Talbot effect

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 fractal pattern of sub images with ever-decreasing size, often referred to as a Talbot carpet. Talbot cavities are used for coherent beam combination of laser sets. 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 (page 204): where a {\displaystyle a} is the period of the diffraction grating and λ {\displaystyle \lambda } is the wavelength of the light incident on the grating. For λ ≪ a {\displaystyle \lambda \ll a} , the Talbot length is approximately given by: The number of Fresnel zones N F {\displaystyle N_{\text{F}}} that form first Talbot self-image of the grating with period p {\displaystyle p} and transverse size N ⋅ p {\displaystyle N\cdot p} is given by exact formula N F = ( N − 1 ) 2 {\displaystyle N_{\text{F}}=(N-1)^{2}} .