berry, D. H.

M. C. King and D. H. Berry, "Photolithographic mask alignment using moire techniques," J. Vac. Sci. Technol. B 11, 2455-2458 (1972).

Bryngdahl, O.

O. Bryngdahl, "Moiré and higher grating harmonics," J. Opt. Soc. Am. A 65, 685-694 (1975).
[CrossRef]

O. Bryngdahl, "Moiré: Formation and interpretation," J. Opt. Soc. Am. A 64, 1287-1294 (1974).
[CrossRef]

Chou, S. Y.

N. H. Li, W. Wu, and S. Y. Chou, "Sub-20-nm alignment in nanoimprint lithography using Moire fringe," Nano Lett. 6, 2626-2629 (2006).
[CrossRef] [PubMed]

Constant, K.

J. H. Lee, C. H. Kim, Y.-S. Kim, K. M. Ho, K. Constant, W. Leung, and C. H. Oh, "Diffracted moire fringes as analysis and alignment tools for multilayer fabrication in soft lithography," Appl. Phys. Lett. 86, 204101-204101 -- 204101-204103 (2005).[DOI: 10. 1063/1.1927268]

Frankel, R. D.

A. Moel, E. E. Moon, R. D. Frankel, and H. I. Smith, "Novel on-axis interferometric alignment method with sub-10 nm precision," J. Vac. Sci. Technol. B 11, 2191-2194 (1993).
[CrossRef]

Glinsner, T.

M. Muhlberger, I. Bergmair, W. Schwinger, M. Gmainer, R. Schoftner, T. Glinsner, C. Hasenfuss, K. Hingerl, M. Vogler, H. Schmidt, and E. B. Kley, "A Moire method for high accuracy alignment in nanoimprint lithography," Microelectron. Eng. 84, 925-927 (2007).
[CrossRef]

Gmainer, M.

M. Muhlberger, I. Bergmair, W. Schwinger, M. Gmainer, R. Schoftner, T. Glinsner, C. Hasenfuss, K. Hingerl, M. Vogler, H. Schmidt, and E. B. Kley, "A Moire method for high accuracy alignment in nanoimprint lithography," Microelectron. Eng. 84, 925-927 (2007).
[CrossRef]

Hara, K.

K. Hara, Y. Uchida, T. Nomura, S. Kimura, D. Sugimoto, A. Yoshida, H. Miyake, T. Iida, and S. Hattori, "An alignment technique using diffracted moire signals," J. Vac. Sci. Technol. B 7, 1977-1979 (1989).
[CrossRef]

Hasenfuss, C.

M. Muhlberger, I. Bergmair, W. Schwinger, M. Gmainer, R. Schoftner, T. Glinsner, C. Hasenfuss, K. Hingerl, M. Vogler, H. Schmidt, and E. B. Kley, "A Moire method for high accuracy alignment in nanoimprint lithography," Microelectron. Eng. 84, 925-927 (2007).
[CrossRef]

Hattori, S.

K. Hara, Y. Uchida, T. Nomura, S. Kimura, D. Sugimoto, A. Yoshida, H. Miyake, T. Iida, and S. Hattori, "An alignment technique using diffracted moire signals," J. Vac. Sci. Technol. B 7, 1977-1979 (1989).
[CrossRef]

Hersch, R. D.

Hingerl, K.

M. Muhlberger, I. Bergmair, W. Schwinger, M. Gmainer, R. Schoftner, T. Glinsner, C. Hasenfuss, K. Hingerl, M. Vogler, H. Schmidt, and E. B. Kley, "A Moire method for high accuracy alignment in nanoimprint lithography," Microelectron. Eng. 84, 925-927 (2007).
[CrossRef]

Ho, K. M.

J. H. Lee, C. H. Kim, Y.-S. Kim, K. M. Ho, K. Constant, W. Leung, and C. H. Oh, "Diffracted moire fringes as analysis and alignment tools for multilayer fabrication in soft lithography," Appl. Phys. Lett. 86, 204101-204101 -- 204101-204103 (2005).[DOI: 10. 1063/1.1927268]

Iida, T.

K. Hara, Y. Uchida, T. Nomura, S. Kimura, D. Sugimoto, A. Yoshida, H. Miyake, T. Iida, and S. Hattori, "An alignment technique using diffracted moire signals," J. Vac. Sci. Technol. B 7, 1977-1979 (1989).
[CrossRef]

Kim, C. H.

