Abstract

Fractional Talbot images of amplitude line gratings, with small opening slits compared to the period, are characterized by an integer multiple of the gratings’ spatial frequency. We investigate the formation of fractional Talbot images analytically within a scalar framework and give a comprehensible insight into the paraxial limits involved. Particular attention is paid to nonparaxial effects on the intensity distribution at fractional Talbot planes and their lateral periodicities. We present a comparison between the measured intensity distributions and a numerical implementation of our analytical method. Both ways reveal the paraxial limits of frequency multiplication on fractional Talbot images. The use of fractional Talbot images for lithography results in ghost diffraction orders. We roughly estimate the ghost orders quantitatively with a simple numerical model for monochromatic and polychromatic illumination.

© 2014 Optical Society of America

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References

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  1. H. F. Talbot, “Facts relating to optical science,” Philos. Mag., Series 3, 9(56), 401–407 (1836).
  2. Rayleigh, “On copying diffraction-gratings, and on some phenomena connected therewith,” Philos. Mag., Series 5, 11(67), 196–205 (1881).
    [CrossRef]
  3. N. H. Salama, D. Patrignani, L. De Pasquale, and E. E. Sicre, “Wavefront sensor using the Talbot effect,” Opt. Laser Technol. 31, 269–272 (1999).
    [CrossRef]
  4. C. Siegel, F. Loewenthal, J. E. Balmer, and H. P. Weber, “Talbot array illuminator for single-shot measurements of laser-induced-damage thresholds of thin-film coatings,” Appl. Opt. 39, 1493–1499 (2000).
    [CrossRef]
  5. M. Wrage, P. Glas, D. Fischer, M. Leitner, N. N. Elkin, D. V. Vysotsky, A. P. Napartovich, and V. N. Troshchieva, “Phase-locking of a multicore fiber laser by wave propagation through an annular waveguide,” Opt. Commun. 205, 367–375 (2002).
    [CrossRef]
  6. W. B. Case, M. Tomandl, S. Deachapunya, and M. Arndt, “Realization of optical carpets in the Talbot and Talbot-Lau configurations,” Opt. Express 17, 20966–20974 (2009).
    [CrossRef]
  7. A. Isoyan, F. Jiang, Y. C. Cheng, F. Cerrina, P. Wachulak, L. Urbanski, J. Rocca, C. Menoni, and M. Marconi, “Talbot lithography: self-imaging of complex structures,” J. Vac. Sci. Technol. B 27, 2931 (2009).
    [CrossRef]
  8. T. J. Suleski, Y.-C. Chuang, P. C. Deguzman, and R. A. Barton, “Fabrication of optical microstructures through fractional Talbot imaging,” Proc. SPIE 5720, 86–93 (2005).
    [CrossRef]
  9. T. Harzendorf, L. Stuerzebecher, U. Vogler, U. D. Zeitner, and R. Voelkel, “Half-tone proximity lithography,” Proc. SPIE 7716, 77160Y (2010).
    [CrossRef]
  10. L. Stuerzebecher, T. Harzendorf, U. Vogler, U. D. Zeitner, and R. Voelkel, “Advanced mask aligner lithography: fabrication of periodic patterns using pinhole array mask and Talbot effect,” Opt. Express 18, 19485–19494 (2010).
    [CrossRef]
  11. D. Thomae, J. Maaß, O. Sandfuchs, A. Gatto, and R. Brunner, “Flexible mask illumination setup for serial multipatterning in Talbot lithography,” Appl. Opt. 53, 1775–1781 (2014).
    [CrossRef]
  12. H. Weisel, “Über die nach Fresnelscher Art beobachteten Beugungserscheinungen der Gitter,” Annalen der Physik 338, 995–1031 (1910).
  13. J. T. Winthrop and C. R. Worthington, “Theory of Fresnel images I plane periodic objects in monochromatic light,” J. Opt. Soc. Am. 55, 373–380 (1965).
    [CrossRef]
  14. M. Testorf and J. Jahns, “Planar-integrated Talbot array illuminators,” Appl. Opt. 37, 5399–5407 (1998).
    [CrossRef]
  15. K.-H. Luo, J. Wen, X.-H. Chen, Q. Liu, M. Xiao, and L.-A. Wu, “Second-order Talbot effect with entangled photon pairs,” Phys. Rev. A 80, 043820 (2009).
    [CrossRef]
  16. P. M. Mejías and R. Martínez Herrero, “Diffraction by one-dimensional Ronchi grids: on the validity of the Talbot effect,” J. Opt. Soc. Am. 8, 266–269 (1991).
    [CrossRef]
  17. E. Noponen and J. Turunen, “Electromagnetic theory of Talbot imaging,” Opt. Commun. 98, 132–140 (1993).
    [CrossRef]
  18. J. W. Goodman, “3.10 the angular spectrum of plane waves,” in Introduction to Fourier Optics, 3rd ed. (Roberts, 2005), pp. 55–61.
  19. E. di Mambro, R. Haïdar, N. Guérineau, and J. Primot, “Sharpness limitations in the projection of thin lines by use of the Talbot experiment,” J. Opt. Soc. Am. 21, 2276–2282 (2004).
    [CrossRef]
  20. E. G. Loewen, “11.2 spectral purity,” in Diffraction Gratings and Applications, Optical Engineering No. 58 (M. Dekker, 1997), pp. 402–413.
  21. E. G. Loewen, “14.4 accuracy requirements,” in Diffraction Gratings and Applications, Optical Engineering No. 58 (Marcel Dekker, 1997), pp. 507–510.

