## Realization of optical carpets in the Talbot and Talbot-Lau configurations

Optics Express, Vol. 17, Issue 23, pp. 20966-20974 (2009)

http://dx.doi.org/10.1364/OE.17.020966

Acrobat PDF (1345 KB)

### 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

## 1. Self-imaging of periodic structures

*L*(Figure 1).

_{T}2. E. Lau, “Beugungserscheinungen an Doppelrastern,” Ann. Phys. **6**, 417 (
1948). [CrossRef]

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

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]

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]

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

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]

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]

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

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]

## 2. The concept of optical carpets in the Talbot and Talbot-Lau arrangements

*z*=0 to span the

*x/y*plane with its periodic modulation in the

*x*direction. A plane wave falls onto the grating with the wave number

*k*=2

*π/λ*. For the Talbot effect we will assume normal incidence but averaging over non-zero angles of incidence

*θ*(Figure 2a) will be required for the Talbot-Lau effect.

*k*=

_{θ}*k*sin(

*θ*) is the projection of the incident wave vector onto the x-axis.

*A*represents the components of the Fourier decomposition of the periodic grating transmission function T(x). We thus see that diffraction at a grating of period

_{n}*d*simply adds multiples of

*k*=2

_{d}*π/d*to the transverse wave vector of the incident plane wave.

*z*from the grating to the screen it acquires an additional phase

*k*where

^{z}_{z}*k*

_{⊥,n}and

*k*are the components of the wave vector perpendicular and parallel to the

_{z}*z*axis, with

*k*=[

_{z}*k*

^{2}-

*k*

^{2}

_{⊥,n}]

^{1/2}. In the paraxial approximation (

*k*

_{⊥,n}≪

*k*) we can expand the phase to second order into

*k*=0 and obtain

_{θ}*L*≡

_{T}*d*

^{2}/

*λ*. The intensity pattern produced by the wave is obtained from

*I*(

*x*,

*z*)=

*ψ*(

*x, z*)

*ψ**(

*x, z*).

*z*=2

*n*·

*LT*, the wave pattern of Eq. (3) reproduces the transmitted amplitude at

*z*=0 as described by Eq. (1). This interferometric self-imaging is the Talbot effect.

*z*=

*L*a similar simplification is possible. Since

_{T}*n*odd implies

*n*

^{2}odd, and

*n*even implies

*n*

^{2}even it follows that

*x*by half the grating constant. For infinitely extended gratings, this shifted self-image will again be repeated at all odd multiples of the Talbot length, i.e. for

*z*=(2

*n*+1)

*L*.

_{T}*z*. Consider the case when

*z*can be represented as a rational multiple of the Talbot length,

*z*=(

*s*+

*p/r*)2

*L*, where

_{T}*s*,

*p*and

*r*are non-negative integers,

*p*<

*r*and

*p*and

*r*are relative primes. The

*z*dependent factor in Eq. (3) becomes

*n*with period

*r*. We expand the expression in Eq. (5) in the discrete Fourier series

*a*can be found using Eq. (7)

_{m}*a*. The expression for

_{m}*a*given in Eq. (8) is a Gauss sum usually discussed in relation to number theory and its value depends on the relation between the integers

_{m}*p*,

*r*and

*m*[36]. For

*r*odd all of the

*a*’s are nonzero and have the same amplitude, independent of

_{m}*m*. Thus for

*r*odd the field represented in Eq. (10) is a superposition of

*r*copies of the original field each of the same amplitude but with different phases. For

*r*even (

*p*odd) half of the

*a*’s are zero while the other half are of the same nonzero amplitude but with different phases. For

_{m}*r*even the field represented in Eq. (10) is a superposition of

*r*/2 copies of the original field each with the same amplitude but with different phases. Combining these cases we see that for