J. H. Lee, C. H. Kim, Y.-S. Kim, K. M. Ho, K. Constant, W. Leung, and C. H. Oh, "Diffracted moire fringes as analysis and alignment tools for multilayer fabrication in soft lithography," Appl. Phys. Lett. 86, 204101-204101 -- 204101-204103 (2005).[DOI: 10. 1063/1.1927268]

Kim, Y.-S.

J. H. Lee, C. H. Kim, Y.-S. Kim, K. M. Ho, K. Constant, W. Leung, and C. H. Oh, "Diffracted moire fringes as analysis and alignment tools for multilayer fabrication in soft lithography," Appl. Phys. Lett. 86, 204101-204101 -- 204101-204103 (2005).[DOI: 10. 1063/1.1927268]

Kimura, S.

K. Hara, Y. Uchida, T. Nomura, S. Kimura, D. Sugimoto, A. Yoshida, H. Miyake, T. Iida, and S. Hattori, "An alignment technique using diffracted moire signals," J. Vac. Sci. Technol. B 7, 1977-1979 (1989).
[CrossRef]

King, M. C.

M. C. King and D. H. Berry, "Photolithographic mask alignment using moire techniques," J. Vac. Sci. Technol. B 11, 2455-2458 (1972).

Kley, E. B.

M. Muhlberger, I. Bergmair, W. Schwinger, M. Gmainer, R. Schoftner, T. Glinsner, C. Hasenfuss, K. Hingerl, M. Vogler, H. Schmidt, and E. B. Kley, "A Moire method for high accuracy alignment in nanoimprint lithography," Microelectron. Eng. 84, 925-927 (2007).
[CrossRef]

Lee, J. H.

J. H. Lee, C. H. Kim, Y.-S. Kim, K. M. Ho, K. Constant, W. Leung, and C. H. Oh, "Diffracted moire fringes as analysis and alignment tools for multilayer fabrication in soft lithography," Appl. Phys. Lett. 86, 204101-204101 -- 204101-204103 (2005).[DOI: 10. 1063/1.1927268]

Leung, W.

J. H. Lee, C. H. Kim, Y.-S. Kim, K. M. Ho, K. Constant, W. Leung, and C. H. Oh, "Diffracted moire fringes as analysis and alignment tools for multilayer fabrication in soft lithography," Appl. Phys. Lett. 86, 204101-204101 -- 204101-204103 (2005).[DOI: 10. 1063/1.1927268]

Li, N. H.

N. H. Li, W. Wu, and S. Y. Chou, "Sub-20-nm alignment in nanoimprint lithography using Moire fringe," Nano Lett. 6, 2626-2629 (2006).
[CrossRef] [PubMed]

Miyake, H.

K. Hara, Y. Uchida, T. Nomura, S. Kimura, D. Sugimoto, A. Yoshida, H. Miyake, T. Iida, and S. Hattori, "An alignment technique using diffracted moire signals," J. Vac. Sci. Technol. B 7, 1977-1979 (1989).
[CrossRef]

Moel, A.

A. Moel, E. E. Moon, R. D. Frankel, and H. I. Smith, "Novel on-axis interferometric alignment method with sub-10 nm precision," J. Vac. Sci. Technol. B 11, 2191-2194 (1993).
[CrossRef]

Moon, E. E.

A. Moel, E. E. Moon, R. D. Frankel, and H. I. Smith, "Novel on-axis interferometric alignment method with sub-10 nm precision," J. Vac. Sci. Technol. B 11, 2191-2194 (1993).
[CrossRef]

Muhlberger, M.

M. Muhlberger, I. Bergmair, W. Schwinger, M. Gmainer, R. Schoftner, T. Glinsner, C. Hasenfuss, K. Hingerl, M. Vogler, H. Schmidt, and E. B. Kley, "A Moire method for high accuracy alignment in nanoimprint lithography," Microelectron. Eng. 84, 925-927 (2007).
[CrossRef]

Nishijima, Y.

Y. Nishijima and G. Oster, "Moiré Patterns: Their Application to Refractive Index and Refractive Index Gradient Measurements," J. Opt. Soc. Am. A 54, 1-5 (1964).
[CrossRef]

Nomura, T.

