Abstract

This paper reports the experimental and theoretical investigation of the Talbot effect beyond the paraxial limit at optical frequencies. Au hole array films with periodicitya0comparable to the wavelength of coherent illumination λ were used to study the non-paraxial Talbot effect. Significant differences from the paraxial (classical) Talbot effect were observed. Depending on the ratio of a0/λ, the interference pattern in the direction perpendicular to the hole array was not necessarily periodic, and the self-image distances deviated from the paraxial Talbot distances. Defects within the hole array film or above the film were healed in the self-images as the light propagated from the surface.

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  1. M. Masud, Classical Optics and its Applications (Cambridge University Press, 2002), Chap. 18.
  2. M. S. Chapman, C. R. Ekstrom, T. D. Hammond, J. Schmiedmayer, B. E. Tannian, S. Wehinger, and D. E. Pritchard, “Near-field imaging of atom diffraction gratings—the atomic Talbot effect,” Phys. Rev. A 51(1), R14–R17 (1995).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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2009 (3)

2007 (3)

M. H. Chowdhury, J. M. Catchmark, and J. R. Lakowicz, “Imaging three-dimensional light propagation through periodic nanohole arrays using scanning aperture microscopy,” Appl. Phys. Lett. 91(10), 103118 (2007).
[CrossRef] [PubMed]

Y. Y. Sun, X. C. Yuan, L. S. Ong, J. Bu, S. W. Zhu, and R. Liu, “Large-scale optical traps on a chip for optical sorting,” Appl. Phys. Lett. 90(3), 031107 (2007).
[CrossRef]

M. R. Dennis, N. I. Zheludev, and F. J. García de Abajo, “The plasmon Talbot effect,” Opt. Express 15(15), 9692–9700 (2007).
[CrossRef] [PubMed]

2006 (1)

J. Henzie, J. E. Barton, C. L. Stender, and T. W. Odom, “Large-area nanoscale patterning: chemistry meets fabrication,” Acc. Chem. Res. 39(4), 249–257 (2006).
[CrossRef] [PubMed]

2004 (1)

1996 (1)

M. V. Berry and S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. 43(10), 2139–2164 (1996).
[CrossRef]

1995 (1)

M. S. Chapman, C. R. Ekstrom, T. D. Hammond, J. Schmiedmayer, B. E. Tannian, S. Wehinger, and D. E. Pritchard, “Near-field imaging of atom diffraction gratings—the atomic Talbot effect,” Phys. Rev. A 51(1), R14–R17 (1995).
[CrossRef] [PubMed]

1993 (1)

E. Noponen and J. Turunen, “Electromagnetic theory of Talbot imaging,” Opt. Commun. 98(1-3), 132–140 (1993).
[CrossRef]

1990 (1)

1985 (1)

1972 (1)

P. B. Johnson and R. W. Christy, “Optical-constants of Noble-metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[CrossRef]

1971 (1)

1967 (1)

1881 (1)

L. Rayleigh, “On copying diffraction gratings, and on some phenomena connected therewith,” Philos. Mag. 11, 196–205 (1881).

Arndt, M.

Barton, J. E.

J. Henzie, J. E. Barton, C. L. Stender, and T. W. Odom, “Large-area nanoscale patterning: chemistry meets fabrication,” Acc. Chem. Res. 39(4), 249–257 (2006).
[CrossRef] [PubMed]

Berry, M. V.

M. V. Berry and S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. 43(10), 2139–2164 (1996).
[CrossRef]

Bu, J.

Y. Y. Sun, X. C. Yuan, L. S. Ong, J. Bu, S. W. Zhu, and R. Liu, “Large-scale optical traps on a chip for optical sorting,” Appl. Phys. Lett. 90(3), 031107 (2007).
[CrossRef]

Case, W. B.

Catchmark, J. M.

M. H. Chowdhury, J. M. Catchmark, and J. R. Lakowicz, “Imaging three-dimensional light propagation through periodic nanohole arrays using scanning aperture microscopy,” Appl. Phys. Lett. 91(10), 103118 (2007).
[CrossRef] [PubMed]

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(6), 2931–2937 (2009).
[CrossRef]

Chapman, M. S.