2014 (1)

2010 (2)

2009 (3)

W. B. Case, M. Tomandl, S. Deachapunya, and M. Arndt, “Realization of optical carpets in the Talbot and Talbot-Lau configurations,” Opt. Express 17, 20966–20974 (2009).
[CrossRef]

A. Isoyan, F. Jiang, Y. C. Cheng, F. Cerrina, P. Wachulak, L. Urbanski, J. Rocca, C. Menoni, and M. Marconi, “Talbot lithography: self-imaging of complex structures,” J. Vac. Sci. Technol. B 27, 2931 (2009).
[CrossRef]

K.-H. Luo, J. Wen, X.-H. Chen, Q. Liu, M. Xiao, and L.-A. Wu, “Second-order Talbot effect with entangled photon pairs,” Phys. Rev. A 80, 043820 (2009).
[CrossRef]

2005 (1)

T. J. Suleski, Y.-C. Chuang, P. C. Deguzman, and R. A. Barton, “Fabrication of optical microstructures through fractional Talbot imaging,” Proc. SPIE 5720, 86–93 (2005).
[CrossRef]

2004 (1)

E. di Mambro, R. Haïdar, N. Guérineau, and J. Primot, “Sharpness limitations in the projection of thin lines by use of the Talbot experiment,” J. Opt. Soc. Am. 21, 2276–2282 (2004).
[CrossRef]

2002 (1)

M. Wrage, P. Glas, D. Fischer, M. Leitner, N. N. Elkin, D. V. Vysotsky, A. P. Napartovich, and V. N. Troshchieva, “Phase-locking of a multicore fiber laser by wave propagation through an annular waveguide,” Opt. Commun. 205, 367–375 (2002).
[CrossRef]

2000 (1)

1999 (1)

N. H. Salama, D. Patrignani, L. De Pasquale, and E. E. Sicre, “Wavefront sensor using the Talbot effect,” Opt. Laser Technol. 31, 269–272 (1999).
[CrossRef]

1998 (1)

1993 (1)

E. Noponen and J. Turunen, “Electromagnetic theory of Talbot imaging,” Opt. Commun. 98, 132–140 (1993).
[CrossRef]

1991 (1)

P. M. Mejías and R. Martínez Herrero, “Diffraction by one-dimensional Ronchi grids: on the validity of the Talbot effect,” J. Opt. Soc. Am. 8, 266–269 (1991).
[CrossRef]

1965 (1)

1910 (1)

H. Weisel, “Über die nach Fresnelscher Art beobachteten Beugungserscheinungen der Gitter,” Annalen der Physik 338, 995–1031 (1910).

1881 (1)

Rayleigh, “On copying diffraction-gratings, and on some phenomena connected therewith,” Philos. Mag., Series 5, 11(67), 196–205 (1881).
[CrossRef]

1836 (1)

H. F. Talbot, “Facts relating to optical science,” Philos. Mag., Series 3, 9(56), 401–407 (1836).

Arndt, M.

Balmer, J. E.

Barton, R. A.

T. J. Suleski, Y.-C. Chuang, P. C. Deguzman, and R. A. Barton, “Fabrication of optical microstructures through fractional Talbot imaging,” Proc. SPIE 5720, 86–93 (2005).
[CrossRef]

Brunner, R.

