Abstract

Talbot and Talbot-Lau effects are frequently used in lensless imaging applications with light, ultrasound, x-rays, atoms and molecules – generally in situations where refractive optical elements are non-existent or not suitable. We here show an experimental visualization of the intriguing wave patterns that are associated with near-field interferometry behind a single periodic diffraction grating under plane wave illumination and which are often referred to as Talbot carpets or quantum carpets. We also show the patterns behind two separated diffraction gratings under nearly-monochromatic but spatially incoherent illumination that illustrate the nature of Talbot-Lau carpets.

© 2009 Optical Society of America

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  1. H. F. Talbot, "Facts Relating to Optical Science," Philos. Mag. 9, 401-407 (1836).
  2. E. Lau, "Beugungserscheinungen an Doppelrastern," Ann. Phys. 6, 417 (1948).
    [CrossRef]
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    [CrossRef]
  5. K. Patorski, "Incoherent Superposition ofMultiple Self-imaging Lau Effect andMoire Fringe Explanation," Opt. Acta 30, 745-758 (1983).
    [CrossRef]
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  9. K. Banaszek, K. Wodkiewicz, and W. P. Schleich, "Fractional Talbot effect in Phase Space: A Compact Summation Formula," Opt. Express 2, 169 - 172 (1998).
    [CrossRef] [PubMed]
  10. O. Friesch, W. Schleich, and I. Marzoli, "Quantum carpets woven by Wigner functions," New J. Phys. 2(1), 4 (2000).
    [CrossRef]
  11. S. Mirza and C. Shakher, "Surface profiling using phase shifting Talbot interferometric technique," Optical Engineering 44, 013,601 (2004).
  12. M. Thakur, C. Tay, and C. Quan, "Surface profiling of a transparent object by use of phase-shifting Talbot interferometry," Appl. Opt. 44(13), 2541-2545 (2005).
    [CrossRef]
  13. J. Bhattacharya, "Measurement of the refractive index using the Talbot effect and a moire technique," Appl. Opt. 28(13), 2600-2604 (1989).
    [CrossRef]
  14. S. Prakash, S. Singh, and A. Verma, "A low cost technique for automated measurement of focal length using Lau effect combined with Moire readout," J. Mod. Opt. 53(14), 2033-2042 (2006).
    [CrossRef]
  15. P. Singh, M. Faridi, and C. Shakher, "Measurement of temperature of an axisymmetric flame using shearing interferometry and Fourier fringe analysis technique," Opt. Eng. 43, 387 (2004).
    [CrossRef]
  16. G. Spagnolo, D. Ambrosini, and D. Paoletti, "Displacement measurement using the Talbot effect with a Ronchi grating," J. Opt. A 4(6), 376-380 (2002).
    [CrossRef]
  17. F. Huang, N. Zheludev, Y. Chen, and F. de Abajo, "Focusing of light by a nanohole array," Appl. Phys. Lett. 90, 091,119 (2007).
    [CrossRef]
  18. M. Dennis, N. Zheludev, and F. García de Abajo, "The plasmon Talbot effect," Opt. Express 15(15), 9692-9700 (2007).
    [CrossRef]
  19. P. Cloetens, J. Guigay, C. De Martino, J. Baruchel, and M. Schlenker, "Fractional Talbot imaging of phase gratings with hard x rays," Opt. Lett. 22(14), 1059-1061 (1997).
    [CrossRef]
  20. F. Pfeiffer, T. Weitkamp, O. Bunk and C. David, "Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources," Nature 2, 258-261 (2006).
  21. J. F. Clauser and S. Li, "‘Heisenberg Microscope’ Decoherence Atom Interferometry," Phys. Rev. A 50, 2430 (1994).
    [CrossRef] [PubMed]
  22. 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, R14 (1995).
    [CrossRef] [PubMed]
  23. S. Nowak, C. Kurtsiefer, T. Pfau, and C. David, "High-Order Talbot Fringes for Atomic Matter Waves," Opt. Lett. 22, 1430-32 (1997).
    [CrossRef]
  24. S. B. Cahn, A. Kumarakrishnan, U. Shim, T. Sleator, P. R. Berman, and B. Dubetsky, "Time-Domain de Broglie Wave Interferometry," Phys. Rev. Lett. 79, 784-787 (1997).
    [CrossRef]
  25. L. Deng, E. W. Hagley, J. Wen, M. Trippenbach, Y. Band, P. S. Julienne, J. E. Simsarian, K. Helmerson, S. L. Rolston, and W. D. Phillips, "Four-wave mixing with matter waves," Nature 398(6724), 218 (1999).
    [CrossRef]
  26. B. Brezger, L. Hackermüller, S. Uttenthaler, J. Petschinka, M. Arndt, and A. Zeilinger, "Matter-Wave Interferometer for Large Molecules," Phys. Rev. Lett. 88, 100,404 (2002).
    [CrossRef]
  27. S. Gerlich, L. Hackermüller, K. Hornberger, A. Stibor, H. Ulbricht, M. Gring, F. Goldfarb, T. Savas, M. Müri, M. Mayor, and M. Arndt, "A Kapitza-Dirac-Talbot-Lau interferometer for highly polarizable molecules," Nat. Phys. 3, 711 - 715 (2007).
    [CrossRef]
  28. B. J. McMorran and A. D. Cronin, "An electron Talbot interferometer," New J. Phys. 11, 033,021 (2009).
  29. I. Marzoli, A. Kaplan, F. Saif, and W. Schleich, "Quantum carpets of a slightly relativistic particle," Fortschr. Phys. 56(10) (2008).
  30. M. Berry, I. Marzoli, and W. Schleich, "Quantum Carpets, Carpets of Light," Phys. World1-6 (2001).
  31. M. Berry and S. Klein, "Integer, Fractional and Fractal Talbot Effects" J. Mod. Opt. 43, 2139-2164 (1996).
    [CrossRef]
  32. J. F. Clauser, "Factoring Integers with Youngs N-Slit Interferometer," Phys. Rev. A 53, 4587-4590 (1996).
    [CrossRef] [PubMed]
  33. S. Wölk, W. P. Schleich, "Quantum carpets: Factorization with degeneracies," Proceedings of the Middleton Festival, Princeton, in print (2009).
  34. J. Yeazell and C. Stroud, "Observation of fractional revivals in the evolution of a Rydberg atomic wave packet," Phys. Rev. A 43(9), 5153-5156 (1991).
    [CrossRef]
  35. M. Vrakking, D. Villeneuve, A. Stolow, "Observation of fractional revivals of a molecular wave packet," Phys. Rev. A 54(1), 37 (1996).
    [CrossRef]
  36. J. H. Hannay, M. V. Berry, "Quantization of linear maps on a torus-fresnel diffraction by a periodic grating," Physica 1D267-290 (1980).