*z*equal to an integer multiple of

*L*the sum in Eq. (10) contains a single copy of the original wave. For

_{T}*z*=

*L*/2 we get a superposition of two waves, for

_{T}*z*=

*L*/3 or 2

_{T}*L*/3 a superposition of three waves, and for

_{T}*z*=

*L*/4 or 3

_{T}*L*/4 four waves. Other rational fractions of

_{T}*L*give similar results generating a very rich and detailed pattern [31

_{T}31. M. Berry and S. Klein, “Integer, Fractional and Fractal Talbot Effects” J. Mod. Opt. **43**, 2139–2164 (
1996). [CrossRef]

*f*, i.e. if the width of the slits is much smaller than the grating period, these patterns will not overlap in lower orders (

*r*small). In this case the resulting intensity pattern will be a sum of the shifted grating transmission function. When the opening fraction is not sufficiently small the patterns will overlap and we expect them to be somewhat washed-out but still with considerable fine structure. As already shown by Berry and Klein [31

**43**, 2139–2164 (
1996). [CrossRef]

*f*=0.1 in our experiments. For such a grating the Fourier components are

*I*=

*ψ*·

*ψ** where is

*ψ*is taken from Eq. 3. It is reproduced using in Fig. 1c where the series has been truncated at

*n*=±25.

*z*=

*L*

_{1}helps refocusing the waves to form again a clear grating image. This pattern can be viewed on a screen a distance

*L*

_{2}behind the second grating. The wave just before the second grating is given by Eq. (2) with

*z*=

*L*

_{1}. If both gratings are identical the second one will impose the same periodic modulation as did the first, and the wave just after the second grating will be

*k*

_{θ}*k*-dependence of Eq. (12) the pattern will be washed-out unless the coefficient of

_{θ}*k*is zero thus we require

_{θ}*A*, the pattern will have a low visibility unless this condition holds among relatively small values of

_{n}*n*,

*n*′,

*m*,

*m*′. Lau’s original choice was

*m*′-

*m*=

*n*-

*n*′ giving

*L*

_{2}=∞. Strong patterns can also be obtained with

*m*′-

*m*=2(

*n*-

*n*′) and

*L*

_{2}=

*L*

_{1}. Both the Talbot and the Talbot-Lau effects can be implemented in a straight-forward way using elementary optical equipment.

## 3. Experimental realization of optical carpets

*λ*=532 nm) which is expanded by an optical telescope to a diameter of 20mm to cover the whole grating. A 10

*µ*m aperture in the focus of the beam expander acts as a spatial filter and mode cleaner to provide a homogeneous illumination. Since the final image covers only a transverse stretch of about one millimeter the Gaussian envelope of the laser beam intensity has negligible influence on the recorded pattern. The gratings are opaque for visible light (chromium on glass, Edmund Optics inc.), have a period of 200

*µ*m and an open fraction of 10%, i.e. an open slit width of 20

*µ*m.

*µ*m size. In order to see all details of wave interference, the Talbot images are magnified by a lens of short focal length (

*f*=15mm) which is mounted in a light-tight tube in front of the CCD-chip. Placed at a distance of 134mm from the detector, the lens realizes a magnification factor of approximately nine. In order to further minimize stray light and reflections at the inner walls an iris is mounted in the middle of the lens-tube. A laser line filter (

_{L}*λ*

_{0}=532 nm, Δ

*λ*

_{1/2}=10±2 nm) effectively reduces the influence of external stray light. A motorized translation stage allows to shift the detector with micrometer resolution for more than 100mm along the z-axis.

*z*=50

*µ*m and taking a CCD snapshot in each position. Care has to be taken to ensure a good alignment between the axis of the translation stage and the k-vector to the incident laser beam: their relative angle must be smaller than 1mrad to avoid a transverse shift of the detected interference pattern when the staged is moved.

*I*(

*x*;

*z*)=∑

_{y}*I*(

*x*,

*y*;

*z*) which we then merge for all z-positions into a single Talbot-carpet

*I*(

*x*,

*z*).