K. Hara, Y. Uchida, T. Nomura, S. Kimura, D. Sugimoto, A. Yoshida, H. Miyake, T. Iida, and S. Hattori, "An alignment technique using diffracted moire signals," J. Vac. Sci. Technol. B 7, 1977-1979 (1989).
[CrossRef]

Oh, C. H.

J. H. Lee, C. H. Kim, Y.-S. Kim, K. M. Ho, K. Constant, W. Leung, and C. H. Oh, "Diffracted moire fringes as analysis and alignment tools for multilayer fabrication in soft lithography," Appl. Phys. Lett. 86, 204101-204101 -- 204101-204103 (2005).[DOI: 10. 1063/1.1927268]

Oster, G.

Y. Nishijima and G. Oster, "Moiré Patterns: Their Application to Refractive Index and Refractive Index Gradient Measurements," J. Opt. Soc. Am. A 54, 1-5 (1964).
[CrossRef]

G. Oster, M. Wasserman, and C. Zwerling, "Theoretical Interpretation of Moiré Patterns," J. Opt. Soc. Am. A 54, 169-175 (1964).
[CrossRef]

Raleigh, L.

L. Raleigh, "On the manufacture and theory of diffraction gratings," Philos. Mag. 4, 81-93 (1874).

Schmidt, H.

M. Muhlberger, I. Bergmair, W. Schwinger, M. Gmainer, R. Schoftner, T. Glinsner, C. Hasenfuss, K. Hingerl, M. Vogler, H. Schmidt, and E. B. Kley, "A Moire method for high accuracy alignment in nanoimprint lithography," Microelectron. Eng. 84, 925-927 (2007).
[CrossRef]

Schoftner, R.

M. Muhlberger, I. Bergmair, W. Schwinger, M. Gmainer, R. Schoftner, T. Glinsner, C. Hasenfuss, K. Hingerl, M. Vogler, H. Schmidt, and E. B. Kley, "A Moire method for high accuracy alignment in nanoimprint lithography," Microelectron. Eng. 84, 925-927 (2007).
[CrossRef]

Schwinger, W.

M. Muhlberger, I. Bergmair, W. Schwinger, M. Gmainer, R. Schoftner, T. Glinsner, C. Hasenfuss, K. Hingerl, M. Vogler, H. Schmidt, and E. B. Kley, "A Moire method for high accuracy alignment in nanoimprint lithography," Microelectron. Eng. 84, 925-927 (2007).
[CrossRef]

Smith, H. I.

A. Moel, E. E. Moon, R. D. Frankel, and H. I. Smith, "Novel on-axis interferometric alignment method with sub-10 nm precision," J. Vac. Sci. Technol. B 11, 2191-2194 (1993).
[CrossRef]

Sugimoto, D.

K. Hara, Y. Uchida, T. Nomura, S. Kimura, D. Sugimoto, A. Yoshida, H. Miyake, T. Iida, and S. Hattori, "An alignment technique using diffracted moire signals," J. Vac. Sci. Technol. B 7, 1977-1979 (1989).
[CrossRef]

Uchida, Y.

K. Hara, Y. Uchida, T. Nomura, S. Kimura, D. Sugimoto, A. Yoshida, H. Miyake, T. Iida, and S. Hattori, "An alignment technique using diffracted moire signals," J. Vac. Sci. Technol. B 7, 1977-1979 (1989).
[CrossRef]

Vogler, M.

M. Muhlberger, I. Bergmair, W. Schwinger, M. Gmainer, R. Schoftner, T. Glinsner, C. Hasenfuss, K. Hingerl, M. Vogler, H. Schmidt, and E. B. Kley, "A Moire method for high accuracy alignment in nanoimprint lithography," Microelectron. Eng. 84, 925-927 (2007).
[CrossRef]

Wasserman, M.

G. Oster, M. Wasserman, and C. Zwerling, "Theoretical Interpretation of Moiré Patterns," J. Opt. Soc. Am. A 54, 169-175 (1964).
[CrossRef]

Wu, W.

N. H. Li, W. Wu, and S. Y. Chou, "Sub-20-nm alignment in nanoimprint lithography using Moire fringe," Nano Lett. 6, 2626-2629 (2006).
[CrossRef] [PubMed]

Yoshida, A.

K. Hara, Y. Uchida, T. Nomura, S. Kimura, D. Sugimoto, A. Yoshida, H. Miyake, T. Iida, and S. Hattori, "An alignment technique using diffracted moire signals," J. Vac. Sci. Technol. B 7, 1977-1979 (1989).
[CrossRef]

Zwerling, C.

