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The Talbot-Effect

1836 William Henry Fox Talbot discovered the lens-less self-imaging of a grating by diffraction in the optical near-field.

When a plane, monochromatic wave falls onto a grating it generates self-images of the grating at regular intervals \(n\cdot L_T (n \in \mathbb{N})\).

This distance, the Talbot length \(L_T\), depends on the wavelength \(\lambda\) and the grating period \(d\): \(L_T = \frac{d^2}{\lambda}\).

The self-images with even \(n\) are shifted by half a period compared to the images with odd \(n\).

If wave and grating are infinitely extended, these self-images recur infinitely often. However, in practice, the size of the grating is limited and the transition to the optical far-field can also be observed in the laboratory.

If the openings of the grating are small compared to the period (small opening fraction) additional periodic structures between the Talbot orders can be identified. At \(\frac{L_T}{n}\) an image of the grating  appears which is smaller by a factor of \(n\); accordingly, at half a Talbot distance an image with half the period and so on. Drawing the interference pattern for each distance behind the grating we obtain a pattern that is often called “carpet of light” or “quantum carpet for matter waves”.

Such a carpet of light can be observed in the experiment below.
Turn on the laser and then move the camera with the help of the arrow buttons. Push the Tempo button to change the speed. At which distance can you find the self-image of the grating? Does it match with the formula for the Talbot length (wavelength of the laser: 532 nm, grating period: 200 µm)?