2009

B. J. McMorran and A. D. Cronin, "An electron Talbot interferometer," New J. Phys. 11, 033,021 (2009).

2008

I. Marzoli, A. Kaplan, F. Saif, and W. Schleich, "Quantum carpets of a slightly relativistic particle," Fortschr. Phys. 56(10) (2008).

2007

F. Huang, N. Zheludev, Y. Chen, and F. de Abajo, "Focusing of light by a nanohole array," Appl. Phys. Lett. 90, 091,119 (2007).
[CrossRef]

S. Gerlich, L. Hackermüller, K. Hornberger, A. Stibor, H. Ulbricht, M. Gring, F. Goldfarb, T. Savas, M. Müri, M. Mayor, and M. Arndt, "A Kapitza-Dirac-Talbot-Lau interferometer for highly polarizable molecules," Nat. Phys. 3, 711 - 715 (2007).
[CrossRef]

M. Dennis, N. Zheludev, and F. García de Abajo, "The plasmon Talbot effect," Opt. Express 15(15), 9692-9700 (2007).
[CrossRef]

2006

F. Pfeiffer, T. Weitkamp, O. Bunk and C. David, "Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources," Nature 2, 258-261 (2006).

S. Prakash, S. Singh, and A. Verma, "A low cost technique for automated measurement of focal length using Lau effect combined with Moire readout," J. Mod. Opt. 53(14), 2033-2042 (2006).
[CrossRef]

2005

2004

P. Singh, M. Faridi, and C. Shakher, "Measurement of temperature of an axisymmetric flame using shearing interferometry and Fourier fringe analysis technique," Opt. Eng. 43, 387 (2004).
[CrossRef]

S. Mirza and C. Shakher, "Surface profiling using phase shifting Talbot interferometric technique," Optical Engineering 44, 013,601 (2004).

2002

B. Brezger, L. Hackermüller, S. Uttenthaler, J. Petschinka, M. Arndt, and A. Zeilinger, "Matter-Wave Interferometer for Large Molecules," Phys. Rev. Lett. 88, 100,404 (2002).
[CrossRef]

G. Spagnolo, D. Ambrosini, and D. Paoletti, "Displacement measurement using the Talbot effect with a Ronchi grating," J. Opt. A 4(6), 376-380 (2002).
[CrossRef]

2001

M. Berry, I. Marzoli, and W. Schleich, "Quantum Carpets, Carpets of Light," Phys. World1-6 (2001).

2000

O. Friesch, W. Schleich, and I. Marzoli, "Quantum carpets woven by Wigner functions," New J. Phys. 2(1), 4 (2000).
[CrossRef]

1999

L. Deng, E. W. Hagley, J. Wen, M. Trippenbach, Y. Band, P. S. Julienne, J. E. Simsarian, K. Helmerson, S. L. Rolston, and W. D. Phillips, "Four-wave mixing with matter waves," Nature 398(6724), 218 (1999).
[CrossRef]

1998

1997

1996

M. Berry and S. Klein, "Integer, Fractional and Fractal Talbot Effects" J. Mod. Opt. 43, 2139-2164 (1996).
[CrossRef]

J. F. Clauser, "Factoring Integers with Youngs N-Slit Interferometer," Phys. Rev. A 53, 4587-4590 (1996).
[CrossRef] [PubMed]

M. Vrakking, D. Villeneuve, A. Stolow, "Observation of fractional revivals of a molecular wave packet," Phys. Rev. A 54(1), 37 (1996).
[CrossRef]

1995

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, R14 (1995).
[CrossRef] [PubMed]

1994

J. F. Clauser and S. Li, "‘Heisenberg Microscope’ Decoherence Atom Interferometry," Phys. Rev. A 50, 2430 (1994).
[CrossRef] [PubMed]

1991

J. Yeazell and C. Stroud, "Observation of fractional revivals in the evolution of a Rydberg atomic wave packet," Phys. Rev. A 43(9), 5153-5156 (1991).
[CrossRef]

1989

1985

1983

K. Patorski, "Incoherent Superposition ofMultiple Self-imaging Lau Effect andMoire Fringe Explanation," Opt. Acta 30, 745-758 (1983).
[CrossRef]

1980

J. H. Hannay, M. V. Berry, "Quantization of linear maps on a torus-fresnel diffraction by a periodic grating," Physica 1D267-290 (1980).