Case, W. B.

Cerrina, F.

A. Isoyan, F. Jiang, Y. C. Cheng, F. Cerrina, P. Wachulak, L. Urbanski, J. Rocca, C. Menoni, and M. Marconi, “Talbot lithography: self-imaging of complex structures,” J. Vac. Sci. Technol. B 27, 2931 (2009).
[CrossRef]

Chen, X.-H.

K.-H. Luo, J. Wen, X.-H. Chen, Q. Liu, M. Xiao, and L.-A. Wu, “Second-order Talbot effect with entangled photon pairs,” Phys. Rev. A 80, 043820 (2009).
[CrossRef]

Cheng, Y. C.

A. Isoyan, F. Jiang, Y. C. Cheng, F. Cerrina, P. Wachulak, L. Urbanski, J. Rocca, C. Menoni, and M. Marconi, “Talbot lithography: self-imaging of complex structures,” J. Vac. Sci. Technol. B 27, 2931 (2009).
[CrossRef]

Chuang, Y.-C.

T. J. Suleski, Y.-C. Chuang, P. C. Deguzman, and R. A. Barton, “Fabrication of optical microstructures through fractional Talbot imaging,” Proc. SPIE 5720, 86–93 (2005).
[CrossRef]

De Pasquale, L.

N. H. Salama, D. Patrignani, L. De Pasquale, and E. E. Sicre, “Wavefront sensor using the Talbot effect,” Opt. Laser Technol. 31, 269–272 (1999).
[CrossRef]

Deachapunya, S.

Deguzman, P. C.

T. J. Suleski, Y.-C. Chuang, P. C. Deguzman, and R. A. Barton, “Fabrication of optical microstructures through fractional Talbot imaging,” Proc. SPIE 5720, 86–93 (2005).
[CrossRef]

di Mambro, E.

E. di Mambro, R. Haïdar, N. Guérineau, and J. Primot, “Sharpness limitations in the projection of thin lines by use of the Talbot experiment,” J. Opt. Soc. Am. 21, 2276–2282 (2004).
[CrossRef]

Elkin, N. N.

M. Wrage, P. Glas, D. Fischer, M. Leitner, N. N. Elkin, D. V. Vysotsky, A. P. Napartovich, and V. N. Troshchieva, “Phase-locking of a multicore fiber laser by wave propagation through an annular waveguide,” Opt. Commun. 205, 367–375 (2002).
[CrossRef]

Fischer, D.

M. Wrage, P. Glas, D. Fischer, M. Leitner, N. N. Elkin, D. V. Vysotsky, A. P. Napartovich, and V. N. Troshchieva, “Phase-locking of a multicore fiber laser by wave propagation through an annular waveguide,” Opt. Commun. 205, 367–375 (2002).
[CrossRef]

Gatto, A.

Glas, P.

M. Wrage, P. Glas, D. Fischer, M. Leitner, N. N. Elkin, D. V. Vysotsky, A. P. Napartovich, and V. N. Troshchieva, “Phase-locking of a multicore fiber laser by wave propagation through an annular waveguide,” Opt. Commun. 205, 367–375 (2002).
[CrossRef]

Goodman, J. W.

J. W. Goodman, “3.10 the angular spectrum of plane waves,” in Introduction to Fourier Optics, 3rd ed. (Roberts, 2005), pp. 55–61.

Guérineau, N.

E. di Mambro, R. Haïdar, N. Guérineau, and J. Primot, “Sharpness limitations in the projection of thin lines by use of the Talbot experiment,” J. Opt. Soc. Am. 21, 2276–2282 (2004).
[CrossRef]

Haïdar, R.

E. di Mambro, R. Haïdar, N. Guérineau, and J. Primot, “Sharpness limitations in the projection of thin lines by use of the Talbot experiment,” J. Opt. Soc. Am. 21, 2276–2282 (2004).
[CrossRef]

Harzendorf, T.

Isoyan, A.

A. Isoyan, F. Jiang, Y. C. Cheng, F. Cerrina, P. Wachulak, L. Urbanski, J. Rocca, C. Menoni, and M. Marconi, “Talbot lithography: self-imaging of complex structures,” J. Vac. Sci. Technol. B 27, 2931 (2009).
[CrossRef]

Jahns, J.

Jiang, F.

