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]
  3. L. Rayleigh, “On Copying Diffraction-Gratings, and some Phenomena Connected Therewith,” Philos. Mag. 11, 196 ( 1881).
  4. W. Montgomery, “Self-imaging objects of infinite aperture,” J. Opt. Soc. Am. 57(6), 772–775 ( 1967).
    [CrossRef]
  5. K. Patorski, “Incoherent Superposition ofMultiple Self-imaging Lau Effect andMoire Fringe Explanation,” Opt. Acta 30, 745–758 ( 1983).
    [CrossRef]
  6. G. J. Swanson and E. N. Leith, “Analysis of the Lau Effect and Generalized Grating Imaging,” J. Opt. Soc. Am. A 2, 789–793 ( 1985).
    [CrossRef]
  7. L. Liu, “Talbot and Lau effects on incident beams of arbitrary wavefront, and their use,” Appl. Opt. 28(21), 4668–4678 ( 1989).
    [CrossRef]
  8. 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.
  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. Worldpp. 1–6 ( 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 and 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, and A. Stolow, “Observation of fractional revivals of a molecular wave packet,” Phys. Rev. A 54(1), 37 ( 1996).
    [CrossRef]
  36. J. H. Hannay and M. V. Berry, “Quantization of linear maps on a torus-fresnel diffraction by a periodic grating,” Physica 1D267–290 ( 1980).

2009 (1)

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

2008 (1)

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

2007 (3)

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]

M. Dennis, N. Zheludev, and F. García de Abajo, “The plasmon Talbot effect,” Opt. Express 15(15), 9692–9700 ( 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]

2006 (2)

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]

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).

2005 (1)

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]

2004 (2)

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]

2002 (2)

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]

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]

2000 (1)

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

1999 (1)

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 (1)

1997 (3)

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

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]

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]

1996 (3)

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, and A. Stolow, “Observation of fractional revivals of a molecular wave packet,” Phys. Rev. A 54(1), 37 ( 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, R14 ( 1995).
[CrossRef] [PubMed]

1994 (1)

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

1991 (1)

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 (2)

J. Bhattacharya, “Measurement of the refractive index using the Talbot effect and a moire technique,” Appl. Opt. 28(13), 2600–2604 ( 1989).
[CrossRef]

L. Liu, “Talbot and Lau effects on incident beams of arbitrary wavefront, and their use,” Appl. Opt. 28(21), 4668–4678 ( 1989).
[CrossRef]

1985 (1)

1983 (1)

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

1980 (1)

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

1967 (1)

W. Montgomery, “Self-imaging objects of infinite aperture,” J. Opt. Soc. Am. 57(6), 772–775 ( 1967).
[CrossRef]

1948 (1)

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

1881 (1)

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

1836 (1)

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.

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]

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 and S. Klein, “Integer, Fractional and Fractal Talbot Effects” J. Mod. Opt. 43, 2139–2164 ( 1996).
[CrossRef]

M. Berry, I. Marzoli, and W. Schleich, “Quantum Carpets, Carpets of Light,” Phys. Worldpp. 1–6 ( 2001).

Berry, M. V.

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

Bhattacharya, J.

J. Bhattacharya, “Measurement of the refractive index using the Talbot effect and a moire technique,” Appl. Opt. 28(13), 2600–2604 ( 1989).
[CrossRef]

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.

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]

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.

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]

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.

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

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.

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

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.

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]

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 and 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.

L. Liu, “Talbot and Lau effects on incident beams of arbitrary wavefront, and their use,” Appl. Opt. 28(21), 4668–4678 ( 1989).
[CrossRef]

Marzoli, I.

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

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

M. Berry, I. Marzoli, and W. Schleich, “Quantum Carpets, Carpets of Light,” Phys. Worldpp. 1–6 ( 2001).

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.

W. Montgomery, “Self-imaging objects of infinite aperture,” J. Opt. Soc. Am. 57(6), 772–775 ( 1967).
[CrossRef]

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]

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.

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.

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]

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).

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

M. Berry, I. Marzoli, and W. Schleich, “Quantum Carpets, Carpets of Light,” Phys. Worldpp. 1–6 ( 2001).

Schleich, W. P.

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]

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

Schlenker, M.

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]

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, and 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.

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]

Thakur, M.

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]

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, and 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, and 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.

Wölk, S.

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

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.

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]

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

Ann. Phys. (1)

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

Appl. Opt. (3)

L. Liu, “Talbot and Lau effects on incident beams of arbitrary wavefront, and their use,” Appl. Opt. 28(21), 4668–4678 ( 1989).
[CrossRef]

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]

J. Bhattacharya, “Measurement of the refractive index using the Talbot effect and a moire technique,” Appl. Opt. 28(13), 2600–2604 ( 1989).
[CrossRef]

Appl. Phys. Lett. (1)

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. (1)

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

J. Mod. Opt. (2)

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 (1)

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. (1)

W. Montgomery, “Self-imaging objects of infinite aperture,” J. Opt. Soc. Am. 57(6), 772–775 ( 1967).
[CrossRef]

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

Nat. Phys. (1)

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 (2)

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).

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]

New J. Phys. (2)

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 (1)

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

Opt. Eng. (1)

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 (2)

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]

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

Opt. Lett. (2)

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]

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

Optical Engineering (1)

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

Philos. Mag. (2)

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

H. F. Talbot, “Facts Relating to Optical Science,” Philos. Mag. 9, 401–407 ( 1836).

Phys. Rev. A (5)

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. F. Clauser, “Factoring Integers with Youngs N-Slit Interferometer,” Phys. Rev. A 53, 4587–4590 ( 1996).
[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, and A. Stolow, “Observation of fractional revivals of a molecular wave packet,” Phys. Rev. A 54(1), 37 ( 1996).
[CrossRef]

Phys. Rev. Lett. (2)

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]

Physica (1)

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

Other (3)

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

M. Berry, I. Marzoli, and W. Schleich, “Quantum Carpets, Carpets of Light,” Phys. Worldpp. 1–6 ( 2001).

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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