M. S. Chapman, C. R. Ekstrom, T. D. Hammond, J. Schmiedmayer, B. E. Tannian, S. Wehinger, and D. E. Pritchard, “Near-field imaging of atom diffraction gratings—the atomic Talbot effect,” Phys. Rev. A 51(1), R14–R17 (1995).
[CrossRef] [PubMed]

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(6), 2931–2937 (2009).
[CrossRef]

Chowdhury, M. H.

M. H. Chowdhury, J. M. Catchmark, and J. R. Lakowicz, “Imaging three-dimensional light propagation through periodic nanohole arrays using scanning aperture microscopy,” Appl. Phys. Lett. 91(10), 103118 (2007).
[CrossRef] [PubMed]

Christy, R. W.

P. B. Johnson and R. W. Christy, “Optical-constants of Noble-metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[CrossRef]

Dammann, H.

Deachapunya, S.

Dennis, M. R.

M. R. Dennis, N. I. Zheludev, and F. J. García de Abajo, “The plasmon Talbot effect,” Opt. Express 15(15), 9692–9700 (2007).
[CrossRef] [PubMed]

J. D. Ring, J. Lindberg, C. J. Howls, and M. R. Dennis, “Aberration-like cusped focusing in the post-paraxial Talbot effect,” Opt. Lett.submitted.

Ekstrom, C. R.

M. S. Chapman, C. R. Ekstrom, T. D. Hammond, J. Schmiedmayer, B. E. Tannian, S. Wehinger, and D. E. Pritchard, “Near-field imaging of atom diffraction gratings—the atomic Talbot effect,” Phys. Rev. A 51(1), R14–R17 (1995).
[CrossRef] [PubMed]

García de Abajo, F. J.

Groh, G.

Hammond, T. D.

M. S. Chapman, C. R. Ekstrom, T. D. Hammond, J. Schmiedmayer, B. E. Tannian, S. Wehinger, and D. E. Pritchard, “Near-field imaging of atom diffraction gratings—the atomic Talbot effect,” Phys. Rev. A 51(1), R14–R17 (1995).
[CrossRef] [PubMed]

Henzie, J.

J. Henzie, J. E. Barton, C. L. Stender, and T. W. Odom, “Large-area nanoscale patterning: chemistry meets fabrication,” Acc. Chem. Res. 39(4), 249–257 (2006).
[CrossRef] [PubMed]

Howls, C. J.

J. D. Ring, J. Lindberg, C. J. Howls, and M. R. Dennis, “Aberration-like cusped focusing in the post-paraxial Talbot effect,” Opt. Lett.submitted.

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(6), 2931–2937 (2009).
[CrossRef]

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(6), 2931–2937 (2009).
[CrossRef]

Johnson, P. B.

P. B. Johnson and R. W. Christy, “Optical-constants of Noble-metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[CrossRef]

Klein, S.

M. V. Berry and S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. 43(10), 2139–2164 (1996).
[CrossRef]

Kock, M.

Lakowicz, J. R.

M. H. Chowdhury, J. M. Catchmark, and J. R. Lakowicz, “Imaging three-dimensional light propagation through periodic nanohole arrays using scanning aperture microscopy,” Appl. Phys. Lett. 91(10), 103118 (2007).
[CrossRef] [PubMed]

Lindberg, J.

J. D. Ring, J. Lindberg, C. J. Howls, and M. R. Dennis, “Aberration-like cusped focusing in the post-paraxial Talbot effect,” Opt. Lett.submitted.

Liu, R.

Y. Y. Sun, X. C. Yuan, L. S. Ong, J. Bu, S. W. Zhu, and R. Liu, “Large-scale optical traps on a chip for optical sorting,” Appl. Phys. Lett. 90(3), 031107 (2007).
[CrossRef]

Lohmann, A. W.

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(6), 2931–2937 (2009).
[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(6), 2931–2937 (2009).
[CrossRef]

Montgomery, W. D.

Murata, K.

Nakano, Y.

Noponen, E.

E. Noponen and J. Turunen, “Electromagnetic theory of Talbot imaging,” Opt. Commun. 98(1-3), 132–140 (1993).
[CrossRef]

Odom, T. W.

J. Henzie, J. E. Barton, C. L. Stender, and T. W. Odom, “Large-area nanoscale patterning: chemistry meets fabrication,” Acc. Chem. Res. 39(4), 249–257 (2006).
[CrossRef] [PubMed]

Ong, L. S.

