Talbot effect

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The optical Talbot Effect for monochromatic light, shown as a "Talbot Carpet". At the bottom of the figure the light can be seen diffracting through a grating, and this exact pattern is reproduced at the top of the picture (one Talbot Length away from the grating). Half way down you see the image shifted to the side, and at regular fractions of the Talbot Length the sub-images are clearly seen.
The optical Talbot Effect for monochromatic light, shown as a "Talbot Carpet". At the bottom of the figure the light can be seen diffracting through a grating, and this exact pattern is reproduced at the top of the picture (one Talbot Length away from the grating). Half way down you see the image shifted to the side, and at regular fractions of the Talbot Length the sub-images are clearly seen.
The Talbot Effect

The Talbot Effect is a near-field diffraction effect first observed in 1836 by Henry Fox Talbot[1] . When a laterally periodic wave distribution is incident upon a diffraction grating, its image is repeated at regular distances away from the grating plane. The regular distance is called the Talbot Length, and the repeated images are called Self Images or Talbot Images. Furthermore, at half the Talbot length, a self image also occurs, but phase-shifted by half a period (the physical meaning of this is that it is laterally shifted by half the width of the grating period). At smaller regular fractions of the Talbot Length, sub-images can also be observed. At one quarter of the Talbot Length, the self image is halved in size, and appears with half the period of the grating (thus twice as many images are seen). At one eighth of the Talbot length, the period and size of the images is halved again, and so forth creating a pattern of sub images with ever decreasing size.

Lord Rayleigh showed that the Talbot Effect was a natural consequence of Fresnel Diffraction and that the Talbot Length can be found by the following formula[2]:

z_T=\frac{2a^2}{\lambda}

where a is the period of the diffraction grating and λ is the wavelength of the light incident on the grating.

The Atomic Talbot Effect

Due to the quantum mechanical wave nature of particles, diffraction effects have also been observed with atoms - effects which are similar to those in the case of light. Chapman et al.[3] carried out an experiment in which a collimated beam of sodium atoms was passed through two diffraction gratings (the second used as a mask) to observe the Talbot effect and measure the Talbot length. The beam had a mean velocity of 1000ms-1 corresponding to a de Broglie wavelength of λdB = 0.17Å. Their experiment was performed with 200- and 300nm gratings which yielded Talbot lengths of 4.7- and 10.6mm respectively. This showed that for an atomic beam of constant velocity, by using λdB, the atomic Talbot length can be found in the same way in as the optical case.

[edit] References

  1. ^ H. F. Talbot 1836 "Facts relating to optical science" No. IV, Philos. Mag. 9
  2. ^ Lord Rayleigh 1881 "On copying diffraction gratings and on some phenomenon connected therewith" Philos. Mag. 11
  3. ^ Michael S. Chapman, Christopher R Ekstrom, Troy D. Hammond, J¨org Schmiedmayer, Bridget E. Tannian, Stefan Wehinger, David E Pritchard 1995 "Near-field imaging of atom diffraction gratings: The atomic Talbot effect" Physical Review A