1967

1948

E. Lau, "Beugungserscheinungen an Doppelrastern," Ann. Phys. 6, 417 (1948).
[CrossRef]

1881

L. Rayleigh, "On Copying Diffraction-Gratings, and some Phenomena Connected Therewith," Philos. Mag. 11, 196 (1881).

1836

H. F. Talbot, "Facts Relating to Optical Science," Philos. Mag. 9, 401-407 (1836).

Ambrosini, D.

G. Spagnolo, D. Ambrosini, and D. Paoletti, "Displacement measurement using the Talbot effect with a Ronchi grating," J. Opt. A 4(6), 376-380 (2002).
[CrossRef]

Arndt, M.

S. Gerlich, L. Hackermüller, K. Hornberger, A. Stibor, H. Ulbricht, M. Gring, F. Goldfarb, T. Savas, M. Müri, M. Mayor, and M. Arndt, "A Kapitza-Dirac-Talbot-Lau interferometer for highly polarizable molecules," Nat. Phys. 3, 711 - 715 (2007).
[CrossRef]

B. Brezger, L. Hackermüller, S. Uttenthaler, J. Petschinka, M. Arndt, and A. Zeilinger, "Matter-Wave Interferometer for Large Molecules," Phys. Rev. Lett. 88, 100,404 (2002).
[CrossRef]

Banaszek, K.

Band, Y.

L. Deng, E. W. Hagley, J. Wen, M. Trippenbach, Y. Band, P. S. Julienne, J. E. Simsarian, K. Helmerson, S. L. Rolston, and W. D. Phillips, "Four-wave mixing with matter waves," Nature 398(6724), 218 (1999).
[CrossRef]

Baruchel, J.

Berman, P. R.

S. B. Cahn, A. Kumarakrishnan, U. Shim, T. Sleator, P. R. Berman, and B. Dubetsky, "Time-Domain de Broglie Wave Interferometry," Phys. Rev. Lett. 79, 784-787 (1997).
[CrossRef]

Berry, M.

M. Berry, I. Marzoli, and W. Schleich, "Quantum Carpets, Carpets of Light," Phys. World1-6 (2001).

M. Berry and S. Klein, "Integer, Fractional and Fractal Talbot Effects" J. Mod. Opt. 43, 2139-2164 (1996).
[CrossRef]

Berry, M. V.

J. H. Hannay, M. V. Berry, "Quantization of linear maps on a torus-fresnel diffraction by a periodic grating," Physica 1D267-290 (1980).

Bhattacharya, J.

Brezger, B.

B. Brezger, L. Hackermüller, S. Uttenthaler, J. Petschinka, M. Arndt, and A. Zeilinger, "Matter-Wave Interferometer for Large Molecules," Phys. Rev. Lett. 88, 100,404 (2002).
[CrossRef]

Bunk, O.

F. Pfeiffer, T. Weitkamp, O. Bunk and C. David, "Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources," Nature 2, 258-261 (2006).

Cahn, S. B.

S. B. Cahn, A. Kumarakrishnan, U. Shim, T. Sleator, P. R. Berman, and B. Dubetsky, "Time-Domain de Broglie Wave Interferometry," Phys. Rev. Lett. 79, 784-787 (1997).
[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, R14 (1995).
[CrossRef] [PubMed]

Chen, Y.

F. Huang, N. Zheludev, Y. Chen, and F. de Abajo, "Focusing of light by a nanohole array," Appl. Phys. Lett. 90, 091,119 (2007).
[CrossRef]

Clauser, J. F.

J. F. Clauser, "Factoring Integers with Youngs N-Slit Interferometer," Phys. Rev. A 53, 4587-4590 (1996).
[CrossRef] [PubMed]

J. F. Clauser and S. Li, "‘Heisenberg Microscope’ Decoherence Atom Interferometry," Phys. Rev. A 50, 2430 (1994).
[CrossRef] [PubMed]

Cloetens, P.

Cronin, A. D.

B. J. McMorran and A. D. Cronin, "An electron Talbot interferometer," New J. Phys. 11, 033,021 (2009).

David, C.

F. Pfeiffer, T. Weitkamp, O. Bunk and C. David, "Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources," Nature 2, 258-261 (2006).

S. Nowak, C. Kurtsiefer, T. Pfau, and C. David, "High-Order Talbot Fringes for Atomic Matter Waves," Opt. Lett. 22, 1430-32 (1997).
[CrossRef]

de Abajo, F.