Y. Y. Sun, X. C. Yuan, L. S. Ong, J. Bu, S. W. Zhu, and R. Liu, “Large-scale optical traps on a chip for optical sorting,” Appl. Phys. Lett. 90(3), 031107 (2007).
[CrossRef]

Pritchard, D. E.

M. S. Chapman, C. R. Ekstrom, T. D. Hammond, J. Schmiedmayer, B. E. Tannian, S. Wehinger, and D. E. Pritchard, “Near-field imaging of atom diffraction gratings—the atomic Talbot effect,” Phys. Rev. A 51(1), R14–R17 (1995).
[CrossRef] [PubMed]

Rayleigh, L.

L. Rayleigh, “On copying diffraction gratings, and on some phenomena connected therewith,” Philos. Mag. 11, 196–205 (1881).

Ring, J. D.

J. D. Ring, J. Lindberg, C. J. Howls, and M. R. Dennis, “Aberration-like cusped focusing in the post-paraxial Talbot effect,” Opt. Lett.submitted.

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(6), 2931–2937 (2009).
[CrossRef]

Saastamoinen, T.

Schmiedmayer, J.

M. S. Chapman, C. R. Ekstrom, T. D. Hammond, J. Schmiedmayer, B. E. Tannian, S. Wehinger, and D. E. Pritchard, “Near-field imaging of atom diffraction gratings—the atomic Talbot effect,” Phys. Rev. A 51(1), R14–R17 (1995).
[CrossRef] [PubMed]

Stender, C. L.

J. Henzie, J. E. Barton, C. L. Stender, and T. W. Odom, “Large-area nanoscale patterning: chemistry meets fabrication,” Acc. Chem. Res. 39(4), 249–257 (2006).
[CrossRef] [PubMed]

Sun, Y. Y.

Y. Y. Sun, X. C. Yuan, L. S. Ong, J. Bu, S. W. Zhu, and R. Liu, “Large-scale optical traps on a chip for optical sorting,” Appl. Phys. Lett. 90(3), 031107 (2007).
[CrossRef]

Tannian, B. E.

M. S. Chapman, C. R. Ekstrom, T. D. Hammond, J. Schmiedmayer, B. E. Tannian, S. Wehinger, and D. E. Pritchard, “Near-field imaging of atom diffraction gratings—the atomic Talbot effect,” Phys. Rev. A 51(1), R14–R17 (1995).
[CrossRef] [PubMed]

Tervo, J.

Thomas, J. A.

Tomandl, M.

Turunen, J.

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(6), 2931–2937 (2009).
[CrossRef]

Vahimaa, P.

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(6), 2931–2937 (2009).
[CrossRef]

Wang, J. Y.

Wehinger, S.

M. S. Chapman, C. R. Ekstrom, T. D. Hammond, J. Schmiedmayer, B. E. Tannian, S. Wehinger, and D. E. Pritchard, “Near-field imaging of atom diffraction gratings—the atomic Talbot effect,” Phys. Rev. A 51(1), R14–R17 (1995).
[CrossRef] [PubMed]

Yuan, X. C.

Y. Y. Sun, X. C. Yuan, L. S. Ong, J. Bu, S. W. Zhu, and R. Liu, “Large-scale optical traps on a chip for optical sorting,” Appl. Phys. Lett. 90(3), 031107 (2007).
[CrossRef]

Zhang, J. S.

Zhang, W. W.

Zhao, C. L.

Zheludev, N. I.

Zhu, S. W.

Y. Y. Sun, X. C. Yuan, L. S. Ong, J. Bu, S. W. Zhu, and R. Liu, “Large-scale optical traps on a chip for optical sorting,” Appl. Phys. Lett. 90(3), 031107 (2007).
[CrossRef]

Acc. Chem. Res. (1)

J. Henzie, J. E. Barton, C. L. Stender, and T. W. Odom, “Large-area nanoscale patterning: chemistry meets fabrication,” Acc. Chem. Res. 39(4), 249–257 (2006).
[CrossRef] [PubMed]

Appl. Opt. (3)

Appl. Phys. Lett. (2)

M. H. Chowdhury, J. M. Catchmark, and J. R. Lakowicz, “Imaging three-dimensional light propagation through periodic nanohole arrays using scanning aperture microscopy,” Appl. Phys. Lett. 91(10), 103118 (2007).
[CrossRef] [PubMed]

Y. Y. Sun, X. C. Yuan, L. S. Ong, J. Bu, S. W. Zhu, and R. Liu, “Large-scale optical traps on a chip for optical sorting,” Appl. Phys. Lett. 90(3), 031107 (2007).
[CrossRef]