A. Isoyan, F. Jiang, Y. C. Cheng, F. Cerrina, P. Wachulak, L. Urbanski, J. Rocca, C. Menoni, and M. Marconi, “Talbot lithography: self-imaging of complex structures,” J. Vac. Sci. Technol. B 27, 2931 (2009).
[CrossRef]

Leitner, M.

M. Wrage, P. Glas, D. Fischer, M. Leitner, N. N. Elkin, D. V. Vysotsky, A. P. Napartovich, and V. N. Troshchieva, “Phase-locking of a multicore fiber laser by wave propagation through an annular waveguide,” Opt. Commun. 205, 367–375 (2002).
[CrossRef]

Liu, Q.

K.-H. Luo, J. Wen, X.-H. Chen, Q. Liu, M. Xiao, and L.-A. Wu, “Second-order Talbot effect with entangled photon pairs,” Phys. Rev. A 80, 043820 (2009).
[CrossRef]

Loewen, E. G.

E. G. Loewen, “11.2 spectral purity,” in Diffraction Gratings and Applications, Optical Engineering No. 58 (M. Dekker, 1997), pp. 402–413.

E. G. Loewen, “14.4 accuracy requirements,” in Diffraction Gratings and Applications, Optical Engineering No. 58 (Marcel Dekker, 1997), pp. 507–510.

Loewenthal, F.

Luo, K.-H.

K.-H. Luo, J. Wen, X.-H. Chen, Q. Liu, M. Xiao, and L.-A. Wu, “Second-order Talbot effect with entangled photon pairs,” Phys. Rev. A 80, 043820 (2009).
[CrossRef]

Maaß, J.

Marconi, M.

A. Isoyan, F. Jiang, Y. C. Cheng, F. Cerrina, P. Wachulak, L. Urbanski, J. Rocca, C. Menoni, and M. Marconi, “Talbot lithography: self-imaging of complex structures,” J. Vac. Sci. Technol. B 27, 2931 (2009).
[CrossRef]

Martínez Herrero, R.

P. M. Mejías and R. Martínez Herrero, “Diffraction by one-dimensional Ronchi grids: on the validity of the Talbot effect,” J. Opt. Soc. Am. 8, 266–269 (1991).
[CrossRef]

Mejías, P. M.

P. M. Mejías and R. Martínez Herrero, “Diffraction by one-dimensional Ronchi grids: on the validity of the Talbot effect,” J. Opt. Soc. Am. 8, 266–269 (1991).
[CrossRef]

Menoni, C.

A. Isoyan, F. Jiang, Y. C. Cheng, F. Cerrina, P. Wachulak, L. Urbanski, J. Rocca, C. Menoni, and M. Marconi, “Talbot lithography: self-imaging of complex structures,” J. Vac. Sci. Technol. B 27, 2931 (2009).
[CrossRef]

Napartovich, A. P.

M. Wrage, P. Glas, D. Fischer, M. Leitner, N. N. Elkin, D. V. Vysotsky, A. P. Napartovich, and V. N. Troshchieva, “Phase-locking of a multicore fiber laser by wave propagation through an annular waveguide,” Opt. Commun. 205, 367–375 (2002).
[CrossRef]

Noponen, E.

E. Noponen and J. Turunen, “Electromagnetic theory of Talbot imaging,” Opt. Commun. 98, 132–140 (1993).
[CrossRef]

Patrignani, D.

N. H. Salama, D. Patrignani, L. De Pasquale, and E. E. Sicre, “Wavefront sensor using the Talbot effect,” Opt. Laser Technol. 31, 269–272 (1999).
[CrossRef]

Primot, J.

E. di Mambro, R. Haïdar, N. Guérineau, and J. Primot, “Sharpness limitations in the projection of thin lines by use of the Talbot experiment,” J. Opt. Soc. Am. 21, 2276–2282 (2004).
[CrossRef]

Rayleigh,

Rayleigh, “On copying diffraction-gratings, and on some phenomena connected therewith,” Philos. Mag., Series 5, 11(67), 196–205 (1881).
[CrossRef]

Rocca, J.

A. Isoyan, F. Jiang, Y. C. Cheng, F. Cerrina, P. Wachulak, L. Urbanski, J. Rocca, C. Menoni, and M. Marconi, “Talbot lithography: self-imaging of complex structures,” J. Vac. Sci. Technol. B 27, 2931 (2009).
[CrossRef]

Salama, N. H.