F. Huang, N. Zheludev, Y. Chen, and F. de Abajo, "Focusing of light by a nanohole array," Appl. Phys. Lett. 90, 091,119 (2007).
[CrossRef]

De Martino, C.

Deng, L.

L. Deng, E. W. Hagley, J. Wen, M. Trippenbach, Y. Band, P. S. Julienne, J. E. Simsarian, K. Helmerson, S. L. Rolston, and W. D. Phillips, "Four-wave mixing with matter waves," Nature 398(6724), 218 (1999).
[CrossRef]

Dennis, M.

Dubetsky, B.

S. B. Cahn, A. Kumarakrishnan, U. Shim, T. Sleator, P. R. Berman, and B. Dubetsky, "Time-Domain de Broglie Wave Interferometry," Phys. Rev. Lett. 79, 784-787 (1997).
[CrossRef]

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, R14 (1995).
[CrossRef] [PubMed]

Faridi, M.

P. Singh, M. Faridi, and C. Shakher, "Measurement of temperature of an axisymmetric flame using shearing interferometry and Fourier fringe analysis technique," Opt. Eng. 43, 387 (2004).
[CrossRef]

Friesch, O.

O. Friesch, W. Schleich, and I. Marzoli, "Quantum carpets woven by Wigner functions," New J. Phys. 2(1), 4 (2000).
[CrossRef]

García de Abajo, F.

Gerlich, S.

S. Gerlich, L. Hackermüller, K. Hornberger, A. Stibor, H. Ulbricht, M. Gring, F. Goldfarb, T. Savas, M. Müri, M. Mayor, and M. Arndt, "A Kapitza-Dirac-Talbot-Lau interferometer for highly polarizable molecules," Nat. Phys. 3, 711 - 715 (2007).
[CrossRef]

Goldfarb, F.

S. Gerlich, L. Hackermüller, K. Hornberger, A. Stibor, H. Ulbricht, M. Gring, F. Goldfarb, T. Savas, M. Müri, M. Mayor, and M. Arndt, "A Kapitza-Dirac-Talbot-Lau interferometer for highly polarizable molecules," Nat. Phys. 3, 711 - 715 (2007).
[CrossRef]

Gring, M.

S. Gerlich, L. Hackermüller, K. Hornberger, A. Stibor, H. Ulbricht, M. Gring, F. Goldfarb, T. Savas, M. Müri, M. Mayor, and M. Arndt, "A Kapitza-Dirac-Talbot-Lau interferometer for highly polarizable molecules," Nat. Phys. 3, 711 - 715 (2007).
[CrossRef]

Guigay, J.

Hackermüller, L.

S. Gerlich, L. Hackermüller, K. Hornberger, A. Stibor, H. Ulbricht, M. Gring, F. Goldfarb, T. Savas, M. Müri, M. Mayor, and M. Arndt, "A Kapitza-Dirac-Talbot-Lau interferometer for highly polarizable molecules," Nat. Phys. 3, 711 - 715 (2007).
[CrossRef]

B. Brezger, L. Hackermüller, S. Uttenthaler, J. Petschinka, M. Arndt, and A. Zeilinger, "Matter-Wave Interferometer for Large Molecules," Phys. Rev. Lett. 88, 100,404 (2002).
[CrossRef]

Hagley, E. W.

L. Deng, E. W. Hagley, J. Wen, M. Trippenbach, Y. Band, P. S. Julienne, J. E. Simsarian, K. Helmerson, S. L. Rolston, and W. D. Phillips, "Four-wave mixing with matter waves," Nature 398(6724), 218 (1999).
[CrossRef]

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, R14 (1995).
[CrossRef] [PubMed]

Hannay, J. H.

J. H. Hannay, M. V. Berry, "Quantization of linear maps on a torus-fresnel diffraction by a periodic grating," Physica 1D267-290 (1980).

Helmerson, K.

L. Deng, E. W. Hagley, J. Wen, M. Trippenbach, Y. Band, P. S. Julienne, J. E. Simsarian, K. Helmerson, S. L. Rolston, and W. D. Phillips, "Four-wave mixing with matter waves," Nature 398(6724), 218 (1999).
[CrossRef]

Hornberger, K.

S. Gerlich, L. Hackermüller, K. Hornberger, A. Stibor, H. Ulbricht, M. Gring, F. Goldfarb, T. Savas, M. Müri, M. Mayor, and M. Arndt, "A Kapitza-Dirac-Talbot-Lau interferometer for highly polarizable molecules," Nat. Phys. 3, 711 - 715 (2007).
[CrossRef]

Huang, F.

F. Huang, N. Zheludev, Y. Chen, and F. de Abajo, "Focusing of light by a nanohole array," Appl. Phys. Lett. 90, 091,119 (2007).
[CrossRef]

Julienne, P. S.

L. Deng, E. W. Hagley, J. Wen, M. Trippenbach, Y. Band, P. S. Julienne, J. E. Simsarian, K. Helmerson, S. L. Rolston, and W. D. Phillips, "Four-wave mixing with matter waves," Nature 398(6724), 218 (1999).
[CrossRef]

Kaplan, A.

I. Marzoli, A. Kaplan, F. Saif, and W. Schleich, "Quantum carpets of a slightly relativistic particle," Fortschr. Phys. 56(10) (2008).

Klein, S.