*n*=±25 and still reproduces the experiment with high accuracy down to the finest details.

*L*=

_{T}*d*

^{2}/

*λ*=75.2mm we identify the grating’s self-image. Rescaled patterns can be found in rational fractions of

*L*, as indicated by the tic-marks in Figure 1c/d.

_{T}_{1}between the gratings as well as the distance L

_{2}between G

_{2}and the image plane can be scanned. Similarly to the Talbot arrangement, we have to align the axes of both translation stages and the laser beam to be parallel within 1mrad. We now use a low-pressure sodium lamp in combination with a frosted glass plate as a spatially incoherent but rather monochromatic light source. The sodium lamp dominantly emits at 589.3 nm. The atomic hyperfine structure cannot be resolved in this experiment.

_{1}and L

_{2}by optimizing the sharpness of the interference pattern on the CCD chip. For

*d*=200

*µ*m and

*λ*=589 nm the Talbot length amounts to 67.9mm. Both the camera and the grating are then scanned in steps Δ

*z*=50

*µ*m to keep the distances balanced.

*n*=±20. Again we find an intricate fine structure, including rational fractions of the Talbot image. Little differences between theory and experiment derive from the infrared and UV lines emitted by the sodium lamp. The additional stripes that show up in the experiment without being reproduced in the simulation are attributed to spurious wavelengths in the UV and IR. The sodium lamp emits dominantly in the yellow spectral range (589 nm, 90% of total intensity). Additional wavelengths create carpets of similar transverse structure but with a different longitudinal scaling behavior. The second-most intense wavelength can be found at 819 nm. It produces a carpet with L

*=48.8mm. The grating self-image in the IR can be seen as an additional feature at 3/4 of the yellow Talbot length.*

_{T}## Acknowledgments

## References and links

1. | H. F. Talbot, “Facts Relating to Optical Science,” Philos. Mag. |

2. | E. Lau, “Beugungserscheinungen an Doppelrastern,” Ann. Phys. |

3. | L. Rayleigh, “On Copying Diffraction-Gratings, and some Phenomena Connected Therewith,” Philos. Mag. |

4. | W. Montgomery, “Self-imaging objects of infinite aperture,” J. Opt. Soc. Am. |

5. | K. Patorski, “Incoherent Superposition ofMultiple Self-imaging Lau Effect andMoire Fringe Explanation,” Opt. Acta |

6. | G. J. Swanson and E. N. Leith, “Analysis of the Lau Effect and Generalized Grating Imaging,” J. Opt. Soc. Am. A |

7. | L. Liu, “Talbot and Lau effects on incident beams of arbitrary wavefront, and their use,” Appl. Opt. |

8. | K. Patorski, “Self-Imaging and its Applications,” in |

9. | K. Banaszek, K. Wodkiewicz, and W. P. Schleich, “Fractional Talbot effect in Phase Space: A Compact Summation Formula,” Opt. Express |

10. | O. Friesch, W. Schleich, and I. Marzoli, “Quantum carpets woven by Wigner functions,” New J. Phys. |

11. | S. Mirza and C. Shakher, “Surface profiling using phase shifting Talbot interferometric technique,” Optical Engineering |

12. | M. Thakur, C. Tay, and C. Quan, “Surface profiling of a transparent object by use of phase-shifting Talbot interferometry,” Appl. Opt. |

13. | J. Bhattacharya, “Measurement of the refractive index using the Talbot effect and a moire technique,” Appl. Opt. |

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. |

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. |

16. | G. Spagnolo, D. Ambrosini, and D. Paoletti, “Displacement measurement using the Talbot effect with a Ronchi grating,” J. Opt. A |

17. | F. Huang, N. Zheludev, Y. Chen, and F. de Abajo, “Focusing of light by a nanohole array,” Appl. Phys. Lett. |

18. | M. Dennis, N. Zheludev, and F. García de Abajo, “The plasmon Talbot effect,” Opt. Express |