N. H. Salama, D. Patrignani, L. De Pasquale, and E. E. Sicre, “Wavefront sensor using the Talbot effect,” Opt. Laser Technol. 31, 269–272 (1999).
[CrossRef]

Sandfuchs, O.

Sicre, E. E.

N. H. Salama, D. Patrignani, L. De Pasquale, and E. E. Sicre, “Wavefront sensor using the Talbot effect,” Opt. Laser Technol. 31, 269–272 (1999).
[CrossRef]

Siegel, C.

Stuerzebecher, L.

Suleski, T. J.

T. J. Suleski, Y.-C. Chuang, P. C. Deguzman, and R. A. Barton, “Fabrication of optical microstructures through fractional Talbot imaging,” Proc. SPIE 5720, 86–93 (2005).
[CrossRef]

Talbot, H. F.

H. F. Talbot, “Facts relating to optical science,” Philos. Mag., Series 3, 9(56), 401–407 (1836).

Testorf, M.

Thomae, D.

Tomandl, M.

Troshchieva, V. N.

M. Wrage, P. Glas, D. Fischer, M. Leitner, N. N. Elkin, D. V. Vysotsky, A. P. Napartovich, and V. N. Troshchieva, “Phase-locking of a multicore fiber laser by wave propagation through an annular waveguide,” Opt. Commun. 205, 367–375 (2002).
[CrossRef]

Turunen, J.

E. Noponen and J. Turunen, “Electromagnetic theory of Talbot imaging,” Opt. Commun. 98, 132–140 (1993).
[CrossRef]

Urbanski, L.

A. Isoyan, F. Jiang, Y. C. Cheng, F. Cerrina, P. Wachulak, L. Urbanski, J. Rocca, C. Menoni, and M. Marconi, “Talbot lithography: self-imaging of complex structures,” J. Vac. Sci. Technol. B 27, 2931 (2009).
[CrossRef]

Voelkel, R.

Vogler, U.

Vysotsky, D. V.

M. Wrage, P. Glas, D. Fischer, M. Leitner, N. N. Elkin, D. V. Vysotsky, A. P. Napartovich, and V. N. Troshchieva, “Phase-locking of a multicore fiber laser by wave propagation through an annular waveguide,” Opt. Commun. 205, 367–375 (2002).
[CrossRef]

Wachulak, P.

A. Isoyan, F. Jiang, Y. C. Cheng, F. Cerrina, P. Wachulak, L. Urbanski, J. Rocca, C. Menoni, and M. Marconi, “Talbot lithography: self-imaging of complex structures,” J. Vac. Sci. Technol. B 27, 2931 (2009).
[CrossRef]

Weber, H. P.

Weisel, H.

H. Weisel, “Über die nach Fresnelscher Art beobachteten Beugungserscheinungen der Gitter,” Annalen der Physik 338, 995–1031 (1910).

Wen, J.

K.-H. Luo, J. Wen, X.-H. Chen, Q. Liu, M. Xiao, and L.-A. Wu, “Second-order Talbot effect with entangled photon pairs,” Phys. Rev. A 80, 043820 (2009).
[CrossRef]

Winthrop, J. T.

Worthington, C. R.

Wrage, M.

M. Wrage, P. Glas, D. Fischer, M. Leitner, N. N. Elkin, D. V. Vysotsky, A. P. Napartovich, and V. N. Troshchieva, “Phase-locking of a multicore fiber laser by wave propagation through an annular waveguide,” Opt. Commun. 205, 367–375 (2002).
[CrossRef]

Wu, L.-A.

K.-H. Luo, J. Wen, X.-H. Chen, Q. Liu, M. Xiao, and L.-A. Wu, “Second-order Talbot effect with entangled photon pairs,” Phys. Rev. A 80, 043820 (2009).
[CrossRef]

Xiao, M.

K.-H. Luo, J. Wen, X.-H. Chen, Q. Liu, M. Xiao, and L.-A. Wu, “Second-order Talbot effect with entangled photon pairs,” Phys. Rev. A 80, 043820 (2009).
[CrossRef]

Zeitner, U. D.

Annalen der Physik (1)

H. Weisel, “Über die nach Fresnelscher Art beobachteten Beugungserscheinungen der Gitter,” Annalen der Physik 338, 995–1031 (1910).