M. Berry and S. Klein, "Integer, Fractional and Fractal Talbot Effects" J. Mod. Opt. 43, 2139-2164 (1996).
[CrossRef]

Kumarakrishnan, A.

S. B. Cahn, A. Kumarakrishnan, U. Shim, T. Sleator, P. R. Berman, and B. Dubetsky, "Time-Domain de Broglie Wave Interferometry," Phys. Rev. Lett. 79, 784-787 (1997).
[CrossRef]

Kurtsiefer, C.

Lau, E.

E. Lau, "Beugungserscheinungen an Doppelrastern," Ann. Phys. 6, 417 (1948).
[CrossRef]

Leith, E. N.

Li, S.

J. F. Clauser and S. Li, "‘Heisenberg Microscope’ Decoherence Atom Interferometry," Phys. Rev. A 50, 2430 (1994).
[CrossRef] [PubMed]

Liu, L.

Marzoli, I.

I. Marzoli, A. Kaplan, F. Saif, and W. Schleich, "Quantum carpets of a slightly relativistic particle," Fortschr. Phys. 56(10) (2008).

M. Berry, I. Marzoli, and W. Schleich, "Quantum Carpets, Carpets of Light," Phys. World1-6 (2001).

O. Friesch, W. Schleich, and I. Marzoli, "Quantum carpets woven by Wigner functions," New J. Phys. 2(1), 4 (2000).
[CrossRef]

Mayor, M.

S. Gerlich, L. Hackermüller, K. Hornberger, A. Stibor, H. Ulbricht, M. Gring, F. Goldfarb, T. Savas, M. Müri, M. Mayor, and M. Arndt, "A Kapitza-Dirac-Talbot-Lau interferometer for highly polarizable molecules," Nat. Phys. 3, 711 - 715 (2007).
[CrossRef]

McMorran, B. J.

B. J. McMorran and A. D. Cronin, "An electron Talbot interferometer," New J. Phys. 11, 033,021 (2009).

Mirza, S.

S. Mirza and C. Shakher, "Surface profiling using phase shifting Talbot interferometric technique," Optical Engineering 44, 013,601 (2004).

Montgomery, W.

Müri, M.

S. Gerlich, L. Hackermüller, K. Hornberger, A. Stibor, H. Ulbricht, M. Gring, F. Goldfarb, T. Savas, M. Müri, M. Mayor, and M. Arndt, "A Kapitza-Dirac-Talbot-Lau interferometer for highly polarizable molecules," Nat. Phys. 3, 711 - 715 (2007).
[CrossRef]

Nowak, S.

Paoletti, D.

G. Spagnolo, D. Ambrosini, and D. Paoletti, "Displacement measurement using the Talbot effect with a Ronchi grating," J. Opt. A 4(6), 376-380 (2002).
[CrossRef]

Patorski, K.

K. Patorski, "Incoherent Superposition ofMultiple Self-imaging Lau Effect andMoire Fringe Explanation," Opt. Acta 30, 745-758 (1983).
[CrossRef]

Petschinka, J.

B. Brezger, L. Hackermüller, S. Uttenthaler, J. Petschinka, M. Arndt, and A. Zeilinger, "Matter-Wave Interferometer for Large Molecules," Phys. Rev. Lett. 88, 100,404 (2002).
[CrossRef]

Pfau, T.

Pfeiffer, F.

F. Pfeiffer, T. Weitkamp, O. Bunk and C. David, "Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources," Nature 2, 258-261 (2006).

Phillips, W. D.

L. Deng, E. W. Hagley, J. Wen, M. Trippenbach, Y. Band, P. S. Julienne, J. E. Simsarian, K. Helmerson, S. L. Rolston, and W. D. Phillips, "Four-wave mixing with matter waves," Nature 398(6724), 218 (1999).
[CrossRef]

Prakash, S.

S. Prakash, S. Singh, and A. Verma, "A low cost technique for automated measurement of focal length using Lau effect combined with Moire readout," J. Mod. Opt. 53(14), 2033-2042 (2006).
[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, R14 (1995).
[CrossRef] [PubMed]

Quan, C.

Rayleigh, L.

L. Rayleigh, "On Copying Diffraction-Gratings, and some Phenomena Connected Therewith," Philos. Mag. 11, 196 (1881).

Rolston, S. L.

L. Deng, E. W. Hagley, J. Wen, M. Trippenbach, Y. Band, P. S. Julienne, J. E. Simsarian, K. Helmerson, S. L. Rolston, and W. D. Phillips, "Four-wave mixing with matter waves," Nature 398(6724), 218 (1999).
[CrossRef]

Saif, F.

I. Marzoli, A. Kaplan, F. Saif, and W. Schleich, "Quantum carpets of a slightly relativistic particle," Fortschr. Phys. 56(10) (2008).

Savas, T.

S. Gerlich, L. Hackermüller, K. Hornberger, A. Stibor, H. Ulbricht, M. Gring, F. Goldfarb, T. Savas, M. Müri, M. Mayor, and M. Arndt, "A Kapitza-Dirac-Talbot-Lau interferometer for highly polarizable molecules," Nat. Phys. 3, 711 - 715 (2007).
[CrossRef]

Schleich, W.