J. Mod. Opt. (1)

M. V. Berry and S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. 43(10), 2139–2164 (1996).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

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(6), 2931–2937 (2009).
[CrossRef]

Opt. Commun. (1)

E. Noponen and J. Turunen, “Electromagnetic theory of Talbot imaging,” Opt. Commun. 98(1-3), 132–140 (1993).
[CrossRef]

Opt. Express (3)

Opt. Lett. (1)

J. D. Ring, J. Lindberg, C. J. Howls, and M. R. Dennis, “Aberration-like cusped focusing in the post-paraxial Talbot effect,” Opt. Lett.submitted.

Philos. Mag. (1)

L. Rayleigh, “On copying diffraction gratings, and on some phenomena connected therewith,” Philos. Mag. 11, 196–205 (1881).

Phys. Rev. A (1)

M. S. Chapman, C. R. Ekstrom, T. D. Hammond, J. Schmiedmayer, B. E. Tannian, S. Wehinger, and D. E. Pritchard, “Near-field imaging of atom diffraction gratings—the atomic Talbot effect,” Phys. Rev. A 51(1), R14–R17 (1995).
[CrossRef] [PubMed]

Phys. Rev. B (1)

P. B. Johnson and R. W. Christy, “Optical-constants of Noble-metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[CrossRef]

Other (1)

M. Masud, Classical Optics and its Applications (Cambridge University Press, 2002), Chap. 18.

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Figures (4)

Fig. 1
Fig. 1

Self-image distances of the non-paraxial Talbot effect are different from the classical Talbot distances. (A-B)Fourier space spectrum of plane waves emitting from the hole array, where the circle with radius a0/λeff separates the propagating waves and evanescent waves. Light patterns from scalar wave calculations with (C) a0/λeff = 1.230 (a0 = 600 nm, λeff = 488 nm), (D)a0/λeff = 1.90 (a0 = 1.2 μm, λeff = 633 nm). Experimental measurement with (E) a0/λeff = 1.230, (F)a0/λeff = 1.90. Light patterns from FDTD simulations with (G) a0/λeff = 1.230, (H)a0/λeff = 1.90.

Fig. 2
Fig. 2

Difference between the measured first self-image distance zR and the Talbot distance zT is related to a0/λeff. The curves indicate the distances where the 1st, 2nd and 3rd nearest neighbors in k space are in phase with the center spatial frequency and approach the Talbot distance with increasing a0/λeff. At the self-image distances (black squares), the deviations from the phase matching curves are relatively small. The error bars indicate the experimental errors in determining the self-image distances due to the noises.

Fig. 3
Fig. 3

2D defects on the film were healed gradually in the self-image planes. Light patterns at self-image distances (A)z = 0 μm, (B)z = 4.4 μm, (C)z = 8.4 μm and (D)z = 13.1 μm. (E) the yz cross-section. As the light propagated in the z direction, the intensity of the defect decreased while the size of the defect increased. (a0 = 1.2 μm, λeff = 543 nm).

Fig. 4
Fig. 4

Defect away from the surface of the film was healed in the self-images of the hole array pattern. The light patterns at (A) the film surface z = 0 μm, (B) the position of the particles z = 11.2 μm(C) the self-image planes at z = 13.5 μm and (D) the self-image planes at z = 29 μm. (E) yz cross-section of the 3D light pattern showed the particle had a very local influence on the light pattern. (a0 = 1.2 μm, λ = 543 nm, n = 1.4)

Equations (5)

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

ψ ( r , λ e f f ) = m 1 = m 2 = A m 1 m 2 ( ω ) × exp [ i k x m 1 x + i k y m 2 y + i k z m 1 , m 2 z ]
k z m 1 , m 2 = ( | k | 2 k x m 1 2 k y m 2 2 ) 1 / 2 2 π [ 1 ( m 1 2 + m 2 2 ) λ e f f 2 / 2 a 0 2 ] / λ e f f
ψ ( x , y , z + s z T , λ e f f ) = exp [ i 2 π s z T / λ e f f ] ψ ( x , y , z , λ e f f )
exp [ i k z m 1 , m 2 z R ] = exp [ i k z 0 , 0 z R ]
z R = λ e f f / ( 1 1 ( λ e f f / a 0 ) 2 )

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