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. |

20. | F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nature |

21. | J. F. Clauser and S. Li, “‘Heisenberg Microscope’ Decoherence Atom Interferometry,” Phys. Rev. A |

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 |

23. | S. Nowak, C. Kurtsiefer, T. Pfau, and C. David, “High-Order Talbot Fringes for Atomic Matter Waves,” Opt. Lett. |

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. |

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 |

26. | B. Brezger, L. Hackermüller, S. Uttenthaler, J. Petschinka, M. Arndt, and A. Zeilinger, “Matter-Wave Interferometer for Large Molecules,” Phys. Rev. Lett. |

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. |

28. | B. J. McMorran and A. D. Cronin, “An electron Talbot interferometer,” New J. Phys. |

29. | I. Marzoli, A. Kaplan, F. Saif, and W. Schleich, “Quantum carpets of a slightly relativistic particle,” Fortschr. Phys. |

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. |

32. | J. F. Clauser, “Factoring Integers with Youngs N-Slit Interferometer,” Phys. Rev. A |

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 |

35. | M. Vrakking, D. Villeneuve, and A. Stolow, “Observation of fractional revivals of a molecular wave packet,” Phys. Rev. A |

36. | J. H. Hannay and M. V. Berry, “Quantization of linear maps on a torus-fresnel diffraction by a periodic grating,” Physica |

**OCIS Codes**

(000.2060) General : Education

(050.1940) Diffraction and gratings : Diffraction

(100.3175) Image processing : Interferometric imaging

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: September 10, 2009

Revised Manuscript: October 16, 2009

Manuscript Accepted: October 28, 2009

Published: November 2, 2009

**Citation**

William B. Case, Mathias Tomandl, Sarayut Deachapunya, and Markus Arndt, "Realization of optical carpets in the Talbot and Talbot-Lau configurations," Opt. Express **17**, 20966-20974 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-23-20966

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### References

- H. F. Talbot, "Facts Relating to Optical Science," Philos. Mag. 9, 401-407 (1836).
- E. Lau, "Beugungserscheinungen an Doppelrastern," Ann. Phys. 6, 417 (1948). [CrossRef]
- L. Rayleigh, "On Copying Diffraction-Gratings, and some Phenomena Connected Therewith," Philos. Mag. 11, 196 (1881).
- W. Montgomery, "Self-imaging objects of infinite aperture," J. Opt. Soc. Am. 57(6), 772-775 (1967). [CrossRef]
- K. Patorski, "Incoherent Superposition ofMultiple Self-imaging Lau Effect andMoire Fringe Explanation," Opt. Acta 30, 745-758 (1983). [CrossRef]
- 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]
- L. Liu, "Talbot and Lau effects on incident beams of arbitrary wavefront, and their use," Appl. Opt. 28(21), 4668-4678 (1989). [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.
- 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]
- O. Friesch, W. Schleich, and I. Marzoli, "Quantum carpets woven by Wigner functions," New J. Phys. 2(1), 4 (2000). [CrossRef]
- S. Mirza and C. Shakher, "Surface profiling using phase shifting Talbot interferometric technique," Optical Engineering 44, 013,601 (2004).
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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).
- 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]
- S. Nowak, C. Kurtsiefer, T. Pfau, and C. David, "High-Order Talbot Fringes for Atomic Matter Waves," Opt. Lett. 22, 1430-32 (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]
- 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]
- 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]
- 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. J. McMorran and A. D. Cronin, "An electron Talbot interferometer," New J. Phys. 11, 033,021 (2009).
- 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).
- 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]
- S. Wölk, W. P. Schleich, "Quantum carpets: Factorization with degeneracies," Proceedings of the Middleton Festival, Princeton, in print (2009).
- 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. H. Hannay, M. V. Berry, "Quantization of linear maps on a torus-fresnel diffraction by a periodic grating," Physica 1D267-290 (1980).

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