I. Marzoli, A. Kaplan, F. Saif, and W. Schleich, "Quantum carpets of a slightly relativistic particle," Fortschr. Phys. 56(10) (2008).

M. Berry, I. Marzoli, and W. Schleich, "Quantum Carpets, Carpets of Light," Phys. World1-6 (2001).

O. Friesch, W. Schleich, and I. Marzoli, "Quantum carpets woven by Wigner functions," New J. Phys. 2(1), 4 (2000).
[CrossRef]

Schleich, W. P.

Schlenker, M.

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, R14 (1995).
[CrossRef] [PubMed]

Shakher, C.

S. Mirza and C. Shakher, "Surface profiling using phase shifting Talbot interferometric technique," Optical Engineering 44, 013,601 (2004).

P. Singh, M. Faridi, and C. Shakher, "Measurement of temperature of an axisymmetric flame using shearing interferometry and Fourier fringe analysis technique," Opt. Eng. 43, 387 (2004).
[CrossRef]

Shim, U.

S. B. Cahn, A. Kumarakrishnan, U. Shim, T. Sleator, P. R. Berman, and B. Dubetsky, "Time-Domain de Broglie Wave Interferometry," Phys. Rev. Lett. 79, 784-787 (1997).
[CrossRef]

Simsarian, J. E.

L. Deng, E. W. Hagley, J. Wen, M. Trippenbach, Y. Band, P. S. Julienne, J. E. Simsarian, K. Helmerson, S. L. Rolston, and W. D. Phillips, "Four-wave mixing with matter waves," Nature 398(6724), 218 (1999).
[CrossRef]

Singh, P.

P. Singh, M. Faridi, and C. Shakher, "Measurement of temperature of an axisymmetric flame using shearing interferometry and Fourier fringe analysis technique," Opt. Eng. 43, 387 (2004).
[CrossRef]

Singh, S.

S. Prakash, S. Singh, and A. Verma, "A low cost technique for automated measurement of focal length using Lau effect combined with Moire readout," J. Mod. Opt. 53(14), 2033-2042 (2006).
[CrossRef]

Sleator, T.

S. B. Cahn, A. Kumarakrishnan, U. Shim, T. Sleator, P. R. Berman, and B. Dubetsky, "Time-Domain de Broglie Wave Interferometry," Phys. Rev. Lett. 79, 784-787 (1997).
[CrossRef]

Spagnolo, G.

G. Spagnolo, D. Ambrosini, and D. Paoletti, "Displacement measurement using the Talbot effect with a Ronchi grating," J. Opt. A 4(6), 376-380 (2002).
[CrossRef]

Stibor, A.

S. Gerlich, L. Hackermüller, K. Hornberger, A. Stibor, H. Ulbricht, M. Gring, F. Goldfarb, T. Savas, M. Müri, M. Mayor, and M. Arndt, "A Kapitza-Dirac-Talbot-Lau interferometer for highly polarizable molecules," Nat. Phys. 3, 711 - 715 (2007).
[CrossRef]

Stolow, A.

M. Vrakking, D. Villeneuve, A. Stolow, "Observation of fractional revivals of a molecular wave packet," Phys. Rev. A 54(1), 37 (1996).
[CrossRef]

Stroud, C.

J. Yeazell and C. Stroud, "Observation of fractional revivals in the evolution of a Rydberg atomic wave packet," Phys. Rev. A 43(9), 5153-5156 (1991).
[CrossRef]

Swanson, G. J.

Talbot, H. F.

H. F. Talbot, "Facts Relating to Optical Science," Philos. Mag. 9, 401-407 (1836).

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, R14 (1995).
[CrossRef] [PubMed]

Tay, C.

Thakur, M.

Trippenbach, M.

L. Deng, E. W. Hagley, J. Wen, M. Trippenbach, Y. Band, P. S. Julienne, J. E. Simsarian, K. Helmerson, S. L. Rolston, and W. D. Phillips, "Four-wave mixing with matter waves," Nature 398(6724), 218 (1999).
[CrossRef]

Ulbricht, H.

S. Gerlich, L. Hackermüller, K. Hornberger, A. Stibor, H. Ulbricht, M. Gring, F. Goldfarb, T. Savas, M. Müri, M. Mayor, and M. Arndt, "A Kapitza-Dirac-Talbot-Lau interferometer for highly polarizable molecules," Nat. Phys. 3, 711 - 715 (2007).
[CrossRef]

Uttenthaler, S.

B. Brezger, L. Hackermüller, S. Uttenthaler, J. Petschinka, M. Arndt, and A. Zeilinger, "Matter-Wave Interferometer for Large Molecules," Phys. Rev. Lett. 88, 100,404 (2002).
[CrossRef]

Verma, A.

S. Prakash, S. Singh, and A. Verma, "A low cost technique for automated measurement of focal length using Lau effect combined with Moire readout," J. Mod. Opt. 53(14), 2033-2042 (2006).
[CrossRef]

Villeneuve, D.

M. Vrakking, D. Villeneuve, A. Stolow, "Observation of fractional revivals of a molecular wave packet," Phys. Rev. A 54(1), 37 (1996).
[CrossRef]

Vrakking, M.