Appl. Opt. (3)

J. Opt. Soc. Am. (3)

J. T. Winthrop and C. R. Worthington, “Theory of Fresnel images I plane periodic objects in monochromatic light,” J. Opt. Soc. Am. 55, 373–380 (1965).
[CrossRef]

P. M. Mejías and R. Martínez Herrero, “Diffraction by one-dimensional Ronchi grids: on the validity of the Talbot effect,” J. Opt. Soc. Am. 8, 266–269 (1991).
[CrossRef]

E. di Mambro, R. Haïdar, N. Guérineau, and J. Primot, “Sharpness limitations in the projection of thin lines by use of the Talbot experiment,” J. Opt. Soc. Am. 21, 2276–2282 (2004).
[CrossRef]

J. Vac. Sci. Technol. B (1)

A. Isoyan, F. Jiang, Y. C. Cheng, F. Cerrina, P. Wachulak, L. Urbanski, J. Rocca, C. Menoni, and M. Marconi, “Talbot lithography: self-imaging of complex structures,” J. Vac. Sci. Technol. B 27, 2931 (2009).
[CrossRef]

Opt. Commun. (2)

M. Wrage, P. Glas, D. Fischer, M. Leitner, N. N. Elkin, D. V. Vysotsky, A. P. Napartovich, and V. N. Troshchieva, “Phase-locking of a multicore fiber laser by wave propagation through an annular waveguide,” Opt. Commun. 205, 367–375 (2002).
[CrossRef]

E. Noponen and J. Turunen, “Electromagnetic theory of Talbot imaging,” Opt. Commun. 98, 132–140 (1993).
[CrossRef]

Opt. Express (2)

Opt. Laser Technol. (1)

N. H. Salama, D. Patrignani, L. De Pasquale, and E. E. Sicre, “Wavefront sensor using the Talbot effect,” Opt. Laser Technol. 31, 269–272 (1999).
[CrossRef]

Philos. Mag. (2)

H. F. Talbot, “Facts relating to optical science,” Philos. Mag., Series 3, 9(56), 401–407 (1836).

Rayleigh, “On copying diffraction-gratings, and on some phenomena connected therewith,” Philos. Mag., Series 5, 11(67), 196–205 (1881).
[CrossRef]

Phys. Rev. A (1)

K.-H. Luo, J. Wen, X.-H. Chen, Q. Liu, M. Xiao, and L.-A. Wu, “Second-order Talbot effect with entangled photon pairs,” Phys. Rev. A 80, 043820 (2009).
[CrossRef]

Proc. SPIE (2)

T. J. Suleski, Y.-C. Chuang, P. C. Deguzman, and R. A. Barton, “Fabrication of optical microstructures through fractional Talbot imaging,” Proc. SPIE 5720, 86–93 (2005).
[CrossRef]

T. Harzendorf, L. Stuerzebecher, U. Vogler, U. D. Zeitner, and R. Voelkel, “Half-tone proximity lithography,” Proc. SPIE 7716, 77160Y (2010).
[CrossRef]

Other (3)

J. W. Goodman, “3.10 the angular spectrum of plane waves,” in Introduction to Fourier Optics, 3rd ed. (Roberts, 2005), pp. 55–61.

E. G. Loewen, “11.2 spectral purity,” in Diffraction Gratings and Applications, Optical Engineering No. 58 (M. Dekker, 1997), pp. 402–413.

E. G. Loewen, “14.4 accuracy requirements,” in Diffraction Gratings and Applications, Optical Engineering No. 58 (Marcel Dekker, 1997), pp. 507–510.

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Figures (8)

Fig. 1.
Fig. 1.

Intensity distribution (“Talbot intensity carpet” [6]) of 200 μm periodic line grating with 5% aperture duty cycle (i.e., an open slit width of 10 μm) illuminated by a plane wave of 100 nm wavelength. The Talbot image is located at zT, a shifted self-image is located at zT/2. Fractional Talbot images with higher frequencies can be found, for example, at zT/4 or zT/8. The carpet was calculated by a numerical implementation of the angular spectrum method (ASM).

Fig. 2.
Fig. 2.

Intensity distribution of 2×2 periods (162×162μm) in different planes behind the mask: Both the measured and the numerical simulated intensity distributions are shown. The mask was illuminated perpendicular by a plane wave of 406 nm wavelength.

Fig. 3.
Fig. 3.