G. Oster, M. Wasserman, and C. Zwerling, "Theoretical Interpretation of Moiré Patterns," J. Opt. Soc. Am. A 54, 169-175 (1964).
[CrossRef]

J. Opt. Soc. Am. A (6)

G. Oster, M. Wasserman, and C. Zwerling, "Theoretical Interpretation of Moiré Patterns," J. Opt. Soc. Am. A 54, 169-175 (1964).
[CrossRef]

Y. Nishijima and G. Oster, "Moiré Patterns: Their Application to Refractive Index and Refractive Index Gradient Measurements," J. Opt. Soc. Am. A 54, 1-5 (1964).
[CrossRef]

O. Bryngdahl, "Moiré: Formation and interpretation," J. Opt. Soc. Am. A 64, 1287-1294 (1974).
[CrossRef]

O. Bryngdahl, "Moiré and higher grating harmonics," J. Opt. Soc. Am. A 65, 685-694 (1975).
[CrossRef]

I. Amidror and R. D. Hersch, "Fourier-based analysis of phase shifts in the superposition of periodic layers and their moire effects," J. Opt. Soc. Am. A 13, 974-987 (1996).
[CrossRef]

I. Amidror and R. D. Hersch, "Fourier-based analysis and synthesis of moirés in the superposition of geometrically transformed periodic structures," J. Opt. Soc. Am. A 15, 1100-1113 (1998).
[CrossRef]

J. Vac. Sci. Technol. B (3)

M. C. King and D. H. Berry, "Photolithographic mask alignment using moire techniques," J. Vac. Sci. Technol. B 11, 2455-2458 (1972).

K. Hara, Y. Uchida, T. Nomura, S. Kimura, D. Sugimoto, A. Yoshida, H. Miyake, T. Iida, and S. Hattori, "An alignment technique using diffracted moire signals," J. Vac. Sci. Technol. B 7, 1977-1979 (1989).
[CrossRef]

A. Moel, E. E. Moon, R. D. Frankel, and H. I. Smith, "Novel on-axis interferometric alignment method with sub-10 nm precision," J. Vac. Sci. Technol. B 11, 2191-2194 (1993).
[CrossRef]

Microelectron. Eng. (1)

M. Muhlberger, I. Bergmair, W. Schwinger, M. Gmainer, R. Schoftner, T. Glinsner, C. Hasenfuss, K. Hingerl, M. Vogler, H. Schmidt, and E. B. Kley, "A Moire method for high accuracy alignment in nanoimprint lithography," Microelectron. Eng. 84, 925-927 (2007).
[CrossRef]

Nano Lett. (1)

N. H. Li, W. Wu, and S. Y. Chou, "Sub-20-nm alignment in nanoimprint lithography using Moire fringe," Nano Lett. 6, 2626-2629 (2006).
[CrossRef] [PubMed]

Philos. Mag. (1)

L. Raleigh, "On the manufacture and theory of diffraction gratings," Philos. Mag. 4, 81-93 (1874).

Other (5)

J. Guid, The Interference Systems of Crossed Diffraction Gratings (Oxford University Press, London, 1956).

J. Guid, Diffraction Gratings as Measuring Scales, (Oxford University Press, London,1960).

I. G. O. Kafri, The Physics of Moire Metrology, (Wiley, New York,1989).

L. Q. T, Physical Optics, (China Machine Press, Beijing, 1986).

J. H. Lee, C. H. Kim, Y.-S. Kim, K. M. Ho, K. Constant, W. Leung, and C. H. Oh, "Diffracted moire fringes as analysis and alignment tools for multilayer fabrication in soft lithography," Appl. Phys. Lett. 86, 204101-204101 -- 204101-204103 (2005).[DOI: 10. 1063/1.1927268]

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Figures (8)

Fig. 1.
Fig. 1.

Standard 1D-grating and the transformed structures: (a) standard 1D-grating with standardized period, (b) rotated 1D-grating with T(x, y)=y-x, (c) ellipse-shaped structures with T(x, y)=y2 /9+x2 /16, (d) parabola-shaped structures with T(x, y)=y2 -x2 .

Fig. 2.
Fig. 2.

(a). The descriptive vector and (b) its corresponding rotated 1D-grating

Fig. 3.
Fig. 3.