M. Vrakking, D. Villeneuve, A. Stolow, "Observation of fractional revivals of a molecular wave packet," Phys. Rev. A 54(1), 37 (1996).
[CrossRef]

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, R14 (1995).
[CrossRef] [PubMed]

Weitkamp, T.

F. Pfeiffer, T. Weitkamp, O. Bunk and C. David, "Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources," Nature 2, 258-261 (2006).

Wen, J.

L. Deng, E. W. Hagley, J. Wen, M. Trippenbach, Y. Band, P. S. Julienne, J. E. Simsarian, K. Helmerson, S. L. Rolston, and W. D. Phillips, "Four-wave mixing with matter waves," Nature 398(6724), 218 (1999).
[CrossRef]

Wodkiewicz, K.

Yeazell, J.

J. Yeazell and C. Stroud, "Observation of fractional revivals in the evolution of a Rydberg atomic wave packet," Phys. Rev. A 43(9), 5153-5156 (1991).
[CrossRef]

Zeilinger, A.

B. Brezger, L. Hackermüller, S. Uttenthaler, J. Petschinka, M. Arndt, and A. Zeilinger, "Matter-Wave Interferometer for Large Molecules," Phys. Rev. Lett. 88, 100,404 (2002).
[CrossRef]

Zheludev, N.

M. Dennis, N. Zheludev, and F. García de Abajo, "The plasmon Talbot effect," Opt. Express 15(15), 9692-9700 (2007).
[CrossRef]

F. Huang, N. Zheludev, Y. Chen, and F. de Abajo, "Focusing of light by a nanohole array," Appl. Phys. Lett. 90, 091,119 (2007).
[CrossRef]

Ann. Phys.

E. Lau, "Beugungserscheinungen an Doppelrastern," Ann. Phys. 6, 417 (1948).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

F. Huang, N. Zheludev, Y. Chen, and F. de Abajo, "Focusing of light by a nanohole array," Appl. Phys. Lett. 90, 091,119 (2007).
[CrossRef]

Fortschr. Phys.

I. Marzoli, A. Kaplan, F. Saif, and W. Schleich, "Quantum carpets of a slightly relativistic particle," Fortschr. Phys. 56(10) (2008).

J. Mod. Opt.

M. Berry and S. Klein, "Integer, Fractional and Fractal Talbot Effects" J. Mod. Opt. 43, 2139-2164 (1996).
[CrossRef]

S. Prakash, S. Singh, and A. Verma, "A low cost technique for automated measurement of focal length using Lau effect combined with Moire readout," J. Mod. Opt. 53(14), 2033-2042 (2006).
[CrossRef]

J. Opt. A

G. Spagnolo, D. Ambrosini, and D. Paoletti, "Displacement measurement using the Talbot effect with a Ronchi grating," J. Opt. A 4(6), 376-380 (2002).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Nat. Phys.

S. Gerlich, L. Hackermüller, K. Hornberger, A. Stibor, H. Ulbricht, M. Gring, F. Goldfarb, T. Savas, M. Müri, M. Mayor, and M. Arndt, "A Kapitza-Dirac-Talbot-Lau interferometer for highly polarizable molecules," Nat. Phys. 3, 711 - 715 (2007).
[CrossRef]

Nature

L. Deng, E. W. Hagley, J. Wen, M. Trippenbach, Y. Band, P. S. Julienne, J. E. Simsarian, K. Helmerson, S. L. Rolston, and W. D. Phillips, "Four-wave mixing with matter waves," Nature 398(6724), 218 (1999).
[CrossRef]

F. Pfeiffer, T. Weitkamp, O. Bunk and C. David, "Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources," Nature 2, 258-261 (2006).

New J. Phys.

B. J. McMorran and A. D. Cronin, "An electron Talbot interferometer," New J. Phys. 11, 033,021 (2009).

O. Friesch, W. Schleich, and I. Marzoli, "Quantum carpets woven by Wigner functions," New J. Phys. 2(1), 4 (2000).
[CrossRef]

Opt. Acta

K. Patorski, "Incoherent Superposition ofMultiple Self-imaging Lau Effect andMoire Fringe Explanation," Opt. Acta 30, 745-758 (1983).
[CrossRef]

Opt. Eng.

P. Singh, M. Faridi, and C. Shakher, "Measurement of temperature of an axisymmetric flame using shearing interferometry and Fourier fringe analysis technique," Opt. Eng. 43, 387 (2004).
[CrossRef]

Opt. Express

Opt. Lett.

Optical Engineering

S. Mirza and C. Shakher, "Surface profiling using phase shifting Talbot interferometric technique," Optical Engineering 44, 013,601 (2004).

Philos. Mag.

H. F. Talbot, "Facts Relating to Optical Science," Philos. Mag. 9, 401-407 (1836).

L. Rayleigh, "On Copying Diffraction-Gratings, and some Phenomena Connected Therewith," Philos. Mag. 11, 196 (1881).

Phys. Rev. A

J. F. Clauser and S. Li, "‘Heisenberg Microscope’ Decoherence Atom Interferometry," Phys. Rev. A 50, 2430 (1994).
[CrossRef] [PubMed]

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, R14 (1995).
[CrossRef] [PubMed]

J. Yeazell and C. Stroud, "Observation of fractional revivals in the evolution of a Rydberg atomic wave packet," Phys. Rev. A 43(9), 5153-5156 (1991).
[CrossRef]

M. Vrakking, D. Villeneuve, A. Stolow, "Observation of fractional revivals of a molecular wave packet," Phys. Rev. A 54(1), 37 (1996).
[CrossRef]

J. F. Clauser, "Factoring Integers with Youngs N-Slit Interferometer," Phys. Rev. A 53, 4587-4590 (1996).
[CrossRef] [PubMed]

Phys. Rev. Lett.