Fractional Talbot image at zT/4 with 2×2 periods (162×162μm) extension and flagged periodicity errors.

Fig. 4.
Fig. 4.

Schematic sketch of generation of virtual grating.

Fig. 5.
Fig. 5.

Intensity distribution close to the quarter-Talbot plane of 50 μm amplitude mask for monochromatic light of 400 nm (dotted curve, z=3,107mm) and polychromatic light with a central wavelength of 400 and 20 nm 1/e-width (thick gray line, z=3,122mm).

Fig. 6.
Fig. 6.

Energy per diffraction order of virtual binary phase grating written with the monochromatic intensity pattern attained at the specified distance behind an amplitude mask. (a) For 50 μm mask period and (b) for 10 μm mask period. The used scaling factor D of Eq. (18) is printed in Fig. 7.

Fig. 7.
Fig. 7.

Scaling factor D for monochromatic (black) and polychromatic (gray) illumination radiation as a function of z. The curves of the 10 μm mask are plotted as a solid line; the curves of the 50 μm mask are plotted as a dotted line. For the 10 μm mask with monochromatic illumination, the optimization algorithm did not find a meaningful D in a small interval around 1.035·zT/4.

Fig. 8.
Fig. 8.

Energy per diffraction order of virtual binary phase grating written with a polychromatic intensity pattern attained at the specified distance behind an amplitude mask. The central wavelength was 400 nm with 1/e spectral width of 20 nm. (a) for 50 μm mask period and (b) for 10 μm mask period. The used scaling factor D of Eq. (18) is printed in Fig. 7.

Tables (3)

Tables Icon

Table 1. Relative Phase Shift φrel,n of Diffraction Orders n=17 with Respect to the Zeroth Order for Different Distances z behind the Mask

Tables Icon

Table 2. Relative Phase Shift at z=zT of Four Diffraction Orders n with respect to the Zeroth Order for Different Mask-Period-To-Wavelength Ratios (Rounded to (1/100)·π)

Tables Icon

Table 3. Amplitude Coefficients B0 to B3 of an Amplitude Line Grating with an Aperture Duty Cycle of 0.25; φrel, cos(φrel) and sinφrel) for a Period of 10 μm and λ=400nm at the Talbot Quarter Plane and the Intensities’ Fourier Coefficients Cn for B0 to B3 and for B0 to B25

Equations (18)

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

zT=λ11λ2p2.
U(x,z=0+)=T(x).
A(fx,0+)=F[T(x)].
A(fx,z>0)=A(fx,0+)·ei2πzλ·1(λfx)2=A(fx,0+)·eiφ(fx,z).
U(x,z>0)=F1[A(fx,z>0)].
φrel(fx,z)=2πzλ[1(λfx)21].
T(x,z=0)=U(x,z=0+)=n=0n=Bncos(nk0x).
U(x,z>0)=F1[eiφrel(kx,z)F(n=0n=Bncos(nk0x))].
U(x,z>0)=n=0n=eiφrel(nk0z)Bncos(nk0x).
U(x,z=zT4)=[B0+B2cos(2k0x)+B4cos(4k0x)+]i[B1cos(1k0x)+B3cos(3k0x)+].
I(z=zT4)=|U|2=Ureal2+Uimag2.
I(z)=C0+C1cos(1k0x)+C2cos(2k0x)+
C0=B02+12(B12+B22+B32)C1=C3=C5=0C2=12B12+2B0B2+B1B3C4=12B22+B1B3C6=12B32.
I(z=zT4)=[B0+B2cosφrel2cos(2k0x)+B3cosφrel3cos(3k0x)+]2+[B1sinφrel1cos(1k0x)+B2sinφrel2cos(2k0x)+]2.
C0=B02+12(B12+B22+B32)C1=B2B3[cosφrel2cosφrel3+sinφrel2sinφrel3]B1B2sinφrel2C2=2B0B2cosφrel2B1B3sinφrel3+12B12C3=2B0B3cosφrel3B1B2sinφrel2C4=12B22B1B3sinφrel3C5=B2B3[cosφrel2cosφrel3+sinφrel2sinφrel3]C6=12B32.
φrel FA(fx,z)=zπfx2λ.
Γ(α)=2(1sin2α1)sin2α=21+cosαwith the diffraction angle α as defined below Eq. (3)withαR&90°<α<90°.
Δφ(x)=exp(i·D·I(x,z)).

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