(a). Standard 1D-grating with standardized frequency of 1. (b): standard 1D-grating with frequency of 0.9. (c) and (d) show two circular gratings that are obtained by application of the transformation: T 1(x, y)=[(x+4)2+y 2]1/2 and T 2(x, y)=[(x-4)2+y 2]1/2 to 1D-gratings in (a) and (b). (e): moiré pattern generated from superposition of these two circular gratings with centers offset by 8 units. (f): the (1,-1) moiré fringe extracted according to conclusion 1.

Fig. 4.
Fig. 4.

(a). 1D-distributed moiré of two superposed standard 1D-gratings in Fig. 1(b), the (1,-1) moiré fringe extracted from two circular gratings with centers superposed according to conclusion 1. (c): Circular moiré fringe directly observed from two circular gratings with centers superposed.

Fig. 5.
Fig. 5.

(a). A 1D-grating denoted by F⃗ 1=2·exp(/6). (b). A 1D-grating denoted by F⃗ 2=exp(/6). (c): moiré fringes generated by 1D-gratings in (a) and (b).

Fig. 6.
Fig. 6.

(a) and (b) are two structures transformed from the same standard 1D-grating with frequency of 1 by T1 (x, y)=x/3+y2 /36 and T2 (x, y)=y2 /36. (c) is the moiré fringes of two superposed structures in (a) and (b). (d) and (e) are two elliptic structures transformed from the same standard 1D-grating by T1 (x, y)=x2 /25+(y-4)2/36 and T2 (x, y)=(x+4) 2 /25+y 2/36. (f) is the extracted moiré fringes of two superposed structures in both (d) and (e).

Fig. 7.
Fig. 7.

the circular moiré pattern of two superposed circular gratings marks with centers offset by (a) 0 units; (b) 1 unit; (c) 2 units; (d) 4 units.

Fig. 8.
Fig. 8.

(a). The grating mark on the wafer. (b) The grating mark on the mask. Moiré patterns of the two marks that are misaligned by (c) paverage /4; (d) paverage /16; (e)paverage /8; (f) 0 or integer times of paverage /2.

Equations (19)

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G ( x ) = n = + rect ( x np d )
G T ( x , y ) = G [ T ( x , y ) ] = n = + rect ( T ( x , y ) np d )
E ( x ) = + E ( f ) exp ( i 2 π f x ) d f = n = + a n exp ( i 2 π n f 0 x )
E T ( x , y ) = n = + a n exp [ i 2 π n f 0 T ( x , y ) ]
E ( x , y ) = E 1 ( x , y ) E 2 ( x , y )
= m = + n = + a n b m exp { i 2 π [ m f 2 T 2 ( x , y ) + n f 1 T 1 ( x , y ) ] }
E ( x , y ) = n = + a nk 1 b bk 2 exp { i 2 π n f [ k 1 . f 1 f . T 1 ( x , y ) + k 2 . f 2 f . T 2 ( x , y ) ] }
E ( x ) = n = + a nk 1 b nk 2 exp ( i 2 π n f x )
T ( x , y ) = k 1 · f 1 f · T 1 ( x , y ) + k 2 · f 2 f · T 2 ( x , y ) ]
T 1 ( x , y ) = x sin θ 1 y cos θ 1
T 2 ( x , y ) = x sin θ 2 y cos θ 2
E ( x , y ) = n = + a nk 1 b nk 2 exp [ i 2 π n f e ( x sin θ e y cos θ e ) ]
f e cos θ e = k 1 f 1 cos θ 1 + k 2 f 2 cos θ 2
f e = F e = ( f 1 k 1 ) 2 + ( f 2 k 2 ) 2 + 2 f 1 f 2 k 1 k 2 cos ( θ 1 θ 2 )
E ( x , y ) = n = + a n b n exp { i 2 π n f [ T 1 ( x , y ) T 2 ( x , y ) ] }
E ( x , y ) = n = + a n b n exp [ i 2 π n ( f 1 f 2 ) T ( x , y ) ]
E p ( x , y ) = n = + a n b n exp [ i 2 π n [ ( f 1 f 2 ) x ]
E a ( x , y ) = n = + a n b n exp { i 2 π n [ ( f 1 f 2 ) x + f 1 · Δ x ] }
Δ L = Δ x p 1 ( p 2 p 1 ) + Δ x p 2 ( p 2 p 1 ) = Δ x ( p 1 + p 2 ) p 2 p 1

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