S. B. Cahn, A. Kumarakrishnan, U. Shim, T. Sleator, P. R. Berman, and B. Dubetsky, "Time-Domain de Broglie Wave Interferometry," Phys. Rev. Lett. 79, 784-787 (1997).
[CrossRef]

B. Brezger, L. Hackermüller, S. Uttenthaler, J. Petschinka, M. Arndt, and A. Zeilinger, "Matter-Wave Interferometer for Large Molecules," Phys. Rev. Lett. 88, 100,404 (2002).
[CrossRef]

Phys. World

M. Berry, I. Marzoli, and W. Schleich, "Quantum Carpets, Carpets of Light," Phys. World1-6 (2001).

Physica

J. H. Hannay, M. V. Berry, "Quantization of linear maps on a torus-fresnel diffraction by a periodic grating," Physica 1D267-290 (1980).

Other

S. Wölk, W. P. Schleich, "Quantum carpets: Factorization with degeneracies," Proceedings of the Middleton Festival, Princeton, in print (2009).

K. Patorski, "Self-Imaging and its Applications," in Progress in Optics XXVII, E. Wolf, ed., (Elsevier Science Publishers B. V., Amsterdam, 1989), pp. 2-108.

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Figures (2)

Fig. 1.
Fig. 1.

(Color online) a) Talbot self-imaging using plane-parallel, monochromatic light waves; b) from left to right: laser at 532 nm, beam expander, grating, imaging lens in blackened lens tube and CCD detector on a micrometer translation stage; c) Theoretical expectation according to Eq. (3); d) Experimental carpet of light. The x-axis counts the grating periods. The z-axis represents the distance between the detector image plane and the grating in multiples of the corresponding Talbot distance. The numbers m indicate where self images of the mth order are observed.

Fig. 2.
Fig. 2.

(Color online) a) Talbot-Lau imaging with spatially incoherent light: A monochromatic but spatially incoherent beam of light can be decomposed into a bundle of plane waves covering a range of incident angles. Since the laterally shifted self-images overlap non-synchronously they wash out. Insertion of a second grating leads to the refocusing of the diverging wavefronts and to the emergence of a Talbot-Lau interference pattern. b) Experimental implementation with a filtered sodium lamp. The frosted glass transforms the incident light into a diffusely scattered, spatially incoherent beam. The first grating prepares spatial coherence, the second grating rephases the outgoing waves which then interfere in the image plane of the camera lens; c) Computational simulation according to Eq. (13); d) Experimental Talbot-Lau carpet. The z-axis now represents the distance L 1 between the gratings G1 and G2 which equals the distance between G2 and the imaging plane; The numbers m indicate where self images of the mth order are observed.

Equations (21)

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

ψ ( x , + 0 ) = ψ ( x , 0 ) · T ( x ) = n A n e i ( k θ + nk d ) x = n A n e ik , n x
ψ ( x , z ) = n A n exp [ i ( k θ + n k d ) x + i ( k ( k θ + n k d ) 2 2 k ) z ] .
ψ ( x , z ) = n A n e ink d x e in 2 π ( z L T ) ,
ψ ( x , z = L T ) = n A n e ink d x e in π = n A n e ink d ( x d 2 ) .
e in 2 π ( z L T ) = e i 2 πn 2 p r
e iπn 2 p r = m = 0 r 1 a m e i 2 πmn .
n = 0 r 1 e i 2 πmn r e i 2 πm n r = r δ mm .
a m = 1 r n = 0 r 1 e i 2 π ( n 2 mn ) p r ,
ψ ( x , z ) = m = 0 r 1 a m n A n e ink d ( x md r )
= m = 0 r 1 a m ψ ( x md r ) .
A 0 = f A n = sin ( n π f ) / ( n f ) ; for     n 0
ψ θ ( x ) = n , m A n A m exp [ i ( k θ + ( n + m ) k d ) x ]
× exp [ i ( k θ + nk d ) 2 L 1 ( 2 k ) ] .
ψ θ ( x ) = n , m A n A m exp [ i ( k θ + ( n + m ) k d ) x ]
× exp [ i ( k θ + nk d ) 2 L 1 ( 2 k ) ]
× exp [ i ( k θ + ( n + m ) k d ) 2 L 2 ( 2 k ) ] .
I θ = ψ θ ψ θ * = n , m , n , m A n A m A n * A m * exp [ ik d ( n + m n m ) x ] .
× exp [ i 2 k θ k d [ ( n n ) L 1 + ( n n + m m ) L 2 ] 2 k ]
× exp [ i k d 2 ( n 2 n 2 ) L 1 2 k ] exp [ i k d 2 ( ( n + m ) 2 ( n + m ) 2 ) L 2 2 k ] .
I Lau = θ I θ .
L 2 = L 1 m m n n 1 .

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